{"problems":[{"id":"1938-A1","year":1938,"session":"A","number":1,"statement":"1. A solid is bounded by two bases in the horizontal planes $ z=h / 2 $ and $ z $ $ =-h / 2 $, and by such a surface that the area of every section in a horizontal plane is given by a formula of the sort\n$$\n\\text { Area }=a_{0} z^{3}+a_{1} z^{2}+a_{2} z+a_{3}\n$$\n(where as special cases some of the coefficients may be 0 ). Show that the volume is given by the formula\n$$\nV=\\frac{1}{6} h\\left[B_{1}+B_{2}+4 M\\right]\n$$\nwhere $ B_{1} $ and $ B_{2} $ are the areas of the bases, and $ M $ is the area of the middle horizontal section. Show that the formulas for the volume of a cone and of a sphere can be included in this formula when $ a_{0}=0 $.","subjects":["analysis","integration","solid geometry","simpson rule","numerical integration","cubic quadrature"]},{"id":"1938-A2","year":1938,"session":"A","number":2,"statement":"2. A can buoy is to be made of three pieces, namely, a cylinder and two equal cones, the altitude of each cone being equal to the altitude of the cylinder. For a given area of surface, what shape will have the greatest volume?","subjects":["analysis","optimization","solid geometry"]},{"id":"1938-A3","year":1938,"session":"A","number":3,"statement":"3. If a particle moves in the plane, we may express its coordinates $ x $ and $ y $ as functions of the time $ t $. If $ x=t^{3}-t $ and $ y=t^{4}+t $, show that the curve has a point of inflection at $ t=0 $ and that the velocity of the moving particle has a maximum at $ t=0 $.","subjects":["analysis","parametric curves","kinematics","local extrema","convexity"],"errata":"Interpret the claimed maximum as a strict local maximum of scalar speed. The squared speed has leading term $16t^6$ and is unbounded on the real line, so no global maximum exists. The canonical solution explicitly concludes a local maximum. The curve has a regular inflection at zero. The printed wording is retained above."},{"id":"1938-A4","year":1938,"session":"A","number":4,"statement":"4. A lumberman wishes to cut down a tree whose trunk is cylindrical and whose material is uniform. He will cut a notch, the two sides of which will be planes intersecting at a dihedral angle $ \\theta $ along a horizontal line through the axis of the cylinder. If $ \\theta $ is given, show that the least volume of material is cut out when the plane bisecting the dihedral angle is horizontal.","subjects":["geometry","solid geometry","optimization"]},{"id":"1938-A5","year":1938,"session":"A","number":5,"statement":"5. Evaluate the following limits:\n(i) $ \\lim _{n \\rightarrow \\infty} \\frac{n^{2}}{e^{n}} $.\n(ii) $ \\lim _{x \\rightarrow 0} \\frac{1}{x} \\int_{0}^{x}(1+\\sin 2 t)^{1 / t} d t $.","subjects":["analysis"]},{"id":"1938-A6","year":1938,"session":"A","number":6,"statement":"6. A swimmer stands at one corner of a square swimming pool and wishes to reach the diagonally opposite corner. If $ w $ is his walking speed and $ s $ is his swimming speed $ (s<w) $, find his path for shortest time. [Consider two cases: (i) $ w / s<\\sqrt{2} $, and (ii) $ w / s>\\sqrt{2} $.]","subjects":["geometry","optimization","path length","time optimal paths","mixed mode motion","calculus of variations"]},{"id":"1938-A7","year":1938,"session":"A","number":7,"statement":"7. Take either (i) or (ii).\n(i) Show that the gravitational attraction exerted by a thin homogeneous spherical shell at an external point is the same as if the material of the shell were concentrated at its center.\n(page 82)\n(ii) Determine all the straight lines which lie upon the surface $ z=x y $, and draw a figure to illustrate your result.","subjects":["analysis","gravitation","spherical symmetry"]},{"id":"1938-B1","year":1938,"session":"B","number":1,"statement":"8. Take either (i) or (ii).\n(i) Let $ A_{i k} $ be the cofactor of $ a_{i k} $ in the determinant\n$$\n\\boldsymbol{d}=\\left|\\begin{array}{llll}\na_{11} & a_{12} & a_{13} & a_{14} \\\\\na_{21} & a_{22} & a_{23} & a_{24} \\\\\na_{31} & a_{32} & a_{33} & a_{34} \\\\\na_{41} & a_{42} & a_{43} & a_{44}\n\\end{array}\\right|\n$$\n\nLet $ D $ be the corresponding determinant with $ a_{i k} $ replaced by $ A_{i k} $. Prove $ D= $ $ d^{3} $.\n(page 86)\n(ii) Let $ P(y)=A y^{2}+B y+C $ be a quadratic polynomial in $ y $. If the roots of the quadratic equation $ P(y)-y=0 $ are $ a $ and $ b(a \\neq b) $, show that $ a $ and $ b $ are roots of the biquadratic equation $ P[P(y)]-y=0 $. Hence write down a quadratic equation which will give the other two roots, $ c $ and $ d $, of the biquadratic. Apply this result to solving the following biquadratic equation:\n$$\n\\left(y^{2}-3 y+2\\right)^{2}-3\\left(y^{2}-3 y+2\\right)+2-y=0\n$$","subjects":["linear algebra","determinants","adjugate","algebra","polynomial composition","quartic equations","cofactor matrices"]},{"id":"1938-B2","year":1938,"session":"B","number":2,"statement":"9. Find all the solutions of the equation\n$$\ny y^{\\prime \\prime}-2\\left(y^{\\prime}\\right)^{2}=0\n$$\nwhich pass through the point $ x=1, y=1 $.","subjects":["analysis","ordinary differential equations","initial value problems"],"errata":"Reading convention, not a correction. Solutions are classified on a connected interval containing the initial point $x=1$, with their interval restrictions allowed. The explicit $C^2$ formulation adds no regularity assumption for the classical solutions through $(1,1)$; this follows from the differential equation. The target also includes the maximal domains."},{"id":"1938-B3","year":1938,"session":"B","number":3,"statement":"10. A horizontal disc of diameter 3 inches is rotating at 4 revolutions per minute. A light is shining at a distant point in the plane of the disc. An insect is placed at the edge of the disc furthest from the light, facing the light. It at once starts crawling, and crawls so as always to face the light, at 1 inch per second. Set up the differential equation of motion, and find at what point the insect again reaches the edge of the disc.","subjects":["analysis","kinematics","ordinary differential equations","rotating disc","parallel ray model","power series"],"errata":"The book's solution treats the distant light as a light at infinity, so its rays have a fixed parallel direction in the stationary plane. The current proofs use this idealization and measure crawling speed relative to the rotating disc. A finite point light produces a different position-dependent velocity field and is not covered by this target. The book explicitly makes the infinity assumption on printed p. 90, PDF p. 102. This is an explicit model interpretation, not a claim that a finite-light theorem has been disproved."},{"id":"1938-B4","year":1938,"session":"B","number":4,"statement":"11. Given the parabola $ y^{2}=2 m x $, what is the length of the shortest chord that is normal to the curve at one end?","subjects":["geometry","parabola","optimization"]},{"id":"1938-B5","year":1938,"session":"B","number":5,"statement":"12. From the center of a rectangular hyperbola a perpendicular is dropped upon a variable tangent. Find the locus of the foot of the perpendicular. Obtain the equation of the locus in polar coordinates, and sketch the curve.","subjects":["geometry","hyperbola","polar coordinates","pedal curve"]},{"id":"1938-B6","year":1938,"session":"B","number":6,"statement":"13. Find the shortest distance between the plane $ A x+B y+C z+1=0 $ and the ellipsoid $ x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1 $. (For brevity, let\n$$\n\\left.h=1 / \\sqrt{A^{2}+B^{2}+C^{2}} \\text { and } m=\\sqrt{a^{2} A^{2}+b^{2} B^{2}+c^{2} C^{2}} .\\right)\n$$\n\nState algebraically the condition that the plane shall lie outside the ellipsoid.","subjects":["geometry","ellipsoid","distance","linear algebra"]},{"id":"1939-A1","year":1939,"session":"A","number":1,"statement":"1. Find the length of the curve $ y^{2}=x^{3} $ from the origin to the point where the tangent makes an angle of $ 45^{\\circ} $ with the $ x $-axis.","subjects":["analysis","arc length","algebraic curves"]},{"id":"1939-A2","year":1939,"session":"A","number":2,"statement":"2. A point $ P $ is taken on the curve $ y=x^{3} $. The tangent at $ P $ meets the curve again at $ Q $. Prove that the slope of the curve at $ Q $ is four times the slope at $ P $.","subjects":["geometry","polynomials","tangent intersection"],"errata":"For a distinct second intersection, exclude the origin. At $P=(t,t^3)$ the other tangent-intersection abscissa is $-2t$, so it merges with $P$ when $t=0$. The book explicitly permits a multiplicity interpretation at that triple intersection. The corrected distinct-point assertion uses $t\\ne0$ and separately describes the limiting case."},{"id":"1939-A3","year":1939,"session":"A","number":3,"statement":"$$\n\\begin{array}{l}\n\\text { 3. Find the cubic equation whose roots are the cubes of the roots of }\\\\\nx^{3}+a x^{2}+b x+c=0\n\\end{array}\n$$","subjects":["algebra","polynomials","symmetric polynomials"]},{"id":"1939-A4","year":1939,"session":"A","number":4,"statement":"4. Find the equations of the two straight lines each of which cuts all the four straight lines\n$$\nx=1, y=0 ; \\quad y=1, z=0 ; \\quad z=1, x=0 ; \\quad x=y=-6 z\n$$","subjects":["geometry","spatial lines","transversals"]},{"id":"1939-A5","year":1939,"session":"A","number":5,"statement":"5. Take either (i) or (ii).\n(i) Solve the system of differential equations\n$$\n\\begin{array}{l}\n\\frac{d x}{d t}=x+y-3 \\\\\n\\frac{d y}{d t}=-2 x+3 y+1\n\\end{array}\n$$\nsubject to the conditions $ x=y=0 $ for $ t=0 $.\n(page 101)\n(ii) A heavy particle is attached to the end $ A $ of a light $ \\operatorname{rod} A B $ of length $ a $. The rod is hinged at $ B $ so that it can turn freely in a vertical plane. The rod is balanced in the vertical position above the hinge and then slightly disturbed. Prove that the time taken to pass from the horizontal position to the lowest position is\n$$\n\\sqrt{\\frac{a}{g}} \\log _{e}(1+\\sqrt{2})\n$$","subjects":["analysis","linear ode","systems of odes"]},{"id":"1939-A6","year":1939,"session":"A","number":6,"statement":"6. Take either (i) or (ii).\n(i) A circle of radius $ a $ rolls on the inner side of the circumference of a circle of radius $ 3 a $. Find the area contained within the closed curve generated by a point on the circumference of the rolling circle.\n(page 105)\n(ii) A shell strikes an airplane flying at a height $ h $ above the ground. It is known that the shell was fired from a gun on the ground with a muzzle velocity of magnitude $ V $, but the position of the gun and its angle of elevation are both unknown. Deduce that the gun is situated within a circle whose center lies directly below the airplane and whose radius is\n$$\n\\frac{V}{g} \\sqrt{V^{2}-2 g h}\n$$\n(Neglect the resistance of the atmosphere.)","subjects":["geometry","mechanics","optimization","plane area","projectiles","rolling curves"]},{"id":"1939-A7","year":1939,"session":"A","number":7,"statement":"7. Take either (i) or (ii).\n(i) Find the curve touched by all the curves of the family\n$$\n\\left(y-k^{2}\\right)^{2}=x^{2}\\left(k^{2}-x^{2}\\right)\n$$\n\nMake a rough sketch showing this curve and two curves of the family.\n(page 109)\n(ii) If the expansion in powers of $ x $ of the function\n$$\n\\frac{1}{(1-a x)(1-b x)}\n$$\nis given by\n$$\nc_{0}+c_{1} x+c_{2} x^{2}+c_{3} x^{3}+\\cdots,\n$$\nprove that the expansion in powers of $ x $ of the function\n$$\n\\frac{1+a b x}{(1-a b x)\\left(1-a^{2} x\\right)\\left(1-b^{2} x\\right)}\n$$\nis given by\n$$\nc_{0}^{2}+c_{1}^{2} x+c_{2}^{2} x^{2}+c_{3}^{2} x^{3}+\\cdots\n$$","subjects":["algebra","power series"]},{"id":"1939-B1","year":1939,"session":"B","number":1,"statement":"8. From the vertex $ (0, c) $ of the catenary\n$$\ny=c \\cosh \\frac{x}{c}\n$$\na line $ L $ is drawn perpendicular to the tangent to the catenary at a point $ P $. Prove that the length of $ L $ intercepted by the axes is equal to the ordinate $ y $ of the point $ P $.","subjects":["geometry","catenary","hyperbolic functions"]},{"id":"1939-B2","year":1939,"session":"B","number":2,"statement":"9. Evaluate the definite integrals\n(i) $ \\int_{1}^{3} \\frac{d x}{\\sqrt{(x-1)(3-x)}} $,\n(ii) $ \\int_{1}^{\\infty} \\frac{d x}{e^{x+1}+e^{3-x}} $.","subjects":["analysis","improper integrals"]},{"id":"1939-B3","year":1939,"session":"B","number":3,"statement":"10. Given the power-series\n$$\na_{0}+a_{1} x+a_{2} x^{2}+\\cdots\n$$\nin which\n$$\na_{n}=\\left(n^{2}+1\\right) 3^{n}\n$$\nshow that there is a relation of the form\n$$\na_{n}+p a_{n+1}+q a_{n+2}+r a_{n+3}=0\n$$\nin which $ p, q, r $ are constants independent of $ n $. Find these constants and the sum of the power-series.","subjects":["algebra","sequences","finite differences"]},{"id":"1939-B4","year":1939,"session":"B","number":4,"statement":"11. Find the equation of the parabola which touches the $ x $-axis at the point $ (1,0) $ and the $ y $-axis at the point $ (0,2) $. Find the equation of the axis of the parabola and the coordinates of its vertex.","subjects":["geometry","conics","parabola","coordinate rotation"]},{"id":"1939-B5","year":1939,"session":"B","number":5,"statement":"12. Take either (i) or (ii).\n(i) Prove that\n$$\n\\int_{1}^{a}[x] f^{\\prime}(x) d x=[a] f(a)-\\{f(1)+\\cdots+f([a])\\}\n$$\nwhere $ a $ is greater than 1 and where [ $ x $ ] denotes the greatest of the integers not exceeding $ x $. Obtain a corresponding expression for\n$$\n\\int_{1}^{a}\\left[x^{2}\\right] f^{\\prime}(x) d x\n$$\n(ii) A particle moves on a straight line, the only force acting on it being a resistance proportional to the velocity. If it started with a velocity of $ 1,000 \\mathrm{ft} $. per sec. and had a velocity of 900 ft . per sec. when it had travelled $ 1,200 \\mathrm{ft} $., calculate to the nearest hundredth of a second the time it took to travel this distance.","subjects":["analysis","floor function"]},{"id":"1939-B6","year":1939,"session":"B","number":6,"statement":"13. Take either (i) or (ii).\n(i) Let $ f(x) $ be defined for $ a \\leq x \\leq b $. Assuming appropriate properties of continuity and derivability, prove for $ a<x<b $ that\n$$\n\\frac{\\frac{f(x)-f(a)}{x-a}-\\frac{f(b)-f(a)}{b-a}}{x-b}=\\frac{1}{2} f^{\\prime \\prime}(\\xi)\n$$\nwhere $ \\boldsymbol{\\xi} $ is some number between $ \\boldsymbol{a} $ and $ \\boldsymbol{b} $.\n(page 123)\n(ii) Calculate the mutual gravitational attraction of two uniform rods, each of mass $ m $ and length $ 2 a $, placed parallel to one another and perpendicular to the line joining their centers at a distance $ b $ apart.\n\nIn your answer let $ a $ approach zero, and comment on the form of the result.","subjects":["newtonian gravitation","analysis","interpolation","multiple integrals"]},{"id":"1939-B7","year":1939,"session":"B","number":7,"statement":"14. Take either (i) or (ii).\n(i) If\n$$\n\\begin{array}{l}\nu=1+\\frac{x^{3}}{3!}+\\frac{x^{6}}{6!}+\\cdots \\\\\nv=\\frac{x}{1!}+\\frac{x^{4}}{4!}+\\frac{x^{7}}{7!}+\\cdots \\\\\nw=\\frac{x^{2}}{2!}+\\frac{x^{5}}{5!}+\\frac{x^{8}}{8!}+\\cdots,\n\\end{array}\n$$\nprove that\n$$\nu^{3}+v^{3}+w^{3}-3 u v w=1 .\n$$\n(ii) Consider the central conics\n$$\n\\begin{array}{l}\n\\left(a x^{2}+b y^{2}\\right)+2(p x+q y)+c=0 \\\\\n\\left(a x^{2}+b y^{2}\\right)+2 \\lambda(p x+q y)+\\lambda^{2} c=0\n\\end{array}\n$$\nwhere $ \\lambda $ is a given positive constant.\nShow that if all radii from the origin to the first conic are changed in the ratio $ \\lambda $ to 1 the tips of these new radii generate the second conic.\n\nLet $ P $ be the point with coordinates\n$$\nx=-\\frac{p}{a} \\frac{2 \\lambda}{1+\\lambda}, \\quad y=-\\frac{q}{b} \\frac{2 \\lambda}{1+\\lambda}\n$$\n\nShow that if all radii from $ P $ to the first conic are changed in the ratio $ \\lambda $ to 1 and then reversed about $ P $ the tips of these new radii generate the second conic.\n\nComment on these results in case $ \\lambda=1 $.","subjects":["algebraic identities","analysis","conics","homothety","power series"]},{"id":"1940-A1","year":1940,"session":"A","number":1,"statement":"1. Prove that if $ f(x) $ is a polynomial with integral coefficients, and there exists an integer $ k $ such that none of the integers $ f(1), f(2), \\ldots, f(k) $ is divisible by $ k $, then $ f(x) $ has no integral root.","subjects":["number theory","polynomial congruences"]},{"id":"1940-A2","year":1940,"session":"A","number":2,"statement":"2. Let $ A $ and $ B $ be two fixed points on the curve $ y=f(x) $, where $ f(x) $ is continuous and has a continuous derivative, and the $ \\operatorname{arc} A B $ is concave to the chord $ A B $. If $ P $ is a point of the $ \\operatorname{arc} A B $ for which $ A P+P B $ is a maximum, prove that $ P A $ and $ P B $ are equally inclined to the tangent to the curve $ y= $ $ f(x) $ at the point $ P $.","subjects":["geometry","optimization","reflection"],"errata":"The equal-inclination conclusion is read at a maximizing point $P$ distinct from $A$ and $B$, where both joining lines are defined. For a nonstraight concave arc every maximizing point is interior. If the straight chord is admitted as a degenerate arc, its endpoints also maximize the distance sum, but an endpoint joining segment has no angle. The target includes the interior straight-chord case and excludes those undefined endpoint angles. The book's complete argument is on printed pp. 133-134, PDF pp. 145-146."},{"id":"1940-A3","year":1940,"session":"A","number":3,"statement":"3. Find $ f(x) $ such that\n$$\n\\int[f(x)]^{n} d x=\\left[\\int f(x) d x\\right]^{n}\n$$\nwhen constants of integration are suitably chosen.","subjects":["analysis","functional identities","antiderivatives","exponential functions"]},{"id":"1940-A4","year":1940,"session":"A","number":4,"statement":"4. The parabola $ y^{2}=-4 p x $ rolls without slipping around the parabola $ y^{2}=4 p x $. Find the equation of the locus of the vertex of the rolling parabola.","subjects":["geometry","rolling curves","parabola","cissoid","rigid motion","kinematics"]},{"id":"1940-A5","year":1940,"session":"A","number":5,"statement":"5. Prove that the simultaneous equations\n$$\nx^{4}-x^{2}=y^{4}-y^{2}=z^{4}-z^{2}\n$$\nare satisfied by the points of four straight lines and six ellipses, and by no other points.","subjects":["algebra","real algebraic geometry","ellipses"]},{"id":"1940-A6","year":1940,"session":"A","number":6,"statement":"6. $ f(x) $ is a polynomial of degree $ n $, such that a power of $ f(x) $ is divisible by a power of its derivative $ f^{\\prime}(x) $; i.e., $ [f(x)]^{p} $ is divisible by $ \\left[f^{\\prime}(x)\\right]^{q} ; p, q $, positive integers. Prove that $ f(x) $ is divisible by $ f^{\\prime}(x) $ and that $ f(x) $ has a single root of multiplicity $ n $.","subjects":["algebra","polynomial divisibility","root multiplicity","complex analysis","topology"]},{"id":"1940-A7","year":1940,"session":"A","number":7,"statement":"7. If $ u_{1}{ }^{2}+u_{2}{ }^{2}+\\cdots $ and $ v_{1}{ }^{2}+v_{2}{ }^{2}+\\cdots $ are convergent series of real constants, prove that\n$$\n\\left(u_{1}-v_{1}\\right)^{p}+\\left(u_{2}-v_{2}\\right)^{p}+\\cdots, p \\text { an integer } \\geq 2\n$$\nis convergent.","subjects":["analysis","series","absolute convergence","sequence spaces"]},{"id":"1940-A8","year":1940,"session":"A","number":8,"statement":"8. A triangle is bounded by the lines\n$$\n\\begin{array}{c}\nA_{1} x+B_{1} y+C_{1}=0, \\quad A_{2} x+B_{2} y+C_{2}=0 \\\\\nA_{3} x+B_{3} y+C_{3}=0\n\\end{array}\n$$\n\nShow that the area, disregarding sign, is $$\\frac{\\Delta^2}{2\\left|(A_2B_3-A_3B_2)(A_3B_1-A_1B_3)(A_1B_2-A_2B_1)\\right|},$$ where $\\Delta=\\det\\begin{pmatrix}A_1&B_1&C_1\\\\A_2&B_2&C_2\\\\A_3&B_3&C_3\\end{pmatrix}$.","subjects":["geometry","triangle area","determinants"]},{"id":"1940-B1","year":1940,"session":"B","number":1,"statement":"9. A projectile, thrown with initial velocity $ v_{0} $ in a direction making angle $ \\alpha $ with the horizontal, is acted on by no force except gravity. Find the length of its path until it strikes a horizontal plane through the starting point. Show that the flight is longest when\n$$\n\\sin \\alpha \\log (\\sec \\alpha+\\tan \\alpha)=1\n$$","subjects":["mechanics","projectiles","arc length","optimization","integral geometry","concavity"]},{"id":"1940-B2","year":1940,"session":"B","number":2,"statement":"10. A cylindrical hole of radius $ r $ is bored through a cylinder of radius $ R $ ( $ r \\leq R $ ) so that the axes intersect at right angles.\n(i) Show that the area of the larger cylinder which is inside the smaller can be expressed in the form\n$$\nS=8 r^{2} \\int_{0}^{1} \\frac{1-v^{2}}{\\sqrt{\\left(1-v^{2}\\right)\\left(1-m^{2} v^{2}\\right)}} d v \\quad \\text { where } \\quad m=\\frac{r}{R}\n$$\n(ii) If\n$$\nK=\\int_{0}^{1} \\frac{d v}{\\sqrt{\\left(1-v^{2}\\right)\\left(1-m^{2} v^{2}\\right)}} \\text { and } E=\\int_{0}^{1} \\sqrt{\\frac{1-m^{2} v^{2}}{1-v^{2}}} d v\n$$\nshow that\n$$\nS=8\\left[R^{2} E-\\left(R^{2}-r^{2}\\right) K\\right]\n$$","subjects":["analysis","surface area","elliptic integrals"],"errata":"At equal positive radii, the first-kind elliptic integral $K$ diverges. The separate expression $(R^2-r^2)K$ is then undefined and must be read as a limit of the combined expression, or after combining its integrands. The geometric area at $r=R>0$ is $8R^2$. This endpoint diagnosis follows from the printed defining integrals. Scholes states the unequal-radius case."},{"id":"1940-B3","year":1940,"session":"B","number":3,"statement":"11. From any point $ (a, b) $ in the Cartesian plane, show that (i) three normals, real or imaginary, can be drawn to the parabola $ y^{2}=4 p x $; (ii) these are real and distinct if $ 4(2 p-a)^{3}+27 p b^{2}<0 $; (iii) two of them coincide if $ (a, b) $ lies on the curve $ 27 p y^{2}=4(x-2 p)^{3} $; (iv) all three coincide only if $ (a, b) $ is the point $ (2 p, 0) $.","subjects":["geometry","parabola","normal lines","cubic discriminant"]},{"id":"1940-B4","year":1940,"session":"B","number":4,"statement":"12. Prove that the locus of the point of intersection of three mutually perpendicular planes tangent to the surface\n$$\na x^{2}+b y^{2}+c z^{2}=1 \\quad(a b c \\neq 0)\n$$\nis the sphere\n$$\nx^{2}+y^{2}+z^{2}=\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}\n$$","subjects":["geometry","quadrics","director sphere","quadratic forms","spectral theory","convex geometry"],"errata":"The full locus claim with only $abc\\ne0$ is false for proper finite real tangent planes. Scholes explicitly flags the missing converse and its tangencies-at-infinity issue. The separately labeled corrected theorem in these tracks assumes $a,b,c>0$, so the quadric is an ellipsoid, and proves the complete director-sphere locus there. This is an explicit restriction, not a proof of the unrestricted printed assertion. The retained counterexample uses $x^2+y^2-z^2=1$."},{"id":"1940-B5","year":1940,"session":"B","number":5,"statement":"$$\n\\begin{array}{l}\n\\text { 13. Determine all rational values for which } a, b, c \\text { are the roots of }\\\\\nx^{3}+a x^{2}+b x+c=0\n\\end{array}\n$$","subjects":["algebra","rational roots","number theory","valuations"]},{"id":"1940-B6","year":1940,"session":"B","number":6,"statement":"14. Prove that\n$$\n\\left|\\begin{array}{lllll}\na_{1}^{2}+k & a_{1} a_{2} & a_{1} a_{3} & \\ldots & a_{1} a_{n} \\\\\na_{2} a_{1} & a_{2}^{2}+k & a_{2} a_{3} & \\ldots & a_{2} a_{n} \\\\\n\\ldots & \\ldots & \\ldots & \\ldots & \\ldots \\\\\na_{n} a_{1} & a_{n} a_{2} & a_{n} a_{3} & \\ldots & a_{n}^{2}+k\n\\end{array}\\right|\n$$\nis divisible by $ \\boldsymbol{k}^{n-1} $ and find its other factor.","subjects":["linear algebra","determinants","rank one matrices"]},{"id":"1940-B7","year":1940,"session":"B","number":7,"statement":"15. Which is greater\n$$\n(\\sqrt{n})^{\\sqrt{n+1}} \\text { or }(\\sqrt{n+1})^{\\sqrt{n}}\n$$\nwhere $ n>8 $ ?","subjects":["analysis","inequalities","logarithms"]},{"id":"1941-A1","year":1941,"session":"A","number":1,"statement":"1. Prove that the polynomial\n$$\n(a-x)^{6}-3 a(a-x)^{5}+\\frac{5}{2} a^{2}(a-x)^{4}-\\frac{1}{2} a^{4}(a-x)^{2}\n$$\ntakes only negative values for $ 0<x<a $.","subjects":["algebra","polynomial inequalities"]},{"id":"1941-A2","year":1941,"session":"A","number":2,"statement":"2. Find the $ n $th derivative with respect to $ x $ of $ \\int_{0}^{x}\\left[1+\\frac{(x-t)}{1!}+\\frac{(x-t)^{2}}{2!}+\\cdots+\\frac{(x-t)^{n-1}}{(n-1)!}\\right] e^{n t} d t $.","subjects":["analysis","higher derivatives","convolution"]},{"id":"1941-A3","year":1941,"session":"A","number":3,"statement":"3. A circle of radius a rolls in its plane along the $ x $-axis. Show that the envelope of a diameter is a cycloid, coinciding with the cycloid traced out by a point on the circumference of a circle of diameter $ a $, likewise rolling in its plane along the $ x $-axis.","subjects":["geometry","envelopes","cycloid","rolling circles"]},{"id":"1941-A4","year":1941,"session":"A","number":4,"statement":"4. Let the roots $ a, b, c $ of\n$$\nf(x) \\equiv x^{3}+p x^{2}+q x+r=0\n$$\nbe real, and let $ a \\leq b \\leq c $. Prove that, if the interval $ (b, c) $ is divided into six equal parts, a root of $ f^{\\prime}(x)=0 $ will lie in the fourth part counting from the end $ b $. What will be the form of $ f(x) $ if the root in question of $ f^{\\prime}(x)=0 $ falls at either end of the fourth part?","subjects":["algebra","real root location","intermediate value theorem"]},{"id":"1941-A5","year":1941,"session":"A","number":5,"statement":"5. Show that the line which moves parallel to the plane $ y=z $ and which intersects the two parabolas $ y^{2}=2 x, z=0 $ and $ z^{2}=3 x, y=0 $ sweeps out the surface\n$$\nx=(y-z)\\left(\\frac{y}{2}-\\frac{z}{3}\\right) .\n$$","subjects":["geometry","ruled surfaces","parabola","degeneracies"],"errata":"Distinguish the actual swept set from its closure. Lines joining distinct points of the two parabolas generate the displayed quadric with the ruling $x=0,\\ y=z$ omitted. Their closure is the full quadric. If the common origin may count as both intersections, the admissible lines also cover the entire plane $y=z$. The canonical solution and Scholes discuss this exceptional plane. The corrected theorem explicitly uses the closure of distinct-point joins."},{"id":"1941-A6","year":1941,"session":"A","number":6,"statement":"6. If the $ x $-coordinate $ \\bar{x} $ of the center of mass of the area lying between the $ x $-axis and the curve $ y=f(x),(f(x)>0) $, and between the lines $ x=0 $ and $ x $ $ =a $ is given by\n$$\n\\bar{x}=g(a)\n$$\nshow that\n$$\nf(x)=A \\frac{g^{\\prime}(x)}{[x-g(x)]^{2}} e^{\\int d x /(x-g(x))}\n$$\nwhere $ A $ is a positive constant.","subjects":["analysis","centroids","inverse problems","ordinary differential equations","integral equations"],"errata":"Positivity alone does not allow pointwise reconstruction of $f$ from its areas and first moments. For example, changing $f=1$ only at $x=1/2$ to $2$ leaves all those integrals unchanged. The current corrected target explicitly assumes a positive continuous density on $[0,L]$ and reconstructs it only for $x>0$. The printed book differentiates the integrals pointwise but does not explicitly supply the continuity hypothesis. This is therefore an explicitly conditional theorem, not an authenticated historical correction. No formula is evaluated at the singular origin, and negative intervals are outside this target."},{"id":"1941-A7","year":1941,"session":"A","number":7,"statement":"7. Take either (i) or (ii).\n(i) Prove that\n$$\n\\left|\\begin{array}{ccc}\n1+a^{2}-b^{2}-c^{2} & 2(a b+c) & 2(c a-b) \\\\\n2(a b-c) & 1+b^{2}-c^{2}-a^{2} & 2(b c+a) \\\\\n2(c a+b) & 2(b c-a) & 1+c^{2}-a^{2}-b^{2}\n\\end{array}\\right|=(1+a^2+b^2+c^2)^3\n$$\n(ii) A semi-ellipsoid of revolution is formed by revolving about the $ x $-axis the area lying within the first quadrant of the ellipse\n$$\n\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\n$$\n\nShow that this semi-ellipsoid will balance in stable equilibrium, with its vertex resting on a horizontal plane, when and only when\n$$\nb \\sqrt{8} \\geq a \\sqrt{5}\n$$","subjects":["linear algebra","determinants","skew symmetric matrices"]},{"id":"1941-B1","year":1941,"session":"B","number":1,"statement":"8. A particle $ (x, y) $ moves so that its angular velocities about $ (1,0) $ and $ (-1,0) $ are equal in magnitude but opposite in sign. Prove that\n$$\ny\\left(x^{2}+y^{2}+1\\right) d x=x\\left(x^{2}+y^{2}-1\\right) d y\n$$\nand verify that this is the differential equation of the family of rectangular hyperbolas passing through $ (1,0) $ and $ (-1,0) $ and having the origin as center.","subjects":["analysis","differential equations","rectangular hyperbola"]},{"id":"1941-B2","year":1941,"session":"B","number":2,"statement":"$$\n\\begin{array}{l}\n\\text { 9. Evaluate the following limits: }\\\\\n\\begin{array}{l}\n\\lim _{n \\rightarrow \\infty}\\left(\\frac{1}{\\sqrt{n^{2}+1^{2}}}+\\frac{1}{\\sqrt{n^{2}+2^{2}}}+\\cdots+\\frac{1}{\\sqrt{n^{2}+n^{2}}}\\right) \\\\\n\\lim _{n \\rightarrow \\infty}\\left(\\frac{1}{\\sqrt{n^{2}+1}}+\\frac{1}{\\sqrt{n^{2}+2}}+\\cdots+\\frac{1}{\\sqrt{n^{2}+n}}\\right) \\\\\n\\lim _{n \\rightarrow \\infty}\\left(\\frac{1}{\\sqrt{n^{2}+1}}+\\frac{1}{\\sqrt{n^{2}+2}}+\\cdots+\\frac{1}{\\sqrt{n^{2}+n^{2}}}\\right)\n\\end{array}\n\\end{array}\n$$","subjects":["analysis","limits","riemann sums"]},{"id":"1941-B3","year":1941,"session":"B","number":3,"statement":"10. Find the differential equation satisfied by the product $ z $ of any two linearly independent integrals of the equation\n$$\ny^{\\prime \\prime}+y^{\\prime} P(x)+y Q(x)=0\n$$","subjects":["analysis","linear ode","symmetric square","wronskians","reduction of order"]},{"id":"1941-B4","year":1941,"session":"B","number":4,"statement":"11. Two perpendicular diameters of the ellipse\n$$\n\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\n$$\nare given, and the two diameters conjugate to them are constructed. Show that the rectangular hyperbola passing through the ends of these conjugate diameters passes through the foci of the ellipse.","subjects":["geometry","ellipse","rectangular hyperbola","conic pencil","complex coordinates"],"errata":"Allow a degenerate rectangular conic consisting of two perpendicular lines. If the original diameters are the axes of a noncircular ellipse, no proper rectangular hyperbola passes through the four prescribed endpoints. The perpendicular axes give the required degenerate conic. For a circle, the corresponding diameter pair is always degenerate. The canonical book solution explicitly discusses both cases. The solutions state which object class is being used."},{"id":"1941-B5","year":1941,"session":"B","number":5,"statement":"12. A car is being driven so that its wheels, all of radius $ a $ feet, have an angular velocity of $ \\omega $ radians per second. A particle is thrown off from the tire of one of these wheels, where it is supposed that $ a \\omega^{2}>g $. Neglecting the resistance of the air, show that the maximum height above the roadway which the particle can reach is\n$$\n\\frac{\\left(a \\omega+g \\omega^{-1}\\right)^{2}}{2 g}\n$$","subjects":["mechanics","projectiles","cycloid","optimization"]},{"id":"1941-B6","year":1941,"session":"B","number":6,"statement":"13. Assuming that $ f(x) $ is continuous in the interval $ (0,1) $, prove that $ \\int_{x=0}^{x=1} \\int_{y=x}^{y=1} \\int_{z=x}^{z=y} f(x) f(y) f(z) d x d y d z=\\frac{1}{3!}\\left(\\int_{t=0}^{t=1} f(t) d t\\right)^{3} $.","subjects":["analysis","multiple integrals","symmetry"],"errata":"Editorial qualification. Read the nested integral in the order $dz\\,dy\\,dx$. The printed differentials do not match the displayed bounds. Continuity only on $(0,1)$ does not guarantee finite integrals, as $f(t)=1/t$ shows. The reviewed arguments prove the explicitly corrected theorem for absolutely integrable $f$, which includes functions continuous on $[0,1]$."},{"id":"1941-B7","year":1941,"session":"B","number":7,"statement":"14. Take either (i) or (ii).\n(i) Show that any solution $ f(t) $ of the functional equation\n$$\nf(x+y) f(x-y)=f(x) f(x)+f(y) f(y)-1, \\quad(x, y, \\text { real })\n$$\nis such that\n$$\nf^{\\prime \\prime}(t)= \\pm m^{2} f(t), \\quad(m \\text { constant and } \\geq 0)\n$$\nassuming the existence and continuity of the second derivative. Deduce that $ f(t) $ is one of the functions\n$$\n\\pm \\cos m t, \\quad \\pm \\cosh m t\n$$\n(ii) With $ n $ constant values $ a_{1}, a_{2}, \\ldots, a_{n} $, supposed all different, let $ n $ constant values $ b_{1}, b_{2}, \\ldots, b_{n} $ be associated, and let a polynomial $ P(x) $ be. defined by the identity in $ \\boldsymbol{x} $\n$$\n\\left|\\begin{array}{cccccc}\n1 & x & x^{2} & \\cdots & x^{n-1} & P(x) \\\\\n1 & a_{1} & a_{1}^{2} & \\cdots & a_{1}{ }^{n-1} & b_{1} \\\\\n1 & a_{2} & a_{2}^{2} & \\cdots & a_{2}{ }^{n-1} & b_{2} \\\\\n\\cdots & \\cdots & \\ldots & \\ldots & \\cdots & \\cdots\n\\end{array}\\right| \\equiv 0 . \\cdots \\cdots .\n$$\n\nGiven a polynomial $ \\phi(t) $, let a polynomial $ Q(x) $ be defined by the identity in $ x $ obtained on replacing $ P(x), b_{1}, b_{2}, \\ldots, b_{n} $ of the identity above by $ Q(x) $, $ \\phi\\left(b_{1}\\right), \\phi\\left(b_{2}\\right), \\ldots ; \\phi\\left(b_{n}\\right) $. Prove that the remainder obtained on dividing $ \\phi(P(x)) $ by $ \\left(x-a_{1}\\right)\\left(x-a_{2}\\right) \\cdots\\left(x-a_{n}\\right) $ is $ Q(x) $.","subjects":["analysis","functional equations","linear ode"]},{"id":"1942-A1","year":1942,"session":"A","number":1,"statement":"1. A square of side $ 2 a $, lying always in the first quadrant of the $ X Y $ plane, moves so that two consecutive vertices are always on the $ X $ - and $ Y $-axes respectively. Find the locus of the midpoint of the square.","subjects":["geometry","square","kinematic locus"]},{"id":"1942-A2","year":1942,"session":"A","number":2,"statement":"2. If a polynomial $ f(x) $ is divided by $ (x-a)^{2}(x-b) $, where $ a \\neq b $, derive a formula for the remainder.","subjects":["algebra","polynomial remainders","hermite interpolation"]},{"id":"1942-A3","year":1942,"session":"A","number":3,"statement":"$$\n\\begin{array}{l}\n\\text { 3. Is the following series convergent or divergent? }\\\\\n1+\\frac{1}{2} \\cdot \\frac{19}{7}+\\frac{2!}{3^{2}}\\left(\\frac{19}{7}\\right)^{2}+\\frac{3!}{4^{3}}\\left(\\frac{19}{7}\\right)^{3}+\\frac{4!}{5^{4}}\\left(\\frac{19}{7}\\right)^{4}+\\cdots\n\\end{array}\n$$","subjects":["analysis","positive series","convergence tests"]},{"id":"1942-A4","year":1942,"session":"A","number":4,"statement":"4. Find the orthogonal trajectories of the family of conics $ (x+2 y)^{2} $ $ =a(x+y) $. At what angle do the curves of one family cut the curves of the other family at the origin?","subjects":["analysis","orthogonal trajectories","singular points"]},{"id":"1942-A5","year":1942,"session":"A","number":5,"statement":"5. A circle of radius $ a $ is revolved through $ 180^{\\circ} $ about a line in its plane, distant $ b $ from the center of the circle, where $ b>a $. For what value of the ratio $ b / a $ does the center of gravity of the solid thus generated lie on the surface of the solid?","subjects":["geometry","centroid","solid of revolution","torus"]},{"id":"1942-A6","year":1942,"session":"A","number":6,"statement":"6. Any circle in the $ X Y $ (horizontal) plane is \"represented\" by a point on the vertical line through the center of the circle and at a distance \"above\" the plane of the circle equal to the radius of the circle.\n\nShow that the locus of the representations of all the circles which cut a fixed circle at a constant angle is a (portion of a) one-sheeted hyperboloid.\n\nBy consideration of suitable families of circles in the plane, demonstrate the existence of two families of rulings on the hyperboloid.","subjects":["geometry","circle intersection","hyperboloids","ruled surfaces"],"errata":"For ordinary unoriented crossing angles $0<\\alpha\\le\\pi/2$, the positive-radius representative locus is the union of $x^2+y^2-(z+\\epsilon a\\cos\\alpha)^2=a^2\\sin^2\\alpha$, with $z>0$ and $\\epsilon=\\pm1$. The two branches coincide at a right angle. Each has two ruling families. At zero angle the tangent-circle loci are cones, with the coincident circle excluded. The book explicitly discusses the two branches."},{"id":"1942-B1","year":1942,"session":"B","number":1,"statement":"7. A square of side $ 2 a $, lying always in the first quadrant of the $ X Y $ plane, moves so that two consecutive vertices are always on the $ X $ - and $ Y $-axes respectively. Prove that a point within or on the boundary of the square will in general describe a (portion of a) conic. For what points of the square does this locus degenerate?","subjects":["geometry","square","conic loci","affine maps"]},{"id":"1942-B2","year":1942,"session":"B","number":2,"statement":"8. For the family of parabolas\n$$\ny=\\frac{a^{3} x^{2}}{3}+\\frac{a^{2} x}{2}-2 a,\n$$\n(i) find the locus of vertices,\n(ii) find the envelope,\n(iii) sketch the envelope and two typical curves of the family.","subjects":["geometry","parabola","envelopes","hyperbola"]},{"id":"1942-B3","year":1942,"session":"B","number":3,"statement":"9. Given\n$$\n\\begin{array}{l}\nx=\\phi(u, v) \\\\\ny=\\psi(u, v)\n\\end{array}\n$$\nwhere $ \\phi $ and $ \\psi $ are solutions of the partial differential equation\n$$\n\\frac{\\partial \\phi}{\\partial u} \\frac{\\partial \\psi}{\\partial v}-\\frac{\\partial \\phi}{\\partial v} \\frac{\\partial \\psi}{\\partial u}=1\n$$\n\nBy assuming that $ x $ and $ v $ are the independent variables, show that (1) may be transformed to\n$$\n\\frac{\\partial y}{\\partial v}=\\frac{\\partial u}{\\partial x} .\n$$\n\nIntegrate (2), and show how this effects in general the solution of (1). What other solutions does (1) possess?","subjects":["analysis","jacobian","partial differential equations"]},{"id":"1942-B4","year":1942,"session":"B","number":4,"statement":"10. A particle moves under a central force inversely proportional to the $ \\boldsymbol{k} $ th power of the distance. If the particle describes a circle (the central force proceeding from a point on the circumference of the circle $ ) $, find $ k $.","subjects":["mechanics","central forces","circular orbits"]},{"id":"1942-B5","year":1942,"session":"B","number":5,"statement":"11. Sketch the curve\n$$\ny=\\frac{x}{1+x^{6} \\sin ^{2} x}\n$$\nand show that\n$$\n\\int_{0}^{\\infty} \\frac{x d x}{1+x^{6} \\sin ^{2} x}\n$$\nexists.","subjects":["analysis","improper integrals","oscillatory functions","spike estimates"]},{"id":"1946-A1","year":1946,"session":"A","number":1,"statement":"1. Suppose that the function $ f(x)=a x^{2}+b x+c $, where $ a, b, c $ are real constants, satisfies the condition $ |f(x)| \\leq 1 $ for $ |x| \\leq 1 $. Prove that $ \\left|f^{\\prime}(x)\\right| $ $ \\leq 4 $ for $ |x| \\leq 1 $.","subjects":["analysis","polynomial inequalities","derivative bounds","chebyshev polynomials"]},{"id":"1946-A2","year":1946,"session":"A","number":2,"statement":"2. If $ a(x), b(x), c(x) $, and $ d(x) $ are polynomials in $ x $, show that\n$$\n\\int_{1}^{x} a(x) c(x) d x \\cdot \\int_{1}^{x} b(x) d(x) d x-\\int_{1}^{x} a(x) d(x) d x \\cdot \\int_{1}^{x} b(x) c(x) d x\n$$\nis divisible by $ (x-1)^{4} $.","subjects":["algebra","polynomial divisibility","integral identities"]},{"id":"1946-A3","year":1946,"session":"A","number":3,"statement":"3. A projectile in flight is observed simultaneously from four radar stations which are situated at the corners of a square of side $ b $. The distances of the projectile from the four stations, taken in order around the square, are found to be $ \\boldsymbol{R}_{1}, \\boldsymbol{R}_{2}, \\boldsymbol{R}_{3}, \\boldsymbol{R}_{4} $. Show that\n$$\nR_{1}^{2}+R_{3}^{2}=R_{2}^{2}+R_{4}^{2}\n$$\n\nShow also that the height $ h $ of the projectile above the ground is given by\n$$\n\\begin{aligned}\nh^{2}= & -\\frac{1}{2} b^{2}+\\frac{1}{4}\\left(R_{1}^{2}+R_{2}^{2}+R_{3}^{2}+R_{4}^{2}\\right) \\\\\n& -\\frac{1}{8 b^{2}}\\left(R_{1}^{4}+R_{2}^{4}+R_{3}^{4}+R_{4}^{4}-2 R_{1}^{2} R_{3}^{2}-2 R_{2}^{2} R_{4}^{2}\\right)\n\\end{aligned}\n$$","subjects":["geometry","distance geometry","square","trilateration"]},{"id":"1946-A4","year":1946,"session":"A","number":4,"statement":"4. Let $ g(x) $ be a function that has a continuous first derivative $ g^{\\prime}(x) $ for all values of $ x $. Suppose that the following conditions hold for every $ x $ : (i) $ g(0)= $ $ 0 ; $ (ii) $ \\left|g^{\\prime}(x)\\right| \\leq|g(x)| $. Prove that $ g(x) $ vanishes identically.","subjects":["analysis","differential inequalities","uniqueness"]},{"id":"1946-A5","year":1946,"session":"A","number":5,"statement":"5. Find the smallest volume bounded by the coordinate planes and by a tangent plane to the ellipsoid\n$$\n\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}+\\frac{z^{2}}{c^{2}}=1\n$$","subjects":["geometry","ellipsoid","tetrahedron","optimization"]},{"id":"1946-A6","year":1946,"session":"A","number":6,"statement":"6. A particle of unit mass moves on a straight line under the action of a force which is a function $ f(v) $ of the velocity $ v $ of the particle, but the form of this function is not known. A motion is observed, and the distance $ x $ covered in time $ t $ is found to be connected with $ t $ by the formula $ x=a t+b t^{2}+c t^{3} $, where $ a, b, c $ have numerical values determined by observation of the motion. Find the function $ f(v) $ for the range of $ v $ covered by the experiment.","subjects":["mechanics","inverse problem","velocity dependent force"]},{"id":"1946-B1","year":1946,"session":"B","number":1,"statement":"1. Let $ K $ denote the circumference of a circular disc of radius one, and let $ k $ denote a circular arc that joins two points $ a, b $ on $ K $ and lies otherwise in the given circular disc. Suppose that $ k $ divides the circular disc into two parts of equal area. Prove that the length of $ k $ exceeds 2 .","subjects":["geometry","circle","area bisectors","arc length"]},{"id":"1946-B2","year":1946,"session":"B","number":2,"statement":"2. Let $ A, B $ be variable points on a parabola $ P $, such that the tangents at $ A $ and $ B $ are perpendicular to each other. Show that the locus of the centroid of the triangle formed by $ A, B $ and the vertex of $ P $ is a parabola $ P_{1} $. Apply the same process to $ P_{1} $, obtaining a parabola $ P_{2} $, and repeat the process, obtaining altogether the sequence of parabolas $ P, P_{1}, P_{2}, \\ldots, P_{n} $. If the equation of $ P $ is $ y^{2}=m x $, find the equation of $ P_{n} $.","subjects":["geometry","parabola","centroid","iteration"]},{"id":"1946-B3","year":1946,"session":"B","number":3,"statement":"3. In a solid sphere of radius $ R $ the density $ \\rho $ is a function of $ r $, the distance from the center of the sphere. If the magnitude of the gravitational force of attraction due to the sphere at any point inside the sphere is $ k r^{2} $, where $ k $ is a constant, find $ \\rho $ as a function of $ r $. Find also the magnitude of the force of attraction at a point outside the sphere at a distance $ r $ from the center. (Assume that the magnitude of the force of attraction at a point $ P $ due to a thin spherical shell is zero if $ P $ is inside the shell, and is $ m / r^{2} $ if $ P $ is outside the shell, $ m $ being the mass of the shell, and $ r $ the distance of $ P $ from the center.)","subjects":["mechanics","gravitation","spherical density"]},{"id":"1946-B4","year":1946,"session":"B","number":4,"statement":"4. For each positive integer $ n $, put\n$$\np_{n}=(1+1 / n)^{n}, P_{n}=(1+1 / n)^{n+1}, h_{n}=\\frac{2 p_{n} P_{n}}{p_{n}+P_{n}}\n$$\n\nProve that $ h_{1}<h_{2}<\\cdots<h_{n}<\\cdots $.","subjects":["analysis","sequences","harmonic mean","monotonicity","discrete inequalities"]},{"id":"1946-B5","year":1946,"session":"B","number":5,"statement":"$$\n\\text { 5. Show that the integer next above }(\\sqrt{3}+1)^{2 n} \\text { is divisible by } 2^{n+1} \\text {. }\n$$","subjects":["number theory","algebraic conjugates","divisibility","algebraic number theory"]},{"id":"1946-B6","year":1946,"session":"B","number":6,"statement":"6. A particle moves on a circle with center $ O $, starting from rest at a point $ P $ and coming to rest again at a point $ Q $, without coming to rest at any intermediate point. Prove that the acceleration vector of the particle does not vanish at any point between $ P $ and $ Q $ and that, at some point $ R $ between $ P $ and $ Q $, the acceleration vector points in along the radius $ R O $.","subjects":["mechanics","circular motion"],"errata":"Interpret coming to rest as having zero instantaneous velocity between the endpoints, rather than remaining stationary for a nonzero interval. The book commentary distinguishes these meanings."},{"id":"1947-A1","year":1947,"session":"A","number":1,"statement":"1. If $ \\left\\{a_{n}\\right\\} $ is a sequence of numbers such that for $ n \\geq 1 $\n$$\n\\left(2-a_{n}\\right) a_{n+1}=1\n$$\nprove that $ \\lim a_{n} $, as $ n \\rightarrow \\infty $, exists and is equal to one.","subjects":["analysis","recurrences","fractional linear transformations"]},{"id":"1947-A2","year":1947,"session":"A","number":2,"statement":"2. A real valued continuous function satisfies for all real $ x $ and $ y $ the functional equation\n$$\nf\\left(\\sqrt{x^{2}+y^{2}}\\right)=f(x) f(y)\n$$\n\nProve that\n$$\nf(x)=[f(1)]^{x^{2}}\n$$","subjects":["analysis","functional equations","continuous characters","multiplicative semigroups"],"errata":"The zero function also satisfies the functional equation, but the displayed expression is undefined at $x=0$ because it becomes $0^0$. The book explicitly flags this exception. The complete conclusion is that either $f$ is identically zero, or $f(x)=b^{x^2}$ for every real $x$, where $b=f(1)>0$. The reviewed proofs establish this classification."},{"id":"1947-A3","year":1947,"session":"A","number":3,"statement":"3. Let $V_1,V_2,V_3$ be the vertices of a triangle and let $V_4$ be a point strictly inside it. Let $s_1,\\ldots,s_6$ be the six straight segments joining the four points in pairs and any two points $ Q_{1}, Q_{2} $ in the plane not lying on any of the segments $ s_{1}, s_{2}, \\ldots, s_{6} $, show that there does not exist a polygonal line $ P $ joining $ Q_{1} $ and $ Q_{2} $ such that:\n(i) $ P $ crosses each $ s_{i}, i=1,2, \\ldots, 6 $, exactly once;\n(ii) $ \\boldsymbol{P} $ does not intersect itself;\n(iii) $ P $ does not pass through any vertex $ V_{1}, V_{2}, V_{3}, V_{4} $.","subjects":["combinatorics","planar graphs","crossing parity"]},{"id":"1947-A4","year":1947,"session":"A","number":4,"statement":"4. A coast artillery gun can fire at any angle of elevation between $ 0^{\\circ} $ and $ 90^{\\circ} $ in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant $ \\left(=v_{0}\\right) $, determine the set $ H $ of points in the plane and above the horizontal which can be hit.","subjects":["mechanics","projectiles","reachable set","envelopes"]},{"id":"1947-A5","year":1947,"session":"A","number":5,"statement":"5. $ a_{1}, b_{1}, c_{1} $ are positive numbers whose sum is 1 , and for $ n=1,2, \\ldots $ we define\n$$\na_{n+1}=a_{n}^{2}+2 b_{n} c_{n}, b_{n+1}=b_{n}^{2}+2 c_{n} a_{n}, c_{n+1}=c_{n}^{2}+2 a_{n} b_{n} .\n$$\n\nShow that $ a_{n}, b_{n}, \\boldsymbol{c}_{n} $ approach limits as $ n \\rightarrow \\infty $ and find these limits.","subjects":["analysis","nonlinear recurrences","simplex","convergence"]},{"id":"1947-A6","year":1947,"session":"A","number":6,"statement":"6. A three-by-three matrix has determinant zero, and has the further property that the cofactor of any element is equal to the square of that element. (The cofactor of $ a_{i j} $ is $ (-1)^{i+j} $ multiplied by the determinant obtained by striking out the $ i $ th row and $ j $ th column.) Show that every element in the matrix is zero.","subjects":["linear algebra","cofactors","matrix rank","polynomial algebra"]},{"id":"1947-B1","year":1947,"session":"B","number":1,"statement":"7. Let $ f(x) $ be a function such that $ f(1)=1 $ and for $ x \\geq 1 $\n$$\nf^{\\prime}(x)=\\frac{1}{x^{2}+f^{2}(x)}\n$$\n\nProve that\n$$\n\\lim _{x \\rightarrow \\infty} f(x)\n$$\nexists and is less than $ 1+\\pi / 4 $","subjects":["analysis","ordinary differential equations","monotone convergence"]},{"id":"1947-B2","year":1947,"session":"B","number":2,"statement":"8. Let $ f(x) $ be a differentiable function defined in the closed interval $ (0,1) $ and such that\n$$\n\\left|f^{\\prime}(x)\\right| \\leq M, \\quad 0<x<1\n$$\n\nProve that\n$$\n\\left|\\int_{0}^{1} f(x) d x-\\frac{1}{n} \\sum_{k=1}^{n} f\\left(\\frac{k}{n}\\right)\\right| \\leq \\frac{M}{n} .\n$$","subjects":["analysis","riemann sums","error bounds","lipschitz functions"],"errata":"Read the interval as $[0,1]$, with $f$ continuous there, differentiable on $(0,1)$, and $|f'|\\le M$ on $(0,1)$, where $M\\ge0$ and $n$ is a positive integer. Endpoint values are determined by continuity. The contradictory closed-interval wording appears in the book."},{"id":"1947-B3","year":1947,"session":"B","number":3,"statement":"9. Let $ x, y $ be Cartesian coordinates in the plane. $ I $ denotes the line segment $ 1 \\leq x \\leq 3, y=1 $. For every point $ P $ on $ I $, let $ P^{*} $ denote that point that lies on the segment joining the origin to $ P $ and such that the distance $ P P^{*} $ is equal to $ 1 / 100 $. As $ P $ describes $ I $, the corresponding point $ P^{*} $ describes a certain curve $ C^{*} $. Let $ l(I), l\\left(C^{*}\\right) $ be the lengths of $ I $ and $ C^{*} $ respectively. Which one of $ l(I), l\\left(C^{*}\\right) $ is greater? Prove your answer.","subjects":["geometry","arc length","polar coordinates"]},{"id":"1947-B4","year":1947,"session":"B","number":4,"statement":"10. Given $ P(z)=z^{2}+a z+b $, a quadratic polynomial of the complex variable $ z $ with complex coefficients $ a, b $. Suppose that $ |P(z)|=1 $ for every $ z $ such that $ |z|=1 $. Prove that $ a=b=0 $.","subjects":["algebra","complex polynomials","unit circle"]},{"id":"1947-B5","year":1947,"session":"B","number":5,"statement":"11. $ a, b, c, d $ are distinct integers such that\n$$\n(x-a)(x-b)(x-c)(x-d)-4=0\n$$\nhas an integral root $ r $. Show that $ 4 r=a+b+c+d $.","subjects":["number theory","integer roots","divisibility"]},{"id":"1947-B6","year":1947,"session":"B","number":6,"statement":"12. $ C $ is a fixed point on $ O Z $ and $ U, V $ are variable points on $ O X, O Y $ respectively, where $ O X, O Y, O Z $ are mutually orthogonal lines. Find the locus of a point $ P $ such that $ P U, P V, P C $ are mutually orthogonal.","subjects":["geometry","solid geometry","locus","orthogonality"]},{"id":"1948-A1","year":1948,"session":"A","number":1,"statement":"1. What is the maximum of $ \\left|z^{3}-z+2\\right| $; where $ z $ is a complex number with $ |z|=1 ? $","subjects":["algebra","complex numbers","optimization"]},{"id":"1948-A2","year":1948,"session":"A","number":2,"statement":"2. Two spheres in contact have a common tangent cone. These three surfaces divide the space into various parts, only one of which is bounded by all three surfaces; it is \"ring-shaped.\" Being given the radii of the spheres, $ r $ and $ \\boldsymbol{R} $, find the volume of the \"ring-shaped\" part. (The desired expression is a rational function of $ r $ and $ R $. )","subjects":["geometry","solid geometry","volume","integration"]},{"id":"1948-A3","year":1948,"session":"A","number":3,"statement":"3. Let $ \\left\\{a_{n}\\right\\} $ be a decreasing sequence of positive numbers with limit 0 such that\n$$\nb_{n}=a_{n}-2 a_{n+1}+a_{n+2} \\geq 0\n$$\nfor all $ n $. Prove that\n$$\n\\sum_{n=1}^{\\infty} n b_{n}=a_{1}\n$$","subjects":["analysis","series","convex sequences"]},{"id":"1948-A4","year":1948,"session":"A","number":4,"statement":"4. Let $ D $ be a plane region bounded by a circle of radius $ r $. Let $ (x, y) $ be a point of $ D $ and consider a circle of radius $ \\delta $ and center at $ (x, y) $. Denote by $ l(x, y) $ the length of that arc of the circle which is outside $ D $. Find\n$$\n\\lim _{\\delta \\rightarrow 0} \\frac{1}{\\delta^{2}} \\iint_{D} l(x, y) d x d y\n$$","subjects":["analysis","geometry","double integrals","asymptotics"]},{"id":"1948-A5","year":1948,"session":"A","number":5,"statement":"$$\n\\begin{array}{l}\n\\text { 5. If } x_{1}, \\ldots, x_{n} \\text { denote the } n \\text {th roots of unity, evaluate }\\\\\n\\prod_{i<j}\\left(x_{i}-x_{j}\\right)^{2} \\quad(i<j)\n\\end{array}\n$$","subjects":["algebra","roots of unity","discriminants"]},{"id":"1948-A6","year":1948,"session":"A","number":6,"statement":"6. Answer either (i) or (ii):\n(i) A force acts on the element $ d s $ of a closed plane curve. The magnitude of this force is $ r^{-1} d s $ where $ r $ is the radius of curvature at the point considered, and the direction of the force is perpendicular to the curve; it points to the convex side. Show that the system of such forces acting on all elements of the curve keep it in equilibrium.\n(page 252)\n(ii) Show that\n$$\nx+\\frac{2}{3} x^{3}+\\frac{2}{3} \\frac{4}{5} x^{5}+\\frac{2}{3} \\frac{4}{5} \\frac{6}{7} x^{7}+\\cdots=\\frac{\\arcsin x}{\\sqrt{1-x^{2}}} .\n$$","subjects":["analysis","mechanics","power series","curvature"]},{"id":"1948-B1","year":1948,"session":"B","number":1,"statement":"1. Let $ f(x) $ be a cubic polynomial with roots $ x_{1}, x_{2} $, and $ x_{3} $. Assume that $ f(2 x) $ is divisible by $ f^{\\prime}(x) $ and compute the ratios $ x_{1}: x_{2}: x_{3} $.","subjects":["algebra","polynomial divisibility","cubic roots"],"errata":"The admissible cubics include $f(x)=Ax^3$, $A\\ne0$, whose three zero roots have no defined ratio. In all other cases, allowing complex roots and suitable ordering, their ratio is $1:(-2+2\\sqrt3\\,i):(-2-2\\sqrt3\\,i)$. The book explicitly identifies the all-zero-root exception."},{"id":"1948-B2","year":1948,"session":"B","number":2,"statement":"2. \"A penny in a corner.\" A circle moves so that it is continually in contact with all three coordinate planes of an ordinary rectangular system. Find the locus of the center of the circle.","subjects":["geometry","solid geometry","locus","orthogonal projection","spatial geometry"]},{"id":"1948-B3","year":1948,"session":"B","number":3,"statement":"3. If $ n $ is a positive integer, prove that\n$$\n[\\sqrt{n}+\\sqrt{n+1}]=[\\sqrt{4 n+2}]\n$$\nwhere $ [x] $ denotes as usual the greatest integer not exceeding $ x . \\quad $","subjects":["number theory","floor function","radicals"]},{"id":"1948-B4","year":1948,"session":"B","number":4,"statement":"4. Let $ \\min (x, y) $ denote the smaller of the numbers $ x $ and $ y $. For what $ \\lambda $ 's does the equation\n$$\n\\int_{0}^{1} \\min (x, y) f(y) d y=\\lambda f(x)\n$$\nhave continuous solutions which do not vanish identically in $ (0,1) $ ? What are these solutions?","subjects":["analysis","integral equations","eigenvalues","boundary value problems"]},{"id":"1948-B5","year":1948,"session":"B","number":5,"statement":"5. The pairs of numbers $ (a, b) $ such that $ \\left|a+b t+t^{2}\\right| \\leq 1 $ for $ 0 \\leq t \\leq 1 $ fill a certain region in the $ (a, b) $-plane. What is the area of this region?","subjects":["analysis","geometry","quadratic inequalities","area"]},{"id":"1948-B6","year":1948,"session":"B","number":6,"statement":"6. Answer either (i) or (ii):\n(i) Let $ V_{1}, V_{2}, V_{3} $, and $ V $ denote four vertices of a cube. $ V_{1}, V_{2} $, and $ V_{3} $ are next neighbors of $ V $, that is, the lines $ V V_{1}, V V_{2} $, and $ V V_{3} $ are edges of the cube. Project the cube orthogonally onto a plane (the $ z $-plane, the Gaussian plane) of which the points are marked with complex numbers. Let the projection of $ V $ fall in the origin and the projections of $ V_{1}, V_{2} $, and $ V_{3} $ in points marked with the complex numbers $ z_{1}, z_{2} $, and $ z_{3} $, respectively. Show that $ z_1^2+z_2^2+z_3^2=0 $.\n(page 261)\n(ii) Let $ a_{i j} $ be a determinant in which each diagonal element exceeds in absolute value the sum of the absolute values of the other elements of its row, that is\n$$\n\\left|a_{i i}\\right|>\\left|a_{i 1}\\right|+\\left|a_{i 2}\\right|+\\cdots+\\left|a_{i, i-1}\\right|+\\left|a_{i, i+1}\\right|+\\cdots+\\left|a_{i n}\\right|\n$$\n\nShow that the determinant is not equal to zero. (Consider the corresponding system of linear homogeneous equations.)","subjects":["linear algebra","geometry","complex numbers","diagonal dominance"]},{"id":"1949-A1","year":1949,"session":"A","number":1,"statement":"1. Answer either (i) or (ii):\n(i) Three straight lines pass through the three points $ (0,-a, a),(a, 0 $, $ -a) $, and $ (-a, a, 0) $, parallel to the $ x $-axis, $ y $-axis, and $ z $-axis, respectively; $ a $ $ >0 $. A variable straight line moves so that it has one point in common with each of the three given straight lines. Find the equation of the surface described by the variable line.\n(page 264)\n(ii) Which planes cut the surface $ x y+x z+y z=0 $ in (1) circles, (2) parabolas?","subjects":["geometry","quadric surfaces","conic sections"]},{"id":"1949-A2","year":1949,"session":"A","number":2,"statement":"2. We consider three vectors drawn from the same initial point $ O $, of lengths $ a, b $, and $ c $, respectively. Let $ E $ be the parallelepiped with vertex $ O $ of which the given vectors are the edges and $ H $ the parallelepiped with vertex $ O $ of which these vectors are the altitudes. Show that the product of the volumes of $ E $ and $ H $ equals ( $ a b c)^{2} $, and generalize the theorem, with proof, to $ n $ dimensions.","subjects":["linear algebra","geometry","determinants","volume","dual bases"]},{"id":"1949-A3","year":1949,"session":"A","number":3,"statement":"3. Assume that the complex numbers $ a_{1}, a_{2}, \\ldots, a_{n}, \\ldots $ are all different from zero, and that $ \\left|a_{r}-a_{s}\\right|>1 $ for $ r \\neq s $. Show that the series\n$$\n\\sum_{n=1}^{\\infty} \\frac{1}{a_{n}{ }^{3}}\n$$\nconverges.","subjects":["analysis","complex series","geometric packing"]},{"id":"1949-A4","year":1949,"session":"A","number":4,"statement":"4. Given that $ P $ is a point inside a tetrahedron with vertices at $ A, B, C $, and $ D $, such that the sum of the distances $ P A+P B+P C+P D $ is a minimum, show that the two angles $ \\angle A P B $ and $ \\angle C P D $ are equal and are bisected by the same straight line. What other pairs of angles must be equal?","subjects":["geometry","optimization","geometric median","vectors"]},{"id":"1949-A5","year":1949,"session":"A","number":5,"statement":"5. How many roots of the equation $ z^{6}+6 z+10=0 $ lie in each quadrant of the complex plane?","subjects":["complex analysis","polynomial roots","root location"]},{"id":"1949-A6","year":1949,"session":"A","number":6,"statement":"$$\n\\begin{array}{l}\n\\text { 6. Prove that for every real or complex } \\boldsymbol{x}\\\\\n\\prod_{k=1}^{\\infty} \\frac{1+2 \\cos \\frac{2 x}{3^{k}}}{3}=\\frac{\\sin x}{x}\n\\end{array}\n$$","subjects":["analysis","infinite products","trigonometric identities"]},{"id":"1949-B1","year":1949,"session":"B","number":1,"statement":"1. Each rational number $ p / q $ ( $ p, q $ relatively prime positive integers) of the open interval $ (0,1) $ is covered by a closed interval of length $ 1 / 2 q^{2} $, whose center is at $ p / q $. Prove that $ \\sqrt{2} / 2 $ is not covered by any of the above closed intervals.","subjects":["number theory","diophantine approximation","quadratic irrationals"]},{"id":"1949-B2","year":1949,"session":"B","number":2,"statement":"2. Answer either (i) or (ii):\n(i) Prove that\n$$\n\\sum_{n=2}^{\\infty} \\frac{\\cos (\\log \\log n)}{\\log n}\n$$\ndiverges.\n(page 277)\n(ii) Assume that $ p>0, a>0 $, and $ a c-b^{2}>0 $, and show that\n$$\n\\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} \\frac{d x d y}{\\left(p+a x^{2}+2 b x y+c y^{2}\\right)^{2}}=\\pi p^{-1}\\left(a c-b^{2}\\right)^{-1/2}\n$$","subjects":["analysis","series divergence","multivariable integration","quadratic forms"]},{"id":"1949-B3","year":1949,"session":"B","number":3,"statement":"3. Let $ K $ be a closed plane curve such that the distance between any two points of $ K $ is always less than 1 . Show that $ K $ lies inside a circle of radius $ 1 / \\sqrt{3} $","subjects":["geometry","convexity","enclosing circles"]},{"id":"1949-B4","year":1949,"session":"B","number":4,"statement":"4. Show that the coefficients $ a_{1}, a_{2}, a_{3}, \\ldots $ in the expansion $ \\frac{1}{4}[1+x- $ $ \\left.\\left(1-6 x+x^{2}\\right)^{1/2}\\right]=a_{1} x+a_{2} x^{2}+a_{3} x^{3}+\\cdots $ are positive integers.","subjects":["algebra","integer coefficients","combinatorics"]},{"id":"1949-B5","year":1949,"session":"B","number":5,"statement":"5. Let $ a_{1}, a_{2}, \\ldots, a_{n}, \\ldots $ be an arbitrary sequence of positive numbers. Show that\n$$\n\\lim _{n \\rightarrow \\infty} \\sup \\left(\\frac{a_{1}+a_{n+1}}{a_{n}}\\right)^{n} \\geq e\n$$","subjects":["analysis","sequences","limsup","inequalities"]},{"id":"1949-B6","year":1949,"session":"B","number":6,"statement":"6. Let $ C $ be a closed convex curve with a continuously turning tangent and let $ O $ be a point inside $ C $. With each point $ P $ on $ C $ we associate the point $ T(P) $ on $ C $ which is defined as follows: Draw the tangent to $ C $ at $ P $ and from $ O $ drop the perpendicular to that tangent. $ T(P) $ is then the point at which this perpendicular intersects the curve $ C $.\n\nStarting now with a point $ P_{0} $ on $ C $, we define points $ P_{n} $ by the formula $ P_{n}= $ $ T\\left(P_{n-1}\\right), n \\geq 1 $. Prove that the points $ P_{n} $ approach a limit, and characterize those points which can be limits of sequences $ P_{n} $. (You may consider the facts that $ T $ is a continuous transformation and that a convex curve lies on one side of each of its tangents as not requiring proofs.)","subjects":["geometry","convex curves","iteration","convergence"],"errata":"Select the intersection on the ray from $O$ toward the perpendicular foot on the tangent. For a closed regular convex $C^1$ planar curve with $O$ strictly interior, all iterates converge. The possible limits are exactly the points whose tangent is perpendicular to $OP$. A full perpendicular line has two boundary intersections and does not specify this transformation. The book uses the ray convention."},{"id":"1950-A1","year":1950,"session":"A","number":1,"statement":"1. For what values of the ratio $ a / b $ is the limacon $ r=a-b \\cos \\theta $ a convex curve? $ (a>b>0) $","subjects":["geometry","polar curves","convexity","curvature"]},{"id":"1950-A2","year":1950,"session":"A","number":2,"statement":"2. Answer both (i) and (ii). Test for convergence the series\n(i) $ \\frac{1}{\\log (2!)}+\\frac{1}{\\log (3!)}+\\frac{1}{\\log (4!)}+\\cdots+\\frac{1}{\\log (n!)}+\\cdots $\n(ii) $ \\frac{1}{3}+\\frac{1}{3 \\sqrt{3}}+\\frac{1}{3 \\sqrt{3} \\sqrt[3]{3}}+\\cdots+\\frac{1}{3 \\sqrt{3} \\sqrt[3]{3} \\cdots \\sqrt[n]{3}}+\\cdots $.","subjects":["analysis","series","convergence tests"]},{"id":"1950-A3","year":1950,"session":"A","number":3,"statement":"3. The sequence $ x_{0}, x_{1}, x_{2}, \\ldots $ is defined by the conditions\n$$\nx_{0}=a, x_{1}=b, x_{n+1}=\\frac{x_{n-1}+(2 n-1) x_{n}}{2 n} \\quad \\text { for } n \\geq 1,\n$$\nwhere $ a $ and $ b $ are given numbers. Express $ \\lim _{n-\\infty} x_{n} $ concisely in terms of $ a $ and $ b $.","subjects":["analysis","recurrences","sequences","exponential series"]},{"id":"1950-A4","year":1950,"session":"A","number":4,"statement":"4. Answer either (i) or (ii).\n(i) In a right prism with triangular base, given the sum of the areas of three mutually adjacent faces (that is, of two lateral faces and one base), show that these faces are of equal area and perpendicular to each other when the volume attains its maximum.\n(page 292)\n(ii) Show that\n$$\n\\frac{\\frac{x}{1}+\\frac{x^{3}}{1 \\cdot 3}+\\frac{x^{5}}{1 \\cdot 3 \\cdot 5}+\\frac{x^{7}}{1 \\cdot 3 \\cdot 5 \\cdot 7}+\\cdots}{1+\\frac{x^{2}}{2}+\\frac{x^{4}}{2 \\cdot 4}+\\frac{x^{6}}{2 \\cdot 4 \\cdot 6}+\\cdots}=\\int_{0}^{x} e^{-t^2/2} d t .\n$$","subjects":["analysis","geometry","optimization","power series"]},{"id":"1950-A5","year":1950,"session":"A","number":5,"statement":"5. A function $ D(n) $ of the positive integral variable $ n $ is defined by the following properties: $ D(1)=0, D(p)=1 $ if $ p $ is a prime, $ D(u v)=u D(v)+ $ $ v D(u) $ for any two positive integers $ u $ and $ v $. Answer all three parts below.\n(i) Show that these properties are compatible and determine uniquely $ \\boldsymbol{D}(n) $. (Derive a formula for $ \\boldsymbol{D}(n) / n $, assuming that $ n=p_{1}^{\\alpha_{1}} p_{2}{ }^{\\alpha_{2}} \\cdots p_{k}{ }^{\\alpha_{k}} $ where $ p_{1}, p_{2}, \\ldots, p_{k} $ are different primes.)\n(ii) For what values of $ n $ is $ D(n)=n $ ?\n(iii) Define $ D^{2}(n)=D[D(n)] $, etc., and find the limit of $ D^{m}(63) $ as $ m $ tends to $ \\infty $.","subjects":["number theory","arithmetic derivative","prime factorization","iteration"]},{"id":"1950-A6","year":1950,"session":"A","number":6,"statement":"6. Each coefficient $ a_{n} $ of the power series\n$$\na_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+\\cdots=f(x)\n$$\nhas either the value 1 or the value 0 . Prove the easier of the two assertions:\n(i) If $ f(0.5) $ is a rational number, $ f(x) $ is a rational function.\n(ii) If $ f(0.5) $ is not a rational number, $ f(x) $ is not a rational function.","subjects":["algebra","analysis","binary expansions"]},{"id":"1950-B1","year":1950,"session":"B","number":1,"statement":"1. In each of $ n $ houses on a straight street are one or more boys. At what point should all the boys meet so that the sum of the distances that they walk is as small as possible?","subjects":["optimization","absolute deviations","weighted median"]},{"id":"1950-B2","year":1950,"session":"B","number":2,"statement":"2. Two obvious approximations to the length of the perimeter of the ellipse with semi-axes $ a $ and $ b $ are $ \\pi(a+b) $ and $ 2 \\pi(a b)^{1/2} $. Which one comes nearer the truth when the ratio $ b / a $ is very close to $ 1 ? $","subjects":["analysis","geometry","ellipse perimeter","asymptotic expansion","inequalities"]},{"id":"1950-B3","year":1950,"session":"B","number":3,"statement":"3. In the Gregorian calendar:\n(i) years not divisible by 4 are common years;\n(ii) years divisible by 4 but not by 100 are leap years;\n(iii) years divisible by 100 but not by 400 are common years;\n(iv) years divisible by 400 are leap years;\n(v) a leap year contains 366 days; a common year 365 days.\n\nProve that the probability that Christmas falls on a Wednesday is not $ 1 / 7 $.","subjects":["combinatorics","calendar arithmetic","periodicity","probability"]},{"id":"1950-B4","year":1950,"session":"B","number":4,"statement":"4. The cross-section of a right cylinder is an ellipse, with semi-axes $ a $ and $ b $, where $ a>b $. The cylinder is very long, made of very light homogeneous material. The cylinder rests on the horizontal ground which it touches along the straight line joining the lower endpoints of the minor axes of its several cross-sections. Along the upper endpoints of these minor axes lies a very heavy homogeneous wire, straight and just as long as the cylinder. The wire and the cylinder are rigidly connected. We neglect the weight of the cylinder, the breadth of the wire, and the friction of the ground.\n\nThe system described is in equilibrium, because of its symmetry. This equilibrium seems to be stable when the ratio $ b / a $ is very small, but unstable when this ratio comes close to 1 . Examine this assertion and find the value of the ratio $ b / a $ which separates the cases of stable and unstable equilibrium.","subjects":["mechanics","geometry","stability","ellipse","potential energy"]},{"id":"1950-B5","year":1950,"session":"B","number":5,"statement":"5. Answer either (i) or (ii).\n(i) Given that the sequence whose $ n $th term is $ \\left(s_{n}+2 s_{n+1}\\right) $ converges, show that the sequence $ \\left\\{s_{n}\\right\\} $ converges also.\n(page 306)\n(ii) A plane varies so that it includes a cone of constant volume equal to $ \\pi a^{3} / 3 $ with the surface the equation of which in rectangular coordinates is $ 2 x y=z^{2} $. Find the equation of the envelope of the various positions of this plane.\n\nState the result so that it applies to a general cone (that is, conic surface) of the second order.","subjects":["analysis","sequences","stable recurrences","geometry","quadratic cones","envelopes","volume"]},{"id":"1950-B6","year":1950,"session":"B","number":6,"statement":"6. Consider the closed plane curves $ C_{i} $ and $ C_{o} $, their respective lengths $ \\left|C_{i}\\right| $ and $ \\left|C_{o}\\right| $, the closed surfaces $ S_{i} $ and $ S_{o} $, and their respective areas $ \\left|S_{i}\\right| $ and $ \\left|S_{o}\\right| $. Assume that $ C_{i} $ lies inside $ C_{o} $ and $ S_{i} $ inside $ S_{o} $. (Subscript $ i $ stands for \"inner,\" $ o $ for \"outer.\") Prove the correct assertions among the following four, and disprove the others.\n(i) If $ C_{i} $ is convex, $ \\left|C_{i}\\right| \\leq\\left|C_{o}\\right| $.\n(ii) If $ S_{i} $ is convex, $ \\left|S_{i}\\right| \\leq\\left|S_{o}\\right| $.\n(iii) If $ C_{o} $ is the smallest convex curve containing $ C_{i} $, then $ \\left|C_{o}\\right| \\leq\\left|C_{i}\\right| $.\n(iv) If $ S_{o} $ is the smallest convex surface containing $ S_{i} $, then $ \\left|S_{o}\\right| \\leq\\left|S_{i}\\right| $.\n\nYou may assume that $ C_{i} $ and $ C_{o} $ are polygons and $ S_{i} $ and $ S_{o} $ polyhedra. (Why?)","subjects":["geometry","convexity","perimeter","surface area","counterexamples","integral geometry"]},{"id":"1951-A1","year":1951,"session":"A","number":1,"statement":"1. Show that the determinant:\n$$\n\\left|\\begin{array}{rrrr}\n0 & a & b & c \\\\\n-a & 0 & d & e \\\\\n-b & -d & 0 & f \\\\\n-c & -e & -f & 0\n\\end{array}\\right|\n$$\nis non-negative, if its elements $ a, b, c $, etc., are real.","subjects":["linear algebra","skew-symmetric matrices","determinants"]},{"id":"1951-A2","year":1951,"session":"A","number":2,"statement":"2. In the plane, what is the locus of points the sum of the squares of whose distances from $ n $ fixed points is a constant? What restrictions, stated in geometric terms, must be put on the constant so that the locus is non-null?","subjects":["geometry","locus","centroid","sum of squared distances"]},{"id":"1951-A3","year":1951,"session":"A","number":3,"statement":"$$\n\\begin{array}{l}\n\\text { 3. Find the sum to infinity of the series: }\\\\\n1-\\frac{1}{4}+\\frac{1}{7}-\\frac{1}{10}+\\cdots+\\frac{(-1)^{n+1}}{3 n-2}+\\cdots . \\quad(\\text { page } 319)\n\\end{array}\n$$","subjects":["analysis","series summation","rational integration"]},{"id":"1951-A4","year":1951,"session":"A","number":4,"statement":"$$\n\\begin{array}{l}\n\\text { 4. Trace the curve whose equation is: }\\\\\ny^{4}-x^{4}-96 y^{2}+100 x^{2}=0\n\\end{array}\n$$","subjects":["geometry","algebraic curves","curve tracing","implicit differentiation"]},{"id":"1951-A5","year":1951,"session":"A","number":5,"statement":"5. Consider in the plane the network of points having integral coordinates. For lines having rational slope show that:\n(i) the line passes through no points of the network or through infinitely many;\n(ii) there exists for each line a positive number $ d $ having the property that no point of the network, except such as may be on the line, is closer to the line than the distance $ d $.","subjects":["number theory","geometry","integer lattice","rational slope","lattices"]},{"id":"1951-A6","year":1951,"session":"A","number":6,"statement":"6. Determine the position of a normal chord of a parabola such that it cuts off of the parabola a segment of minimum area.","subjects":["geometry","parabolas","optimization","area"]},{"id":"1951-A7","year":1951,"session":"A","number":7,"statement":"7. Show that if the series $ a_{1}+a_{2}+a_{3}+\\cdots+a_{n}+\\cdots $ converges, then the series $ a_1+\\frac{a_2}{2}+\\frac{a_3}{3}+\\cdots+\\frac{a_n}{n}+\\cdots $ converges also.","subjects":["analysis","series","dirichlet test","probability"]},{"id":"1951-B1","year":1951,"session":"B","number":1,"statement":"1. Find the condition that the functions $ M(x, y) $ and $ N(x, y) $ must satisfy in order that the differential equation $ M d x+N d y=0 $ shall have an integrating factor of the form $ f(x y) $. You may assume that $ M $ and $ N $ have continuous partial derivatives of all orders.","subjects":["analysis","differential equations","integrating factors","differential forms"]},{"id":"1951-B2","year":1951,"session":"B","number":2,"statement":"2. Two functions of $ x $ are differentiable and not identically zero. Find an example of two such functions having the property that the derivative of their quotient is the quotient of their derivatives.","subjects":["analysis","differentiation","functional identities"]},{"id":"1951-B3","year":1951,"session":"B","number":3,"statement":"$$\n\\begin{array}{l}\n\\text { 3. Show that if } x \\text { is positive, then }\\\\\n\\log _{e}(1+1 / x)>\\frac{1}{1+x}\n\\end{array}\n$$","subjects":["analysis","logarithmic inequalities"]},{"id":"1951-B4","year":1951,"session":"B","number":4,"statement":"4. Investigate, in any way which yields significant results, the existence, in the plane, of the configuration consisting of an ellipse simultaneously tangent to four distinct concentric circles.","subjects":["geometry","ellipses","concentric circles","normals"]},{"id":"1951-B5","year":1951,"session":"B","number":5,"statement":"5. A plane through the center of a torus is tangent to the torus. Prove that the intersection of the plane and the torus consists of two circles.","subjects":["geometry","torus","plane sections","villarceau circles"]},{"id":"1951-B6","year":1951,"session":"B","number":6,"statement":"6. Assuming that all the roots of the cubic equation $ x^{3}+a x^{2}+b x+c= $ 0 are real, show that the difference between the greatest and the least roots is not less than $ \\sqrt{a^2-3b} $ or greater than $ 2\\sqrt{a^2-3b}/\\sqrt{3} $.","subjects":["algebra","real-rooted polynomials","root spacing","inequalities"]},{"id":"1951-B7","year":1951,"session":"B","number":7,"statement":"7. Find the volume of the four-dimensional hypersphere $ x^{2}+y^{2}+z^{2}+ $ $ t^{2}=r^{2} $. and also the hypervolume of its interior $ x^{2}+y^{2}+z^{2}+t^{2}<r^{2} $.","subjects":["analysis","geometry","four-dimensional volume","gaussian integrals"]},{"id":"1952-A1","year":1952,"session":"A","number":1,"statement":"1. Let\n$$\nf(x)=\\sum_{i=0}^{i=n} a_{i} x^{n-i}\n$$\nbe a polynomial of degree $ n $ with integral coefficients. If $ a_{0}, a_{n} $, and $ f(1) $ are odd, prove that $ f(x)=0 $ has no rational roots.","subjects":["number theory","integer polynomials","rational roots","parity"]},{"id":"1952-A2","year":1952,"session":"A","number":2,"statement":"2. Show that the equation\n$$\n\\left(9-x^{2}\\right)\\left(\\frac{d y}{d x}\\right)^{2}=\\left(9-y^{2}\\right)\n$$\ncharacterizes a family of conics touching the four sides of a fixed square.","subjects":["analysis","differential equations","conic sections","singular solutions"],"errata":"The conic characterization is local on connected regular graph arcs where $(9-x^2)(9-y^2)>0$. These arcs lie in $x^2+y^2-2\\lambda xy=9(1-\\lambda^2)$, including the diagonal degenerate cases. Nondegenerate conics touch the four side lines, with hyperbolic contacts on their extensions. The unrestricted equation also has $y=\\pm3$ and $C^1$ glued solutions. The book discusses these failures of a global conic characterization."},{"id":"1952-A3","year":1952,"session":"A","number":3,"statement":"3. Develop necessary and sufficient conditions which ensure that $ r_{1}, r_{2}, r_{3} $ and $ r_{1}{ }^{2}, r_{2}{ }^{2}, r_{3}{ }^{2} $ are simultaneously roots of the equation $ x^{3}+a x^{2}+b x+c $ $ =0 $","subjects":["algebra","symmetric polynomials","root multisets","polynomial divisibility","discrete dynamics"],"errata":"The squared roots can mean the same full root multiset or merely roots from the same distinct root set. These readings differ, for example for $x(x-1)(x+1)$. The book discusses both interpretations. The solutions give necessary and sufficient coefficient tests for both readings over $\\mathbb C$, with real coefficients obtained by restriction."},{"id":"1952-A4","year":1952,"session":"A","number":4,"statement":"4. The flag of the United Nations consists of a polar map of the world, with the North Pole as center, extending approximately to $ 45^{\\circ} $ South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude $ 30^{\\circ} \\mathrm{S} $. In the very close vicinity of this parallel how much are East and West distances exaggerated as compared to North and South distances?","subjects":["geometry","spherical geometry","map projections","local scale","cartography"]},{"id":"1952-A5","year":1952,"session":"A","number":5,"statement":"5. Let $ a_{j}(j=1,2, \\ldots, n) $ be entirely arbitrary numbers except that no one is equal to unity. Prove\n$$\n\\left.a_{1}+\\sum_{i=2}^{n} a_{i} \\prod_{j=1}^{i-1}\\left(1-a_{j}\\right)=1-\\prod_{j=1}^{n}\\left(1-a_{j}\\right) \\quad \\quad \\text { (page } 350\\right)\n$$","subjects":["algebra","finite products","probability","polynomial algebra"]},{"id":"1952-A6","year":1952,"session":"A","number":6,"statement":"6. A man has a rectangular block of wood $ m $ by $ n $ by $ r $ inches $ (m, n $, and $ r $ are integers). He paints the entire surface of the block, cuts the block into inch cubes, and notices that exactly half the cubes are completely unpainted. Prove that the number of essentially different blocks with this property is finite. (Do not attempt to enumerate them.)","subjects":["number theory","diophantine equations","finiteness","integer boxes"]},{"id":"1952-A7","year":1952,"session":"A","number":7,"statement":"7. Directed lines are drawn from the center of a circle, making angles of 0 , $ \\pm 1, \\pm 2, \\pm 3, \\ldots $ (measured in radians from a prime direction). If these lines meet the circle in points $ P_{0}, P_{1}, P_{-1}, P_{2}, P_{-2}, \\ldots $, show that there is no interval on the circumference of the circle which does not contain some $ P_{ \\pm i} $. (You may assume that $ \\pi $ is irrational.)","subjects":["number theory","irrational rotation","density"]},{"id":"1952-B1","year":1952,"session":"B","number":1,"statement":"1. A mathematical moron is given two sides and the included angle of a triangle and attempts to use the Law of Cosines: $ a^{2}=b^{2}+c^{2}-2 b c \\cos A $, to find the third side $ a $. He uses logarithms as follows. He finds $ \\log b $ and doubles it; adds to that the double of $ \\log c $; subtracts the sum of the logarithms of $ 2, b, c $, and $ \\cos A $; divides the result by 2 ; and takes the antilogarithm. Although his method may be open to suspicion, his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result?","subjects":["geometry","trigonometry","logarithms","law of cosines"]},{"id":"1952-B2","year":1952,"session":"B","number":2,"statement":"2. Find the surface generated by the solutions of\n$$\n\\frac{d x}{y z}=\\frac{d y}{z x}=\\frac{d z}{x y}\n$$\nwhich intersects the circle $ y^{2}+z^{2}=1, x=0 $.","subjects":["analysis","differential equations","first integrals","quadric surfaces"]},{"id":"1952-B3","year":1952,"session":"B","number":3,"statement":"3. Develop necessary and sufficient conditions that the equation\n$$\n\\left|\\begin{array}{ccc}\n0 & a_{1}-x & a_{2}-x \\\\\n-a_{1}-x & 0 & a_{3}-x \\\\\n-a_{2}-x & -a_{3}-x & 0\n\\end{array}\\right|=0 \\quad\\left(a_{i} \\neq 0\\right)\n$$\nshall have a multiple root.","subjects":["algebra","determinants","multiple roots","linear algebra"]},{"id":"1952-B4","year":1952,"session":"B","number":4,"statement":"4. A homogeneous solid body is made by joining a base of a circular cylinder of height $ h $ and radius $ r $, and the base of a hemisphere of radius $ r $. This body is placed with the hemispherical end on a horizontal table, with the axis of the cylinder in a vertical position, and then slightly oscillated. It is intuitively evident that if $ r $ is large as compared to $ h $, the equilibrium will be stable; but if $ r $ is small as compared to $ h $, the equilibrium will be unstable. What is the critical value of the ratio $ r / h $ which enables the body to rest in neutral equilibrium in any position?","subjects":["mechanics","centroid","neutral equilibrium","solid geometry"]},{"id":"1952-B5","year":1952,"session":"B","number":5,"statement":"5. If the terms of a sequence, $ a_{n} $, are monotonic, and if $ \\sum_{1}^{\\infty} a_{n} $ converges, show that $ \\sum_{1}^{\\infty} n\\left(a_{n}-a_{n+1}\\right) $ converges.","subjects":["analysis","monotone sequences","series"]},{"id":"1952-B6","year":1952,"session":"B","number":6,"statement":"6. Prove the necessary and sufficient condition that a triangle inscribed in an ellipse shall have maximum area is that its centroid coincide with the center of the ellipse.","subjects":["geometry","ellipses","inscribed triangles","affine geometry","area optimization"]},{"id":"1952-B7","year":1952,"session":"B","number":7,"statement":"7. Given any real number $ N_{0} $, if $ N_{j+1}=\\cos N_{j} $, prove that $ \\lim _{j-\\infty} N_{j} $ exists and is independent of $ N_{0} $.","subjects":["analysis","iteration","fixed points","contraction"]},{"id":"1953-A1","year":1953,"session":"A","number":1,"statement":"1. Prove that, for every positive integer $ n $,\n$$\n\\sqrt{1}+\\sqrt{2}+\\cdots+\\sqrt{n}\n$$\nis more than $ \\frac{2}{3} n \\sqrt{n} $ and less than\n$$\n\\frac{4 n+3}{6} \\sqrt{n} .\n$$","subjects":["analysis","sum estimates","concavity","integral inequalities"]},{"id":"1953-A2","year":1953,"session":"A","number":2,"statement":"2. Six points are in general position in space (no three in a line, no four in a plane). The fifteen line segments joining them in pairs are drawn and then painted, some segments red, some blue. Prove that some triangle has all its sides the same color.","subjects":["combinatorics","ramsey theory","edge coloring","graph theory"]},{"id":"1953-A3","year":1953,"session":"A","number":3,"statement":"3. If $ x_{1}, x_{2}, x_{3} $ are real numbers and the sum of any two is greater than the third, show that\n$$\n\\frac{2}{3} \\sum_{i=1}^{3} x_{i} \\sum_{i=1}^{3} x_{i}^{2}>\\sum_{i=1}^{3} x_{i}^{3}+x_{1} x_{2} x_{3}\n$$","subjects":["algebra","triangle inequalities","symmetric inequalities"]},{"id":"1953-A4","year":1953,"session":"A","number":4,"statement":"4. From the identity\n$$\n\\begin{aligned}\n\\int_{0}^{\\pi / 2} \\log \\sin 2 x d x & =\\int_{0}^{\\pi / 2} \\log \\sin x d x \\\\\n& +\\int_{0}^{\\pi / 2} \\log \\cos x d x+\\int_{0}^{\\pi / 2} \\log 2 d x\n\\end{aligned}\n$$\ndeduce the value of\n$$\n\\int_{0}^{\\pi / 2} \\log \\sin x d x\n$$","subjects":["analysis","improper integrals","trigonometric integrals"]},{"id":"1953-A5","year":1953,"session":"A","number":5,"statement":"5. Let $ P $ be a point from which three distinct normals can be drawn to a parabola. Show that the sum of the angles which these three normals make with the axis exceeds by a multiple of $ \\pi $ the angle which the line joining $ P $ to the focus makes with the axis.","subjects":["geometry","parabola normals","complex numbers"]},{"id":"1953-A6","year":1953,"session":"A","number":6,"statement":"6. Show that the sequence\n$$\n\\sqrt{7}, \\sqrt{7-\\sqrt{7}}, \\sqrt{7-\\sqrt{7+\\sqrt{7}}} \\sqrt{7-\\sqrt{7+\\sqrt{7-\\sqrt{7}}}} \\ldots\n$$\nconverges, and evaluate the limit.","subjects":["analysis","nested radicals","recurrences","contraction"],"errata":"The canonical dataset transcription loses the separator between the third and fourth radicals. The book prints them separately. Use $x_0=\\sqrt7$, $x_1=\\sqrt{7-\\sqrt7}$ and $x_{n+2}=\\sqrt{7-\\sqrt{7+x_n}}$ for $n\\ge0$, with principal real square roots. This sequence converges to $2$. This is a dataset-rendering correction, not an established error in the printed book or examination."},{"id":"1953-A7","year":1953,"session":"A","number":7,"statement":"7. Assuming that the roots of $ x^{3}+p x^{2}+q x+r=0 $ are all real and positive, find the relation between $ p, q $, and $ r $ which is a necessary and sufficient condition that the roots may be the cosines of the angles of a triangle.","subjects":["algebra","geometry","cubic roots","triangle cosine identity"]},{"id":"1953-B1","year":1953,"session":"B","number":1,"statement":"1. Is the infinite series\n$$\n\\sum_{n=1}^{\\infty} \\frac{1}{n^{(n+1) / n}}\n$$\nconvergent? Prove your statement.","subjects":["analysis","series","harmonic comparison"]},{"id":"1953-B2","year":1953,"session":"B","number":2,"statement":"2. Let $ a_{0}, a_{1}, \\ldots, a_{n} $ be real numbers and let $ f(x)=a_{0}+a_{1} x+\\ldots+ $ $ a_{n} x^{n} $. Suppose that, for every integer $ i, f(i) $ is an integer. Prove that $ n!\\cdot a_{k} $ is an integer for each $ k $.","subjects":["algebra","integer-valued polynomials","finite differences","interpolation"]},{"id":"1953-B3","year":1953,"session":"B","number":3,"statement":"3. Solve the equations\n$$\n\\frac{d y}{d x}=z(y+z)^{n} \\quad \\frac{d z}{d x}=y(y+z)^{n}\n$$\ngiven the initial conditions $ y=1 $ and $ z=0 $ when $ x=0 $.","subjects":["analysis","differential equations","nonlinear systems","diagonalization"]},{"id":"1953-B4","year":1953,"session":"B","number":4,"statement":"4. Determine the equation of a surface in three dimensional cartesian space which has the following properties: (a) it passes through the point $ (1,1 $, 1); and (b) if the tangent plane be drawn at any point $ P $, and $ A, B $ and $ C $ are the intersections of this plane with the $ x, y $ and $ z $ axes respectively, then $ P $ is the orthocenter (intersection of the altitudes) of the triangle $ A B C $.","subjects":["geometry","differential geometry","tangent planes","orthocenter"]},{"id":"1953-B5","year":1953,"session":"B","number":5,"statement":"5. Show that the roots of $ x^{4}+a x^{3}+b x^{2}+c x+d=0 $, if suitably numbered, satisfy the relation $ r_1/r_2=r_3/r_4 $. provided $ a^{2} d=c^{2} \\neq 0 $","subjects":["algebra","quartic polynomials","root pairing","symmetric functions","polynomial roots"]},{"id":"1953-B6","year":1953,"session":"B","number":6,"statement":"6. $ P $ and $ Q $ are any points inside a circle $ (C) $ with center $ C $, such that $ C P $ $ =C Q $. Determine the location of a point $ Z $ on $ (C) $ such that $ P Z=Q Z $ shall be a minimum.","subjects":["geometry","circle optimization","sum of distances","inversion","geometric optimization"],"errata":"The equals sign in $PZ=QZ$ is a misprint in the original contest wording, explicitly corrected by the book's note on printed p. 40, PDF p. 52. The intended objective is $PZ+QZ$, with $P,Q$ still strictly inside the circle and $CP=CQ$. The corrected review target is the following. Let $P,Q$ be points inside a circle with center $C$, with $CP=CQ$. Determine all points $Z$ on the circle minimizing $PZ+QZ$. The canonical statement is retained for provenance. This note records the target for review and does not certify any proof."},{"id":"1953-B7","year":1953,"session":"B","number":7,"statement":"7. Let $ w $ be an irrational number with $ 0<w<1 $. Prove that $ w $ has a unique convergent expansion of the form\n$$\nw=\\frac{1}{p_{0}}-\\frac{1}{p_{0} p_{1}}+\\frac{1}{p_{0} p_{1} p_{2}}-\\frac{1}{p_{0} p_{1} p_{2} p_{3}}+\\cdots\n$$\nwhere $ p_{0}, p_{1}, p_{2}, \\ldots $ are integers and $ 1 \\leq p_{0}<p_{1}<p_{2}<\\cdots $. If $ w= $ $ \\frac{1}{2} \\sqrt{2} $, find $ p_{0}, p_{1}, p_{2} $.","subjects":["number theory","alternating expansions","irrational numbers","real analysis"]},{"id":"1954-A1","year":1954,"session":"A","number":1,"statement":"1. Let $ n $ be an odd integer greater than 1 . Let $ A $ be an $ n $ by $ n $ symmetric matrix such that each row and each column of $ A $ consists of some permutation of the integers $ 1, \\ldots, n $. Show that each one of the integers $ 1, \\ldots, n $ must appear in the main diagonal of $ A $.","subjects":["combinatorics","matrices","parity","symmetric latin squares"]},{"id":"1954-A2","year":1954,"session":"A","number":2,"statement":"2. Consider any five points $ P_{1}, P_{2}, P_{3}, P_{4}, P_{5} $ in the interior of a square $ S $ of side-length 1. Denote by $ d_{i j} $ the distance between the points $ P_{i} $ and $ P_{j} $. Prove that at least one of the distances $ d_{i j} $ is less than $ \\sqrt{2} / 2 $. Can $ \\sqrt{2} / 2 $ be replaced by a smaller number in this statement?","subjects":["geometry","point packing","sharp bounds"]},{"id":"1954-A3","year":1954,"session":"A","number":3,"statement":"3. Prove that if the family of integral curves of the differential equation\n$$\n\\frac{d y}{d x}+p(x) y=q(x) \\quad p(x) \\cdot q(x) \\neq 0\n$$\nis cut by the line $ x=k $, the tangents at the points of intersection are concurrent.","subjects":["analysis","differential equations","tangent lines","concurrency"]},{"id":"1954-A4","year":1954,"session":"A","number":4,"statement":"4. A uniform rod of length $ 2 k $ and weight $ w $ rests with the end $ A $ against a smooth vertical wall, while to the lower end $ B $ is fastened a string $ B C $ of length $ 2 b $ coming from a point $ C $ in the wall directly above $ A $. If the system is in equilibrium, determine the angle $ A B C $.","subjects":["mechanics","statics","rod and string","equilibrium"],"errata":"Reading convention, not a correction. The endpoint ordering $B$ below $A$ and $C$ strictly above $A$ is already stated in the text and confirmed by the book's solution figure. Lengths and weight have their ordinary positive physical meanings. The explicit target retains every geometrically feasible equilibrium; no extra hypothesis is inferred from the drawing's proportions."},{"id":"1954-A5","year":1954,"session":"A","number":5,"statement":"5. If $ f(x) $ is a real-valued function defined for $ 0<x<1 $, then the formula $ f(x)=o(x) $ is an abbreviation for the statement that\n$$\n\\frac{f(x)}{x} \\rightarrow 0 \\quad \\text { as } x \\rightarrow 0\n$$\n\nKeeping this in mind, prove the following: if\n$$\n\\lim _{x \\rightarrow 0} f(x)=0 \\text { and } f(x)-f\\left(\\frac{x}{2}\\right)=o(x)\n$$\nthen $ f(x)=o(x) $.","subjects":["analysis","little-o estimates","dyadic scaling"]},{"id":"1954-A6","year":1954,"session":"A","number":6,"statement":"6. Suppose that $ u_{0}, u_{1}, u_{2}, \\ldots $ is a sequence of real numbers such that\n$$\nu_{n}=\\sum_{k=1}^{\\infty} u_{n+k}^{2} \\quad \\text { for } n=0,1,2, \\ldots\n$$\n\nProve that if $ \\Sigma u_{n} $ converges then $ u_{k}=0 $ for all $ k $.","subjects":["analysis","nonlinear recurrences","series","rigidity"]},{"id":"1954-A7","year":1954,"session":"A","number":7,"statement":"7. Prove that there are no integers $ x $ and $ y $ for which\n$$\nx^{2}+3 x y-2 y^{2}=122\n$$","subjects":["number theory","diophantine equations","quadratic residues"]},{"id":"1954-B1","year":1954,"session":"B","number":1,"statement":"1. Show that the equation $ x^{2}-y^{2}=a^{3} $ has always integral solutions for $ x $ and $ y $ whenever $ a $ is a positive integer.","subjects":["number theory","diophantine equations","difference of squares"]},{"id":"1954-B2","year":1954,"session":"B","number":2,"statement":"2. Assume as known the (true) fact that the alternating harmonic series\n(1) $ \\quad 1-1 / 2+1 / 3-1 / 4+1 / 5-1 / 6+1 / 7-1 / 8+\\cdots $ is convergent, and denote its sum by $ s $. Rearrange the series (1) as follows:\n$$\n1+1 / 3-1 / 2+1 / 5+1 / 7-1 / 4+1 / 9+1 / 11-1 / 6+\\cdots\n$$\n\nAssume as known the (true) fact that the series (2) is also convergent, and denote its sum by $ S $. Denote by $ s_{k}, S_{k} $ the $ k $ th partial sum of the series (1) and (2) respectively. Prove the following statements.\n(i) $ \\quad S_{3 n}=s_{4 n}+\\frac{1}{2} s_{2 n} $,\n(ii) $ \\quad S \\neq s $.","subjects":["analysis","infinite series","rearrangements"]},{"id":"1954-B3","year":1954,"session":"B","number":3,"statement":"3. Let $ a $ and $ b $ denote real numbers such that $ a<b $. The symbol $ (a, b) $ will denote the closed interval with the end points $ a, b $. Let there be given a collection of closed intervals $ \\left(a_{1}, b_{1}\\right), \\ldots,\\left(a_{n}, b_{n}\\right) $ such that any two of these closed intervals have at least one point in common. Prove that there exists then a point which is contained in every one of these intervals.","subjects":["geometry","intervals","convexity"]},{"id":"1954-B4","year":1954,"session":"B","number":4,"statement":"4. Given the focus $ f $ and the directrix $ D $ of a parabola $ P $ and a line $ L $, describe (with proof) a Euclidean (i.e., ruler and compass) construction of the point or points of intersection of $ L $ and $ P $. Be sure to identify the case for which there are no points of intersection.","subjects":["geometry","conic sections","euclidean constructions"]},{"id":"1954-B5","year":1954,"session":"B","number":5,"statement":"5. Let $ f(x) $ be a real-valued function, defined for $ -1<x<1 $, such that $ f^{\\prime}(0) $ exists. Let $ a_{n}, b_{n} $ be two sequences such that\n$$\n-1<a_{n}<0<b_{n}<1, \\lim _{n \\rightarrow \\infty} a_{n}=0, \\lim _{n \\rightarrow \\infty} b_{n}=0\n$$\n\nProve that\n$$\n\\lim _{n \\rightarrow \\infty} \\frac{f\\left(b_{n}\\right)-f\\left(a_{n}\\right)}{b_{n}-a_{n}}=f^{\\prime}(0) . \\quad \\quad \\quad \\text { page 397) }\n$$","subjects":["analysis","derivatives","limits"]},{"id":"1954-B6","year":1954,"session":"B","number":6,"statement":"6. Prove that every positive rational number is the sum of a finite number of distinct terms of the series\n$$\n1+\\frac{1}{2}+\\frac{1}{3}+\\cdots+\\frac{1}{n}+\\cdots\n$$","subjects":["number theory","egyptian fractions","rational numbers"]},{"id":"1954-B7","year":1954,"session":"B","number":7,"statement":"7. Show that\n$$\n\\lim _{n \\rightarrow \\infty} \\sum_{s=1}^{n}\\left(\\frac{a+s}{n}\\right)^{n} \\quad(a>0)\n$$\nlies between $ e^{a} $ and $ e^{a+1} $.","subjects":["analysis","sequences and series","asymptotics"]},{"id":"1955-A1","year":1955,"session":"A","number":1,"statement":"1. Prove that there is no set of integers $ m, n, p $ except $ 0,0,0 $ for which $ m $ $ +n \\sqrt{2}+p \\sqrt{3}=0 $.","subjects":["number theory","irrationality","radicals"]},{"id":"1955-A2","year":1955,"session":"A","number":2,"statement":"2. $ A_{1} A_{2} \\ldots A_{n} $ is a regular polygon inscribed in a circle of radius $ r $ and center $ O . P $ is a point on line $ O A_{1} $ extended beyond $ A_{1} $. Show that\n$$\n\\prod_{i=1}^{n} \\overline{P A}_{i}=\\overline{O P}^{n}-r^{n}\n$$","subjects":["geometry","regular polygons","complex numbers"]},{"id":"1955-A3","year":1955,"session":"A","number":3,"statement":"3. Suppose that $ \\sum_{i=1}^{\\infty} x_{i} $ is a convergent series of positive terms which monotonically decrease (that is, $ x_{1} \\geq x_{2} \\geq x_{3} \\geq \\cdots $ ). Let $ P $ denote the set of all numbers which are sums of some (finite or infinite) subseries of $ \\sum_{i=1}^{\\infty} x_{i} $. Show that $ P $ is an interval if and only if\n$$\nx_{n} \\leq \\sum_{i=n+1}^{\\infty} x_{i} \\quad \\text { for every integer } n . \\quad \\text { (page 403) }\n$$","subjects":["analysis","subseries","achievement sets"]},{"id":"1955-A4","year":1955,"session":"A","number":4,"statement":"4. On a circle, $ n $ points are selected and the chords joining them in pairs are drawn. Assuming that no three of these chords are concurrent (except at the endpoints), how many points of intersection are there?","subjects":["combinatorics","geometry","counting intersections"]},{"id":"1955-A5","year":1955,"session":"A","number":5,"statement":"5. If a parabola is given in the plane, find a geometric construction (ruler and compass) for the focus.","subjects":["geometry","parabolas","euclidean constructions"]},{"id":"1955-A6","year":1955,"session":"A","number":6,"statement":"6. Find a necessary and sufficient condition on the positive integer $ n $ that the equation\n$$\nx^{n}+(2+x)^{n}+(2-x)^{n}=0\n$$\nhave a rational root.","subjects":["algebra","rational roots","number theory"]},{"id":"1955-A7","year":1955,"session":"A","number":7,"statement":"7. Consider the function $ f $ defined by the differential equation\n$$\nf^{\\prime \\prime}(x)=\\left(x^{3}+a x\\right) f(x)\n$$\nand the initial conditions $ f(0)=1, f^{\\prime}(0)=0 $. Prove that the roots of $ f $ are bounded above but unbounded below.","subjects":["analysis","ordinary differential equations","oscillation"]},{"id":"1955-B1","year":1955,"session":"B","number":1,"statement":"1. A sphere rolls along two intersecting straight lines. Find the locus of its center.","subjects":["geometry","solid geometry","loci"]},{"id":"1955-B2","year":1955,"session":"B","number":2,"statement":"2. Suppose that $ f $ is a function with two continuous derivatives and $ f(0)= $ 0 . Prove that the function $ g $, defined by $ g(0)=f^{\\prime}(0), g(x)=f(x)/x $ for $ x \\neq $ 0 , has a continuous derivative.","subjects":["analysis","smoothness","removable singularities"]},{"id":"1955-B3","year":1955,"session":"B","number":3,"statement":"3. Prove that there exists no distance-preserving map of a spherical cap into the plane. (Distances on the sphere are to be measured along great circles on the surface.)","subjects":["geometry","spherical geometry","isometries"]},{"id":"1955-B4","year":1955,"session":"B","number":4,"statement":"4. Do there exist $ 1,000,000 $ consecutive integers each of which contains a repeated prime factor?","subjects":["number theory","squarefree integers"]},{"id":"1955-B5","year":1955,"session":"B","number":5,"statement":"5. Given an infinite sequence of 0's and 1's and a fixed integer $ k $, suppose that there are no more than $ k $ distinct blocks of $ k $ consecutive terms. Show that the sequence is eventually periodic. (For example, the sequence 11011010101 followed by alternating 0 's and 1 's indefinitely, which is periodic beginning with the fifth term.)","subjects":["combinatorics","words","eventual periodicity"]},{"id":"1955-B6","year":1955,"session":"B","number":6,"statement":"6. Prove: If $ f(x)>0 $ for all $ x $ and $ f(x) \\rightarrow 0 $ as $ x \\rightarrow \\infty $, then there exists at most a finite number of solutions of\n$$\nf(m)+f(n)+f(p)=1\n$$\nin positive integers $ m, n $, and $ p $.","subjects":["analysis","sequences","diophantine finiteness"]},{"id":"1955-B7","year":1955,"session":"B","number":7,"statement":"7. Four forces acting on a body are in equilibrium. Prove that, if their lines of action are mutually skew, they are rulings of a hyperboloid.","subjects":["geometry","statics","ruled quadrics"],"errata":"Assume all four forces are nonzero. The conclusion is a nonsingular real doubly ruled projective quadric. In affine terminology it can be a one-sheeted hyperboloid or a hyperbolic paraboloid. Equilibrium forces along skew rulings of $z=xy$ refute a central-hyperboloid-only reading. The book explicitly discusses this terminology and example."},{"id":"1956-A1","year":1956,"session":"A","number":1,"statement":"1. Evaluate\n$$\n\\lim _{x \\rightarrow \\infty}\\left[\\frac{1}{x} \\frac{a^{x}-1}{a-1}\\right]^{1 / x}\n$$\nwhere $ a>0, a \\neq 1 $.","subjects":["analysis","limits","exponential growth"]},{"id":"1956-A2","year":1956,"session":"A","number":2,"statement":"2. Prove that every positive integer has a multiple whose decimal representation involves all ten digits.","subjects":["number theory","decimal expansions","divisibility"]},{"id":"1956-A3","year":1956,"session":"A","number":3,"statement":"3. A particle falls in a vertical plane from rest under the influence of gravity and a force perpendicular to and proportional to its velocity. Obtain the equations of the trajectory and identify the curve.","subjects":["analysis","mechanics","differential equations","cycloids"]},{"id":"1956-A4","year":1956,"session":"A","number":4,"statement":"4. Suppose the $ n $ times differentiable real function $ f(x) $ has at least $ n+1 $ distinct zeros in the closed interval $ [a, b] $ and that the polynomial $ P(z) \\equiv z^n $ $ +C_{n-1} z^{n-1}+\\cdots+C_{0} $ has only real zeros. Show that ( $ D^{n}+C_{n-1} D^{n-1} $ $ \\left.+\\cdots+C_{0}\\right) f(x) $ has at least one zero in the interval $ [a, b] $ where $ D^{n} $ denotes, as usual, $ d^{n} / d x^{n} $.","subjects":["analysis","differential operators","zeros of functions"]},{"id":"1956-A5","year":1956,"session":"A","number":5,"statement":"5. Given $ n $ objects arranged in a row. A subset of these objects is called unfriendly if no two of its elements are consecutive. Show that the number of unfriendly subsets each having $ k $ elements is\n$$\n\\binom{n-k+1}{k}\n$$","subjects":["combinatorics","subsets","nonadjacent selections"]},{"id":"1956-A6","year":1956,"session":"A","number":6,"statement":"6. (i) A transformation of the plane into itself preserves all rational distances. Prove that it preserves all distances.\n(ii) Show that the corresponding theorem for the line is false.","subjects":["geometry","isometries","rational distances"]},{"id":"1956-A7","year":1956,"session":"A","number":7,"statement":"7. Prove that the number of odd binomial coefficients in any finite binomial expansion is a power of 2 .","subjects":["number theory","combinatorics","binomial parity"]},{"id":"1956-B1","year":1956,"session":"B","number":1,"statement":"1. Show that if the differential equation\n$$\nM(x, y) d x+N(x, y) d y=0\n$$\nis both homogeneous and exact then the solution $ y=f(x) $ satisfies $ x M+y N $ $ =C( $ constant $ ) $.","subjects":["analysis","exact differential equations","homogeneous functions"]},{"id":"1956-B2","year":1956,"session":"B","number":2,"statement":"2. Suppose that each set $ X $ of points in the plane has an associated set $ \\bar{X} $ of points called its cover. Suppose further that\n(1) $ \\overline{X\\cup Y}\\supset\\overline{\\bar X}\\cup\\bar Y\\cup Y $, where $ \\cup $ designates point set sum (or union) and $ \\supset $ denotes set inclusion.\nProve: (i) $ \\bar{X} \\supset X $, (ii) $ \\overline{\\bar{X}}=\\bar{X} $, (iii) $ X \\supset Y $ implies $ \\bar{X} \\supset \\bar{Y} $.\nProve conversely that (i), (ii) and (iii) imply (1).","subjects":["set theory","closure operators","order theory"]},{"id":"1956-B3","year":1956,"session":"B","number":3,"statement":"3. A sphere is inscribed in a tetrahedron and each point of contact of the sphere with the four faces is joined to the vertices of the face containing the point. Show that the four sets of three angles so formed are identical.","subjects":["geometry","tetrahedra","tangent lengths","linear algebra"]},{"id":"1956-B4","year":1956,"session":"B","number":4,"statement":"4. Prove that if $ A, B $, and $ C $ are angles of a triangle measured in radians then $ A \\cos B+\\sin A \\cos C>0 $.","subjects":["analysis","triangle inequalities","trigonometry"]},{"id":"1956-B5","year":1956,"session":"B","number":5,"statement":"5. Consider a set of $ 2 n $ points in space, $ n>1 $. Suppose they are joined by at least $ n^{2}+1 $ segments. Show that at least one triangle is formed. Show that for each $ n $ it is possible to have $ 2 n $ points joined by $ n^{2} $ segments without any triangles being formed.","subjects":["combinatorics","graph theory","extremal graphs"]},{"id":"1956-B6","year":1956,"session":"B","number":6,"statement":"6. Given $ T_{1}=2, T_{n+1}=T_{n}{ }^{2}-T_{n}+1, n>0 $, Prove:\n(i) If $ m \\neq n, T_{m} $ and $ T_{n} $ have no common factor greater than 1 .\n$$\n\\sum_{i=1}^{\\infty} \\frac{1}{T_{i}}=1\n$$","subjects":["number theory","recursive sequences"]},{"id":"1956-B7","year":1956,"session":"B","number":7,"statement":"7. The polynomials $ P(z) $ and $ Q(z) $ with complex coefficients have the same set of numbers for their zeros but possibly different multiplicities. The same is true of the polynomials\n$$\nP(z)+1 \\text { and } Q(z)+1\n$$\n\nProve that $ P(z) \\equiv Q(z) $.","subjects":["algebra","complex polynomials","root multiplicity"],"errata":"The intended assertion excludes constant polynomials. The book solution makes this restriction. Distinct nonzero constant polynomials other than $-1$ give counterexamples to the unrestricted wording."},{"id":"1957-A1","year":1957,"session":"A","number":1,"statement":"1. The normals to a surface all intersect a fixed straight line. Show that the surface is a portion of a surface of revolution.","subjects":["geometry","differential geometry","surfaces of revolution"],"errata":"For an embedded $C^1$ surface without boundary, the conclusion is local. Near each point, the surface agrees with a regular surface of revolution about the given line. If the surface is closed as a subset of $\\mathbb R^3$, it is globally invariant under rotations about that line. Arbitrary nonclosed patches need not be globally invariant or have one regular rotational completion. The book distinguishes these assertions."},{"id":"1957-A2","year":1957,"session":"A","number":2,"statement":"2. A uniform wire is bent into a form coinciding with the portion of the curve $ y=e^{x}, 0 \\leq x \\leq a, a>1 $, and the line segment $ a-1 \\leq x \\leq a, y= $ $ e^{a} $. The wire is then suspended from the point $ \\left(a-1, e^{a}\\right) $ and a horizontal force $ F $ is applied at the point $ (0,1) $ to hold the wire in coincidence with the curve and segment. Assuming the $ \\boldsymbol{x} $ axis is horizontal, show that the force $ F $ is directed to the right.","subjects":["analysis","mechanics","centroids","integral inequalities"]},{"id":"1957-A3","year":1957,"session":"A","number":3,"statement":"3. $ A $ and $ B $ are real numbers and $ k $ an integer greater than 1. Show that\n$$\n\\left|\\frac{\\cos k B \\cos A-\\cos k A \\cos B}{\\cos B-\\cos A}\\right|<k^{2}-1\n$$\nwhenever the left side is defined.","subjects":["analysis","trigonometric inequalities","multiple angles"]},{"id":"1957-A4","year":1957,"session":"A","number":4,"statement":"4. $ P(z) $ is a complex polynomial whose roots (as points in the Argand plane) can be covered by a closed circular disc of radius $ R $. Show that the roots of $ n P(z)-k P^{\\prime}(z) $ can be covered by a closed circular disc of radius $ R $ $ +|k| $, where $ n $ is the degree of $ P(z), k $ is any complex number, and $ P^{\\prime}(z) $ is the derivative of $ P(z) $.","subjects":["algebra","complex polynomials","root location"],"errata":"Assume $P$ has positive degree $n$. For a nonzero constant $P$, $nP-kP'$ is the zero polynomial, whose zero set is the whole plane. The positive-degree theorem includes $k=0$, $R=0$ and repeated roots. The constant-polynomial counterexample is an explanatory boundary qualification, not a warning attributed to a printed official erratum."},{"id":"1957-A5","year":1957,"session":"A","number":5,"statement":"5. Given $ n $ points in the plane, show that the largest distance determined by these points cannot occur more than $ n $ times.","subjects":["geometry","extremal distances","diameter graphs"]},{"id":"1957-A6","year":1957,"session":"A","number":6,"statement":"6. $ S_{1}=\\ln a $ and $ S_{n}=\\sum_{i=1}^{n-1} \\ln \\left(a-S_{i}\\right), n>1 $.\n\nShow that\n$$\n\\lim _{n \\rightarrow \\infty} S_{n}=a-1\n$$","subjects":["analysis","recursive sequences","monotone convergence"],"errata":"Retain the displayed sum definition and use real natural logarithms. Assume $a>0$ and $a-\\ln(a-\\ln a)>0$, exactly the condition for real continuation. Then $S_n\\to a-1$. The one-step recurrence begins at $n=2$ because $S_2=\\ln(a-\\ln a)$. The literal book definition gives the exceptional second term, although its solution starts the recurrence one step too early. Scholes separately warns that sufficiently small positive parameters fail to give a real sequence."},{"id":"1957-A7","year":1957,"session":"A","number":7,"statement":"7. Each member of a set of circles in the $ x y $ plane is tangent to the $ x $ axis and no two of the circles intersect. Show that:\n(i) the points of tangency can include all the rational points on the axis, but\n(ii) the points of tangency cannot include all the irrational points.","subjects":["geometry","countability","tangent circles"]},{"id":"1957-B1","year":1957,"session":"B","number":1,"statement":"1. Consider the determinant $ \\left|a_{i j}\\right| $ of order 100 with $ a_{i j}=i \\times j $. Prove that if the absolute value of each of the 100 ! terms in the expansion of this determinant is divided by 101 then the remainder in each case is 1.","subjects":["number theory","determinants","modular arithmetic"]},{"id":"1957-B2","year":1957,"session":"B","number":2,"statement":"2. If facilities for division are not available, it is sometimes convenient in determining the decimal expansion of $ 1 / A, A>0 $ to use the iteration $ X_{k+1} $ $ =X_{k}\\left(2-A X_{k}\\right), k=0,1,2, \\ldots $, where $ X_{0} $ is a selected \"starting\" value. Find the limitations, if any, on the starting value $ X_{0} $ in order that the above iteration converges to the desired value $ 1 / A $.","subjects":["analysis","iteration","newton method"]},{"id":"1957-B3","year":1957,"session":"B","number":3,"statement":"3. For $ f(x) $ a positive, monotone decreasing function defined in $ 0 \\leq x \\leq 1 $ prove that\n$$\n\\frac{\\int_{0}^{1} x f^{2}(x) d x}{\\int_{0}^{1} x f(x) d x} \\leq \\frac{\\int_{0}^{1} f^{2}(x) d x}{\\int_{0}^{1} f(x) d x}\n$$","subjects":["analysis","integral inequalities","monotone functions"]},{"id":"1957-B4","year":1957,"session":"B","number":4,"statement":"4. Let $ a(n) $ be the number of representations of the positive integer $ n $ as the sums of 1's and 2's taking order into account. For example, since\n$$\n\\begin{array}{c}\n4=1+1+2=1+2+1=2+1+1 \\\\\n=2+2=1+1+1+1\n\\end{array}\n$$\nthen $ a(4)=5 $. Let $ b(n) $ be the number of representations of $ n $ as the sum of integers greater than 1 , again taking order into account and counting the summand $ n $. For example, since $ 6=4+2=2+4=3+3=2+2+2 $, we have $ b(6)=5 $. Show that for each $ n, a(n)=b(n+2) $.","subjects":["combinatorics","compositions","fibonacci numbers"]},{"id":"1957-B5","year":1957,"session":"B","number":5,"statement":"5. With each subset $ X $ of a set is associated a second subset $ f(X) $. The association is such that whenever $ X $ contains $ Y $ then $ f(X) $ contains $ f(Y) $. Show that for some set $ A, f(A)=A $.","subjects":["set theory","fixed points","order theory"]},{"id":"1957-B6","year":1957,"session":"B","number":6,"statement":"6. The curve $ y=f(x) $ passes through the origin with a slope of 1 . It satisfies the differential equation $ \\left(x^{2}+9\\right) y^{\\prime \\prime}+\\left(x^{2}+4\\right) y=0 $. Show that it crosses the $ x $ axis between\n$$\nx=\\frac{3}{2} \\pi \\quad \\text { and } \\quad x=\\sqrt{\\frac{63}{53}} \\pi\n$$","subjects":["analysis","ordinary differential equations","zero location"]},{"id":"1957-B7","year":1957,"session":"B","number":7,"statement":"7. Let $ C $ be a closed convex planar disc bounded by a regular polygon. Show that for each positive integer $ n $ there exists a set of points $ S(n) $ in the plane such that each $ n $ points of $ S(n) $ can be covered by $ C $, but $ S(n) $ itself cannot be covered by $ \\boldsymbol{C} $.","subjects":["geometry","covering","regular polygons"]},{"id":"1958-2-A1","year":1958,"session":"A","number":1,"statement":"1. Let $ f(m, 1)=f(1, n)=1 $ for $ m \\geq 1, n \\geq 1 $, and let $ f(m, n)= $ $ f(m-1, n)+f(m, n-1)+f(m-1, n-1) $ for $ m>1 $ and $ n>1 $. Also let\n$$\nS(n)=\\sum_{a+b=n} f(a, b), \\quad a \\geq 1 \\text { and } b \\geq 1\n$$\n\nProve that\n$$\nS(n+2)=S(n)+2 S(n+1) \\quad \\text { for } n \\geq 2\n$$","subjects":["combinatorics","recurrences","lattice paths"]},{"id":"1958-2-A2","year":1958,"session":"A","number":2,"statement":"2. Let\n$$\nR_{1}=1, \\quad R_{n+1}=1+n / R_{n}, \\quad n \\geq 1\n$$\n\nShow that for $ n \\geq 1 $,\n$$\n\\sqrt{n} \\leq R_{n} \\leq \\sqrt{n}+1\n$$","subjects":["analysis","recursive sequences"]},{"id":"1958-2-A3","year":1958,"session":"A","number":3,"statement":"3. Under the assumption that the following set of relations has a unique solution for $ u(t) $, determine it.\n$$\n\\begin{aligned}\n\\frac{d u(t)}{d t} & =u(t)+\\int_{0}^{1} u(s) d s \\\\\nu(0) & =1\n\\end{aligned}\n$$","subjects":["analysis","integro-differential equations","linear ode"]},{"id":"1958-2-A4","year":1958,"session":"A","number":4,"statement":"4. In assigning dormitory rooms, a college gives preference to pairs of students in this order:\n$$\nA A, A B, A C, B B, B C, A D, C C, B D, C D, D D\n$$\nin which $ A A $ means two seniors, $ A B $ means a senior and a junior, etc. Determine numerical values to assign to $ A, B, C, D $ so that the set of numbers $ A $ $ +A, A+B, A+C, B+B $, etc., corresponding to the order above will be in descending magnitude. Find the general solution and the solution in least positive integers.","subjects":["algebra","linear inequalities","integer optimization"]},{"id":"1958-2-A5","year":1958,"session":"A","number":5,"statement":"5. Show that the number of non-zero terms in the expansion of the $ n $th order determinant having zeros in the main diagonal and ones elsewhere is\n$$\nn!\\left[1-\\frac{1}{1!}+\\frac{1}{2!}-\\frac{1}{3!}+\\cdots+\\frac{(-1)^{n}}{n!}\\right] . \\quad(\\text { page 481) }\n$$","subjects":["combinatorics","derangements","determinants"]},{"id":"1958-2-A6","year":1958,"session":"A","number":6,"statement":"6. Let $ a(x) $ and $ b(x) $ be continuous functions on $ 0 \\leq x \\leq 1 $ and let $ 0 \\leq $ $ a(x) \\leq a<1 $ on that range. Under what other conditions (if any) is the solution of the equation for $ u $,\n$$\nu=\\underset{0 \\leq x \\leq 1}{\\operatorname{maximum}}[b(x)+a(x) \\cdot u],\n$$\ngiven by\n$$\nu=\\operatorname{maximum}_{0 \\leq x \\leq 1}\\left[\\frac{b(x)}{1-a(x)}\\right] ?\n$$","subjects":["analysis","maxima","fixed points"]},{"id":"1958-2-A7","year":1958,"session":"A","number":7,"statement":"7. Let $ a $ and $ b $ be relatively prime positive integers, $ b $ even. For each positive integer $ q $ let $ p=p(q) $ be chosen so that\n$$\n\\left|\\frac{p}{q}-\\frac{a}{b}\\right|\n$$\nis a minimum. Prove that\n$$\n\\lim _{n-\\infty} \\sum_{y=1}^{n} \\frac{q\\left|\\frac{p}{q}-\\frac{a}{b}\\right|}{n}=\\frac{1}{4} .\n$$","subjects":["number theory","rational approximation","periodic averages"]},{"id":"1958-2-B1","year":1958,"session":"B","number":1,"statement":"1. Given\n$$\nb_{n}=\\sum_{k=0}^{n}\\binom{n}{k}^{-1}, \\quad n \\geq 1,\n$$\nprove that\n$$\nb_{n}=\\frac{n+1}{2 n} b_{n-1}+1, \\quad n \\geq 2\n$$\nand hence, as a corollary,\n$$\n\\lim _{n \\rightarrow \\infty} b_{n}=2\n$$","subjects":["analysis","binomial coefficients","recurrences"]},{"id":"1958-2-B2","year":1958,"session":"B","number":2,"statement":"2. Given a set of $ n+1 $ positive integers, none of which exceeds $ 2 n $, show that at least one member of the set must divide another member of the set.","subjects":["number theory","divisibility"]},{"id":"1958-2-B3","year":1958,"session":"B","number":3,"statement":"3. If a square of unit side be partitioned into two sets, then the diameter (least upper bound of the distances between pairs of points) of one of the sets is not less than $ \\sqrt{5} / 2 $. Show also that no larger number will do.","subjects":["geometry","diameter","two coloring"]},{"id":"1958-2-B4","year":1958,"session":"B","number":4,"statement":"4. Let $ C $ be a real number, and let $ f $ be a function such that\n$$\n\\lim _{x \\rightarrow \\infty} f(x)=C, \\quad \\lim _{x \\rightarrow \\infty} f^{\\prime \\prime \\prime}(x)=0\n$$\n\nProve that\n$$\n\\lim _{x \\rightarrow \\infty} f^{\\prime}(x)=0 \\quad \\text { and } \\quad \\lim _{x \\rightarrow \\infty} f^{\\prime \\prime}(x)=0\n$$\nwhere superscripts denote derivatives.","subjects":["analysis","derivative limits"]},{"id":"1958-2-B5","year":1958,"session":"B","number":5,"statement":"5. The lengths of successive segments of a broken line are represented by the successive terms of the harmonic progression $ 1,1 / 2,1 / 3, \\ldots, 1 / n, \\ldots $. Each segment makes with the preceding segment a given angle $ \\theta $. What is the distance and what is the direction of the limiting point (if there is one) from the initial point of the first segment?","subjects":["analysis","complex series","fourier series","statement ambiguity"],"errata":"The unsigned angle alone does not determine the path. At $\\theta=\\pi/2$, alternating left and right turns produce divergent harmonic sums in both coordinates, while constant left turns converge. The source adopts a planar fixed signed turn. The proposed qualified review target states that convention explicitly; the unqualified historical data do not have a unique answer."},{"id":"1958-2-B6","year":1958,"session":"B","number":6,"statement":"6. Let a complete oriented graph on $ n $ points be given, i.e., a set of $ n $ points $ 1,2,3, \\ldots, n $, and between any two points $ i $ and $ j $ a direction, $ i \\rightarrow j $. Show that there exists a permutation of the points, $ \\left[a_{1}, a_{2}, a_{3}, \\ldots, a_{n}\\right] $, such that $ a_{1} \\rightarrow a_{2} \\rightarrow a_{3} \\rightarrow \\cdots \\rightarrow a_{n} $.","subjects":["combinatorics","tournaments","hamiltonian paths"]},{"id":"1958-2-B7","year":1958,"session":"B","number":7,"statement":"7. Let $ a_{1}, a_{2}, \\ldots, a_{n} $ be a permutation of the integers $ 1,2, \\ldots, n $. Call $ a_{i} $ a \"big\" integer if $ a_{i}>a_{j} $ for all $ j>i $. Find the mean number of \"big\" integers over all permutations on the first $ n $ positive integers.","subjects":["combinatorics","probability","permutation records"]},{"id":"1958-A1","year":1958,"session":"A","number":1,"statement":"1. If $ a_{0}, a_{1}, \\ldots, a_{n} $ are real numbers satisfying\n$$\n\\frac{a_{0}}{1}+\\frac{a_{1}}{2}+\\cdots+\\frac{a_{n}}{n+1}=0\n$$\nshow that the equation $ a_{0}+a_{1} x+a_{2} x^{2}+\\cdots+a_{n} x^{n}=0 $ has at least one real root.","subjects":["analysis","polynomials","real roots"]},{"id":"1958-A2","year":1958,"session":"A","number":2,"statement":"2. Two uniform solid spheres of equal radii are so placed that one is directly above the other. The bottom sphere is fixed, and the top sphere, initially at rest, rolls off. At what point will contact between the two spheres be \"lost\"? Assume the coefficient of friction is such that no slipping occurs.","subjects":["analysis","mechanics","rolling motion","energy conservation"],"errata":"An upper sphere at exact rest directly above the fixed sphere remains in its unstable equilibrium. The book explicitly corrects the question to the limit of slightly displaced rest releases. Under the stated ideal rolling-without-slipping constraint, a release at angle $\\theta_0>0$ first separates at $\\cos\\theta_s=(10/17)\\cos\\theta_0$. The limiting angle is $\\arccos(10/17)$ as $\\theta_0\\to0^+$. The solutions prove this release-limit theorem, not spontaneous departure from exact rest. The ideal no-slip constraint is distinguished from a fixed finite Coulomb-friction bound at separation."},{"id":"1958-A3","year":1958,"session":"A","number":3,"statement":"3. Real numbers are chosen at random from the interval $ (0 \\leq x \\leq 1) $. If after choosing the $ n $th number the sum of the numbers so chosen first exceeds 1 , show that the expected or average value for $ n $ is $ e $.","subjects":["probability","stopping times","uniform random variables"]},{"id":"1958-A4","year":1958,"session":"A","number":4,"statement":"4. If $ a_{1}, a_{2}, \\ldots, a_{n} $ are complex numbers such that\n$$\n\\left|a_{1}\\right|=\\left|a_{2}\\right|=\\cdots=\\left|a_{n}\\right|=r \\neq 0\n$$\nand if $ { }_{n} T_{s} $ denotes the sum of all products of these $ n $ numbers taken $ s $ at a time, prove that\n$$\n\\left|\\frac{{ }_{n} T_{s}}{{ }_{n} T_{n-s}}\\right|=r^{2 s-n}\n$$\nwhenever the denominator of the left-hand side is different from zero.","subjects":["algebra","complex numbers","symmetric polynomials"]},{"id":"1958-A5","year":1958,"session":"A","number":5,"statement":"5. Show that the integral equation\n$$\nf(x, y)=1+\\int_{0}^{x} \\int_{0}^{y} f(u, v) d u d v\n$$\nhas at most one solution continuous for $ 0 \\leq x \\leq 1,0 \\leq y \\leq 1 $.","subjects":["analysis","integral equations","uniqueness"]},{"id":"1958-A6","year":1958,"session":"A","number":6,"statement":"6. What is the smallest amount that may be invested at interest rate $ i $, compounded annually, in order that one may withdraw 1 dollar at the end of the first year, 4 dollars at the end of the second year, $ \\ldots, n^{2} $ dollars at the end of the $ n $th year, in perpetuity?","subjects":["analysis","power series","annuity valuation"]},{"id":"1958-A7","year":1958,"session":"A","number":7,"statement":"7. Show that ten equal-sized squares cannot be placed on a plane in such a way that no two have an interior point in common and the first touches each of the others.","subjects":["geometry","packing","equal squares"]},{"id":"1958-B1","year":1958,"session":"B","number":1,"statement":"1. (i) Given line segments $ A, B, C, D $, with $ A $ the longest, construct a quadrilateral with these sides and with $ A $ and $ B $ parallel, when possible.\n(ii) Given any acute-angled triangle $ A B C $ and one altitude $ A H $, select any point $ D $ on $ A H $, then draw $ B D $ and extend until it intersects $ A C $ in $ E $, and draw $ C D $ and extend until it intersects $ A B $ in $ F $. Prove angle $ A H E= $ angle $ A H F $.","subjects":["geometry","euclidean constructions","trapezoids","triangle geometry"]},{"id":"1958-B2","year":1958,"session":"B","number":2,"statement":"2. Prove that the product of four consecutive positive integers cannot be a perfect square or cube.","subjects":["number theory","perfect powers","consecutive integers"]},{"id":"1958-B3","year":1958,"session":"B","number":3,"statement":"3. In a round-robin tournament with $ n $ players (each pair of players plays one game) in which there are no draws, the numbers of wins scored by the players are $ s_{1}, s_{2}, \\ldots, s_{n} $. Prove that a necessary and sufficient condition for the existence of 3 players, $ A, B, C $, such that $ A $ beat $ B, B $ beat $ C $, and $ C $ beat $ A $ is\n$$\ns_{1}^{2}+s_{2}^{2}+\\cdots+s_{n}^{2}<\\frac{(n-1)n(2n-1)}{6}\n$$","subjects":["combinatorics","tournaments","score sequences"]},{"id":"1958-B4","year":1958,"session":"B","number":4,"statement":"4. What is the average straight line distance between two points on a sphere of radius $ 1 ? $","subjects":["probability","spherical geometry","expected distance"]},{"id":"1958-B5","year":1958,"session":"B","number":5,"statement":"5. Given an infinite number of points in a plane, prove that if all the distances determined between them are integers then the points are all in a straight line.","subjects":["geometry","integer distances","anning-erdos theorem","discrete geometry"]},{"id":"1958-B6","year":1958,"session":"B","number":6,"statement":"6. A projectile moves in a resisting medium. The resisting force is a function of the velocity and is directed along the velocity vector. The equation $ x $ $ =f(t) $ gives the horizontal distance in terms of the time $ t $. Show that the vertical distance $ y $ is given by\n$$\ny=-g f(t) \\int \\frac{d t}{f^{\\prime}(t)}+g \\int \\frac{f(t)}{f^{\\prime}(t)} d t+A f(t)+B\n$$\nwhere $ A $ and $ B $ are constants and $ g $ is the acceleration due to gravity.","subjects":["analysis","mechanics","differential equations","variation of parameters"],"errata":"The displayed quotient formula applies on connected time intervals where $f'(t)\\ne0$. A trajectory with identically constant horizontal position is a separate vertical-motion case, for which these quotients are undefined and the horizontal data do not determine vertical motion. The book explicitly notes this exception. Its uniqueness discussion assumes a regular drag law, which is sufficient to keep a nonzero horizontal velocity from vanishing. The solutions state the local nonzero-velocity theorem rather than silently imposing that extra regularity globally."},{"id":"1958-B7","year":1958,"session":"B","number":7,"statement":"7. Prove that if $ f(x) $ is continuous for $ a \\leq x \\leq b $ and $ \\int_{a}^{b} x^{n} f(x) d x=0 $ for $ n=0,1,2, \\ldots $ then $ f(x) $ is identically zero on $ a \\leq x \\leq b $.","subjects":["analysis","moment problems","polynomial approximation"],"errata":"Editorial clarification. Require $a<b$. If $a=b=0$ and $f(0)=1$, all the displayed integrals are zero, but $f$ is not zero. The three solution tracks state this exception and prove the nondegenerate-interval theorem. This is a model-reviewed correction, not an attributed official erratum."},{"id":"1959-A1","year":1959,"session":"A","number":1,"statement":"1. Let $ n $ be a positive integer. Prove that $ x^{n}-\\left(1 / x^{n}\\right) $ is expressible as a polynomial in $ x-(1 / x) $ with real coefficients if and only if $ n $ is odd.","subjects":["algebra","laurent polynomials","polynomial representations"]},{"id":"1959-A2","year":1959,"session":"A","number":2,"statement":"2. Prove that if the points in the complex plane corresponding to two distinct complex numbers $ z_{1} $ and $ z_{2} $ are two vertices of an equilateral triangle, then the third vertex corresponds to $ -\\omega z_{1}-\\omega^{2} z_{2} $, where $ \\omega $ is an imaginary cube root of unity.","subjects":["geometry","complex numbers","equilateral triangles"]},{"id":"1959-A3","year":1959,"session":"A","number":3,"statement":"3. Find all complex-valued functions $ f $ of a complex variable such that $ f(z)+z f(1-z)=1+z $ for all $ z $.","subjects":["algebra","functional equations","complex functions"]},{"id":"1959-A4","year":1959,"session":"A","number":4,"statement":"4. If $ f $ and $ g $ are real-valued functions of one real variable, show that there exist numbers $ x $ and $ y $ such that $ 0 \\leq x \\leq 1,0 \\leq y \\leq 1 $, and $ \\mid x y-f(x)- $ $ g(y) \\mid \\geq 1 / 4 $","subjects":["analysis","functional inequalities","uniform approximation"]},{"id":"1959-A5","year":1959,"session":"A","number":5,"statement":"5. A sparrow, flying horizontally in a straight line, is 50 feet directly below an eagle and 100 feet directly above a hawk. Both hawk and eagle fly directly toward the sparrow, reaching it simultaneously. The hawk flies twice as fast as the sparrow. How far does each bird fly? At what rate does the eagle fly?","subjects":["analysis","mechanics","pursuit curves","differential equations"],"errata":"Errata. The source book explicitly supplies the omitted assumption that every bird has uniform speed. All three tracks prove that intended constant-positive-speed problem. Without the assumption, the eagle's distance and rate are not determined. For a literal counterexample, let the sparrow be $q(t)=(t,0)$, keep the speed-two hawk, and put $T=200/3$. Instead of the constant-speed eagle, take $p(t)=(t^2/T,50(1-t/T)^2)$ for $0\\leq t\\leq T$. Then $p'(t)=2(q(t)-p(t))/(T(1-t/T))$ for $t<T$, so this positive-variable-speed eagle always pursues the sparrow and catches it at the same $T$. Its length is $2(455+144\\log6)/15$, different from $25(3+\\sqrt{73})/3$ for constant speed. The exact verification is in the companion notes. The canonical historical wording is unchanged. The numerical eagle rate is necessarily relative to the sparrow or hawk speed, not an absolute rate."},{"id":"1959-A6","year":1959,"session":"A","number":6,"statement":"6. Let $ m $ and $ n $ be integers greater than 1 , and $ a_{1}, \\ldots, a_{m+1} $ real numbers. Prove that there exist real $ n $ by $ n $ matrices $ A_{1}, \\ldots, A_{m} $ such that (i) $ \\operatorname{Det}\\left(A_{j}\\right)=a_{j} $ for $ j=1, \\ldots, m $, and (ii) $ \\operatorname{Det}\\left(A_{1}+\\cdots+A_{m}\\right)=a_{m+1} $.","subjects":["algebra","linear algebra","determinants"]},{"id":"1959-A7","year":1959,"session":"A","number":7,"statement":"7. If $ f $ is a real-valued function of one real variable which has a continuous derivative on the closed interval $ [a, b] $ and for which there is no $ x \\in[a, b] $ such that $ f(x)=f^{\\prime}(x)=0 $, then prove that there is a function $ g $ with continuous first derivative on $ [a, b] $ such that $ f g^{\\prime}-f^{\\prime} g $ is positive on $ [a, b] $.","subjects":["analysis","wronskians","c1 functions"]},{"id":"1959-B1","year":1959,"session":"B","number":1,"statement":"1. Let each of $ m $ distinct points on the positive part of the $ X $-axis be joined to $ n $ distinct points on the positive part of the $ \\boldsymbol{Y} $-axis. Obtain a formula for the number of intersection points of these segments (exclusive of endpoints), assuming that no three of the segments are concurrent.","subjects":["combinatorics","geometry","segment intersections"]},{"id":"1959-B2","year":1959,"session":"B","number":2,"statement":"2. Let $ c $ be a positive real number. Prove that $ c $ can be expressed in infinitely many ways as a sum of infinitely many distinct terms selected from the sequence\n$$\n1 / 10,1 / 20, \\ldots, 1 / 10 n, \\ldots\n$$","subjects":["analysis","subseries","harmonic series"]},{"id":"1959-B3","year":1959,"session":"B","number":3,"statement":"3. Give an example of a continuous real-valued function $ f $ from $ [0,1] $ to $ [0,1] $ which takes on every value in $ [0,1] $ an infinite number of times.","subjects":["analysis","continuity","cantor set","topology"]},{"id":"1959-B4","year":1959,"session":"B","number":4,"statement":"4. Given the following matrix of 25 elements\n$$\n\\left(\\begin{array}{rrrrr}\n11 & 17 & 25 & 19 & 16 \\\\\n24 & 10 & 13 & 15 & 3 \\\\\n12 & 5 & 14 & 2 & 18 \\\\\n23 & 4 & 1 & 8 & 22 \\\\\n6 & 20 & 7 & 21 & 9\n\\end{array}\\right),\n$$\nchoose five of these elements, no two coming from the same row or column, in such a way that the minimum of these five elements is as large as possible. Prove that your answer is correct.","subjects":["combinatorics","assignment problem"]},{"id":"1959-B5","year":1959,"session":"B","number":5,"statement":"5. Find the equation of the smallest sphere which is tangent to both of the lines: (i) $ x=t+1, y=2 t+4, z=-3 t+5 $, and (ii) $ x=4 t-12, y= $ $ -t+8, z=t+17 $.","subjects":["geometry","analytic geometry","skew lines","optimization"]},{"id":"1959-B6","year":1959,"session":"B","number":6,"statement":"6. Prove that, if $ x $ and $ y $ are positive irrationals such that $ 1 / x+1 / y=1 $, then the sequences $ [x],[2 x], \\ldots,[n x], \\ldots $ and $ [y],[2 y], \\ldots,[n y], \\ldots $ together include every positive integer exactly once. (The notation $ [x] $ means the largest integer not exceeding $ x $.)","subjects":["number theory","beatty sequences","floor function"]},{"id":"1959-B7","year":1959,"session":"B","number":7,"statement":"7. For each positive integer $ n $, let $ f_{n} $ be a real-valued symmetric function of $ n $ real variables. Suppose that for all $ n $ and for all real numbers $ x_{1}, \\ldots $, $ x_{n+1}, y $, it is true that\n(1) $ f_{n}\\left(x_{1}+y, \\ldots, x_{n}+y\\right)=f_{n}\\left(x_{1}, \\ldots, x_{n}\\right)+y $,\n(2) $ f_{n}\\left(-x_{1}, \\ldots,-x_{n}\\right)=-f_{n}\\left(x_{1}, \\ldots x_{n}\\right) $,\n(3) $ f_{n+1}\\left(f_{n}\\left(x_{1}, \\ldots, x_{n}\\right), \\ldots, f_{n}\\left(x_{1}, \\ldots, x_{n}\\right), x_{n+1}\\right)=f_{n+1}\\left(x_{1}, \\ldots, x_{n+1}\\right) $.\n\nProve that\n(4) $ f_{n}\\left(x_{1}, \\ldots, x_{n}\\right)=\\left(x_{1}+\\cdots+x_{n}\\right) / n $.","subjects":["algebra","functional equations","symmetric functions"]},{"id":"1960-A1","year":1960,"session":"A","number":1,"statement":"1. Let $ n $ be a given positive integer. How many solutions are there in ordered positive integer pairs $ (x, y) $ to the equation\n$$\n\\frac{x y}{x+y}=n ? \\quad \\quad(\\text { page } 516)\n$$","subjects":["number theory","diophantine equations","divisors"]},{"id":"1960-A2","year":1960,"session":"A","number":2,"statement":"2. Show that if three points are inside a closed square of unit side, then some pair of them are within $ \\sqrt{6}-\\sqrt{2} $ units apart.","subjects":["geometry","packing","square"]},{"id":"1960-A3","year":1960,"session":"A","number":3,"statement":"$$\n\\begin{array}{l}\n\\text { 3. Show that if } t_{1}, t_{2}, t_{3}, t_{4}, t_{5} \\text { are real numbers, then }\\\\\n\\sum_{j=1}^{5}\\left(1-t_{j}\\right) \\exp \\left(\\sum_{k=1}^{j} t_{k}\\right) \\leq e^{e^{e^{e}}}\n\\end{array}\n$$","subjects":["analysis","inequalities","exponential function"]},{"id":"1960-A4","year":1960,"session":"A","number":4,"statement":"4. Given two points in the plane, $ P $ and $ Q $, at fixed distances from a line $ L $, and on the same side of the line, the problem is to find a third point $ R $ so that $ P R+R Q+R S $ is a minimum, where $ R S $ is perpendicular to $ L $. Consider all cases.","subjects":["geometry","convex optimization","geometric medians"]},{"id":"1960-A5","year":1960,"session":"A","number":5,"statement":"5. Consider a polynomial $ f(x) $ with real coefficients having the property $ f(g(x))=g(f(x)) $ for every polynomial $ g(x) $ with real coefficients. Determine and prove the nature of $ f(x) $.","subjects":["algebra","polynomials","functional equations"]},{"id":"1960-A6","year":1960,"session":"A","number":6,"statement":"6. A player throwing a die scores as many points as on the top face of the die and is to play until his score reaches or passes a total $ n $. Denote by $ p(n) $ the probability of making exactly the total $ n $, and find the value of $ \\lim _{n \\rightarrow \\infty} p(n) $.","subjects":["probability","recurrences"]},{"id":"1960-A7","year":1960,"session":"A","number":7,"statement":"7. Let $ N(n) $ denote the smallest positive integer $ N $ such that $ x^{N}=1 $ for every permutation $ \\boldsymbol{x} $ on $ \\boldsymbol{n} $ symbols, where 1 denotes the identity permutation. Prove that if $ n>1 $,\n$$\n\\begin{aligned}\n\\frac{N(n)}{N(n-1)} & =1 \\text { if } n \\text { is divisible by } 2 \\text { distinct primes, } \\\\\n& =p \\text { if } n \\text { is a power of a prime } p\n\\end{aligned}\n$$","subjects":["algebra","permutations","least common multiple","prime powers"]},{"id":"1960-B1","year":1960,"session":"B","number":1,"statement":"1. Find all solutions of $ n^{m}=m^{n} $ in integers $ n $ and $ m(n \\neq m) $. Prove that you have obtained all of them.","subjects":["number theory","integer powers","diophantine equations"]},{"id":"1960-B2","year":1960,"session":"B","number":2,"statement":"$$\n\\begin{array}{l}\n\\text { 2. Evaluate the double series }\\\\\n\\sum_{j=0}^{\\infty} \\sum_{k=0}^{\\infty} 2^{-3k-j-(k+j)^2} .\n\\end{array}\n$$","subjects":["analysis","double series","enumeration","geometric series"]},{"id":"1960-B3","year":1960,"session":"B","number":3,"statement":"3. The motion of the particles of a fluid in the plane is specified by the following components of velocity\n$$\n\\frac{d x}{d t}=y+2 x\\left(1-x^{2}-y^{2}\\right)\n$$\n$$\n\\frac{d y}{d t}=-x\n$$\n\nSketch the shape of the trajectories near the origin. Discuss what happens to an individual particle as $ t \\rightarrow+\\infty $, and justify your conclusion. (page 529)\n\nNote. The second $ y $ in the first equation above was broken and looked like a $ v $ on the examination as it was distributed to the contestants.","subjects":["analysis","differential equations","dynamical systems","limit cycles"]},{"id":"1960-B4","year":1960,"session":"B","number":4,"statement":"4. Consider the arithmetic progression $ a, a+d, a+2 d, \\ldots $, where $ a $ and $ d $ are positive integers. For any positive integer $ k $, prove that the progression has either no exact $ k $ th powers or infinitely many.","subjects":["number theory","arithmetic progressions","congruences"]},{"id":"1960-B5","year":1960,"session":"B","number":5,"statement":"5. Define a sequence as follows:\n$$\n\\begin{array}{l}\na_{0}=0 \\\\\na_{1}=1+\\sin (-1) \\\\\n\\vdots \\\\\na_{n}=1+\\sin \\left(a_{n-1}-1\\right) \\\\\n\\vdots\n\\end{array}\n$$\n\nEvaluate\n$$\n\\lim _{n \\rightarrow \\infty} \\frac{1}{n} \\sum_{k=1}^{n} a_{k} .\n$$","subjects":["analysis","recursive sequences","cesàro means"]},{"id":"1960-B6","year":1960,"session":"B","number":6,"statement":"6. Any positive integer may be written in the form $ n=2^{k}(2 l+1) $. Let $ a_{n} $ $ =e^{-k} $ and $ b_{n}=a_{1} a_{2} a_{3} \\cdots a_{n} $. Prove that $ \\Sigma b_{n} $ converges.","subjects":["analysis","infinite series"]},{"id":"1960-B7","year":1960,"session":"B","number":7,"statement":"7. Let $ g(t) $ and $ h(t) $ be real, continuous functions for $ t \\geq 0 $. Show that any function $ v(t) $ satisfying the differential inequality\n$$\n\\frac{d v}{d t}+g(t) v \\geq h(t), \\quad v(0)=c\n$$\nsatisfies the further inequality $ v(t) \\geq u(t) $ where\n$$\n\\frac{d u}{d t}+g(t) u=h(t), \\quad u(0)=c\n$$\n\nFrom this, conclude that for sufficiently small $ t>0 $, the solution of\n$$\n\\frac{d v}{d t}+g(t) v=v^{2}, \\quad v(0)=c_{1}\n$$\nmay be written\n$$\nv=\\max _{w}\\left[c_{1} e^{-\\int_{0}^{t}[g(s)-2 w(s)] d s}-\\int_{0}^{t} e^{-\\int_{s}^{t}\\left[g\\left(s_{1}\\right)-2 w\\left(s_{1}\\right)\\right] d s_{1}} w^{2}(s) d s\\right]\n$$\nwhere the maximization is over all continuous functions $ \\boldsymbol{w}(t) $ defined over some $ t $-interval $ \\left[0, t_{0}\\right] $.","subjects":["analysis","differential inequalities","riccati equation","variational formula"]},{"id":"1961-A1","year":1961,"session":"A","number":1,"statement":"1. The graph of the equation $ x^{y}=y^{x} $ in the first quadrant (i.e., the region where $ x>0 $ and $ y>0 $ ) consists of a straight line and a curve. Find the coordinates of the intersection point of the line and the curve.","subjects":["analysis","implicit curves","logarithms"]},{"id":"1961-A2","year":1961,"session":"A","number":2,"statement":"2. For a real-valued function $ f(x, y) $ of the two positive real variables $ x $ and $ y $, define $ f(x, y) $ to be linearly bounded if and only if there exists a positive number $ K $ such that $ |f(x, y)|<(x+y) K $ for all positive $ x $ and $ y $. Find necessary and sufficient conditions on the real numbers $ \\alpha $ and $ \\beta $ such that $ x^{\\alpha} y^{\\beta} $ is linearly bounded.","subjects":["analysis","homogeneity","power functions","inequalities"]},{"id":"1961-A3","year":1961,"session":"A","number":3,"statement":"3. Evaluate\n$$\n\\lim _{n \\rightarrow \\infty} \\sum_{j=1}^{n^{2}} \\frac{n}{n^{2}+j^{2}} .\n$$","subjects":["analysis","limits","riemann sums","improper integrals"]},{"id":"1961-A4","year":1961,"session":"A","number":4,"statement":"4. Define a function $ f $ over the domain of positive integers as follows: $ f(1) $ $ =1 $, and for $ n>1, f(n)=(-1)^{k} $ where $ k $ is the total number of prime factors of $ n $. For example $ f(9)=(-1)^{2}, f(20)=(-1)^{3} $. Define $ F(n) $ as $ \\Sigma f(d) $ where the sum ranges over all positive integer divisors of $ n $. Prove that for every positive integer $ n, F(n)=0 $ or $ F(n)=1 $. For which integers $ n $ is $ F(n) $ $ =1 ? \\quad $","subjects":["number theory","multiplicative functions","liouville function","divisor sums"]},{"id":"1961-A5","year":1961,"session":"A","number":5,"statement":"5. Let $ \\Omega $ be a set of $ n $ points, where $ n>2 $. Let $ \\Sigma $ be a nonempty subcollection of the $ 2^{n} $ subsets of $ \\Omega $ that is closed with respect to unions, intersections, and complements (that is, if $ A $ and $ B $ are members of $ \\Sigma $, then so are $ A \\cup B $, $ A \\cap B, \\Omega-A $ and $ \\Omega-B $, where $ \\Omega-B $ denotes all points in $ \\Omega $ but not in $ B) $. If $ k $ is the number of members of $ \\Sigma $, what are the possible values of $ k $ ? Give a proof.","subjects":["algebra","boolean algebras","finite sets"]},{"id":"1961-A6","year":1961,"session":"A","number":6,"statement":"6. If $ J_{2}=\\{0,1\\} $ is the field of integers modulo 2 , and if $ J_{2}[x] $ is the integral domain of polynomials in one indeterminate with coefficients in $ J_{2} $, prove that $ p(x)=1+x+x^{2}+\\cdots+x^{n} $ is reducible (factorable) in case $ n $ +1 is composite. Is the converse true? That is, if $ n+1 $ is prime, is $ p(x) $ irreducible?","subjects":["algebra","polynomials over finite fields","irreducibility"]},{"id":"1961-A7","year":1961,"session":"A","number":7,"statement":"7. Let $ S $ be a nonempty closed set in the Euclidean plane for which there is a closed disk $ D $ (a circle together with its interior) containing $ S $ such that $ D $ is a subset of every.closed disk that contains $ S $. Prove that every point inside $ D $ is the midpoint of a segment joining two points of $ S $.","subjects":["geometry","convexity","closed sets","disks"]},{"id":"1961-B1","year":1961,"session":"B","number":1,"statement":"1. Let $ \\alpha_{1}, \\alpha_{2}, \\alpha_{3}, \\ldots $ be a sequence of positive real numbers; define $ s_{n} $ as $ \\left(\\alpha_{1}+\\alpha_{2}+\\cdots+\\alpha_{n}\\right) / n $ and $ r_{n} $ as $ \\left(\\alpha_{1}^{-1}+\\alpha_{2}^{-1}+\\cdots+\\alpha_{n}^{-1}\\right) / n $. Given that $ \\lim s_{n} $ and $ \\lim r_{n} $ exist as $ n \\rightarrow \\infty $, prove that the product of these limits is not less than 1.","subjects":["analysis","inequalities","arithmetic and harmonic means"]},{"id":"1961-B2","year":1961,"session":"B","number":2,"statement":"2. Let $ \\alpha $ and $ \\beta $ be given positive real numbers, with $ \\alpha<\\beta $. If two points are selected at random from a straight line segment of length $ \\beta $, what is the probability that the distance between them is at least $ \\alpha $ ?","subjects":["probability","uniform distribution","geometric probability"]},{"id":"1961-B3","year":1961,"session":"B","number":3,"statement":"3. Consider four points in a plane, no three of which are collinear, and such that the circle through three of them does not pass through the fourth. Prove that one of the four points can be selected having the property that it lies inside the circle determined by the other three.","subjects":["geometry","circles","convex hulls"]},{"id":"1961-B4","year":1961,"session":"B","number":4,"statement":"4. For a fixed positive integer $ n $ let $ x_{1}, x_{2}, \\ldots, x_{n} $ be real numbers satisfying $ 0 \\leq x_{k} \\leq 1 $ for $ k=1,2, \\ldots, n $. Determine the maximum value, as a function of $ n $, of the sum of the $ n(n-1) / 2 $ terms:\n$$\n\\sum_{\\substack{i . j=1 \\\\ i<j}}^{n}\\left|x_{i}-x_{j}\\right|\n$$","subjects":["analysis","optimization","absolute values","combinatorics"]},{"id":"1961-B5","year":1961,"session":"B","number":5,"statement":"5. Let $ k $ be a positive integer, and $ n $ a positive integer greater than 2 . Define\n$$\nf_{1}(n)=n, f_{2}(n)=n^{f_{1}(n)}, \\ldots, f_{j+1}(n)=n^{f_{j}(n)}, \\text { etc. }\n$$\n\nProve either part of the inequality\n$$\nf_{k}(n)<n!!!\\cdots!<f_{k+1}(n)\n$$\nwhere the middle term has $ k $ factorial symbols.","subjects":["analysis","factorials","power towers","growth inequalities"]},{"id":"1961-B6","year":1961,"session":"B","number":6,"statement":"6. Consider the function $ y(x) $ satisfying the differential equation $ y^{\\prime \\prime}=- $ $ (1+\\sqrt{x}) y $ with $ y(0)=1 $ and $ y^{\\prime}(0)=0 $. Prove that $ y(x) $ vanishes exactly once on the interval $ 0<x<\\pi / 2 $, and find a positive lower bound for the zero.","subjects":["analysis","differential equations","oscillation"]},{"id":"1961-B7","year":1961,"session":"B","number":7,"statement":"7. Given a sequence $ \\left\\{a_{n}\\right\\} $ of non-negative real numbers such that $ a_{n+m} \\leq $ $ a_{n} a_{m} $ for all pairs of positive integers, $ m $ and $ n $, prove that the sequence $ \\left\\{\\sqrt[n]{a_{n}}\\right\\} $ has a limit as $ n \\rightarrow \\infty $.","subjects":["analysis","submultiplicative sequences","limits"]},{"id":"1962-A1","year":1962,"session":"A","number":1,"statement":"1. Given five points in a plane, no three of which lie on a straight line, show that some four of these points form the vertices of a convex quadrilateral.","subjects":["geometry","convexity"]},{"id":"1962-A2","year":1962,"session":"A","number":2,"statement":"2. Find every real-valued function $ f $ whose domain is an interval $ I $ (finite or infinite) having 0 as a left-hand endpoint, such that for every positive member $ x $ of $ I $ the average of $ f $ over the closed interval $ [0, x] $ is equal to the geometric mean of the numbers $ f(0) $ and $ f(x) $.","subjects":["analysis","integral equations","separable differential equations"],"errata":"Use the book's strictly positive geometric-mean convention. Let $I$ be an interval in $[0,\\infty)$ containing $0$, and require $f:I\\to(0,\\infty)$ to be integrable on every $[0,x]$ used. Impose the equality at every positive $x\\in I$. The solutions are $f(x)=a/(1-cx)^2$, with $a>0$ and $1-cx>0$ throughout $I$, including restrictions to shorter intervals. A separately admitted singleton interval has any positive value under this convention. Allowing zero as a geometric-mean argument gives additional solutions."},{"id":"1962-A3","year":1962,"session":"A","number":3,"statement":"3. In a triangle $ A B C $ in the Euclidean plane, let $ A^{\\prime} $ be a point on the segment from $ B $ to $ C, B^{\\prime} $ a point on the segment from $ C $ to $ A $, and $ C^{\\prime} $ a point on the segment from $ A $ to $ B $ such that\n$$\n\\frac{A B^{\\prime}}{B^{\\prime} C}=\\frac{B C^{\\prime}}{C^{\\prime} A}=\\frac{C A^{\\prime}}{A^{\\prime} B}=k\n$$\nwhere $ k $ is a positive constant. Let $ \\Delta $ be the triangle formed by parts of the segments obtained by joining $ A $ and $ A^{\\prime}, B $ and $ B^{\\prime} $, and $ C $ and $ C^{\\prime} $. Prove that the areas of the triangles $ \\Delta $ and $ A B C $ are in the ratio.\n$$\n\\frac{(k-1)^{2}}{k^{2}+k+1}\n$$","subjects":["geometry","triangle geometry","area ratios","barycentric coordinates"]},{"id":"1962-A4","year":1962,"session":"A","number":4,"statement":"4. Assume that $ |f(x)| \\leq 1 $ and $ \\left|f^{\\prime \\prime}(x)\\right| \\leq 1 $ for all $ x $ on an interval of length at least 2 . Show that $ \\left|f^{\\prime}(x)\\right| \\leq 2 $ on the interval.","subjects":["analysis","derivative bounds"]},{"id":"1962-A5","year":1962,"session":"A","number":5,"statement":"5. Evaluate in closed form\n$$\n\\sum_{k=1}^{n}\\binom{n}{k} k^{2}\n$$\n(page 560)\n\nNote:\n$$\n\\binom{n}{k}=\\frac{n(n-1) \\cdots(n-k+1)}{1 \\cdot 2 \\cdots k}\n$$","subjects":["combinatorics","binomial sums"]},{"id":"1962-A6","year":1962,"session":"A","number":6,"statement":"6. Let $ S $ be a set of rational numbers such that whenever $ a $ and $ b $ are members of $ S $, so are $ a+b $ and $ a b $, and having the property that for every rational number $ r $ exactly one of the following three statements is true:\n$$\nr \\in S, \\quad-r \\in S, \\quad r=0\n$$\n\nProve that $ S $ is the set of all positive rational numbers.","subjects":["algebra"]},{"id":"1962-B1","year":1962,"session":"B","number":1,"statement":"1. Let $ x^{(n)}=x(x-1) \\cdots(x-n+1) $ for $ n $ a positive integer and let $ x^{(0)} $ $ =1 $. Prove that\n$$\n(x+y)^{(n)}=\\sum_{k=0}^{n}\\binom{n}{k} x^{(k)} y^{(n-k)}\n$$\n\nNote:\n$$\n\\binom{n}{k}=\\frac{n(n-1) \\cdots(n-k+1)}{1 \\cdot 2 \\cdots k}\n$$","subjects":["algebra","falling factorials","binomial identities"]},{"id":"1962-B2","year":1962,"session":"B","number":2,"statement":"2. Let $ R $ be the set of all real numbers and $ S $ the set of all subsets of the positive integers. Construct a function $ f $ whose domain is $ R $ and whose range is in $ S $, such that $ f(a) $ is a proper subset of $ f(b) $ whenever $ a<b $.","subjects":["set theory","countability","dense orders"]},{"id":"1962-B3","year":1962,"session":"B","number":3,"statement":"3. Let $ S $ be a convex region in the Euclidean plane containing the origin. Assume that every ray (that is, half-line) from the origin has at least one point outside $ S $. Prove that $ S $ is bounded. (A region in the plane is defined to be convex if and only if the line segment joining every pair of its points lies entirely within the region.)","subjects":["geometry","convexity","boundedness","compactness"],"errata":"Here region must mean an open set, as required in the book solution. For arbitrary convex sets, the open strip $0<x<1$ together with the origin is a counterexample."},{"id":"1962-B4","year":1962,"session":"B","number":4,"statement":"4. The Euclidean plane is divided into regions by drawing a finite number of circles. Show that it is possible to color each of these regions either red or blue in such a way that no two adjacent regions have the same color. (Two such regions are said to be adjacent if and only if their boundaries have an arc of a circle in common.)","subjects":["combinatorics","planar coloring","parity"]},{"id":"1962-B5","year":1962,"session":"B","number":5,"statement":"$$\n\\begin{array}{l}\n\\text { 5. Prove that for every integer } n \\text { greater than 1: }\\\\\n\\frac{3 n+1}{2 n+2}<\\left(\\frac{1}{n}\\right)^{n}+\\left(\\frac{2}{n}\\right)^{n}+\\cdots+\\left(\\frac{n}{n}\\right)^{n}<2 .(\\text { page } 566)\n\\end{array}\n$$","subjects":["analysis","power sums","convexity","integral comparison"]},{"id":"1962-B6","year":1962,"session":"B","number":6,"statement":"6. Let\n$$\nf(x)=\\sum_{k=0}^{n} a_{k} \\sin k x+b_{k} \\cos k x\n$$\nwhere $ a_{k} $ and $ b_{k} $ are constants. Show that, if $ |f(x)| \\leq 1 $ for $ 0 \\leq x \\leq 2 \\pi $ and $ \\left|f\\left(x_{i}\\right)\\right|=1 $ for $ 0 \\leq x_{1}<x_{2}<\\cdots<x_{2 n}<2 \\pi $, then $ f(x)=\\cos (n x+a) $ for some constant $ a $.","subjects":["analysis","trigonometric polynomials","zero counting","rigidity"],"errata":"Assume $n\\ge1$ and that $f$ is a nonconstant real trigonometric polynomial of degree at most $n$. Count the $2n$ contact points as distinct modulo $2\\pi$. Constants and complex coefficients otherwise give counterexamples. Under these hypotheses the conclusion is $f(x)=\\cos(nx+a)$ with real $a$. The book explicitly flags the constant and complex-coefficient omissions."},{"id":"1963-A1","year":1963,"session":"A","number":1,"statement":"1. (i) Show that a regular hexagon, six squares, and six equilateral triangles can be assembled without overlapping to form a regular dodecagon.\n(ii) Let $ P_{1}, P_{2}, \\ldots, P_{12} $ be the successive vertices of a regular dodecagon. Explain how the three diagonals $ P_{1} P_{9}, P_{2} P_{11} $, and $ P_{4} P_{12} $ intersect.","subjects":["geometry","regular polygons","tilings","concurrency"]},{"id":"1963-A2","year":1963,"session":"A","number":2,"statement":"2. Let $ \\{f(n)\\} $ be a strictly increasing sequence of positive integers such that $ f(2)=2 $ and $ f(m n)=f(m) f(n) $ for every relatively prime pair of positive integers $ m $ and $ n $ (the greatest common divisor of $ m $ and $ n $ is equal to 1). Prove that $ f(n)=n $ for every positive integer $ n $.","subjects":["number theory","multiplicative functions","monotone sequences"]},{"id":"1963-A3","year":1963,"session":"A","number":3,"statement":"3. Find an integral formula for the solution of the differential equation\n$$\n\\delta(\\delta-1)(\\delta-2) \\cdots(\\delta-n+1) y=f(x), \\quad x \\geq 1\n$$\nfor $ y $ as a function of $ x $ satisfying the initial conditions $ y(1)=y^{\\prime}(1)=\\ldots= $ $ y^{(n-1)}(1)=0 $, where $ f $ is continuous and\n$$\n\\delta \\equiv x \\frac{d}{d x}\n$$","subjects":["analysis"]},{"id":"1963-A4","year":1963,"session":"A","number":4,"statement":"4. Let $ \\left\\{a_{n}\\right\\} $ be a sequence of positive real numbers. Show that\n$$\n\\lim _{n \\rightarrow \\infty} \\sup n\\left(\\frac{1+a_{n+1}}{a_{n}}-1\\right) \\geq 1\n$$\n\nShow that the number 1 on the right-hand side of this inequality cannot be replaced by any larger number. (The symbol lim sup is sometimes written lim.)","subjects":["analysis","limsup","positive sequences","harmonic series"]},{"id":"1963-A5","year":1963,"session":"A","number":5,"statement":"5. (i) Prove that if a function $ f $ is continuous on the closed interval $ [0, \\pi] $ and if\n$$\n\\int_{0}^{\\pi} f(\\theta) \\cos \\theta d \\theta=\\int_{0}^{\\pi} f(\\theta) \\sin \\theta d \\theta=0\n$$\nthen there exist points $ \\alpha $ and $ \\beta $ such that\n$$\n0<\\alpha<\\beta<\\pi \\quad \\text { and } f(\\alpha)=f(\\beta)=0\n$$\n(ii) Let $ R $ be any bounded convex open region in the Euclidean plane (that is, $ R $ is a connected open set contained in some circular disk, and the line segment joining any two points of $ R $ lies entirely in $ R $ ). Prove with the help of part (i) that the centroid (center of gravity) of $ R $ bisects at least three distinct chords of the boundary of $ R $.","subjects":["analysis","orthogonality","convex geometry","centroids","chords"]},{"id":"1963-A6","year":1963,"session":"A","number":6,"statement":"6. Let $ U $ and $ V $ be any two distinct points on an ellipse, let $ M $ be the midpoint of the chord $ U V $, and let $ A B $ and $ C D $ be any two other chords through $ M $. If the line $ U V $ meets the line $ A C $ in the point $ P $ and the line $ B D $ in the point $ Q $, prove that $ M $ is the midpoint of the segment $ P Q $.","subjects":["geometry","conics","butterfly theorem","polynomial pencils"]},{"id":"1963-B1","year":1963,"session":"B","number":1,"statement":"$$\n\\text { 1. For what integer } a \\text { does } x^{2}-x+a \\text { divide } x^{13}+x+90 \\text { ? (page 577) }\n$$","subjects":["algebra","polynomial divisibility","integer coefficients"]},{"id":"1963-B2","year":1963,"session":"B","number":2,"statement":"2. Let $ S $ be the set of all numbers of the form $ 2^{m} 3^{n} $, where $ m $ and $ n $ are integers, and let $ P $ be the set of all positive real numbers. Is $ S $ dense in $ P $ ?","subjects":["number theory","diophantine approximation","dense subgroups"]},{"id":"1963-B3","year":1963,"session":"B","number":3,"statement":"3. Find every twice-differentiable real-valued function $ f $ with domain the set of all real numbers and satisfying the functional equation\n$$\n(f(x))^{2}-(f(y))^{2}=f(x+y) f(x-y)\n$$\nfor all real numbers $ x $ and $ y $.","subjects":["analysis","functional equations","differential equations"]},{"id":"1963-B4","year":1963,"session":"B","number":4,"statement":"4. Let $ C $ be a closed plane curve that has a continuously turning tangent and bounds a convex region. If $ T $ is a triangle inscribed in $ C $ with maximum perimeter, show that the normal to $ C $ at each vertex of $ T $ bisects the angle of $ T $ at that vertex. If a triangle $ T $ has the property just described, does it necessarily have maximum perimeter? What is the situation if $ C $ is a circle? (A convex region is a connected open set such that the line segment joining any two points of the set lies entirely in the set.)","subjects":["geometry","optimization","convex curves","angle bisectors"]},{"id":"1963-B5","year":1963,"session":"B","number":5,"statement":"5. Let $ \\left\\{a_{n}\\right\\} $ be a sequence of real numbers satisfying the inequalities\n$$\n0 \\leq a_{k} \\leq 100 a_{n} \\text { for } n \\leq k \\leq 2 n \\text { and } n=1,2, \\ldots,\n$$\nand such that the series\n$$\n\\sum_{n=0}^{\\infty} a_{n}\n$$\nconverges. Prove that\n$$\n\\lim _{n \\rightarrow \\infty} n a_{n}=0\n$$","subjects":["analysis","convergent series","tauberian estimates"]},{"id":"1963-B6","year":1963,"session":"B","number":6,"statement":"6. Let $ E $ be a Euclidean space of at most three dimensions. If $ A $ is a nonempty subset of $ E $, define $ S(A) $ to be the set of all points that lie on closed segments joining pairs of points of $ A $. For a given nonempty set $ A_{0} $, define $ A_{n} \\equiv S\\left(A_{n-1}\\right) $ for $ n=1,2, \\ldots $. Prove that $ A_{2}=A_{3}=\\cdots $. (A one-point set should be considered to be a special case of a closed segment.)","subjects":["geometry","convex hulls","carathéodory theorem","affine dependence","euclidean geometry"]},{"id":"1964-A1","year":1964,"session":"A","number":1,"statement":"1. Given a set of 6 points in the plane, prove that the ratio of the longest distance between any pair to the shortest is at least $ \\sqrt{3} $.","subjects":["geometry","distance ratios","convex hulls","cosine rule"]},{"id":"1964-A2","year":1964,"session":"A","number":2,"statement":"2. Find all continuous positive functions $ f(x) $, for $ 0 \\leq x \\leq 1 $, such that\n$$\n\\begin{array}{c}\n\\int_{0}^{1} f(x) d x=1 \\\\\n\\int_{0}^{1} f(x) x d x=\\alpha \\\\\n\\int_{0}^{1} f(x) x^{2} d x=\\alpha^{2}\n\\end{array}\n$$\nwhere $ \\alpha $ is a given real number.","subjects":["analysis","algebra"]},{"id":"1964-A3","year":1964,"session":"A","number":3,"statement":"3. Let $ P_{1}, P_{2}, \\ldots $ be a sequence of distinct points which is dense in the interval $ (0,1) $. The points $ P_{1}, P_{2}, \\ldots, P_{n-1} $ decompose the interval into $ n $ parts, and $ P_{n} $ decomposes one of these into two parts. Let $ a_{n} $ and $ b_{n} $ be the lengths of these two intervals. Prove that\n$$\n\\sum_{n=1}^{\\infty} a_{n} b_{n}\\left(a_{n}+b_{n}\\right)=1 / 3\n$$\n(A sequence of points in an interval is said to be dense when every subinterval contains at least one point of the sequence.)","subjects":["analysis","algebra"]},{"id":"1964-A4","year":1964,"session":"A","number":4,"statement":"4. Let $ p_{n}(n=1,2, \\ldots) $ be a bounded sequence of integers which satisfies the recursion\n$$\np_{n}=\\frac{p_{n-1}+p_{n-2}+p_{n-3} p_{n-4}}{p_{n-1} p_{n-2}+p_{n-3}+p_{n-4}}\n$$\n\nShow that the sequence eventually becomes periodic.","subjects":["combinatorics","integer sequences","recurrences"]},{"id":"1964-A5","year":1964,"session":"A","number":5,"statement":"5. Prove that there is a constant $ K $ such that the following inequality holds for any sequence of positive numbers $ a_{1}, a_{2}, a_{3}, \\ldots $ :\n$$\n\\sum_{n=1}^{\\infty} \\frac{n}{a_{1}+a_{2}+\\cdots+a_{n}} \\leq K \\sum_{n=1}^{\\infty} \\frac{1}{a_{n}} . \\quad(\\text { page 589) }\n$$","subjects":["analysis","series inequalities","hardy-type inequalities"]},{"id":"1964-A6","year":1964,"session":"A","number":6,"statement":"6. Let $ S $ be a finite subset of a straight line. Say that $ S $ has the repeated distance property when every value of the distance between pairs of points of $ S $ (except for the longest) occurs at least twice. Show that if $ S $ has the repeated distance property then the ratio of any two distances between two points of $ S $ is a rational number.","subjects":["algebra","rational linear dependence","distance sets"]},{"id":"1964-B1","year":1964,"session":"B","number":1,"statement":"1. Let $ u_{k}(k=1,2, \\ldots) $ be a sequence of positive integers, and let $ V_{n} $ be the number of those which are less than or equal to $ n $. Show that if\n$$\n\\sum_{k=1}^{\\infty} 1 / u_{k}<\\infty\n$$\nthen\n$$\n\\lim _{n \\rightarrow \\infty} V_{n} / n=0\n$$","subjects":["analysis","counting functions","convergent series"]},{"id":"1964-B2","year":1964,"session":"B","number":2,"statement":"2. Let $ S $ be a set of $ n>0 $ elements, and let $ A_{1}, A_{2}, \\ldots, A_{k} $ be a family of distinct subsets, with the property that any two of these subsets meet. Assume that no other subset of $ S $ meets all of the $ A_{i} $.\n\nProve that $ k=2^{n-1} $.","subjects":["combinatorics","set systems","intersecting families"],"errata":"Require every family member to be nonempty, or interpret the intersection condition as also testing each member against itself. Under the distinct-pair-only reading, the singleton family $\\{\\varnothing\\}$ is a counterexample for $n>1$. This qualification is justified by that explicit counterexample, not presented as a printed official correction."},{"id":"1964-B3","year":1964,"session":"B","number":3,"statement":"3. Let $ f(x) $ be a real continuous function defined for all real $ x $. Assume that for every $ \\epsilon>0 $\n$$\n\\lim _{n \\rightarrow \\infty} f(n \\epsilon)=0, \\quad \\text { (where } n \\text { is a positive integer) }\n$$\n\nProve that\n$$\n\\lim _{x \\rightarrow \\infty} f(x)=0\n$$","subjects":["analysis","continuity","baire category","limits"]},{"id":"1964-B4","year":1964,"session":"B","number":4,"statement":"4. Into how many regions do $ n $ great circles (no three concurrent) decompose the surface of the sphere on which they lie?","subjects":["combinatorics","spherical geometry","arrangements"]},{"id":"1964-B5","year":1964,"session":"B","number":5,"statement":"5. Let $ u_{n}(n=1,2,3, \\ldots) $ denote the least common multiple of the first $ n $ terms of a strictly increasing sequence of positive integers (for example, the sequence $ 1,2,3,4,5,6,10,12, \\ldots) $. Prove that the series\n$$\n\\sum_{n=1}^{\\infty} 1 / u_{n}\n$$\nis convergent.","subjects":["number theory","least common multiples","series convergence"]},{"id":"1964-B6","year":1964,"session":"B","number":6,"statement":"6. Show that the unit disk in the plane cannot be partitioned into two disjoint congruent subsets.","subjects":["geometry","congruence","disk partitions"]},{"id":"1965-A1","year":1965,"session":"A","number":1,"statement":"A-1. Let $ A B C $ be a triangle with angle $ A< $ angle $ C<90^{\\circ}< $ angle $ B $. Consider the bisectors of the external angles at $ A $ and $ B $, each measured from the vertex to the opposite side (extended). Suppose both of these line-segments are equal to $ A B $. Compute the angle $ A $.","subjects":["geometry","triangle geometry","external angle bisectors"]},{"id":"1965-A2","year":1965,"session":"A","number":2,"statement":"A-2. Show that, for any positive integer $ n $,\n$$\n\\sum_{r=0}^{[(n-1) / 2]}\\left\\{\\frac{n-2 r}{n}\\binom{n}{r}\\right\\}^{2}=\\frac{1}{n}\\binom{2 n-2}{n-1}\n$$\nwhere $ [x] $ means the greatest integer not exceeding $ x $, and $ \\left({ }_{r}^{n}\\right) $ is the binomial coefficient","subjects":["combinatorics","binomial identities","catalan numbers"]},{"id":"1965-A3","year":1965,"session":"A","number":3,"statement":"A-3. Show that, for any sequence $ a_{1}, a_{2}, \\cdots $ of real numbers, the two conditions\n(A)\n$$\n\\lim _{n \\rightarrow \\infty} \\frac{e^{\\left(i a_{1}\\right)}+e^{\\left(i a_{2}\\right)}+\\cdots+e^{\\left(i a_{n}\\right)}}{n}=\\alpha\n$$\nand\n(B)\n$$\n\\lim _{n \\rightarrow \\infty} \\frac{\\sum_{k=1}^{n^2} e^{ia_k}}{n^2}=\\alpha\n$$\nare equivalent.","subjects":["analysis","cesàro means","subsequences","bounded sequences"]},{"id":"1965-A4","year":1965,"session":"A","number":4,"statement":"A-4. At a party, assume that no boy dances with every girl but each girl dances with at least one boy. Prove that there are two couples $ g b $ and $ g^{\\prime} b^{\\prime} $ which dance whereas $ b $ does not dance with $ g^{\\prime} $ nor does $ g $ dance with $ b^{\\prime} $.","subjects":["combinatorics","bipartite graphs","induced matchings"],"errata":"Assume a finite party with at least one girl. Without these conventions the empty party is vacuous, and infinitely many nested finite partner sets give a counterexample. The source proof uses a participant with a maximum finite number of partners. Finiteness is an intended-setting interpretation, not an explicitly quoted official erratum."},{"id":"1965-A5","year":1965,"session":"A","number":5,"statement":"A-5. In how many ways can the integers from 1 to $ n $ be ordered subject to the condition that, except for the first integer on the left, every integer differs by 1 from some integer to the left of it?","subjects":["combinatorics"]},{"id":"1965-A6","year":1965,"session":"A","number":6,"statement":"A-6. In the plane with orthogonal Cartesian coordinates $ x $ and $ y $, prove that the line whose equation is $ u x+v y=1 $ will be tangent to the curve $ x^{m}+y^{m}=1 $ (where $ m>1 $ ) if and only if $ u^{n}+v^{n}=1 $ and $ m^{-1}+n^{-1}=1 $.","subjects":["analysis","convexity","tangency"],"errata":"The printed problem omits necessary domain conventions. The book's solution on printed pp. 48-49, PDF pp. 56-57, explicitly restricts the discussion to nonnegative $x,y,u,v$. Use ordinary real powers on nonnegative bases and the following corrected review target. Let $m>1$, put $n=m/(m-1)$, and take $u,v\\ge0$. Prove that the line $ux+vy=1$ is tangent to the arc $x^m+y^m=1$ in the closed first quadrant if and only if $u^n+v^n=1$, including the limiting tangents at the axis endpoints. At $u=v=0$ the equation defines no line and is not tangent. For comparison, $m=3$ has a tangent at $(-1,2^{1/3})$ with $u=1$, $v=2^{2/3}$ and $u^{3/2}+v^{3/2}=3$, so the unrestricted statement is false. The canonical statement is retained. This is a source-documented domain qualification, not an independently authenticated official erratum or proof acceptance."},{"id":"1965-B1","year":1965,"session":"B","number":1,"statement":"$$\n\\begin{array}{l}\n\\text { B-1. Evaluate }\\\\\n\\lim _{n \\rightarrow \\infty} \\int_{0}^{1} \\int_{0}^{1} \\cdots \\int_{0}^{1} \\cos ^{2}\\left\\{\\frac{\\pi}{2 n}\\left(x_{1}+x_{2}+\\cdots+x_n\\right)\\right\\} d x_{1} d x_{2} \\cdots d x_{n}\n\\end{array}\n$$","subjects":["analysis","multiple integrals","symmetry"]},{"id":"1965-B2","year":1965,"session":"B","number":2,"statement":"B-2. In a round-robin tournament with $ n $ players $ P_{1}, P_{2}, \\cdots, P_{n} $ (where $ n>1 $ ), each player plays one game with each of the other players and the rules are such that no ties can occur. Let $ w_r $ and $ l_{r} $ be the number of games won and lost, respectively, by $ P_{r} $. Show that\n$$\n\\sum_{r=1}^{n} w_{r}^{2}=\\sum_{r=1}^{n} l_{r}^{2}\n$$","subjects":["combinatorics"]},{"id":"1965-B3","year":1965,"session":"B","number":3,"statement":"B-3. Prove that there are exactly three right-angled triangles whose sides are integers while the area is numerically equal to twice the perimeter.","subjects":["number theory","pythagorean triples","integer triangles"]},{"id":"1965-B4","year":1965,"session":"B","number":4,"statement":"B-4. Consider the function\n$$\nf(x, n)=\\frac{\\binom{n}{0}+\\binom{n}{2} x+\\binom{n}{4} x^{2}+\\cdots}{\\binom{n}{1}+\\binom{n}{3} x+\\binom{n}{5} x^{2}+\\cdots}\n$$\nwhere $ n $ is a positive integer. Express $ f(x, n+1) $ rationally in terms of $ f(x, n) $ and $ x $. Hence, or otherwise, evaluate $ \\lim _{n \\rightarrow \\infty} f(x, n) $ for suitable fixed values of $ x $. (The symbols $ \\binom{n}{r} $ represent the binomial coefficients.)","subjects":["algebra","binomial sums","möbius iteration","limits"]},{"id":"1965-B5","year":1965,"session":"B","number":5,"statement":"B-5. Consider collections of unordered pairs of $ V $ different objects $ a, b, c, \\cdots, k $. Three pairs such as $ b c, c a, a b $ are said to form a triangle. Prove that, if $ 4 E \\leqq V^{2} $, it is possible to choose $ E $ pairs so that no triangle is formed.","subjects":["combinatorics","graph theory","triangle-free graphs"]},{"id":"1965-B6","year":1965,"session":"B","number":6,"statement":"B-6. If $ A, B, C, D $ are four distinct points such that every circle through $ A $ and $ B $ intersects (or coincides with) every circle through $ C $ and $ D $, prove that the four points are either collinear (all of one line) or concyclic (all on one circle).","subjects":["geometry","circle pencils","concentric circles","analytic geometry"]},{"id":"1966-A1","year":1966,"session":"A","number":1,"statement":"A-1. Let $ f(n) $ be the sum of the first $ n $ terms of the sequence $ 0,1,1,2,2,3,3,4, \\cdots $, where the $ n $th term is given by\n$$\na_{n}=\\left\\{\\begin{array}{cc}\nn / 2 & \\text { if } n \\text { is even } \\\\\n(n-1) / 2 & \\text { if } n \\text { is odd }\n\\end{array}\\right.\n$$\n\nShow that if $ x $ and $ y $ are positive integers and $ x>y $ then $ x y=f(x+y)-f(x-y) $.","subjects":["algebra","finite sums","floor function"]},{"id":"1966-A2","year":1966,"session":"A","number":2,"statement":"A-2. Let $ a, b, c $ be the lengths of the sides of a triangle, let $ p=(a+b+c) / 2 $, and $ r $ be the radius of the inscribed cricle. Show that\n$$\n\\frac{1}{(p-a)^{2}}+\\frac{1}{(p-b)^{2}}+\\frac{1}{(p-c)^{2}} \\geqq \\frac{1}{r^{2}}\n$$","subjects":["geometry","triangle inequalities","inradius"]},{"id":"1966-A3","year":1966,"session":"A","number":3,"statement":"$$\n\\begin{array}{l}\n\\text { A-3. Let } 0<x_{1}<1 \\text { and } x_{n+1}=x_{n}\\left(1-x_{n}\\right), n=1,2,3, \\cdots \\text {. Show that }\\\\\n\\lim _{n \\rightarrow \\infty} n x_{n}=1\n\\end{array}\n$$","subjects":["analysis","nonlinear recurrences","asymptotics"]},{"id":"1966-A4","year":1966,"session":"A","number":4,"statement":"A-4. Prove that after deleting the perfect squares from the list of positive integers the number we find in the $ n $th position is equal to $ n+\\{\\sqrt{ } n\\} $, where $ \\{\\sqrt{ } n\\} $ denotes the integer closest to $ \\sqrt{ } n $.","subjects":["number theory","enumeration","perfect squares","nearest integer"]},{"id":"1966-A5","year":1966,"session":"A","number":5,"statement":"A-5. Let $ C $ denote the family of continuous functions on the real axis. Let $ T $ be a mapping of $ C $ into $ C $ which has the following properties:\n1. $ T $ is linear, i.e. $ T\\left(c_{1} \\psi_{1}+c_{2} \\psi_{2}\\right)=c_{1} T \\psi_{1}+c_{2} T \\psi_{2} $, for $ c_{1} $ and $ c_{2} $ real and $ \\psi_{1} $ and $ \\psi_{2} $ in $ C $.\n2. $ T $ is local, i.e. if $ \\psi_{1} \\equiv \\psi_{2} $ in some interval $ I $ then also $ T \\psi_{1} \\equiv T \\psi_{2} $ holds in $ I $.\n\nShow that $ T $ must necessarily be of the form $ T \\psi(x)=f(x) \\psi(x) $ where $ f(x) $ is a suitable continuous function.","subjects":["analysis","linear operators","locality","continuous functions"]},{"id":"1966-A6","year":1966,"session":"A","number":6,"statement":"$$\n\\begin{array}{l}\n\\text{A-6} \\text {. Justify the statement that }\\\\\n3=\\sqrt{1+2 \\sqrt{1+3 \\sqrt{1+4 \\sqrt{1+5 \\sqrt{1+\\cdots}}}}}\n\\end{array}\n$$","subjects":["analysis","nested radicals","convergence","ramanujan radical"]},{"id":"1966-B1","year":1966,"session":"B","number":1,"statement":"B-1. Let a convex polygon $ P $ be contained in a square of side one. Show that the sum of the squares of the sides of $ P $ is less than or equal to 4.","subjects":["geometry","convex polygons","quadratic perimeter"]},{"id":"1966-B2","year":1966,"session":"B","number":2,"statement":"B-2. Prove that among any ten consecutive integers at least one is relatively prime to each of the others.","subjects":["number theory","coprimality","consecutive integers"]},{"id":"1966-B3","year":1966,"session":"B","number":3,"statement":"B-3. Show that if the series\n$$\n\\sum_{n=1}^{\\infty} \\frac{1}{p_{n}}\n$$\nis convergent, where $ p_{1}, p_{2}, p_{3}, \\cdots, p_{n}, \\cdots $ are positive real numbers, then the series\n$$\n\\sum_{n=1}^{\\infty} \\frac{n^{2}}{\\left(p_{1}+p_{2}+\\cdots+p_{n}\\right)^{2}} p_{n}\n$$\nis also convergent.","subjects":["analysis","series","inequalities"]},{"id":"1966-B4","year":1966,"session":"B","number":4,"statement":"B-4. Let $ 0<a_{1}<a_{2}<\\cdots<a_{m n+1} $ be $ m n+1 $ integers. Prove that you can select either $ m+1 $ of them no one of which divides any other, or $ n+1 $ of them each dividing the following one.","subjects":["combinatorics","partially ordered sets","divisibility"]},{"id":"1966-B5","year":1966,"session":"B","number":5,"statement":"B-5. Given $ n(\\geqq 3) $ distinct points in the plane, no three of which are on the same straight line, prove that there exists a simple closed polygon with these points as vertices.","subjects":["geometry","discrete geometry","polygons"]},{"id":"1966-B6","year":1966,"session":"B","number":6,"statement":"$$\n\\text { B-6. Show that all solutions of the differential equation } y^{\\prime \\prime}+e^{x} y=0 \\text { remain bounded as } x \\rightarrow \\infty \\text {. }\n$$","subjects":["analysis","ordinary differential equations","boundedness"]},{"id":"1967-A1","year":1967,"session":"A","number":1,"statement":"A-1. Let $ f(x)=a_{1} \\sin x+a_{2} \\sin 2 x+\\cdots+a_{n} \\sin n x $, where $ a_{1}, a_{2}, \\cdots, a_{n} $ are real numbers and where $ n $ is a positive integer. Given that $ |f(x)| \\leqq|\\sin x| $ for all real $ x $, prove that\n$$\n\\left|a_{1}+2 a_{2}+\\cdots+n a_{n}\\right| \\leqq 1\n$$","subjects":["analysis","trigonometric polynomials","inequalities"]},{"id":"1967-A2","year":1967,"session":"A","number":2,"statement":"A-2. Define $ S_{0} $ to be 1 . For $ n \\geqq 1 $, let $ S_{n} $ be the number of $ n \\times n $ matrices whose elements are nonnegative integers with the property that $ a_{i j}=a_{j i},(i, j=1,2, \\cdots, n) $ and where $ \\sum_{i=1}^{n} a_{i j} $ $ =1,(j=1,2, \\cdots, n) $. Prove\n(a) $ S_{n+1}=S_{n}+n S_{n-1} $,\n(b) $ \\sum_{n=0}^{\\infty} S_{n} \\frac{x^{n}}{n!}=\\exp \\left(x+x^{2} / 2\\right), \\quad $ where $ \\exp (x)=e^{x} $.","subjects":["combinatorics","involutions"]},{"id":"1967-A3","year":1967,"session":"A","number":3,"statement":"A-3. Consider polynomial forms $ a x^{2}-b x+c $ with integer coefficients which have two distinct zeros in the open interval $ 0<x<1 $. Exhibit with a proof the least positive integer value of $ a $ for which such a polynomial exists.","subjects":["algebra","integer polynomials","quadratic polynomials"]},{"id":"1967-A4","year":1967,"session":"A","number":4,"statement":"A-4. Show that if $ \\lambda>\\frac{1}{2} $ there does not exist a real-valued function $ u $ such that for all $ x $ in the closed interval $ 0 \\leqq x \\leqq 1, u(x)=1+\\lambda \\int_{x}^{1} u(y) u(y-x) d y $.","subjects":["analysis","integral equations","nonexistence"],"errata":"Integration convention, not a historical correction. The explicit target uses finite Lebesgue integrals and includes ordinary Riemann candidates. The displayed equation supplies the square integrability needed by the proofs; no continuity or sign assumption is added. The primary source does not specify a historical choice of integration theory."},{"id":"1967-A5","year":1967,"session":"A","number":5,"statement":"A-5. Show that in a convex region in the plane whose boundary contains at most a finite number of straight line segments and whose area is greater than $ \\pi / 4 $ there is at least one pair of points a unit distance apart.","subjects":["geometry","convexity","area","isodiametric inequality"]},{"id":"1967-A6","year":1967,"session":"A","number":6,"statement":"A-6. Given real numbers $ \\left\\{a_{i}\\right\\} $ and $ \\left\\{b_{i}\\right\\},(i=1,2,3,4) $, such that $ a_{1} b_{2}-a_{2} b_{1} \\neq 0 $. Consider the set of all solutions ( $ x_{1}, x_{2}, x_3, x_{4} $ ) of the simultaneous equations\n$$\na_{1} x_{1}+a_{2} x_{2}+a_3 x_3+a_{4} x_{4}=0 \\text { and } b_{1} x_{1}+b_{2} x_{2}+b_3 x_3+b_{4} x_{4}=0\n$$\nfor which no $ x_{i}(i=1,2,3,4) $ is zero. Each such solution generates a 4-tuple of plus and minus signs (signum $ x_{1} $, signum $ x_{2} $, signum $ x_{3} $, signum $ x_{4} $ ).\n(a) Determine, with a proof, the maximum number of distinct 4-tuples possible.\n(b) Investigate necessary and sufficient conditions on the real numbers $ \\left\\{a_{i}\\right\\} $ and $ \\left\\{b_{i}\\right\\} $ such that the above maximum number of 4 -tuples is attained.","subjects":["linear algebra","geometry","sign patterns"]},{"id":"1967-B1","year":1967,"session":"B","number":1,"statement":"B-1. Let $ (A B C D E F) $ be a hexagon inscribed in a circle of radius $ r $. Show that if $ \\overline{A B}=\\overline{CD} $ $ =\\overline{E F}=r $, then the midpoints of $ \\overline{B C}, \\overline{D E}, \\overline{F A} $ are the vertices of an equilateral triangle.","subjects":["geometry","complex numbers","cyclic polygons"]},{"id":"1967-B2","year":1967,"session":"B","number":2,"statement":"B-2. Let $ 0 \\leqq p \\leqq 1 $ and $ 0 \\leqq r \\leqq 1 $ and consider the identities\n(a) $ (p x+(1-p) y)^{2}=A x^{2}+B x y+C y^{2} $,\n(b) $ (p x+(1-p) y)(r x+(1-r) y)=\\alpha x^{2}+\\beta x y+\\gamma y^{2} $.\n\nShow that (with respect to $ p $ and $ r $ )\n(a) $ \\max \\{A, B, C\\} \\geqq 4 / 9 $,\n(b) $ \\max \\{\\alpha, \\beta, \\gamma\\} \\geqq 4 / 9 $.","subjects":["algebra","inequalities","quadratic polynomials"]},{"id":"1967-B3","year":1967,"session":"B","number":3,"statement":"B-3. If $ f $ and $ g $ are continuous and periodic functions with period 1 on the real line, then $ \\lim _{n \\rightarrow \\infty} \\int_{0}^{1} f(x) g(n x) d x=\\left(\\int_{0}^{1} f(x) d x\\right)\\left(\\int_{0}^{1} g(x) d x\\right) $.","subjects":["analysis","periodic functions","integration","limits"]},{"id":"1967-B4","year":1967,"session":"B","number":4,"statement":"B-4. (a) A certain locker room contains $ n $ lockers numbered $ 1,2,3, \\cdots, n $ and all are originally locked. An attendant performs a sequence of operations $ T_{1}, T_{2}, \\cdots, T_{n} $ whereby with the operation $ T_{k}, 1 \\leqq k \\leqq n $, the condition of being locked or unlocked is changed for all those lockers and only those lockers whose numbers are multiples of $ k $. After all the $ n $ operations have been performed it is observed that all lockers whose numbers are perfect squares (and only those lockers) are now open or unlocked. Prove this mathematically.\n(b) Investigate in a meaningful mathematical way a procedure or set of operations similar to those above which will produce the set of cubes, or the set of numbers of the form $ 2 \\mathrm{~m}^{2} $, or the set of numbers of the form $ m^{2}+1 $, or some nontrivial similar set of your own selection.","subjects":["number theory","divisors","parity","incidence algebra","linear algebra"]},{"id":"1967-B5","year":1967,"session":"B","number":5,"statement":"B-5. Show that the sum of the first $ n $ terms in the binomial expansion of $ (2-1)^{-n} $ is $ \\frac{1}{2} $, where $ n $ is a positive integer.","subjects":["combinatorics","binomial coefficients","probability"]},{"id":"1967-B6","year":1967,"session":"B","number":6,"statement":"B-6. Let $ f $ beareal-valued function having partial derivatives and which is defined for $ x^{2}+y^{2} \\leq 1 $ and is such that $ |f(x, y)| \\leqq 1 $. Show that there exists a point ( $ x_{0}, y_{0} $ ) in the interior of the unit circle such that\n$$\n\\left(\\frac{\\partial f}{\\partial x}\\left(x_{0}, y_{0}\\right)\\right)^{2}+\\left(\\frac{\\partial f}{\\partial y}\\left(x_{0}, y_{0}\\right)\\right)^{2} \\leqq 16\n$$","subjects":["analysis","partial derivatives","continuity","gradient bounds"],"errata":"Hypothesis clarification. Add joint continuity on the closed disk for the standard minimum argument. Existence of the two partial derivatives alone does not suffice. A separately smooth bounded counterexample to the literal formulation, and a proof of the continuity-qualified theorem, have passed two independent model audits. The research text and audit records are linked below. These are advisory mathematical reviews, not an official erratum or three approved solution tracks."},{"id":"1968-A1","year":1968,"session":"A","number":1,"statement":"$$\n\\begin{array}{l}\n\\text { A-1. Prove }\\\\\n\\frac{22}{7}-\\pi=\\int_{0}^{1} \\frac{x^{4}(1-x)^{4}}{1+x^{2}} d x\n\\end{array}\n$$","subjects":["analysis","definite integrals","rational functions"]},{"id":"1968-A2","year":1968,"session":"A","number":2,"statement":"A-2. Given integers $ a, b, e, c, d $, and $ f $ with $ a d \\neq b c $, and given a real number $ \\epsilon>0 $, show that there exist rational numbers $ r $ and $ s $ for which\n$$\n\\begin{array}{l}\n0<|r a+s b-e|<\\epsilon \\\\\n0<|r c+s d-f|<\\epsilon .\n\\end{array}\n$$","subjects":["linear algebra","rational numbers","approximation"]},{"id":"1968-A3","year":1968,"session":"A","number":3,"statement":"A-3. Prove that a list can be made of all the subsets of a finite set in such a way that (i) the empty set is first in the list, (ii) each subset occurs exactly once, (iii) each subset in the list is obtained either by adding one element to the preceding subset or by deleting one element of the preceding subset.","subjects":["combinatorics","subsets","gray codes"]},{"id":"1968-A4","year":1968,"session":"A","number":4,"statement":"A-4. Given $ n $ points on the sphere $ \\left\\{(x, y, z): x^{2}+y^{2}+z^{2}=1\\right\\} $, demonstrate that the sum of the squares of the distances between them does not exceed $ n^{2} $.","subjects":["geometry","vectors","distance sums"]},{"id":"1968-A5","year":1968,"session":"A","number":5,"statement":"A-5. Let $ V $ be the collection of all quadratic polynomials $ P $ with real coefficients such that $ |P(x)| \\leqq 1 $ for all $ x $ on the closed interval $ [0,1] $. Determine\n$$\n\\sup \\left\\{\\left|P^{\\prime}(0)\\right|: P \\in V\\right\\}\n$$","subjects":["analysis","polynomials","extremal problems"]},{"id":"1968-A6","year":1968,"session":"A","number":6,"statement":"A-6. Determine all polynomials of the form $ \\sum_{0}^{n} a_{i} x^{n-i} $ with $ a_{i}= \\pm 1(0 \\leqq i \\leqq n, 1 \\leqq n<\\infty) $ such that each has only real zeros.","subjects":["algebra","polynomial roots","integer polynomials"]},{"id":"1968-B1","year":1968,"session":"B","number":1,"statement":"B-1. The temperatures in Chicago and Detroit are $ x^{\\circ} $ and $ y^{\\circ} $, respectively. These temperatures are not assumed to be independent; namely, we are given:\n(i) $ P\\left(x^{\\circ}=70^{\\circ}\\right) $, the probability that the temperature in Chicago is $ 70^{\\circ} $,\n(ii) $ P\\left(y^{\\circ}=70^{\\circ}\\right) $, and\n(iii) $ P\\left(\\max \\left(x^{\\circ}, y^{\\circ}\\right)=70^{\\circ}\\right) $.\n\nDetermine $ P\\left(\\min \\left(x^{\\circ}, y^{\\circ}\\right)=70^{\\circ}\\right) $.","subjects":["probability","dependent random variables","order statistics"]},{"id":"1968-B2","year":1968,"session":"B","number":2,"statement":"B-2. $ A $ is a subset of a finite group $ G $ (with group operation called multiplication), and $ A $contains more than one half of the elements of $ G $. Prove that each element of $ G $ is the product of two elements of $ A $.","subjects":["algebra","finite groups","product sets"]},{"id":"1968-B3","year":1968,"session":"B","number":3,"statement":"B-3. Assume that a $ 60^{\\circ} $ angle cannot be trisected with ruler and compass alone. Prove that if $ n $ is a positive multiple of 3 , then no angle of $ 360 / n $ degrees can be trisected with ruler and compass alone.","subjects":["algebra","constructibility","cyclotomic fields","local fields"]},{"id":"1968-B4","year":1968,"session":"B","number":4,"statement":"B-4. Show that if $ f $ is real-valued and continuous on $ (-\\infty, \\infty) $ and $ \\int_{-\\infty}^{\\infty} f(x) d x $ exists, then $ \\int_{-\\infty}^{\\infty} f\\left(x-\\frac{1}{x}\\right) d x=\\int_{-\\infty}^{\\infty} f(x) d x $.","subjects":["analysis","improper integrals"]},{"id":"1968-B5","year":1968,"session":"B","number":5,"statement":"B-5. Let $ p $ be a prime number. Let $ J $ be the set of all $ 2 \\times 2 $ matrices $ \\binom{a b}{c d} $ whose entries are chosen from $ \\{0,1,2, \\cdots, p-1\\} $ and satisfy the conditions $ a+d \\equiv 1(\\bmod p), a d-b c \\equiv 0(\\bmod p) $.\n\nDetermine how many members $ J $ has.","subjects":["linear algebra","finite fields","matrix enumeration"]},{"id":"1968-B6","year":1968,"session":"B","number":6,"statement":"B-6. A set of real numbers is called compact if it is closed and bounded. Show that there does not exist a sequence $ \\left\\{K_{n}\\right\\}_{n=0}^{\\infty} $ of compact sets of rational numbers such that each compact set of rationals is contained in at least one $ K_{n} $.","subjects":["analysis","topology","compactness","rational numbers","countable compact spaces"]},{"id":"1969-A1","year":1969,"session":"A","number":1,"statement":"A-1. Let $ f(x, y) $ be a polynomial with real coefficients in the real variables $ x $ and $ y $ defined over the entire $ x-y $ plane. What are the possibilities for the range of $ f(x, y) $ ?","subjects":["algebra","real polynomials","ranges","connectedness","interpolation","linear algebra"]},{"id":"1969-A2","year":1969,"session":"A","number":2,"statement":"A-2. Let $ D_{n} $ be the determinant of order $ n $ of which the element in the $ i $ th row and the $ j $ th column is the absolute value of the difference of $ i $ and $ j $. Show that $ D_{n} $ is equal to\n$$\n(-1)^{n-1}(n-1) 2^{n-2}\n$$","subjects":["linear algebra","determinants","distance matrices"]},{"id":"1969-A3","year":1969,"session":"A","number":3,"statement":"A-3. Let $ P $ be a non-self-intersecting closed polygon with $ n $ sides. Let its vertices be $ P_{1} $, $ P_{2}, \\cdots, P_{n} $. Let $ m $ other points, $ Q_{1}, Q_{2}, \\cdots, Q_{m} $ interior to $ P $ be given. Let the figure be triangulated. This means that certain pairs of the $ (n+m) $ points $ P_{1}, \\cdots, Q_{m} $ are connected by line segments such that (i) the resulting figure consists exclusively of a set $ T $ of triangles, (ii) if two different triangles in $ T $ have more than a vertex in common then they have exactly a side in common, and (iii) the set of vertices of the triangles in $ T $ is precisely the set of ( $ n+m $ ) points $ P_{1}, \\cdots, Q_{m} $. How many triangles in $ T $ ?","subjects":["combinatorics","geometry","triangulations","planar graphs"],"errata":"Require the triangles to cover and subdivide exactly the closed region bounded by $P$, with the stated compatible intersections and no additional vertices. The book's comment on printed p. 62, PDF p. 70, explicitly identifies the missing interior-triangulation requirement. Allowing exterior triangles under the literal listed conditions can change the count. The original wording is retained, and the reviewed target states this interior-subdivision correction explicitly."},{"id":"1969-A4","year":1969,"session":"A","number":4,"statement":"A-4. Show that\n$$\n\\int_{0}^{1} x^{x} d x=\\sum_{n=1}^{\\infty}(-1)^{n+1} n^{-n}\n$$\n(The integrand is taken to be 1 at $ x=0 $.)","subjects":["analysis","integration","power series"]},{"id":"1969-A5","year":1969,"session":"A","number":5,"statement":"A-5. Let $ u(t) $ be a continuous function in the system of differential equations\n$$\n\\frac{d x}{d t}=-2 y+u(t), \\quad \\frac{d y}{d t}=-2 x+u(t)\n$$\n\nShow that, regardless of the choice of $ u(t) $, the solution of the system which satisfies $ x=x_{0}, y=y_{0} $ at $ t=0 $ will never pass through $ (0,0) $ unless $ x_{0}=y_{0} $. When $ x_{0}=y_{0} $, show that, for any positive value $ t_{0} $ of $ t $, it is possible to choose $ u(t) $ so the solution is at $ (0,0) $ when $ t=t_{0} $.","subjects":["analysis","ordinary differential equations","control"]},{"id":"1969-A6","year":1969,"session":"A","number":6,"statement":"A-6. Let a sequence $ \\left\\{x_{n}\\right\\} $ be given, and let $ y_{n}=x_{n-1}+2 x_{n}, n=2,3,4, \\cdots $ Suppose that the sequence $ \\left\\{y_{n}\\right\\} $ converges. Prove that the sequence $ \\left\\{x_{n}\\right\\} $ also converges.","subjects":["analysis","sequences","linear recurrences"]},{"id":"1969-B1","year":1969,"session":"B","number":1,"statement":"B-1. Let $ n $ be a positive integer such that $ n+1 $ is divisible by 24 . Prove that the sum of all the divisors of $ n $ is divisible by 24 .","subjects":["number theory","divisor sums","congruences","representations by quadratic forms"]},{"id":"1969-B2","year":1969,"session":"B","number":2,"statement":"B-2. Show that a finite group can not be the union of two of its proper subgroups. Does the statement remain true if \"two\" is replaced by \"three\"?","subjects":["algebra","finite groups","subgroup covers"]},{"id":"1969-B3","year":1969,"session":"B","number":3,"statement":"B-3. The terms of a sequence $ T_{n} $ satisfy\n$$\nT_{n} T_{n+1}=n \\quad(n=1,2,3, \\cdots) \\text { and } \\lim _{n \\rightarrow \\infty} \\frac{T_{n}}{T_{n+1}}=1\n$$\n\nShow that $ \\pi T_{1}^{2}=2 $.","subjects":["analysis","sequences","wallis product","gaussian integrals"]},{"id":"1969-B4","year":1969,"session":"B","number":4,"statement":"$$\n\\text { B-4. Show that any curve of unit length can be covered by a closed rectangle of area } 1 / 4 \\text {. }\n$$","subjects":["geometry","rectifiable curves","bounding rectangles"]},{"id":"1969-B5","year":1969,"session":"B","number":5,"statement":"B-5. Let $ a_{1}<a_{2}<a_{3}<\\cdots $ be an increasing sequence of positive integers. Let the series\n$$\n\\sum_{n=1}^{\\infty} 1 / a_{n}\n$$\nbe convergent. For any number $ x $, let $ k(x) $ be the number of the $ a_{n} $ 's which do not exceed $ x $. Show that $ \\lim _{x \\rightarrow \\infty} k(x) / x=0 $.","subjects":["analysis","series","asymptotic density"]},{"id":"1969-B6","year":1969,"session":"B","number":6,"statement":"B-6. Let $ A $ and $ B $ be matrices of size $ 3 \\times 2 $ and $ 2 \\times 3 $ respectively. Suppose that their product in the order $ A B $ is given by\n$$\nA B=\\left[\\begin{array}{rrr}\n8 & 2 & -2 \\\\\n2 & 5 & 4 \\\\\n-2 & 4 & 5\n\\end{array}\\right]\n$$\n\nShow that the product $ B A $ is given by\n$$\nB A=\\left[\\begin{array}{ll}\n9 & 0 \\\\\n0 & 9\n\\end{array}\\right]\n$$","subjects":["linear algebra","matrix products","rank"]},{"id":"1970-A1","year":1970,"session":"A","number":1,"statement":"A-1. Show that the power series for the function\n$$\ne^{a x} \\cos b x \\quad(a>0, b>0)\n$$\nin powers of $ x $ has either no zero coefficients or infinitely many zero coefficients.","subjects":["analysis","power series","zero coefficients"]},{"id":"1970-A2","year":1970,"session":"A","number":2,"statement":"A-2. Consider the locus given by the real polynomial equation\n$$\nA x^{2}+B x y+C y^{2}+D x^{3}+E x^{2} y+F x y^{2}+G y^{3}=0\n$$\nwhere $ B^2-4 A C<0 $. Prove that there is a positive number $ \\delta $ such that there are no points of the locus in the punctured disk\n$$\n0<x^{2}+y^{2}<\\delta^{2} .\n$$","subjects":["analysis","local extrema","real algebraic curves","quadratic forms"]},{"id":"1970-A3","year":1970,"session":"A","number":3,"statement":"A-3. Find the length of the longest sequence of equal nonzero digits in which an integral square can terminate (in base 10) and find the smallest square which terminates in such a sequence.","subjects":["number theory","decimal expansions","quadratic residues"]},{"id":"1970-A4","year":1970,"session":"A","number":4,"statement":"A-4. Given a sequence $ \\left\\{x_{n}\\right\\}, n=1,2, \\cdots $, such that $ \\operatorname{limit}_{n \\rightarrow \\infty}\\left\\{x_{n}-x_{n-2}\\right\\}=0 $. Prove that\n$$\n\\operatorname{limit}_{n \\rightarrow \\infty} \\frac{x_{n}-x_{n-1}}{n}=0\n$$","subjects":["analysis","sequences","cesaro limits"]},{"id":"1970-A5","year":1970,"session":"A","number":5,"statement":"$$\n\\begin{array}{l}\n\\text { A-5. Determine the radius of the largest circle which can lie on the ellipsoid }\\\\\n\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}+\\frac{z^{2}}{c^{2}}=1 \\quad(a>b>c)\n\\end{array}\n$$","subjects":["geometry","quadratic forms","ellipsoids","plane sections"]},{"id":"1970-A6","year":1970,"session":"A","number":6,"statement":"A-6. Three numbers are chosen independently at random, one from each of the three intervals $ \\left[0, L_{i}\\right](i=1,2,3) $. If the distribution of each random number is uniform with respect to length in the interval it is chosen from, determine the expected value of the smallest of the three numbers chosen.","subjects":["probability","expectation","order statistics"]},{"id":"1970-B1","year":1970,"session":"B","number":1,"statement":"$$\n\\begin{array}{l}\n\\text { B-1. Evaluate }\\\\\n\\operatorname{limit}_{n \\rightarrow \\infty} \\frac{1}{n^{4}} \\prod_{i=1}^{2 n}\\left(n^{2}+i^{2}\\right)^{1 / n}\n\\end{array}\n$$","subjects":["analysis","product limits","special functions"]},{"id":"1970-B2","year":1970,"session":"B","number":2,"statement":"B-2. The time-varying temperature of a certain body is given by a polynomial in the time of degree at most three. Show that the average temperature of the body between 9 A.m. and 3 p.m. can always be found by taking the average of the temperatures at two fixed times, which are independent of which polynomial occurs. Also, show that these two times are 10:16 A.M. and 1:44 P.M. to the nearest minute.","subjects":["analysis","quadrature","cubic polynomials"]},{"id":"1970-B3","year":1970,"session":"B","number":3,"statement":"$$\n\\text { B-3. A closed subset } S \\text { of } R^{2} \\text { lies in } a<x<b \\text {. Show that its projection on the } y \\text {-axis is closed. }\n$$","subjects":["analysis","topology","closed sets","projections"]},{"id":"1970-B4","year":1970,"session":"B","number":4,"statement":"B-4. An automobile starts from rest and ends at rest, traversing a distance of one mile in one minute, along a straight road. If a governor prevents the speed of the car from exceeding ninety miles per hour, show that at some time of the traverse the acceleration or deceleration of the car was at least $ 6.6 \\mathrm{ft} . / \\mathrm{sec} .^{2} $","subjects":["analysis","kinematics","optimization"]},{"id":"1970-B5","year":1970,"session":"B","number":5,"statement":"B-5. Let $ u_{n} $ denote the \"ramp\" function\n$$\nu_{n}(x)=\\left\\{\\begin{aligned}\n-n & \\text { for } \\quad x \\leqq-n \\\\\nx & \\text { for }-n<x \\leqq n, \\\\\nn & \\text { for } \\quad x>n,\n\\end{aligned}\\right.\n$$\nand let $ F $ denote a real function of a real variable. Show that $ F $ is continuous if and only if $ u_{n} \\circ F $ is continuous for all $ n $. (Note: $ \\left(u_{n} \\circ F\\right)(x)=u_{n}[F(x)] $.)","subjects":["analysis","continuity","composition","truncation"]},{"id":"1970-B6","year":1970,"session":"B","number":6,"statement":"B-6. A quadrilateral which can be inscribed in a circle is said to be inscribable or cyclic. A quadrilateral which can be circumscribed to a circle is said to be circumscribable. Show that if a circumscribable quadrilateral of sides $ a, b, c, d $ has area $ A=\\sqrt{a b c d} $, then it is also inscribable.","subjects":["geometry","quadrilaterals","cyclicity","tangency"]},{"id":"1971-A1","year":1971,"session":"A","number":1,"statement":"A-1. Let there be given nine lattice points (points with integral coordinates) in three dimensional Euclidean space. Show that there is a lattice point on the interior of one of the line segments joining two of these points.","subjects":["number theory","lattice points","euclidean geometry"]},{"id":"1971-A2","year":1971,"session":"A","number":2,"statement":"$$\n\\text { A-2. Determine all polynomials } P(x) \\text { such that } P\\left(x^{2}+1\\right)=(P(x))^{2}+1 \\text { and } P(0)=0 \\text {. }\n$$","subjects":["algebra","polynomial functional equations","iteration"]},{"id":"1971-A3","year":1971,"session":"A","number":3,"statement":"A-3. The three vertices of a triangle of sides $ a, b $, and $ c $ are lattice points and lie on a circle of radius $ R $. Show that $ a b c \\geqq 2 R $. (Lattice points are points in the Euclidean plane with integral coordinates.)","subjects":["geometry","number theory","lattice triangles","circumradius"]},{"id":"1971-A4","year":1971,"session":"A","number":4,"statement":"A-4. Show that for $ 0<\\varepsilon<1 $ the expression $ (x+y)^{n}\\left(x^{2}-(2-\\varepsilon) x y+y^{2}\\right) $ is a polynomial with positive coefficients for $ n $ sufficiently large and integral. For $ \\varepsilon=.002 $ find the smallest admissible value of $ n $.","subjects":["algebra","binomial coefficients","positive coefficients"]},{"id":"1971-A5","year":1971,"session":"A","number":5,"statement":"A-5. A game of solitaire is played as follows. After each play, according to the outcome, the player receives either $ a $ or $ b $ points ( $ a $ and $ b $ are positive integers with $ a $ greater than $ b) $, and his score accumulates from play to play. It has been noticed that there are thirtyfive non-attainable scores and that one of these is 58 . Find $ a $ and $ b $.","subjects":["number theory","numerical semigroups","frobenius problem"]},{"id":"1971-A6","year":1971,"session":"A","number":6,"statement":"A-6. Let $ c $ be a real number such that $ n^{c} $ is an integer for every positive integer $ n $. Show that $ c $ is a non-negative integer.","subjects":["analysis","number theory","finite differences"]},{"id":"1971-B1","year":1971,"session":"B","number":1,"statement":"B-1. Let $ S $ be a set and let $ \\circ $ be a binary operation on $ S $ satisfying the two laws\n$$\n\\begin{array}{l}\nx \\circ x=x \\text { for all } x \\text { in } S \\text {, and } \\\\\n(x \\circ y) \\circ z=(y \\circ z) \\circ x \\text { for all } x, y, z \\text { in } S .\n\\end{array}\n$$\n\nShow that $ \\circ $ is associative and commutative.","subjects":["algebra","binary operations","semilattices"]},{"id":"1971-B2","year":1971,"session":"B","number":2,"statement":"B-2. Let $ F(x) $ be a real valued function defined for all real $ x $ except for $ x=0 $ and $ x=1 $ and satisfying the functional equation $ F(x)+F\\{(x-1) / x\\}=1+x $. Find all functions $ F(x) $ satisfying these conditions.","subjects":["algebra","functional equations","mobius transformations"]},{"id":"1971-B3","year":1971,"session":"B","number":3,"statement":"B-3. Two cars travel around a track at equal and constant speeds, each completing a lap every hour. From a common starting point, the first starts at time $ t=0 $ and the second at an arbitrary later time $ t=T>0 $. Prove that there is a total period of exactly one hour during the motion in which the first has completed twice as many laps as the second.","subjects":["combinatorics","floor functions","time intervals"],"errata":"Completed laps means completed whole laps, with zero allowed for both cars. Measure total duration only at times at or after the second car starts. The resulting set can have two components and has total length one hour. The book explicitly restricts the intended claim to times after the second start."},{"id":"1971-B4","year":1971,"session":"B","number":4,"statement":"B-4. A \"spherical ellipse\" with foci $ A, B $ on a given sphere is defined as the set of all points $ P $ on the sphere such that $ \\overset{\\frown}{P A}+\\overset{\\frown}{P B}= $ constant. Here $ \\overset{\\frown}{P A} $ denotes the shortest distance on the sphere between $ P $ and $ A $. Determine the entire class of real spherical ellipses which are circles.","subjects":["geometry","spherical geometry","circles","loci"]},{"id":"1971-B5","year":1971,"session":"B","number":5,"statement":"B-5. Show that the graphs in the $ x-y $ plane of all solutions of the system of differential equations\n$$\nx^{\\prime \\prime}+y^{\\prime}+6 x=0, y^{\\prime \\prime}-x^{\\prime}+6 y=0 \\quad\\left({ }^{\\prime}=d / d t\\right)\n$$\nwhich satisfy $ x^{\\prime}(0)=y^{\\prime}(0)=0 $ are hypocycloids, and find the radius of the fixed circle and the two possible values of the radius of the rolling circle for each such solution. (A hypocycloid is the path described by a fixed point on the circumference of a circle which rolls on the inside of a given fixed circle.)","subjects":["analysis","ordinary differential equations","hypocycloids"],"errata":"The identically zero solution is the stationary degenerate case. For a nonzero solution let $r_0=\\sqrt{x(0)^2+y(0)^2}$. Its fixed-circle radius is $r_0$, and its two rolling-circle radii are $2r_0/5$ and $3r_0/5$. This states the zero-amplitude case separately rather than claiming it is a positive-radius hypocycloid."},{"id":"1971-B6","year":1971,"session":"B","number":6,"statement":"B-6. Let $ \\delta(x) $ be the greatest odd divisor of the positive integer $ x $. Show that $ \\left|\\sum_{n=1}^{x}\\delta(n)/n-2x/3\\right|<1 $, for all positive integers $ x $.","subjects":["number theory","binary expansion","summatory functions"]},{"id":"1972-A1","year":1972,"session":"A","number":1,"statement":"A-1. Show that there are no four consecutive binomial coefficients $ \\binom nr,\\binom n{r+1},\\binom n{r+2},\\binom n{r+3} $ ( $ n, r $ integers $ >0 $ and $ r+3 \\leqq n $ ) which are in arithmetic progression.","subjects":["combinatorics","binomial coefficients","arithmetic progressions"]},{"id":"1972-A2","year":1972,"session":"A","number":2,"statement":"A-2. Let $ S $ be a set and let * be a binary operation on $ S $ satisfying the laws\n$$\n\\begin{array}{l}\nx *(x * y)=y \\text { for all } x, y \\text { in } S, \\\\\n(y * x) * x=y \\text { for all } x, y \\text { in } S .\n\\end{array}\n$$\n\nShow that * is commutative but not necessarily associative.","subjects":["algebra","binary operations","quasigroups"]},{"id":"1972-A3","year":1972,"session":"A","number":3,"statement":"A-3. If for a sequence $ x_{1}, x_{2}, x_{3}, \\cdots, \\lim _{n \\rightarrow \\infty}\\left(x_{1}+x_{2}+\\cdots+x_{n}\\right) / n $ exists, call this limit the $ C $-limit of the sequence. A function $ f(x) $ from $ [0,1] $ to the reals is called a supercontinuous function on the interval $ [0,1] $ if the $ C $-limit exists for the sequence $ f\\left(x_{1}\\right), f\\left(x_{2}\\right), f\\left(x_{3}\\right), \\cdots $ whenever the $ C $-limit exists for the sequence $ x_{1}, x_{2}, x_{3} \\cdots $. Find all supercontinuous functions on [0,1].","subjects":["analysis","functional equations","cesaro limits","affine functions"]},{"id":"1972-A4","year":1972,"session":"A","number":4,"statement":"$$\n\\text { A-4. Of all ellipses inscribed in a square, show that the circle has the maximum perimeter. }\n$$","subjects":["geometry","ellipses","perimeter","convexity"]},{"id":"1972-A5","year":1972,"session":"A","number":5,"statement":"$$\n\\text { A-5. Show that if } n \\text { is an integer greater than } 1 \\text {, then } n \\text { does not divide } 2^{n}-1 \\text {. }\n$$","subjects":["number theory","multiplicative order","divisibility"]},{"id":"1972-A6","year":1972,"session":"A","number":6,"statement":"A-6. Let $ f(x) $ be an integrable function in $ 0 \\leqq x \\leqq 1 $ and suppose $ \\int_{0}^{1} f(x) d x=0, \\int_{0}^{1} x f(x) d x=0, \\cdots $, $ \\int_{0}^{1} x^{n-1} f(x) d x=0 $ and $ \\int_{0}^{1} x^{n} f(x) d x=1 $. Show that $ |f(x)| \\geqq 2^{n}(n+1) $ in a set of positive measure.","subjects":["analysis","moments","integral inequalities","positive measure"]},{"id":"1972-B1","year":1972,"session":"B","number":1,"statement":"B-1. Show that the power series representation for the series $ \\sum_{n=0}^{\\infty}\\left(x^{n}(x-1)^{2 n}\\right) / n $ ! cannot have three consecutive zero coefficients.","subjects":["analysis","power series","exponential functions"]},{"id":"1972-B2","year":1972,"session":"B","number":2,"statement":"B-2. A particle moving on a straight line starts from rest and attains a velocity $ v_{0} $ after traversing a distance $ s_{0} $. If the motion is such that the acceleration was never increasing, find the maximum time for the traverse.","subjects":["analysis","mechanics","monotone acceleration","optimization"]},{"id":"1972-B3","year":1972,"session":"B","number":3,"statement":"B-3. Let $ A $ and $ B $ be two elements in a group such that $ A B A=B A^{2} B, A^{3}=1 $ and $ B^{2 n-1}=1 $ for some positive integer $ n $. Prove $ B=1 $.","subjects":["algebra","group identities","odd order"]},{"id":"1972-B4","year":1972,"session":"B","number":4,"statement":"B-4. Let $ n $ be an integer greater than 1. Show that there exists a polynomial $ P(x, y, z) $ with integral coefficients such that $ x \\equiv P\\left(x^{n}, x^{n+1}, x+x^{n+2}\\right) $.","subjects":["algebra","integer polynomials","polynomial identities"]},{"id":"1972-B5","year":1972,"session":"B","number":5,"statement":"B-5. If the opposite angles of a skew (non-planar) quadrilateral are equal in pairs, prove that the opposite sides are equal in pairs.","subjects":["geometry","space geometry","quadrilaterals","ptolemy inequality"]},{"id":"1972-B6","year":1972,"session":"B","number":6,"statement":"B-6. Let $ n_{1}<n_{2}<n_{3}<\\cdots<n_{k} $ be a set of positive integers. Prove that the polynomial $ 1+z^{n_{1}} $ $ +z^{n_{2}}+\\cdots+z^{n_{k}} $ has no roots inside the circle $ |z|<(\\sqrt{5}-1) / 2 $.","subjects":["analysis","complex polynomials","root bounds"]},{"id":"1973-A1","year":1973,"session":"A","number":1,"statement":"A-1. (a) Let $ A B C $ be any triangle. Let $ X, Y, Z $ be points on the sides $ B C, C A, A B $ respectively. Suppose the distances $ \\overline{B X} \\leqq \\overline{X C}, \\overline{C Y} \\leqq \\overline{Y A}, \\overline{A Z} \\leqq \\overline{Z B} $. Show that the area of the triangle $ X Y Z $ is $ \\geqq(1 / 4) $ (area of triangle $ A B C $ ).\n(b) Let $ A B C $ be any triangle, and let $ X, Y, Z $ be points on the sides $ B C, C A, A B $ respectively (but without any assumption about the ratios of the distances $ \\overline{B X} / \\overline{X C} $, etc.). Using (a) or by any other method, show: One of the three corner triangles $ A Z Y $, $ B X Z, C Y X $ has an area $ \\leqq $ area of triangle $ X Y Z $.","subjects":["geometry","triangle area","points on sides","affine geometry"]},{"id":"1973-A2","year":1973,"session":"A","number":2,"statement":"A-2. Consider an infinite series whose $ n $th term is $ \\pm(1 / n) $, the $ \\pm $ signs being determined according to a pattern that repeats periodically in blocks of eight. [There are $ 2^{8} $ possible patterns of which two examples are:\n$$\n\\begin{array}{l}\n++----++ \\\\\n+---+---\n\\end{array}\n$$\n\nThe first example would generate the series\n$$\n\\begin{array}{l}\n1+(1 / 2)-(1 / 3)-(1 / 4)-(1 / 5)-(1 / 6)+(1 / 7)+(1 / 8) \\\\\n+(1 / 9)+(1 / 10)-(1 / 11)-(1 / 12)-\\cdots .]\n\\end{array}\n$$\n(a) Show that a sufficient condition for the series to be conditionally convergent is that there be four \"+\" signs and four \"-\" signs in the block of eight.\n(b) Is this sufficient condition also necessary?\n[Here \"convergent\" means \"convergent to a finite limit.\"]","subjects":["analysis","series","periodic sequences","conditional convergence"]},{"id":"1973-A3","year":1973,"session":"A","number":3,"statement":"A-3. Let $ n $ be a fixed positive integer and let $ b(n) $ be the minimum value of\n$$\nk+\\frac{n}{k}\n$$\nas $ k $ is allowed to range through all positive integers. Prove that $ b(n) $ and $ \\sqrt{4 n+1} $ have the same integer part. [The \"integer part\" of a real number is the greatest integer which does not exceed it, e.g. for $ \\pi $ it is 3 , for $ \\sqrt{21} $ it is 4 , for 5 it is 5 , etc.]","subjects":["algebra","integer optimization","floor functions"]},{"id":"1973-A4","year":1973,"session":"A","number":4,"statement":"A-4. How many zeros does the function $ f(x)=2^{x}-1-x^{2} $ have on the real line? [By a \"zero\" of a function $ f $, we mean a value $ x_{0} $ in the domain of $ f $ (here the set of all real numbers) such that $ \\left.f\\left(x_{0}\\right)=0.\\right] $","subjects":["analysis","real zeros","exponential functions"]},{"id":"1973-A5","year":1973,"session":"A","number":5,"statement":"A-5. A particle moves in 3-space according to the equations:\n$$\n\\frac{d x}{d t}=y z, \\quad \\frac{d y}{d t}=z x, \\quad \\frac{d z}{d t}=x y .\n$$\n[Here $ x(t), y(t), z(t) $ are real-valued functions of the real variable $ t $.] Show that:\n(a) If two of $ x(0), y(0), z(0) $ equal zero, then the particle never moves.\n(b) If $ x(0)=y(0)=1, z(0)=0 $, then the solution is:\n$$\nx=\\sec t, y=\\sec t, z=\\tan t\n$$\nwhereas if $ x(0)=y(0)=1, z(0)=-1 $, then\n$$\nx=1 /(t+1), y=1 /(t+1), z=-1 /(t+1)\n$$\n(c) If at least two of the values $ x(0), y(0), z(0) $ are different from zero, then either the particle moves to infinity at some finite time in the future, or it came from infinity at some finite time in the past. [A point $ (x, y, z) $ in 3-space \"moves to infinity\" if its distance from the origin approaches infinity.]","subjects":["analysis","ordinary differential equations","finite time blowup"]},{"id":"1973-A6","year":1973,"session":"A","number":6,"statement":"A-6. Prove that it is impossible for seven distinct straight lines to be situated in the euclidean plane so as to have at least six points where exactly three of these lines intersect and at least four points where exactly two of these lines intersect.","subjects":["combinatorics","incidence geometry"]},{"id":"1973-B1","year":1973,"session":"B","number":1,"statement":"B-1. Let $ a_{1}, a_{2}, \\ldots, a_{2 n+1} $ be a set of integers such that, if any one of them is removed, the remaining ones can be divided into two sets of $ n $ integers with equal sums. Prove $ a_{1}=a_{2}=\\cdots=a_{2 n+1} $.","subjects":["number theory","parity","linear algebra","spectral methods"]},{"id":"1973-B2","year":1973,"session":"B","number":2,"statement":"B-2. Let $ z=x+i y $ be a complex number with $ x $ and $ y $ rational and with $ |z|=1 $. Show that the number $ \\left|z^{2 n}-1\\right| $ is rational for every integer $ n $.","subjects":["algebra","complex numbers","rational points","unit circle"]},{"id":"1973-B3","year":1973,"session":"B","number":3,"statement":"B-3. Consider an integer $ p>1 $ with the property that the polynomial $ x^{2}-x+p $ takes prime values for all integers $ x $ in the range $ 0 \\leqq x<p $. (Examples: $ p=5 $ and $ p=41 $ have this property.) Show that there is exactly one triple of integers $ a, b, c $ satisfying the conditions:\n$$\n\\begin{array}{r}\nb^{2}-4 a c=1-4 p \\\\\n0<a \\leqq c \\\\\n-a \\leqq b<a\n\\end{array}\n$$","subjects":["number theory","quadratic forms","prime values"]},{"id":"1973-B4","year":1973,"session":"B","number":4,"statement":"B-4. (a) On [0,1], let $ f $ have a continuous derivative satisfying $ 0<f^{\\prime}(x) \\leqq 1 $. Also suppose that $ f(0)=0 $. Prove that\n$$\n\\left[\\int_{0}^{1} f(x) d x\\right]^{2} \\geqq \\int_{0}^{1}[f(x)]^{3} d x\n$$\n[Hint: Replace the inequality by one involving the inverse function to $ f $.]\n(b) Show an example in which equality occurs.","subjects":["analysis","integral inequalities","inverse functions","lipschitz functions"]},{"id":"1973-B5","year":1973,"session":"B","number":5,"statement":"B-5. (a) Let $ z $ be a solution of the quadratic equation\n$$\na z^{2}+b z+c=0\n$$\nand let $ n $ be a positive integer. Show that $ z $ can be expressed as a rational function of $ z^{n}, a, b, c $.\n(b) Using (a) or by any other means, express $ x $ as a rational function of $ x^{3} $ and $ x+(1 / x) $. (Display your answer explicitly in a clearly visible form.)\n[By a rational function of several variables, we mean a quotient of polynomials in those variables, the polynomials having rational numbers as coefficients, and the denominator being not identically zero. Thus to obtain $ x $ as a rational function of $ u=x^{2} $ and $ v= $ $ x+(1 / x) $, we could write $ x=(u+1) / v $.","subjects":["algebra","field extensions","polynomials","rational functions","root recovery","statement erratum"],"errata":"Editorial qualification. Part (a) cannot mean pointwise recovery for every numerical specialization. For $z=\\sqrt2$, $a=1$, $b=0$, $c=-2$ and $n=2$, all the proposed inputs are rational, so a defined rational-coefficient quotient cannot recover $z$. The revised drafts distinguish a generic rational-function identity from pointwise specializations and state the exceptional cases. The repaired arguments passed two completed independent model reviews for the explicitly corrected scope. The unqualified pointwise claim remains false."},{"id":"1973-B6","year":1973,"session":"B","number":6,"statement":"B-6. On the domain $ 0 \\leqq \\theta \\leqq 2 \\pi $ :\n(a) Prove that $ \\sin ^{2} \\theta \\cdot \\sin (2 \\theta) $ takes its maximum at $ \\pi / 3 $ and $ 4 \\pi / 3 $ (and hence its minimum at $ 2 \\pi / 3 $ and $ 5 \\pi / 3 $ ).\n(b) Show that\n$$\n\\left|\\sin ^{2} \\theta\\left\\{\\sin ^{3}(2 \\theta) \\cdot \\sin^3(4\\theta) \\cdots \\sin ^{3}\\left(2^{n-1} \\theta\\right)\\right\\} \\sin \\left(2^{n} \\theta\\right)\\right|\n$$\ntakes its maximum at $ \\theta=\\pi / 3 $. (The maximum may also be attained at other points.)\n(c) Derive the inequality:\n$$\n\\sin ^{2} \\theta \\cdot \\sin ^{2}(2 \\theta) \\cdot \\sin ^{2}(4 \\theta) \\cdots \\sin ^{2}\\left(2^{n} \\theta\\right) \\leqq(3 / 4)^{n}\n$$","subjects":["analysis","trigonometry","inequalities"]},{"id":"1974-A1","year":1974,"session":"A","number":1,"statement":"A-1. Call a set of positive integers \"conspiratorial\" if no three of them are pairwise relatively prime. (A set of integers is \"pairwise relatively prime\" if no pair of them has a common divisor greater than 1 .) What is the largest number of elements in any \"conspiratorial\" subset of the integers 1 through 16?","subjects":["combinatorics","number theory","coprimality","extremal sets","finite combinatorics","parity","extremal combinatorics"]},{"id":"1974-A2","year":1974,"session":"A","number":2,"statement":"A-2. A circle stands in a plane perpendicular to the ground and a point $ A $ lies in this plane exterior to the circle and higher than its bottom. A particle starting from rest at $ A $ slides without friction down an inclined straight line until it reaches the circle. Which straight line allows descent in the shortest time? [Assume that the force of gravity is constant over the region involved, there are no relativistic effects, etc.]\n\nThe starting point A and the circle are fixed; the stopping point $ B $ is allowed to vary over the circle.\n\nNote. The answer may be given in any form which specifies the line of descent in an unambiguous manner; it is not required to find the coordinates of the point $ B $.","subjects":["mechanics","geometry","optimization"]},{"id":"1974-A3","year":1974,"session":"A","number":3,"statement":"A-3. A well-known theorem asserts that a prime $ p>2 $ can be written as the sum of two perfect squares ( $ p=m^{2}+n^{2} $, with $ m $ and $ n $ integers) if and only if $ p \\equiv 1(\\bmod 4) $. Assuming this result, find which primes $ p>2 $ can be written in each of the following forms, using (not necessarily positive) integers $ \\boldsymbol{x} $ and $ y $ :\n(a) $ x^{2}+16 y^{2} $;\n(b) $ 4 x^{2}+4 x y+5 y^{2} $.","subjects":["number theory","quadratic forms","congruences","geometry of numbers","lattices"]},{"id":"1974-A4","year":1974,"session":"A","number":4,"statement":"A-4. An unbiased coin is tossed $ n $ times. What is the expected value of $ |H-T| $, where $ H $ is the number of heads and $ T $ is the number of tails? In other words, evaluate in closed form:\n$$\n\\frac{1}{2^{n-1}} \\sum_{k<n / 2}(n-2 k)\\binom{n}{k} .\n$$\n(In this problem, \"closed form\" means a form not involving a series. The given series can be reduced to a single term involving only binomial coefficients, rational functions of $ n $ and $ 2^{n} $, and the greatest integer function $ [x] $.)","subjects":["probability","combinatorics","random walks"]},{"id":"1974-A5","year":1974,"session":"A","number":5,"statement":"A-5. Consider the two mutually tangent parabolas $ y=x^{2} $ and $ y=-x^{2} $. [These have foci at $ (0,1 / 4) $ and $ (0,-1 / 4) $, and directrices $ y=-1 / 4 $ and $ y=1 / 4 $, respectively.] The upper parabola rolls without slipping around the fixed lower parabola. Find the locus of the focus of the moving parabola.","subjects":["geometry","kinematics","parabolas"]},{"id":"1974-A6","year":1974,"session":"A","number":6,"statement":"A-6. It is well known that the value of the polynomial $ (x+1)(x+2) \\cdots(x+n) $ is exactly divisible by $ n $ for every integer $ x $. Given $ n $, let $ \\boldsymbol{k}=\\boldsymbol{k}(\\boldsymbol{n}) $ be the minimal degree of any monic integral polynomial\n$$\nf(x)=x^{k}+a_{1} x^{k-1}+\\cdots+a_{k}\n$$\n(with integer coefficients and leading coefficient 1) such that the value of $ f(x) $ is exactly divisible by $ n $ for every integer $ \\boldsymbol{x} $.\n\nFind the relationship between $ n $ and $ k=k(n) $. In particular, find the value of $ k $ corresponding to $ n=1000000 $.","subjects":["number theory","polynomials","divisibility","finite fields","prime adic valuations"]},{"id":"1974-B1","year":1974,"session":"B","number":1,"statement":"B-1. Which configurations of five (not necessarily distinct) points $ p_{1}, \\cdots, p_5 $ on the circle $ x^{2}+y^{2}=1 $ maximize the sum of the ten distances\n$$\n\\sum_{i<j} d\\left(p_i,p_j\\right) ?\n$$\n[Here $ d(p, q) $ denotes the straight line distance between $ p $ and $ q $.]","subjects":["geometry","optimization","circle geometry"]},{"id":"1974-B2","year":1974,"session":"B","number":2,"statement":"B-2. Let $ y(x) $ be a continuously differentiable real-valued function of a real variable $ x $. Show that if $ \\left(y^{\\prime}\\right)^{2}+y^{3} \\rightarrow 0 $ as $ x \\rightarrow+\\infty $, then $ y(x) $ and $ y^{\\prime}(x) \\rightarrow 0 $ as $ x \\rightarrow+\\infty $.","subjects":["analysis","differential inequalities","asymptotics","variational methods"]},{"id":"1974-B3","year":1974,"session":"B","number":3,"statement":"B-3. Prove that if $ \\alpha $ is a real number such that\n$$\n\\cos \\pi \\alpha=1 / 3,\n$$\nthen $ \\alpha $ is irrational. (The angle $ \\pi \\alpha $ is in radians.)","subjects":["number theory","trigonometry","irrationality"]},{"id":"1974-B4","year":1974,"session":"B","number":4,"statement":"B-4. In the standard definition, a real -valued function of two real variables $ g: R^{2} \\rightarrow R^{1} $ is continuous if, for every point $ \\left(x_{0}, y_{0}\\right) \\in R^{2} $ and every $ \\varepsilon>0 $, there is a corresponding $ \\delta>0 $ such that $ \\left[\\left(x-x_{0}\\right)^{2}+\\left(y-y_{0}\\right)^{2}\\right]^{1 / 2}<\\delta $ implies $ \\left|g(x, y)-g\\left(x_{0}, y_{0}\\right)\\right|<\\varepsilon $.\n\nBy contrast, $ f: R^{\\mathbf{2}} \\rightarrow R $ is said to be continuous in each variable separately if, for each fixed value $ y_{0} $ of $ y $, the function $ f\\left(x, y_{0}\\right) $ is continuous in the usual sense as a function of $ x $, and similarly $ f\\left(x_{0}, y\\right) $ is continuous as a function of $ y $ for each fixed $ x_{0} $.\n\nLet $ f: R^{\\mathbf{2}} \\rightarrow R $ be continuous in each variable separately. Show that there exists a sequence of continuous functions $ g_{n}: R^{2} \\rightarrow R $ such that\n$$\nf(x, y)=\\lim _{n \\rightarrow \\infty} g_{n}(x, y) \\text { for all }(x, y) \\in R^{2} .\n$$","subjects":["analysis","topology","separate continuity","approximation"]},{"id":"1974-B5","year":1974,"session":"B","number":5,"statement":"B-5. Show that $ 1+(n / 1!)+\\left(n^{2} / 2!\\right)+\\cdots+\\left(n^{n} / n!\\right)>e^{n} / 2 $ for every integer $ n \\geqq 0 $.\nRemarks You may assume as known Taylor's remainder formula:\n$$\ne^{x}-\\sum_{k=0}^{n} \\frac{x^{k}}{k!}=\\frac{1}{n!} \\int_{0}^{x}(x-t)^{n} e^t d t\n$$\nas well as the fact that\n$$\nn!=\\int_{0}^{\\infty} t^{n} e^{-t} d t\n$$","subjects":["analysis","inequalities","exponential series"]},{"id":"1974-B6","year":1974,"session":"B","number":6,"statement":"B-6. For a set with $ n $ elements, how many subsets are there whose cardinality (the number of elements in the subset) is respectively $ \\equiv 0(\\bmod 3), \\equiv 1(\\bmod 3), \\equiv 2(\\bmod 3) $ ? In other words, calculate\n$$\ns_{i, n}=\\sum_{\\substack{0\\leq k\\leq n\\\\k\\equiv i\\pmod 3}}\\binom{n}{k} \\quad \\text { for } i=0,1,2 .\n$$\n\nYour result should be strong enough to permit direct evaluation of the numbers $ s_{i,n} $ and to show clearly the relationship of $ s_{0, n} $ and $ s_{1, n} $ and $ s_{2, n} $ to each other for all positive integers $ n $. In particular, show the relationships among these three sums for $ n=1000 $. [An illustration of the definition of $ s_{i,n} $ is $ s_{0.6}=\\binom{6}{0}+\\binom{6}{3}+\\binom{6}{6}=22.\\,] $","subjects":["combinatorics","binomial coefficients"]},{"id":"1975-A1","year":1975,"session":"A","number":1,"statement":"A-1. Supposing that an integer $ n $ is the sum of two triangular numbers,\n$$\nn=\\frac{a^{2}+a}{2}+\\frac{b^{2}+b}{2}\n$$\nwrite $ 4 n+1 $ as the sum of two squares, $ 4 n+1=x^{2}+y^{2} $, and show how $ x $ and $ y $ can be expressed in terms of $ a $ and $ b $.\n\nShow that, conversely, if $ 4 n+1=x^{2}+y^{2} $, then $ n $ is the sum of two triangular numbers.\n[Of course, $ a, b, x, y $ are understood to be integers.]","subjects":["number theory","triangular numbers","sums of squares"]},{"id":"1975-A2","year":1975,"session":"A","number":2,"statement":"A-2. For which ordered pairs of real numbers $ b, c $ do both roots of the quadratic equation\n$$\nz^{2}+b z+c=0\n$$\nlie inside the unit disk $ \\{|z|<1\\} $ in the complex plane?\nDraw a reasonably accurate picture (i.e., 'graph') of the region in the real bc-plane for which the above condition holds. Identify precisely the boundary curves of this region.","subjects":["algebra","complex analysis","quadratic polynomials"]},{"id":"1975-A3","year":1975,"session":"A","number":3,"statement":"A-3. Let $ a, b, c $ be constants with $ 0<a<b<c $. At what points of the set\n$$\n\\left\\{x^{b}+y^{b}+z^{b}=1, x \\geqq 0, y \\geqq 0, z \\geqq 0\\right\\}\n$$\nin three-dimensional space $ R^{3} $ does the function $ f(x, y, z)=x^{a}+y^{b}+z^{c} $ assume its maximum and minimum values?","subjects":["analysis","optimization","power functions"]},{"id":"1975-A4","year":1975,"session":"A","number":4,"statement":"A-4. Let $ n=2 m $, where $ m $ is an odd integer greater than 1 . Let $ \\theta=e^{2\\pi i/n} $. Express $ (1-\\theta)^{-1} $ explicitly as a polynomial in $ \\boldsymbol{\\theta} $,\n$$\na_{k} \\theta^{k}+a_{k-1} \\theta^{k-1}+\\cdots+a_{1} \\theta+a_{0}\n$$\nwith integer coefficients $ a_{1} $.\n[Note that $ \\boldsymbol{\\theta} $ is a primitive $ \\boldsymbol{n} $-th root of unity, and thus it satisfies all of the identities which hold for such roots.]","subjects":["algebra","roots of unity"]},{"id":"1975-A5","year":1975,"session":"A","number":5,"statement":"A-5. On some interval $ I $ of the real line, let $ y_{1}(x) $ and $ y_{2}(x) $ be linearly independent solutions of the differential equation\n$$\ny^{\\prime \\prime}=f(x) y\n$$\nwhere $ f(x) $ is a continuous real-valued function. Suppose that $ y_{1}(x)>0 $ and $ y_{2}(x)>0 $ on $ I $. Show that there exists a positive constant $ c $ such that, on $ I $, the function\n$$\nz(x)=c \\sqrt{y_{1}(x) y_{2}(x)}\n$$\nsatisfies the equation\n$$\nz^{\\prime \\prime}+\\frac{1}{z^{3}}=f(x) z\n$$\n\nState clearly the manner in which $ c $ depends on $ y_{1}(x) $ and $ y_{2}(x) $.","subjects":["analysis","ordinary differential equations","wronskians"]},{"id":"1975-A6","year":1975,"session":"A","number":6,"statement":"A-6. Let $ P_{1}, P_{2}, P_{3} $ be the vertices of an acute-angled triangle situated in three-dimensional space. Show that it is always possible to locate two additional points $ P_{4} $ and $ P_{5} $ in such a way that no three of the points are collinear and so that the line through any two of the five points is perpendicular to the plane determined by the other three.\n\nIn writing your answer, state clearly the locations at which you place the points $ P_{4} $ and $ P_{5} $.","subjects":["geometry","solid geometry","orthocentric systems","tetrahedra"]},{"id":"1975-B1","year":1975,"session":"B","number":1,"statement":"B-1. In the additive group of ordered pairs of integers ( $ m, n $ ) [with addition defined componentwise: $ \\left.(m, n)+\\left(m^{\\prime}, n^{\\prime}\\right)=\\left(m+m^{\\prime}, n+n^{\\prime}\\right)\\right] $ consider the subgroup $ H $ generated by the three elements\n$$\n(3,8),\\quad(4,-1),\\quad(5,4)\n$$\n\nThen $ \\boldsymbol{H} $ has another set of generators of the form\n$$\n(1, b), \\quad(0, a)\n$$\nfor some integers $ a, b $ with $ a>0 $. Find $ a $.\n[Elements $ g_{1}, \\ldots, g_{k} $ are said to generate a subgroup $ H $ if (i) each $ g_{1} \\in H $, and (ii) every $ h \\in H $ can be written as a sum $ h=n_{1} g_{1}+\\cdots+n_{k} g_{k} $ where the $ n_{1} $ are integers (and where, for example, $ 3 g_{1}-2 g_{2} $ means $ \\left.\\left.g_{1}+g_{1}+g_{1}-g_{2}-g_{2}\\right).\\right] $","subjects":["algebra","number theory","integer lattices"]},{"id":"1975-B2","year":1975,"session":"B","number":2,"statement":"B-2. In three-dimensional Euclidean space, define a slab to be the open set of points lying between two parallel planes. The distance between the planes is called the thickness of the slab. Given an infinite sequence $ S_{1}, S_{2}, \\ldots $ of slabs of thicknesses $ d_{1}, d_{2}, \\ldots $, respectively, such that $ \\sum_{i=1}^{\\infty} d_i $ converges, prove that there is some point in the space which is not contained in any of the slabs.","subjects":["geometry","measure theory","covering"]},{"id":"1975-B3","year":1975,"session":"B","number":3,"statement":"B-3. Let $ s_{k}\\left(a_{1}, \\ldots, a_{n}\\right) $ denote the $ k $-th elementary symmetric function of $ a_{1}, \\ldots, a_{n} $. With $ k $ held fixed, find the supremum (or least upper bound) $ M_{k} $ of\n$$\ns_{k}\\left(a_{1}, \\ldots, a_{n}\\right) /\\left[s_{1}\\left(a_{1}, \\ldots, a_{n}\\right)\\right]^{k}\n$$\nfor arbitrary $ n \\geqq k $ and arbitrary $ n $-tuples $ a_{1}, \\ldots, a_{n} $ of positive real numbers.\n[The symmetric function $ s_{k}\\left(a_{1}, \\ldots, a_{n}\\right) $ is the sum of all $ k $-fold products of the variables $ a_{1}, \\ldots, a_{n} $. Thus, for example:\n$$\n\\begin{array}{c}\ns_{1}\\left(a_{1}, \\ldots, a_{n}\\right)=a_{1}+a_{2}+\\ldots+a_{n} \\\\\ns_{3}\\left(a_{1}, a_{2}, a_{3}, a_{4}\\right)=a_{1} a_{2} a_{3}+a_{1} a_{2} a_{4}+a_{1} a_{3} a_{4}+a_{2} a_{3} a_{4}\n\\end{array}\n$$\n\nIt should be remarked that the supremum $ \\boldsymbol{M}_{\\boldsymbol{k}} $ is never attained; it is approached arbitrarily closely when, for fixed $ k $, the number $ n $ of variables increases without bound, and the values $ a_{i}>0 $ are suitably chosen.]","subjects":["algebra","inequalities","symmetric polynomials"]},{"id":"1975-B4","year":1975,"session":"B","number":4,"statement":"B-4. Does there exist a subset $ B $ of the unit circle $ x^{2}+y^{2}=1 $ such that (i) $ B $ is topologically closed, and (ii) $ B $ contains exactly one point from each pair of diametrically opposite points on the circle?\n[A set B is topologically closed if it contains the limit of every convergent sequence of points in B.]","subjects":["topology","connectedness","circle"]},{"id":"1975-B5","year":1975,"session":"B","number":5,"statement":"$$\n\\begin{array}{l}\n\\text { B-5. Let } f_{0}(x)=e^{x} \\text { and } f_{n+1}(x)=x f_{n}^{\\prime}(x) \\text { for } n=0,1,2, \\ldots \\text {. Show that }\\\\\n\\sum_{n=0}^{\\infty} \\frac{f_{n}(1)}{n!}=e^{e}\n\\end{array}\n$$","subjects":["analysis","combinatorics"]},{"id":"1975-B6","year":1975,"session":"B","number":6,"statement":"B-6. Show that if $ s_{n}=1+\\frac{1}{2}+\\frac{1}{3}+\\cdots+1 / n $, then\n(a) $ n(n+1)^{1 / n}<n+s_{n} $ for $ n>1 $, and\n(b) $ (n-1) n^{-1 /(n-1)}<n-s_{n} $ for $ n>2 $.","subjects":["analysis","inequalities","harmonic numbers","monotonicity"]},{"id":"1976-A1","year":1976,"session":"A","number":1,"statement":"A-1. $ P $ is an interior point of the angle whose sides are the rays $ O A $ and $ O B $. Locate $ X $ on $ O A $ and $ Y $ on $ O B $ so that the line segment $ \\overline{X Y} $ contains $ P $ and so that the product of distances $ (P X)(P Y) $ is a minimum.","subjects":["geometry","optimization","angle geometry"]},{"id":"1976-A2","year":1976,"session":"A","number":2,"statement":"A-2. Let $ P(x, y)=x^{2} y+x y^{2} $ and $ Q(x, y)=x^{2}+x y+y^{2} $. For $ n=1,2,3, \\ldots $, let $ F_{n}(x, y)=(x+y)^{n}-x^{n}-y^{n} $ and $ G_{n}(x, y)=(x+y)^{n}+x^{n}+y^{n} $. One observes that $ G_{2}=2 Q, F_{3}=3 P, G_{4}=2 Q^{2}, F_{5}=5 P Q, G_{6}=2 Q^{3}+3 P^{2} $. Prove that, in fact, for each $ n $ either $ F_{n} $ or $ G_{n} $ is expressible as a polynomial in $ P $ and $ Q $ with integer coefficients.","subjects":["algebra","symmetric polynomials","recurrences"]},{"id":"1976-A3","year":1976,"session":"A","number":3,"statement":"A-3. Find all integral solutions of the equation\n$$\n\\left|p^r-q^{s}\\right|=1,\n$$\nwhere $ p $ and $ q $ are prime numbers and $ r $ and $ s $ are positive integers larger than unity. Prove that there are no other solutions.","subjects":["number theory","diophantine equations","prime powers"]},{"id":"1976-A4","year":1976,"session":"A","number":4,"statement":"A-4. Let $ r $ be a root of $ P(x)=x^{3}+a x^{2}+b x-1=0 $ and $ r+1 $ be a root of $ y^{3}+c y^{2}+d y+1=0 $, where $ a, b, c $, and $ d $ are integers. Also let $ P(x) $ be irreducible over the rational numbers. Express another root $ s $ of $ P(x)=0 $ as a function of $ r $ which does not explicitly involve $ a, b, c $, or $ d $.","subjects":["algebra","irreducible polynomials","mobius transformations"]},{"id":"1976-A5","year":1976,"session":"A","number":5,"statement":"A-5. In the $ (x, y) $-plane, if $ R $ is the set of points inside and on a convex polygon, let $ D(x, y) $ be the distance from $ (x, y) $ to the nearest point of $ R $. (a) Show that there exist constants $ a, b $, and $ c $, independent of $ R $, such that\n$$\n\\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} e^{-D(x, y)} d x d y=a+b L+c A\n$$\nwhere $ L $ is the perimeter of $ R $ and $ A $ is the area of $ R $. (b) Find the values of $ a, b $, and $ c $.","subjects":["geometry","integration","convex polygons","convex geometry","integral geometry"]},{"id":"1976-A6","year":1976,"session":"A","number":6,"statement":"A-6. Suppose $ f(x) $ is a twice continuously differentiable real valued function defined for all real numbers $ x $ and satisfying $ |f(x)| \\leqq 1 $ for all $ x $ and $ (f(0))^{2}+\\left(f^{\\prime}(0)\\right)^{2}=4 $. Prove that there exists a real number $ x_{0} $ such that $ f\\left(x_{0}\\right)+f^{\\prime \\prime}\\left(x_{0}\\right)=0 $.","subjects":["analysis","ordinary differential equations","energy methods"]},{"id":"1976-B1","year":1976,"session":"B","number":1,"statement":"B-1. Evaluate\n$$\n\\lim _{n \\rightarrow \\infty} \\frac{1}{n} \\sum_{k=1}^{n}\\left(\\left[\\frac{2 n}{k}\\right]-2\\left[\\frac{n}{k}\\right]\\right)\n$$\nand express your answer in the form $ \\log a-b $, with $ a $ and $ b $ positive integers.\nHere $ [x] $ is defined to be the integer such that $ [x] \\leqq x<[x]+1 $ and $ \\log x $ is the logarithm of $ x $ to base $ e $.","subjects":["analysis","number theory","floor functions","limits"]},{"id":"1976-B2","year":1976,"session":"B","number":2,"statement":"B-2. Suppose that $ G $ is a group generated by elements $ A $ and $ B $, that is, every element of $ G $ can be written as a finite \"word\" $ A^{n_1}B^{n_2}A^{n_3} \\cdots B^{n_{k}} $, where $ n_{1}, \\ldots, n_{k} $ are any integers, and $ A^{0}=B^0=1 $ as usual. Also, suppose that $ A^{4}=B^{7}=A B A^{-1} B=1, A^{2} \\neq 1 $, and $ B \\neq 1 $.\n(a) How many elements of $ G $ are of the form $ C^{2} $ with $ C $ in $ G $ ?\n(b) Write each such square as a word in $ A $ and $ B $.","subjects":["algebra","group theory","semidirect products"]},{"id":"1976-B3","year":1976,"session":"B","number":3,"statement":"B-3. Suppose that we have $ n $ events $ A_{1}, \\ldots, A_{n} $, each of which has probability at least $ 1-a $ of occurring, where $ a<1 / 4 $. Further suppose that $ A_i $ and $ A_j $ are mutually independent if $ |i-j|>1 $, although $ A_i $ and $ A_{i+1} $ may be dependent. Assume as known that the recurrence $ u_{k+1}=u_{k}-a u_{k-1}, u_{0}=1, u_{1}=1-a $, defines positive real numbers $ u_{k} $ for $ k=0,1, \\ldots $. Show that the probability of all of $ A_{1}, \\ldots, A_{n} $ occurring is at least $ u_{n} $.","subjects":["probability","finite fields","independence","counterexamples"],"errata":"Pairwise independence alone is insufficient for $n\\ge5$, as the book commentary explains with a counterexample. The intended theorem needs each event to be independent of the conjunction of its preceding non-neighbors. The solutions must specify the stronger hypothesis they use."},{"id":"1976-B4","year":1976,"session":"B","number":4,"statement":"B-4. For a point $ P $ on an ellipse, let $ d $ be the distance from the center of the ellipse to the line tangent to the ellipse at $ P $. Prove that $ \\left(P F_{1}\\right)\\left(P F_{2}\\right) d^{2} $ is constant as $ P $ varies on the ellipse, where $ P F_{1} $ and $ P F_{2} $ are the distances from $ P $ to the foci $ F_{1} $ and $ F_{2} $ of the ellipse.","subjects":["geometry","ellipses","analytic geometry"]},{"id":"1976-B5","year":1976,"session":"B","number":5,"statement":"$$\n\\begin{array}{l}\n\\text { B-5. Evaluate }\\\\\n\\sum_{k=0}^{n}(-1)^{k}\\binom{n}{k}(x-k)^{n} .\n\\end{array}\n$$","subjects":["algebra","combinatorics","finite differences"]},{"id":"1976-B6","year":1976,"session":"B","number":6,"statement":"B-6. As usual, let $ \\sigma(N) $ denote the sum of all the (positive integral) divisors of $ N $. (Included among these divisors are $ I $ and $ N $ itself.) For example, if $ p $ is a prime, then $ \\sigma(p)=p+1 $. Motivated by the notion of a \"perfect\" number, a positive integer $ N $ is called \"quasiperfect\" if $ \\sigma(N)=2 N+1 $. Prove that every quasiperfect number is the square of an odd integer.","subjects":["number theory","divisor sums","quadratic residues"]},{"id":"1977-A1","year":1977,"session":"A","number":1,"statement":"Problem A-1\nConsider all lines which meet the graph of\n$$\ny=2 x^{4}+7 x^{3}+3 x-5\n$$\nin four distinct points, say $ \\left(x_{i}, y_{i}\\right), i=1,2,3,4 $. Show that\n$$\n\\frac{x_{1}+x_{2}+x_{3}+x_{4}}{4}\n$$\nis independent of the line and find its value.","subjects":["algebra","polynomials"]},{"id":"1977-A2","year":1977,"session":"A","number":2,"statement":"Problem A-2\nDetermine all solutions in real numbers $ x, y, z, w $ of the system\n$$\n\\begin{aligned}\nx+y+z & =w \\\\\n\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z} & =\\frac{1}{w}\n\\end{aligned}\n$$","subjects":["algebra","systems of equations","symmetric identities"]},{"id":"1977-A3","year":1977,"session":"A","number":3,"statement":"Problem A-3\nLet $ u, f $, and $ g $ be functions, defined for all real numbers $ x $, such that\n$$\n\\frac{u(x+1)+u(x-1)}{2}=f(x) \\text { and } \\frac{u(x+4)+u(x-4)}{2}=g(x) .\n$$\n\nDetermine $ u(x) $ in terms of $ f $ and $ g $.","subjects":["algebra","functional equations","shift operators"]},{"id":"1977-A4","year":1977,"session":"A","number":4,"statement":"Problem A-4\nFor $ 0<x<1 $, express\n$$\n\\sum_{n=0}^{\\infty} \\frac{x^{2^{n}}}{1-x^{2^{n+1}}}\n$$\nas a rational function of $ x $.","subjects":["analysis","series","binary decomposition"]},{"id":"1977-A5","year":1977,"session":"A","number":5,"statement":"Problem A-5\nProve that\n$$\n\\binom{p a}{p b} \\equiv\\binom{a}{b}(\\bmod p)\n$$\nfor all integers $ p, a $, and $ b $ with $ p $ a prime, $ p>0 $, and $ a>b>0 $.\nNotation: $ \\binom{m}{n} $ denotes the binomial coefficient $ \\frac{m!}{n!(m-n)!} $.","subjects":["number theory","combinatorics","binomial congruences"]},{"id":"1977-A6","year":1977,"session":"A","number":6,"statement":"Problem A-6\nLet $ f(x, y) $ be a continuous function on the square\n$$\nS=\\{(x, y): 0\\leq x\\leq 1,0\\leq y\\leq 1\\} .\n$$\n\nFor each point $ (a, b) $ in the interior of $ S $, let $ S_{(a, b)} $ be the largest square that is contained in $ S $, is centered at $ (a, b) $, and has sides parallel to those of $ S $. If the double integral $ \\iint f(x, y) d x d y $ is zero when taken over each square $ S_{(a, b)} $ must $ f(x, y) $ be identically zero on $ S $ ?","subjects":["analysis","multivariable calculus","integral geometry"]},{"id":"1977-B1","year":1977,"session":"B","number":1,"statement":"Problem B-1\nEvaluate the infinite product\n$$\n\\prod_{n=2}^{\\infty} \\frac{n^{3}-1}{n^{3}+1}\n$$","subjects":["analysis","infinite products","factorization"]},{"id":"1977-B2","year":1977,"session":"B","number":2,"statement":"Problem B-2\nGiven a convex quadrilateral $ A B C D $ and a point $ O $ not in the plane of $ A B C D $, locate point $ A^{\\prime} $ on line $ O A $, point $ B^{\\prime} $ on line $ O B $, point $ C^{\\prime} $ on line $ O C $, and point $ D^{\\prime} $ on line $ O D $ so that $ A^{\\prime} B^{\\prime} C^{\\prime} D^{\\prime} $ is a parallelogram.","subjects":["geometry","solid geometry","affine geometry"]},{"id":"1977-B3","year":1977,"session":"B","number":3,"statement":"Problem B-3\nAn (ordered) triple ( $ x_{1}, x_{2}, x_{3} $ ) of positive irrational numbers with $ x_{1}+x_{2}+x_{3}=1 $ is called \"balanced\" if each $ x_{i}<1 / 2 $. If a triple is not balanced, say if $ x_{j}>1 / 2 $, one performs the following \"balancing act\"\n$$\nB\\left(x_{1}, x_{2}, x_{3}\\right)=\\left(x_{1}^{\\prime}, x_{2}^{\\prime}, x_{3}^{\\prime}\\right),\n$$\nwhere $ x_{i}^{\\prime}=2 x_{i} $ if $ i \\neq j $ and $ x_{j}^{\\prime}=2 x_{j}-1 $. If the new triple is not balanced, one performs the balancing act on it. Does continuation of this process always lead to a balanced triple after a finite number of performances of the balancing act?","subjects":["analysis","dynamical systems","binary expansions"]},{"id":"1977-B4","year":1977,"session":"B","number":4,"statement":"Problem B-4\nLet $ C $ be a continuous closed curve in the plane which does not cross itself and let $ Q $ be a point inside $ C $. Show that there exist points $ P_{1} $ and $ P_{2} $ on $ C $ such that $ Q $ is the midpoint of the line segment $ P_{1} P_{2} $.","subjects":["topology","plane geometry","jordan curves"]},{"id":"1977-B5","year":1977,"session":"B","number":5,"statement":"Problem B-5\nSuppose that $ a_{1}, a_{2}, \\ldots, a_{n} $ are real $ (n>1) $ and\n$$\nA+\\sum_{i=1}^{n} a_{i}^{2}<\\frac{1}{n-1}\\left(\\sum_{i=1}^{n} a_{i}\\right)^{2}\n$$\n\nProve that $ A<2 a_{i} a_{j} $ for $ 1\\leq i<j\\leq n $.","subjects":["algebra","inequalities","quadratic forms"]},{"id":"1977-B6","year":1977,"session":"B","number":6,"statement":"Problem B-6\nLet $ H $ be a subgroup with $ h $ elements in a group $ G $. Suppose that $ G $ has an element $ a $ such that for all $ x $ in $ H $, $ (x a)^{3}=1 $, the identity. In $ G $, let $ P $ be the subset of all products $ x_{1} a x_{2} a \\cdots x_{n} a $, with $ n $ a positive integer and the $ x_{i} $ in H.\n(a) Show that $ P $ is a finite set.\n(b) Show that, in fact, $ P $ has no more than $ 3 h^{2} $ elements.","subjects":["algebra","group theory"]},{"id":"1978-A1","year":1978,"session":"A","number":1,"statement":"Problem A-1\nLet $ A $ be any set of 20 distinct integers chosen from the arithmetic progression $ 1,4,7, \\ldots, 100 $. Prove that there must be two distinct integers in $ A $ whose sum is 104.","subjects":["combinatorics","additive combinatorics"]},{"id":"1978-A2","year":1978,"session":"A","number":2,"statement":"Problem A-2\nLet $ a, b, p_{1}, p_{2}, \\ldots, p_{n} $ be real numbers with $ a \\neq b $. Define $ f(x)=\\left(p_{1}-x\\right)\\left(p_{2}-x\\right)\\left(p_{3}-x\\right) \\cdots\\left(p_{n}-x\\right) $. Show that\n$$\n\\operatorname{det}\\left(\\begin{array}{lllllll}\np_{1} & a & a & a & \\cdots & a & a \\\\\nb & p_{2} & a & a & \\cdots & a & a \\\\\nb & b & p_{3} & a & \\cdots & a & a \\\\\nb & b & b & p_{4} & \\cdots & a & a \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\\nb & b & b & b & \\cdots & p_{n-1} & a \\\\\nb & b & b & b & \\cdots & b & p_{n}\n\\end{array}\\right)=\\frac{b f(a)-a f(b)}{b-a} .\n$$","subjects":["linear algebra","determinants","rank-one perturbations"]},{"id":"1978-A3","year":1978,"session":"A","number":3,"statement":"Problem A-3\nLet $ p(x)=2+4 x+3 x^{2}+5 x^{3}+3 x^{4}+4 x^{5}+2 x^{6} $. For $ k $ with $ 0<k<5 $, define\n$$\nI_{k}=\\int_{0}^{\\infty} \\frac{x^{k}}{p(x)} d x\n$$\n\nFor which $ k $ is $ I_{k} $ smallest?","subjects":["analysis","integrals","reciprocal polynomials","inequalities"]},{"id":"1978-A4","year":1978,"session":"A","number":4,"statement":"Problem A-4\nA \"bypass\" operation on a set $ S $ is a mapping from $ S \\times S $ to $ S $ with the property\n$$\nB(B(w, x), B(y, z))=B(w, z) \\quad \\text { for all } \\quad w, x, y, z \\text { in } S .\n$$\n(a) Prove that $ B(a, b)=c $ implies $ B(c, c)=c $ when $ B $ is a bypass.\n(b) Prove that $ B(a, b)=c $ implies $ B(a, x)=B(c, x) $ for all $ x $ in $ S $ when $ B $ is a bypass.\n(c) Construct a table for a bypass operation $ B $ on a finite set $ S $ with the following three properties:\n(i) $ B(x, x)=x $ for all $ x $ in $ S $.\n(ii) There exist $ d $ and $ e $ in $ S $ with $ B(d, e)=d \\neq e $.\n(iii) There exist $ f $ and $ g $ in $ S $ with $ B(f, g) \\neq f $.","subjects":["algebra","binary operations","rectangular bands"]},{"id":"1978-A5","year":1978,"session":"A","number":5,"statement":"Problem A-5\nLet $ 0<x_{i}<\\pi $ for $ i=1,2, \\ldots, n $ and set\n$$\nx=\\frac{x_{1}+x_{2}+\\cdots+x_{n}}{n}\n$$\n\nProve that\n$$\n\\prod_{i=1}^{n} \\frac{\\sin x_{i}}{x_{i}}<\\left(\\frac{\\sin x}{x}\\right)^{n}\n$$","subjects":["analysis","inequalities","log concavity","trigonometry"],"errata":"Replace the unconditional strict inequality by $\\leq$. Equality holds exactly when $x_1=\\cdots=x_n$, including the case $n=1$. Equal arguments disprove the printed strict claim. Scholes states the non-strict inequality, and the separately labeled corrected theorem includes its equality case."},{"id":"1978-A6","year":1978,"session":"A","number":6,"statement":"Problem A-6\nLet $ n $ distinct points in the plane be given. Prove that fewer than $ 2 n^{3 / 2} $ pairs of them are unit distance apart.","subjects":["combinatorics","discrete geometry","unit distances"]},{"id":"1978-B1","year":1978,"session":"B","number":1,"statement":"Problem B-1\nFind the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remaining four sides of length 2 units. Give the answer in the form $ r+s \\sqrt{t} $ with $ r, s $, and $ t $ positive integers.","subjects":["geometry","cyclic polygons","area"]},{"id":"1978-B2","year":1978,"session":"B","number":2,"statement":"Problem B-2\nExpress\n$$\n\\sum_{n=1}^{\\infty} \\sum_{m=1}^{\\infty} \\frac{1}{m^{2} n+m n^{2}+2 m n}\n$$\nas a rational number.","subjects":["analysis","algebra","double series"]},{"id":"1978-B3","year":1978,"session":"B","number":3,"statement":"Problem B-3\nThe sequence $ \\left\\{Q_{n}(x)\\right\\} $ of polynomials is defined by\n$$\nQ_{1}(x)=1+x, Q_{2}(x)=1+2 x\n$$\nand, for $ m\\geq 1 $, by\n$$\n\\begin{array}{l}\nQ_{2 m+1}(x)=Q_{2 m}(x)+(m+1) x Q_{2 m-1}(x) \\\\\nQ_{2 m+2}(x)=Q_{2 m+1}(x)+(m+1) x Q_{2 m}(x)\n\\end{array}\n$$\n\nLet $ x_{n} $ be the largest real solution of $ Q_{n}(x)=0 $. Prove that $ \\left\\{x_{n}\\right\\} $ is an increasing sequence and that $ \\lim _{n \\rightarrow \\infty} x_{n}=0 $.","subjects":["algebra","polynomial roots","recurrences","spectral theory","real-rooted polynomials","polynomial zeros"],"errata":"The supplied book prints $m>1$, which leaves $Q_3$ and $Q_4$ undefined. This archive uses the source-corroborated correction $m\\ge1$ in the displayed recurrence. Both initial polynomials and the corrected starting index are needed for the intended sequence."},{"id":"1978-B4","year":1978,"session":"B","number":4,"statement":"Problem B-4\nProve that for every real number $ N $, the equation\n$$\nx_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1} x_{2} x_{3}+x_{1} x_{2} x_{4}+x_{1} x_{3} x_{4}+x_{2} x_{3} x_{4}\n$$\nhas a solution for which $ x_{1}, x_{2}, x_{3}, x_{4} $ are all integers larger than $ N $.","subjects":["number theory","diophantine equations","vieta jumping"]},{"id":"1978-B5","year":1978,"session":"B","number":5,"statement":"Problem B-5\nFind the largest $ A $ for which there exists a polynomial\n$$\nP(x)=A x^{4}+B x^{3}+C x^{2}+D x+E\n$$\nwith real coefficients, which satisfies\n$$\n0<P(x)<1 \\quad \\text { for } \\quad-1<x<1 .\n$$","subjects":["algebra","polynomial inequalities","chebyshev polynomials","polynomial optimization","nonattainment"],"errata":"With the strict inequalities as printed, no largest $A$ exists. The supremum is $4$, and it is not attained. The source solution claims an extremizer that has interior zeros and an interior value of one, so it violates the stated strict bounds. Under non-strict inequalities the maximum would be $4$. The proof tracks identify whether they establish nonattainment alone or also the numerical supremum."},{"id":"1978-B6","year":1978,"session":"B","number":6,"statement":"Problem B-6\nLet $ p $ and $ n $ be positive integers. Suppose that the numbers $ c_{h, k}(h=1,2, \\ldots, n ; k=1,2, \\ldots, p h) $ satisfy $ 0<c_{h, k}<1 $. Prove that\n$$\n\\left(\\sum \\frac{c_{h, k}}{h}\\right)^{2}<2 p \\sum c_{h, k}\n$$\nwhere each summation is over all admissible ordered pairs $ (h, k) $.","subjects":["algebra","inequalities"]},{"id":"1979-A1","year":1979,"session":"A","number":1,"statement":"Problem A-1\nFind positive integers $ n $ and $ a_{1}, a_{2}, \\ldots, a_{n} $ such that\n$$\na_{1}+a_{2}+\\cdots+a_{n}=1979\n$$\nand the product $ a_{1} a_{2} \\cdots a_{n} $ is as large as possible.","subjects":["number theory","optimization","integer partitions"]},{"id":"1979-A2","year":1979,"session":"A","number":2,"statement":"Problem A-2\nEstablish necessary and sufficient conditions on the constant $ \\boldsymbol{k} $ for the existence of a continuous real valued function $ f(x) $ satisfying $ f(f(x))=k x^{9} $ for all real $ x $.","subjects":["analysis","functional equations","monotonicity"]},{"id":"1979-A3","year":1979,"session":"A","number":3,"statement":"Problem A-3\nLet $ x_{1}, x_{2}, x_{3}, \\ldots $ be a sequence of nonzero real numbers satisfying\n$$\nx_{n}=\\frac{x_{n-2} x_{n-1}}{2 x_{n-2}-x_{n-1}} \\text { for } n=3,4,5, \\ldots\n$$\n\nEstablish necessary and sufficient conditions on $ x_{1} $ and $ x_{2} $ for $ x_{n} $ to be an integer for infinitely many values of $ n $.","subjects":["algebra","recurrences","integrality"]},{"id":"1979-A4","year":1979,"session":"A","number":4,"statement":"Problem A-4\nLet $ A $ be a set of $ 2 n $ points in the plane, no three of which are collinear. Suppose that $ n $ of them are colored red and the remaining $ n $ blue. Prove or disprove: there are $ n $ closed straight line segments, no two with a point in common, such that the endpoints of each segment are points of $ A $ having different colors.","subjects":["geometry","combinatorics","noncrossing matchings"]},{"id":"1979-A5","year":1979,"session":"A","number":5,"statement":"Problem A-5\nDenote by $ [x] $ the greatest integer less than or equal to $ x $ and by $ S(x) $ the sequence $ [x],[2 x],[3 x], \\ldots $. Prove that there are distinct real solutions $ \\alpha $ and $ \\beta $ of the equation $ x^{3}-10 x^{2}+29 x-25=0 $ such that infinitely many positive integers appear both in $ S(\\alpha) $ and in $ S(\\beta) $.","subjects":["number theory","beatty sequences","density","algebraic numbers"]},{"id":"1979-A6","year":1979,"session":"A","number":6,"statement":"Problem A-6\nLet $ 0<p_{i}<1 $ for $ i=1,2, \\ldots, n $. Show that\n$$\n\\sum_{i=1}^{n} \\frac{1}{\\left|x-p_{i}\\right|}<8 n\\left(1+\\frac{1}{3}+\\frac{1}{5}+\\cdots+\\frac{1}{2 n-1}\\right)\n$$\nfor some $ x $ satisfying $ 0<x<1 $.","subjects":["analysis","inequalities","averaging","discrete geometry"]},{"id":"1979-B1","year":1979,"session":"B","number":1,"statement":"Problem B-1\nProve or disprove: there is at least one straight line normal to the graph of $ y=\\cosh x $ at a point $ (a, \\cosh a) $ and also normal to the graph of $ y=\\sinh x $ at a point $ (c, \\sinh c) $.\n[At a point on a graph, the normal line is the perpendicular to the tangent at that point. Also, $ \\cosh x=\\left(e^{x}+\\right. $ $ \\left.e^{-x}\\right) / 2 $ and $ \\left.\\sinh x=\\left(e^{x}-e^{-x}\\right) / 2.\\right] $","subjects":["analysis","geometry","hyperbolic functions","normal lines"]},{"id":"1979-B2","year":1979,"session":"B","number":2,"statement":"Problem B-2\nLet $ 0<a<b $. Evaluate\n$$\n\\lim _{t \\rightarrow 0}\\left\\{\\int_{0}^{1}[b x+a(1-x)]^t d x\\right\\}^{1 / t}\n$$\n[The final answer should not involve any operations other than addition, subtraction, multiplication, division, and exponentiation.]","subjects":["analysis","limits","power means"]},{"id":"1979-B3","year":1979,"session":"B","number":3,"statement":"Problem B-3\nLet $ F $ be a finite field having an odd number $ m $ of elements. Let $ p(x) $ be an irreducible (i.e., nonfactorable) polynomial over $ F $ of the form\n$$\nx^{2}+b x+c, \\quad b, c \\in F\n$$\n\nFor how many elements $ k $ in $ F $ is $ p(x)+k $ irreducible over $ F $ ?","subjects":["algebra","finite fields","irreducible quadratics"]},{"id":"1979-B4","year":1979,"session":"B","number":4,"statement":"Problem B-4\n(a) Find a solution that is not identically zero, of the homogeneous linear differential equation\n$$\n\\left(3 x^{2}+x-1\\right) y^{\\prime \\prime}-\\left(9 x^{2}+9 x-2\\right) y^{\\prime}+(18 x+3) y=0\n$$\n\nIntelligent guessing of the form of a solution may be helpful.\n(b) Let $ y=f(x) $ be the solution of the nonhomogeneous differential equation\n$$\n\\left(3 x^{2}+x-1\\right) y^{\\prime \\prime}-\\left(9 x^{2}+9 x-2\\right) y^{\\prime}+(18 x+3) y=6(6 x+1)\n$$\nthat has $ f(0)=1 $ and $ (f(-1)-2)(f(1)-6)=1 $. Find integers $ a, b, c $ such that $ (f(-2)-a)(f(2)-b)=c $.","subjects":["analysis","ordinary differential equations","singular coefficients"]},{"id":"1979-B5","year":1979,"session":"B","number":5,"statement":"Problem B-5\nIn the plane, let $ C $ be a closed convex set that contains $ (0,0) $ but no other point with integer coordinates. Suppose that $ A(C) $, the area of $ C $, is equally distributed among the four quadrants. Prove that $ A(C)<4 $.","subjects":["geometry","convexity","lattice points","area"]},{"id":"1979-B6","year":1979,"session":"B","number":6,"statement":"Problem B-6\nFor $ k=1,2, \\ldots, n $ let $ z_{k}=x_{k}+i y_{k} $, where the $ x_{k} $ and $ y_{k} $ are real and $ i=\\sqrt{-1} $. Let $ r $ be the absolute value of the real part of\n$$\n\\pm \\sqrt{z_{1}^{2}+z_{2}^{2}+\\cdots+z_{n}^{2}}\n$$\n\nProve that $ r<\\left|x_{1}\\right|+\\left|x_{2}\\right|+\\cdots+\\left|x_{n}\\right| $.","subjects":["algebra","complex numbers","norm inequalities","complex analysis","asymptotic analysis"],"errata":"The strict sign in the dataset transcription is incorrect. The supplied book prints the non-strict bound on printed p. 34, PDF p. 43, and concludes with the same bound on printed p. 112, PDF p. 120. This is a transcription correction, not an official book erratum. The corrected review target is the following. For every integer $n\\ge1$ and complex numbers $z_k=x_k+iy_k$ with real $x_k,y_k$, let $r$ be the absolute value of the real part of either square root of $\\sum_{k=1}^n z_k^2$. Prove $r\\le\\sum_{k=1}^n|x_k|$. For $n=1,z_1=1$, both sides are $1$, so unconditional strictness is impossible. The canonical statement is retained for provenance. This note records the intended target and does not certify any proof."},{"id":"1980-A1","year":1980,"session":"A","number":1,"statement":"Problem A-1\nLet $ b $ and $ c $ be fixed real numbers and let the ten points $ \\left(j, y_{j}\\right) . j=1.2 \\ldots .10 $, lie on the parabola $ y=x^{2}+b x+c $. For $ j=1,2 \\ldots, 9 $, let $ I $, be the point of intersection of the tangents to the given parabola at $ \\left(j, y_{j}\\right) $ and $ \\left(\\jmath+1, y_{j+1}\\right) $. Determine the polynomial function $ y=g(x) $ of least degree whose graph passes through all nine points $ I_{f} $.","subjects":["algebra","analytic geometry","parabolas","interpolation"]},{"id":"1980-A2","year":1980,"session":"A","number":2,"statement":"Problem A-2\nLet $ r $ and $ s $ be positive integers. Derive a formula for the number of ordered quadruples $ (a, b, c, d) $ of positive integers such that\n$$\n3^{r} \\cdot 7^{s}=\\operatorname{lcm}[a, b, c]=\\operatorname{lcm}[a, b, d]=\\operatorname{lcm}[a, c, d]=\\operatorname{lcm}[b, c, d]\n$$\n\nThe answer should be a function of $ r $ and $ s $.\n(Note that $ 1 \\mathrm{~cm}[x, y, z] $ denotes the least common multiple of $ x, y, z $. )","subjects":["number theory","combinatorics","least common multiples"]},{"id":"1980-A3","year":1980,"session":"A","number":3,"statement":"Problem A-3\nEvaluate\n$$\n\\int_{0}^{\\pi / 2} \\frac{d x}{1+(\\tan x)^{\\sqrt{2}}}\n$$","subjects":["analysis","integration"]},{"id":"1980-A4","year":1980,"session":"A","number":4,"statement":"Problem A-4\n(a) Prove that there exist integers $ a, b, c $, not all zero and each of absolute value less than one million, such that\n$$\n|a+b \\sqrt{2}+c \\sqrt{3}|<10^{-11}\n$$\n(b) Let $ a, b, c $ be integers, not all zero and each of absolute value less than one million. Prove that\n$$\n|a+b \\sqrt{2}+c \\sqrt{3}|>10^{-21}\n$$","subjects":["number theory","diophantine approximation","algebraic norms"]},{"id":"1980-A5","year":1980,"session":"A","number":5,"statement":"Problem A-5\nLet $ P(t) $ be a nonconstant polynomial with real coefficients. Prove that the system of simultaneous equations\n$$\n0=\\int_{0}^{x} P(t) \\sin t d t=\\int_{0}^{x} P(t) \\cos t d t\n$$\nhas only finitely many real solutions $ x $.","subjects":["analysis","polynomials","oscillatory integrals","complex analysis","monotone functions"]},{"id":"1980-A6","year":1980,"session":"A","number":6,"statement":"Problem A-6\nLet $ C $ be the class of all real valued continuously differentiable functions $ f $ on the interval $ 0\\leq x\\leq 1 $ with $ f(0)=0 $ and $ f(1)=1 $. Determine the largest real number $ u $ such that\n$$\nu \\leqslant \\int_{0}^{1}\\left|f^{\\prime}(x)-f(x)\\right| d x\n$$\nfor all $ f $ in $ C $.","subjects":["analysis","calculus of variations","sharp inequalities"]},{"id":"1980-B1","year":1980,"session":"B","number":1,"statement":"Problem B-1\nFor which real numbers $ c $ is $ \\left(e^{x}+e^{-x}\\right) / 2 \\leqslant e^{c x^{2}} $ for all real $ x $ ?","subjects":["analysis","inequalities","hyperbolic functions"]},{"id":"1980-B2","year":1980,"session":"B","number":2,"statement":"Problem B-2\nLet $ S $ be the solid in three-dimensional space consisting of all points ( $ x, y, z $ ) satisfying the following system of six simultaneous conditions:\n$$\n\\begin{array}{l}\nx \\geqslant 0, \\quad y \\geqslant 0, \\quad z \\geqslant 0 \\\\\nx+y+z \\leqslant 11, \\\\\n2 x+4 y+3 z \\leqslant 36 \\\\\n2 x+3 z \\leqslant 24 .\n\\end{array}\n$$\n(a) Determine the number $ v $ of vertices of $ S $.\n(b) Determine the number $ e $ of edges of $ S $.\n(c) Sketch in the $ b c $-plane the set of points $ (b, c) $ such that $ (2,5,4) $ is one of the points $ (x, y, z) $ at which the linear function $ b x+c y+z $ assumes its maximum value on $ S $.","subjects":["geometry","linear algebra","polyhedra","linear programming"]},{"id":"1980-B3","year":1980,"session":"B","number":3,"statement":"Problem B-3\nFor which real numbers $ a $ does the sequence defined by the initial condition $ u_{0}=a $ and the recursion $ u_{n+1}=2 u_{n}-n^{2} $ have $ u_{n}>0 $ for all $ n>0 $ ?\n(Express the answer in the simplest form.)","subjects":["algebra","recurrences","positivity","power series"]},{"id":"1980-B4","year":1980,"session":"B","number":4,"statement":"Problem B-4\nLet $ A_{1}, A_{2}, \\ldots, A_{1066} $ be subsets of a finite set $ X $ such that $ \\left|A_{i}\\right|>\\frac{1}{2}|X| $ for $ 1 \\leqslant i \\leqslant 1066 $. Prove there exist ten elements $ x_{1}, \\ldots, x_{10} $ of $ X $ such that every $ A_{i} $ contains at least one of $ x_{1}, \\ldots, x_{10} $.\n(Here $ |S| $ means the number of elements in the set $ S $.)","subjects":["combinatorics","set covers"]},{"id":"1980-B5","year":1980,"session":"B","number":5,"statement":"Problem B-5\nFor each $ t \\geqslant 0 $, let $ S_t $ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $ f(x) $ defined on the closed interval $ [0,1] $ for which\n$$\nf(1)-2 f(2 / 3)+f(1 / 3) \\geqslant t[f(2 / 3)-2 f(1 / 3)+f(0)] .\n$$\n\nDevelop necessary and sufficient conditions on $ \\boldsymbol{t} $ for $ S_t $ to be closed under multiplication.\n(This closure means that, if the functions $ f(x) $ and $ g(x) $ are in $ S_{t} $, so is their product $ f(x) g(x) $. A function $ f(x) $ is convex if and only if $ f(su+(1-s)v)\\leq s f(u)+(1-s)f(v) $ whenever $ 0<s \\leqslant 1 $.)","subjects":["analysis","convex functions","closure under multiplication"]},{"id":"1980-B6","year":1980,"session":"B","number":6,"statement":"Problem B-6\nAn infinite array of rational numbers $ G(d, n) $ is defined for integers $ d $ and $ n $ with $ 1 \\leqslant d \\leqslant n $ as follows:\n$$\nG(1, n)=\\frac{1}{n}, \\quad G(d, n)=\\frac{d}{n} \\sum_{i=d}^{n} G(d-1, i-1) \\quad \\text { for } \\quad d>1 .\n$$\n\nFor $ 1<d \\leqslant p $ and $ p $ prime, prove that $ G(d, p) $ is expressible as a quotient $ s / t $ of integers $ s $ and $ t $ with $ t $ not an integral multiple of $ p $.\n(For example, $ G(3,5)=7 / 4 $ with the denominator 4 not a multiple of 5 .)","subjects":["number theory","denominators","stirling numbers","permutations","group actions","modular arithmetic"]},{"id":"1981-A1","year":1981,"session":"A","number":1,"statement":"Problem A-1\nLet $ E(n) $ denote the largest integer $ k $ such that $ 5^{k} $ is an integral divisor of the product $ 1^{1} 2^{2} 3^{3} \\cdots n^{n} $. Calculate $ \\lim _{n \\rightarrow \\infty} \\frac{E(n)}{n^{2}} $.","subjects":["number theory","prime-adic valuations","asymptotics"]},{"id":"1981-A2","year":1981,"session":"A","number":2,"statement":"Problem A-2\nTwo distinct squares of the 8 by 8 chessboard $ C $ are said to be adjacent if they have a vertex or side in common. Also, $ g $ is called a $ C $-gap if for every numbering of the squares of $ C $ with all the integers $ 1,2, \\ldots, 64 $ there exist two adjacent squares whose numbers differ by at least $ g $. Determine the largest $ C $-gap $ g $.","subjects":["combinatorics","graph theory","bandwidth","chessboard"]},{"id":"1981-A3","year":1981,"session":"A","number":3,"statement":"Problem A-3\nFind\n$$\n\\lim _{t \\rightarrow \\infty}\\left[e^{-t} \\int_{0}^{t} \\int_{0}^{t} \\frac{e^{x}-e^{y}}{x-y} d x d y\\right]\n$$\nor show that the limit does not exist.","subjects":["analysis","improper integrals","asymptotic limits"]},{"id":"1981-A4","year":1981,"session":"A","number":4,"statement":"Problem A-4\nA point $ P $ moves inside a unit square in a straight line at unit speed. When it meets a corner it escapes. When it meets an edge its line of motion is reflected so that the angle of incidence equals the angle of reflection.\n\nLet $ N(T) $ be the number of starting directions from a fixed interior point $ P_{0} $ for which $ P $ escapes within $ T $ units of time. Find the least constant $ a $ for which constants $ b $ and $ c $ exist such that\n$$\nN(T) \\leq a T^{2}+b T+c\n$$\nfor all $ T>0 $ and all initial points $ P_{0} $.","subjects":["geometry","lattice points","billiards"]},{"id":"1981-A5","year":1981,"session":"A","number":5,"statement":"Problem A-5\nLet $ P(x) $ be a polynomial with real coefficients and form the polynomial\n$$\nQ(x)=\\left(x^{2}+1\\right) P(x) P^{\\prime}(x)+x\\left([P(x)]^{2}+\\left[P^{\\prime}(x)\\right]^{2}\\right)\n$$\n\nGiven that the equation $ P(x)=0 $ has $ n $ distinct real roots exceeding 1 , prove or disprove that the equation $ Q(x)=0 $ has at least $ 2 n-1 $ distinct real roots.","subjects":["algebra","real polynomials","real roots"]},{"id":"1981-A6","year":1981,"session":"A","number":6,"statement":"Problem A-6\nSuppose that each of the vertices of $ \\triangle A B C $ is a lattice point in the $ (x, y) $-plane and that there is exactly one lattice point $ P $ in the interior of the triangle. The line $ A P $ is extended to meet $ B C $ at $ E $. Determine the largest possible value for the ratio of lengths of segments\n$$\n\\frac{|A P|}{|P E|}\n$$\n[A lattice point is a point whose coordinates $ x $ and $ y $ are integers.]","subjects":["geometry","lattice points","barycentric coordinates"]},{"id":"1981-B1","year":1981,"session":"B","number":1,"statement":"Problem B-I\nFind\n$$\n\\lim _{n \\rightarrow \\infty}\\left[\\frac{1}{n^{5}} \\sum_{h=1}^{n} \\sum_{k=1}^{n}\\left(5 h^{4}-18 h^{2} k^{2}+5 k^{4}\\right)\\right] .\n$$","subjects":["analysis","finite sums","asymptotic limits"]},{"id":"1981-B2","year":1981,"session":"B","number":2,"statement":"Problem B-2\nDetermine the minimum value of\n$$\n(r-1)^{2}+\\left(\\frac{s}{r}-1\\right)^{2}+\\left(\\frac{t}{s}-1\\right)^{2}+\\left(\\frac{4}{t}-1\\right)^{2}\n$$\nfor all real numbers $ r, s, t $ with $ 1 \\leq r \\leq s \\leq t \\leq 4 $.","subjects":["inequalities","optimization","real variables"]},{"id":"1981-B3","year":1981,"session":"B","number":3,"statement":"Problem B-3\nProve that there are infinitely many positive integers $ n $ with the property that if $ p $ is a prime divisor of $ n^{2}+3 $, then $ p $ is also a divisor of $ k^{2}+3 $ for some integer $ k $ with $ k^{2}<n $.","subjects":["number theory","prime divisors","polynomial identities"]},{"id":"1981-B4","year":1981,"session":"B","number":4,"statement":"Problem B-4\nLet $ V $ be a set of 5 by 7 matrices, with real entries and with the property that $ r A+s B \\in V $ whenever $ A, B \\in V $ and $ r $ and $ s $ are scalars (i.e., real numbers). Prove or disprove the following assertion: If $ V $ contains matrices of ranks 0,1 . 2,4 , and 5 , then it also contains a matrix of rank 3.\n[The rank of a nonzero matrix $ M $ is the largest $ k $ such that the entries of some $ k $ rows and some $ k $ columns form a $ k $ by $ k $ matrix with a nonzero determinant.]","subjects":["linear algebra","matrix rank","counterexamples"]},{"id":"1981-B5","year":1981,"session":"B","number":5,"statement":"Problem B-5\nLet $ B(n) $ be the number of ones in the base two expression for the positive integer $ n $. For example, $ B(6)=B\\left(110_{2}\\right) $ $ =2 $ and $ B(15)=B\\left(1111_{2}\\right)=4 $. Determine whether or not\n$$\n\\exp \\left(\\sum_{n=1}^{\\infty} \\frac{B(n)}{n(n+1)}\\right)\n$$\nis a rational number. Here $ \\exp (x) $ denotes $ e^{x} $.","subjects":["number theory","binary digits","infinite series"]},{"id":"1981-B6","year":1981,"session":"B","number":6,"statement":"Problem B-6\nLet $ C $ be a fixed unit circle in the Cartesian plane. For any convex polygon $ P $ each of whose sides is tangent to $ C $. let $ N(P, h, k) $ be the number of points common to $ P $ and the unit circle with center at (h.k). Let $ H(P) $ be the region of all points $ (x, y) $ for which $ N(P, x, y) \\geq 1 $ and $ F(P) $ be the area of $ H(P) $. Find the smallest number $ u $ with\n$$\n\\frac{1}{F(P)} \\iint N(P, x, y) d x d y<u\n$$\nfor all polygons $ P $, where the double integral is taken over $ H(P) $.","subjects":["geometry","convex polygons","geometric probability"],"errata":"Here $P$ denotes the polygonal boundary, and the fixed unit circle is an incircle of its bounded convex region. Intersections with a filled polygon would not generally be finite. The canonical solution and Scholes count side intersections and use the polygon interior together with its exterior unit-width strips and sectors. The solutions explicitly state this boundary-and-incircle interpretation."},{"id":"1982-A1","year":1982,"session":"A","number":1,"statement":"Problem A-1\nLet $ V $ be the region in the cartesian plane consisting of all points $ (x, y) $ satisfying the simultaneous conditions\n$$\n|x| \\leqslant y \\leqslant|x|+3 \\text { and } y \\leqslant 4\n$$\n\nFind the centroid $ (\\bar{x}, \\bar{y}) $ of $ V $.","subjects":["geometry","centroids","planar integration"]},{"id":"1982-A2","year":1982,"session":"A","number":2,"statement":"Problem A-2\nFor positive real $ x $, let\n$$\nB_{n}(x)=1^{x}+2^{x}+3^{x}+\\cdots+n^{x}\n$$\n\nProve or disprove the convergence of\n$$\n\\sum_{n=2}^{\\infty} \\frac{B_{n}\\left(\\log _{n} 2\\right)}{\\left(n \\log _{2} n\\right)^{2}}\n$$","subjects":["analysis","series convergence","comparison tests"]},{"id":"1982-A3","year":1982,"session":"A","number":3,"statement":"Problem A-3\nEvaluate\n$$\n\\int_{0}^{\\infty} \\frac{\\operatorname{Arctan}(\\pi x)-\\operatorname{Arctan} x}{x} d x\n$$","subjects":["analysis","improper integrals","inverse trigonometric functions"]},{"id":"1982-A4","year":1982,"session":"A","number":4,"statement":"Problem A-4\nAssume that the system of simultaneous differential equations\n$$\ny^{\\prime}=-z^{3}, \\quad z^{\\prime}=y^{3}\n$$\nwith the initial conditions $ y(0)=1, z(0)=0 $ has a unique solution $ y=f(x), z=g(x) $ defined for all real $ x $. Prove that there exists a positive constant $ L $ such that for all real $ x $.\n$$\nf(x+L)=f(x), \\quad g(x+L)=g(x)\n$$","subjects":["analysis","ordinary differential equations","periodicity"]},{"id":"1982-A5","year":1982,"session":"A","number":5,"statement":"Problem A-5\nLet $ a, b, c $, and $ d $ be positive integers and\n$$\nr=1-\\frac{a}{b}-\\frac{c}{d}\n$$\n\nGiven that $ a+c \\leqslant 1982 $ and $ r>0 $, prove that\n$$\nr>\\frac{1}{1983^{3}} .\n$$","subjects":["number theory","rational inequalities","integer bounds"]},{"id":"1982-A6","year":1982,"session":"A","number":6,"statement":"Problem A-6\nLet $ \\sigma $ be a bijection of the positive integers, that is, a one-to-one function from ( $ 1,2,3, \\ldots $ ) onto itself. Let $ x_{1}, x_{2}, x_{3}, \\ldots $ be a sequence of real numbers with the following three properties:\n(i) $ \\left|x_{n}\\right| $ is a strictly decreasing function of $ n $;\n(ii) $ |\\sigma(n)-n| \\cdot\\left|x_{n}\\right| \\rightarrow 0 $ as $ n \\rightarrow \\infty $;\n(iii) $ \\lim _{n \\rightarrow \\infty} \\sum_{k=1}^{n} x_{k}=1 $.\n\nProve or disprove that these conditions imply that\n$$\n\\lim _{n \\rightarrow \\infty} \\sum_{k=1}^{n} x_{\\sigma(k)}=1\n$$","subjects":["analysis","conditionally convergent series","rearrangements"]},{"id":"1982-B1","year":1982,"session":"B","number":1,"statement":"Problem B-1\nLet $ M $ be the midpoint of side $ B C $ of a general $ \\triangle A B C $. Using the smallest possible $ n $, describe a method for cutting $ \\triangle A M B $ into $ n $ triangles which can be reassembled to form a triangle congruent to $ \\triangle A M C $.","subjects":["geometry","dissections","triangle congruence"]},{"id":"1982-B2","year":1982,"session":"B","number":2,"statement":"Problem B-2\nLet $ A(x, y) $ denote the number of points $ (m, n) $ in the plane with integer coordinates $ m $ and $ n $ satisfying $ m^{2}+n^{2} \\leqslant x^{2}+y^{2} $. Let $ g=\\sum_{k=0}^{\\infty} e^{-k^{2}} $. Express\n$$\n\\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} A(x, y) e^{-x^2-y^2} d x d y\n$$\nas a polynomial in $ g $.","subjects":["analysis","gaussian integrals","lattice points","fourier analysis"]},{"id":"1982-B3","year":1982,"session":"B","number":3,"statement":"Problem B-3\nLet $ p_{n} $ be the probability that $ c+d $ is a perfect square when the integers $ c $ and $ d $ are selected independently at random from the set $ \\{1,2,3, \\ldots, n\\} $. Show that $ \\lim _{n \\rightarrow \\infty}\\left(p_{n} \\sqrt{n}\\right) $ exists and express this limit in the form $ r(\\sqrt{s}-1) $. where $ s $ and $ t $ are integers and $ r $ is a rational number.","subjects":["probability","lattice counting","asymptotic limits"]},{"id":"1982-B4","year":1982,"session":"B","number":4,"statement":"Problem B-4\nLet $ n_{1}, n_{2}, \\ldots, n_{s} $ be distinct integers such that\n$$\n\\left(n_{1}+k\\right)\\left(n_{2}+k\\right) \\cdots\\left(n_s+k\\right)\n$$\nis an integral multiple of $ n_{1} n_{2} \\cdots n_{s} $ for every integer $ k $. For each of the following assertions. give a proof or a counterexample:\n(a) $ \\left|n_{i}\\right|=1 $ for some $ i $.\n(b) If further all $ n $, are positive, then\n$$\n\\left\\{n_{1}, n_{2}, \\ldots, n_{s}\\right\\}=\\{1,2, \\ldots, s\\} .\n$$","subjects":["number theory","integer-valued polynomials","finite differences"]},{"id":"1982-B5","year":1982,"session":"B","number":5,"statement":"Problem B-5\nFor each $ x>e^{e} $ define a sequence $ S_x=u_{0}, u_{1}, u_{2}, \\ldots $ recursively as follows: $ u_{0}=e $, while for $ n \\geqslant 0, u_{n+1} $ is the logarithm of $ x $ to the base $ u_{n} $. Prove that $ S_x $ converges to a number $ g(x) $ and that the function $ g $ defined in this way is continuous for $ x>e^e $.","subjects":["analysis","recursive sequences","fixed points"]},{"id":"1982-B6","year":1982,"session":"B","number":6,"statement":"Problem B-6\nLet $ K(x, y, z) $ denote the area of a triangle whose sides have lengths $ x, y $, and $ z $. For any two triangles with sides $ a, b, c $ and $ a^{\\prime}, b^{\\prime}, c^{\\prime} $, respectively, prove that\n$$\n\\sqrt{K(a, b, c)}+\\sqrt{K\\left(a^{\\prime}, b^{\\prime}, c^{\\prime}\\right)} \\leqslant \\sqrt{K\\left(a+a^{\\prime}, b+b^{\\prime}, c+c^{\\prime}\\right)}\n$$\nand determine the cases of equality.","subjects":["geometry","triangle area","inequalities"]},{"id":"1983-A1","year":1983,"session":"A","number":1,"statement":"Problem A-1\nHow many positive integers $ n $ are there such that $ n $ is an exact divisor of at least one of the numbers $ 10^{40}, 20^{30} $ ?","subjects":["number theory","divisors"]},{"id":"1983-A2","year":1983,"session":"A","number":2,"statement":"Problem A-2\n\nThe hands of an accurate clock have lengths 3 and 4 . Find the distance between the tips of the hands when that distance is increasing most rapidly.","subjects":["geometry","calculus","optimization"]},{"id":"1983-A3","year":1983,"session":"A","number":3,"statement":"Problem A-3\n\nLet $ p $ be in the set $ \\{3,5,7,11, \\ldots\\} $ of odd primes and let\n$$\nF(n)=1+2 n+3 n^{2}+\\cdots+(p-1) n^{p-2} .\n$$\n\nProve that if $ a $ and $ b $ are distinct integers in $ \\{0,1,2, \\ldots, p-1\\} $ then $ F(a) $ and $ F(b) $ are not congruent modulo $ p $, that is, $ F(a)-F(b) $ is not exactly divisible by $ p $.","subjects":["algebra","finite fields","permutation polynomials"]},{"id":"1983-A4","year":1983,"session":"A","number":4,"statement":"Problem A-4\n\nLet $ k $ be a positive integer and let $ m=6 k-1 $. Let\n$$\nS(m)=\\sum_{j=1}^{2 k-1}(-1)^{j+1}\\binom{m}{3 j-1}\n$$\n\nFor example with $ k=3 $,\n$$\nS(17)=\\binom{17}{2}-\\binom{17}{5}+\\binom{17}{8}-\\binom{17}{11}+\\binom{17}{14} .\n$$\n\nProve that $ S(m) $ is never zero. $ \\left[\\right. $ As usual, $ \\binom{m}{r}=\\frac{m!}{r!(m-r)!} $.]","subjects":["combinatorics","binomial coefficients","congruences"]},{"id":"1983-A5","year":1983,"session":"A","number":5,"statement":"Problem A-5\nProve or disprove that there exists a positive real number $ u $ such that $ \\left[u^{n}\\right]-n $ is an even integer for all positive integers $ n $.\n\nHere $ [x] $ denotes the greatest integer less than or equal to $ x $.","subjects":["analysis","integer parts","algebra"]},{"id":"1983-A6","year":1983,"session":"A","number":6,"statement":"Problem A-6\n\nLet $ \\exp (t) $ denote $ e^{t} $ and\n$$\nF(x)=\\frac{x^{4}}{\\exp \\left(x^{3}\\right)} \\int_{0}^{x} \\int_{0}^{x-u} \\exp \\left(u^{3}+v^{3}\\right) d v d u\n$$\n\nFind $ \\lim _{x \\rightarrow \\infty} F(x) $ or prove that it does not exist.","subjects":["analysis","asymptotic integrals"]},{"id":"1983-B1","year":1983,"session":"B","number":1,"statement":"Problem B-1\n\nLet $ v $ be a vertex (corner) of a cube $ C $ with edges of length 4. Let $ S $ be the largest sphere that can be inscribed in $ C $. Let $ R $ be the region consisting of all points $ p $ between $ S $ and $ C $ such that $ p $ is closer to $ v $ than to any other vertex of the cube. Find the volume of $ R $.","subjects":["geometry","solid geometry","volume"]},{"id":"1983-B2","year":1983,"session":"B","number":2,"statement":"Problem B-2\n\nFor positive integers $ n $, let $ C(n) $ be the number of representations of $ n $ as a sum of nonincreasing powers of 2 , where no power can be used more than three times. For example, $ C(8)=5 $ since the representations for 8 are:\n$$\n8,4+4,4+2+2,4+2+1+1, \\text { and } 2+2+2+1+1\n$$\n\nProve or disprove that there is a polynomial $ P(x) $ such that $ C(n)=[P(n)] $ for all positive integers $ n $; here [u] denotes the greatest integer less than or equal to $ u $.","subjects":["combinatorics","binary partitions"]},{"id":"1983-B3","year":1983,"session":"B","number":3,"statement":"Problem B-3\n\nAssume that the differential equation\n$$\ny^{\\prime \\prime \\prime}+p(x) y^{\\prime \\prime}+q(x) y^{\\prime}+r(x) y=0\n$$\nhas solutions $ y_{1}(x), y_{2}(x) $, and $ y_{3}(x) $ on the whole real line such that\n$$\ny_{1}^{2}(x)+y_{2}^{2}(x)+y_{3}^{2}(x)=1\n$$\nfor all real $ x $. Let\n$$\nf(x)=\\left(y_{1}^{\\prime}(x)\\right)^{2}+\\left(y_{2}^{\\prime}(x)\\right)^{2}+\\left(y_{3}^{\\prime}(x)\\right)^{2} .\n$$\n\nFind constants $ A $ and $ B $ such that $ f(x) $ is a solution to the differential equation\n$$\ny^{\\prime}+A p(x) y=B r(x)\n$$","subjects":["analysis","ordinary differential equations","differential identities"]},{"id":"1983-B4","year":1983,"session":"B","number":4,"statement":"Problem B-4\nLet $ f(n)=n+[\\sqrt{n}] $ where $ [x] $ is the largest integer less than or equal to $ x $. Prove that, for every positive integer $ m $, the sequence\n$$\nm, f(m), f(f(m)), f(f(f(m))), \\ldots\n$$\ncontains at least one square of an integer.","subjects":["number theory","integer iteration"]},{"id":"1983-B5","year":1983,"session":"B","number":5,"statement":"Problem B-5\nLet $ \\|u\\| $ denote the distance from the real number $ u $ to the nearest integer. (For example, $ \\|0.8\\|=0.2=\\|3.2\\| $ ) For positive integers $ n $, let\n$$\na_{n}=\\frac{1}{n} \\int_{1}^{n}\\left\\|\\frac{n}{x}\\right\\| d x\n$$\n\nDetermine $ \\lim _{n \\rightarrow \\infty} a_{n} $. You may assume the identity\n$$\n\\frac{2}{1} \\cdot \\frac{2}{3} \\cdot \\frac{4}{3} \\cdot \\frac{4}{5} \\cdot \\frac{6}{5} \\cdot \\frac{6}{7} \\cdot \\frac{8}{7} \\cdot \\frac{8}{9} \\cdot \\cdots=\\frac{\\pi}{2}\n$$","subjects":["analysis","improper integrals","wallis product"]},{"id":"1983-B6","year":1983,"session":"B","number":6,"statement":"Problem B-6\nLet $ k $ be a positive integer, let $ m=2^{k}+1 $, and let $ r \\neq 1 $ be a complex root of $ z^{m}-1=0 $. Prove that there exist polynomials $ P(z) $ and $ Q(z) $ with integer coefficients such that\n$$\n(P(r))^{2}+(Q(r))^{2}=-1\n$$","subjects":["algebra","roots of unity","sums of squares"]},{"id":"1984-A1","year":1984,"session":"A","number":1,"statement":"Problem A-1\nLet $ A $ be a solid $ a \\times b \\times c $ rectangular brick in three dimensions, where $ a, b, c>0 $. Let $ B $ be the set of all points which are a distance at most one from some point of $ A $ (in particular, $ B $ contains $ A $ ). Express the volume of $ B $ as a polynomial in $ a, b $, and $ c $.","subjects":["geometry","solid geometry","parallel bodies"]},{"id":"1984-A2","year":1984,"session":"A","number":2,"statement":"Problem A-2\nExpress $ \\sum_{k=1}^{\\infty}\\frac{6^k}{(3^{k+1}-2^{k+1})(3^k-2^k)} $ as a rational number.","subjects":["analysis","infinite series"]},{"id":"1984-A3","year":1984,"session":"A","number":3,"statement":"Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \\neq b$, and let $M_n$ denote the $2n \\times 2n$ matrix whose $(i,j)$ entry $m_{ij}$ is given by\n$$\nm_{ij}=\\begin{cases}\nx & \\text{if }i=j, \\\\\na & \\text{if }i \\neq j\\text{ and }i+j\\text{ is even}, \\\\\nb & \\text{if }i \\neq j\\text{ and }i+j\\text{ is odd}.\n\\end{cases}\n$$\nThus, for example, $M_2=\\begin{pmatrix} x & b & a & b \\\\ b & x & b & a \\\\ a & b & x & b \\\\ b & a & b & x \\end{pmatrix}$. Express $\\lim_{x \\to a} \\det M_n/(x-a)^{2n-2}$ as a polynomial in $a$, $b$, and $n$, where $\\det M_n$ denotes the determinant of $M_n$.","subjects":["linear algebra","determinants","eigenvalues"]},{"id":"1984-A4","year":1984,"session":"A","number":4,"statement":"Problem A-4\nA convex pentagon $ P=A B C D E $, with vertices labeled consecutively, is inscribed in a circle of radius 1 . Find the maximum area of $ P $ subject to the condition that the chords $ A C $ and $ B D $ be perpendicular.","subjects":["geometry","cyclic polygons","area optimization"]},{"id":"1984-A5","year":1984,"session":"A","number":5,"statement":"Problem A-5\nLet $ R $ be the region consisting of all triples ( $ x, y, z $ ) of nonnegative real numbers satisfying $ x+y+z \\leqslant 1 $. Let $ w=1-x-y-z $. Express the value of the triple integral\n$$\n\\iiint_{R} x^{1} y^{9} z^{8} w^{4} d x d y d z\n$$\nin the form $ a!b!c!d!/ n! $, where $ a, b, c, d $, and $ n $ are positive integers.","subjects":["analysis","multiple integrals","simplex integrals"]},{"id":"1984-A6","year":1984,"session":"A","number":6,"statement":"Problem A-6. Let $ n $ be a positive integer, and let $ f(n) $ denote the last nonzero digit in the decimal expansion of $ n! $. For instance, $ f(5)=2 $.\n(a) Show that if $ a_{1}, a_{2}, \\ldots, a_{k} $ are distinct nonnegative integers, then $ f\\left(5^{a_{1}}+5^{a_{2}}+\\cdots+5^{a_{k}}\\right) $ depends only on the sum $ a_{1}+a_{2}+\\cdots+a_{k} $.\n(b) Assuming part (a), we can define\n$$\ng(s)=f\\left(5^{a_{1}}+5^{a_{2}}+\\cdots+5^{a_{k}}\\right)\n$$\nwhere $ s=a_{1}+a_{2}+\\cdots+a_{k} $. Find the least positive integer $ p $ for which\n$$\ng(s)=g(s+p), \\quad \\text { for all } s \\geqslant 1\n$$\nor else show that no such $ p $ exists.","subjects":["number theory","factorials","last nonzero digits"]},{"id":"1984-B1","year":1984,"session":"B","number":1,"statement":"Problem B-1\nLet $ n $ be a positive integer, and define\n$$\nf(n)=1!+2!+\\cdots+n!\n$$\n\nFind polynomials $ P(x) $ and $ Q(x) $ such that\n$$\nf(n+2)=P(n) f(n+1)+Q(n) f(n)\n$$\nfor all $ n \\geqslant 1 $.","subjects":["algebra","factorial sums","recurrences"]},{"id":"1984-B2","year":1984,"session":"B","number":2,"statement":"Problem B-2. Find the minimum value of\n$$\n(u-v)^{2}+\\left(\\sqrt{2-u^{2}}-\\frac{9}{v}\\right)^{2}\n$$\nfor $ 0<u<\\sqrt{2} $ and $ v>0 $.","subjects":["inequalities","euclidean distance","optimization"]},{"id":"1984-B3","year":1984,"session":"B","number":3,"statement":"Problem B-3\nProve or disprove the following statement: If $ F $ is a finite set with two or more elements, then there exists a binary operation * on $ F $ such that for all $ x, y, z $ in $ F $.\n(i) $ x * z=y * z $ implies $ x=y $ (right cancellation holds). and\n(ii) $ x *(y * z) \\neq(x * y) * z $ (no case of associativity holds).","subjects":["algebra","binary operations","nonassociativity"]},{"id":"1984-B4","year":1984,"session":"B","number":4,"statement":"Problem B-4\nFind, with proof, all real-valued functions $ y=g(x) $ defined and continuous on $ [0, \\infty) $, positive on $ (0, \\infty) $, such that for all $ x>0 $ the $ y $-coordinate of the centroid of the region\n$$\nR_{\\mathrm{r}}=\\{(s, t) \\mid 0 \\leqslant s \\leqslant x, \\quad 0 \\leqslant t \\leqslant g(s)\\}\n$$\nis the same as the average value of $ g $ on $ [0, x] $.","subjects":["analysis","centroids","functional equations"]},{"id":"1984-B5","year":1984,"session":"B","number":5,"statement":"Problem B-5\nFor each nonnegative integer $ k $, let $ d(k) $ denote the number of 1 's in the binary expansion of $ k $ (for example. $ d(0)=0 $ and $ d(5)=2 $ ). Let $ m $ be a positive integer. Express\n$$\n\\sum_{k=0}^{2^{m}-1}(-1)^{d(k)} k^{m}\n$$\nin the form $ (-1)^{m} a^{f(m)}(g(m))! $, where $ a $ is an integer and $ f $ and $ g $ are polynomials.","subjects":["combinatorics","binary digits","signed power sums"]},{"id":"1984-B6","year":1984,"session":"B","number":6,"statement":"Problem B-6\nA sequence of convex polygons $ \\left\\{P_{n}\\right\\}, n \\geqslant 0 $, is defined inductively as follows. $ P_{0} $ is an equilateral triangle with sides of length 1 . Once $ P_{n} $ has been determined, its sides are trisected; the vertices of $ P_{n+1} $ are the interior trisection points of the sides of $ P_{n} $. Thus, $ P_{n+1} $ is obtained by cutting corners off $ P_{n} $, and $ P_{n} $ has $ 3 \\cdot 2^{n} $ sides. ( $ P_{1} $ is a regular hexagon with sides of length $ 1 / 3 $.)\n\nExpress $ \\lim _{n \\rightarrow \\infty} \\operatorname{Area}\\left(P_{n}\\right) $ in the form $ \\sqrt{a} / b $, where $ a $ and $ b $ are positive integers.","subjects":["geometry","polygon iteration","area limits"]},{"id":"1985-A1","year":1985,"session":"A","number":1,"statement":"Determine, with proof, the number of ordered triples $(A_1, A_2, A_3)$\nof sets which have the property that\n\n(i) $A_1 \\cup A_2 \\cup A_3 = \\{1,2,3,4,5,6,7,8,9,10\\}$, and\n\n(ii) $A_1 \\cap A_2 \\cap A_3 = \\emptyset$.\n\nExpress your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,c,d$\nare nonnegative integers.","subjects":["combinatorics","sets","counting"]},{"id":"1985-A2","year":1985,"session":"A","number":2,"statement":"Let $T$ be an acute triangle. Inscribe a rectangle $R$ in $T$ with one\nside along a side of $T$. Then inscribe a rectangle $S$ in the triangle\nformed by the side of $R$ opposite the side on the boundary of $T$,\nand the other two sides of $T$, with one side along the side of\n$R$. For any polygon $X$, let $A(X)$ denote the area of $X$. Find the\nmaximum value, or show that no maximum exists, of\n$\\frac{A(R)+A(S)}{A(T)}$, where $T$ ranges over all triangles and\n$R,S$ over all rectangles as above.","subjects":["geometry","inscribed rectangles","area optimization"]},{"id":"1985-A3","year":1985,"session":"A","number":3,"statement":"Let $d$ be a real number. For each integer $m \\geq 0$, define a\nsequence $\\{a_m(j)\\}$, $j=0,1,2,\\dots$ by the condition\n$$\n\\begin{aligned}\na_m(0) &= d/2^m, \\\\\na_m(j+1) &= (a_m(j))^2 + 2a_m(j), \\qquad j \\geq 0.\n\\end{aligned}\n$$\nEvaluate $\\lim_{n \\to \\infty} a_n(n)$.","subjects":["analysis","nonlinear recurrence","exponential limits"]},{"id":"1985-A4","year":1985,"session":"A","number":4,"statement":"Define a sequence $\\{a_i\\}$ by $a_1=3$ and $a_{i+1}=3^{a_i}$ for\n$i\\geq 1$. Which integers between 00 and 99 inclusive occur as the\nlast two digits in the decimal expansion of infinitely many $a_i$?","subjects":["number theory","power towers","modular arithmetic"]},{"id":"1985-A5","year":1985,"session":"A","number":5,"statement":"Let $I_m = \\int_0^{2\\pi} \\cos(x)\\cos(2x)\\cdots \\cos(mx)\\,dx$. For\nwhich integers $m$, $1 \\leq m \\leq 10$ is $I_m \\neq 0$?","subjects":["analysis","trigonometric integrals","fourier coefficients"]},{"id":"1985-A6","year":1985,"session":"A","number":6,"statement":"If $p(x)= a_0 + a_1 x + \\cdots + a_m x^m$ is a polynomial with real\ncoefficients $a_i$, then set\n$$\n\\Gamma(p(x)) = a_0^2 + a_1^2 + \\cdots + a_m^2.\n$$\nLet $F(x) = 3x^2+7x+2$. Find, with proof, a polynomial $g(x)$ with\nreal coefficients such that\n\n(i) $g(0)=1$, and\n\n(ii) $\\Gamma(f(x)^n) = \\Gamma(g(x)^n)$\n\nfor every integer $n \\geq 1$.","subjects":["algebra","coefficient norms","laurent polynomials"],"errata":"In condition (ii), read the undefined lowercase $f$ as the polynomial $F(x)=3x^2+7x+2$ defined just above it. The supplied book uses a consistent lowercase name, and Scholes gives an unambiguous equivalent formulation. The archived prompt is preserved, with this notation correction appended."},{"id":"1985-B1","year":1985,"session":"B","number":1,"statement":"Let $k$ be the smallest positive integer for which there exist\ndistinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial\n$$\np(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5)\n$$\nhas exactly $k$ nonzero coefficients. Find, with proof, a set of\nintegers $m_1, m_2, m_3, m_4, m_5$ for which this minimum $k$ is achieved.","subjects":["algebra","sparse polynomials","real roots"]},{"id":"1985-B2","year":1985,"session":"B","number":2,"statement":"Define polynomials $f_n(x)$ for $n \\geq 0$ by $f_0(x)=1$, $f_n(0)=0$\nfor $n \\geq 1$, and\n$$\n\\frac{d}{dx} f_{n+1}(x) = (n+1)f_n(x+1)\n$$\nfor $n \\geq 0$. Find, with proof, the explicit factorization of\n$f_{100}(1)$ into powers of distinct primes.","subjects":["algebra","polynomial recurrences","prime factorization"]},{"id":"1985-B3","year":1985,"session":"B","number":3,"statement":"Let\n$$\n\\begin{array}{cccc} a_{1,1} & a_{1,2} & a_{1,3} & \\dots \\\\\na_{2,1} & a_{2,2} & a_{2,3} & \\dots \\\\\na_{3,1} & a_{3,2} & a_{3,3} & \\dots \\\\\n\\vdots & \\vdots & \\vdots & \\ddots\n\\end{array}\n$$\nbe a doubly infinite array of positive integers, and suppose each\npositive integer appears exactly eight times in the array. Prove that\n$a_{m,n} > mn$ for some pair of positive integers $(m,n)$.","subjects":["combinatorics","infinite arrays","counting contradiction"]},{"id":"1985-B4","year":1985,"session":"B","number":4,"statement":"Let $C$ be the unit circle $x^2+y^2=1$. A point $p$ is chosen randomly\non the circumference $C$ and another point $q$ is chosen randomly from\nthe interior of $C$ (these points are chosen independently and\nuniformly over their domains). Let $R$ be the rectangle with sides\nparallel to the $x$ and $y$-axes with diagonal $pq$. What is the\nprobability that no point of $R$ lies outside of $C$?","subjects":["probability","geometric probability","random rectangles"]},{"id":"1985-B5","year":1985,"session":"B","number":5,"statement":"Evaluate $\\int_0^\\infty t^{-1/2}e^{-1985(t+t^{-1})}\\,dt$. You may\nassume that $\\int_{-\\infty}^\\infty e^{-x^2}\\,dx = \\sqrt{\\pi}$.","subjects":["analysis","gaussian integrals","improper integrals"]},{"id":"1985-B6","year":1985,"session":"B","number":6,"statement":"Let $G$ be a finite set of real $n\\times n$ matrices $\\{M_i\\}$, $1\n\\leq i \\leq r$, which form a group under matrix\nmultiplication. Suppose that $\\sum_{i=1}^r \\mathrm{tr}(M_i)=0$, where\n$\\mathrm{tr}(A)$\ndenotes the trace of the matrix $A$. Prove that $\\sum_{i=1}^r M_i$ is\nthe $n \\times n$ zero matrix.","subjects":["linear algebra","finite matrix groups","trace"]},{"id":"1986-A1","year":1986,"session":"A","number":1,"statement":"Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the\nset of all real numbers $x$ satisfying $x^4+36\\leq 13x^2$.","subjects":["algebra","calculus","constrained optimization"]},{"id":"1986-A2","year":1986,"session":"A","number":2,"statement":"What is the units (i.e., rightmost) digit of\n$$\n\\left\\lfloor \\frac{10^{20000}}{10^{100}+3}\\right\\rfloor ?\n$$","subjects":["number theory","decimal digits","euclidean division"]},{"id":"1986-A3","year":1986,"session":"A","number":3,"statement":"Evaluate $\\sum_{n=0}^\\infty \\mathrm{Arccot}(n^2+n+1)$, where\n$\\mathrm{Arccot}\\,t$ for $t \\geq 0$ denotes the number $\\theta$ in the\ninterval $0 < \\theta \\leq \\pi/2$ with $\\cot \\theta = t$.","subjects":["analysis","inverse trigonometric series"]},{"id":"1986-A4","year":1986,"session":"A","number":4,"statement":"A transversal of an $n\\times n$ matrix $A$ consists of $n$\nentries of $A$, no two in the same row or column. Let $f(n)$ be the\nnumber of $n \\times n$ matrices $A$ satisfying the following two\nconditions:\n\n(a) Each entry $\\alpha_{i,j}$ of $A$ is in the set\n$\\{-1,0,1\\}$.\n\n(b) The sum of the $n$ entries of a transversal is the same for\nall transversals of $A$.\n\nAn example of such a matrix $A$ is\n$$\nA = \\left( \\begin{array}{ccc} -1 & 0 & -1 \\\\ 0 & 1 & 0 \\\\ 0 & 1 & 0\n\\end{array}\n\\right).\n$$\nDetermine with proof a formula for $f(n)$ of the form\n$$\nf(n) = a_1 b_1^n + a_2 b_2^n + a_3 b_3^n + a_4,\n$$\nwhere the $a_i$'s and $b_i$'s are rational numbers.","subjects":["combinatorics","matrices","transversal sums","recurrences"]},{"id":"1986-A5","year":1986,"session":"A","number":5,"statement":"Suppose $f_1(x), f_2(x), \\dots, f_n(x)$ are functions of $n$ real\nvariables $x = (x_1, \\dots, x_n)$ with continuous second-order partial\nderivatives everywhere on $\\mathbb{R}^n$. Suppose further that there are\nconstants $c_{ij}$ such that\n$$\n\\frac{\\partial f_i}{\\partial x_j} - \\frac{\\partial f_j}{\\partial x_i}\n= c_{ij}\n$$\nfor all $i$ and $j$, $1\\leq i \\leq n$, $1 \\leq j \\leq n$. Prove that\nthere is a function $g(x)$ on $\\mathbb{R}^n$ such that $f_i + \\partial\ng/\\partial x_i$ is linear for all $i$, $1 \\leq i \\leq n$. (A linear\nfunction is one of the form\n$$\na_0 + a_1 x_1 + a_2 x_2 + \\cdots + a_n x_n.)\n$$","subjects":["analysis","multivariable calculus","potential functions"]},{"id":"1986-A6","year":1986,"session":"A","number":6,"statement":"Let $a_1, a_2, \\dots, a_n$ be real numbers, and let $b_1, b_2, \\dots,\nb_n$ be distinct positive integers. Suppose that there is a polynomial\n$f(x)$ satisfying the identity\n$$\n(1-x)^n f(x) = 1 + \\sum_{i=1}^n a_i x^{b_i}.\n$$\nFind a simple expression (not involving any sums) for $f(1)$ in terms\nof $b_1, b_2, \\dots, b_n$ and $n$ (but independent of $a_1, a_2,\n\\dots, a_n$).","subjects":["algebra","polynomial divisibility","differential operators"]},{"id":"1986-B1","year":1986,"session":"B","number":1,"statement":"Inscribe a rectangle of base $b$ and height $h$ in a circle of radius\none, and inscribe an isosceles triangle in the region of the circle\ncut off by one base of the rectangle (with that side as the base of\nthe triangle).\nFor what\nvalue of $h$ do the rectangle and triangle have the same area?","subjects":["geometry","inscribed figures","area equality"]},{"id":"1986-B2","year":1986,"session":"B","number":2,"statement":"Prove that there are only a finite number of possibilities for the\nordered triple $T=(x-y,y-z,z-x)$, where $x,y,z$ are complex numbers\nsatisfying the simultaneous equations\n$$\nx(x-1)+2yz = y(y-1)+2zx = z(z-1)+2xy,\n$$\nand list all such triples $T$.","subjects":["algebra","complex polynomial systems","finite classification"]},{"id":"1986-B3","year":1986,"session":"B","number":3,"statement":"Let $\\Gamma$ consist of all polynomials in $x$ with integer\ncoefficients. For $f$ and $g$ in $\\Gamma$ and $m$ a positive integer,\nlet $f \\equiv g \\pmod{m}$ mean that every coefficient of $f-g$ is an\nintegral multiple of $m$. Let $n$ and $p$ be positive integers with\n$p$ prime. Given that $f,g,h,r$ and $s$ are in $\\Gamma$ with\n$rf+sg\\equiv 1 \\pmod{p}$ and $fg \\equiv h \\pmod{p}$, prove that there\nexist $F$ and $G$ in $\\Gamma$ with $F \\equiv f \\pmod{p}$, $G \\equiv g\n\\pmod{p}$, and $FG \\equiv h \\pmod{p^n}$.","subjects":["algebra","polynomial congruences","hensel lifting"]},{"id":"1986-B4","year":1986,"session":"B","number":4,"statement":"For a positive real number $r$, let $G(r)$ be the minimum value of $|r\n- \\sqrt{m^2+2n^2}|$ for all integers $m$ and $n$. Prove or disprove\nthe assertion that $\\lim_{r\\to \\infty}G(r)$ exists and equals 0.","subjects":["number theory","quadratic forms","approximation"]},{"id":"1986-B5","year":1986,"session":"B","number":5,"statement":"Let $f(x,y,z) = x^2+y^2+z^2+xyz$. Let $p(x,y,z), q(x,y,z)$, $r(x,y,z)$\nbe polynomials with real coefficients satisfying\n$$\nf(p(x,y,z), q(x,y,z), r(x,y,z)) = f(x,y,z).\n$$\nProve or disprove the assertion that the sequence $p,q,r$ consists of\nsome permutation of $\\pm x, \\pm y, \\pm z$, where the number of minus\nsigns is 0 or 2.","subjects":["algebra","polynomial invariants","counterexamples"]},{"id":"1986-B6","year":1986,"session":"B","number":6,"statement":"Suppose $A,B,C,D$ are $n \\times n$ matrices with entries in a field\n$F$, satisfying the conditions that $AB^T$ and $CD^T$ are symmetric and\n$AD^T - BC^T = I$. Here $I$ is the $n \\times n$ identity matrix, and\nif $M$ is an $n \\times n$ matrix, $M^T$ is its transpose. Prove that\n$A^T D - C^T B = I$.","subjects":["linear algebra","block matrices","symplectic identities"]},{"id":"1987-A1","year":1987,"session":"A","number":1,"statement":"Curves $A,B,C$ and $D$ are defined in the plane as follows:\n$$\n\\begin{aligned}\nA &= \\left\\{ (x,y): x^2-y^2 = \\frac{x}{x^2+y^2} \\right\\}, \\\\\nB &= \\left\\{ (x,y): 2xy + \\frac{y}{x^2+y^2} = 3 \\right\\}, \\\\\nC &= \\left\\{ (x,y): x^3-3xy^2+3y=1 \\right\\}, \\\\\nD &= \\left\\{ (x,y): 3x^2 y - 3x - y^3 = 0\\right\\}.\n\\end{aligned}\n$$\nProve that $A \\cap B = C \\cap D$.","subjects":["algebra","complex numbers","plane curves"]},{"id":"1987-A2","year":1987,"session":"A","number":2,"statement":"The sequence of digits\n$$\n1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 \\dots\n$$\nis obtained by writing the positive integers in order. If the\n$10^n$-th digit in this sequence occurs in the part of the sequence in\nwhich the $m$-digit numbers are placed, define $f(n)$ to be $m$. For\nexample, $f(2)=2$ because the 100th digit enters the sequence in the\nplacement of the two-digit integer 55. Find, with proof, $f(1987)$.","subjects":["combinatorics","digit counting","decimal concatenation"]},{"id":"1987-A3","year":1987,"session":"A","number":3,"statement":"For all real $x$, the real-valued function $y=f(x)$ satisfies\n$$\ny''-2y'+y=2e^x.\n$$\n\n(a) If $f(x)>0$ for all real $x$, must $f'(x) > 0$ for all real\n$x$? Explain.\n\n(b) If $f'(x)>0$ for all real $x$, must $f(x) > 0$ for all real\n$x$? Explain.","subjects":["analysis","ordinary differential equations","positivity"]},{"id":"1987-A4","year":1987,"session":"A","number":4,"statement":"Let $P$ be a polynomial, with real coefficients, in three variables\nand $F$ be a function of two variables such that\n$$\nP(ux, uy, uz) = u^2 F(y-x,z-x) \\quad \\text{for all real }x,y,z,u,\n$$\nand such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let\n$A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$. Find $|C-A|$.","subjects":["algebra","homogeneous polynomials","complex quadratic roots"]},{"id":"1987-A5","year":1987,"session":"A","number":5,"statement":"Let\n$$\n\\vec{G}(x,y) = \\left( \\frac{-y}{x^2+4y^2}, \\frac{x}{x^2+4y^2},0\n\\right).\n$$\nProve or disprove that there is a vector-valued function\n$$\n\\vec{F}(x,y,z) = (M(x,y,z), N(x,y,z), P(x,y,z))\n$$\nwith the following properties:\n\n(i) $M,N,P$ have continuous partial derivatives for all\n$(x,y,z) \\neq (0,0,0)$;\n\n(ii) $\\mathrm{Curl}\\,\\vec{F} = \\vec{0}$ for all $(x,y,z) \\neq (0,0,0)$;\n\n(iii) $\\vec{F}(x,y,0) = \\vec{G}(x,y)$.","subjects":["analysis","vector calculus","topological obstruction"]},{"id":"1987-A6","year":1987,"session":"A","number":6,"statement":"For each positive integer $n$, let $a(n)$ be the number of zeroes in\nthe base 3 representation of $n$. For which positive real numbers $x$\ndoes the series\n$$\n\\sum_{n=1}^\\infty \\frac{x^{a(n)}}{n^3}\n$$\nconverge?","subjects":["analysis","digit-weighted series","base-three expansions"]},{"id":"1987-B1","year":1987,"session":"B","number":1,"statement":"Evaluate\n$$\n\\int_2^4 \\frac{\\sqrt{\\ln(9-x)}\\,dx}{\\sqrt{\\ln(9-x)}+\\sqrt{\\ln(x+3)}}.\n$$","subjects":["analysis","definite integrals","symmetry"]},{"id":"1987-B2","year":1987,"session":"B","number":2,"statement":"Let $r,s$ and $t$ be integers with $0 \\leq r$, $0 \\leq s$ and $r+s\n\\leq t$. Prove that\n$$\n\\frac{\\binom s0}{\\binom tr}\n+ \\frac{\\binom s1}{\\binom{t}{r+1}} + \\cdots\n+ \\frac{\\binom ss}{\\binom{t}{r+s}}\n= \\frac{t+1}{(t+1-s)\\binom{t-s}{r}}.\n$$","subjects":["combinatorics","binomial identities","reciprocal binomial sums"]},{"id":"1987-B3","year":1987,"session":"B","number":3,"statement":"Let $F$ be a field in which $1+1 \\neq 0$. Show that the set of\nsolutions to the equation $x^2+y^2=1$ with $x$ and $y$ in $F$ is given\nby $(x,y)=(1,0)$ and\n$$\n(x,y) = \\left( \\frac{r^2-1}{r^2+1}, \\frac{2r}{r^2+1} \\right)\n$$\nwhere $r$ runs through the elements of $F$ such that $r^2\\neq -1$.","subjects":["algebra","fields","rational parametrization"]},{"id":"1987-B4","year":1987,"session":"B","number":4,"statement":"Let $(x_1,y_1) = (0.8, 0.6)$ and let $x_{n+1} = x_n \\cos y_n - y_n\n\\sin y_n$ and $y_{n+1}= x_n \\sin y_n + y_n \\cos y_n$ for\n$n=1,2,3,\\dots$. For each of $\\lim_{n\\to \\infty} x_n$ and $\\lim_{n \\to\n\\infty} y_n$, prove that the limit exists and find it or prove that\nthe limit does not exist.","subjects":["analysis","recursive sequences","dynamical systems"]},{"id":"1987-B5","year":1987,"session":"B","number":5,"statement":"Let $O_n$ be the $n$-dimensional vector $(0,0,\\cdots, 0)$. Let $M$ be\na $2n \\times n$ matrix of complex numbers such that whenever $(z_1,\nz_2, \\dots, z_{2n})M = O_n$, with complex $z_i$, not all zero, then at\nleast one of the $z_i$ is not real. Prove that for arbitrary real\nnumbers $r_1, r_2, \\dots, r_{2n}$, there are complex numbers $w_1,\nw_2, \\dots, w_n$ such that\n$$\n\\mathrm{re}\\left[ M \\left( \\begin{array}{c} w_1 \\\\ \\vdots \\\\ w_n \\end{array}\n\\right) \\right] = \\left( \\begin{array}{c} r_1 \\\\ \\vdots \\\\ r_n\n\\end{array} \\right).\n$$\n(Note: if $C$ is a matrix of complex numbers, $\\mathrm{re}(C)$ is the matrix\nwhose entries are the real parts of the entries of $C$.)","subjects":["linear algebra","complex matrices","realification"],"errata":"The final entry of the right-hand column is $r_{2n}$, not $r_n$. The product on the left has $2n$ rows, and the quantified real data are $r_1,\\ldots,r_{2n}$. The supplied book prints the dimensionally consistent column. Scholes also states the target as a real $2n$-component vector."},{"id":"1987-B6","year":1987,"session":"B","number":6,"statement":"Let $F$ be the field of $p^2$ elements, where $p$ is an odd\nprime. Suppose $S$ is a set of $(p^2-1)/2$ distinct nonzero elements\nof $F$ with the property that for each $a\\neq 0$ in $F$, exactly one\nof $a$ and $-a$ is in $S$. Let $N$ be the number of elements in the\nintersection $S \\cap \\{2a: a \\in S\\}$. Prove that $N$ is even.","subjects":["abstract algebra"]},{"id":"1988-A1","year":1988,"session":"A","number":1,"statement":"Let $R$ be the region consisting of the points $(x,y)$ of the\ncartesian plane satisfying both $|x|-|y| \\leq 1$ and $|y| \\leq 1$.\nSketch the region $R$ and find its area.","subjects":["geometry"]},{"id":"1988-A2","year":1988,"session":"A","number":2,"statement":"A not uncommon calculus mistake is to believe that the product rule\nfor derivatives says that $(fg)' = f'g'$. If $f(x)=e^{x^2}$,\ndetermine, with proof, whether there exists an open interval $(a,b)$\nand a nonzero function $g$ defined on $(a,b)$ such that this wrong\nproduct rule is true for $x$ in $(a,b)$.","subjects":["analysis","product rule","ordinary differential equations"]},{"id":"1988-A3","year":1988,"session":"A","number":3,"statement":"Determine, with proof, the set of real numbers $x$ for which\n$$\n\\sum_{n=1}^\\infty \\left( \\frac{1}{n} \\csc \\frac{1}{n} - 1 \\right)^x\n$$\nconverges.","subjects":["analysis","series convergence","taylor asymptotics"]},{"id":"1988-A4","year":1988,"session":"A","number":4,"statement":"(a) If every point of the plane is painted one of three colors,\ndo there necessarily exist two points of the same color exactly one\ninch apart?\n\n(b) What if ``three'' is replaced by ``nine''?","subjects":["combinatorics","plane coloring","unit-distance graphs"]},{"id":"1988-A5","year":1988,"session":"A","number":5,"statement":"Prove that there exists a unique function $f$ from the set\n$\\mathrm{R}^+$ of positive real numbers to $\\mathrm{R}^+$ such that\n$$\nf(f(x)) = 6x-f(x)\n$$\nand\n$$\nf(x)>0\n$$\nfor all $x>0$.","subjects":["analysis"]},{"id":"1988-A6","year":1988,"session":"A","number":6,"statement":"If a linear transformation $A$ on an $n$-dimensional vector space has\n$n+1$ eigenvectors such that any $n$ of them are linearly independent,\ndoes it follow that $A$ is a scalar multiple of the identity? Prove\nyour answer.","subjects":["linear algebra","eigenvectors","scalar operators"]},{"id":"1988-B1","year":1988,"session":"B","number":1,"statement":"A composite (positive integer) is a product $ab$ with $a$ and\n$b$ not necessarily distinct integers in $\\{2,3,4,\\dots\\}$. Show that\nevery composite is expressible as $xy+xz+yz+1$, with $x,y,z$ positive\nintegers.","subjects":["number theory","composite integers","factorization"]},{"id":"1988-B2","year":1988,"session":"B","number":2,"statement":"Prove or disprove: If $x$ and $y$ are real numbers with $y\\geq0$ and\n$y(y+1) \\leq (x+1)^2$, then $y(y-1)\\leq x^2$.","subjects":["algebra","inequalities","real numbers"]},{"id":"1988-B3","year":1988,"session":"B","number":3,"statement":"For every $n$ in the set $\\mathrm{N} = \\{1,2,\\dots \\}$ of positive integers,\nlet $r_n$ be the minimum value of $|c-d\\sqrt{3}|$ for all nonnegative\nintegers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest\npositive real number $g$ with $r_n \\leq g$ for all $n \\in \\mathrm{N}$.","subjects":["number theory","diophantine approximation","irrational rotations","pell equations"]},{"id":"1988-B4","year":1988,"session":"B","number":4,"statement":"Prove that if $\\sum_{n=1}^\\infty a_n$ is a convergent series of\npositive real numbers, then so is $\\sum_{n=1}^\\infty (a_n)^{n/(n+1)}$.","subjects":["analysis","infinite series","convergence"]},{"id":"1988-B5","year":1988,"session":"B","number":5,"statement":"For positive integers $n$, let $M_n$ be the $2n+1$ by $2n+1$\nskew-symmetric matrix for which each entry in the first $n$\nsubdiagonals below the main diagonal is 1 and each of the remaining\nentries below the main diagonal is -1. Find, with proof, the rank of\n$M_n$. (According to one definition, the rank of a matrix is the\nlargest $k$ such that there is a $k \\times k$ submatrix with nonzero\ndeterminant.)\n\nOne may note that\n$$\n\\begin{aligned}\nM_1 &= \\left( \\begin{array}{ccc} 0 & -1 & 1 \\\\ 1 & 0 & -1 \\\\ -1 & 1 & 0\n\\end{array}\\right) \\\\\nM_2 &= \\left( \\begin{array}{ccccc} 0 & -1 & -1 & 1\n& 1 \\\\ 1 & 0 & -1 & -1 & 1 \\\\ 1 & 1 & 0 & -1 & -1 \\\\ -1 & 1 & 1 & 0 &\n-1 \\\\ -1 & -1 & 1 & 1 & 0 \\end{array} \\right).\n\\end{aligned}\n$$","subjects":["linear algebra","matrix rank","skew-symmetric matrices","circulant matrices"]},{"id":"1988-B6","year":1988,"session":"B","number":6,"statement":"Prove that there exist an infinite number of ordered pairs $(a,b)$ of\nintegers such that for every positive integer $t$, the number $at+b$\nis a triangular number if and only if $t$ is a triangular number. (The\ntriangular numbers are the $t_n = n(n+1)/2$ with $n$ in $\\{0,1,2,\\dots\\}$.)","subjects":["number theory","triangular numbers","perfect squares"]},{"id":"1989-A1","year":1989,"session":"A","number":1,"statement":"How many primes among the positive integers, written as usual in base 10,\nare alternating 1's and 0's, beginning and ending with 1?","subjects":["number theory","decimal expansions","prime numbers","factorization"]},{"id":"1989-A2","year":1989,"session":"A","number":2,"statement":"Evaluate\n$\\displaystyle{\\int_0^a\\int_0^b e^{{\\rm max}\\{b^2x^2, a^2y^2\\}}\\,dy\\,dx}$\nwhere $a$ and $b$ are positive.","subjects":["analysis","double integrals","probability","order statistics"]},{"id":"1989-A3","year":1989,"session":"A","number":3,"statement":"Prove that if\n$$\n11z^{10}+10iz^9+10iz-11=0,\n$$\nthen $|z|=1.$ (Here $z$ is a complex number and $i^2=-1$.)","subjects":["complex numbers","polynomial roots","unit circle"]},{"id":"1989-A4","year":1989,"session":"A","number":4,"statement":"If $\\alpha$ is an irrational number, $0 < \\alpha < 1$, is there a\nfinite game with an honest coin such that the probability of one player\nwinning the game is $\\alpha$? (An honest coin is one for which the\nprobability of heads and the probability of tails are both $\\frac12$.\nA game is finite if with probability 1 it must end in a finite number of moves.)","subjects":["probability","fair coins","binary expansions","stopping times"]},{"id":"1989-A5","year":1989,"session":"A","number":5,"statement":"Let $m$ be a positive integer and let $\\mathcal{G}$ be a regular $(2m+1)$-gon\ninscribed in the unit circle. Show that there is a positive constant $A$,\nindependent of $m$, with the following property. For any points $p$ inside\n$\\cal G$ there are two distinct vertices $v_1$ and $v_2$ of $\\cal G$\nsuch that\n$$\n\\left|\\,|p-v_1| - |p-v_2|\\,\\right| < \\frac1{m} - \\frac{A}{m^3}.\n$$\nHere $|s-t|$ denotes the distance between the points $s$ and $t$.","subjects":["geometry","regular polygons","distance inequalities"]},{"id":"1989-A6","year":1989,"session":"A","number":6,"statement":"Let $\\alpha=1+a_1x+a_2x^2+\\cdots$ be a formal power series with coefficients\nin the field of two elements. Let\n$$\na_n =\n\\begin{cases}\n1 & \\text{if every block of zeros in the binary expansion of }n\\text{\nhas an even number of zeros in the block} \\\\[.3in]\n0 & \\text{otherwise.}\n\\end{cases}\n$$\n(For example, $a_{36}=1$ because $36=100100_2$ and $a_{20}=0$ because\n$20=10100_2.$)\nProve that $\\alpha^3+x\\alpha+1=0.$","subjects":["combinatorics","formal power series","binary expansions","finite fields"]},{"id":"1989-B1","year":1989,"session":"B","number":1,"statement":"A dart, thrown at random, hits a square target. Assuming that any two\nparts of the target of equal area are equally likely to be hit, find\nthe probability that the point hit is nearer to the center than to any\nedge. Express your answer in the form $\\displaystyle{\\frac{a\\sqrt{b} + c}{d}}$,\nwhere $a,\\,b,\\,c,\\,d$ are integers.","subjects":["geometry","probability","areas","parabolas"]},{"id":"1989-B2","year":1989,"session":"B","number":2,"statement":"Let $S$ be a non-empty set with an associative operation that is left and\nright cancellative ($xy=xz$ implies $y=z$, and $yx=zx$ implies $y=z$).\nAssume that for every $a$ in $S$ the set $\\{a^n:\\,n=1, 2, 3, \\ldots\\}$ is\nfinite. Must $S$ be a group?","subjects":["algebra","semigroups","groups","cancellation"]},{"id":"1989-B3","year":1989,"session":"B","number":3,"statement":"Let $f$ be a function on $[0,\\infty)$, differentiable and satisfying\n$$\nf'(x)=-3f(x)+6f(2x)\n$$\nfor $x>0$. Assume that $|f(x)|\\le e^{-\\sqrt{x}}$ for $x\\ge 0$ (so that\n$f(x)$ tends rapidly to $0$ as $x$ increases).\nFor $n$ a non-negative integer, define\n$$\n\\mu_n=\\int_0^\\infty x^n f(x)\\,dx\n$$\n(sometimes called the $n$th moment of $f$).\n\na) Express $\\mu_n$ in terms of $\\mu_0$.\n\nb) Prove that the sequence $\\{\\mu_n \\frac{3^n}{n!}\\}$ always converges,\nand that the limit is $0$ only if $\\mu_0=0$.","subjects":["analysis","moments","functional differential equations","infinite products"]},{"id":"1989-B4","year":1989,"session":"B","number":4,"statement":"Can a countably infinite set have an uncountable collection of\nnon-empty subsets such that the intersection of any two of them is\nfinite?","subjects":["set theory","countability","almost disjoint families"]},{"id":"1989-B5","year":1989,"session":"B","number":5,"statement":"Label the vertices of a trapezoid $T$ (quadrilateral with two parallel sides)\ninscribed in the unit circle as $A,\\,B,\\,C,\\,D$ so that $AB$ is parallel to\n$CD$ and $A,\\,B,\\,C,\\,D$ are in counterclockwise order. Let\n$s_1,\\,s_2$, and $d$ denote the lengths of the line segments\n$AB,\\, CD$, and $OE$, where E  is the point of intersection of the diagonals\nof $T$, and $O$ is the center of the circle. Determine the least upper bound of\n$\\frac{s_1-s_2}{d}$ over all such $T$ for which $d\\ne 0$, and describe all\ncases, if any, in which it is attained.","subjects":["geometry","cyclic trapezoids","extremal geometry","circle chords"]},{"id":"1989-B6","year":1989,"session":"B","number":6,"statement":"Let $(x_1,\\,x_2,\\,\\ldots\\,x_n)$ be a point chosen at random from the\n$n$-dimensional region defined by\n$0<x_1<x_2<\\cdots < x_n<1.$  Let $f$ be a continuous function on\n$[0,1]$ with $f(1)=0$.\nSet $x_0=0$ and $x_{n+1}=1$. Show that the expected value of the\nRiemann sum\n$$\n\\sum_{i=0}^n (x_{i+1}-x_i) f(x_{i+1})\n$$\nis $\\int_0^1 f(t)P(t)\\, dt$, where $P$ is a polynomial of degree $n$,\nindependent of $f$, with $0\\le P(t)\\le 1$ for $0\\le t \\le 1$.","subjects":["probability","order statistics","riemann sums","polynomials"]},{"id":"1990-A1","year":1990,"session":"A","number":1,"statement":"Let\n$$\nT_0 = 2, T_1 = 3, T_2 = 6,\n$$\nand for $n \\geq 3$,\n$$\nT_n = (n+4)T_{n-1} - 4n T_{n-2} + (4n-8) T_{n-3}.\n$$\nThe first few terms are\n$$\n2, 3, 6, 14, 40, 152, 784, 5168, 40576.\n$$\nFind, with proof, a formula for $T_n$ of the form $T_n = A_n + B_n$,\nwhere $\\{A_n\\}$ and $\\{B_n\\}$ are well-known sequences.","subjects":["algebra","recurrences","factorials","powers of two"]},{"id":"1990-A2","year":1990,"session":"A","number":2,"statement":"Is $\\sqrt{2}$ the limit of a sequence of numbers of the form\n$\\sqrt[3]{n} - \\sqrt[3]{m}$ ($n,m = 0, 1, 2, \\dots$)?","subjects":["analysis","sequences","cube roots","approximation"]},{"id":"1990-A3","year":1990,"session":"A","number":3,"statement":"Prove that any convex pentagon whose vertices (no three of\nwhich are collinear) have integer coordinates must have area greater than\nor equal to 5/2.","subjects":["geometry","lattice polygons","area","parity"]},{"id":"1990-A4","year":1990,"session":"A","number":4,"statement":"Consider a paper punch that can be centered at any point of\nthe plane and that, when operated, removes from the plane precisely those\npoints whose distance from the center is irrational. How many punches are\nneeded to remove every point?","subjects":["geometry","countability","rational distances","irrational numbers"]},{"id":"1990-A5","year":1990,"session":"A","number":5,"statement":"If $\\mathbf{A}$ and $\\mathbf{B}$ are square matrices of the\nsame size such that $\\mathbf{ABAB = 0}$, does it follow that\n$\\mathbf{BABA = 0}$?","subjects":["linear algebra","matrix products","counterexamples","nilpotent matrices"]},{"id":"1990-A6","year":1990,"session":"A","number":6,"statement":"If $X$ is a finite set, let $X$ denote the number of elements\nin $X$. Call an ordered pair $(S, T)$ of subsets of $\\{1, 2, \\dots, n\\}$\nadmissible if $s > |T|$ for each $s \\in S$, and $t > |S|$ for each\n$t \\in T$. How many admissible ordered pairs of subsets of $\\{1, 2,\n\\dots, 10\\}$ are there? Prove your answer.","subjects":["combinatorics","admissible subsets","fibonacci numbers","tilings"]},{"id":"1990-B1","year":1990,"session":"B","number":1,"statement":"Find all real-valued continuously differentiable functions $f$\non the real line such that for all $x$,\n$$\n(f(x))^2 = \\int_0^x [(f(t))^2 + (f'(t))^2]\\,dt + 1990.\n$$","subjects":["analysis","integral equations","differential equations"]},{"id":"1990-B2","year":1990,"session":"B","number":2,"statement":"Prove that for $|x| < 1$, $|z| > 1$,\n$$\n1 + \\sum_{j=1}^\\infty (1 + x^j)P_j = 0,\n$$\nwhere $P_j$ is\n$$\n\\frac{(1 - z)(1 - zx)(1 - zx^2) \\cdots (1 - zx^{j-1})}\n{(z - x)(z - x^2)(z - x^3) \\cdots (z - x^j)}.\n$$","subjects":["analysis","infinite series"]},{"id":"1990-B3","year":1990,"session":"B","number":3,"statement":"Let $S$ be a set of $2 \\times 2$ integer matrices whose\nentries $a_{ij}$ (1) are all squares of integers and, (2) satisfy $a_{ij}\n\\leq 200$. Show that if $S$ has more than 50387 ($= 15^4 - 15^2 - 15 +\n2$) elements, then it has two elements that commute.","subjects":["linear algebra","combinatorics","commuting matrices"]},{"id":"1990-B4","year":1990,"session":"B","number":4,"statement":"Let $G$ be a finite group of order $n$ generated by $a$ and\n$b$. Prove or disprove: there is a sequence\n$$\ng_1, g_2, g_3, \\dots, g_{2n}\n$$\nsuch that\n\n(1) every element of $G$ occurs exactly twice, and\n\n(2) $g_{i+1}$ equals $g_i a$ or $g_i b$ for $i = 1, 2, \\dots,\n2n$. (Interpret $g_{2n+1}$ as $g_1$.)","subjects":["algebra","graph theory","finite groups","euler circuits","cayley graphs"]},{"id":"1990-B5","year":1990,"session":"B","number":5,"statement":"Is there an infinite sequence $a_0, a_1, a_2, \\dots$ of\nnonzero real numbers such that for $n = 1, 2, 3, \\dots$ the polynomial\n$$\np_n(x) = a_0 + a_1x + a_2x^2 + \\cdots + a_nx^n\n$$\nhas exactly $n$ distinct real roots?","subjects":["algebra","real-rooted polynomials","construction","intermediate value theorem"]},{"id":"1990-B6","year":1990,"session":"B","number":6,"statement":"Let $S$ be a nonempty closed bounded convex set in the plane.\nLet $K$ be a line and $t$ a positive number. Let $L_1$ and $L_2$ be\nsupport lines for $S$ parallel to $K_1$, and let $\\overline{L}$ be the\nline parallel to $K$ and midway between $L_1$ and $L_2$. Let $B_S(K, t)$\nbe the band of points whose distance from $\\overline{L}$ is at most\n$(t/2)w$, where $w$ is the distance between $L_1$ and $L_2$. What is the\nsmallest $t$ such that\n$$\nS \\cap \\bigcap_K B_S(K, t) \\neq \\emptyset\n$$\nfor all $S$? ($K$ runs over all lines in the plane.)","subjects":["geometry","convex sets","centroids","support lines","extremal triangles"],"errata":"The supporting lines are parallel to $K$, not the undefined $K_1$ in the archived exam and imported statement. The supplied book prints $K$ on printed pp. 14 and 128, PDF pp. 28 and 142, agreeing with the subsequent definition of the $K$-parallel midline. The corrected target makes this notation clarification and retains the original statement for provenance."},{"id":"1991-A1","year":1991,"session":"A","number":1,"statement":"A $2 \\times 3$ rectangle has vertices as $(0, 0), (2,0), (0,3),$ and $(2,\n3)$. It rotates $90^\\circ$ clockwise about the point $(2, 0)$. It then\nrotates $90^\\circ$ clockwise about the point $(5, 0)$, then $90^\\circ$\nclockwise about the point $(7, 0)$, and finally, $90^\\circ$ clockwise\nabout the point $(10, 0)$. (The side originally on the $x$-axis is now\nback on the $x$-axis.) Find the area of the region above the $x$-axis and\nbelow the curve traced out by the point whose initial position is (1,1).","subjects":["geometry","area","rotations","circular arcs"]},{"id":"1991-A2","year":1991,"session":"A","number":2,"statement":"Let $\\mathbf{A}$ and $\\mathbf{B}$ be different $n \\times n$ matrices with real entries.\nIf $\\mathbf{A}^3 = \\mathbf{B}^3$ and $\\mathbf{A}^2 \\mathbf{B} = \\mathbf{B}^2 \\mathbf{A}$, can $\\mathbf{A}^2 + \\mathbf{B}^2$ be\ninvertible?","subjects":["linear algebra","matrices"]},{"id":"1991-A3","year":1991,"session":"A","number":3,"statement":"Find all real polynomials $p(x)$ of degree $n \\geq 2$ for which there\nexist real numbers $r_1 < r_2 < \\cdots < r_n$ such that\n\n- $p(r_i) = 0, \\qquad i = 1, 2, \\dots, n,$ and\n\n- $p' \\left( \\frac{r_i + r_{i+1}}{2} \\right) = 0 \\qquad i = 1, 2,\n\\dots, n-1,$\n\nwhere $p'(x)$ denotes the derivative of $p(x)$.","subjects":["algebra","polynomials","real roots","critical points"]},{"id":"1991-A4","year":1991,"session":"A","number":4,"statement":"Does there exist an infinite sequence of closed discs $D_1, D_2, D_3,\n\\dots$ in the plane, with centers $c_1, c_2, c_3, \\dots$, respectively,\nsuch that\n\n- the $c_i$ have no limit point in the finite plane,\n\n- the sum of the areas of the $D_i$ is finite, and\n\n- every line in the plane intersects at least one of the $D_i$?","subjects":["geometry","infinite constructions","convergent series","line transversals"]},{"id":"1991-A5","year":1991,"session":"A","number":5,"statement":"Find the maximum value of\n$$\n\\int_0^y \\sqrt{x^4 + (y-y^2)^2}\\,dx\n$$\nfor $0 \\leq y \\leq 1$.","subjects":["analysis","integral inequalities","optimization"]},{"id":"1991-A6","year":1991,"session":"A","number":6,"statement":"Let $A(n)$ denote the number of sums of positive integers\n$$\na_1 + a_2 + \\cdots + a_r\n$$\nwhich add up to $n$ with\n$$\n\\begin{gathered}\na_1 > a_2 + a_3, a_2 > a_3 + a_4, \\ldots, \\\\\na_{r-2} > a_{r-1} + a_r, a_{r-1}  > a_r.\n\\end{gathered}\n$$\nLet $B(n)$ denote the number of $b_1 + b_2 + \\cdots + b_s$ which add up\nto $n$, with\n\n- $b_1 \\geq b_2 \\geq \\dots \\geq b_s,$\n\n- each $b_i$ is in the sequence $1, 2, 4, \\dots, g_j, \\dots$\ndefined by $g_1 = 1$, $g_2 = 2$, and $g_j = g_{j-1} + g_{j-2} + 1,$ and\n\n- if $b_1 = g_k$ then every element in $\\{1, 2, 4, \\dots, g_k\\}$\nappears at least once as a $b_i$.\n\nProve that $A(n) = B(n)$ for each $n \\geq 1$.\n\n(For example, $A(7) = 5$ because the relevant sums are $7, 6+1, 5+2, 4+3,\n4+2+1,$ and $B(7) = 5$ because the relevant sums are $4+2+1, 2+2+2+1,\n2+2+1+1+1, 2+1+1+1+1+1, 1+1+1+1+1+1+1.$)","subjects":["combinatorics","partitions","fibonacci numbers","bijections"]},{"id":"1991-B1","year":1991,"session":"B","number":1,"statement":"For each integer $n \\geq 0$, let $S(n) = n - m^2$, where $m$ is the\ngreatest integer with $m^2 \\leq n$. Define a sequence\n$(a_k)_{k=0}^\\infty$ by $a_0 = A$ and $a_{k+1} = a_k + S(a_k)$ for $k\n\\geq 0$. For what positive integers $A$ is this sequence eventually constant?","subjects":["number theory","integer sequences","perfect squares","parity"]},{"id":"1991-B2","year":1991,"session":"B","number":2,"statement":"Suppose $f$ and $g$ are non-constant, differentiable, real-valued\nfunctions defined on $(-\\infty, \\infty)$. Furthermore, suppose that for\neach pair of real numbers $x$ and $y$,\n$$\n\\begin{aligned}\nf(x+y) &= f(x)f(y) - g(x)g(y), \\\\\ng(x+y) &= f(x)g(y) + g(x)f(y).\n\\end{aligned}\n$$\nIf $f'(0) = 0$, prove that $(f(x))^2 + (g(x))^2 = 1$ for all $x$.","subjects":["analysis","functional equations","addition formulas","differential equations"]},{"id":"1991-B3","year":1991,"session":"B","number":3,"statement":"Does there exist a real number $L$ such that, if $m$ and $n$ are integers\ngreater than $L$, then an $m \\times n$ rectangle may be expressed as a\nunion of $4 \\times 6$ and $5 \\times 7$ rectangles, any two of which\nintersect at most along their boundaries?","subjects":["number theory","tilings","coin problem","rectangles"]},{"id":"1991-B4","year":1991,"session":"B","number":4,"statement":"Suppose $p$ is an odd prime. Prove that\n$$\n\\sum_{j=0}^p \\binom{p}{j} \\binom{p+j}{j} \\equiv 2^p + 1\\pmod{p^2}.\n$$","subjects":["number theory","binomial congruences"]},{"id":"1991-B5","year":1991,"session":"B","number":5,"statement":"Let $p$ be an odd prime and let $\\mathbb{Z}_p$ denote (the field of) integers\nmodulo $p$. How many elements are in the set\n$$\n\\{x^2: x \\in \\mathbb{Z}_p\\} \\cap \\{y^2 + 1 : y \\in \\mathbb{Z}_p\\}?\n$$","subjects":["number theory","finite fields","quadratic residues","character sums"]},{"id":"1991-B6","year":1991,"session":"B","number":6,"statement":"Let $a$ and $b$ be positive numbers. Find the largest number $c$, in\nterms of $a$ and $b$, such that\n$$\na^x b^{1-x} \\leq a \\frac{\\sinh ux}{\\sinh u} + b \\frac{\\sinh u(1-x)}{\\sinh u}\n$$\nfor all $u$ with $0 < |u| \\leq c$ and for all $x$, $0 < x < 1$. (Note:\n$\\sinh u = (e^u - e^{-u})/2$.)","subjects":["analysis","hyperbolic functions","convexity","differential inequalities"]},{"id":"1992-A1","year":1992,"session":"A","number":1,"statement":"Prove that $f(n) = 1-n$ is the only integer-valued function\ndefined on the integers that satisfies the following conditions.\n\n(i) $f(f(n)) = n$, for all integers $n$;\n\n(ii) $f(f(n+2)+2) = n$ for all integers $n$;\n\n(iii) $f(0) = 1$.","subjects":["algebra","functional equations","integer functions","involutions"]},{"id":"1992-A2","year":1992,"session":"A","number":2,"statement":"Define $C(\\alpha)$ to be the coefficient of $x^{1992}$ in the\npower series about $x=0$ of $(1 + x)^\\alpha$. Evaluate\n$$\n\\int_0^1 \\left( C(-y-1) \\sum_{k=1}^{1992} \\frac{1}{y+k} \\right)\\,dy.\n$$","subjects":["analysis","integration","binomial coefficients","geometric probability"]},{"id":"1992-A3","year":1992,"session":"A","number":3,"statement":"For a given positive integer $m$, find all triples $(n, x, y)$\nof positive integers, with $n$ relatively prime to $m$, which satisfy\n$$\n(x^2 + y^2)^m = (xy)^n.\n$$","subjects":["number theory","diophantine equations","prime valuations","coprimality"]},{"id":"1992-A4","year":1992,"session":"A","number":4,"statement":"Let $f$ be an infinitely differentiable real-valued function\ndefined on the real numbers. If\n$$\nf\\left( \\frac{1}{n} \\right) = \\frac{n^2}{n^2 + 1}, \\qquad n = 1, 2, 3, \\dots,\n$$\ncompute the values of the derivatives $f^{(k)}(0), k = 1, 2, 3, \\dots$.","subjects":["analysis","smooth functions","taylor coefficients","accumulating zeros"]},{"id":"1992-A5","year":1992,"session":"A","number":5,"statement":"For each positive integer $n$, let $a_n = 0$ (or 1) if the\nnumber of 1's in the binary representation of $n$ is even (or odd),\nrespectively. Show that there do not exist positive integers $k$ and $m$\nsuch that\n$$\na_{k+j} = a_{k+m+j} = a_{k+2m+j},\n$$\nfor $0 \\leq j \\leq m-1$.","subjects":["combinatorics","binary expansions","thue-morse sequence","words"]},{"id":"1992-A6","year":1992,"session":"A","number":6,"statement":"Four points are chosen at random on the surface of a sphere.\nWhat is the probability that the center of the sphere lies inside the\ntetrahedron whose vertices are at the four points? (It is understood that\neach point is independently chosen relative to a uniform distribution on\nthe sphere.)","subjects":["probability","geometry","random tetrahedra","convex hulls"]},{"id":"1992-B1","year":1992,"session":"B","number":1,"statement":"Let $S$ be a set of $n$ distinct real numbers. Let $A_S$ be\nthe set of numbers that occur as averages of two distinct elements of\n$S$. For a given $n \\geq 2$, what is the smallest possible number of\nelements in $A_S$?","subjects":["combinatorics","additive combinatorics","ordered real sets"]},{"id":"1992-B2","year":1992,"session":"B","number":2,"statement":"For nonnegative integers $n$ and $k$, define $Q(n, k)$ to be\nthe coefficient of $x^k$ in the expansion of $(1 + x + x^2 + x^3)^n$.\nProve that\n$$\nQ(n, k) = \\sum_{j=0}^k \\binom{n}{j} \\binom{n}{k-2j},\n$$\nwhere $\\binom{a}{b}$ is the standard binomial coefficient. (Reminder: For\nintegers $a$ and $b$ with $a \\geq 0$, $\\binom{a}{b} =\n\\frac{a!}{b!(a-b)!}$ for $0 \\leq b \\leq a$, with $\\binom{a}{b} = 0$ otherwise.)","subjects":["combinatorics","binomial coefficients"]},{"id":"1992-B3","year":1992,"session":"B","number":3,"statement":"For any pair $(x, y)$ of real numbers, a sequence\n$(a_n(x,y))_{n\\geq 0}$ is defined as follows:\n$$\n\\begin{aligned}\na_0(x, y) &= x, \\\\\na_{n+1}(x, y) &= \\frac{(a_n(x, y))^2 + y^2}{2}, \\qquad \\text{for }n \\geq 0.\n\\end{aligned}\n$$\nFind the area of the region\n$$\n\\{ (x, y) | (a_n(x, y))_{n \\geq 0}\\ \\text{converges}\\}.\n$$","subjects":["analysis","dynamical systems","fixed points","area"]},{"id":"1992-B4","year":1992,"session":"B","number":4,"statement":"Let $p(x)$ be a nonzero polynomial of degree less than 1992\nhaving no nonconstant factor in common with $x^3 - x$. Let\n$$\n\\frac{d^{1992}}{dx^{1992}} \\left( \\frac{p(x)}{x^3 - x} \\right) =\n\\frac{f(x)}{g(x)}\n$$\nfor polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$.","subjects":["analysis","rational functions","high derivatives","polynomial degree"]},{"id":"1992-B5","year":1992,"session":"B","number":5,"statement":"Let $D_n$ denote the value of the $(n-1) \\times (n-1)$ determinant\n$$\n\\left[\n\\begin{array}{cccccc}\n3 & 1 & 1 & 1 & \\cdots & 1 \\\\\n1 & 4 & 1 & 1 & \\cdots & 1 \\\\\n1 & 1 & 5 & 1 & \\cdots & 1 \\\\\n1 & 1 & 1 & 6 & \\cdots & 1 \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n1 & 1 & 1 & 1 & \\cdots & n+1\n\\end{array}\n\\right].\n$$\nIs the set $\\left\\{ \\frac{D_n}{n!} \\right\\}_{n \\geq 2}$ bounded?","subjects":["linear algebra","determinants","harmonic series","rank-one perturbations"]},{"id":"1992-B6","year":1992,"session":"B","number":6,"statement":"Let $\\mathcal{M}$ be a set of real $n \\times n$ matrices such that\n\n(i) $I \\in \\mathcal{M}$, where $I$ is the $n \\times n$ identity matrix;\n\n(ii) if $A \\in \\mathcal{M}$ and $B \\in \\mathcal{M}$, then either $AB \\in \\mathcal{M}$ or\n$-AB \\in \\mathcal{M}$, but not both;\n\n(iii) if $A \\in \\mathcal{M}$ and $B \\in \\mathcal{M}$, then either $AB = BA$ or\n$AB = -BA$;\n\n(iv) if $A \\in \\mathcal{M}$ and $A \\neq I$, there is at least one $B \\in\n\\mathcal{M}$ such that $AB = - BA$.\n\nProve that $\\mathcal{M}$ contains at most $n^2$ matrices.","subjects":["linear algebra","anticommuting matrices","trace","linear independence"]},{"id":"1993-A1","year":1993,"session":"A","number":1,"statement":"The horizontal line $y=c$ intersects the curve $y = 2x - 3x^3$\nin the first quadrant. Find $c$ so that the areas of the\ntwo shaded regions are equal. The first region is\nbounded by the $y$-axis, the line $y=c$ and the curve; the other lies\nunder the curve and above the line $y=c$ between their two points of\nintersection.","subjects":["analysis","geometry","integration","equal areas","cubic curves"]},{"id":"1993-A2","year":1993,"session":"A","number":2,"statement":"Let $(x_n)_{n \\geq 0}$ be a sequence of nonzero real numbers\nsuch that $x_n^2 - x_{n-1}x_{n+1} = 1$ for $n=1,2,3,\\dots$. Prove there\nexists a real number $a$ such that $x_{n+1} = ax_n - x_{n-1}$ for all $n\n\\geq 1$.","subjects":["algebra","recurrences","invariants","nonzero sequences"]},{"id":"1993-A3","year":1993,"session":"A","number":3,"statement":"Let ${\\cal P}_n$ be the set of subsets of $\\{1, 2, \\dots,\nn\\}$. Let $c(n, m)$ be the number of functions $f: {\\cal P}_n \\to \\{1, 2,\n\\dots, m\\}$ such that $f(A \\cap B) = \\min\\{f(A), f(B)\\}$. Prove that\n$$\nc(n, m) = \\sum_{j=1}^m j^n.\n$$","subjects":["combinatorics","boolean lattice","meet homomorphisms","enumeration"]},{"id":"1993-A4","year":1993,"session":"A","number":4,"statement":"Let $x_1, x_2, \\dots, x_{19}$ be positive integers each of\nwhich is less than or equal to 93. Let $y_1, y_2, \\dots, y_{93}$ be\npositive integers each of which is less than or equal to 19. Prove that\nthere exists a (nonempty) sum of some $x_i$'s equal to a sum of some $y_j$'s.","subjects":["combinatorics","subset sums","integer walks"]},{"id":"1993-A5","year":1993,"session":"A","number":5,"statement":"Show that\n$$\n\\begin{gathered}\n\\int_{-100}^{-10} \\left( \\frac{x^2 - x}{x^3 - 3x + 1} \\right)^2\\,dx + \\\\\n\\int_{\\frac{1}{101}}^{\\frac{1}{11}} \\left( \\frac{x^2 - x}{x^3 - 3x + 1} \\right)^2\\,dx + \\\\\n\\int_{\\frac{101}{100}}^{\\frac{11}{10}} \\left( \\frac{x^2 - x}{x^3 - 3x + 1} \\right)^2\\,dx\n\\end{gathered}\n$$\nis a rational number.","subjects":["analysis","rational integrals","mobius transformations"]},{"id":"1993-A6","year":1993,"session":"A","number":6,"statement":"The infinite sequence of 2's and 3's\n$$\n\\begin{aligned}\n&2,3,3,2,3,3,3,2,3,3,3,2,3,3,2,3,3, \\\\\n&3,2,3,3,3,2,3,3,3,2,3,3,2,3,3,3,2,\\dots\n\\end{aligned}\n$$\nhas the property that, if one forms a second sequence that records the\nnumber of 3's between successive 2's, the result is identical to the\ngiven sequence. Show that there exists a real number $r$ such that, for\nany $n$, the $n$th term of the sequence is 2 if and only if $n = 1 +\n\\lfloor rm \\rfloor$ for some nonnegative integer $m$. (Note: $\\lfloor x\n\\rfloor$ denotes the largest integer less than or equal to $x$.)","subjects":["number theory","beatty sequences","self-generating words","floor functions"]},{"id":"1993-B1","year":1993,"session":"B","number":1,"statement":"Find the smallest positive integer $n$ such that for every integer $m$\nwith $0 < m < 1993$, there exists an integer $k$ for which\n$$\n\\frac{m}{1993} < \\frac{k}{n} < \\frac{m+1}{1994}.\n$$","subjects":["number theory","rational approximation","farey fractions"]},{"id":"1993-B2","year":1993,"session":"B","number":2,"statement":"Consider the following game played with a deck of $2n$ cards numbered\nfrom 1 to $2n$. The deck is randomly shuffled and $n$ cards are dealt to\neach of two players. Beginning with $A$, the players take turns\ndiscarding one of their remaining cards and announcing its number. The\ngame ends as soon as the sum of the numbers on the discarded cards is\ndivisible by $2n+1$. The last person to discard wins the game. Assuming\noptimal strategy by both $A$ and $B$, what is the probability that $A$ wins?","subjects":["combinatorics","games","modular arithmetic"]},{"id":"1993-B3","year":1993,"session":"B","number":3,"statement":"Two real numbers $x$ and $y$ are chosen at random in the interval (0,1)\nwith respect to the uniform distribution. What is the probability that\nthe closest integer to $x/y$ is even? Express the answer in the form\n$r+s\\pi$, where $r$ and $s$ are rational numbers.","subjects":["probability","geometric probability","arctangent series"]},{"id":"1993-B4","year":1993,"session":"B","number":4,"statement":"The function $K(x,y)$ is positive and continuous for $0 \\leq x \\leq 1, 0\n\\leq y \\leq 1$, and the functions $f(x)$ and $g(x)$ are positive and\ncontinuous for $0 \\leq x \\leq 1$. Suppose that for all $x$, $0 \\leq x \\leq 1$,\n$$\n\\int_0^1 f(y)K(x,y)\\,dy = g(x)\n$$\nand\n$$\n\\int_0^1 g(y)K(x,y)\\,dy = f(x).\n$$\nShow that $f(x) = g(x)$ for $0 \\leq x \\leq 1$.","subjects":["analysis","positive integral operators","functional equations","maximum principle"]},{"id":"1993-B5","year":1993,"session":"B","number":5,"statement":"Show there do not exist four points in the Euclidean plane such that the\npairwise distances between the points are all odd integers.","subjects":["geometry","number theory","distance geometry","gram matrices"]},{"id":"1993-B6","year":1993,"session":"B","number":6,"statement":"Let $S$ be a set of three, not necessarily distinct, positive integers.\nShow that one can transform $S$ into a set containing 0 by a finite\nnumber of applications of the following rule: Select two of the three\nintegers, say $x$ and $y$, where $x \\leq y$ and replace them with $2x$ and $y-x$.","subjects":["number theory","integer dynamics","binary expansions"]},{"id":"1994-A1","year":1994,"session":"A","number":1,"statement":"Suppose that a sequence $a_1, a_2, a_3, \\dots$ satisfies\n$0 < a_n \\leq a_{2n} + a_{2n+1}$ for all $n \\geq 1$. Prove that the series\n$\\sum_{n=1}^{\\infty} a_n$ diverges.","subjects":["analysis","positive series","binary trees","divergence"]},{"id":"1994-A2","year":1994,"session":"A","number":2,"statement":"Let $A$ be the area of the region in the first quadrant bounded by the\nline $y = \\frac{1}{2} x$, the $x$-axis, and the ellipse $\\frac{1}{9} x^2\n+ y^2 = 1$. Find the positive number $m$ such that $A$ is equal to the\narea of the region in the first quadrant bounded by the line $y = mx$,\nthe $y$-axis, and the ellipse $\\frac{1}{9} x^2 + y^2 = 1$.","subjects":["geometry","ellipse","area-preserving transformations","sectors"]},{"id":"1994-A3","year":1994,"session":"A","number":3,"statement":"Show that if the points of an isosceles right triangle of side length\n1 are each colored with one of four colors, then there must be two points\nof the same color whch are at least a distance $2 - \\sqrt{2}$ apart.","subjects":["geometry","coloring","unit triangles","chromatic obstruction"]},{"id":"1994-A4","year":1994,"session":"A","number":4,"statement":"Let $A$ and $B$ be $2 \\times 2$ matrices with integer entries such\nthat $A, A+B, A+2B, A+3B$, and $A+4B$ are all invertible matrices whose\ninverses have integer entries. Show that $A+5B$ is invertible and that\nits inverse has integer entries.","subjects":["linear algebra","integer matrices","unimodular matrices","polynomials"]},{"id":"1994-A5","year":1994,"session":"A","number":5,"statement":"Let $(r_n)_{n \\geq 0}$ be a sequence of positive real numbers such that\n$\\lim_{n \\to \\infty} r_n = 0$. Let $S$ be the set of numbers representable\nas a sum\n$$\nr_{i_1} + r_{i_2} + \\cdots + r_{i_{1994}},\n$$\nwith $i_1 < i_2 < \\cdots < i_{1994}$. Show that every nonempty interval\n$(a,b)$ contains a nonempty subinterval $(c,d)$ that does not intersect $S$.","subjects":["analysis","topology","nowhere dense sets","finite sums"]},{"id":"1994-A6","year":1994,"session":"A","number":6,"statement":"Let $f_1, \\dots, f_{10}$ be bijections of the set of integers such that for\neach integer $n$, there is some composition $f_{i_1} \\circ f_{i_2}\n\\circ \\cdots \\circ f_{i_m}$ of these functions (allowing repetitions)\nwhich maps 0 to $n$. Consider the set of 1024 functions\n$$\n\\mathcal{F} = \\{f_1^{e_1} \\circ f_2^{e_2} \\circ \\cdots \\circ f_{10}^{e_{10}}\\},\n$$\n$e_i = 0$ or 1 for $1 \\leq i \\leq 10$. ($f_i^0$ is the identity function\nand $f_i^1 = f_i$.) Show that if $A$ is any nonempty finite set of\nintegers, then at most 512 of the functions in $\\mathcal{F}$ map $A$ to\nitself.","subjects":["combinatorics","linear algebra","permutation groups","group actions"]},{"id":"1994-B1","year":1994,"session":"B","number":1,"statement":"Find all positive integers $n$ that are within 250 of exactly 15 perfect\nsquares.","subjects":["number theory","perfect squares","interval counting"]},{"id":"1994-B2","year":1994,"session":"B","number":2,"statement":"For which real numbers $c$ is there a straight line that intersects the curve\n$$ x^4 + 9x^3 + cx^2 + 9x + 4\n$$\nin four distinct points?","subjects":["algebra","quartic polynomials","real roots","convexity"]},{"id":"1994-B3","year":1994,"session":"B","number":3,"statement":"Find the set of all real numbers $k$ with the following property: For any\npositive, differentiable function $f$ that satisfies $f'(x) > f(x)$\nfor all $x$, there is some number $N$ such that\n$f(x) > e^{kx}$ for all $x > N$.","subjects":["analysis","growth rates","differential inequalities","exponential functions"]},{"id":"1994-B4","year":1994,"session":"B","number":4,"statement":"For $n \\geq 1$, let $d_n$ be the greatest common divisor of the entries of\n$A^n - I$, where\n$$\nA = \\begin{pmatrix} 3 & 2 \\\\ 4 & 3 \\end{pmatrix}\n\\quad \\text{ and } \\quad\nI = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}.\n$$\nShow that $\\lim_{n \\to \\infty} d_n = \\infty$.","subjects":["number theory","linear algebra","matrix powers","pell equations","greatest common divisors"]},{"id":"1994-B5","year":1994,"session":"B","number":5,"statement":"For any real number $\\alpha$, define the function $f_{\\alpha}(x)\n= \\lfloor \\alpha x \\rfloor$. Let $n$ be a positive integer. Show that\nthere exists an $\\alpha$ such that for $1 \\leq k \\leq n$,\n$$\nf_\\alpha^k(n^2) = n^2 - k = f_{\\alpha^k}(n^2).\n$$","subjects":["number theory","floor functions","iteration","bernoulli inequalities"]},{"id":"1994-B6","year":1994,"session":"B","number":6,"statement":"For any integer $n$, set\n$$\nn_a = 101a - 100\\cdot 2^a.\n$$\nShow that for $0 \\leq a,b,c,d \\leq 99$, $n_a + n_b \\equiv\nn_c + n_d \\pmod{10100}$ implies $\\{a,b\\} = \\{c,d\\}$.","subjects":["number theory","sidon sets","finite fields","primitive roots"]},{"id":"1995-A1","year":1995,"session":"A","number":1,"statement":"Let $S$ be a set of real numbers which is closed under\nmultiplication (that is, if $a$ and $b$ are in $S$, then so is $ab$).\nLet $T$ and $U$ be disjoint subsets of $S$ whose union is $S$. Given\nthat the product of any three (not necessarily distinct)\nelements of $T$ is in $T$ and that the product of any three elements\nof $U$ is in $U$, show that at least one of the two subsets $T,U$ is\nclosed under multiplication.","subjects":["algebra","semigroups","multiplicative closure","two-color partitions"]},{"id":"1995-A2","year":1995,"session":"A","number":2,"statement":"For what pairs $(a,b)$ of positive real numbers does the\nimproper integral\n$$\n\\int_{b}^{\\infty} \\left( \\sqrt{\\sqrt{x+a}-\\sqrt{x}} -\n\\sqrt{\\sqrt{x}-\\sqrt{x-b}} \\right)\\,dx\n$$\nconverge?","subjects":["analysis"]},{"id":"1995-A3","year":1995,"session":"A","number":3,"statement":"The number $d_{1}d_{2}\\dots d_{9}$ has nine (not\nnecessarily distinct) decimal digits. The number $e_{1}e_{2}\\dots\ne_{9}$ is such that each of the nine 9-digit numbers formed by\nreplacing just one of the digits $d_{i}$ is $d_{1}d_{2}\\dots d_{9}$\nby the corresponding digit $e_{i}$ ($1 \\leq i \\leq 9$) is divisible\nby 7. The number $f_{1}f_{2}\\dots f_{9}$ is related to\n$e_{1}e_{2}\\dots e_{9}$ is the same way: that is, each of the nine\nnumbers formed by replacing one of the $e_{i}$ by the corresponding\n$f_{i}$ is divisible by 7. Show that, for each $i$, $d_{i}-f_{i}$ is\ndivisible by 7. [For example, if $d_{1}d_{2}\\dots d_{9} = 199501996$,\nthen $e_{6}$ may be 2 or 9, since $199502996$ and $199509996$ are\nmultiples of 7.]","subjects":["number theory","decimal digits","congruences","modular linear relations"]},{"id":"1995-A4","year":1995,"session":"A","number":4,"statement":"Suppose we have a necklace of $n$ beads. Each bead is\nlabeled with an integer and the sum of all these labels is $n-1$.\nProve that we can cut the necklace to form a string whose\nconsecutive labels $x_{1},x_{2},\\dots,x_{n}$ satisfy\n$$\n\\sum_{i=1}^{k} x_{i} \\leq k-1 \\qquad \\text{for} \\quad k=1,2,\\dots,n.\n$$","subjects":["combinatorics","cyclic sequences","partial sums","cycle lemma"]},{"id":"1995-A5","year":1995,"session":"A","number":5,"statement":"Let $x_{1},x_{2},\\dots,x_{n}$ be differentiable\n(real-valued) functions of a single variable $t$ which satisfy\n$$\n\\begin{aligned}\n\\frac{dx_{1}}{dt} &= a_{11}x_{1} + a_{12}x_{2} + \\cdots +\na_{1n}x_{n} \\\\\n\\frac{dx_{2}}{dt} &= a_{21}x_{1} + a_{22}x_{2} + \\cdots +\na_{2n}x_{n} \\\\\n\\vdots && \\vdots \\\\\n\\frac{dx_{n}}{dt} &= a_{n1}x_{1} + a_{n2}x_{2} + \\cdots +\na_{nn}x_{n}\n\\end{aligned}\n$$\nfor some constants $a_{ij}>0$. Suppose that for all $i$, $x_{i}(t)\n\\to 0$ as $t \\to \\infty$. Are the functions $x_{1},x_{2},\\dots,x_{n}$\nnecessarily linearly dependent?","subjects":["linear algebra","analysis"]},{"id":"1995-A6","year":1995,"session":"A","number":6,"statement":"Suppose that each of $n$ people writes down the numbers\n1,2,3 in random order in one column of a $3 \\times n$ matrix, with\nall orders equally likely and with the orders for different columns\nindependent of each other. Let the row sums $a,b,c$ of the resulting\nmatrix be rearranged (if necessary) so that $a \\leq b \\leq c$. Show\nthat for some $n \\geq 1995$, it is at least four times as likely that\nboth $b=a+1$ and $c=a+2$ as that $a=b=c$.","subjects":["probability","random walks","multinomial coefficients","return probabilities"]},{"id":"1995-B1","year":1995,"session":"B","number":1,"statement":"For a partition $\\pi$ of $\\{1, 2, 3, 4, 5, 6, 7, 8, 9\\}$,\nlet $\\pi(x)$ be the number of elements in the part containing $x$.\nProve that for any two partitions $\\pi$ and $\\pi'$, there are two\ndistinct numbers $x$ and $y$ in $\\{1, 2, 3, 4, 5, 6, 7, 8, 9\\}$\nsuch that $\\pi(x) = \\pi(y)$ and $\\pi'(x) = \\pi'(y)$. [A partition of a set $S$ is a collection of disjoint subsets (parts)\nwhose union is $S$.]","subjects":["combinatorics","set partitions","block sizes"]},{"id":"1995-B2","year":1995,"session":"B","number":2,"statement":"An ellipse, whose semi-axes have lengths $a$ and $b$,\nrolls without slipping on the curve $y = c \\sin \\left( \\frac{x}{a}\n\\right)$. How are $a,b,c$ related, given that the ellipse completes\none revolution when it traverses one period of the curve?","subjects":["geometry","analysis","arc length","ellipses","rolling curves"]},{"id":"1995-B3","year":1995,"session":"B","number":3,"statement":"To each positive integer with $n^{2}$ decimal digits, we\nassociate the determinant of the matrix obtained by writing the\ndigits in order across the rows. For example, for $n=2$, to the\ninteger 8617 we associate $\\det \\left(\n\\begin{array}{cc} 8 & 6 \\\\\n1 & 7 \\end{array} \\right) = 50$. Find, as a function of $n$, the\nsum of all the determinants associated with $n^{2}$-digit\nintegers. (Leading digits are assumed to be nonzero; for example,\nfor $n=2$, there are 9000 determinants.)","subjects":["linear algebra","combinatorics","determinant sums","involutions","expectation"]},{"id":"1995-B4","year":1995,"session":"B","number":4,"statement":"Evaluate\n$$\n\\sqrt[8]{2207 - \\frac{1}{2207-\\frac{1}{2207-\\dots}}}.\n$$\nExpress your answer in the form $\\frac{a+b\\sqrt{c}}{d}$, where\n$a,b,c,d$ are integers.","subjects":["algebra","continued fractions","radicals"]},{"id":"1995-B5","year":1995,"session":"B","number":5,"statement":"A game starts with four heaps of beans, containing 3,4,5\nand 6 beans. The two players move alternately. A move consists of\ntaking either\n\na) one bean from a heap, provided at least two beans are\nleft behind in that heap, or\n\nb) a complete heap of two or three beans.\n\nThe player who takes the last heap wins. To win the game, do you\nwant to move first or second? Give a winning strategy.","subjects":["combinatorics","impartial games","combinatorial games"]},{"id":"1995-B6","year":1995,"session":"B","number":6,"statement":"For a positive real number $\\alpha$, define\n$$\nS(\\alpha) = \\{ \\lfloor n\\alpha \\rfloor : n = 1,2,3,\\dots \\}.\n$$\nProve that $\\{1,2,3,\\dots\\}$ cannot be expressed as the disjoint\nunion of three sets $S(\\alpha), S(\\beta)$ and $S(\\gamma)$. [As\nusual, $\\lfloor x \\rfloor$ is the greatest integer $\\leq x$.]","subjects":["number theory","beatty sequences","floor functions"]},{"id":"1996-A1","year":1996,"session":"A","number":1,"statement":"Find the least number $A$ such that for any two squares of combined\narea 1, a rectangle of area $A$ exists such that the two squares can\nbe packed in the rectangle (without interior overlap). You may assume\nthat the sides of the squares are parallel to the sides of the\nrectangle.","subjects":["geometry","optimization","rectangles"]},{"id":"1996-A2","year":1996,"session":"A","number":2,"statement":"Let $C_1$ and $C_2$ be circles whose centers are 10 units apart, and\nwhose radii are 1 and 3. Find, with proof, the locus of all points $M$\nfor which there exists points $X$ on $C_1$ and $Y$ on $C_2$ such that\n$M$ is the midpoint of the line segment $XY$.","subjects":["geometry","circles","loci"]},{"id":"1996-A3","year":1996,"session":"A","number":3,"statement":"Suppose that each of 20 students has made a choice of anywhere from 0\nto 6 courses from a total of 6 courses offered. Prove or disprove:\nthere are 5 students and 2 courses such that all 5 have chosen both\ncourses or all 5 have chosen neither course.","subjects":["combinatorics","extremal set theory","counterexamples"]},{"id":"1996-A4","year":1996,"session":"A","number":4,"statement":"Let $S$ be the set of ordered triples $(a, b, c)$ of distinct elements\nof a finite set $A$. Suppose that\n\n- $(a,b,c) \\in S$ if and only if $(b,c,a) \\in S$;\n\n- $(a,b,c) \\in S$ if and only if $(c,b,a) \\notin S$;\n\n- $(a,b,c)$ and $(c,d,a)$ are both in $S$ if and only if $(b,c,d)$\nand $(d,a,b)$ are both in $S$.\n\nProve that there exists a one-to-one function $g$ from $A$ to $R$ such\nthat $g(a) < g(b) < g(c)$ implies $(a,b,c) \\in S$. Note: $R$ is the\nset of real numbers.","subjects":["combinatorics","order theory","finite algebra","cyclic orders"],"errata":"Read $S$ as a subset of the ordered triples of pairwise distinct elements of $A$. Apply the reversal axiom only to pairwise distinct elements. The corrected assertion includes sets with fewer than three elements. The original all-triples reading has inconsistent hypotheses when $|A|\\ge3$, so that reading is vacuous rather than a true-premise counterexample to the implication."},{"id":"1996-A5","year":1996,"session":"A","number":5,"statement":"If $p$ is a prime number greater than 3 and $k = \\lfloor 2p/3\n\\rfloor$, prove that the sum\n$$\n\\binom p1 + \\binom p2 + \\cdots + \\binom pk\n$$\nof binomial coefficients is divisible by $p^2$.","subjects":["number theory","binomial coefficients"]},{"id":"1996-A6","year":1996,"session":"A","number":6,"statement":"Let $c>0$ be a constant. Give a complete description, with proof, of\nthe set of all continuous functions $f: R \\to R$ such that $f(x) =\nf(x^2+c)$ for all $x \\in R$. Note that $R$ denotes the set of real numbers.","subjects":["analysis","dynamical systems"]},{"id":"1996-B1","year":1996,"session":"B","number":1,"statement":"Define a selfish set to be a set which has its own\ncardinality (number of elements) as an element. Find, with proof, the\nnumber of subsets of $\\{1, 2, \\ldots, n\\}$ which are minimal\nselfish sets, that is, selfish sets none of whose proper subsets is selfish.","subjects":["combinatorics"]},{"id":"1996-B2","year":1996,"session":"B","number":2,"statement":"Show that for every positive integer $n$,\n$$\n\\left( \\frac{2n-1}{e} \\right)^{\\frac{2n-1}{2}} < 1 \\cdot 3 \\cdot 5\n\\cdots (2n-1) < \\left( \\frac{2n+1}{e} \\right)^{\\frac{2n+1}{2}}.\n$$","subjects":["analysis","inequalities"]},{"id":"1996-B3","year":1996,"session":"B","number":3,"statement":"Given that $\\{x_1, x_2, \\ldots, x_n\\} = \\{1, 2, \\ldots, n\\}$, find,\nwith proof, the largest possible value, as a function of $n$ (with $n\n\\geq 2$), of\n$$\nx_1x_2 + x_2x_3 + \\cdots + x_{n-1}x_n + x_nx_1.\n$$","subjects":["combinatorics","optimization","cyclic permutations"]},{"id":"1996-B4","year":1996,"session":"B","number":4,"statement":"For any square matrix $A$, we can define $\\sin A$ by the usual power\nseries:\n$$\n\\sin A = \\sum_{n=0}^\\infty \\frac{(-1)^n}{(2n+1)!} A^{2n+1}.\n$$\nProve or disprove: there exists a $2 \\times 2$ matrix $A$ with real\nentries such that\n$$\n\\sin A = \\left( \\begin{array}{cc} 1 & 1996 \\\\ 0 & 1 \\end{array} \\right).\n$$","subjects":["linear algebra","matrix functions","jordan form"]},{"id":"1996-B5","year":1996,"session":"B","number":5,"statement":"Given a finite string $S$ of symbols $X$ and $O$, we write $\\Delta(S)$\nfor the number of $X$'s in $S$ minus the number of $O$'s. For example,\n$\\Delta(XOOXOOX) = -1$. We call a string $S$ balanced if every\nsubstring $T$ of (consecutive symbols of) $S$ has $-2 \\leq \\Delta(T)\n\\leq 2$. Thus, $XOOXOOX$ is not balanced, since it contains the\nsubstring $OOXOO$. Find, with proof, the number of balanced strings of\nlength $n$.","subjects":["combinatorics","binary strings","lattice paths"]},{"id":"1996-B6","year":1996,"session":"B","number":6,"statement":"Let $(a_1, b_1), (a_2, b_2), \\ldots, (a_n, b_n)$ be the vertices of a\nconvex polygon which contains the origin in its interior. Prove that\nthere exist positive real numbers $x$ and $y$ such that\n$$\n\\begin{gathered}\n(a_1, b_1)x^{a_1} y^{b_1} + (a_2, b_2)x^{a_2}y^{b_2} + \\cdots \\\\\n+ (a_n, b_n)x^{a_n}y^{b_n} = (0,0).\n\\end{gathered}\n$$","subjects":["analysis","convex geometry","exponential functions"]},{"id":"1997-A1","year":1997,"session":"A","number":1,"statement":"A rectangle, $HOMF$, has sides $HO=11$ and $OM=5$.  A triangle\n$ABC$ has $H$ as the intersection of the altitudes, $O$ the center of\nthe circumscribed circle, $M$ the midpoint of $BC$, and $F$ the foot of the\naltitude from $A$.  What is the length of $BC$?","subjects":["geometry","triangle centers","rectangles"]},{"id":"1997-A2","year":1997,"session":"A","number":2,"statement":"Players $1,2,3,\\ldots,n$ are seated around a table, and each has\na single penny.  Player 1 passes a penny to player 2, who then passes\ntwo pennies to player 3.  Player 3 then passes one penny to Player 4,\nwho passes two pennies to Player 5, and so on, players alternately\npassing one penny or two to the next player who still has some\npennies.  A player who runs out of pennies drops out of the game and\nleaves the table.  Find an infinite set of numbers $n$ for which some\nplayer ends up with all $n$ pennies.","subjects":["combinatorics","games"]},{"id":"1997-A3","year":1997,"session":"A","number":3,"statement":"Evaluate\n$$\n\\begin{gathered}\n\\int_0^\\infty \\left(x-\\frac{x^3}{2}+\\frac{x^5}{2\\cdot\n4}-\\frac{x^7}{2\\cdot 4\\cdot 6}+\\cdots\\right) \\\\\n\\left(1+\\frac{x^2}{2^2}+\n\\frac{x^4}{2^2\\cdot 4^2}+\\frac{x^6}{2^2\\cdot 4^2 \\cdot 6^2}+\\cdots\\right)\\,dx.\n\\end{gathered}\n$$","subjects":["analysis","improper integrals","power series"]},{"id":"1997-A4","year":1997,"session":"A","number":4,"statement":"Let $G$ be a group with identity $e$ and $\\phi:G\\rightarrow G$\na function such that\n$$\\phi(g_1)\\phi(g_2)\\phi(g_3)=\\phi(h_1)\\phi(h_2)\\phi(h_3)$$\nwhenever $g_1g_2g_3=e=h_1h_2h_3$.  Prove that there exists an element\n$a\\in G$ such that $\\psi(x)=a\\phi(x)$ is a homomorphism (i.e.\n$\\psi(xy)=\\psi(x)\\psi(y)$ for all $x,y\\in G$).","subjects":["algebra","group theory"]},{"id":"1997-A5","year":1997,"session":"A","number":5,"statement":"Let $N_n$ denote the number of ordered $n$-tuples of positive\nintegers $(a_1,a_2,\\ldots,a_n)$ such that $1/a_1 + 1/a_2 +\\ldots +\n1/a_n=1$.  Determine whether $N_{10}$ is even or odd.","subjects":["combinatorics","number theory"]},{"id":"1997-A6","year":1997,"session":"A","number":6,"statement":"For a positive integer $n$ and any real number $c$, define\n$x_k$ recursively by $x_0=0$, $x_1=1$, and for $k\\geq 0$,\n$$x_{k+2}=\\frac{cx_{k+1}-(n-k)x_k}{k+1}.$$\nFix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$.\nFind $x_k$ in terms of $n$ and $k$, $1\\leq k\\leq n$.","subjects":["algebra","recurrences","eigenvalues"]},{"id":"1997-B1","year":1997,"session":"B","number":1,"statement":"Let $\\{x\\}$ denote the distance between the real number $x$ and the\nnearest integer.  For each positive integer $n$, evaluate\n$$F_n=\\sum_{m=1}^{6n-1} \\min(\\{\\frac{m}{6n}\\},\\{\\frac{m}{3n}\\}).$$\n(Here $\\min(a,b)$ denotes the minimum of $a$ and $b$.)","subjects":["analysis","combinatorics"]},{"id":"1997-B2","year":1997,"session":"B","number":2,"statement":"Let $f$ be a twice-differentiable real-valued function satisfying\n$$f(x)+f''(x)=-xg(x)f'(x),$$\nwhere $g(x)\\geq 0$ for all real $x$.  Prove that $|f(x)|$ is bounded.","subjects":["analysis","differential equations"]},{"id":"1997-B3","year":1997,"session":"B","number":3,"statement":"For each positive integer $n$, write the sum $\\sum_{m=1}^n\n1/m$ in the form $p_n/q_n$, where $p_n$ and $q_n$ are relatively prime\npositive integers.  Determine all $n$ such that 5 does not divide $q_n$.","subjects":["number theory"]},{"id":"1997-B4","year":1997,"session":"B","number":4,"statement":"Let $a_{m,n}$ denote the coefficient of $x^n$ in the expansion of\n$(1+x+x^2)^m$.  Prove that for all [integers] $k\\geq 0$,\n$$0\\leq \\sum_{i=0}^{\\lfloor \\frac{2k}{3}\\rfloor} (-1)^i a_{k-i,i}\\leq\n1.$$","subjects":["combinatorics","algebra","signed enumeration","linear algebra","graph theory"]},{"id":"1997-B5","year":1997,"session":"B","number":5,"statement":"Prove that for $n\\geq 2$,\n$$\n\\overbrace{2^{2^{\\cdots^{2}}}}^{n\\text{ terms}} \\equiv\n\\overbrace{2^{2^{\\cdots^{2}}}}^{n-1\\text{ terms}} \\quad \\pmod{n}.\n$$","subjects":["number theory"]},{"id":"1997-B6","year":1997,"session":"B","number":6,"statement":"The dissection of the 3--4--5 triangle formed by joining the midpoints of its sides (into four\ncongruent right triangles similar to the original)\nhas diameter $5/2$.\nFind the least diameter of a dissection of this triangle into four parts.\n(The diameter of a dissection is the least upper bound of the distances\nbetween pairs of points belonging to the same part.)","subjects":["geometry"]},{"id":"1998-A1","year":1998,"session":"A","number":1,"statement":"A right circular cone has base of radius 1 and height 3.  A\ncube is inscribed in the cone so that one face of the cube is\ncontained in the base of the cone.  What is the side-length of\nthe cube?","subjects":["geometry"]},{"id":"1998-A2","year":1998,"session":"A","number":2,"statement":"Let $s$ be any arc of the unit circle lying entirely in the first\nquadrant.  Let $A$ be the area of the region lying below $s$ and\nabove the $x$-axis and let $B$ be the area of the region lying to the\nright of the $y$-axis and to the left of $s$.  Prove that $A+B$ depends\nonly on the arc length, and not on the position, of $s$.","subjects":["geometry","analysis","circle arcs","area","calculus"]},{"id":"1998-A3","year":1998,"session":"A","number":3,"statement":"Let $f$ be a real function on the real line with continuous third\nderivative.  Prove that there exists a point $a$ such that\n$$f(a)\\cdot f'(a) \\cdot f''(a) \\cdot f'''(a)\\geq 0 .$$","subjects":["analysis","real analysis","differential inequalities"]},{"id":"1998-A4","year":1998,"session":"A","number":4,"statement":"Let $A_1=0$ and $A_2=1$.  For $n>2$, the number $A_n$ is defined by\nconcatenating the decimal expansions of $A_{n-1}$ and $A_{n-2}$ from\nleft to right.  For example $A_3=A_2 A_1=10$, $A_4=A_3 A_2 = 101$,\n$A_5=A_4 A_3 = 10110$, and so forth.  Determine all $n$ such that\n$11$ divides $A_n$.","subjects":["number theory","combinatorics"]},{"id":"1998-A5","year":1998,"session":"A","number":5,"statement":"Let $\\mathcal F$ be a finite collection of open discs in $\\mathbb R^2$\nwhose union contains a set $E\\subseteq \\mathbb R^2$.  Show that there\nis a pairwise disjoint subcollection $D_1,\\ldots, D_n$ in $\\mathcal F$\nsuch that\n$$E\\subseteq \\cup_{j=1}^n 3D_j.$$\nHere, if $D$ is the disc of radius $r$ and center $P$, then $3D$ is the\ndisc of radius $3r$ and center $P$.","subjects":["geometry","analysis"]},{"id":"1998-A6","year":1998,"session":"A","number":6,"statement":"Let $A, B, C$ denote distinct points with integer coordinates in $\\mathbb\nR^2$.  Prove that if\n$$(|AB|+|BC|)^2<8\\cdot [ABC]+1$$\nthen $A, B, C$ are three vertices of a square.  Here $|XY|$ is the length\nof segment $XY$ and $[ABC]$ is the area of triangle $ABC$.","subjects":["geometry","number theory"]},{"id":"1998-B1","year":1998,"session":"B","number":1,"statement":"Find the minimum value of\n$$\\frac{(x+1/x)^6-(x^6+1/x^6)-2}{(x+1/x)^3+(x^3+1/x^3)}$$\nfor $x>0$.","subjects":["algebra","inequalities"]},{"id":"1998-B2","year":1998,"session":"B","number":2,"statement":"Given a point $(a,b)$ with $0<b<a$, determine the minimum perimeter of a\ntriangle with one vertex at $(a,b)$, one on the $x$-axis, and one on the\nline $y=x$.  You may assume that a triangle of minimum perimeter exists.","subjects":["geometry","optimization"]},{"id":"1998-B3","year":1998,"session":"B","number":3,"statement":"let $H$ be the unit hemisphere $\\{(x,y,z):x^2+y^2+z^2=1,z\\geq 0\\}$, $C$\nthe unit circle $\\{(x,y,0):x^2+y^2=1\\}$, and $P$ the regular pentagon\ninscribed in $C$.  Determine the surface area of that portion of $H$ lying\nover the planar region inside $P$, and write your answer in the form\n$A \\sin\\alpha + B \\cos\\beta$, where $A,B,\\alpha,\\beta$ are real numbers.","subjects":["geometry","surface area","sphere","spherical caps"]},{"id":"1998-B4","year":1998,"session":"B","number":4,"statement":"Find necessary and sufficient conditions on positive integers $m$ and $n$\nso that\n$$\\sum_{i=0}^{mn-1} (-1)^{\\lfloor i/m \\rfloor +\\lfloor i/n\\rfloor}=0.$$","subjects":["number theory","combinatorics"]},{"id":"1998-B5","year":1998,"session":"B","number":5,"statement":"Let $N$ be the positive integer with 1998 decimal digits, all of them 1;\nthat is,\n$$N=1111\\cdots 11.$$\nFind the thousandth digit after the decimal point of $\\sqrt N$.","subjects":["number theory","analysis"]},{"id":"1998-B6","year":1998,"session":"B","number":6,"statement":"Prove that, for any integers $a, b, c$, there exists a positive integer\n$n$ such that $\\sqrt{n^3+an^2+bn+c}$ is not an integer.","subjects":["number theory","polynomial values","perfect squares"]},{"id":"1999-A1","year":1999,"session":"A","number":1,"statement":"Find polynomials $f(x)$,$g(x)$, and $h(x)$, if they exist, such\nthat for all $x$,\n$$\n|f(x)|-|g(x)|+h(x) = \\begin{cases} -1 & \\text{if }x<-1 \\\\\n3x+2 & \\text{if }-1 \\leq x \\leq 0 \\\\\n-2x+2 & \\text{if }x>0\\text{.}\n\\end{cases}\n$$","subjects":["algebra","analysis"]},{"id":"1999-A2","year":1999,"session":"A","number":2,"statement":"Let $p(x)$ be a polynomial that is nonnegative for all real $x$.  Prove that\nfor some $k$, there are polynomials $f_1(x),\\dots,f_k(x$) such that\n$$p(x) =  \\sum_{j=1}^k (f_j(x))^2.$$","subjects":["algebra","real polynomials","sums of squares"]},{"id":"1999-A3","year":1999,"session":"A","number":3,"statement":"Consider the power series expansion\n$$\\frac{1}{1-2x-x^2} = \\sum_{n=0}^\\infty a_n x^n.$$\nProve that, for each integer $n\\geq 0$, there is an integer $m$ such that\n$$a_n^2 + a_{n+1}^2 = a_m .$$","subjects":["algebra","recurrences","pell numbers"]},{"id":"1999-A4","year":1999,"session":"A","number":4,"statement":"Sum the series\n$$\\sum_{m=1}^\\infty \\sum_{n=1}^\\infty \\frac{m^2 n}{3^m(n3^m+m3^n)}.$$","subjects":["analysis"]},{"id":"1999-A5","year":1999,"session":"A","number":5,"statement":"Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial\nof degree 1999, then\n$$|p(0)|\\leq C \\int_{-1}^1 |p(x)|\\,dx.$$","subjects":["analysis","polynomials","norms","finite dimensional spaces"]},{"id":"1999-A6","year":1999,"session":"A","number":6,"statement":"The sequence $(a_n)_{n\\geq 1}$ is defined by $a_1=1, a_2=2, a_3=24,$ and, for $n\\geq 4$,\n$$a_n = \\frac{6a_{n-1}^2a_{n-3} -\n8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}.$$\nShow that, for all n, $a_n$ is an integer multiple of $n$.","subjects":["number theory","nonlinear recurrences","divisibility"]},{"id":"1999-B1","year":1999,"session":"B","number":1,"statement":"Right triangle $ABC$ has right angle at $C$ and $\\angle BAC =\\theta$;\nthe point $D$ is chosen on $AB$ so that $|AC|=|AD|=1$; the point $E$\nis chosen on $BC$ so that $\\angle CDE = \\theta$.  The perpendicular\nto $BC$ at $E$ meets $AB$ at $F$.  Evaluate $\\lim_{\\theta\\rightarrow 0}\n|EF|$.","subjects":["geometry","analysis"]},{"id":"1999-B2","year":1999,"session":"B","number":2,"statement":"Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P''(x)$,\nwhere $Q(x)$ is a quadratic polynomial and $P''(x)$ is the second\nderivative of $P(x)$.  Show that if $P(x)$ has at least two distinct\nroots then it must have $n$ distinct roots.","subjects":["algebra","polynomial roots","multiplicity","differential equations"]},{"id":"1999-B3","year":1999,"session":"B","number":3,"statement":"Let $A=\\{(x,y):0\\leq x,y<1\\}$.  For $(x,y)\\in A$, let\n$$S(x,y) = \\sum_{\\frac{1}{2}\\leq \\frac{m}{n}\\leq 2} x^m y^n,$$\nwhere the sum ranges over all pairs $(m,n)$ of positive integers\nsatisfying the indicated inequalities.  Evaluate\n$$\\lim_{(x,y)\\rightarrow (1,1), (x,y)\\in A} (1-xy^2)(1-x^2y)S(x,y).$$","subjects":["analysis","lattice cones"]},{"id":"1999-B4","year":1999,"session":"B","number":4,"statement":"Let $f$ be a real function with a continuous third derivative such that $f(x),\nf'(x), f''(x), f'''(x)$ are positive for all $x$.  Suppose that\n$f'''(x)\\leq f(x)$ for all $x$.  Show that $f'(x)<2f(x)$ for all $x$.","subjects":["analysis","differential inequalities","derivative bounds"]},{"id":"1999-B5","year":1999,"session":"B","number":5,"statement":"For an integer $n\\geq 3$, let $\\theta=2\\pi/n$.  Evaluate the determinant of the\n$n\\times n$ matrix $I+A$, where $I$ is the $n\\times n$ identity matrix and\n$A=(a_{jk})$ has entries $a_{jk}=\\cos(j\\theta+k\\theta)$ for all $j,k$.","subjects":["linear algebra","determinants","trigonometric matrices"]},{"id":"1999-B6","year":1999,"session":"B","number":6,"statement":"Let $S$ be a finite set of integers, each greater than 1.  Suppose that\nfor each integer $n$ there is some $s\\in S$ such that $\\gcd(s,n)=1$ or\n$\\gcd(s,n)=s$.  Show that there exist $s,t\\in S$ such that $\\gcd(s,t)$\nis prime.","subjects":["number theory","gcd","prime divisors"]},{"id":"2000-A1","year":2000,"session":"A","number":1,"statement":"Let $A$ be a positive real number.  What are the possible values of\n$\\sum_{j=0}^\\infty x_j^2$, given that $x_0,x_1,\\ldots$ are positive\nnumbers\nfor which $\\sum_{j=0}^\\infty x_j=A$?","subjects":["analysis"]},{"id":"2000-A2","year":2000,"session":"A","number":2,"statement":"Prove that there exist infinitely many integers $n$ such that\n$n,n+1,n+2$ are each the sum of the squares of two integers.\n[Example: $0=0^2+0^2$, $1=0^2+1^2$, $2=1^2+1^2$.]","subjects":["number theory"]},{"id":"2000-A3","year":2000,"session":"A","number":3,"statement":"The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle, with\nthe\nvertices around the circumference in the given order.  Given that the\npolygon\n$P_1P_3P_5P_7$ is a square of area 5, and the polygon $P_2P_4P_6P_8$ is a\nrectangle of area 4, find the maximum possible area of the octagon.","subjects":["geometry","optimization"]},{"id":"2000-A4","year":2000,"session":"A","number":4,"statement":"Show that the improper integral\n$$ \\lim_{B\\to\\infty}\\int_{0}^B \\sin(x) \\sin(x^2)\\,dx$$\nconverges.","subjects":["analysis","oscillatory integrals","improper integrals","alternating series","infinite series","conditional convergence"]},{"id":"2000-A5","year":2000,"session":"A","number":5,"statement":"Three distinct points with integer coordinates lie in the plane on a\ncircle of radius $r>0$.  Show that two of these points are separated by a\ndistance of at least $r^{1/3}$.","subjects":["geometry","number theory"]},{"id":"2000-A6","year":2000,"session":"A","number":6,"statement":"Let $f(x)$ be a polynomial with integer coefficients.  Define a\nsequence $a_0,a_1,\\ldots$ of integers such that $a_0=0$ and\n$a_{n+1}=f(a_n)$\nfor all $n\\geq 0$.  Prove that if there exists a positive integer $m$ for\nwhich $a_m=0$ then either $a_1=0$ or $a_2=0$.","subjects":["number theory","polynomial iteration","integer cycles"]},{"id":"2000-B1","year":2000,"session":"B","number":1,"statement":"Let $a_j,b_j,c_j$ be integers for $1\\leq j\\leq N$.  Assume for each\n$j$, at least one of $a_j,b_j,c_j$ is odd.  Show that there exist integers\n$r$, $s$, $t$ such that $ra_j+sb_j+tc_j$ is odd for at least $4N/7$ values\nof $j$, $1\\leq j\\leq N$.","subjects":["combinatorics","algebra"]},{"id":"2000-B2","year":2000,"session":"B","number":2,"statement":"Prove that the expression\n$$ \\frac{gcd(m,n)}{n}\\binom{n}{m} $$\nis an integer for all pairs of integers $n\\geq m\\geq 1$.","subjects":["number theory","combinatorics"]},{"id":"2000-B3","year":2000,"session":"B","number":3,"statement":"Let $f(t)=\\sum_{j=1}^N a_j \\sin(2\\pi jt)$, where each $a_j$ is real\nand\n$a_N$ is not equal to 0.  Let $N_k$ denote the number of zeroes (including\nmultiplicities) of $\\frac{d^k f}{dt^k}$.\nProve that\n$$N_0\\leq N_1\\leq N_2\\leq \\cdots \\text{ and } \\lim_{k\\to\\infty} N_k =\n2N.$$\n[Editorial clarification: only zeroes in $[0, 1)$ should be counted.]","subjects":["analysis"]},{"id":"2000-B4","year":2000,"session":"B","number":4,"statement":"Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for\nall $x$.  Show that $f(x)=0$ for $-1\\leq x\\leq 1$.","subjects":["analysis","functional equations","chebyshev dynamics","harmonic analysis","polynomial approximation"]},{"id":"2000-B5","year":2000,"session":"B","number":5,"statement":"Let $S_0$ be a finite set of positive integers.  We define finite\nsets\n$S_1,S_2,\\ldots$ of positive integers as follows:\nthe integer $a$ is in $S_{n+1}$ if and only if exactly one of $a-1$ or $a$ is\nin\n$S_n$.\nShow that there exist infinitely many integers $N$ for which\n$S_N=S_0\\cup\\{N+a: a\\in S_0\\}$.","subjects":["combinatorics","algebra"]},{"id":"2000-B6","year":2000,"session":"B","number":6,"statement":"Let $B$ be a set of more than $2^{n+1}/n$ distinct points with\ncoordinates\nof the form $(\\pm 1,\\pm 1,\\ldots,\\pm 1)$ in $n$-dimensional space with\n$n\\geq 3$.\nShow that there are three distinct points in $B$ which are the vertices of\nan\nequilateral triangle.","subjects":["combinatorics","geometry","linear algebra"]},{"id":"2001-A1","year":2001,"session":"A","number":1,"statement":"Consider a set $S$ and a binary operation $*$, i.e., for each $a,b\\in S$,\n$a*b\\in S$.  Assume $(a*b)*a=b$ for all $a,b\\in S$.  Prove that\n$a*(b*a)=b$ for all $a,b\\in S$.","subjects":["algebra","binary operations","identities"]},{"id":"2001-A2","year":2001,"session":"A","number":2,"statement":"You have coins $C_1,C_2,\\ldots,C_n$.  For each $k$, $C_k$ is biased so\nthat, when tossed, it has probability $1/(2k+1)$ of falling heads.\nIf the $n$ coins are tossed, what is the probability that the number of\nheads is odd?  Express the answer as a rational function of $n$.","subjects":["probability","parity","independent trials","permutation records","involutions"],"errata":"Interpret the tosses as mutually independent, as in the Scholes statement and the intended solution. The marginal probabilities alone do not determine the answer. For $n=2$, nested heads events $U<1/3$ and $U<1/5$ for one uniform $U$ have odd-parity probability $2/15$, not the independent value $2/5$. Under mutual independence the answer is $n/(2n+1)$."},{"id":"2001-A3","year":2001,"session":"A","number":3,"statement":"For each integer $m$, consider the polynomial\n$$P_m(x)=x^4-(2m+4)x^2+(m-2)^2.$$ For what values of $m$ is $P_m(x)$\nthe product of two non-constant polynomials with integer coefficients?","subjects":["algebra","number theory"]},{"id":"2001-A4","year":2001,"session":"A","number":4,"statement":"Triangle $ABC$ has an area 1.  Points $E,F,G$ lie, respectively,\non sides $BC$, $CA$, $AB$ such that $AE$ bisects $BF$ at point $R$,\n$BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$.\nFind the area of the triangle $RST$.","subjects":["geometry"]},{"id":"2001-A5","year":2001,"session":"A","number":5,"statement":"Prove that there are unique positive integers $a$, $n$ such that\n$a^{n+1}-(a+1)^n=2001$.","subjects":["number theory","exponential sequences"]},{"id":"2001-A6","year":2001,"session":"A","number":6,"statement":"Can an arc of a parabola inside a circle of radius 1 have a length\ngreater than 4?","subjects":["analysis","geometry"]},{"id":"2001-B1","year":2001,"session":"B","number":1,"statement":"Let $n$ be an even positive integer.  Write the numbers\n$1,2,\\ldots,n^2$ in the squares of an $n\\times n$ grid so that the\n$k$-th row, from left to right, is\n$$(k-1)n+1,(k-1)n+2,\\ldots, (k-1)n+n.$$\nColor the squares of the grid so that half of the squares in each\nrow and in each column are red and the other half are black (a\ncheckerboard coloring is one possibility).  Prove that for each\ncoloring, the sum of the numbers on the red squares is equal to\nthe sum of the numbers on the black squares.","subjects":["combinatorics","arrays","balanced colorings","graph theory","matchings"]},{"id":"2001-B2","year":2001,"session":"B","number":2,"statement":"Find all pairs of real numbers $(x,y)$ satisfying the system\nof equations\n$$\n\\begin{aligned}\n\\frac{1}{x} + \\frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2) \\\\\n\\frac{1}{x} - \\frac{1}{2y} &= 2(y^4-x^4).\n\\end{aligned}\n$$","subjects":["algebra","polynomial systems","binomial theorem","convexity"]},{"id":"2001-B3","year":2001,"session":"B","number":3,"statement":"For any positive integer $n$, let $\\langle n\\rangle$ denote\nthe closest integer to $\\sqrt{n}$.  Evaluate\n$$\\sum_{n=1}^\\infty \\frac{2^{\\langle n\\rangle}+2^{-\\langle n\\rangle}}\n{2^n}.$$","subjects":["analysis","infinite series","nearest integer"]},{"id":"2001-B4","year":2001,"session":"B","number":4,"statement":"Let $S$ denote the set of rational numbers different from\n$\\{-1,0,1\\}$.  Define $f:S\\rightarrow S$ by $f(x)=x-1/x$.  Prove\nor disprove that\n$$\\bigcap_{n=1}^\\infty f^{(n)}(S) = \\emptyset,$$\nwhere $f^{(n)}$ denotes $f$ composed with itself $n$ times.","subjects":["number theory","rational dynamics","height functions"]},{"id":"2001-B5","year":2001,"session":"B","number":5,"statement":"Let $a$ and $b$ be real numbers in the interval $(0,1/2)$, and\nlet $g$ be a continuous real-valued function such that\n$g(g(x))= ag(x)+bx$ for all real $x$.  Prove that\n$g(x)=cx$ for some constant $c$.","subjects":["analysis","functional equations"]},{"id":"2001-B6","year":2001,"session":"B","number":6,"statement":"Assume that $(a_n)_{n\\geq 1}$ is an increasing sequence of\npositive real numbers such that\n$\\lim a_n/n=0$.  Must there exist infinitely many positive integers\n$n$ such that $a_{n-i}+a_{n+i}<2a_n$ for $i=1,2,\\ldots,n-1$?","subjects":["analysis","sequences"]},{"id":"2002-A1","year":2002,"session":"A","number":1,"statement":"Let $k$ be a fixed positive integer. The $n$-th derivative of\n$\\frac{1}{x^k - 1}$ has the form $\\frac{P_n(x)}{(x^k - 1)^{n+1}}$\nwhere $P_n(x)$ is a polynomial. Find $P_n(1)$.","subjects":["analysis","derivatives","rational functions","enumerative combinatorics","power series"]},{"id":"2002-A2","year":2002,"session":"A","number":2,"statement":"Given any five points on a sphere, show that some four of them\nmust lie on a closed hemisphere.","subjects":["geometry","sphere","hemispheres","convex geometry","linear algebra"]},{"id":"2002-A3","year":2002,"session":"A","number":3,"statement":"Let $n \\geq 2$ be an integer and $T_n$ be the number of non-empty\nsubsets $S$ of $\\{1, 2, 3, \\dots, n\\}$ with the property that the\naverage of the elements of $S$ is an integer. Prove that\n$T_n - n$ is always even.","subjects":["combinatorics","number theory"]},{"id":"2002-A4","year":2002,"session":"A","number":4,"statement":"In Determinant Tic-Tac-Toe, Player 1 enters a 1 in an empty\n$3 \\times 3$ matrix. Player 0 counters with a 0 in a vacant position,\nand play continues in turn until the $3 \\times 3$ matrix is\ncompleted with five 1's and four 0's. Player 0 wins if the\ndeterminant is 0 and player 1 wins otherwise. Assuming both\nplayers pursue optimal strategies, who will win and how?","subjects":["linear algebra","combinatorics","games"]},{"id":"2002-A5","year":2002,"session":"A","number":5,"statement":"Define a sequence by $a_0=1$, together with the rules\n$a_{2n+1} = a_n$ and $a_{2n+2} = a_n + a_{n+1}$ for each\ninteger $n \\geq 0$. Prove that every positive rational number\nappears in the set\n$$\n\\left\\{ \\frac{a_{n-1}}{a_n}: n \\geq 1 \\right\\} =\n\\left\\{ \\frac{1}{1}, \\frac{1}{2}, \\frac{2}{1}, \\frac{1}{3},\n\\frac{3}{2}, \\dots \\right\\}.\n$$","subjects":["number theory","sequences"]},{"id":"2002-A6","year":2002,"session":"A","number":6,"statement":"Fix an integer $b \\geq 2$. Let $f(1) = 1$, $f(2) = 2$, and for each\n$n \\geq 3$, define $f(n) = n f(d)$, where $d$ is the number of\nbase-$b$ digits of $n$. For which values of $b$ does\n$$\n\\sum_{n=1}^\\infty \\frac{1}{f(n)}\n$$\nconverge?","subjects":["analysis","sequences"]},{"id":"2002-B1","year":2002,"session":"B","number":1,"statement":"Shanille O'Keal shoots free throws on a basketball court. She hits\nthe first and misses the second, and thereafter the probability that\nshe hits the next shot is equal to the proportion of shots she\nhas hit so far. What is the probability she hits exactly 50 of\nher first 100 shots?","subjects":["probability","combinatorics"]},{"id":"2002-B2","year":2002,"session":"B","number":2,"statement":"Consider a polyhedron with at least five faces such that exactly three\nedges emerge from each of its vertices. Two players play the following\ngame:\n\nEach player, in turn, signs his or her name on a previously\nunsigned face. The winner is the player who first succeeds in\nsigning three faces that share a common vertex.\n\nShow that the player who signs first will always win by playing\nas well as possible.","subjects":["combinatorics","polyhedral graphs","positional games","topology","linear algebra"],"errata":"Errata. The book's solution requires connectedness and the usual no-multiple-shared-edge incidence condition. The unrestricted wording permits Walter Stromquist's tetrahedral-notch counterexample. Label an outer tetrahedron $A,B,C,D$ and take a small homothetic tetrahedron $a,b,c,d$ centered on an interior point of edge $AB$, with $a,b$ on $AB$, $c$ inside face $ABC$, and $d$ inside face $ABD$. Removing this small tetrahedron creates two hexagonal faces $H_C=AacbBC$ and $H_D=AadbBD$, two unchanged large triangles $L_A=ACD$, $L_B=BCD$, and two new small triangles $T_a=acd$, $T_b=bcd$. Its eight vertices have incident face triples\n$$\\begin{array}{c|c}\nA&H_C,H_D,L_A\\\\ B&H_C,H_D,L_B\\\\\na&H_C,H_D,T_a\\\\ b&H_C,H_D,T_b\\\\\nC&H_C,L_A,L_B\\\\ D&H_D,L_A,L_B\\\\\nc&H_C,T_a,T_b\\\\ d&H_D,T_a,T_b\n\\end{array}$$\nEvery vertex contains a complete pair of one of the three types, not necessarily the two hexagons. Pair $H_C$ with $H_D$, $L_A$ with $L_B$, and $T_a$ with $T_b$. The second player always signs the other member of the pair just chosen by the first. That response is available because every touched pair previously has one face per player. The first player can never own a whole pair and hence never a winning triple. This blocks a first-player forced win on a six-face cubic polyhedral sphere. The two hexagons share multiple edges, so this is excluded by the precise incidence-qualified reviewTarget proved in the saved tracks. The literal historical statement has not been replaced."},{"id":"2002-B3","year":2002,"session":"B","number":3,"statement":"Show that, for all integers $n > 1$,\n$$\n\\frac{1}{2ne} < \\frac{1}{e} - \\left( 1 - \\frac{1}{n} \\right)^n\n< \\frac{1}{ne}.\n$$","subjects":["analysis","inequalities","exponential limits","integral inequalities","convexity"]},{"id":"2002-B4","year":2002,"session":"B","number":4,"statement":"An integer $n$, unknown to you, has been randomly chosen in the\ninterval $[1, 2002]$ with uniform probability. Your objective is\nto select $n$ in an odd number of guesses. After\neach incorrect guess, you are informed whether $n$ is higher\nor lower, and you must guess an integer on your next turn\namong the numbers that are still feasibly correct. Show that you\nhave a strategy so that the chance of winning is greater than $2/3$.","subjects":["combinatorics","search strategies","probability","dynamic programming","binary trees","comparison search"]},{"id":"2002-B5","year":2002,"session":"B","number":5,"statement":"A palindrome in base $b$ is a positive integer whose base-$b$\ndigits read the same backwards and forwards; for example,\n$2002$ is a 4-digit palindrome in base 10. Note that 200 is not\na palindrome in base 10, but it is the 3-digit palindrome\n242 in base 9, and 404 in base 7. Prove that there is an integer\nwhich is a 3-digit palindrome in base $b$ for at least 2002\ndifferent values of $b$.","subjects":["number theory","palindromes","numeral bases"]},{"id":"2002-B6","year":2002,"session":"B","number":6,"statement":"Let $p$ be a prime number. Prove that the determinant of the matrix\n$$\n\\begin{pmatrix}\nx & y & z \\\\\nx^p & y^p & z^p \\\\\nx^{p^2} & y^{p^2} & z^{p^2}\n\\end{pmatrix}\n$$\nis congruent modulo $p$ to a product of polynomials of the form\n$ax+by+cz$, where $a,b,c$ are integers. (We say two integer\npolynomials are congruent modulo $p$ if corresponding coefficients\nare congruent modulo $p$.)","subjects":["algebra","number theory"]},{"id":"2003-A1","year":2003,"session":"A","number":1,"statement":"Let $n$ be a fixed positive integer. How many ways are there to write  $n$\nas a sum of positive integers,  \n$$\nn = a_1 + a_2 + \\cdots + a_k,\n$$\nwith  $k$  an\narbitrary positive integer and  $a_1 \\le a_2 \\le \\cdots \\le a_k \\le a_1  + 1$?\nFor example, with $n=4$ there are four ways: 4, 2+2, 1+1+2, 1+1+1+1.","subjects":["combinatorics","integer partitions","division algorithm"]},{"id":"2003-A2","year":2003,"session":"A","number":2,"statement":"Let $a_1, a_2, \\dots, a_n$  and  $b_1, b_2, \\dots, b_n$\nbe nonnegative real numbers.\nShow that\n$$\n\\begin{aligned}\n& (a_1 a_2 \\cdots a_n)^{1/n} + (b_1 b_2 \\cdots b_n)^{1/n} \\\\\n&\\leq [(a_1+b_1) (a_2+b_2) \\cdots (a_n + b_n) ]^{1/n}.\n\\end{aligned}\n$$","subjects":["inequalities","arithmetic-geometric mean","convexity"]},{"id":"2003-A3","year":2003,"session":"A","number":3,"statement":"Find the minimum value of\n$$\n| \\sin x + \\cos x + \\tan x + \\cot x + \\sec x + \\csc x |\n$$\nfor real numbers  $x$.","subjects":["trigonometry","inequalities","optimization"]},{"id":"2003-A4","year":2003,"session":"A","number":4,"statement":"Suppose that $a,b,c,A,B,C$  are real numbers, $a\\ne 0$ and $A \\ne 0$, such that\n$$\n| a x^2 + b x + c | \\leq | A x^2 + B x + C |\n$$\nfor all real numbers  $x$. Show that\n$$\n| b^2 - 4 a c | \\leq | B^2 - 4 A C |.\n$$","subjects":["algebra","quadratic polynomials","discriminants"]},{"id":"2003-A5","year":2003,"session":"A","number":5,"statement":"A Dyck $n$-path is a lattice path of  $n$  upsteps $(1,1)$ and  $n$\ndownsteps $(1,-1)$\nthat starts at the origin  $O$  and never dips below the  $x$-axis.\nA return is a maximal sequence of contiguous downsteps that terminates\non the  $x$-axis. For example, the Dyck 5-path with successive vertices $(0,0),(1,1),(2,2),(3,1),(4,2),(5,3),(6,2),(7,1),(8,0),(9,1),(10,0)$ has two returns, of length 3 and 1 respectively.\n\nShow that there is a one-to-one correspondence between the Dyck  $n$-paths\nwith no return of even length and the Dyck $(n-1)$-paths.","subjects":["combinatorics","dyck paths","catalan numbers"]},{"id":"2003-A6","year":2003,"session":"A","number":6,"statement":"For a set  $S$  of nonnegative integers, let  $r_S(n)$  denote the number of\nordered pairs  $(s_1, s_2)$  such that  $s_1 \\in S$, $s_2 \\in S$, $s_1 \\ne\ns_2$,\nand  $s_1 + s_2 = n$.  Is it possible to partition the nonnegative\nintegers into two sets  $A$  and  $B$  in such a way that  $r_A(n) = r_B(n)$\nfor all  $n$ ?","subjects":["combinatorics","number theory","binary expansions","thue-morse sequence"]},{"id":"2003-B1","year":2003,"session":"B","number":1,"statement":"Do there exist polynomials  $a(x), b(x), c(y), d(y)$  such that\n$$\n1 + x y + x^2 y^2 = a(x) c(y) + b(x) d(y)\n$$\nholds identically?","subjects":["algebra","polynomials","linear algebra"]},{"id":"2003-B2","year":2003,"session":"B","number":2,"statement":"Let  $n$  be a positive integer. Starting with the sequence\n$1, \\frac{1}{2}, \\frac{1}{3}, \\dots, \\frac{1}{n}$,\nform a new sequence of  $n-1$  entries\n$\\frac{3}{4}, \\frac{5}{12}, \\dots, \\frac{2n-1}{2n(n-1)}$\nby taking the averages of\ntwo consecutive entries in the first sequence. Repeat the\naveraging of neighbors on the second sequence to obtain a third\nsequence of  $n-2$  entries, and continue until the final sequence produced\nconsists of a single number  $x_n$.  Show that  $x_n < 2/n$.","subjects":["combinatorics","binomial coefficients","sequences"]},{"id":"2003-B3","year":2003,"session":"B","number":3,"statement":"Show that for each positive integer  n,\n$$\nn! = \\prod_{i=1}^n  \\mathrm{lcm}\\{1, 2, \\dots, \\lfloor n/i\\rfloor\\} .\n$$\n(Here $\\mathrm{lcm}$ denotes the least common multiple, and\n$\\lfloor x \\rfloor$ denotes the greatest integer $\\leq x$.)","subjects":["number theory","least common multiples","factorials"]},{"id":"2003-B4","year":2003,"session":"B","number":4,"statement":"Let $f(z) = a z^4 + b z^3 + c z^2 + d z + e = a(z-r_1)(z-r_2)(z-r_3)(z-r_4)$\nwhere  $a,b,c,d,e$  are integers, $a \\ne 0$.  Show that if  $r_1 + r_2$  is a\nrational number and $r_1 + r_2 \\ne r_3 + r_4$,  then  $r_1 r_2$  is a\nrational number.","subjects":["algebra","polynomial roots","galois theory"]},{"id":"2003-B5","year":2003,"session":"B","number":5,"statement":"Let  $A,B$, and $C$  be equidistant points on the circumference of a circle\nof unit radius centered at  $O$,  and let  $P$  be any point in the circle's\ninterior.  Let  $a, b, c$  be the distance from  $P$  to $A, B, C$,\nrespectively.\nShow that there is a triangle with side lengths  $a, b, c$,  and that the\narea of this triangle depends only on the distance from  $P$  to  $O$.","subjects":["geometry","complex numbers","triangle area"]},{"id":"2003-B6","year":2003,"session":"B","number":6,"statement":"Let  $f(x)$  be  a continuous real-valued function defined on the interval\n$[0,1]$. Show that\n$$\n\\int_0^1 \\int_0^1 | f(x) + f(y) |\\,dx\\,dy \\geq \\int_0^1 |f(x)|\\,dx.\n$$","subjects":["analysis","integral inequalities","probability"]},{"id":"2004-A1","year":2004,"session":"A","number":1,"statement":"Basketball star Shanille O'Keal's team statistician\nkeeps track of the number, $S(N)$, of successful free throws she has made\nin her first $N$ attempts of the season.\nEarly in the season, $S(N)$  was less than 80\\% of  $N$,\nbut by the end of the season, $S(N)$ was more than 80\\% of $N$.\nWas there necessarily a moment in between when $S(N)$ was exactly 80\\% of\n$N$?","subjects":["number theory","discrete intermediate value","lattice paths","algebra","combinatorics"]},{"id":"2004-A2","year":2004,"session":"A","number":2,"statement":"For $i = 1,2$ let $T_i$ be a triangle with side lengths $a_i, b_i, c_i$,\nand area $A_i$.  Suppose that $a_1 \\le a_2,  b_1 \\le b_2,  c_1 \\le\nc_2$,\nand that $T_2$ is an acute triangle. Does it follow that $A_1\n\\le A_2$?","subjects":["geometry","triangle area","inequalities","quadratic forms"]},{"id":"2004-A3","year":2004,"session":"A","number":3,"statement":"Define a sequence $\\{ u_n \\}_{n=0}^\\infty$\nby  $u_0 = u_1 = u_2 = 1$, and thereafter by\nthe\ncondition that\n$$\n\\det\\begin{pmatrix}\nu_n &   u_{n+1}\\\\\nu_{n+2} & u_{n+3}\n\\end{pmatrix}\n= n!\n$$\nfor all $n \\ge 0$. Show that $u_n$ is an integer for all $n$.\n(By convention, $0! = 1$.)","subjects":["algebra","sequences"]},{"id":"2004-A4","year":2004,"session":"A","number":4,"statement":"Show that for any positive integer $n$ there is an integer $N$ such that\nthe product $x_1 x_2 \\cdots x_n$ can be expressed identically in the form\n$$\nx_1 x_2 \\cdots x_n =\n\\sum_{i=1}^N  c_i\n( a_{i1} x_1 + a_{i2} x_2 + \\cdots + a_{in} x_n )^n\n$$\nwhere the $c_i$ are rational numbers and each $a_{ij}$ is one of the\nnumbers $-1, 0, 1$.","subjects":["algebra"]},{"id":"2004-A5","year":2004,"session":"A","number":5,"statement":"An $m \\times n$ checkerboard is colored randomly: each square is\nindependently\nassigned red or black with probability $1/2$. We say that two squares,\n$p$ and $q$,  are in the same connected monochromatic region if there is\na sequence of squares, all of the same color, starting at $p$ and ending\nat $q$, in which successive squares in the sequence share a common side.\nShow that the expected number of connected monochromatic regions is\ngreater than $m n / 8$.","subjects":["combinatorics","probability","planar graphs"]},{"id":"2004-A6","year":2004,"session":"A","number":6,"statement":"Suppose that $f(x,y)$ is a continuous real-valued function on the unit\nsquare $0 \\le x \\le 1, 0 \\le y \\le 1$.  Show that\n$$\n\\begin{aligned}\n& \\int_0^1 \\left( \\int_0^1  f(x,y) dx \\right)^2 dy +\n\\int_0^1 \\left( \\int_0^1  f(x,y) dy \\right)^2 dx \\\\\n&\\leq\n\\left( \\int_0^1 \\int_0^1  f(x,y) dx\\, dy \\right)^2 +\n\\int_0^1 \\int_0^1 \\left[ f(x,y) \\right]^2 dx\\,dy.\n\\end{aligned}\n$$","subjects":["analysis","linear algebra"]},{"id":"2004-B1","year":2004,"session":"B","number":1,"statement":"Let $P(x) = c_n x^n + c_{n-1} x^{n-1} + \\cdots + c_0$ be a polynomial with\ninteger coefficients. Suppose that $r$ is a rational number such that\n$P(r) = 0$.  Show that the $n$ numbers\n$$\n\\begin{gathered}\nc_n r, \\, c_n r^2 + c_{n-1} r, \\, c_n r^3 + c_{n-1} r^2 + c_{n-2} r, \\\\\n\\dots, \\, c_n r^n + c_{n-1} r^{n-1} + \\cdots + c_1 r\n\\end{gathered}\n$$\nare integers.","subjects":["number theory","polynomials","rational roots"]},{"id":"2004-B2","year":2004,"session":"B","number":2,"statement":"Let $m$ and $n$ be positive integers.  Show that\n$$\n\\frac{(m+n)!}{(m+n)^{m+n}}\n< \\frac{m!}{m^m} \\frac{n!}{n^n}.\n$$","subjects":["inequalities","factorials","binomial coefficients"]},{"id":"2004-B3","year":2004,"session":"B","number":3,"statement":"Determine all real numbers $a > 0$ for which there exists a nonnegative\ncontinuous function $f(x)$ defined on $[0,a]$ with the property that the\nregion\n$$\nR = \\{ (x,y) ; 0\n\\le x \\le a, 0 \\le y \\le\nf(x) \\}\n$$\nhas perimeter $k$ units and area $k$ square units for some real number $k$.","subjects":["analysis","geometry"]},{"id":"2004-B4","year":2004,"session":"B","number":4,"statement":"Let $n$ be a positive integer, $n \\ge\n2$, and put  $\\theta = 2 \\pi / n$.\nDefine points $P_k = (k,0)$ in the $xy$-plane, for $k = 1, 2\n, \\dots, n$.\nLet $R_k$ be the map that rotates the plane counterclockwise by the\nangle $\\theta$ about the point $P_k$.  Let $R$ denote the map obtained\nby applying, in order, $R_1$,  then $R_2, \\dots$,\nthen $R_n$.\nFor an arbitrary point $(x,y)$, find, and simplify, the coordinates\nof $R(x,y)$.","subjects":["geometry","complex numbers","plane isometries"]},{"id":"2004-B5","year":2004,"session":"B","number":5,"statement":"Evaluate\n$$\n\\lim_{x \\to 1^-} \\prod_{n=0}^\\infty \\left(\\frac{1 + x^{n+1}}{1 +\nx^n}\\right)^{x^n}.\n$$","subjects":["analysis","infinite products","limits"]},{"id":"2004-B6","year":2004,"session":"B","number":6,"statement":"Let $\\mathcal{A}$\nbe a non-empty set of positive integers, and let $N(x)$ denote\nthe number of elements of $\\mathcal{A}$ not exceeding $x$.\nLet $\\mathcal{B}$ denote the set\nof positive integers $b$ that can be written in the form $b = a - a'$ with\n$a \\in \\mathcal{A}$  and $a' \\in  \\mathcal{A}$. Let $b_1 < b_2 < \\cdots$\nbe the members of $\\mathcal{B}$,\nlisted in increasing order. Show that if the sequence $b_{i+1} - b_i$ is\nunbounded, then\n$$\n\\lim_{x \\to\\infty}  N(x)/x  = 0.\n$$","subjects":["number theory","additive combinatorics","asymptotic density"]},{"id":"2005-A1","year":2005,"session":"A","number":1,"statement":"Show that every positive integer is a sum of one or more numbers of the\nform $2^r 3^s$, where $r$ and $s$ are nonnegative integers and no\nsummand divides another.\n(For example, 23 = 9 + 8 + 6.)","subjects":["number theory","divisibility"]},{"id":"2005-A2","year":2005,"session":"A","number":2,"statement":"Let $\\mathbf{S} = \\{(a,b) | a = 1, 2, \\dots,n, b = 1,2,3\\}$.\nA rook tour of $\\mathbf{S}$ is a polygonal path made up of line\nsegments connecting points $p_1, p_2, \\dots, p_{3n}$ in sequence such that\n\n(i) $p_i \\in \\mathbf{S}$,\n\n(ii) $p_i$ and $p_{i+1}$ are a unit distance apart, for\n$1 \\leq i <3n$,\n\n(iii) for each $p \\in \\mathbf{S}$ there is a unique $i$ such that\n$p_i = p$. How many rook tours are there that begin at $(1,1)$\nand end at $(n,1)$?\n\n(An example of such a rook tour for $n=5$ was depicted in the original.)","subjects":["combinatorics","hamiltonian paths","grid graphs"]},{"id":"2005-A3","year":2005,"session":"A","number":3,"statement":"Let $p(z)$ be a polynomial of degree $n$ all of whose zeros have absolute\nvalue 1 in the complex  plane. Put $g(z) = p(z)/z^{n/2}$. Show that all zeros\nof $g'(z) = 0$ have absolute value 1.","subjects":["complex polynomials","root location","unitary matrices","statement erratum"],"errata":"Require positive degree $n\\ge1$ and work away from $z=0$ on a local branch of $z^{-n/2}$. The unrestricted constant case is false: $p(z)=1$ has $g'(2)=0$. The source explicitly treats nonconstancy and the exclusion of zero as implicit. These assumptions are part of the proposed qualified review target, rather than silently inserted into the historical statement."},{"id":"2005-A4","year":2005,"session":"A","number":4,"statement":"Let $H$ be an $n \\times n$ matrix all of whose entries are $\\pm 1$ and\nwhose rows are mutually orthogonal. Suppose $H$ has an $a \\times b$ submatrix\nwhose entries are all $1$. Show that $ab \\leq n$.","subjects":["linear algebra","hadamard matrices","orthogonality"]},{"id":"2005-A5","year":2005,"session":"A","number":5,"statement":"Evaluate $\\int_0^1 \\frac{\\ln(x+1)}{x^2+1}\\,dx$.","subjects":["analysis","definite integrals","logarithms"]},{"id":"2005-A6","year":2005,"session":"A","number":6,"statement":"Let $n$ be given, $n \\geq 4$, and suppose that $P_1, P_2, \\dots, P_n$\nare $n$ randomly, independently and uniformly, chosen points on a circle.\nConsider the convex $n$-gon whose vertices are the $P_i$. What is the\nprobability that at least one of the vertex angles of this polygon is\nacute?","subjects":["probability","random polygons","circular spacings"]},{"id":"2005-B1","year":2005,"session":"B","number":1,"statement":"Find a nonzero polynomial $P(x,y)$ such that $P(\\lfloor a \\rfloor,\n\\lfloor 2a \\rfloor) = 0$ for all real numbers $a$.\n(Note: $\\lfloor \\nu \\rfloor$ is the greatest integer less than\nor equal to $\\nu$.)","subjects":["algebra","floor function","polynomial construction"]},{"id":"2005-B2","year":2005,"session":"B","number":2,"statement":"Find all positive integers $n, k_1, \\dots, k_n$ such that\n$k_1 + \\cdots + k_n = 5n-4$ and\n$$\n\\frac{1}{k_1} + \\cdots + \\frac{1}{k_n} = 1.\n$$","subjects":["number theory","egyptian fractions","diophantine equations","inequalities"]},{"id":"2005-B3","year":2005,"session":"B","number":3,"statement":"Find all differentiable functions $f: (0, \\infty) \\to (0, \\infty)$ for which\nthere is a positive real number $a$ such that\n$$\nf' \\left( \\frac{a}{x} \\right) = \\frac{x}{f(x)}\n$$\nfor all $x > 0$.","subjects":["analysis","functional-differential equations","power functions"]},{"id":"2005-B4","year":2005,"session":"B","number":4,"statement":"For positive integers $m$ and $n$, let $f(m,n)$ denote the number of\n$n$-tuples $(x_1,x_2,\\dots,x_n)$ of integers such that\n$|x_1| + |x_2| + \\cdots + |x_n| \\leq m$.\nShow that $f(m,n) = f(n,m)$.","subjects":["combinatorics","lattice points","delannoy numbers"]},{"id":"2005-B5","year":2005,"session":"B","number":5,"statement":"Let $P(x_1,\\dots,x_n)$ denote a polynomial with real coefficients in the\nvariables $x_1, \\dots, x_n$, and suppose that\n$$\n\\left( \\frac{\\partial^2}{\\partial x_1^2} + \\cdots + \\frac{\\partial^2}{\\partial\nx_n^2}\\right) P(x_1, \\dots,x_n) = 0 \\quad \\text{(identically)}\n$$ \nand that\n$$\nx_1^2 + \\cdots + x_n^2 \\text{ divides } P(x_1, \\dots, x_n).\n$$ \nShow that $P=0$ identically.","subjects":["algebra","harmonic polynomials","partial differential equations"]},{"id":"2005-B6","year":2005,"session":"B","number":6,"statement":"Let $S_n$ denote the set of all permutations of the numbers $1,2,\\dots,n$.\nFor $\\pi \\in S_n$, let $\\sigma(\\pi) = 1$ if $\\pi$ is an even permutation\nand $\\sigma(\\pi) = -1$ if $\\pi$ is an odd permutation.\nAlso, let $\\nu(\\pi)$ denote the number of fixed points of $\\pi$.\nShow that\n$$\n\\sum_{\\pi \\in S_n} \\frac{\\sigma(\\pi)}{\\nu(\\pi) + 1} = (-1)^{n+1}\n\\frac{n}{n+1}.\n$$","subjects":["combinatorics","permutations","determinants"]},{"id":"2006-A1","year":2006,"session":"A","number":1,"statement":"Find the volume of the region of points $(x,y,z)$ such that\n$$\n(x^2 + y^2 + z^2 + 8)^2 \\leq 36(x^2 + y^2).\n$$","subjects":["geometry","multivariable calculus","volume","torus"]},{"id":"2006-A2","year":2006,"session":"A","number":2,"statement":"Alice and Bob play a game in which they take turns removing stones from\na heap that initially has $n$ stones. The number of stones removed at\neach turn must be one less than a prime number. The winner is the player\nwho takes the last stone. Alice plays first. Prove that there are\ninfinitely many $n$ such that Bob has a winning strategy.\n(For example, if $n=17$, then Alice might take 6 leaving 11; then\nBob might take 1 leaving 10; then Alice can take the remaining stones\nto win.)","subjects":["combinatorics","impartial games","prime numbers"]},{"id":"2006-A3","year":2006,"session":"A","number":3,"statement":"Let $1, 2, 3, \\dots, 2005, 2006, 2007, 2009, 2012, 2016, \\dots$\nbe a sequence defined by $x_k = k$ for $k=1, 2, \\dots, 2006$ and\n$x_{k+1} = x_k + x_{k-2005}$ for $k \\geq 2006$. Show that the sequence has\n2005 consecutive terms each divisible by 2006.","subjects":["number theory","sequences"]},{"id":"2006-A4","year":2006,"session":"A","number":4,"statement":"Let $S = \\{1, 2, \\dots, n\\}$ for some integer $n > 1$. Say a permutation\n$\\pi$ of $S$ has a local maximum at $k \\in S$ if\n\n(i) $\\pi(k) > \\pi(k+1)$ for $k=1$;\n\n(ii) $\\pi(k-1) < \\pi(k)$ and $\\pi(k) > \\pi(k+1)$ for $1 < k < n$;\n\n(iii) $\\pi(k-1) < \\pi(k)$ for $k=n$.\n\n(For example, if $n=5$ and $\\pi$ takes values at $1, 2, 3, 4, 5$ of\n$2, 1, 4, 5, 3$, then $\\pi$ has a local maximum of 2 at $k=1$,\nand a local maximum of 5 at $k=4$.)\nWhat is the average number of local maxima of a permutation of $S$,\naveraging over all permutations of $S$?","subjects":["combinatorics","probability","permutation peaks"]},{"id":"2006-A5","year":2006,"session":"A","number":5,"statement":"Let $n$ be a positive odd integer and let $\\theta$ be a real number such\nthat $\\theta/\\pi$ is irrational. Set $a_k = \\tan (\\theta + k \\pi/n)$,\n$k=1,2,\\dots,n$. Prove that\n$$\n\\frac{a_1 + a_2 + \\cdots + a_n}{a_1 a_2 \\cdots a_n}\n$$\nis an integer, and determine its value.","subjects":["trigonometry","polynomial roots","symmetric functions"]},{"id":"2006-A6","year":2006,"session":"A","number":6,"statement":"Four points are chosen uniformly and independently at random in the interior\nof a given circle. Find the probability that they are the vertices\nof a convex quadrilateral.","subjects":["probability","geometric probability","random triangle area"]},{"id":"2006-B1","year":2006,"session":"B","number":1,"statement":"Show that the curve $x^3 + 3xy + y^3 = 1$ contains only one set of three\ndistinct points, $A$, $B$, and $C$, which are vertices of an equilateral\ntriangle, and find its area.","subjects":["algebra","plane curves","equilateral triangles","linear algebra","complex numbers"]},{"id":"2006-B2","year":2006,"session":"B","number":2,"statement":"Prove that, for every set $X = \\{x_1, x_2, \\dots, x_n\\}$ of $n$\nreal numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$\nsuch that\n$$\n\\left| m + \\sum_{s \\in S} s \\right| \\leq \\frac{1}{n+1}.\n$$","subjects":["number theory","diophantine approximation","measure","probability"]},{"id":"2006-B3","year":2006,"session":"B","number":3,"statement":"Let $S$ be a finite set of points in the plane. A linear partition of $S$\nis an unordered pair $\\{A,B\\}$ of subsets of $S$ such that $A \\cup B = S$,\n$A \\cap B = \\emptyset$, and $A$ and $B$ lie on opposite sides of some\nstraight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_S$ be the\nnumber of linear partitions of $S$. For each positive integer $n$, find the\nmaximum of $L_S$ over all sets $S$ of $n$ points.","subjects":["combinatorics","discrete geometry","hyperplane arrangements"]},{"id":"2006-B4","year":2006,"session":"B","number":4,"statement":"Let $Z$ denote the set of points in $\\mathbb{R}^n$ whose coordinates are 0\nor 1. (Thus $Z$ has $2^n$ elements, which are the vertices of a unit\nhypercube in $\\mathbb{R}^n$.) Given a vector subspace $V$\nof $\\mathbb{R}^n$, let $Z(V)$\ndenote the number of members of $Z$ that lie in $V$. Let $k$ be given,\n$0 \\leq k \\leq n$. Find the maximum, over all vector subspaces $V\n\\subseteq \\mathbb{R}^n$ of dimension $k$, of the number of points in\n$V \\cap Z$. [Editorial note: the proposers probably intended to write\n$Z(V)$ instead of\n``the number of points in $V \\cap Z$'', but this changes nothing.]","subjects":["linear algebra","hypercubes","finite fields"]},{"id":"2006-B5","year":2006,"session":"B","number":5,"statement":"For each continuous function $f: [0,1] \\to \\mathbb{R}$, let $I(f) =\n\\int_0^1 x^2 f(x)\\,dx$ and $J(x) = \\int_0^1 x \\left(f(x)\\right)^2\\,dx$.\nFind the maximum value of $I(f) - J(f)$ over all such functions $f$.","subjects":["analysis"]},{"id":"2006-B6","year":2006,"session":"B","number":6,"statement":"Let $k$ be an integer greater than 1. Suppose $a_0 > 0$, and define\n$$\na_{n+1} = a_n + \\frac{1}{\\sqrt[k]{a_n}}\n$$\nfor $n > 0$. Evaluate\n$$\n\\lim_{n \\to \\infty} \\frac{a_n^{k+1}}{n^k}.\n$$","subjects":["analysis","asymptotic sequences","nonlinear recurrences"]},{"id":"2007-A1","year":2007,"session":"A","number":1,"statement":"Find all values of $\\alpha$ for which the curves $y = \\alpha x^2 +\n\\alpha x + \\frac{1}{24}$ and $x = \\alpha y^2 + \\alpha y + \\frac{1}{24}$\nare tangent to each other.","subjects":["algebra","analytic geometry","tangency"]},{"id":"2007-A2","year":2007,"session":"A","number":2,"statement":"Find the least possible area of a convex set in the plane that\nintersects both branches of the hyperbola $xy = 1$ and both branches of\nthe hyperbola $xy = -1$. (A set $S$ in the plane is called convex\nif for any two points in $S$ the line segment connecting them is\ncontained in $S$.)","subjects":["geometry","convex sets","area inequalities","support lines","hyperbolas"]},{"id":"2007-A3","year":2007,"session":"A","number":3,"statement":"Let $k$ be a positive integer. Suppose that the integers $1, 2, 3,\n\\dots, 3k+1$ are written down in random order. What is the probability\nthat at no time during this process, the sum of the integers that have\nbeen written up to that time is a positive integer divisible by 3? Your\nanswer should be in closed form, but may include factorials.","subjects":["probability","combinatorics"]},{"id":"2007-A4","year":2007,"session":"A","number":4,"statement":"A repunit is a positive integer whose digits in base 10 are\nall ones. Find all polynomials $f$ with real coefficients such that if\n$n$ is a repunit, then so is $f(n)$.","subjects":["algebra","polynomial rigidity","repunit numbers","diophantine equations"]},{"id":"2007-A5","year":2007,"session":"A","number":5,"statement":"Suppose that a finite group has exactly $n$ elements of order $p$,\nwhere $p$ is a prime. Prove that either $n=0$ or $p$ divides $n+1$.","subjects":["group theory","finite groups","group actions","sylow subgroups","subgroup counting"]},{"id":"2007-A6","year":2007,"session":"A","number":6,"statement":"A triangulation $\\mathcal{T}$ of a polygon $P$ is a finite\ncollection of triangles whose union is $P$, and such that the\nintersection of any two triangles is either empty, or a shared vertex,\nor a shared side. Moreover, each side is a side of exactly one triangle\nin $\\mathcal{T}$. Say that $\\mathcal{T}$ is admissible if every\ninternal vertex is shared by 6 or more triangles. Prove that there is an integer $M_n$, depending only on $n$,\nsuch that any admissible triangulation of a polygon $P$ with $n$ sides\nhas at most $M_n$ triangles.","subjects":["geometry","combinatorics"]},{"id":"2007-B1","year":2007,"session":"B","number":1,"statement":"Let $f$ be a polynomial with positive integer coefficients. Prove\nthat if $n$ is a positive integer, then $f(n)$ divides $f(f(n)+1)$ if\nand only if $n=1$. [Editor's note: one must assume $f$ is nonconstant.]","subjects":["number theory","polynomial congruences","group actions","combinatorial counting"]},{"id":"2007-B2","year":2007,"session":"B","number":2,"statement":"Suppose that $f: [0,1] \\to \\mathbb{R}$ has a continuous derivative\nand that $\\int_0^1 f(x)\\,dx = 0$. Prove that for every $\\alpha \\in (0,1)$,\n$$\n\\left| \\int_0^\\alpha f(x)\\,dx \\right| \\leq \\frac{1}{8} \\max_{0 \\leq x\n\\leq 1} |f'(x)|.\n$$","subjects":["analysis","integral inequalities","lipschitz functions"]},{"id":"2007-B3","year":2007,"session":"B","number":3,"statement":"Let $x_0 = 1$ and for $n \\geq 0$, let $x_{n+1} = 3x_n + \\lfloor x_n\n\\sqrt{5} \\rfloor$. In particular, $x_1 = 5$, $x_2 = 26$, $x_3 = 136$,\n$x_4 = 712$. Find a closed-form expression for $x_{2007}$. ($\\lfloor a\n\\rfloor$ means the largest integer $\\leq a$.)","subjects":["number theory","floor recurrences","linear recurrences"]},{"id":"2007-B4","year":2007,"session":"B","number":4,"statement":"Let $n$ be a positive integer. Find the number of pairs $P, Q$ of\npolynomials with real coefficients such that\n$$\n(P(X))^2 + (Q(X))^2 = X^{2n} + 1\n$$\nand $\\deg P > \\deg Q$.","subjects":["algebra","analysis"]},{"id":"2007-B5","year":2007,"session":"B","number":5,"statement":"Let $k$ be a positive integer. Prove that there exist polynomials\n$P_0(n), P_1(n), \\dots, P_{k-1}(n)$ (which may depend on $k$) such that\nfor any integer $n$,\n$$\n\\left\\lfloor \\frac{n}{k} \\right\\rfloor^k = P_0(n) + P_1(n) \\left\\lfloor\n\\frac{n}{k} \\right\\rfloor + \\cdots + P_{k-1}(n) \\left\\lfloor \\frac{n}{k}\n\\right\\rfloor^{k-1}.\n$$\n($\\lfloor a \\rfloor$ means the largest integer $\\leq a$.)","subjects":["algebra"]},{"id":"2007-B6","year":2007,"session":"B","number":6,"statement":"For each positive integer $n$, let $f(n)$ be the number of ways to\nmake $n!$ cents using an unordered collection of coins, each worth $k!$\ncents for some $k$, $1 \\leq k \\leq n$. Prove that for some constant $C$,\nindependent of $n$,\n$$\nn^{n^2/2 - Cn} e^{-n^2/4} \\leq f(n) \\leq n^{n^2/2 + Cn}e^{-n^2/4}.\n$$","subjects":["combinatorics","analysis"],"errata":"Replace positive integer $n$ with integer $n\\ge2$. At $n=1$, $f(1)=1$, while the proposed upper bound is $e^{-1/4}<1$ for every $C$."},{"id":"2008-A1","year":2008,"session":"A","number":1,"statement":"Let $f: \\mathbb{R}^2 \\to \\mathbb{R}$ be a function such that $f(x,y) + f(y,z)\n+ f(z,x) = 0$ for all real numbers $x$, $y$, and $z$. Prove that there exists\na function $g: \\mathbb{R} \\to \\mathbb{R}$ such that $f(x,y) = g(x) - g(y)$\nfor all real numbers $x$ and $y$.","subjects":["algebra","functional equations","cocycles","graph theory"]},{"id":"2008-A2","year":2008,"session":"A","number":2,"statement":"Alan and Barbara play a game in which they take turns filling entries of\nan initially empty $2008 \\times 2008$ array. Alan plays first. At each\nturn, a player chooses a real number and places it in a vacant entry.\nThe game ends when all the entries are filled. Alan wins if the\ndeterminant of the resulting matrix is nonzero; Barbara wins if it is zero.\nWhich player has a winning strategy?","subjects":["combinatorics","strategy games","determinants","determinant games","permutation symmetry"]},{"id":"2008-A3","year":2008,"session":"A","number":3,"statement":"Start with a finite sequence $a_1, a_2, \\dots, a_n$ of positive integers.\nIf possible, choose two indices $j < k$ such that $a_j$ does not divide\n$a_k$, and replace $a_j$ and $a_k$ by $\\mathrm{gcd}(a_j, a_k)$\nand $\\mathrm{lcm}(a_j, a_k)$, respectively. Prove that if this process is\nrepeated, it must eventually stop and the final sequence does not depend\non the choices made. (Note: gcd means greatest common divisor and lcm\nmeans least common multiple.)","subjects":["number theory","algorithms","gcd and lcm","distributive lattices","rewriting systems","gcd-lcm sorting"]},{"id":"2008-A4","year":2008,"session":"A","number":4,"statement":"Define $f: \\mathbb{R} \\to \\mathbb{R}$ by\n$$\nf(x) = \\begin{cases} x & \\text{if }x \\leq e \\\\ x f(\\ln x) &\n\\text{if }x > e\\text{.} \\end{cases}\n$$\nDoes $\\sum_{n=1}^\\infty \\frac{1}{f(n)}$ converge?","subjects":["analysis","sequences"]},{"id":"2008-A5","year":2008,"session":"A","number":5,"statement":"Let $n \\geq 3$ be an integer. Let $f(x)$ and $g(x)$ be polynomials\nwith real coefficients such that the points\n$(f(1), g(1)), (f(2), g(2)), \\dots, (f(n), g(n))$\nin $\\mathbb{R}^2$ are the vertices of a regular $n$-gon in\ncounterclockwise order. Prove that at least one of $f(x)$\nand $g(x)$ has degree greater than or equal to $n-1$.","subjects":["algebra","geometry"]},{"id":"2008-A6","year":2008,"session":"A","number":6,"statement":"Prove that there exists a constant $c>0$ such that in every\nnontrivial finite group $G$ there exists a sequence of length\nat most $c \\log |G|$ with the property that each element of $G$\nequals the product of some subsequence. (The elements of $G$ in the\nsequence are not required to be distinct. A subsequence\nof a sequence is obtained by selecting some of the terms,\nnot necessarily consecutive, without reordering them; for\nexample, $4, 4, 2$ is a subsequence of $2, 4, 6, 4, 2$, but\n$2, 2, 4$ is not.)","subjects":["group theory","combinatorial group theory","logarithmic constructions"]},{"id":"2008-B1","year":2008,"session":"B","number":1,"statement":"What is the maximum number of rational points that can lie on a circle\nin $\\mathbb{R}^2$ whose center is not a rational point? (A rational\npoint is a point both of whose coordinates are rational numbers.)","subjects":["geometry","rational points","circle centers","linear algebra"]},{"id":"2008-B2","year":2008,"session":"B","number":2,"statement":"Let $F_0(x) = \\ln x$. For $n \\geq 0$ and $x > 0$, let\n$F_{n+1}(x) = \\int_0^x F_n(t)\\,dt$. Evaluate\n$$\n\\lim_{n \\to \\infty} \\frac{n! F_n(1)}{\\ln n}.\n$$","subjects":["analysis","repeated integration","harmonic numbers"]},{"id":"2008-B3","year":2008,"session":"B","number":3,"statement":"What is the largest possible radius of a circle contained in a 4-dimensional\nhypercube of side length 1?","subjects":["geometry","linear algebra"]},{"id":"2008-B4","year":2008,"session":"B","number":4,"statement":"Let $p$ be a prime number. Let $h(x)$ be a polynomial with integer coefficients\nsuch that $h(0), h(1), \\dots, h(p^2-1)$ are distinct modulo $p^2$.\nShow that $h(0), h(1), \\dots, h(p^3-1)$ are distinct modulo $p^3$.","subjects":["number theory","algebra"]},{"id":"2008-B5","year":2008,"session":"B","number":5,"statement":"Find all continuously differentiable functions $f: \\mathbb{R} \\to \\mathbb{R}$\nsuch that for every rational number $q$, the number $f(q)$ is rational\nand has the same denominator as $q$. (The denominator of a rational number\n$q$ is the unique positive integer $b$ such that $q = a/b$\nfor some integer $a$ with $\\mathrm{gcd}(a,b) = 1$.)\n(Note: gcd means greatest common\ndivisor.)","subjects":["analysis","number theory"]},{"id":"2008-B6","year":2008,"session":"B","number":6,"statement":"Let $n$ and $k$ be positive integers. Say that a permutation $\\sigma$\nof $\\{1,2,\\dots,n\\}$ is $k$-limited if $|\\sigma(i) - i| \\leq k$\nfor all $i$. Prove that the number of $k$-limited permutations of\n$\\{1,2,\\dots,n\\}$ is odd if and only if $n \\equiv 0$ or $1$\n(mod $2k+1$).","subjects":["combinatorics","permutation parity","banded matrices","schur complements"]},{"id":"2009-A1","year":2009,"session":"A","number":1,"statement":"Let $f$ be a real-valued function on the plane such that for every\nsquare $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that\n$f(P)=0$ for all points $P$ in the plane?","subjects":["algebra","functional equations","plane geometry","finite differences","recurrences"]},{"id":"2009-A2","year":2009,"session":"A","number":2,"statement":"Functions $f,g,h$ are differentiable on some open interval around $0$\nand satisfy the equations and initial conditions\n$$\n\\begin{gathered}\nf' = 2f^2gh+\\frac{1}{gh},\\quad f(0)=1, \\\\\ng'=fg^2h+\\frac{4}{fh}, \\quad g(0)=1, \\\\\nh'=3fgh^2+\\frac{1}{fg}, \\quad h(0)=1.\n\\end{gathered}\n$$\nFind an explicit formula for $f(x)$, valid in some open interval around $0$.","subjects":["analysis","differential equations","riccati equations","conserved quantities"]},{"id":"2009-A3","year":2009,"session":"A","number":3,"statement":"Let $d_n$ be the determinant of the $n \\times n$ matrix whose entries, from\nleft to right and then from top to bottom, are $\\cos 1, \\cos 2, \\dots, \\cos\nn^2$. (For example,\n$$\nd_3 = \\left| \\begin{matrix} \\cos 1 & \\cos 2 & \\cos 3 \\\\\n\\cos 4 & \\cos 5 & \\cos 6 \\\\\n\\cos 7 & \\cos 8 & \\cos 9\n\\end{matrix} \\right|.\n$$\nThe argument of $\\cos$ is always in radians, not degrees.) Evaluate\n$\\lim_{n\\to\\infty} d_n$.","subjects":["linear algebra"]},{"id":"2009-A4","year":2009,"session":"A","number":4,"statement":"Let $S$ be a set of rational numbers such that\n\n(a) $0 \\in S$;\n\n(b) If $x \\in S$ then $x+1\\in S$ and $x-1\\in S$; and\n\n(c) If $x\\in S$ and $x\\not\\in\\{0,1\\}$, then $\\frac{1}{x(x-1)}\\in S$.\n\nMust $S$ contain all rational numbers?","subjects":["number theory"]},{"id":"2009-A5","year":2009,"session":"A","number":5,"statement":"Is there a finite abelian group $G$ such that the product of the\norders of all its elements is $2^{2009}$?","subjects":["group theory","finite abelian groups","element orders"]},{"id":"2009-A6","year":2009,"session":"A","number":6,"statement":"Let $f:[0,1]^2 \\to \\mathbb{R}$ be a continuous function on the closed unit\nsquare such that $\\frac{\\partial f}{\\partial x}$ and $\\frac{\\partial f}{\\partial y}$ exist\nand are continuous on the interior $(0,1)^2$. Let $a = \\int_0^1 f(0,y)\\,dy$,\n$b = \\int_0^1 f(1,y)\\,dy$, $c = \\int_0^1 f(x,0)\\,dx$, $d = \\int_0^1 f(x,1)\\,dx$.\nProve or disprove: There must be a point $(x_0,y_0)$ in $(0,1)^2$ such that\n$$\n\\frac{\\partial f}{\\partial x} (x_0,y_0) = b - a\n\\quad \\text{and} \\quad\n\\frac{\\partial f}{\\partial y} (x_0,y_0) = d - c.\n$$","subjects":["analysis","multivariable calculus","mean value counterexamples"]},{"id":"2009-B1","year":2009,"session":"B","number":1,"statement":"Show that every positive rational number can be written as a quotient of products of factorials\nof (not necessarily distinct) primes. For example,\n$$\n\\frac{10}{9} = \\frac{2!\\cdot 5!}{3!\\cdot 3! \\cdot 3!}.\n$$\n\\,","subjects":["number theory","factorials","multiplicative generation","linear algebra","combinatorial cancellation"]},{"id":"2009-B2","year":2009,"session":"B","number":2,"statement":"A game involves jumping to the right on the real number line. If $a$ and $b$ are real numbers\nand $b > a$, the cost of jumping from $a$ to $b$ is $b^3-ab^2$. For what real numbers\n$c$ can one travel from $0$ to $1$ in a finite number of jumps with total cost exactly $c$?","subjects":["analysis","riemann sums","continuous interpolation","optimization","sequences"]},{"id":"2009-B3","year":2009,"session":"B","number":3,"statement":"Call a subset $S$ of $\\{1, 2, \\dots, n\\}$ mediocre if it has the following property:\nWhenever $a$ and $b$ are elements of $S$ whose average is an integer, that average is also\nan element of $S$. Let $A(n)$ be the number of mediocre subsets of $\\{1,2,\\dots,n\\}$.\n[For instance, every subset of $\\{1,2,3\\}$ except $\\{1,3\\}$ is mediocre, so $A(3) =7$.]\nFind all positive integers $n$ such that $A(n+2) - 2A(n+1) + A(n) = 1$.","subjects":["combinatorics","number theory"]},{"id":"2009-B4","year":2009,"session":"B","number":4,"statement":"Say that a polynomial with real coefficients in two variables, $x,y$, is balanced if\nthe average value of the polynomial on each circle centered at the origin is $0$.\nThe balanced polynomials of degree at most $2009$ form a vector space $V$ over $\\mathbb{R}$.\nFind the dimension of $V$.","subjects":["linear algebra","polynomial spaces","circle averages","differential operators","rotation invariance"]},{"id":"2009-B5","year":2009,"session":"B","number":5,"statement":"Let $f: (1, \\infty) \\to \\mathbb{R}$ be a differentiable function such that\n$$\nf'(x) = \\frac{x^2 - f(x)^2}{x^2 (f(x)^2 + 1)}\n\\qquad \\text{for all }x>1\\text{.}\n$$\nProve that $\\lim_{x \\to \\infty} f(x) = \\infty$.","subjects":["analysis","differential equations","asymptotic growth"]},{"id":"2009-B6","year":2009,"session":"B","number":6,"statement":"Prove that for every positive integer $n$, there is a sequence of integers\n$a_0, a_1, \\dots, a_{2009}$ with $a_0 = 0$ and $a_{2009} = n$ such that each term\nafter $a_0$ is either an earlier term plus $2^k$ for some nonnegative integer $k$,\nor of the form $b\\,\\mathrm{mod}\\,c$ for some earlier positive terms $b$ and $c$.\n[Here $b\\,\\mathrm{mod}\\,c$ denotes the remainder when $b$ is divided by $c$,\nso $0 \\leq (b\\,\\mathrm{mod}\\,c) < c$.]","subjects":["number theory","constructive algorithms","modular arithmetic"]},{"id":"2010-A1","year":2010,"session":"A","number":1,"statement":"Given a positive integer $n$, what is the largest $k$ such that the\nnumbers $1,2,\\dots,n$ can be put into $k$ boxes so that the sum of the numbers\nin each box is the same? [When $n=8$, the example $\\{1,2,3,6\\}, \\{4,8\\}, \\{5,7\\}$\nshows that the largest $k$ is at least 3.]","subjects":["combinatorics","equal-sum partitions","extremal constructions"]},{"id":"2010-A2","year":2010,"session":"A","number":2,"statement":"Find all differentiable functions $f:\\mathbb{R} \\to \\mathbb{R}$ such that\n$$\nf'(x) = \\frac{f(x+n)-f(x)}{n}\n$$\nfor all real numbers $x$ and all positive integers $n$.","subjects":["analysis","functional-differential equations","affine rigidity","maximum principles","asymptotic comparison"]},{"id":"2010-A3","year":2010,"session":"A","number":3,"statement":"Suppose that the function $h:\\mathbb{R}^2\\to \\mathbb{R}$ has continuous partial\nderivatives and satisfies the equation\n$$\nh(x,y) = a \\frac{\\partial h}{\\partial x}(x,y) +\nb \\frac{\\partial h}{\\partial y}(x,y)\n$$\nfor some constants $a,b$. Prove that if there is a constant $M$ such that\n$|h(x,y)|\\leq M$ for all $(x,y) \\in \\mathbb{R}^2$, then $h$ is identically zero.","subjects":["analysis","partial differential equations","bounded solutions"]},{"id":"2010-A4","year":2010,"session":"A","number":4,"statement":"Prove that for each positive integer $n$, the number\n$10^{10^{10^n}} + 10^{10^n} + 10^n - 1$\nis not prime.","subjects":["number theory"]},{"id":"2010-A5","year":2010,"session":"A","number":5,"statement":"Let $G$ be a group, with operation $*$. Suppose that\n\n(i) $G$ is a subset of $\\mathbb{R}^3$ (but $*$ need not be related to addition of vectors);\n\n(ii) For each $\\mathbf{a},\\mathbf{b} \\in G$, either $\\mathbf{a}\\times \\mathbf{b} = \\mathbf{a}*\\mathbf{b}$\nor $\\mathbf{a}\\times \\mathbf{b} = 0$ (or\nboth), where $\\times$ is the usual cross product in $\\mathbb{R}^3$.\n\nProve that $\\mathbf{a} \\times \\mathbf{b} = 0$ for all $\\mathbf{a}, \\mathbf{b} \\in G$.","subjects":["algebra","geometry"]},{"id":"2010-A6","year":2010,"session":"A","number":6,"statement":"Let $f:[0,\\infty)\\to \\mathbb{R}$ be a strictly decreasing continuous function\nsuch that $\\lim_{x\\to\\infty} f(x) = 0$. Prove that\n$\\int_0^\\infty \\frac{f(x)-f(x+1)}{f(x)}\\,dx$ diverges.","subjects":["analysis","improper integrals","monotone functions"]},{"id":"2010-B1","year":2010,"session":"B","number":1,"statement":"Is there an infinite sequence of real numbers $a_1, a_2, a_3, \\dots$ such that\n$$\na_1^m + a_2^m + a_3^m + \\cdots = m\n$$\nfor every positive integer $m$?","subjects":["analysis"]},{"id":"2010-B2","year":2010,"session":"B","number":2,"statement":"Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates\nsuch that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$?","subjects":["number theory","lattice geometry","integer triangles"]},{"id":"2010-B3","year":2010,"session":"B","number":3,"statement":"There are 2010 boxes labeled $B_1, B_2, \\dots, B_{2010}$, and $2010n$ balls have been distributed\namong them, for some positive integer $n$. You may redistribute the balls by a sequence of moves,\neach of which consists of choosing an $i$ and moving exactly $i$ balls from box $B_i$ into any\none other box. For which values of $n$ is it possible to reach the distribution with exactly $n$ balls\nin each box, regardless of the initial distribution of balls?","subjects":["combinatorics"]},{"id":"2010-B4","year":2010,"session":"B","number":4,"statement":"Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which\n$$\np(x) q(x+1) - p(x+1) q(x) = 1.\n$$","subjects":["algebra","polynomials","finite differences"]},{"id":"2010-B5","year":2010,"session":"B","number":5,"statement":"Is there a strictly increasing function $f: \\mathbb{R} \\to \\mathbb{R}$ such that $f'(x) = f(f(x))$ for all $x$?","subjects":["analysis","differential equations","monotone functions"]},{"id":"2010-B6","year":2010,"session":"B","number":6,"statement":"Let $A$ be an $n \\times n$ matrix of real numbers for some $n \\geq 1$.\nFor each positive integer $k$, let $A^{[k]}$ be the matrix obtained by raising each entry to the $k$th\npower. Show that if $A^k = A^{[k]}$ for $k=1,2,\\dots,n+1$, then $A^k = A^{[k]}$ for all $k \\geq 1$.","subjects":["linear algebra","matrix powers","polynomials","entrywise matrix powers","finite moment problems"]},{"id":"2011-A1","year":2011,"session":"A","number":1,"statement":"Define a growing spiral in the plane to be a sequence\nof points with integer coordinates $P_0 = (0,0), P_1, \\dots, P_n$ such\nthat $n \\geq 2$ and:\n\n- the directed line segments $P_0 P_1, P_1 P_2, \\dots, P_{n-1} P_n$\nare in the successive coordinate directions east (for $P_0 P_1$), north,\nwest, south, east, etc.;\n\n- the lengths of these line segments are positive and strictly\nincreasing.\nHow many of the points $(x,y)$ with\ninteger coordinates $0\\leq x\\leq 2011, 0\\leq y\\leq 2011$ cannot\nbe the last point, $P_n$ of any growing spiral?","subjects":["combinatorics","lattice geometry","spirals","integer reachability"]},{"id":"2011-A2","year":2011,"session":"A","number":2,"statement":"Let $a_1,a_2,\\dots$ and $b_1,b_2,\\dots$ be sequences of positive\nreal numbers such that $a_1 = b_1 = 1$ and $b_n = b_{n-1} a_n - 2$ for\n$n=2,3,\\dots$. Assume that the sequence $(b_j)$ is bounded. Prove that\n$$\nS = \\sum_{n=1}^\\infty \\frac{1}{a_1...a_n}\n$$\nconverges, and evaluate $S$.","subjects":["analysis","sequences","series","probability"]},{"id":"2011-A3","year":2011,"session":"A","number":3,"statement":"Find a real number $c$ and a positive number $L$ for which\n$$\n\\lim_{r\\to\\infty} \\frac{r^c \\int_0^{\\pi/2} x^r \\sin x \\,dx}{\\int_0^{\\pi/2} x^r \\cos x \\,dx} = L.\n$$","subjects":["analysis","asymptotic analysis","integrals","endpoint asymptotics","moment integrals"]},{"id":"2011-A4","year":2011,"session":"A","number":4,"statement":"For which positive integers $n$ is there an $n \\times n$ matrix\nwith integer entries such that every dot product of a row with itself is\neven, while every dot product of two different rows is odd?","subjects":["linear algebra","combinatorics"]},{"id":"2011-A5","year":2011,"session":"A","number":5,"statement":"Let $F : \\mathbb{R}^2 \\to \\mathbb{R}$ and $g : \\mathbb{R} \\to \\mathbb{R}$ be twice\ncontinuously differentiable functions with the following properties:\n\n- $F(u,u) = 0$ for every $u \\in \\mathbb{R}$;\n\n- for every $x \\in \\mathbb{R}$, $g(x) > 0$ and $x^2 g(x) \\leq 1$;\n\n- for every $(u,v) \\in \\mathbb{R}^2$, the vector $\\nabla F(u,v)$ is either $\\mathbf{0}$ or parallel to the vector $\\langle g(u), -g(v) \\rangle$.\n\nProve that there exists a constant $C$ such that for every $n\\geq 2$ and any $x_1,\\dots,x_{n+1} \\in \\mathbb{R}$, we have\n$$\n\\min_{i \\neq j} |F(x_i,x_j)| \\leq \\frac{C}{n}.\n$$","subjects":["analysis","multivariable calculus"]},{"id":"2011-A6","year":2011,"session":"A","number":6,"statement":"Let $G$ be an abelian group with $n$ elements, and let\n$$ \\{g_1=e,g_2,\\dots,g_k\\} \\subsetneqq G $$\nbe a (not necessarily minimal) set of distinct generators of $G$. A special\ndie, which randomly selects one of the elements $g_1,g_2,...,g_k$ with equal\nprobability, is rolled $m$ times and the selected elements are multiplied\nto produce an element $g \\in G$.  Prove that there exists a real number\n$b \\in (0,1)$ such that\n\n$$ \\lim_{m\\to\\infty} \\frac{1}{b^{2m}} \\sum_{x\\in G} \\left(\\mathrm{Prob}(g=x)\n- \\frac{1}{n}\\right)^2 $$\nis positive and finite.","subjects":["algebra","probability"]},{"id":"2011-B1","year":2011,"session":"B","number":1,"statement":"Let $h$ and $k$ be positive integers. Prove that for every\n$\\epsilon > 0$, there are positive integers $m$ and $n$ such that\n$$ \\epsilon < |h \\sqrt{m} - k \\sqrt{n}| < 2\\epsilon.  $$","subjects":["number theory","approximation","real analysis","radical approximation","irrational rotations"]},{"id":"2011-B2","year":2011,"session":"B","number":2,"statement":"Let $S$ be the set of all ordered triples $(p,q,r)$ of prime\nnumbers for which at least one rational number $x$ satisfies $px^2 + qx +\nr =0$. Which primes appear in seven or more elements of $S$?","subjects":["number theory","primes","quadratic equations","rational roots","prime occurrence counting"]},{"id":"2011-B3","year":2011,"session":"B","number":3,"statement":"Let $f$ and $g$ be (real-valued) functions defined on an open\ninterval containing $0$, with $g$ nonzero and continuous at $0$.  If $fg$\nand $f/g$ are differentiable at $0$, must $f$ be differentiable at 0?","subjects":["analysis","differentiability"]},{"id":"2011-B4","year":2011,"session":"B","number":4,"statement":"In a tournament, 2011 players meet 2011 times to play a\nmultiplayer game. Every game is played by all 2011 players together and\nends with each of the players either winning or losing. The standings\nare kept in two $2011 \\times 2011$ matrices, $T = (T_{hk})$ and $W =\n(W_{hk})$. Initially, $T=W=0$. After every game, for every $(h,k)$ (including\nfor $h=k$), if players $h$ and $k$ tied (that is, both won or both lost),\nthe entry $T_{hk}$ is increased by 1, while if player $h$ won and player $k$\nlost, the entry $W_{hk}$ is increased by 1 and $W_{kh}$ is decreased by 1.\n\nProve that at the end of the tournament, $\\det(T+iW)$ is a non-negative\ninteger divisible by $2^{2010}$.","subjects":["linear algebra"]},{"id":"2011-B5","year":2011,"session":"B","number":5,"statement":"Let $a_1, a_2, \\dots$ be real numbers. Suppose that there is\na constant $A$ such that for all $n$,\n$$\n\\int_{-\\infty}^\\infty \\left( \\sum_{i=1}^n \\frac{1}{1 + (x-a_i)^2} \\right)^2\\,dx \\leq An.\n$$\nProve there is a constant $B>0$ such that for all $n$,\n$$\n\\sum_{i,j=1}^n (1 + (a_i - a_j)^2) \\geq Bn^3.\n$$","subjects":["analysis","integral inequalities","sequences"]},{"id":"2011-B6","year":2011,"session":"B","number":6,"statement":"Let $p$ be an odd prime. Show that for at least $(p+1)/2$ values of $n$ in $\\{0,1,2,\\dots,p-1\\}$,\n$$\n\\sum_{k=0}^{p-1} k! n^k \\qquad \\text{is not divisible by }p\\text{.}\n$$","subjects":["number theory","finite fields","polynomial roots"]},{"id":"2012-A1","year":2012,"session":"A","number":1,"statement":"Let $d_1, d_2, \\dots, d_{12}$ be real numbers in the open\ninterval $(1, 12)$. Show that there exist distinct indices $i, j, k$\nsuch that $d_i, d_j, d_k$ are the side lengths of an acute triangle.","subjects":["combinatorics","inequalities","triangle geometry"]},{"id":"2012-A2","year":2012,"session":"A","number":2,"statement":"Let $*$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$\nand $y$ in $S$, there exists $z$ in $S$ such that $x * z = y$. (This $z$ may depend on $x$ and $y$.)\nShow that if $a,b,c$ are in $S$ and $a*c = b*c$, then $a=b$.","subjects":["algebra","semigroups"]},{"id":"2012-A3","year":2012,"session":"A","number":3,"statement":"Let $f: [-1, 1] \\to \\mathbb{R}$ be a continuous function such that\n\n(i) $f(x) = \\frac{2-x^2}{2} f \\left( \\frac{x^2}{2-x^2} \\right)$ for every $x$ in $[-1, 1]$,\n\n(ii) $f(0) = 1$, and\n\n(iii) $\\lim_{x \\to 1^-} \\frac{f(x)}{\\sqrt{1-x}}$ exists and is finite.\n\nProve that $f$ is unique, and express $f(x)$ in closed form.","subjects":["analysis","functional equations"]},{"id":"2012-A4","year":2012,"session":"A","number":4,"statement":"Let $q$ and $r$ be integers with $q > 0$, and let $A$ and $B$ be intervals on the real line.\nLet $T$ be the set of all $b+mq$ where $b$ and $m$ are integers with $b$ in $B$,\nand let $S$ be the set of all integers $a$ in $A$ such that $ra$ is in $T$. Show that if the\nproduct of the lengths of $A$ and $B$ is less than $q$, then $S$ is the intersection of $A$\nwith some arithmetic progression.","subjects":["number theory","geometry"],"errata":"Arithmetic progressions here may be finite. The source solution uses this convention. Requiring an infinite progression makes the assertion false."},{"id":"2012-A5","year":2012,"session":"A","number":5,"statement":"Let $\\mathbb{F}_p$ denote the field of integers modulo a prime $p$, and let $n$ be a positive integer.\nLet $v$ be a fixed vector in $\\mathbb{F}_p^n$, let $M$ be an $n \\times n$ matrix with entries of $\\mathbb{F}_p$,\nand define $G: \\mathbb{F}_p^n \\to \\mathbb{F}_p^n$ by $G(x) = v + Mx$. Let $G^{(k)}$ denote the $k$-fold\ncomposition of $G$ with itself, that is, $G^{(1)}(x) = G(x)$ and $G^{(k+1)}(x) = G(G^{(k)}(x))$.\nDetermine all pairs $p, n$ for which there exist $v$ and $M$ such that the $p^n$ vectors\n$G^{(k)}(0)$, $k=1,2,\\dots,p^n$ are distinct.","subjects":["linear algebra","finite fields","affine transformations"]},{"id":"2012-A6","year":2012,"session":"A","number":6,"statement":"Let $f(x,y)$ be a continuous, real-valued function on $\\mathbb{R}^2$. Suppose that, for every\nrectangular region $R$ of area $1$, the double integral of $f(x,y)$ over $R$ equals $0$.\nMust $f(x,y)$ be identically 0?","subjects":["analysis","multivariable calculus","integral geometry"]},{"id":"2012-B1","year":2012,"session":"B","number":1,"statement":"Let $S$ be a class of functions from $[0, \\infty)$ to $[0, \\infty)$ that satisfies:\n\n(i) The functions $f_1(x) = e^x - 1$ and $f_2(x) = \\ln(x+1)$ are in $S$;\n\n(ii) If $f(x)$ and $g(x)$ are in $S$, the functions $f(x) + g(x)$ and $f(g(x))$ are in $S$;\n\n(iii) If $f(x)$ and $g(x)$ are in $S$ and $f(x) \\geq g(x)$ for all $x \\geq 0$, then the function\n$f(x) - g(x)$ is in $S$.\n\nProve that if $f(x)$ and $g(x)$ are in $S$, then the function $f(x) g(x)$ is also in $S$.","subjects":["algebra","functional closure","exponential and logarithm"]},{"id":"2012-B2","year":2012,"session":"B","number":2,"statement":"Let $P$ be a given (non-degenerate) polyhedron. Prove that there is a constant $c(P) > 0$\nwith the following property: If a collection of $n$ balls whose volumes sum to $V$ contains\nthe entire surface of $P$, then $n > c(P) / V^2$.","subjects":["geometry","inequalities","covering"]},{"id":"2012-B3","year":2012,"session":"B","number":3,"statement":"A round-robin tournament of $2n$ teams lasted for $2n-1$ days, as follows.\nOn each day, every team played one game against another team, with one team winning\nand one team losing in each of the $n$ games. Over the course of the tournament,\neach team played every other team exactly once. Can one necessarily choose\none winning team from each day without choosing any team more than once?","subjects":["combinatorics","matching theory","tournaments"]},{"id":"2012-B4","year":2012,"session":"B","number":4,"statement":"Suppose that $a_0 = 1$ and that $a_{n+1} = a_n + e^{-a_n}$ for $n=0,1,2,\\dots$. Does $a_n - \\log n$\nhave a finite limit as $n \\to \\infty$? (Here $\\log n = \\log_e n = \\ln n$.)","subjects":["analysis","recurrences","asymptotic analysis"]},{"id":"2012-B5","year":2012,"session":"B","number":5,"statement":"Prove that, for any two bounded functions $g_1, g_2: \\mathbb{R} \\to [1, \\infty)$,\nthere exist functions $h_1, h_2: \\mathbb{R} \\to \\mathbb{R}$ such that, for every $x \\in \\mathbb{R}$,\n$$\n\\sup_{s \\in \\mathbb{R}} (g_1(s)^x g_2(s))  = \\max_{t \\in \\mathbb{R}} (x h_1(t) + h_2(t)).\n$$","subjects":["analysis","convexity","supporting lines"]},{"id":"2012-B6","year":2012,"session":"B","number":6,"statement":"Let $p$ be an odd prime number such that $p \\equiv 2 \\pmod{3}$. Define a permutation $\\pi$ of the\nresidue classes modulo $p$ by $\\pi(x) \\equiv x^3 \\pmod{p}$. Show that $\\pi$ is an even permutation\nif and only if $p \\equiv 3 \\pmod{4}$.","subjects":["algebra","number theory","permutation parity"]},{"id":"2013-A1","year":2013,"session":"A","number":1,"statement":"Recall that a regular icosahedron is a convex polyhedron\nhaving 12 vertices and 20 faces;\nthe faces are congruent equilateral triangles.\nOn each face of a regular icosahedron is written a nonnegative integer\nsuch that the sum of all 20 integers is 39. Show that there are \ntwo faces that share a vertex and have the same integer written on them.","subjects":["combinatorics","polyhedra","graph theory","linear algebra"]},{"id":"2013-A2","year":2013,"session":"A","number":2,"statement":"Let $S$ be the set of all positive integers that are not perfect squares. For $n$ in $S$, consider choices of integers\n$a_1, a_2, \\dots, a_r$ such that $n < a_1<  a_2 < \\cdots < a_r$\nand $n \\cdot a_1 \\cdot a_2 \\cdots a_r$ is a perfect square, and\nlet $f(n)$ be the minumum of $a_r$ over all such choices. For example,\n$2 \\cdot 3 \\cdot 6$ is a perfect square, while $2 \\cdot 3$, $2 \\cdot 4$, \n$2 \\cdot 5$, $2 \\cdot 3 \\cdot 4$, $2 \\cdot 3 \\cdot 5$, $2 \\cdot 4 \\cdot 5$, and $2 \\cdot 3 \\cdot 4 \\cdot 5$ are not, and so $f(2) = 6$.\nShow that the function $f$ from $S$ to the integers is one-to-one.","subjects":["number theory","combinatorics","perfect squares","linear algebra"]},{"id":"2013-A3","year":2013,"session":"A","number":3,"statement":"Suppose that the real numbers $a_0, a_1, \\dots, a_n$ and\n$x$, with $0 < x < 1$, satisfy\n$$\n\\frac{a_0}{1-x} + \\frac{a_1}{1-x^2} + \\cdots + \\frac{a_n}{1 - x^{n+1}} = 0.\n$$\nProve that there exists a real number $y$ with $0 < y < 1$ such that\n$$\na_0 + a_1 y + \\cdots + a_n y^n = 0.\n$$","subjects":["analysis","polynomials"]},{"id":"2013-A4","year":2013,"session":"A","number":4,"statement":"A finite collection of digits $0$ and $1$ is written around a circle.\nAn arc of length $L \\geq 0$ consists of $L$ consecutive digits around the circle. For each arc $w$, let $Z(w)$ and $N(w)$ denote the number of $0$'s in $w$ and the number of $1$'s in $w$, respectively.\nAssume that $\\left| Z(w) - Z(w') \\right| \\leq 1$ for any two arcs $w, w'$ of the same length. Suppose that some arcs $w_1,\\dots,w_k$ have the property that\n$$\nZ = \\frac{1}{k} \\sum_{j=1}^k Z(w_j) \\text{ and }\nN = \\frac{1}{k} \\sum_{j=1}^k N(w_j)\n$$\nare both integers. Prove that there exists an arc $w$ with $Z(w) = Z$\nand $N(w) = N$.","subjects":["combinatorics","words","balanced sequences"]},{"id":"2013-A5","year":2013,"session":"A","number":5,"statement":"For $m \\geq 3$, a list of $\\binom{m}{3}$ real numbers $a_{ijk}$ ($1 \\leq i < < j < k \\leq m$) is said to be area definite for $\\mathbb{R}^n$ if the inequality\n$$\n\\sum_{1 \\leq i < j < k \\leq m} a_{ijk} \\cdot \\mathrm{Area}(\\Delta A_i A_j A_k) \\geq 0\n$$\nholds for every choice of $m$ points $A_1,\\dots,A_m$ in $\\mathbb{R}^n$.\nFor example, the list of four numbers $a_{123} = a_{124} = a_{134} = 1$, $a_{234} = -1$ is area definite for $\\mathbb{R}^2$. Prove that if a list of $\\binom{m}{3}$ numbers is area definite for $\\mathbb{R}^2$,\nthen it is area definite for $\\mathbb{R}^3$.","subjects":["geometry","integral geometry","inequalities"]},{"id":"2013-A6","year":2013,"session":"A","number":6,"statement":"Define a function $w: \\mathbb{Z} \\times \\mathbb{Z} \\to \\mathbb{Z}$\nas follows. For $\\left| a \\right|, \\left| b \\right| \\leq 2$,\nlet $w(a,b)$ be as in the table shown; otherwise, let $w(a,b) = 0$.\n$$\\begin{array}{c|rrrrr}a\\backslash b&-2&-1&0&1&2\\\\\\hline -2&-1&-2&2&-2&-1\\\\-1&-2&4&-4&4&-2\\\\0&2&-4&12&-4&2\\\\1&-2&4&-4&4&-2\\\\2&-1&-2&2&-2&-1\\end{array}$$\nFor every finite subset $S$ of $\\mathbb{Z} \\times \\mathbb{Z}$,\ndefine\n$$\nA(S) = \\sum_{(\\mathbf{s}, \\mathbf{s}') \\in S \\times S} w(\\mathbf{s} - \\mathbf{s}').\n$$\nProve that if $S$ is any finite nonempty subset of $\\mathbb{Z} \\times \\mathbb{Z}$, then $A(S) > 0$.\n(For example, if $S = \\{(0,1), (0,2), (2,0), (3,1)\\}$, then the terms in $A(S)$ are $12, 12, 12, 12, 4, 4, 0, 0, 0,0,-1,-1,-2,-2,-4,-4$.)","subjects":["algebra","harmonic analysis","quadratic forms"]},{"id":"2013-B1","year":2013,"session":"B","number":1,"statement":"For positive integers $n$, let the numbers $c(n)$ be determined by \nthe rules $c(1) = 1$, $c(2n) = c(n)$, and $c(2n+1) = (-1)^n c(n)$.\nFind the value of \n$$\n\\sum_{n=1}^{2013} c(n) c(n+2).\n$$","subjects":["combinatorics","binary sequences","polynomials","finite sums"]},{"id":"2013-B2","year":2013,"session":"B","number":2,"statement":"Let $C = \\bigcup_{N=1}^\\infty C_N$, where $C_N$ denotes the set of those `cosine polynomials' of the form\n$$\nf(x) = 1 + \\sum_{n=1}^N a_n \\cos(2 \\pi n x)\n$$\nfor which:\n\n(i) $f(x) \\geq 0$ for all real $x$, and\n\n(ii) $a_n = 0$ whenever $n$ is a multiple of $3$.\n\nDetermine the maximum value of $f(0)$ as $f$ ranges through $C$, and \nprove that this maximum is attained.","subjects":["analysis","trigonometric polynomials","fourier analysis","extremal problems"]},{"id":"2013-B3","year":2013,"session":"B","number":3,"statement":"Let $\\mathcal{P}$ be a nonempty collection of subsets of $\\{1,\\dots, n\\}$ such that:\n\n(i) if $S, S' \\in \\mathcal{P}$, then $S \\cup S' \\in \\mathcal{P}$ and $S \\cap S' \\in \\mathcal{P}$, and\n\n(ii) if $S \\in \\mathcal{P}$ and $S \\neq \\emptyset$, then there is a subset $T \\subset S$\nsuch that $T \\in \\mathcal{P}$ and $T$ contains exactly one fewer element than $S$.\n\nSuppose that $f: \\mathcal{P} \\to \\mathbb{R}$ is a function such that\n$f(\\emptyset) = 0$ and\n$$\nf(S \\cup S') = f(S) + f(S') - f(S \\cap S') \\text{ for all }S,S' \\in \\mathcal{P}\\text{.}\n$$\nMust there exist real numbers $f_1,\\dots,f_n$ such that\n$$\nf(S) = \\sum_{i \\in S} f_i\n$$\nfor every $S \\in \\mathcal{P}$?","subjects":["combinatorics","set systems","valuations"]},{"id":"2013-B4","year":2013,"session":"B","number":4,"statement":"For any continuous real-valued function $f$ defined on the interval $[0,1]$, let\n$$\n\\begin{gathered}\n\\mu(f) = \\int_0^1 f(x)\\,dx, \\,\n\\mathrm{Var}(f) = \\int_0^1 (f(x) - \\mu(f))^2\\,dx, \\\\\nM(f) = \\max_{0 \\leq x \\leq 1} \\left| f(x) \\right|.\n\\end{gathered}\n$$\nShow that if $f$ and $g$ are continuous real-valued functions\ndefined on the interval $[0,1]$,\nthen\n$$\n\\mathrm{Var}(fg) \\leq 2 \\mathrm{Var}(f) M(g)^2 + 2 \\mathrm{Var}(g) M(f)^2.\n$$","subjects":["analysis","variance","integral inequalities"]},{"id":"2013-B5","year":2013,"session":"B","number":5,"statement":"Let $X = \\{1, 2, \\dots, n\\}$, and let $k \\in X$. Show that there are exactly $k \\cdot n^{n-1}$ functions $f: X \\to X$ such that for every $x \\in X$ there is a $j \\geq 0$ such that $f^{(j)}(x) \\leq k$.\n[Here $f^{(j)}$ denotes the $j$th iterate of $f$, so that $f^{(0)}(x) = x$ and $f^{(j+1)}(x) = f(f^{(j)}(x))$.]","subjects":["combinatorics","functional digraphs","spanning forests"]},{"id":"2013-B6","year":2013,"session":"B","number":6,"statement":"Let $n \\geq 1$ be an odd integer. Alice and Bob play the following game,\ntaking alternating turns, with Alice playing first. \nThe playing area consists of $n$ spaces, arranged in a line.\nInitially all spaces are empty.\nAt each turn, a player either\n\n- places a stone in an empty space, or\n\n- removes a stone from a nonempty space $s$, \nplaces a stone in the nearest empty space to the left of $s$\n(if such a space exists),\nand places a stone in the nearest empty space to the right of $s$\n(if such a space exists).\n\nFurthermore, a move is permitted only if the resulting position has not occurred previously in the game. A player loses if he or she is unable to move. Assuming that both players play optimally throughout the game, what moves may Alice make on her first turn?","subjects":["combinatorics","game theory","modular invariants"]},{"id":"2014-A1","year":2014,"session":"A","number":1,"statement":"Prove that every nonzero coefficient of the Taylor series of\n$$\n(1 - x + x^2)e^x\n$$\nabout $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.","subjects":["number theory","taylor series","divisibility","combinatorics","group actions"]},{"id":"2014-A2","year":2014,"session":"A","number":2,"statement":"Let $A$ be the $n \\times n$ matrix whose entry in the $i$-th row and $j$-th column is\n$$\n\\frac{1}{\\min(i,j)}\n$$\nfor $1 \\leq i,j \\leq n$. Compute $\\det(A)$.","subjects":["linear algebra","determinants"]},{"id":"2014-A3","year":2014,"session":"A","number":3,"statement":"Let $a_0 = 5/2$ and $a_k = a_{k-1}^2 - 2$ for $k \\geq 1$. Compute\n$$\n\\prod_{k=0}^\\infty \\left(1 - \\frac{1}{a_k} \\right)\n$$\nin closed form.","subjects":["analysis","infinite products","recurrences"]},{"id":"2014-A4","year":2014,"session":"A","number":4,"statement":"Suppose $X$ is a random variable that takes on only nonnegative integer values,\nwith $E\\left[ X \\right] = 1$, $E\\left[ X^2 \\right] = 2$, and $E \\left[ X^3 \\right] = 5$.\n(Here $E\\left[ y \\right]$ denotes the expectation of the random variable $Y$.)\nDetermine the smallest possible value of the probability of the event $X=0$.","subjects":["probability","moment problems","polynomial inequalities"]},{"id":"2014-A5","year":2014,"session":"A","number":5,"statement":"Let\n$$\nP_n(x) = 1 + 2 x  + 3 x^2 + \\cdots + n x^{n-1}.\n$$\nProve that the polynomials $P_j(x)$\nand $P_k(x)$ are relatively prime\nfor all positive integers $j$ and $k$ with $j \\neq k$.","subjects":["algebra","polynomial roots","complex numbers"]},{"id":"2014-A6","year":2014,"session":"A","number":6,"statement":"Let $n$ be a positive integer. What is the largest $k$ for which there exist $n \\times n$ matrices $M_1, \\dots, M_k$ and $N_1, \\dots, N_k$ with real entries such that for all $i$ and $j$, the matrix product $M_i N_j$ has a zero entry somewhere on its diagonal if and only if $i \\neq j$?","subjects":["linear algebra","extremal problems"]},{"id":"2014-B1","year":2014,"session":"B","number":1,"statement":"A base $10$ over-expansion of a positive integer $N$ is an expression of the form\n$$\nN = d_k 10^k + d_{k-1} 10^{k-1} + \\cdots + d_0 10^0\n$$\nwith $d_k \\neq 0$ and $d_i \\in \\{0,1,2,\\dots,10\\}$ for all $i$.\nFor instance, the integer $N = 10$ has two base 10 over-expansions: $10 = 10 \\cdot 10^0$\nand the usual base 10 expansion $10 = 1 \\cdot 10^1 + 0 \\cdot 10^0$.\nWhich positive integers have a unique base 10 over-expansion?","subjects":["number theory","positional notation"]},{"id":"2014-B2","year":2014,"session":"B","number":2,"statement":"Suppose that $f$ is a function on the interval $[1,3]$ such that $-1 \\leq f(x) \\leq 1$ for all $x$ and $\\int_1^3 f(x)\\,dx = 0$. How large can $\\int_1^3 \\frac{f(x)}{x}\\,dx$ be?\n\n\\,","subjects":["analysis","extremal integrals","rearrangement"]},{"id":"2014-B3","year":2014,"session":"B","number":3,"statement":"Let $A$ be an $m \\times n$ matrix with rational entries.\nSuppose that there are at least $m+n$ distinct prime numbers among the absolute values of the entries of $A$. Show that the rank of $A$ is at least 2.","subjects":["linear algebra","number theory","graph theory"]},{"id":"2014-B4","year":2014,"session":"B","number":4,"statement":"Show that for each positive integer $n$, all the roots of the polynomial\n$$\n\\sum_{k=0}^n 2^{k(n-k)} x^k\n$$\nare real numbers.","subjects":["algebra","real-rooted polynomials"]},{"id":"2014-B5","year":2014,"session":"B","number":5,"statement":"In the 75th annual Putnam Games, participants compete at mathematical games.\nPatniss and Keeta play a game in which they take turns choosing an element \nfrom the group of invertible $n \\times n$ matrices with entries in the field\n$\\mathbb{Z}/p \\mathbb{Z}$ of integers modulo $p$, where $n$ is a fixed positive integer and $p$ is a fixed prime number. The rules of the game are:\n\n(1) A player cannot choose an element that has been chosen by either player on any previous turn.\n\n(2) A player can only choose an element that commutes with all previously chosen elements.\n\n(3) A player who cannot choose an element on his/her turn loses the game.\n\nPatniss takes the first turn. Which player has a winning strategy?\n(Your answer may depend on $n$ and $p$.)","subjects":["algebra","game theory","finite fields"]},{"id":"2014-B6","year":2014,"session":"B","number":6,"statement":"Let $f: [0,1] \\to \\mathbb{R}$ be a function for which there exists a constant $K>0$\nsuch that $\\left| f(x) - f(y) \\right| \\leq K \\left| x - y \\right|$ for all $x,y \\in [0,1]$.\nSuppose also that for each rational number $r \\in [0,1]$, there exist integers $a$ and $b$\nsuch that $f(r) = a + br$. Prove that there exist finitely many intervals $I_1, \\dots, I_n$\nsuch that $f$ is a linear function on each $I_i$ and $[0,1] = \\bigcup_{i=1}^n I_i$.","subjects":["analysis","number theory","piecewise linear functions"]},{"id":"2015-A1","year":2015,"session":"A","number":1,"statement":"Let $A$ and $B$ be points on the same branch of the hyperbola $xy=1$. Suppose that $P$ is a point lying between $A$ and $B$ on this hyperbola, such that the area of the triangle $APB$ is as large as possible. Show that the region bounded by the hyperbola and the chord $AP$ has the same area as the region bounded by the hyperbola and the chord $PB$.","subjects":["geometry","calculus","hyperbolas"]},{"id":"2015-A2","year":2015,"session":"A","number":2,"statement":"Let $a_0=1$, $a_1=2$, and $a_n=4a_{n-1}-a_{n-2}$ for $n\\geq 2$. Find an odd prime factor of $a_{2015}$.","subjects":["number theory","algebra"]},{"id":"2015-A3","year":2015,"session":"A","number":3,"statement":"Compute\n$$\n\\log_2 \\left( \\prod_{a=1}^{2015} \\prod_{b=1}^{2015} (1+e^{2\\pi i a b/2015}) \\right)\n$$\nHere $i$ is the imaginary unit (that is, $i^2=-1$).","subjects":["number theory","roots of unity","multiplicative functions"]},{"id":"2015-A4","year":2015,"session":"A","number":4,"statement":"For each real number $x$, let\n$$\nf(x) = \\sum_{n\\in S_x} \\frac{1}{2^n},\n$$\nwhere $S_x$ is the set of positive integers $n$ for which $\\lfloor nx \\rfloor$ is even. What is the largest real number $L$ such that $f(x) \\geq L$ for all $x \\in [0,1)$? (As usual, $\\lfloor z \\rfloor$ denotes the greatest integer less than or equal to $z$.)","subjects":["analysis","binary expansions","floor functions"]},{"id":"2015-A5","year":2015,"session":"A","number":5,"statement":"Let $q$ be an odd positive integer, and let $N_q$ denote the number of integers $a$ such that $0 < a < q/4$ and $\\gcd(a,q) = 1$. Show that $N_q$ is odd if and only if $q$ is of the form $p^k$ with $k$ a positive integer and $p$ a prime congruent to $5$ or $7$ modulo $8$.","subjects":["number theory","parity"]},{"id":"2015-A6","year":2015,"session":"A","number":6,"statement":"Let $n$ be a positive integer. Suppose that $A$, $B$, and $M$ are $n\\times n$ matrices with real entries such that $AM = MB$, and such that $A$ and $B$ have the same characteristic polynomial. Prove that $\\det(A-MX) = \\det(B-XM)$ for every $n\\times n$ matrix $X$ with real entries.","subjects":["linear algebra","determinants","intertwining operators"]},{"id":"2015-B1","year":2015,"session":"B","number":1,"statement":"Let $f$ be a three times differentiable function (defined on $\\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeros. Prove that $f + 6f' + 12f'' + 8f'''$ has at least two distinct real zeros.","subjects":["analysis","differential operators","real analysis","integration"]},{"id":"2015-B2","year":2015,"session":"B","number":2,"statement":"Given a list of the positive integers $1,2,3,4,\\dots$, take the first three numbers\n$1,2,3$ and their sum $6$ and cross all four numbers off the list. Repeat with the three \nsmallest remaining numbers $4,5,7$ and their sum $16$. Continue in this way, crossing off the three smallest remaining numbers and their sum, and consider the sequence of sums produced: $6, 16, 27, 36, \\dots$. Prove or disprove that there is some number in the sequence whose base 10 representation ends with $2015$.\n\n\\,","subjects":["combinatorics","recursive algorithms","decimal congruences","automatic sequences"]},{"id":"2015-B3","year":2015,"session":"B","number":3,"statement":"Let $S$ be the set of all $2 \\times 2$ real matrices\n$$\nM = \\begin{pmatrix} a & b \\\\\nc & d \\end{pmatrix}\n$$\nwhose entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S$.","subjects":["linear algebra","matrix powers","arithmetic progressions"]},{"id":"2015-B4","year":2015,"session":"B","number":4,"statement":"Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c$. Express\n$$\n\\sum_{(a,b,c) \\in T} \\frac{2^a}{3^b 5^c} \n$$\nas a rational number in lowest terms.","subjects":["analysis","integer triangles"]},{"id":"2015-B5","year":2015,"session":"B","number":5,"statement":"Let $P_n$ be the number of permutations $\\pi$ of $\\{1,2,\\dots,n\\}$ such that\n$$\n|i-j| = 1 \\text{ implies } |\\pi(i) -\\pi(j)| \\leq 2\n$$\nfor all $i,j$ in $\\{1,2,\\dots,n\\}$. Show that for $n \\geq 2$, the quantity\n$$\nP_{n+5} - P_{n+4} - P_{n+3} + P_n\n$$\ndoes not depend on $n$, and find its value.","subjects":["combinatorics","permutations","linear recurrences"]},{"id":"2015-B6","year":2015,"session":"B","number":6,"statement":"For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \\sqrt{2k})$. Evaluate\n$$\n\\sum_{k=1}^\\infty (-1)^{k-1} \\frac{A(k)}{k}.\n$$","subjects":["analysis","conditionally convergent series","divisor sums"]},{"id":"2016-A1","year":2016,"session":"A","number":1,"statement":"Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k$, the integer\n$$\np^{(j)}(k) = \\left. \\frac{d^j}{dx^j} p(x) \\right|_{x=k} \n$$ \n(the $j$-th derivative of $p(x)$ at $k$) is divisible by 2016.","subjects":["number theory","polynomials","derivatives"]},{"id":"2016-A2","year":2016,"session":"A","number":2,"statement":"Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that \n$$\n\\binom{m}{n-1} > \\binom{m-1}{n}.\n$$\nEvaluate \n$$\n\\lim_{n \\to \\infty} \\frac{M(n)}{n}.\n$$","subjects":["analysis","binomial coefficients","asymptotic analysis"]},{"id":"2016-A3","year":2016,"session":"A","number":3,"statement":"Suppose that $f$ is a function from $\\mathbb{R}$ to $\\mathbb{R}$ such that\n$$\nf(x) + f\\left( 1 - \\frac{1}{x} \\right) = \\arctan x\n$$\nfor all real $x \\neq 0$. (As usual, $y = \\arctan x$ means $-\\pi/2 < y < \\pi/2$ and $\\tan y = x$.) Find \n$$\n\\int_0^1 f(x)\\,dx.\n$$","subjects":["analysis","functional equations","integration"]},{"id":"2016-A4","year":2016,"session":"A","number":4,"statement":"Consider a $(2m-1) \\times (2n-1)$ rectangular region, where $m$ and $n$ are integers such that $m, n \\geq 4$. This region is to be tiled using two types of tile. One is the union of the three closed unit squares whose lower-left corners are $(0,0),(0,1),(1,1)$. The other is the union of the four closed unit squares whose lower-left corners are $(0,0),(1,0),(1,1),(2,1)$.\nThe tiles may be rotated and reflected, as long as their sides are parallel to the sides\nof the rectangular region. They must all fit within the region, and they must cover it completely without overlapping.\n\nWhat is the minimum number of tiles required to tile the region?","subjects":["combinatorics","tilings","coloring invariants"]},{"id":"2016-A5","year":2016,"session":"A","number":5,"statement":"Suppose that $G$ is a finite group generated by the two elements $g$ and $h$, where the order of $g$ is odd. Show that every element of $G$ can be written in the form\n$$\ng^{m_1} h^{n_1} g^{m_2} h^{n_2} \\cdots g^{m_r} h^{n_r}\n$$\nwith $1 \\leq r \\leq |G|$ and $m_1, n_1, m_2, n_2, \\ldots, m_r, n_r \\in \\{-1, 1\\}$. \n(Here $|G|$ is the number of elements of $G$.)\n\n\\,","subjects":["algebra","finite groups","cayley graphs"]},{"id":"2016-A6","year":2016,"session":"A","number":6,"statement":"Find the smallest constant $C$ such that for every real polynomial $P(x)$ of degree 3 that has a root in the interval $[0,1]$,\n$$\n\\int_0^1 \\left| P(x) \\right|\\,dx \\leq C \\max_{x \\in [0,1]} \\left| P(x) \\right|.\n$$","subjects":["analysis","polynomial inequalities","quadrature"]},{"id":"2016-B1","year":2016,"session":"B","number":1,"statement":"Let $x_0,x_1,x_2,\\dots$ be the sequence such that $x_0=1$ and for $n \\geq 0$,\n$$\nx_{n+1} = \\ln(e^{x_n} - x_n)\n$$\n(as usual, the function $\\ln$ is the natural logarithm). Show that the infinite series\n$$\nx_0 + x_1 + x_2 + \\cdots\n$$\nconverges and find its sum.","subjects":["analysis","recurrences","dynamical systems"]},{"id":"2016-B2","year":2016,"session":"B","number":2,"statement":"Define a positive integer $n$ to be squarish if either $n$ is itself a perfect square or the distance from $n$ to the nearest perfect square is a perfect square. For example, 2016 is squarish, because the nearest perfect square to 2016 is $45^2 = 2025$ and $2025-2016=9$ is a perfect square. (Of the positive integers between 1 and 10, only 6 and 7 are not squarish.)\n\nFor a positive integer $N$, let $S(N)$ be the number of squarish integers between 1 and $N$,\ninclusive. Find positive constants $\\alpha$ and $\\beta$ such that\n$$\n\\lim_{N \\to \\infty} \\frac{S(N)}{N^\\alpha} = \\beta,\n$$\nor show that no such constants exist.","subjects":["number theory","counting","asymptotic analysis"]},{"id":"2016-B3","year":2016,"session":"B","number":3,"statement":"Suppose that $S$ is a finite set of points in the plane such that the area of triangle\n$\\triangle ABC$ is at most 1 whenever $A$, $B$, and $C$ are in $S$. Show that there exists a triangle of area 4 that (together with its interior) covers the set $S$.","subjects":["geometry"]},{"id":"2016-B4","year":2016,"session":"B","number":4,"statement":"Let $A$ be a $2n \\times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be 0 or 1, each with probability $1/2$. Find the expected value of $\\det(A-A^t)$ (as a function of $n$), where $A^t$ is the transpose of $A$.","subjects":["probability","determinants","random matrices"]},{"id":"2016-B5","year":2016,"session":"B","number":5,"statement":"Find all functions $f$ from the interval $(1, \\infty)$ to $(1, \\infty)$ with the following property:\nif $x,y \\in (1, \\infty)$ and $x^2 \\leq y \\leq x^3$, then $(f(x))^2 \\leq f(y) \\leq (f(x))^3$.","subjects":["analysis","functional inequalities"]},{"id":"2016-B6","year":2016,"session":"B","number":6,"statement":"Evaluate\n$$\n\\sum_{k=1}^\\infty \\frac{(-1)^{k-1}}{k} \\sum_{n=0}^\\infty \\frac{1}{k2^n + 1}.\n$$","subjects":["analysis","double series","binary factorization"]},{"id":"2017-A1","year":2017,"session":"A","number":1,"statement":"Let $S$ be the smallest set of positive integers such that\n\n(a) $2$ is in $S$,\n\n(b) $n$ is in $S$ whenever $n^2$ is in $S$, and\n\n(c) $(n+5)^2$ is in $S$ whenever $n$ is in $S$.\n\nWhich positive integers are not in $S$?\n\n(The set $S$ is ``smallest'' in the sense that $S$ is contained in any other such set.)","subjects":["number theory","closure operations","congruences"]},{"id":"2017-A2","year":2017,"session":"A","number":2,"statement":"Let $Q_0(x) = 1$, $Q_1(x) = x$, and\n$$\nQ_n(x) = \\frac{(Q_{n-1}(x))^2 - 1}{Q_{n-2}(x)}\n$$\nfor all $n \\geq 2$. Show that, whenever $n$ is a positive integer, $Q_n(x)$ is equal to a polynomial with integer coefficients.","subjects":["algebra","polynomial recurrences","determinant identities"]},{"id":"2017-A3","year":2017,"session":"A","number":3,"statement":"Let $a$ and $b$ be real numbers with $a<b$, and let $f$ and $g$ be continuous functions from $[a,b]$ to $(0, \\infty)$\nsuch that $\\int_a^b f(x)\\,dx = \\int_a^b g(x)\\,dx$ but $f \\neq g$. For every positive integer $n$, define\n$$\nI_n = \\int_a^b \\frac{(f(x))^{n+1}}{(g(x))^n}\\,dx.\n$$\nShow that $I_1, I_2, I_3, \\dots$ is an increasing sequence with $\\lim_{n \\to \\infty} I_n = \\infty$.","subjects":["analysis","moment inequalities","integral sequences"]},{"id":"2017-A4","year":2017,"session":"A","number":4,"statement":"A class with $2N$ students took a quiz, on which the possible scores were $0,1,\\dots,10$. Each of these scores\noccurred at least once, and the average score was exactly $7.4$. Show that the class can be divided into two groups of $N$ students in such a way that the average score for each group was exactly $7.4$.","subjects":["combinatorics","partition problems","integer sums","discrete mathematics","integer partitions"]},{"id":"2017-A5","year":2017,"session":"A","number":5,"statement":"Each of the integers from $1$ to $n$ is written on a separate card, and then the cards are combined into a deck and shuffled. Three players, $A$, $B$, and $C$, take turns in the order $A,B,C,A,\\dots$ choosing one card at random from the deck. (Each card in the deck is equally likely to be chosen.) After a card is chosen, that card and all higher-numbered cards are removed from the deck, and the remaining cards are reshuffled before the next turn. Play continues until one of the three players wins the game by drawing the card numbered $1$.\n\nShow that for each of the three players, there are arbitrarily large values of $n$ for which that player has the highest probability among the three players of winning the game.","subjects":["probability","complex numbers"]},{"id":"2017-A6","year":2017,"session":"A","number":6,"statement":"The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,\\dots,30$. How many different ways \nare there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? [Note: the top matter on each exam paper included the logo of the Mathematical Association of America, which is itself an icosahedron.]","subjects":["combinatorics","finite fields","polyhedral graphs"]},{"id":"2017-B1","year":2017,"session":"B","number":1,"statement":"Let $L_1$ and $L_2$ be distinct lines in the plane. Prove that $L_1$ and $L_2$ intersect if and only if, for every\nreal number $\\lambda\\neq 0$ and every point $P$ not on $L_1$ or $L_2$, there exist points $A_1$ on $L_1$ and $A_2$\non $L_2$ such that $\\overrightarrow{PA_2} = \\lambda \\overrightarrow{PA_1}$.","subjects":["geometry","affine lines","linear algebra"]},{"id":"2017-B2","year":2017,"session":"B","number":2,"statement":"Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers\n$$\nN = a + (a+1) +(a+2) + \\cdots + (a+k-1)\n$$\nfor $k=2017$ but for no other values of $k>1$. Considering all positive integers $N$ with this property,\nwhat is the smallest positive integer $a$ that occurs in any of these expressions?","subjects":["number theory","consecutive sums","divisibility"]},{"id":"2017-B3","year":2017,"session":"B","number":3,"statement":"Suppose that $f(x) = \\sum_{i=0}^\\infty c_i x^i$ is a power series for which each coefficient $c_i$ is $0$ or $1$.\nShow that if $f(2/3) = 3/2$, then $f(1/2)$ must be irrational.","subjects":["number theory","power series","binary expansions"]},{"id":"2017-B4","year":2017,"session":"B","number":4,"statement":"Evaluate the sum\n$$\n\\begin{gathered}\n\\sum_{k=0}^\\infty \\left( 3 \\cdot \\frac{\\ln(4k+2)}{4k+2} - \\frac{\\ln(4k+3)}{4k+3} - \\frac{\\ln(4k+4)}{4k+4} - \\frac{\\ln(4k+5)}{4k+5} \\right) \\\\\n= 3 \\cdot \\frac{\\ln 2}{2} - \\frac{\\ln 3}{3} - \\frac{\\ln 4}{4} - \\frac{\\ln 5}{5}\n+ 3 \\cdot \\frac{\\ln 6}{6} - \\frac{\\ln 7}{7} \\\\ - \\frac{\\ln 8}{8} - \\frac{\\ln 9}{9}\n+ 3 \\cdot \\frac{\\ln 10}{10} - \\cdots .\n\\end{gathered}\n$$\n(As usual, $\\ln x$ denotes the natural logarithm of $x$.)","subjects":["analysis","infinite series","logarithms"]},{"id":"2017-B5","year":2017,"session":"B","number":5,"statement":"A line in the plane of a triangle $T$ is called an equalizer if it divides $T$ into two regions having equal area and equal perimeter. Find positive integers $a>b>c$, with $a$ as small as possible, such that there exists a triangle with side lengths $a, b, c$ that has exactly two distinct equalizers.","subjects":["geometry","triangles","diophantine equations"]},{"id":"2017-B6","year":2017,"session":"B","number":6,"statement":"Find the number of ordered $64$-tuples $(x_0,x_1,\\dots,x_{63})$ such that $x_0,x_1,\\dots,x_{63}$ are distinct elements of $\\{1,2,\\dots,2017\\}$ and \n$$\nx_0 + x_1 + 2x_2 + 3x_3 + \\cdots + 63 x_{63}\n$$\nis divisible by 2017.","subjects":["combinatorics","finite fields","enumeration"]},{"id":"2018-A1","year":2018,"session":"A","number":1,"statement":"Find all ordered pairs $(a,b)$ of positive integers for which\n$$\n\\frac{1}{a} + \\frac{1}{b} = \\frac{3}{2018}.\n$$","subjects":["number theory","egyptian fractions","divisibility"]},{"id":"2018-A2","year":2018,"session":"A","number":2,"statement":"Let $S_1, S_2, \\dots, S_{2^n-1}$ be the nonempty subsets of $\\{1,2,\\dots,n\\}$ in some order, and let\n$M$ be the $(2^n-1) \\times (2^n-1)$ matrix whose $(i,j)$ entry is\n$$\nm_{ij} = \\begin{cases} 0 & \\text{if }S_i \\cap S_j = \\emptyset; \\\\\n1 & \\text{otherwise.}\n\\end{cases}\n$$\nCalculate the determinant of $M$.","subjects":["linear algebra","determinants","subsets"]},{"id":"2018-A3","year":2018,"session":"A","number":3,"statement":"Determine the greatest possible value of $\\sum_{i=1}^{10} \\cos(3x_i)$ for real numbers $x_1,x_2,\\dots,x_{10}$\nsatisfying $\\sum_{i=1}^{10} \\cos(x_i) = 0$.","subjects":["analysis","optimization","trigonometric identities","moment inequalities"]},{"id":"2018-A4","year":2018,"session":"A","number":4,"statement":"Let $m$ and $n$ be positive integers with $\\gcd(m,n) = 1$, and let\n$$\na_k = \\left\\lfloor \\frac{mk}{n} \\right\\rfloor - \\left\\lfloor \\frac{m(k-1)}{n} \\right\\rfloor\n$$\nfor $k=1,2,\\dots,n$.\nSuppose that $g$ and $h$ are elements in a group $G$ and that \n$$\ngh^{a_1} gh^{a_2} \\cdots gh^{a_n} = e,\n$$\nwhere $e$ is the identity element. Show that $gh= hg$. (As usual, $\\lfloor x \\rfloor$ denotes the greatest integer\nless than or equal to $x$.)","subjects":["algebra","group theory","floor functions"]},{"id":"2018-A5","year":2018,"session":"A","number":5,"statement":"Let $f: \\mathbb{R} \\to \\mathbb{R}$ be an infinitely differentiable function satisfying $f(0) = 0$, $f(1)= 1$,\nand $f(x) \\geq 0$ for all $x \\in \\mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$\nsuch that $f^{(n)}(x) < 0$.","subjects":["analysis","smooth functions","derivative inequalities"]},{"id":"2018-A6","year":2018,"session":"A","number":6,"statement":"Suppose that $A,B,C,$ and $D$ are distinct points, no three of which lie on a line,\nin the Euclidean plane. Show\nthat if the squares of the lengths of the line segments $AB$, $AC$, $AD$, $BC$, $BD$, and $CD$\nare rational numbers, then\nthe quotient\n$$\n\\frac{\\mathrm{area}(\\triangle ABC)}{\\mathrm{area}(\\triangle ABD)}\n$$\nis a rational number.","subjects":["geometry","rational distances","linear algebra"]},{"id":"2018-B1","year":2018,"session":"B","number":1,"statement":"Let $\\mathcal{P}$ be the set of vectors defined by\n$$\n\\mathcal{P} = \\left\\{ \\left. \\begin{pmatrix} a \\\\ b \\end{pmatrix} \\right| 0 \\leq a \\leq 2, 0 \\leq b \\leq 100, \\text{ and } a,b \\in \\mathbb{Z} \\right\\}.\n$$\nFind all $\\mathbf{v} \\in \\mathcal{P}$ such that the set $\\mathcal{P} \\setminus \\{ \\mathbf{v} \\}$ obtained by omitting\nvector $\\mathbf{v}$ from $\\mathcal{P}$ can be partitioned into two sets of equal size and equal sum.","subjects":["combinatorics","lattice points","equal-sum partitions"]},{"id":"2018-B2","year":2018,"session":"B","number":2,"statement":"Let $n$ be a positive integer, and let $f_n(z) = n + (n-1) z + (n-2)z^2 + \\cdots + z^{n-1}$. Prove that\n$f_n$ has no roots in the closed unit disk $\\{z \\in \\mathbb{C}\\colon |z| \\leq 1 \\}$.","subjects":["complex analysis","polynomial roots","unit disk"]},{"id":"2018-B3","year":2018,"session":"B","number":3,"statement":"Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n-1$,\nand $n-2$ divides $2^n - 2$.","subjects":["number theory","multiplicative orders","divisibility"]},{"id":"2018-B4","year":2018,"session":"B","number":4,"statement":"Given a real number $a$, we define a sequence by $x_0 = 1$, $x_1 = x_2 = a$, and $x_{n+1} = 2x_n x_{n-1} - x_{n-2}$ for $n \\geq 2$. Prove that if $x_n = 0$ for some $n$, then the sequence is periodic.","subjects":["algebra","recurrences"]},{"id":"2018-B5","year":2018,"session":"B","number":5,"statement":"Let $f = (f_1, f_2)$ be a function from $\\mathbb{R}^2$ to $\\mathbb{R}^2$ with continuous partial derivatives\n$\\frac{\\partial f_i}{\\partial x_j}$ that are positive everywhere. Suppose that\n$$\n\\frac{\\partial f_1}{\\partial x_1} \\frac{\\partial f_2}{\\partial x_2} - \\frac{1}{4} \\left( \\frac{\\partial f_1}{\\partial x_2}  + \\frac{\\partial f_2}{\\partial x_1} \\right)^2 > 0\n$$\neverywhere. Prove that $f$ is one-to-one.","subjects":["analysis"]},{"id":"2018-B6","year":2018,"session":"B","number":6,"statement":"Let $S$ be the set of sequences of length $2018$ whose terms  are in the set $\\{1,2,3,4,5,6,10\\}$ and sum to $3860$.\nProve that the cardinality of $S$ is at most\n$$\n2^{3860} \\cdot \\left( \\frac{2018}{2048} \\right)^{2018}.\n$$","subjects":["combinatorics","coefficient bounds"]},{"id":"2019-A1","year":2019,"session":"A","number":1,"statement":"Determine all possible values of the expression\n$$\nA^3+B^3+C^3-3ABC\n$$\nwhere $A, B$, and $C$ are nonnegative integers.","subjects":["number theory","polynomial values","modular arithmetic"]},{"id":"2019-A2","year":2019,"session":"A","number":2,"statement":"In the triangle $\\triangle ABC$, let $G$ be the centroid, and let $I$ be the center of the inscribed circle.\nLet $\\alpha$ and $\\beta$ be the angles at the vertices $A$ and $B$, respectively.\nSuppose that the segment $IG$ is parallel to $AB$ and that $\\beta = 2 \\tan^{-1} (1/3)$. Find $\\alpha$.","subjects":["geometry","triangle centers","half-angle formulas"]},{"id":"2019-A3","year":2019,"session":"A","number":3,"statement":"Given real numbers $b_0, b_1, \\dots, b_{2019}$ with $b_{2019} \\neq 0$, let $z_1,z_2,\\dots,z_{2019}$ be \nthe roots in the complex plane of the polynomial \n$$\nP(z) = \\sum_{k=0}^{2019} b_k z^k.\n$$\nLet $\\mu = (|z_1| + \\cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\\dots,z_{2019}$ to the origin. Determine the largest constant $M$ such that $\\mu \\geq M$ for all choices of $b_0,b_1,\\dots, b_{2019}$ that satisfy\n$$\n1 \\leq b_0 < b_1 < b_2 < \\cdots < b_{2019} \\leq 2019.\n$$","subjects":["algebra","polynomial roots","inequalities"]},{"id":"2019-A4","year":2019,"session":"A","number":4,"statement":"Let $f$ be a continuous real-valued function on $\\mathbb{R}^3$. Suppose that for every sphere $S$ of radius 1,\nthe integral of $f(x,y,z)$ over the surface of $S$ equals 0. Must $f(x,y,z)$ be identically 0?","subjects":["analysis","surface integrals","counterexamples"]},{"id":"2019-A5","year":2019,"session":"A","number":5,"statement":"Let $p$ be an odd prime number, and let $\\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\\mathbb{F}_p[x]$ be the ring of polynomials over $\\mathbb{F}_p$, and let $q(x) \\in \\mathbb{F}_p[x]$ be given by \n$$\nq(x) = \\sum_{k=1}^{p-1} a_k x^k,\n$$\nwhere\n$$\na_k = k^{(p-1)/2} \\mod{p}. \n$$\nFind the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\\mathbb{F}_p[x]$.","subjects":["algebra","finite fields","root multiplicity"]},{"id":"2019-A6","year":2019,"session":"A","number":6,"statement":"Let $g$ be a real-valued function that is continuous on the closed interval $[0,1]$ and twice differentiable on \nthe open interval $(0,1)$. Suppose that for some real number $r>1$, \n$$\n\\lim_{x \\to 0^+} \\frac{g(x)}{x^r} = 0.\n$$\nProve that either\n$$\n\\lim_{x \\to 0^+} g'(x) = 0 \\qquad \\text{or} \\qquad \\limsup_{x \\to 0^+} x^r |g''(x)| = \\infty.\n$$","subjects":["analysis","asymptotic estimates","derivative inequalities"]},{"id":"2019-B1","year":2019,"session":"B","number":1,"statement":"Denote by $\\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \\geq 0$, let $P_n$ be the subset of $\\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2 + y^2 = 2^k$ for some integer $k \\leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square.","subjects":["combinatorics","lattice geometry","squares","powers of two"]},{"id":"2019-B2","year":2019,"session":"B","number":2,"statement":"For all $n \\geq 1$, let\n$$\na_n = \\sum_{k=1}^{n-1} \\frac{\\sin \\left( \\frac{(2k-1)\\pi}{2n} \\right)}{\\cos^2 \\left( \\frac{(k-1)\\pi}{2n} \\right) \\cos^2 \\left( \\frac{k\\pi}{2n} \\right)}.\n$$\nDetermine\n$$\n\\lim_{n \\to \\infty} \\frac{a_n}{n^3}.\n$$","subjects":["analysis","trigonometric sums","limits"]},{"id":"2019-B3","year":2019,"session":"B","number":3,"statement":"Let $Q$ be an $n$-by-$n$ real orthogonal matrix, and let $u \\in \\mathbb{R}^n$ be a unit column vector (that is,\n$u^T u = 1$). Let $P = I - 2uu^T$, where $I$ is the $n$-by-$n$ identity matrix. Show that if $1$ is not an eigenvalue of $Q$, then $1$ is an eigenvalue of $PQ$.","subjects":["linear algebra","orthogonal matrices","reflections"]},{"id":"2019-B4","year":2019,"session":"B","number":4,"statement":"Let $\\mathcal{F}$ be the set of functions $f(x,y)$ that are twice continuously differentiable for $x \\geq 1$, $y \\geq 1$ and that satisfy the following two equations (where subscripts denote partial derivatives):\n$$\n\\begin{gathered}\nxf_x + yf_y = xy \\ln(xy), \\\\\nx^2 f_{xx} + y^2 f_{yy} = xy.\n\\end{gathered}\n$$\nFor each $f \\in \\mathcal{F}$, let\n$$\nm(f) = \\min_{s \\geq 1} \\left(f(s+1,s+1) - f(s+1,s) - f(s,s+1) + f(s,s) \\right).\n$$\nDetermine $m(f)$, and show that it is independent of the choice of $f$.","subjects":["analysis","partial differential equations","finite differences"]},{"id":"2019-B5","year":2019,"session":"B","number":5,"statement":"Let $F_m$ be the $m$th Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_m = F_{m-1} + F_{m-2}$ for all $m \\geq 3$.\nLet $p(x)$ be the polynomial of degree $1008$ such that $p(2n+1) = F_{2n+1}$ for $n=0,1,2,\\dots,1008$. Find integers $j$ and $k$ such that $p(2019) = F_j - F_k$.","subjects":["algebra","polynomial interpolation","fibonacci numbers"]},{"id":"2019-B6","year":2019,"session":"B","number":6,"statement":"Let $\\mathbb{Z}^n$ be the integer lattice in $\\mathbb{R}^n$. Two points in $\\mathbb{Z}^n$ are called \nneighbors if they differ by exactly $1$ in one coordinate and are equal in all other coordinates. \nFor which integers $n \\geq 1$ does there exist a set of points $S \\subset \\mathbb{Z}^n$ satisfying the following two conditions?\n\n(1) If $p$ is in $S$, then none of the neighbors of $p$ is in $S$.\n\n(2) If $p \\in \\mathbb{Z}^n$ is not in $S$, then exactly one of the neighbors of $p$ is in $S$.","subjects":["combinatorics","algebra","integer lattices"]},{"id":"2020-A1","year":2020,"session":"A","number":1,"statement":"How many positive integers $N$ satisfy all of the following three conditions?\n\n(i) $N$ is divisible by 2020.\n\n(ii) $N$ has at most 2020 decimal digits.\n\n(iii) The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros.","subjects":["number theory","decimal digits","enumeration"]},{"id":"2020-A2","year":2020,"session":"A","number":2,"statement":"Let $k$ be a nonnegative integer. Evaluate\n$$\n\\sum_{j=0}^k 2^{k-j} \\binom{k+j}{j}.\n$$","subjects":["combinatorics","binomial sums","probability"]},{"id":"2020-A3","year":2020,"session":"A","number":3,"statement":"Let $a_0 = \\pi/2$, and let $a_n = \\sin(a_{n-1})$ for $n \\geq 1$. Determine whether\n$$\n\\sum_{n=1}^\\infty a_n^2\n$$\nconverges.","subjects":["analysis","iterated functions","series convergence"]},{"id":"2020-A4","year":2020,"session":"A","number":4,"statement":"Consider a horizontal strip of $N+2$ squares in which the first and the last square are black and the remaining $N$ squares are all white. Choose a white square uniformly at random, choose one of its two neighbors with equal probability,\nand color this neighboring square black if it is not already black. Repeat this process until all the remaining white squares have only black neighbors. Let $w(N)$ be the expected number of white squares remaining. Find\n$$\n\\lim_{N \\to \\infty} \\frac{w(N)}{N}.\n$$","subjects":["probability","random processes"]},{"id":"2020-A5","year":2020,"session":"A","number":5,"statement":"Let $a_n$ be the number of sets $S$ of positive integers for which\n$$\n\\sum_{k \\in S} F_k = n,\n$$\nwhere the Fibonacci sequence $(F_k)_{k \\geq 1}$ satisfies $F_{k+2} = F_{k+1} + F_k$ and begins $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$. Find the largest integer $n$ such that $a_n = 2020$.","subjects":["combinatorics","fibonacci representations","extremal counting"]},{"id":"2020-A6","year":2020,"session":"A","number":6,"statement":"For a positive integer $N$, let $f_N$ [Archive note. Corrected from $F_N$ in the source.] be the function defined by \n$$\nf_N(x) = \\sum_{n=0}^N \\frac{N+1/2-n}{(N+1)(2n+1)} \\sin((2n+1)x).\n$$\nDetermine the smallest constant $M$ such that $f_N(x) \\leq M$ for all $N$ and all real $x$.","subjects":["analysis","trigonometric polynomials","fourier series"]},{"id":"2020-B1","year":2020,"session":"B","number":1,"statement":"For a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary (for example, $d(13) = 1+1+0+1=3)$. Let\n$$\nS = \\sum_{k=1}^{2020} (-1)^{d(k)} k^3.\n$$\nDetermine $S$ modulo 2020.","subjects":["combinatorics","binary digits","weighted sums","congruences"]},{"id":"2020-B2","year":2020,"session":"B","number":2,"statement":"Let $k$ and $n$ be integers with $1 \\leq k < n$. Alice and Bob play a game with $k$ pegs in a line of $n$ holes. At the beginning of the game, the pegs occupy the $k$ leftmost holes. A legal move consists of moving a single peg\nto any vacant hole that is further to the right. The players alternate moves, with Alice playing first. The game ends when the pegs are in the $k$ rightmost holes, so whoever is next to play cannot move and therefore loses. For what values\nof $n$ and $k$ does Alice have a winning strategy?","subjects":["combinatorial game theory","peg games","winning strategies","binary arithmetic"]},{"id":"2020-B3","year":2020,"session":"B","number":3,"statement":"Let $x_0 = 1$, and let $\\delta$ be some constant satisfying $0 < \\delta < 1$. Iteratively, for $n=0,1,2,\\dots$, a point $x_{n+1}$ is chosen uniformly from the interval $[0, x_n]$. Let $Z$ be the smallest value of $n$ for which $x_n < \\delta$.\nFind the expected value of $Z$, as a function of $\\delta$.","subjects":["probability","stopping times","products of uniforms"]},{"id":"2020-B4","year":2020,"session":"B","number":4,"statement":"Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\\mathbf{v} = (s_0, s_1, \\cdots, s_{2n-1}, s_{2n})$ for which $s_0 = s_{2n} = 0$ and $|s_j - s_{j-1}| = 1$ for $j=1,2,\\cdots,2n$. Define\n$$\nq(\\mathbf{v}) = 1 + \\sum_{j=1}^{2n-1} 3^{s_j},\n$$\nand let $M(n)$ be the average of $\\frac{1}{q(\\mathbf{v})}$ over all $\\mathbf{v} \\in V_n$. Evaluate $M(2020)$.","subjects":["combinatorics","probability","lattice paths"]},{"id":"2020-B5","year":2020,"session":"B","number":5,"statement":"For $j \\in \\{1, 2, 3, 4\\}$, let $z_j$ be a complex number with $|z_j| = 1$ and $z_j \\neq 1$. Prove that\n$$\n3 - z_1 - z_2 - z_3 - z_4 + z_1 z_2 z_3 z_4 \\neq 0.\n$$","subjects":["complex analysis","unit circle","polynomial inequalities"]},{"id":"2020-B6","year":2020,"session":"B","number":6,"statement":"Let $n$ be a positive integer. Prove that\n$$\n\\sum_{k=1}^n (-1)^{\\lfloor k(\\sqrt{2}-1) \\rfloor} \\geq 0.\n$$\n(As usual, $\\lfloor x \\rfloor$ denotes the greatest integer less than or equal to $x$.)","subjects":["number theory","floor sequences","self similarity"]},{"id":"2021-A1","year":2021,"session":"A","number":1,"statement":"A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops.\nEach hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the first hop.\nWhat is the smallest number of hops needed for the grasshopper to reach the point $(2021, 2021)$?","subjects":["geometry","lattice walks","optimization"]},{"id":"2021-A2","year":2021,"session":"A","number":2,"statement":"For every positive real number $x$, let\n$$\ng(x) = \\lim_{r \\to 0} ((x+1)^{r+1} - x^{r+1})^{\\frac{1}{r}}.\n$$\nFind $\\lim_{x \\to \\infty} \\frac{g(x)}{x}$.","subjects":["analysis","limits","exponential functions"]},{"id":"2021-A3","year":2021,"session":"A","number":3,"statement":"Determine all positive integers $N$ for which the sphere\n$$\nx^2 + y^2 + z^2 = N\n$$\nhas an inscribed regular tetrahedron whose vertices have integer coordinates.","subjects":["geometry","number theory","regular tetrahedra","integer coordinates"]},{"id":"2021-A4","year":2021,"session":"A","number":4,"statement":"Let\n$$\nI(R) = \\iint_{x^2+y^2 \\leq R^2} \\left( \\frac{1+2x^2}{1+x^4+6x^2y^2+y^4} - \\frac{1+y^2}{2+x^4+y^4} \\right)\\,dx\\,dy.\n$$\nFind\n$$\n\\lim_{R \\to \\infty} I(R),\n$$\nor show that this limit does not exist.","subjects":["analysis","integration","double integrals"]},{"id":"2021-A5","year":2021,"session":"A","number":5,"statement":"Let $A$ be the set of all integers $n$ such that $1 \\leq n \\leq 2021$ and $\\gcd(n, 2021) = 1$.\nFor every nonnegative integer $j$, let\n$$\nS(j) = \\sum_{n \\in A} n^j.\n$$\nDetermine all values of $j$ such that $S(j)$ is a multiple of 2021.","subjects":["number theory","power sums","finite fields"]},{"id":"2021-A6","year":2021,"session":"A","number":6,"statement":"Let $P(x)$ be a polynomial whose coefficients are all either $0$ or $1$.\nSuppose that $P(x)$ can be written as a product of two nonconstant polynomials with integer coefficients. Does it follow that $P(2)$ is a composite integer?","subjects":["algebra","number theory","irreducibility"]},{"id":"2021-B1","year":2021,"session":"B","number":1,"statement":"Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard?","subjects":["probability","lattice geometry","random squares","harmonic analysis"]},{"id":"2021-B2","year":2021,"session":"B","number":2,"statement":"Determine the maximum value of the sum\n$$\nS = \\sum_{n=1}^\\infty \\frac{n}{2^n} (a_1 a_2 \\cdots a_n)^{1/n}\n$$\nover all sequences $a_1, a_2, a_3, \\cdots$ of nonnegative real numbers satisfying\n$$\n\\sum_{k=1}^\\infty a_k = 1.\n$$","subjects":["analysis","infinite series","inequalities","optimization"]},{"id":"2021-B3","year":2021,"session":"B","number":3,"statement":"Let $h(x,y)$ be a real-valued function that is twice continuously differentiable throughout $\\mathbb{R}^2$, and define\n$$\n\\rho(x,y) = yh_x - xh_y.\n$$\nProve or disprove: For any positive constants $d$ and $r$ with $d>r$, there is a circle $\\mathcal{S}$ of radius $r$ whose center is a distance $d$ away from the origin such that the integral of $\\rho$ over the interior of $\\mathcal{S}$ is zero.","subjects":["analysis","geometry"]},{"id":"2021-B4","year":2021,"session":"B","number":4,"statement":"Let $F_0, F_1, \\dots$ be the sequence of Fibonacci numbers, with $F_0 = 0$, $F_1 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n \\geq 2$. For $m > 2$, let $R_m$ be the remainder when the product $\\prod_{k=1}^{F_m-1} k^k$ is divided by $F_m$. Prove that $R_m$ is also a Fibonacci number.","subjects":["number theory","fibonacci numbers","finite fields"]},{"id":"2021-B5","year":2021,"session":"B","number":5,"statement":"Say that an $n$-by-$n$ matrix $A = (a_{ij})_{1 \\leq i,j \\leq n}$ with integer entries is very odd if, for every nonempty subset $S$ of $\\{1,2,\\dots,n\\}$, the $|S|$-by-$|S|$ submatrix $(a_{ij})_{i,j \\in S}$ has odd determinant. Prove that if $A$ is very odd, then $A^k$ is very odd for every $k \\geq 1$.","subjects":["linear algebra","finite fields","principal minors"]},{"id":"2021-B6","year":2021,"session":"B","number":6,"statement":"Given an ordered list of $3N$ real numbers, we can trim it to form a list of $N$ numbers as follows: We divide the list into $N$ groups of $3$ consecutive numbers, and within each group, discard the highest and lowest numbers, keeping only the median.\n\nConsider generating a random number $X$ by the following procedure: Start with a list of $3^{2021}$ numbers, drawn independently and uniformly at random between 0 and 1. Then trim this list as defined above, leaving a list of $3^{2020}$ numbers. Then trim again repeatedly until just one number remains; let $X$ be this number. Let $\\mu$ be the expected value of $|X - \\frac{1}{2}|$. Show that\n$$\n\\mu \\geq \\frac{1}{4} \\left( \\frac{2}{3} \\right)^{2021}.\n$$","subjects":["probability","order statistics","concentration bounds"]},{"id":"2022-A1","year":2022,"session":"A","number":1,"statement":"Determine all ordered pairs of real numbers $(a,b)$ such that the line $y = ax+b$ intersects the curve $y = \\ln(1+x^2)$ in exactly one point.","subjects":["analysis","curve intersections","logarithms","root counting"]},{"id":"2022-A2","year":2022,"session":"A","number":2,"statement":"Let $n$ be an integer with $n \\geq 2$. Over all real polynomials $p(x)$ of degree $n$, what is the largest possible number of negative coefficients of $p(x)^2$?","subjects":["algebra","polynomial coefficients","squares of polynomials"]},{"id":"2022-A3","year":2022,"session":"A","number":3,"statement":"Let $p$ be a prime number greater than 5. Let $f(p)$ denote the number of infinite sequences $a_1, a_2, a_3, \\dots$ such that\n$a_n \\in \\{1, 2, \\dots, p-1\\}$ and $a_n a_{n+2} \\equiv 1 + a_{n+1} \\pmod{p}$ for all $n \\geq 1$. Prove that $f(p)$ is congruent to 0 or 2 $\\pmod{5}$.","subjects":["number theory","combinatorics"]},{"id":"2022-A4","year":2022,"session":"A","number":4,"statement":"Suppose that $X_1, X_2, \\dots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S = \\sum_{i=1}^k X_i/2^i$, where $k$ is the least positive integer such that $X_k < X_{k+1}$, or $k = \\infty$ if there is no such integer. Find the expected value of $S$.","subjects":["probability","stopping rules","order statistics"]},{"id":"2022-A5","year":2022,"session":"A","number":5,"statement":"Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule. Alice's goal is to maximize the number of uncovered squares when the game ends; Bob's goal is to minimize it. What is the greatest number of uncovered squares that Alice can ensure at the end of the game, no matter how Bob plays?","subjects":["combinatorics","games","domino placements"]},{"id":"2022-A6","year":2022,"session":"A","number":6,"statement":"Let $n$ be a positive integer. Determine, in terms of $n$, the largest integer $m$ with the following property: There exist real numbers $x_1,\\dots,x_{2n}$ with $-1 < x_1 < x_2 < \\cdots < x_{2n} < 1$ such that the sum of the lengths of the $n$ intervals\n$$\n[x_1^{2k-1}, x_2^{2k-1}], [x_3^{2k-1},x_4^{2k-1}], \\dots, [x_{2n-1}^{2k-1}, x_{2n}^{2k-1}]\n$$\nis equal to 1 for all integers $k$ with $1 \\leq k \\leq m$.","subjects":["algebra","power sums","roots of unity"]},{"id":"2022-B1","year":2022,"session":"B","number":1,"statement":"Suppose that $P(x) = a_1 x + a_2 x^2 + \\cdots + a_n x^n$ is a polynomial with integer coefficients, with $a_1$ odd. Suppose that $e^{P(x)} = b_0 + b_1 x + b_2 x^2 + \\cdots$ for all $x$. Prove that $b_k$ is nonzero for all $k \\geq 0$.","subjects":["algebra","power series","parity"]},{"id":"2022-B2","year":2022,"session":"B","number":2,"statement":"Let $\\times$ represent the cross product in $\\mathbb{R}^3$. For what positive integers $n$ does there exist a set $S \\subset \\mathbb{R}^3$ with exactly $n$ elements such that \n$$\nS = \\{v \\times w: v, w \\in S\\}?\n$$","subjects":["linear algebra","cross products","finite sets","classification"]},{"id":"2022-B3","year":2022,"session":"B","number":3,"statement":"Assign to each positive real number a color, either red or blue. Let $D$ be the set of all distances $d > 0$ such that there are two points of the same color at distance $d$ apart. Recolor the positive reals so that the numbers in $D$ are red and the numbers not in $D$ are blue. If we iterate this recoloring process, will we always end up with all the numbers red after a finite number of steps?","subjects":["combinatorics"]},{"id":"2022-B4","year":2022,"session":"B","number":4,"statement":"Find all integers $n$ with $n \\geq 4$ for which there exists a sequence of distinct real numbers $x_1,\\dots,x_n$ such that each of the sets\n$$\n\\begin{gathered}\n\\{x_1,x_2,x_3\\}, \\{x_2,x_3,x_4\\}, \\dots, \\\\\n\\{x_{n-2},x_{n-1},x_n\\}, \\{x_{n-1},x_n, x_1\\}, \\text{ and } \\{x_n, x_1, x_2\\}\n\\end{gathered}\n$$\nforms a 3-term arithmetic progression when arranged in increasing order.","subjects":["combinatorics","arithmetic progressions","cyclic sequences"]},{"id":"2022-B5","year":2022,"session":"B","number":5,"statement":"For $0 \\leq p \\leq 1/2$, let $X_1, X_2, \\dots$ be independent random variables such that\n$$\nX_i = \\begin{cases} 1 & \\text{with probability }p\\text{,} \\\\\n-1 & \\text{with probability }p\\text{,} \\\\\n0 & \\text{with probability }1-2p\\text{,}\n\\end{cases}\n$$\nfor all $i \\geq 1$. Given a positive integer $n$ and integers $b, a_1, \\dots, a_n$, let $P(b, a_1, \\dots, a_n)$ denote the probability that $a_1 X_1 + \\cdots + a_n X_n = b$. For which values of $p$ is it the case that\n$$\nP(0, a_1, \\dots, a_n) \\geq P(b, a_1, \\dots, a_n)\n$$\nfor all positive integers $n$ and all integers $b, a_1, \\dots, a_n$?","subjects":["probability","fourier analysis","concentration functions"]},{"id":"2022-B6","year":2022,"session":"B","number":6,"statement":"Find all continuous functions $f: \\mathbb{R}^+ \\to \\mathbb{R}^+$ such that\n$$\nf(xf(y)) + f(yf(x)) = 1 + f(x+y)\n$$\nfor all $x,y > 0$.","subjects":["analysis","functional equations","continuity"]},{"id":"2023-A1","year":2023,"session":"A","number":1,"statement":"For a positive integer $n$, let $f_n(x) = \\cos(x) \\cos(2x) \\cos(3x) \\cdots \\cos(nx)$. Find the smallest $n$ such that $|f_n''(0)| > 2023$.","subjects":["analysis"]},{"id":"2023-A2","year":2023,"session":"A","number":2,"statement":"Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n$; that is to say, $p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \\cdots + a_1 x + a_0$ for some real coefficients $a_0, \\dots, a_{2n-1}$. Suppose that $p(1/k) = k^2$ for all integers $k$ such that $1 \\leq |k| \\leq n$. Find all other real numbers $x$ for which $p(1/x) = x^2$.","subjects":["algebra","polynomial interpolation","reciprocal values"]},{"id":"2023-A3","year":2023,"session":"A","number":3,"statement":"Determine the smallest positive real number $r$ such that there exist differentiable functions $f\\colon \\mathbb{R} \\to \\mathbb{R}$ and\n$g\\colon \\mathbb{R} \\to \\mathbb{R}$ satisfying\n\n(a) $f(0) > 0$,\n\n(b) $g(0) = 0$,\n\n(c) $|f'(x)| \\leq |g(x)|$ for all $x$,\n\n(d) $|g'(x)| \\leq |f(x)|$ for all $x$, and\n\n(e) $f(r) = 0$.","subjects":["analysis","differential inequalities","optimal control","functional inequalities","sobolev energy estimates"]},{"id":"2023-A4","year":2023,"session":"A","number":4,"statement":"Let $v_1, \\dots, v_{12}$ be unit vectors in $\\mathbb{R}^3$ from the origin to the vertices of a regular icosahedron. Show that for every vector $v \\in \\mathbb{R}^3$ and every $\\varepsilon > 0$, there exist integers $a_1,\\dots,a_{12}$ such that $\\| a_1 v_1 + \\cdots + a_{12} v_{12} - v \\| < \\varepsilon$.","subjects":["geometry","diophantine approximation","icosahedral symmetry"]},{"id":"2023-A5","year":2023,"session":"A","number":5,"statement":"For a nonnegative integer $k$, let $f(k)$ be the number of ones in the base 3 representation of $k$. Find all complex numbers $z$ such that\n$$\n\\sum_{k=0}^{3^{1010}-1} (-2)^{f(k)} (z+k)^{2023} = 0.\n$$","subjects":["algebra","analysis","digit sums","finite differences"]},{"id":"2023-A6","year":2023,"session":"A","number":6,"statement":"Alice and Bob play a game in which they take turns choosing integers from $1$ to $n$. Before any integers are chosen, Bob selects a goal of ``odd'' or ``even''. On the first turn, Alice chooses one of the $n$ integers. On the second turn, Bob chooses one of the remaining integers. They continue alternately choosing one of the integers that has not yet been chosen, until the $n$th turn, which is forced and ends the game. Bob wins if the parity of $\\{k\\colon \\text{the number }k\\text{ was chosen on the }k\\text{th turn}\\}$ matches his goal. For which values of $n$ does Bob have a winning strategy?","subjects":["combinatorics","games","permutation fixed points"]},{"id":"2023-B1","year":2023,"session":"B","number":1,"statement":"Consider an $m$-by-$n$ grid of unit squares, indexed by $(i,j)$ with $1 \\leq i \\leq m$ and $1 \\leq j \\leq n$. There are $(m-1)(n-1)$ coins, which are initially placed in the squares $(i,j)$ with $1 \\leq i \\leq m-1$ and $1 \\leq j \\leq n-1$. If a coin occupies the square $(i,j)$ with $i \\leq m-1$ and $j \\leq n-1$ and the squares $(i+1,j), (i,j+1)$, and $(i+1,j+1)$ are unoccupied, then a legal move is to slide the coin from $(i,j)$ to $(i+1,j+1)$. How many distinct configurations of coins can be reached starting from the initial configuration by a (possibly empty) sequence of legal moves?","subjects":["combinatorics","coin-moving dynamics","enumeration"]},{"id":"2023-B2","year":2023,"session":"B","number":2,"statement":"For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of $2023 \\cdot n$. What is the minimum value of $k(n)$?","subjects":["number theory","binary digits","modular exponentiation","finite groups","quadratic residues"]},{"id":"2023-B3","year":2023,"session":"B","number":3,"statement":"A sequence $y_1,y_2,\\dots,y_k$ of real numbers is called zigzag if $k=1$, or if $y_2-y_1, y_3-y_2, \\dots, y_k-y_{k-1}$ are nonzero and alternate in sign. Let $X_1,X_2,\\dots,X_n$ be chosen independently from the uniform distribution on $[0,1]$. Let $a(X_1,X_2,\\dots,X_n)$ be the largest value of $k$ for which there exists an increasing sequence of integers $i_1,i_2,\\dots,i_k$ such that $X_{i_1},X_{i_2},\\dots,X_{i_k}$ is zigzag. Find the expected value of $a(X_1,X_2,\\dots,X_n)$ for $n \\geq 2$.","subjects":["probability","combinatorics","alternating subsequences"]},{"id":"2023-B4","year":2023,"session":"B","number":4,"statement":"For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\\dots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \\geq t_0$ by the following properties:\n\n(a) $f(t)$ is continuous for $t \\geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1,\\dots,t_n$;\n\n(b) $f(t_0) = 1/2$;\n\n(c) $\\lim_{t \\to t_k^+} f'(t) = 0$ for $0 \\leq k \\leq n$;\n\n(d) For $0 \\leq k \\leq n-1$, we have $f''(t) = k+1$ when $t_k < t< t_{k+1}$, and $f''(t) = n+1$ when $t>t_n$.\n\nConsidering all choices of $n$ and $t_0,t_1,\\dots,t_n$ such that $t_k \\geq t_{k-1}+1$ for $1 \\leq k \\leq n$, what is the least possible value of $T$ for which $f(t_0+T) = 2023$?","subjects":["analysis","optimization","piecewise quadratic functions"]},{"id":"2023-B5","year":2023,"session":"B","number":5,"statement":"Determine which positive integers $n$ have the following property:\nFor all integers $m$ that are relatively prime to $n$, there exists a permutation $\\pi\\colon \\{1,2,\\dots,n\\} \\to \\{1,2,\\dots,n\\}$ such that $\\pi(\\pi(k)) \\equiv mk \\pmod{n}$ for all $k \\in \\{1,2,\\dots,n\\}$.","subjects":["algebra","number theory"]},{"id":"2023-B6","year":2023,"session":"B","number":6,"statement":"Let $n$ be a positive integer. For $i$ and $j$ in $\\{1,2,\\dots,n\\}$, let $s(i,j)$ be the number of pairs $(a,b)$ of nonnegative integers satisfying $ai +bj=n$. Let $S$ be the $n$-by-$n$ matrix whose $(i,j)$ entry is $s(i,j)$. For example, when $n=5$, we have\n$S = \\begin{bmatrix} 6 & 3 & 2 & 2 & 2 \\\\\n3 & 0 & 1 & 0 & 1 \\\\\n2 & 1 & 0 & 0 & 1 \\\\\n2 & 0 & 0 & 0 & 1 \\\\\n2 & 1 & 1 & 1 & 2\n\\end{bmatrix}$. \nCompute the determinant of $S$.","subjects":["linear algebra","determinants","divisibility matrices"]},{"id":"2024-A1","year":2024,"session":"A","number":1,"statement":"Determine all positive integers $n$ for which there exist positive integers $a$, $b$, and $c$ satisfying\n$$\n2a^n + 3b^n = 4c^n.\n$$","subjects":["number theory","exponential diophantine equations","congruences"]},{"id":"2024-A2","year":2024,"session":"A","number":2,"statement":"For which real polynomials $p$ is there a real polynomial $q$ such that\n$$\np(p(x)) - x = (p(x) - x)^2 q(x)\n$$\nfor all real $x$?","subjects":["algebra","polynomial divisibility","composition"]},{"id":"2024-A3","year":2024,"session":"A","number":3,"statement":"Let $S$ be the set of bijections\n$$\nT \\colon \\{1,2,3\\} \\times \\{1,2,\\dots,2024\\} \\to \\{1,2,\\dots,6072\\}\n$$\nsuch that $T(1,j) < T(2,j) < T(3,j)$ for all $j \\in \\{1,2,\\dots,2024\\}$ and $T(i,j) < T(i,j+1)$ for all $i \\in \\{1,2,3\\}$ and $j \\in \\{1,2,\\dots,2023\\}$. Do there exist $a$ and $c$ in $\\{1,2,3\\}$ and $b$ and $d$ in $\\{1,2,\\dots,2024\\}$ such that the fraction of elements $T$ in $S$ for which $T(a,b) < T(c,d)$ is at least $1/3$ and at most $2/3$?","subjects":["combinatorics","young tableaux","linear extensions"]},{"id":"2024-A4","year":2024,"session":"A","number":4,"statement":"Find all primes $p > 5$ for which there exists an integer $a$ and an integer $r$ satisfying $1 \\leq r \\leq p-1$ with the following property: the sequence $1,a,a^2,\\dots,a^{p-5}$ can be rearranged to form a sequence $b_0,b_1,b_2,\\dots,b_{p-5}$ such that $b_n-b_{n-1}-r$ is divisible by $p$ for $1 \\leq n \\leq p-5$.","subjects":["number theory","finite fields","geometric progressions"]},{"id":"2024-A5","year":2024,"session":"A","number":5,"statement":"Consider a circle $\\Omega$ with radius 9 and center at the origin $(0,0)$, and a disc $\\Delta$ with radius 1 and center at $(r,0)$, where $0 \\leq r \\leq 8$. Two points $P$ and $Q$ are chosen independently and uniformly at random on $\\Omega$. Which value(s) of $r$ minimize the probability that the chord $\\overline{PQ}$ intersects $\\Delta$?","subjects":["probability","random chords","geometric optimization","integral geometry","potential theory"]},{"id":"2024-A6","year":2024,"session":"A","number":6,"statement":"Let $c_0,c_1,c_2,\\dots$ be the sequence defined so that \n$$\n\\frac{1-3x-\\sqrt{1-14x+9x^2}}{4} = \\sum_{k=0}^\\infty c_k x^k\n$$\nfor sufficiently small $x$. For a positive integer $n$, let $A$ be the $n$-by-$n$ matrix with $i,j$-entry $c_{i+j-1}$ for $i$ and $j$ in $\\{1,\\dots,n\\}$. Find the determinant of $A$.","subjects":["linear algebra","combinatorics","hankel determinants"]},{"id":"2024-B1","year":2024,"session":"B","number":1,"statement":"Let $n$ and $k$ be positive integers. The square in the $i$th row and $j$th column of an $n$-by-$n$ grid contains the number $i+j-k$. For which $n$ and $k$ is it possible to select $n$ squares from the grid, no two in the same row or column, such that the numbers contained in the selected squares are exactly $1,2,\\dots,n$?","subjects":["combinatorics","permutations","grid transversals","combinatorial designs","graph theory"]},{"id":"2024-B2","year":2024,"session":"B","number":2,"statement":"Two convex quadrilaterals are called partners if they have three vertices in common and they can be labeled $ABCD$ and $ABCE$ so that $E$ is the reflection of $D$ across the perpendicular bisector of the diagonal $\\overline{AC}$. Is there an infinite sequence of convex quadrilaterals such that each quadrilateral is a partner of its successor and no two elements of the sequence are congruent?","subjects":["geometry","quadrilaterals","invariants"]},{"id":"2024-B3","year":2024,"session":"B","number":3,"statement":"Let $r_n$ be the $n$th smallest positive solution to $\\tan x = x$, where the argument of tangent is in radians. Prove that\n$$\n0 < r_{n+1} - r_n - \\pi < \\frac{1}{(n^2+n)\\pi}\n$$\nfor $n \\geq 1$.","subjects":["analysis","transcendental roots","root spacing"]},{"id":"2024-B4","year":2024,"session":"B","number":4,"statement":"Let $n$ be a positive integer. Set $a_{n,0} = 1$. For $k \\geq 0$, choose an integer $m_{n,k}$ uniformly at random from the set $\\{1,\\dots,n\\}$, and let\n$$\na_{n,k+1} = \\begin{cases} a_{n,k} + 1, & \\text{if }m_{n,k} > a_{n,k}; \\\\\na_{n,k}, & \\text{if }m_{n,k} = a_{n,k}\\text{;} \\\\\na_{n,k}-1, & \\text{if }m_{n,k} < a_{n,k}\\text{.}\n\\end{cases}\n$$\nLet $E(n)$ be the expected value of $a_{n,n}$. Determine $\\lim_{n\\to \\infty} E(n)/n$.","subjects":["probability","markov chains","asymptotic expectations"]},{"id":"2024-B5","year":2024,"session":"B","number":5,"statement":"Let $k$ and $m$ be positive integers. For a positive integer $n$, let $f(n)$ be the number of integer sequences $x_1,\\dots,x_k,y_1,\\dots,y_m,z$ satisfying $1 \\leq x_1 \\leq \\cdots \\leq x_k \\leq z \\leq n$ and $1 \\leq y_1 \\leq \\cdots \\leq y_m \\leq z \\leq n$. Show that $f(n)$ can be expressed as a polynomial in $n$ with nonnegative coefficients.","subjects":["combinatorics","polynomial positivity","real rootedness","enumeration"]},{"id":"2024-B6","year":2024,"session":"B","number":6,"statement":"For a real number $a$, let $F_a(x) = \\sum_{n \\geq 1} n^a e^{2n} x^{n^2}$ for $0 \\leq x < 1$.\nFind a real number $c$ such that\n$$\n\\begin{aligned}\n& \\lim_{x \\to 1^-} F_a(x) e^{-1/(1-x)} = 0 \\qquad \\text{for all }a < c\\text{, and} \\\\\n& \\lim_{x \\to 1^-} F_a(x) e^{-1/(1-x)} = \\infty \\qquad \\text{for all }a > c\\text{.}\n\\end{aligned}\n$$","subjects":["analysis","asymptotics","infinite series"]},{"id":"2025-A1","year":2025,"session":"A","number":1,"statement":"Let $m_0$ and $n_0$ be distinct positive integers. For every positive integer $k$,\ndefine $m_k$ and $n_k$ to be the relatively prime positive integers such that\n$$\n\\frac{m_k}{n_k} = \\frac{2m_{k-1} + 1}{2n_{k-1}+1}.\n$$\nProve that $2m_k+1$ and $2n_k+1$ are relatively prime for all but finitely many positive integers $k$.","subjects":["number theory","recurrences"]},{"id":"2025-A2","year":2025,"session":"A","number":2,"statement":"Find the largest real number $a$ and the smallest real number $b$ such that\n$$\nax(\\pi-x) \\leq \\sin x \\leq bx(\\pi-x)\n$$\nfor all $x$ in the interval $[0, \\pi]$.","subjects":["analysis","inequalities"]},{"id":"2025-A3","year":2025,"session":"A","number":3,"statement":"Alice and Bob play a game with a string of $n$ digits, each of which is restricted to be $0$, $1$, or $2$. Initially all the digits are $0$. A legal move is to add or subtract $1$ from one digit to create a new string that has not appeared before. A player with no legal move loses, and the other player wins.\nAlice goes first, and the players alternate moves. For each $n \\geq 1$, determine which player has a strategy that guarantees winning.","subjects":["combinatorics","game theory"]},{"id":"2025-A4","year":2025,"session":"A","number":4,"statement":"Find the minimal value of $k$ such that there exist $k$-by-$k$ real matrices $A_1, \\dots, A_{2025}$ with the property that $A_i A_j = A_j A_i$ if and only if $|i-j| \\in \\{0,1,2024\\}$.","subjects":["linear algebra","graph representations"]},{"id":"2025-A5","year":2025,"session":"A","number":5,"statement":"Let $n$ be an integer with $n \\geq 2$. For a sequence $s = (s_1,\\dots,s_{n-1})$ where each $s_i = \\pm 1$, let $f(s)$ be the number of permutations $(a_1,\\dots,a_n)$ of $\\{1,2,\\dots,n\\}$ such that $s_i(a_{i+1}-a_i) > 0$ for all $i$. For each $n$, determine the sequences $s$ for which $f(s)$ is maximal.","subjects":["combinatorics","permutations"]},{"id":"2025-A6","year":2025,"session":"A","number":6,"statement":"Let $b_0 = 0$ and, for $n \\geq 0$, define $b_{n+1} = 2b_n^2 + b_n + 1$. For each $k \\geq 1$,\nshow that $b_{2^{k+1}} - 2b_{2^k}$ is divisible by $2^{2k+2}$ but not by $2^{2k+3}$.","subjects":["number theory","recurrences"]},{"id":"2025-B1","year":2025,"session":"B","number":1,"statement":"Suppose that each point in the plane is colored either red or green, subject to the following condition: For every three noncollinear points $A,B,C$ of the same color, the center of the circle passing through $A,B$ and $C$ is also this color. Prove that all points of the plane are the same color.","subjects":["geometry","colorings"]},{"id":"2025-B2","year":2025,"session":"B","number":2,"statement":"Let $f\\colon [0,1] \\to [0, \\infty)$ be strictly increasing and continuous. Let $R$ be the region bounded by $x=0$, $x=1$, $y=0$, and $y=f(x)$. Let $x_1$ be the $x$-coordinate of the centroid of $R$. Let $x_2$ be the $x$-coordinate of the centroid of the solid generated by rotating $R$ around the $x$-axis. Prove that $x_1 < x_2$.","subjects":["analysis","geometry","probability"]},{"id":"2025-B3","year":2025,"session":"B","number":3,"statement":"Suppose $S$ is a nonempty set of positive integers with the property that if $n$ is in $S$, then every positive divisor of $2025^n -15^n$ is in $S$. Must $S$ contain all positive integers?","subjects":["number theory","closure properties"]},{"id":"2025-B4","year":2025,"session":"B","number":4,"statement":"For $n \\geq 2$, let $A = [a_{i,j}]_{i,j=1}^n$ be an $n$-by-$n$ matrix of nonnegative integers such that\n\n(a) $a_{i,j} = 0$ when $i+j\\leq n$;\n\n(b) $a_{i+1,j} \\in \\{a_{i,j}, a_{i,j}+1\\}$ when $1 \\leq i \\leq n-1$ and $1 \\leq j \\leq n$; and\n\n(c) $a_{i,j+1} \\in \\{a_{i,j}, a_{i,j}+1\\}$ when $1 \\leq i \\leq n$ and $1 \\leq j \\leq n-1$.\n\nLet $S$ be the sum of the entries of $A$, and let $N$ be the number of nonzero entries of $A$. Prove that\n$$\nS \\leq \\frac{(n+2)N}{3}.\n$$","subjects":["combinatorics","matrix inequalities"]},{"id":"2025-B5","year":2025,"session":"B","number":5,"statement":"Let $p$ be a prime number greater than $3$. For each $k \\in \\{1,\\dots,p-1\\}$, let $I(k) \\in \\{1,2,\\dots,p-1\\}$ be such that $k \\cdot I(k) \\equiv 1 \\pmod{p}$. Prove that the number of integers $k \\in \\{1,\\dots,p-2\\}$ such that $I(k+1) < I(k)$ is greater than $p/4-1$.","subjects":["number theory","permutations"]},{"id":"2025-B6","year":2025,"session":"B","number":6,"statement":"Let $\\mathbb{N} = \\{1,2,3,\\dots\\}$. Find the largest real constant $r$ such that there exists a function $g\\colon \\mathbb{N} \\to \\mathbb{N}$ such that\n$$\ng(n+1)-g(n) \\geq (g(g(n)))^r\n$$\nfor all $n \\in \\mathbb{N}$.","subjects":["analysis","functional inequalities","growth"]}],"methodChoices":["pigeonhole principle","generating functions","induction","Cauchy-Schwarz inequality","telescoping","Vieta formulas","AM-GM inequality","double counting","geometric series","parity","contradiction","mean value theorem","triangle inequality","finite differences","integral comparison","intermediate value theorem","explicit construction","factorization","rolle theorem","inclusion-exclusion","sum of squares","bijection","compactness","integration by parts","monotonicity","polynomial factorization","tonelli theorem"]}