32nd Putnam 1971

A1.  Given any 9 lattice points in space, show that we can find two which have a lattice point on the interior of the segment joining them.
A2.  Find all possible polynomials f(x) such that f(0) = 0 and f(x2 + 1) = f(x)2 + 1.
A3.  The vertices of a triangle are lattice points in the plane. Show that the diameter of its circumcircle does not exceed the product of its side lengths.
A4.  α lies in the open interval (1, 2). Show that the polynomial formed by expanding (x + y)n(x2 - α xy + y2) has positive coefficients for sufficiently large n. Find the smallest such n for α = 1.998.
A5.  A player scores either A or B at each turn, where A and B are unequal positive integers. He notices that his cumulative score can take any positive integer value except for those in a finite set S, where |S| =35, and 58 ∈ S. Find A and B.
A6.  α is a real number such that 1α, 2α, 3α, ... are all integers. Show that α ≥ 0 and that α is an integer.
B1.  S is a set with a binary operation * such that (1) a * a = a for all a ∈ S, and (2) (a * b) * c = (b * c) * a for all a, b, c ∈ S. Show that * is associative and commutative.
B2.  Let X be the set of all reals except 0 and 1. Find all real valued functions f(x) on X which satisfy f(x) + f(1 - 1/x) = 1 + x for all x in X.
B3.  Car A starts at time t = 0 and, traveling at a constant speed, completes 1 lap every hour. Car B starts at time t = α > 0 and also completes 1 lap every hour, traveling at a constant speed. Let a(t) be the number of laps completed by A at time t, so that a(t) = 0 for t < 1, a(t) = 1 for 1 ≤ t < 2 and so on. Similarly, let b(t) be the number of laps completed by B at time t. Let S = {t : a(t) = 2 b(t) }. Show that S is made up of intervals of total length 1.
B4.  A and B are two points on a sphere. S(A, B, k) is defined to be the set {P : AP + BP = k}, where XY denotes the great-circle distance between points X and Y on the sphere. Determine all sets S(A, B, k) which are circles.
B5.  A hypocycloid is the path traced out by a point on the circumference of a circle rolling around the inside circumference of a larger fixed circle. Show that the plots in the (x, y) plane of the solutions ( x(t), y(t) ) of the differential equations x'' + y' + 6x = 0, y'' - x' + 6y = 0 with initial conditions x'(0) = y'(0) = 0 are hypocycloids. Find the possible radii of the circles.
B6.  Prove that: |f(1)/1 + f(2)/2 + ... + f(n)/n - 2n/3| < 1, where f(n) is the largest odd divisor of n.


To avoid possible copyright problems, I have changed the wording, but not the substance, of all the problems. The original text and official solutions were published in American Mathematical Monthly 80 (1973) 172-9. They are also available (with the solutions expanded) in: Gerald L Alexanderson et al, The William Lowell Putnam Mathematical Competition, 1965-1984. Out of print, but in some university libraries.

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© John Scholes
25 Aug 2001