### 56th Putnam 1995

 A1.  S is a set of real numbers which is closed under multiplication. S = A ∪ B, and A ∩ B = ∅. If a, b, c ∈ A, then abc ∈ A. Similarly, if a, b, c ∈ B, then abc ∈ B. Show that at least one of the A, B is closed under multiplication. A2.  For what positive reals α, β does ∫β∞ √(√(x + α) - √x) - √(√x - √(x - β) ) dx converge? A3.  d, e and f each have nine digits when written in base 10. Each of the nine numbers formed from d by replacing one of its digits by the corresponding digit of e is divisible by 7. Similarly, each of the nine numbers formed from e by replacing one of its digits by the corresponding digit of f is divisible by 7. Show that each of the nine differences between corresponding digits of d and f is divisible by 7. A4.  n integers totalling n - 1 are arranged in a circle. Prove that we choose one of the integers x1, so that the other integers going around the circle are, in order, x2, ... , xn and we have ∑1k xi ≤ k - 1 for k = 1, 2, ... , n. A5.  R is the reals. xi : R → R are differentiable for i = 1, 2, ... , n and satisfy xi' = ai1x1 + ... + ainxn for some constants aij ≥ 0. Also limt→∞ xi(t) = 0. Can the functions xi be linearly independent? A6.  Each of the n triples (ri, si, ti) is a randomly chosen permutation of (1, 2, 3) and each triple is chosen independently. Let p be the probability that each of the three sums ∑ ri, ∑ si, ∑ ti equals 2n, and let q be the probability that they are 2n - 1, 2n, 2n + 1 in some order. Show that for some n ≥ 1995, 4p ≤ q. B1.  Let X be a set with 9 elements. Given a partition π of X, let π(h) be the number of elements in the part containing h. Given any two partitions π1 and π2 of X, show that we can find h ≠ k such that π1(h) = π1(k) and π2(h) = π2(k). B2.  An ellipse with semi-axes a and b rolls without slipping on the curve y = c sin (x/a) and completes one revolution in one period of the sine curve. What conditions do a, b, c satisfy? B3.  For each positive integer k with n2 decimal digits (and leading digit non-zero), let d(k) be the determinant of the matrix formed by writing the digits in order across the rows (so if k has decimal form a1a2 ... an, then the matrix has elements bij = an(i-1)+j). Find f(n) = ∑d(k), where the sum is taken over all 9·10n2-1 such integers. B4.  Express (2207-1/(2207-1/(2207-1/(2207- ... ))))1/8 in the form (a + b√c)/d, where a, b, c, d are integers. B5.  A game starts with four heaps, containing 3, 4, 5 and 6 items respectively. The two players move alternately. A player may take a complete heap of two or three items or take one item from a heap provided that leaves more than one item in that heap. The player who takes the last item wins. Give a winning strategy for the first or second player. B6.  Let N be the positive integers. For any α > 0, define Sα = { [nα]: n ∈ N}. Prove that we cannot find α, β, γ such that N = Sα ∪ Sβ ∪ Sγ and Sα, Sβ, Sγ are (pairwise) disjoint.

To avoid possible copyright problems, I have changed the wording, but not the substance of all the problems. The original text of the problems and the official solutions are in American Mathematical Monthly 103 (1996) 667.

Putnam home