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.  
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© John Scholes
16 Dec 1998