

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 x_{1}, so that the other integers going around the circle are, in order, x_{2}, ... , x_{n} and we have ∑_{1}^{k} x_{i} ≤ k  1 for k = 1, 2, ... , n.


A5. R is the reals. x_{i} : R → R are differentiable for i = 1, 2, ... , n and satisfy x_{i}' = a_{i1}x_{1} + ... + a_{in}x_{n} for some constants a_{ij} ≥ 0. Also lim_{t→∞} x_{i}(t) = 0. Can the functions x_{i} be linearly independent?


A6. Each of the n triples (r_{i}, s_{i}, t_{i}) 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 ∑ r_{i}, ∑ s_{i}, ∑ t_{i} 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 π_{1} and π_{2} of X, show that we can find h ≠ k such that π_{1}(h) = π_{1}(k) and π_{2}(h) = π_{2}(k).


B2. An ellipse with semiaxes 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 n^{2} decimal digits (and leading digit nonzero), 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 a_{1}a_{2} ... a_{n}, then the matrix has elements b_{ij} = a_{n(i1)+j}). Find f(n) = ∑d(k), where the sum is taken over all 9·10^{n21} such integers.


B4. Express (22071/(22071/(22071/(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 S_{α} = { [nα]: n ∈ N}. Prove that we cannot find α, β, γ such that N = S_{α} ∪ S_{β} ∪ S_{γ} and S_{α}, S_{β}, S_{γ} are (pairwise) disjoint.

