Version: English

Second day

Mar del Plata, Argentina - July 25, 1997

4. An n × n matrix (square array) whose entries come from the set S = {1, 2, …, 2n−1} is called a silver matrix if, for each i = 1, …, n , the ith row and the ith column together contain all elements of S. Show that

(a) there is no silver matrix for n = 1997;

(b) silver matrices exist for infinitely many values of n.

5. Find all pairs (a, b) of integers a ≥ 1, b ≥ 1 that satisfy the equation

a^{b^{2}} = b^{a}.

6. For each positive integer n, let f(n) denote the number of ways of representing n as a sum of powers of 2 with nonnegative integer exponents.

Representations which differ only in the ordering of their summands are considered to be the same. For instance, f(4) = 4 because the number 4 can be represented in the following four ways: 4; 2 + 2; 2 + 1 + 1; 1 + 1 + 1 + 1.

Prove that, for any integer n ≥ 3,

2^{n^{2}/4} < f(2^{n}) < 2^{n^{2}/2}.

Each problem is worth 7 points. Time: 4\frac{1}{2} hours.