Version: English

Second day
Mar del Plata, Argentina - July 25, 1997


An n x 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.


Find all pairs (a,b) of integers a =< 1, b =< 1 that satisfy the equation

a(b2) = ba.


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:

2n2/4 < f(2n) < 2n2/2.

Each problem is worth 7 points.
Time: 4 1/2 hours.