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Version: English

Second day

Mar del Plata, Argentina - July 25, 1997

4

An *n* x *n* matrix (square array) whose entries come from the set
*S* = {1, 2, ... , 2*n* - 1},
is called a *silver* matrix if, for each
*i* = 1, ... , *n*,
the *i*th row and the *i*th 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*^{(b2)} = *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^{n2/4} < *f*(2^{n}) < 2^{n2/2}.

Each problem is worth 7 points.

Time: 4 1/2 hours.