IMO 1997

In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white (as on a chessboard). For any pair of positive integers $m$ and $n$ , consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $m$ and $n$ , lie along edges of the squares. Let $S_1$ be the total area of the black part of the triangle and $S_2$ be the total area of the white part. Let $f(m,n) = | S_1 - S_2 |$ .

a) Calculate $f(m,n)$ for all positive integers $m$ and $n$ which are either both even or both odd.

b) Prove that $f(m,n) \leq \frac 12 \max \{m,n \}$ for all $m$ and $n$ .

c) Show that there is no constant $C\in\mathbb{R}$ such that $f(m,n) < C$ for all $m$ and $n$ .

It is known that $\angle BAC$ is the smallest angle in the triangle $ABC$ . The points $B$ and $C$ divide the circumcircle of the triangle into two arcs. Let $U$ be an interior point of the arc between $B$ and $C$ which does not contain $A$ . The perpendicular bisectors of $AB$ and $AC$ meet the line $AU$ at $V$ and $W$ , respectively. The lines $BV$ and $CW$ meet at $T$ .

Show that $AU = TB + TC$ .

Alternative formulation:

Four different points $A,B,C,D$ are chosen on a circle $\Gamma$ such that the triangle $BCD$ is not right-angled. Prove that:

(a) The perpendicular bisectors of $AB$ and $AC$ meet the line $AD$ at certain points $W$ and $V,$ respectively, and that the lines $CV$ and $BW$ meet at a certain point $T.$

(b) The length of one of the line segments $AD, BT,$ and $CT$ is the sum of the lengths of the other two.

Let $x_1$ , $x_2$ , $\ldots$ , $x_n$ be real numbers satisfying the conditions: $$\left\{\begin{array}{cccc} |x_1 + x_2 + \cdots + x_n | & = & 1 & \ |x_i| & \leq & \frac {n + 1}{2} & \ \textrm{ for }i = 1, 2, \ldots , n. \end{array} \right.$$ Show that there exists a permutation $y_1$ , $y_2$ , $\ldots$ , $y_n$ of $x_1$ , $x_2$ , $\ldots$ , $x_n$ such that $$| y_1 + 2 y_2 + \cdots + n y_n | \leq \frac {n + 1}{2}.$$

An $n \times n$ matrix whose entries come from the set $S = \{1, 2, \ldots , 2n - 1\}$ is called a silver matrix if, for each $i = 1, 2, \ldots , n$ , 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 $n$ .

Find all pairs $(a,b)$ of positive integers that satisfy the equation: $a^{b^2} = b^a$ .

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 $n \geq 3$ we have $2^{\frac {n^2}{4}} < f(2^n) < 2^{\frac {n^2}2}$ .