International Mathematical Olympiad · 1997

Problems

Problem 1In the plane, points with integer coordinates are vertices of unit squares, colored alternately black and white as on a chessboard. For positive integers $m,n$ , consider a right triangle with integer-coordinate vertices and legs of lengths $m,n$ along square edges. Let $S_1$ be the black area and $S_2$ the white area. Define $f(m,n)=|S_1-S_2|$ . (a) Calculate $f(m,n)$ when $m,n$ are both even or both odd. (b) Prove $f(m,n)\le\max(m,n)/2$ . (c) Show that no constant $C$ satisfies $f(m,n)<C$ for all $m,n$ .Solutions: 1
Problem 2The angle at $A$ is the smallest angle in triangle $ABC$ . The points $B,C$ divide its circumcircle into two arcs. Let $U$ be an interior point of the arc $BC$ not containing $A$ . The perpendicular bisectors of $AB$ and $AC$ meet line $AU$ at $V$ and $W$ , respectively. The lines $BV$ and $CW$ meet at $T$ . Show that $AU=TB+TC$ .Solutions: 1
Problem 3Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $|x_1+x_2+\cdots+x_n|=1$ and $|x_i|\le(n+1)/2$ for every $i$ . Show that there is a permutation $y_1,\ldots,y_n$ of the $x_i$ such that $|y_1+2y_2+\cdots+ny_n|\le(n+1)/2$ .Solutions: 1
Problem 4An $n\times n$ matrix whose entries belong to $S=\{1,2,\ldots,2n-1\}$ is called a silver matrix if, for each $i$ , the $i$ th row together with the $i$ th column contains all elements of $S$ . Show that (a) there is no silver matrix for $n=1997$ ; (b) silver matrices exist for infinitely many $n$ .Solutions: 1
Problem 5Find all pairs $(a,b)$ of positive integers satisfying $a^{b^2}=b^a$ .Solutions: 1
Problem 6For each positive integer $N$ , let $f(N)$ be the number of ways to represent $N$ as a sum of powers of $2$ with non-negative integer exponents; order of summands is ignored. For example, $f(4)=4$ . Prove that for every integer $n\ge3$ , $2^{n^2/4}<f(2^n)<2^{n^2/2}$ .Solutions: 1