International Mathematical Olympiad · 1998

Problems

Problem 1In the convex quadrilateral $ABCD$ , the diagonals $AC$ and $BD$ are perpendicular and the opposite sides $AB$ and $DC$ are not parallel. Suppose that the point $P$ , where the perpendicular bisectors of $AB$ and $DC$ meet, is inside $ABCD$ . Prove that $ABCD$ is a cyclic quadrilateral if and only if the triangles $ABP$ and $CDP$ have equal areas.Solutions: 1
Problem 2In a competition, there are $a$ contestants and $b$ judges, where $b\ge3$ is an odd integer. Each judge rates each contestant as either “pass” or “fail”. Suppose $k$ is a number such that, for any two judges, their ratings coincide for at most $k$ contestants. Prove that $\frac{k}{a}\ge\frac{b-1}{2b}$ .Solutions: 1
Problem 3For any positive integer $n$ , let $d(n)$ denote the number of positive divisors of $n$ (including $1$ and $n$ itself). Determine all positive integers $k$ such that $\frac{d(n^2)}{d(n)}=k$ for some $n$ .Solutions: 1
Problem 4Determine all pairs $(a,b)$ of positive integers such that $ab^{2}+b+7$ divides $a^{2}b+a+b$ .Solutions: 1
Problem 5Let $I$ be the incenter of triangle $ABC$ . Let the incircle of $ABC$ touch the sides $BC$ , $CA$ , and $AB$ at $K$ , $L$ , and $M$ , respectively. The line through $B$ parallel to $MK$ meets the lines $LM$ and $LK$ at $R$ and $S$ , respectively. Prove that angle $RIS$ is acute.Solutions: 1
Problem 6Determine the least possible value of $f(1998)$ , where $f:\mathbb{N}\to\mathbb{N}$ is a function such that for all $m,n\in\mathbb{N}$ , $f\left(n^{2}f(m)\right)=m\left(f(n)\right)^{2}.$ Solutions: 1