International Mathematical Olympiad · 1992

Problems

Problem 1Find all integers a,b,c satisfying $1<a<b<c$ such that $(a-1)(b-1)(c-1)$ is a divisor of $abc-1$ .Solutions: 1
Problem 2Find all functions $f:\mathbb R\to\mathbb R$ such that $f(x^2+f(y))=y+f(x)^2$ for all real $x,y$ .Solutions: 1
Problem 3Consider 9 points in space, no 4 coplanar. Each pair is joined by an edge colored red, blue, or left uncolored. Find the smallest $n$ such that whenever exactly $n$ edges are colored, there is a triangle whose three edges have the same color.Solutions: 1
Problem 4Let $\mathcal C$ be a circle tangent to a line $L$ , and let $M\in L$ . Find the locus of points $P$ for which there are distinct points $Q,R\in L$ equidistant from $M$ such that $\mathcal C$ is the incircle of triangle $PQR$ .Solutions: 1
Problem 5Let $S$ be a finite set of points in space, and let $S_x,S_y,S_z$ be its orthogonal projections onto the $yz$ -, $zx$ -, and $xy$ -planes. Prove that $|S|^2\le |S_x||S_y||S_z|$ .Solutions: 1
Problem 6For each positive integer $n$ , let $S(n)$ be the greatest integer such that every $k\le S(n)$ permits writing $n^2$ as a sum of $k$ positive squares. (a) Prove $S(n)\le n^2-14$ for $n\ge4$ ; (b) find $n$ with equality; (c) prove infinitely many such $n$ .Solutions: 1