International Mathematical Olympiad · 1990

Problems

Problem 1Chords $AB$ and $CD$ of a circle intersect at $E$ inside the circle. Let $M$ be an interior point of $EB$ . The tangent at $E$ to the circle through $D,E,M$ meets $BC$ and $AC$ at $F,G$ . If $t=AM/AB$ , find $EF/EG$ .Solutions: 1
Problem 2Let $n\ge3$ and consider a set $E$ of $2n-1$ distinct points on a circle. Exactly $k$ points are black. Call the coloring good if some pair of black points has an arc whose interior contains exactly $n$ points of $E$ . Find the least $k$ for which every coloring of $k$ points is good.Solutions: 1
Problem 3Determine all integers $n>1$ such that $(2^n+1)/n^2$ is an integer.Solutions: 1
Problem 4Construct a function $f:\mathbb Q^+\to\mathbb Q^+$ such that $f(xf(y))=f(x)/y$ for all positive rational numbers $x,y$ .Solutions: 1
Problem 5Given $n_0>1$ , players A and B choose integers alternately. Knowing $n_{2k}$ , A chooses $n_{2k+1}$ with $n_{2k}\le n_{2k+1}\le n_{2k}^2$ . Knowing $n_{2k+1}$ , B chooses $n_{2k+2}$ such that $n_{2k+1}/n_{2k+2}=p^r$ for a prime $p$ and integer $r\ge1$ . A wins by choosing $1990$ , and B wins by choosing $1$ . Classify the initial values according to which player has a winning strategy or neither does.Solutions: 1
Problem 6Prove that there exists a convex 1990-gon whose angles are all equal and whose side lengths are $1^2,2^2,\ldots,1990^2$ in some order.Solutions: 1