International Mathematical Olympiad · 2009

Problems

Problem 1Let $n, k \ge 2$ be positive integers and let $a_1, a_2, \dots, a_k$ be distinct integers in the set $\{1, 2, \dots, n\}$ such that $n$ divides $a_i(a_{i+1} - 1)$ for $i = 1, 2, \dots, k - 1$ . Prove that $n$ does not divide $a_k(a_1 - 1)$ .Solutions: 1
Problem 2Let $ABC$ be a triangle with circumcenter $O$ . The points $P$ and $Q$ are interior points of the sides $CA$ and $AB$ , respectively. Let $K$ , $L$ , and $M$ be the midpoints of the segments $BP$ , $CQ$ , and $PQ$ , respectively, and let $\Gamma$ be the circle passing through $K$ , $L$ , and $M$ . Suppose that the line $PQ$ is tangent to the circle $\Gamma$ . Prove that $OP = OQ$ .Solutions: 1
Problem 3Suppose that $s_1, s_2, s_3, \dots$ is a strictly increasing sequence of positive integers such that the subsequences $s_{s_1}, s_{s_2}, s_{s_3}, \dots$ and $s_{s_1+1}, s_{s_2+1}, s_{s_3+1}, \dots$ are both arithmetic progressions. Prove that the sequence $s_1, s_2, s_3, \dots$ is itself an arithmetic progression.Solutions: 1
Problem 4Let $ABC$ be a triangle with $AB = AC$ . The angle bisectors of $\angle CAB$ and $\angle ABC$ meet the sides $BC$ and $CA$ at $D$ and $E$ , respectively. Let $K$ be the incenter of triangle $ADC$ . Suppose that $\angle BEK = 45^\circ$ . Find all possible values of $\angle CAB$ .Solutions: 1
Problem 5Determine all functions $f$ from the set of positive integers to the set of positive integers such that, for all positive integers $a$ and $b$ , there exists a non-degenerate triangle with sides of lengths $a$ , $f(b)$ and $f(b+f(a)-1)$ . (A triangle is non-degenerate if its vertices are not collinear.)Solutions: 1
Problem 6Let $a_1,\ldots,a_n$ be distinct positive integers and let $M$ be a set of $n-1$ positive integers not containing $s=a_1+\cdots+a_n$ . A grasshopper starts at $0$ and makes $n$ jumps to the right, with lengths $a_1,\ldots,a_n$ in some order. Prove that the order can be chosen so that the grasshopper never lands on a point in $M$ .Solutions: 1