2009 IMO Problems

Problems of the 50th IMO 2009 in Bremen, Germany.

Day I

Problem 1.

Let $n$ be a positive integer and let $a_1,\ldots,a_k (k\ge2)$ be distinct integers in the set $\{1,\ldots,n\}$ such that $n$ divides $a_i(a_{i+1}-1)$ for $i=1,\ldots,k-1$ . Prove that $n$ doesn't divide $a_k(a_1-1)$ .

Author: Ross Atkins, Australia

Problem 2.

Let $ABC$ be a triangle with circumcentre $O$ . 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 $OP=OQ$ .

Author: Sergei Berlov, Russia

Problem 3.

Suppose that $s_1,s_2,s_3,\ldots$ is a strictly increasing sequence of positive integers such that the subsequences

$s_{s_1},s_{s_2},s_{s_3},\ldots$ and $s_{s_1+1},s_{s_2+1},s_{s_3+1},\ldots$

are both arithmetic progressions. Prove that the sequence $s_1,s_2,s_3,\ldots$ is itself an arithmetic progression.

Author: Gabriel Carroll, USA

Day 2

Problem 4.

Let $ABC$ be a triangle with $AB=AC$ . The angle bisectors of $\angle CAB$ and $\angle CBA$ meet the sides $BC$ and $CA$ at $D$ and $E$ , respectively. Let $K$ be the incentre of triangle $ADC$ . Suppose that $\angle BEK=45^\circ$ . Find all possible values of $\angle CAB$ .

Authors: Jan Vonk and Peter Vandendriessche, Belgium, and Hojoo Lee, South Korea

Problem 5.

Determine 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.)

Author: Bruno Le Floch, France

Problem 6.

Let $a_1,a_2,\ldots,a_n$ be distinct positive integers and let $M$ be a set of $n-1$ positive integers not containing $s=a_1+a_2+\ldots+a_n$ . A grasshopper is to jump along the real axis, starting at the point $0$ and making $n$ jumps to the right with lengths $a_1,a_2,\ldots,a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $M$ .

Author: Dmitry Khramtsov, Russia

--Bugi 10:46, 23 July 2009 (UTC)Bugi

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