IMO 2009

Problem 1. Let $n$ be a positive integer and let $a_1,\dots ,a_k$ ($k\ge 2$) be distinct integers in the set $\{1,\dots ,n\}$ such that $n$ divides $a_i(a_{i+1}-1)$ for $i=1,\dots ,k-1$. Prove that $n$ does not divide $a_k(a_1-1)$.

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 $OP=OQ$.

Problem 3. Suppose that $s_1,s_2,s_3,\dots$ is a strictly increasing sequence of positive integers such that the subsequences

$s_{s_1},s_{s_2},s_{s_3},\dots \text{and}{s}_{s_1+1},s_{s_2+1},s_{s_3+1},\dots$

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

Problem 4. Let $ABC$ 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 incentre of triangle $ADC$. Suppose that $\angle BEK=45^{\circ }$. Find all possible values of $\angle CAB$.

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

$a,f(b)\text{and}f(b+f(a)-1).$

(A triangle is non-degenerate if its vertices are not collinear.)

Problem 6. Let $a_1,a_2,\dots ,a_n$ be distinct positive integers and let $M$ be a set of $n-1$ positive integers not containing $s=a_1+a_2+\cdots +a_n$. A grasshopper is to jump along the real axis, starting at the point $0$ and making $n$ jumps to the right with lengths $a_1,a_2,\dots ,a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $M$.