IMO 2009

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 $a_k(a_1 - 1)$.

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 $OP = OQ.$

Proposed by Sergei Berlov, Russia

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

Let $\triangle ABC$ 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 $\angle CAB$.

(This problem was suggested by Peter Vandendriessche and Jan Vonk from Belgium and by Hojoo Lee from South Korea.)

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.)

Proposed by Bruno Le Floch, France

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 $M.$

Proposed by Dmitry Khramtsov, Russia