IMO 2005

Problem 1. Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$; $B_1$, $B_2$ on $CA$; $C_1$, $C_2$ on $AB$. These points are the vertices of a convex hexagon $A_1{A}_2{B}_1{B}_2{C}_1{C}_2$ with equal side lengths. Prove that the lines $A_1{B}_2$, $B_1{C}_2$ and $C_1{A}_2$ are concurrent.

Problem 2. Let $a_1$, $a_2$, ... be a sequence of integers with infinitely many positive terms and infinitely many negative terms. Suppose that for each positive integer $n$, the numbers $a_1$, $a_2$, ..., $a_n$ leave $n$ different remainders on division by $n$. Prove that each integer occurs exactly once in the sequence.

Problem 3. Let $x$, $y$ and $z$ be positive real numbers such that $xyz\ge 1$. Prove that

Problem 4. Consider the sequence $a_1$, $a_2$, ... defined by

$a_n=2^n+3^n+6^n-1(n=1,2,\dots ).$

Determine all positive integers that are relatively prime to every term of the sequence.

Problem 5. Let $ABCD$ be a given convex quadrilateral with sides $BC$ and $AD$ equal in length and not parallel. Let $E$ and $F$ be interior points of the sides $BC$ and $AD$ respectively such that $BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$. Consider all the triangles $PQR$ as $E$ and $F$ vary. Show that the circumcircles of these triangles have a common point other than $P$.

Problem 6. In a mathematical competition 6 problems were posed to the contestants. Each pair of problems was solved by more than $\frac{2}{5}$ of the contestants. Nobody solved all 6 problems. Show that there were at least 2 contestants who each solved exactly 5 problems.