IMO 2005

Six points are chosen on the sides of an equilateral triangle $ABC$ : $A_1$ , $A_2$ on $BC$ , $B_1$ , $B_2$ on $CA$ and $C_1$ , $C_2$ on $AB$ , such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths.

Prove that the lines $A_1B_2$ , $B_1C_2$ and $C_1A_2$ are concurrent.

Bogdan Enescu, Romania

Let $a_1,a_2,\ldots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\ldots,a_n$ leave $n$ different remainders upon division by $n$ .

Prove that every integer occurs exactly once in the sequence $a_1,a_2,\ldots$ .

Let $x,y,z$ be three positive reals such that $xyz\geq 1$ . Prove that $$\frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 .$$ Hojoo Lee, Korea

Determine all positive integers relatively prime to all the terms of the infinite sequence

$$a_{n} = 2^{n} + 3^{n} + 6^{n} - 1, n\geq 1.$$

Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$ . Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$ , respectively and satisfy $BE=DF$ . The lines $AC$ and $BD$ meet at $P$ , the lines $BD$ and $EF$ meet at $Q$ , the lines $EF$ and $AC$ meet at $R$ .

Prove that the circumcircles of the triangles $PQR$ , as $E$ and $F$ vary, have a common point other than $P$ .

In a mathematical competition, in which $6$ problems were posed to the participants, every two of these problems were solved by more than $\frac 25$ of the contestants. Moreover, no contestant solved all the $6$ problems. Show that there are at least $2$ contestants who solved exactly $5$ problems each.

Radu Gologan and Dan Schwartz