International Mathematical Olympiad · 2005

Problems

Problem 1Six 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.Solutions: 1
Problem 2Let $a_1, a_2, \dots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1, a_2, \dots, a_n$ leave $n$ different remainders upon division by $n$ . Prove that every integer occurs exactly once in the sequence.Solutions: 1
Problem 3Let $x, y, z > 0$ satisfy $xyz\ge 1$ . Prove that $\frac{x^5-x^2}{x^5+y^2+z^2} + \frac{y^5-y^2}{x^2+y^5+z^2} + \frac{z^5-z^2}{x^2+y^2+z^5} \ge 0.$ Solutions: 1
Problem 4Determine all positive integers relatively prime to all the terms of the infinite sequence $a_n=2^n+3^n+6^n -1,\ n\geq 1.$ Solutions: 1
Problem 5Let $ABCD$ be a fixed convex quadrilateral with $BC = DA$ and $BC \nparallel DA$ . Let two variable points $E$ and $F$ lie on 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 $P$ .Solutions: 1
Problem 6In a mathematical competition, in which $6$ problems were posed to the participants, every two of these problems were solved by more than $\frac25$ 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.Solutions: 1