

News: IMO Paper 1 (13 July 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 ≥ 1. Prove that (x^{5}x^{2})/(x^{5}+y^{2}+z^{2}) + (y^{5}y^{2})/(y^{5}+z^{2}+x^{2}) + (z^{5}z^{2})/(z^{5}+x^{2}+y^{2}) ≥ 0. Previous news item: Bulletin 1 from Adrian (07:53 BST, 2 July 2005). Next news item: IMO Paper 2 (14 July 2005). 