British Mathematical Olympiad

News: IMO Paper 1 (13 July 2005)

Problem 1. Six points are chosen on the sides of an equilateral triangle ABC: A1, A2 on BC; B1, B2 on CA; C1, C2 on AB. These points are the vertices of a convex hexagon A1A2B1B2C1C2 with equal side lengths. Prove that the lines A1B2, B1C2 and C1A2 are concurrent.

Problem 2. Let a1, a2, ... be a sequence of integers with infinitely many positive terms and infinitely many negative terms. Suppose that for each positive integer n, the numbers a1, a2, ..., an 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 (x5-x2)/(x5+y2+z2) + (y5-y2)/(y5+z2+x2) + (z5-z2)/(z5+x2+y2) ≥ 0.

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