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, ..., 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 (x⁵-x²)/(x⁵+y²+z²) + (y⁵-y²)/(y⁵+z²+x²) + (z⁵-z²)/(z⁵+x²+y²) ≥ 0.

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