International Math Olympiad Questions
IMO Problems 2005
Day 1
IMO 2005 Problem 1
Six points are chosen on the sides of an equilateral triangle ABC. A₁, A₂ on BC; B₁, B₂ on CA; and C₁, C₂ on AB, such that they are the vertices of a convex hexagon A₁, A₂, B₁, B₂, C₁, C₂ with equal side lengths.
Prove that the lines A₁B₂, B₁C₂, and C₁A₂ are concurrent.
IMO 2005 Problem 2
Let a₁, a₂, a₃, … be a sequence of integers with infinitely many positive and negative term. Suppose that for every positive integer k the numbers a₁, a₂, a₃, …, aₖ leaves k different remainders upon division by k.
Prove that every integer occurs exactly once in the sequence a₁, a₂, a₃, …
IMO 2005 Problem 3
Let x, y, z be three positive real numbers such that xyz ≥ 1. Prove that
(x⁵ – x²)/(x⁵ + y² + z²) + (y⁵ – y²)/(y⁵ + x² + z²) + (z⁵ – z²)/(z⁵ + y² + x²) ≥ 0
Day 2
IMO 2005 Problem 4
Determine all positive integers relative prime to all the terms of The Infinite sequence
aₖ = 2ᵏ + 3ᵏ + 6ᵏ – 1, k ≥ 1
IMO 2005 Problem 5
Let ABCD be a fixed convex quadrilateral with BC = DA and BC not parallel with 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 triangle PQR, as E and F vary, have a common point other than P.
IMO 2005 Problem 6
In a mathematical competition, in which 6 problems were posed to the participants. Every two of these problems were solved by more than 2/5 of the contestants. Moreover, no contestant solved all the six problems. Show that there are at least 2 contestants who solved exactly 5 problems each.