2005 IMO Problems
Problems of the 46th IMO 2005 Mérida, Mexico.
Contents
Day 1
Problem 1
Six points are chosen on the sides of an equilateral triangle : on , , on and , on , such that they are the vertices of a convex hexagon with equal side lengths. Prove that the lines and are concurrent.
Problem 2
Let be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer the numbers leave different remainders upon division by . Prove that every integer occurs exactly once in the sequence.
Problem 3
Let satisfy . Prove that
Day 2
Problem 4
Determine all positive integers relatively prime to all the terms of the infinite sequence
Problem 5
Let be a fixed convex quadrilateral with and . Let two variable points and lie of the sides and , respectively, and satisfy . The lines and meet at , the lines and meet at , the lines and meet at . Prove that the circumcircles of the triangles , as and vary, have a common point other than .
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 6 problems. Show that there are at least 2 contestants who solved exactly 5 problems each.
Resources
2005 IMO (Problems) • Resources | ||
Preceded by 2004 IMO Problems |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2006 IMO Problems |
All IMO Problems and Solutions |