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![\[\frac{x^5-x^2}{x^5+y^2+z^2} + \frac{y^5-y^2}{x^2+y^5+z^2} + \frac{z^5-z^2}{x^2+y^2+z^5} \ge 0.\]](http://latex.artofproblemsolving.com/7/9/a/79af15ecc7549604600f49baf6f88c10a4fe01a3.png)
Day 2
Problem 4
Determine all positive integers relatively prime to all the terms of the infinite sequence![\[a_n=2^n+3^n+6^n -1,\ n\geq 1.\]](http://latex.artofproblemsolving.com/a/3/f/a3f3c15ee9d92c2eb2747f800d908cfa7aa6a08b.png)
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 | ||
