2010 IMO Problems
Problems of the 51st IMO 2010 in Astana, Kazakhstan.
Contents
Day 1
Problem 1.
Find all functions such that for all the following equality holds
where is greatest integer not greater than
Author: Pierre Bornsztein, France
Problem 2.
Given a triangle , with as its incenter and as its circumcircle, intersects again at . Let be a point on arc , and a point on the segment , such that . If is the midpoint of , prove that the intersection of lines and lies on .
Authors: Tai Wai Ming and Wang Chongli, Hong Kong
Problem 3.
Find all functions such that is a perfect square for all
Author: Gabriel Carroll, USA
Day 2
Problem 4.
Let be a point interior to triangle (with ). The lines , and meet again its circumcircle at , , respectively . The tangent line at to meets the line at . Show that from follows .
Author: Unknown currently
Problem 5.
Each of the six boxes , , , , , initially contains one coin. The following operations are allowed
Type 1) Choose a non-empty box , , remove one coin from and add two coins to ;
Type 2) Choose a non-empty box , , remove one coin from and swap the contents (maybe empty) of the boxes and .
Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes , , , , become empty, while box contains exactly coins.
Author: Hans Zantema, Netherlands
Problem 6.
Let be a sequence of positive real numbers, and be a positive integer, such that Prove there exist positive integers and , such that
Author: Morteza Saghafiyan, Iran
Resources
2010 IMO (Problems) • Resources | ||
Preceded by 2009 IMO Problems |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2012 IMO Problems |
All IMO Problems and Solutions |