2010 IMO Problems
Problems of the 51st IMO 2010 in Astana, Kazakhstan.
Find all functions such that for all the following equality holds
where is greatest integer not greater than
Author: Pierre Bornsztein, France
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
Find all functions such that is a perfect square for all
Author: Gabriel Carroll, USA
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
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
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
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