2009 IMO Problems
Problems of the 50th IMO 2009 in Bremen, Germany.
Let be a positive integer and let be distinct integers in the set such that divides for . Prove that doesn't divide .
Author: Ross Atkins, Australia
Let be a triangle with circumcentre . The points and are interior points of the sides and respectively. Let and be the midpoints of the segments and , respectively, and let be the circle passing through and . Suppose that the line is tangent to the circle . Prove that .
Author: Sergei Berlov, Russia
Suppose that is a strictly increasing sequence of positive integers such that the subsequences
are both arithmetic progressions. Prove that the sequence is itself an arithmetic progression.
Author: Gabriel Carroll, USA
Let be a triangle with . The angle bisectors of and meet the sides and at and , respectively. Let be the incentre of triangle . Suppose that . Find all possible values of .
Authors: Jan Vonk and Peter Vandendriessche, Belgium, and Hojoo Lee, South Korea
Determine all functions from the set of positive integers to the set of positive integers such that, for all positive integers and , there exists a non-degenerate triangle with sides of lengths
(A triangle is non-degenerate if its vertices are not collinear.)
Author: Bruno Le Floch, France
Let be distinct positive integers and let be a set of positive integers not containing . A grasshopper is to jump along the real axis, starting at the point and making jumps to the right with lengths in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in .
Author: Dmitry Khramtsov, Russia
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