Difference between revisions of "2009 IMO Problems"
m (→Problem 4.: angle BAC is the same as CAB... the bisector of CBA, however WOULD intersect CA. (correct if wrong, please)) |
m (→Problem 4.: fixed from the actual problem PDF) |
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=== Problem 4. === | === Problem 4. === | ||
− | Let <math>ABC</math> be a triangle with <math>AB=AC</math>. The angle bisectors of <math>\angle CAB</math> and <math>\angle | + | Let <math>ABC</math> be a triangle with <math>AB=AC</math>. The angle bisectors of <math>\angle CAB</math> and <math>\angle ABC</math> meet the sides <math>BC</math> and <math>CA</math> at <math>D</math> and <math>E</math>, respectively. Let <math>K</math> be the incentre of triangle <math>ADC</math>. Suppose that <math>\angle BEK=45^\circ</math>. Find all possible values of <math>\angle CAB</math>. |
''Authors: Jan Vonk and Peter Vandendriessche, Belgium, and Hojoo Lee, South Korea'' | ''Authors: Jan Vonk and Peter Vandendriessche, Belgium, and Hojoo Lee, South Korea'' |
Revision as of 15:35, 7 July 2010
Problems of the 50th IMO 2009 in Bremen, Germany.
Contents
Day I
Problem 1.
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
Problem 2.
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
Problem 3.
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
Day 2
Problem 4.
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
Problem 5.
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
Problem 6.
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
--Bugi 10:46, 23 July 2009 (UTC)Bugi