IMO 2009 Shortlist A1
Find the largest possible integer k, such that the following statement is true: Let 2009 arbitrary non-degenerated trian...
Category: Algebra
Problem
Find the largest possible integer k, such that the following statement is true: Let 2009 arbitrary non-degenerated triangles be given. In every triangle the three sides are colored, such that one is blue, one is red and one is white. Now, for every color separately, let us sort the lengths of the sides. We obtain b1 ≤ b2 ≤ ... ≤ b2009 the lengths of the blue sides, r1 ≤ r2 ≤ ... ≤ r2009 the lengths of the red sides, and w1 ≤ w2 ≤ ... ≤ w2009 the lengths of the white sides. Then there exist k indices j such that we can form a non-degenerated triangle with side lengths bj, rj, wj.