MathDB
IMO Shortlist 2009 - Problem A1

Source:

July 5, 2010
algebraIMO Shortlisttriangle inequality

Problem Statement

Find the largest possible integer kk, such that the following statement is true: Let 20092009 arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain b1≤b2≤…≤b2009the lengths of the blue sides r1≤r2≤…≤r2009the lengths of the red sides and w1≤w2≤…≤w2009the lengths of the white sides  \left. \begin{array}{rcl} & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\ & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\ \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\ \end{array}\right. Then there exist kk indices jj such that we can form a non-degenerated triangle with side lengths bjb_j, rjr_j, wjw_j.
Proposed by Michal Rolinek, Czech Republic
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