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There exists a polygonal line

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August 29, 2010
combinatoricscombinatorial geometrygraph theoryExtremal combinatoricsExtremal Graph TheoryIMO Shortlist

Problem Statement

Let AA be a set of nn points in the space. From the family of all segments with endpoints in AA, qq segments have been selected and colored yellow. Suppose that all yellow segments are of different length. Prove that there exists a polygonal line composed of mm yellow segments, where m≥2qnm \geq \frac{2q}{n}, arranged in order of increasing length.
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