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Points and segments in a different way

Source: All-Russian 2021/9.3

April 19, 2021
combinatoricsRussiaAll Russian Olympiad

Problem Statement

On a line n+1n+1 segments are marked such that one of the points of the line is contained in all of them. Prove that one can find 22 distinct segments I,JI, J which intersect at a segment of length at least n−1nd\frac{n-1}{n}d, where dd is the length of the segment II.
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