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Prove that the segment s_0 exists

Source: 2009 Peru Iberoamerican TST problem 6

May 9, 2023
combinatoricscombinatorics unsolved

Problem Statement

Let PP be a set of n≥2n \ge 2 distinct points in the plane, which does not contain any triplet of aligned points. Let SS be the set of all segments whose endpoints are points of PP. Given two segments s1,s2∈Ss_1, s_2 \in S, we write s1⊗s2s_1 \otimes s_2 if the intersection of s1s_1 with s2s_2 is a point other than the endpoints of s1s_1 and s2s_2. Prove that there exists a segment s0∈Ss_0 \in S such that the set {s∈S∣s0⊗s}\{s \in S | s_0 \otimes s\} has at least 115(n−22)\frac{1}{15}\dbinom{n-2}{2} elements
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