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Common intersection of arcs

Source: 2020 CMO P4

November 27, 2019
combinatorics

Problem Statement

Find the largest positive constant CC such that the following is satisfied: Given nn arcs (containing their endpoints) A1,A2,…,AnA_1,A_2,\ldots ,A_n on the circumference of a circle, where among all sets of three arcs (Ai,Aj,Ak)(A_i,A_j,A_k) (1≤i<j<k≤n)(1\le i< j< k\le n), at least half of them has Ai∩Aj∩AkA_i\cap A_j\cap A_k nonempty, then there exists l>Cnl>Cn, such that we can choose ll arcs among A1,A2,…,AnA_1,A_2,\ldots ,A_n, whose intersection is nonempty.
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