2020 CMO 第 5 题

Find the largest positive constant $C$ such that the following is satisfied: Given $n$ arcs (containing their endpoints) $A_1,A_2,\ldots ,A_n$ on the circumference of a circle, where among all sets of three arcs $(A_i,A_j,A_k)$ $(1\le i< j< k\le n)$ , at least half of them has $A_i\cap A_j\cap A_k$ nonempty, then there exists $l>Cn$ , such that we can choose $l$ arcs among $A_1,A_2,\ldots ,A_n$ , whose intersection is nonempty.

找到满足以下条件的最大正常数 $C$：给定圆周上的 $n$ 弧(包含其端点)$A_1,A_2,\ldots ,A_n$，其中在所有三组弧 $(A_i,A_j,A_k)$ $(1\le i< j< k\le n)$ 中，至少有一半具有 $A_i\cap A_j\cap A_k$非空，则存在 $l>Cn$ ，这样我们就可以在 $A_1,A_2,\ldots ,A_n$ 中选择 $l$ 弧，其交集非空。