2020 IMO Shortlist G9

Prove that there exists a positive constant $c$ such that the following statement is true: Assume that $n$ is an integer with $n \geq 2$, and let $\mathcal{S}$ be a set of $n$ points in the plane such that the distance between any two distinct points in $\mathcal{S}$ is at least 1 . Then there is a line $\ell$ separating $\mathcal{S}$ such that the distance from any point of $\mathcal{S}$ to $\ell$ is at least $c n^{-1 / 3}$. (A line $\ell$ separates a point set $\mathcal{S}$ if some segment joining two points in $\mathcal{S}$ crosses $\ell$.)

证明存在正常数 $c$ ，使得以下陈述成立：假设 $n$ 是 $n \geq 2$ 的整数，并令 $\mathcal{S}$ 为平面中 $n$ 个点的集合，使得 $\mathcal{S}$ 中任意两个不同点之间的距离至少为 1 。然后有一条线 $\ell$ 分隔 $\mathcal{S}$，使得 $\mathcal{S}$ 中的任意点到 $\ell$ 的距离至少为 $c n^{-1 / 3}$。 (如果 $\mathcal{S}$ 中连接两个点的某些线段穿过 $\ell$，则线 $\ell$ 分隔点集 $\mathcal{S}$。)