IMO 2020 Problem 6

Prove that there exists a positive constant $c$ such that the following statement is true: Consider an integer $n > 1$ , and a set $\mathcal S$ of $n$ points in the plane such that the distance between any two different points in $\mathcal S$ is at least 1. It follows that there is a line $\ell$ separating $\mathcal S$ such that the distance from any point of $\mathcal S$ to $\ell$ is at least $cn^{-1/3}$ .

(A line $\ell$ separates a set of points S if some segment joining two points in $\mathcal S$ crosses $\ell$ .)

Note. Weaker results with $cn^{-1/3}$ replaced by $cn^{-\alpha}$ may be awarded points depending on the value of the constant $\alpha > 1/3$ .

Proposed by Ting-Feng Lin and Hung-Hsun Hans Yu, Taiwan