Prove that there exists a positive constant $c$ such that the following statement is true:
Consider an integer $n>1$ and a set $S$ of $n$ points in the plane such that the distance between any two different points in $S$ is at least 1. It follows that there is a line $l$ separating $S$ such that the distance from any point of $S$ to $l$ is at least $cn^{-\frac{1}{3}}$.
I have been trying to plug numbers into this question, so as to develop a strategy to solve it, however the results were completely inconsistent. I have been looking at this question, but am out of ideas on how to attack it. Could you please explain to me how to solve it, as well as the way you thought of the particular solution? I have looked everywhere for a solution to help my thought, but can't find it anywhere. I have found the solutions for all the rest of IMO 2020 question (mainly on this youtube channel https://www.youtube.com/channel/UCBoH0u68QUCdGs9Ber_z1mQ) but can't seem to find this particular question anywhere