IMO 2020 #6
Posted: Fri Dec 04, 2020 12:49 pm
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
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