IMO题目已经出来了

https://artofproblemsolving.com/community/c6h2278656_combinatorial_geometry_for_the_win
IMO 2020 Problem 6
naman12
1071 posts
#1Today at 2:33 AM• 1 Y
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 $\alpha > 1/3$.
This post has been edited 1 time. Last edited by naman12, Today at 2:33 AM
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Kamran011
494 posts
#2Today at 2:36 AM• 6 Y
So this year it's basically GACCCC , sad
This post has been edited 1 time. Last edited by Kamran011, Today at 3:09 AM
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USJL
306 posts
#3Today at 2:38 AM• 15 Y
This problem is proposed by ltf0501and me!
Solution
Some interesting stories behind this problem
Attachments:
2020IMOP6.pdf (56kb)

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