IMO 2020 Shortlist G9
Prove that there exists a positive constant c such that the following statement is true: Assume that n is an integer wit...
Category: Geometry
Problem
Prove that there exists a positive constant c such that the following statement is true: Assume that n is an integer with n ě 2, and let S be a set of n points in the plane such that the distance between any two distinct points in S is at least 1. Then there is a line ℓ separating S such that the distance from any point of S to ℓ is at least cn´1{3 . (A line ℓ separates a point set S if some segment joining two points in S crosses ℓ.) (Taiwan) 10 Saint-Petersburg — Russia, 18th–28th September 2020