2020 IMO Problems/Problem 6
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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 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 $cn^−1/3$ (Error compiling LaTeX. Unknown error_msg) . (A line ℓ separates a set of points S if some segment joining two points in S crosses ℓ.) Note. Weaker results with $cn^−1/3$ (Error compiling LaTeX. Unknown error_msg) replaced by $cn^−α$ (Error compiling LaTeX. Unknown error_msg) may be awarded points depending on the value of the constant α > 1/3.
