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Romania TST 2022 Day 2 P5

Source: Romania TST 2022

May 15, 2022
combinatoricslattice pointsromaniaRomanian TST

Problem Statement

Let m,n≥2m,n\geq 2 be positive integers and S⊆[1,m]×[1,n]S\subseteq [1,m]\times [1,n] be a set of lattice points. Prove that if ∣S∣≥m+n+⌊m+n4−12⌋|S|\geq m+n+\bigg\lfloor\frac{m+n}{4}-\frac{1}{2}\bigg\rfloorthen there exists a circle which passes through at least four distinct points of S.S.
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