IMO 2021 Shortlist G3
Version 1. Let n be a fixed positive integer, and let S be the set of points px,yq on the Cartesian plane such that both...
Category: Geometry
Problem
Version 1. Let n be a fixed positive integer, and let S be the set of points px,yq on the Cartesian plane such that both coordinates x and y are nonnegative integers smaller than 2n (thus |S| “ 4n2 ). Assume that F is a set consisting of n2 quadrilaterals such that all their vertices lie in S, and each point in S is a vertex of exactly one of the quadrilaterals in F. Determine the largest possible sum of areas of all n2 quadrilaterals in F. Version 2. Let n be a fixed positive integer, and let S be the set of points px,yq on the Cartesian plane such that both coordinates x and y are nonnegative integers smaller than 2n (thus |S|“4n2 ). Assume that F is a set of polygons such that all vertices of polygons in F lie in S, and each point in S is a vertex of exactly one of the polygons in F. Determine the largest possible sum of areas of all polygons in F.