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19 lattice points with no 3 collinear, always exist 3 with lattice centroid

Source: Mexican Mathematical Olympiad 1990 OMM P5

July 29, 2018
combinatorial geometrylattice pointsCentroidgeometry

Problem Statement

Given 1919 points in the plane with integer coordinates, no three collinear, show that we can always find three points whose centroid has integer coordinates.