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9 lattice points in 3D, at least one midpoint is lattice point

Source: 1976 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p1

November 20, 2022
combinatoricscombinatorial geometrylattice points

Problem Statement

Nine lattice points (i.e. with integer coordinates) P1,P2,...,P9P_1,P_2,...,P_9 are given in space. Show that the midpoint of at least one of the segments PiPjP_iP_j , where 1≤i<j≤91 \le i < j \le 9, is a lattice point as well.
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