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in every 5 lattice points there is pair of lattice points that contains a third

Source: 2004 Indonedia MO Province P2 q5 OSP

December 11, 2020
geometrylattice points

Problem Statement

The lattice point on the plane is a point that has coordinates in the form of a pair of integers. Let P1,P2,P3,P4,P5P_1, P_2, P_3, P_4, P_5 be five different lattice points on the plane. Prove that there is a pair of points (Pi,Pj),i≠j(P_i, P_j), i \ne j, so that the line segment PiPjP_iP_j contains a lattice point other than PiP_i and PjP_j.
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