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n+1 points in the plane with integer differences

Source: China TST 1988, problem 7

June 27, 2005
geometrygeometric transformationvectoranalytic geometrycombinatorics unsolvedcombinatorics

Problem Statement

A polygon ∏\prod is given in the OXYOXY plane and its area exceeds n.n. Prove that there exist n+1n+1 points P1(x1,y1),P2(x2,y2),…,Pn+1(xn+1,yn+1)P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1}) in ∏\prod such that ∀i,j∈{1,2,…,n+1}\forall i,j \in \{1, 2, \ldots, n+1\}, xj−xix_j - x_i and yj−yiy_j - y_i are all integers.
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