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Centroid of four points

Source: Baltic Way 2003

November 6, 2010
calculusintegrationanalytic geometrycombinatorics unsolvedcombinatorics

Problem Statement

A lattice point in the plane is a point with integral coordinates. The centroid of four points (xi,yi)(x_i,y_i ), i=1,2,3,4i = 1, 2, 3, 4, is the point (x1+x2+x3+x44,y1+y2+y3+y44)\left(\frac{x_1 +x_2 +x_3 +x_4}{4},\frac{y_1 +y_2 +y_3 +y_4 }{4}\right). Let nn be the largest natural number for which there are nn distinct lattice points in the plane such that the centroid of any four of them is not a lattice point. Prove that n=12n = 12.
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