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n coloured lattice points , same color with ther center of gravity, bw

Source: 11th QEDMO p5 Seniors (6-8. 12. 2012) https://artofproblemsolving.com/community/c1512515_qedmo_200507

May 16, 2021
latticeColoringcombinatoricscombinatorial geometry

Problem Statement

Let nn be a natural number and L=Z2L = Z^2 the set of points on the plane with integer coordinates. Every point in LL is colored now in one of the colors red or green. Show that there are nn different points x1,...,xn∈Lx_1,...,x_n \in L all of which have the same color and whose center of gravity is also in LL and is of the same color.
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