MathDB

centroids lattice points, 70 lattice points

Source: 1989 Greece MO Grade XII p2

September 6, 2024

geometryweightsanalytic geometryCentroidcombinatorial geometrylattice points

Problem Statement

On the plane we consider

70

points

A_1,A_2,...,A_{70}

with integer coodinates. Suppose each pooints has weight

1

and the centers of gravity of the triangles

A_1A_2A_3

A_2A_3A_4

..

A_{68}A_{69}A_{70}

A_{69}A_{70}A_{1}

A_{70}A_{1}A_{2}

have integer coodinates. Prove that the centers of gravity of any triple

A_i,A_j,...,A_{k}

has integer coodinates.