IMO 1985 LL CZS20

Origin: CZS

Problem

Let T be the set of all lattice points (i.e., all points with integer coordinates) in three-dimensional space. Two such points (x, y, z) and (u, v, w) are called neighbors if |x −u| + |y −v| + |z −w| = 1. Show that there exists a subset S of T such that for each p \inT , there is exactly one point of S among p and its neighbors.