IMO 1977 LL ROM39

Origin: ROM

Problem

Consider 37 distinct points in space, all with integer coordi- nates. Prove that we may find among them three distinct points such that their barycenter has integers coordinates.

Solution

By the pigeonhole principle, we can find 5 distinct points among the given 37 such that their x-coordinates are congruent and their y-coordinates are congruent modulo 3. Now among these 5 points either there exist three with z-coordinates congruent modulo 3, or there exist three whose z- coordinates are congruent to 0, 1, 2 modulo 3. These three points are the desired ones. Remark. The minimum number n such that among any n integer points in space one can find three points whose barycenter is an integer point is n = 19. Each proof of this result seems to consist in studying a great number of cases.