IMO 1971 SL 16

Origin: USS

Problem

Given a convex polyhedron P1 with 9 vertices A1, . . . , A9, let us denote by P2, P3, . . . , P9 the images of P1 under the translations mapping the vertex A1 to A2, A3, . . . , A9 respectively. Prove that among the polyhedra P1, . . . , P9 at least two have a common interior point.

Solution

Denote by P ′ the polyhedron defined as the image of P under the homo- thety with center at A1 and coefficient of similarity 2. It is easy to see that all Pi, i = 1, . . . , 9, are contained in P ′ (indeed, if M \inPk, then −−−\to A1M = 1 2(−−−\to A1Ak + −−−\to A1M ′) for some M ′ \inP, and the claim follows from

the convexity of P). But the volume of P ′ is exactly 8 times the volume of P, while the volumes of Pi add up to 9 times that volume. We conclude that not all Pi have disjoint interiors.