International Mathematical Olympiad 1971 Problem 2

Consider a convex polyhedron $P_1$ with nine vertices $A_1A_2, \ldots, A_9$; let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves vertex $A_1$ to $A_i$ $(i = 2, 3, \ldots, 9)$. Prove that at least two of the polyhedra $P_1, P_2, \ldots, P_9$ have an interior point in common.