IMO Shortlist 1971 problem 16
Let

be a convex polyhedron with vertices

. Let

be the polyhedron obtained from

by a translation that moves

to

. Prove that at least two of the polyhedra

have an interior point in common.
%V0
Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.
Source: Međunarodna matematička olimpijada, shortlist 1971