1971 IMO Problems/Problem 2
Consider a convex polyhedron with nine vertices let be the polyhedron obtained from by a translation that moves vertex to Prove that at least two of the polyhedra have an interior point in common.
WLOG let be the origin . Take any point , then , lies in , the polyhedron stretched by the factor on . More general: take any in any convex shape . Then . Prove: since is convex, , thus .
Now all these nine polyhedrons lie inside . Let be the volume of . Then some polyhedrons with total sum of volumes lie in a shape of volume , thus they must overlap, meaning that they have an interior point in common.
The above solution was posted by ZetaX. The original thread for this problem can be found here: 
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