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A set S of points from the space will be called completely symmetric if it has at least three elements and fulfills the condition that for every two distinct points A and B from S, the perpendicular bisector plane of the segment AB is a plane of symmetry for S. Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, or a regular tetrahedron or a regular octahedron.

Slični zadaci

1995IMO Shortlist 1998 problem G16
2080IMO Shortlist 2001 problem G27
2132IMO Shortlist 2003 problem G118
2165IMO Shortlist 2004 problem G46
2217IMO Shortlist 2006 problem G123
2252IMO Shortlist 2007 problem G49