A set 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 and from , the perpendicular bisector plane of the segment is a plane of symmetry for . 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.
|1995||IMO Shortlist 1998 problem G1||6|
|2080||IMO Shortlist 2001 problem G2||7|
|2132||IMO Shortlist 2003 problem G1||18|
|2165||IMO Shortlist 2004 problem G4||6|
|2217||IMO Shortlist 2006 problem G1||23|
|2252||IMO Shortlist 2007 problem G4||9|