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Set meets any plane at a finite but nonzero number of points

Source: 4th German TST 2006, written on 1 April 2006, problem 3; Engel, PSS, 12.3.2, problem 29

April 3, 2006
geometry proposedgeometry

Problem Statement

Does there exist a set M M of points in space such that every plane intersects M M at a finite but nonzero number of points?
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