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set of 7 planes

Source: Czech and Slovak Olympiad 1980, National Round, Problem 6

September 13, 2024
combinatoricscombinatorial geometrygeometry3D geometry

Problem Statement

Let MM be the set of five points in space, none of which four do not lie in a plane. Let RR be a set of seven planes with properties: a) Each plane from the set RR contains at least one point of the setM M. b) None of the points of the set M lie in the five planes of the set RR. Prove that there are also two distinct points PP, QQ, P∈M P \in M, Q∈MQ \in M, that the line PQPQ is not the intersection of any two planes from the set RR.
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