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A point inside a regular tetrahedron T of unit volume

Source: IMO ShortList 1990, Problem 19 (POL 1)

August 15, 2008
geometry3D geometrytetrahedroncombinatoricsIMO Shortlist

Problem Statement

Let P P be a point inside a regular tetrahedron T T of unit volume. The four planes passing through P P and parallel to the faces of T T partition T T into 14 pieces. Let f(P) f(P) be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for f(P) f(P) as P P varies over T. T.
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