IMO LongList 1967, Mongolia 5
Source: IMO LongList 1967, Mongolia 5
December 16, 2004
geometry3D geometryspherepolyhedronIMO ShortlistIMO Longlist
Problem Statement
Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.