IMO 1967 LL MON34

Origin: MON

Problem

The faces of a convex polyhedron are six squares and eight 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 this polyhedron and compute the ratio of the squares of the volumes of the polyhedron and of the ball whose boundary is the circumscribed sphere.

Solution

Each vertex of the polyhedron is a vertex of exactly two squares and triangles (more than two is not possible; otherwise, the sum of angles at a vertex exceeds 360◦). By using the condition that the trihedral angles are equal it is easy to see that such a polyhedron is uniquely determined by its side length.

The polyhedron obtained from a cube by “cutting” its vertices, as shown in the figure, satisfies the conditions. Now it is easy to calculate that the ratio of the squares of vol- umes of that polyhedron and of the ball whose boundary is the circum- scribed sphere is equal to 25/(8\pi2).