Problem

Source: Romania TST 1995 3.2

Tags: parallelepiped, combinatorial geometry, combinatorics, Volume, cube, partition



A cube is partitioned into finitely many rectangular parallelepipeds with the edges parallel to the edges of the cube. Prove that if the sum of the volumes of the circumspheres of these parallelepipeds equals the volume of the circumscribed sphere of the cube, then all the parallelepipeds are cubes.