IMO 1974 LL BUL5

Origin: BUL

Problem

A straight cone is given inside a rectangular parallelepiped B, with the apex at one of the vertices, say T , of the parallelepiped, and the base touching the three faces opposite to T . Its axis lies at the long diagonal through T . If V1 and V2 are the volumes of the cone and the parallelepiped respectively, prove that V1 \leq \sqrt 3\piV2 .