NL-NWO 1972 Problem 3

$ABCD$ is a regular tetrahedron. The points $P,Q,R$ and $S$ lie outside this tetrahedron in such a way that $ABCP$ , $ABDQ$ , $ACDR$ and $BCDS$ are regular tetrahedra. Prove that the volume of the tetrahedron $PQRS$ is less than the sum of the volumes of $ABCP$ , $ABDQ$ , $ACDR$ , $BCDS$ and $ABCD$ .