ILL 1972 Problem 14

$(a)$ A plane $\pi$ passes through the vertex $O$ of the regular tetrahedron $OPQR$ . We define $p, q, r$ to be the signed distances of $P,Q,R$ from $\pi$ measured along a directed normal to $\pi$ . Prove that $$p^2 + q^2 + r^2 + (q - r)^2 + (r - p)^2 + (p - q)^2 = 2a^2,$$ where $a$ is the length of an edge of a tetrahedron. $(b)$ Given four parallel planes not all of which are coincident, show that a regular tetrahedron exists with a vertex on each plane.

Note: Part $(b)$ is IMO 1972 Problem 6