BG-MO 1972 Problem 6

It is given a tetrahedron $ABCD$ for which two points of opposite edges are mutually perpendicular. Prove that:

(a) the four altitudes of $ABCD$ intersects at a common point $H$ ; (b) $AH+BH+CH+DH<p+2R$ , where $p$ is the sum of the lengths of all edges of $ABCD$ and $R$ is the radii of the sphere circumscribed around $ABCD$ .

H. Lesov