IMO 1971 Problem 4

All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$ , and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$ .

a.) If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$ , then prove none of the closed paths $XYZTX$ has minimal length;

b.) If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$ , then there are infinitely many shortest paths $XYZTX$ , each with length $2AC\sin k$ , where $2k=\angle BAC+\angle CAD+\angle DAB$ .