1971 IMO Problems/Problem 4

All the faces of tetrahedron ABCD are acute-angled triangles. We consider all closed polygonal paths of the form XY ZTX defined as follows: X is a point on edge AB distinct from A and B; similarly, Y;Z; T are interior points of edges BCCD;DA; respectively. Prove: (a) If DAB + BCD is not equal to CDA + ABC; then among the polygonal paths, there is none of minimal length. (b) If DAB + BCD = CDA + ABC; then there are infinitely many shortest polygonal paths, their common length being 2AC sin(®=2); where ® = BAC + CAD + DAB. (triads of three letters represent angles, except ZTX)

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