10th Vietnamese Mathematical Olympiad 1971 Problems

10th Vietnamese Mathematical Olympiad 1971 Problems

A1.  m, n, r, s are positive integers such that: (1) m < n and r < s; (2) m and n are relatively prime, and r and s are relatively prime; and (3) tan^-1m/n + tan^-1r/s = π/4. Given m and n, find r and s. Given n and s, find m and r. Given m and s, find n and r.

B1.  ABCDA'B'C'D' is a cube (with ABCD and A'B'C'D' faces, and AA', BB', CC', DD' edges). L is a line which intersects or is parallel to the lines AA', BC and DB'. L meets the line BC at M (which may be the point at infinity). Let m = |BM|. The plane MAA' meets the line B'C' at E. Show that |B'E| = m. The plane MDB' meets the line A'D' at F. Show that |D'F| = m. Hence or otherwise show how to construct the point P at the intersection of L and the plane A'B'C'D'. Find the distance between P and the line A'B' and the distance between P and the line A'D' in terms of m. Find a relation between these two distances that does not depend on m. Find the locus of M. Let S be the envelope of the line L as M varies. Find the intersection of S with the faces of the cube.