20th British Mathematical Olympiad 1984 Problems

1.  In the triangle ABC, ∠C = 90o. Find all points D such that AD·BC = AC·BD = AB·CD/√2. 2.  ABCD is a tetrahedron such that DA = DB = DC = d and AB = BC = CA = e. M and N are the midpoints of AB and CD. A variable plane through MN meets AD at P and BC at Q. Show that AP/AD = BQ/BC. Find the value of this ratio in terms of d and e which minimises the area of MQNP