VMO 1984 Problem 3

A cube side 2a has ABCD as one face. S is the other vertex (apart from B and D) adjacent to A. M, N are variable points on the lines BC, CD respectively. (1) Find the positions of M and N such that the planes SMA and SMN are perpendicular, BM + DN ≥ 3a/2, and BM·DN has the smallest value possible. (2) Find the positions of M and N such that angle NAM = 45 deg, and the volume of SAMN is (a) a maximum, (b) a minimum, and find the maximum and minimum. (3) Q is a variable point (in space) such that ∠AQB = ∠AQD = $90^o$. p is the plane ABS. Q' is the intersection of DQ and p. Find the locus K of Q'. Let CQ meet K again at R. Let R' be the intersection of DR and p. Show that $sin^2$Q'DB + $sin^2$R'DB is constant.