37th Eötvös 1933 Problem 1 If x 2 + y 2 = u 2 + v 2 = 1 and xu + yv = 0 for real x, y, u, v, find xy + uv.
If u = 0, then v = ±1, and yv = 0, so y = 0. Hence xy + uv = 0. So assume u ≠ 0. Then x = yv/u. So 1 = x 2 + y 2 = y 2 (v 2 /u 2 + 1) = y 2 /u 2 , so y = ±u. If y = u, then x + v = 0, so x = -v and hence xy + uv = 0. If y = -u, then -x + v = 0, so x = v and xy + uv = 0. So in all cases xy + uv = 0. 37th Eötvös 1933 © John Scholes jscholes@kalva.demon.co.uk 1 Nov 2003 Last corrected/updated 1 Nov 03