37th Eötvös Competition Problems 1933

1.  If x2 + y2 = u2 + v2 = 1 and xu + yv = 0 for real x, y, u, v, find xy + uv. 2.  S is a set of 16 squares on an 8 x 8 chessboard such that there are just 2 squares of S in each row and column. Show that 8 black pawns and 8 white pawns can be placed on these squares so that there is just one white pawn and one black pawn in each row and column.