28th Putnam 1967 Problem A6 a i and b i are reals such that a 1 b 2 ≠ a 2 b 1 . What is the maximum number of possible 4-tuples (sign x 1 , sign x 2 , sign x 3 , sign x 4 ) for which all x i are non-zero and x i is a simultaneous solution of a 1 x 1 + a 2 x 2 + a 3 x 3 + a 4 x 4 = 0 and b 1 x 1 + b 2 x 2 + b 3 x 3 + b 4 x 4 = 0. Find necessary and sufficient conditions on a i and b i for this maximum to be achieved.