IMO 1965 Problem B1 Find all sets of four real numbers such that the sum of any one and the product of the other three is 2.
1,1,1,1 or 3,-1,-1,-1. Solution Let the numbers be x 1 , ... , x 4 . Let t = x 1 x 2 x 3 x 4 . Then x 1 + t/x 1 = 2. So all the x i are roots of the quadratic x 2 - 2x + t = 0. This has two roots, whose product is t. If all x i are equal to x, then x 3 + x = 2, and we must have x = 1. If not, then if x 1 and x 2 are unequal roots, we have x 1 x 2 = t and x 1 x 2 x 3 x 4 = t, so x 3 x 4 = 1. But x 3 and x 4 are still roots of x 2 - 2x + t = 0. They cannot be unequal, otherwise x 3 x 4 = t, which gives t = 1 and hence all x i = 1. Hence they are equal, and hence both 1 or both -1. Both 1 gives t = 1 and all x i = 1. Both -1 gives t = -3 and hence x i = 3, -1, -1, -1 (in some order). Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48- X. 7th IMO 1965 © John Scholes jscholes@kalva.demon.co.uk 28 Sep 1998 Last corrected/updated 26 Sep 2003