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.
Answer
1,1,1,1 or 3,-1,-1,-1.
Solution
Let the numbers be x1, ... , x4. Let t = x1x2x3x4. Then x1 + t/x1 = 2. So all the xi are roots of the quadratic x2 - 2x + t = 0. This has two roots, whose product is t.
If all xi are equal to x, then x3 + x = 2, and we must have x = 1. If not, then if x1 and x2 are unequal roots, we have x1x2 = t and x1x2x3x4 = t, so x3x4 = 1. But x3 and x4 are still roots of x2 - 2x + t = 0. They cannot be unequal, otherwise x3x4 = t, which gives t = 1 and hence all xi = 1. Hence they are equal, and hence both 1 or both -1. Both 1 gives t = 1 and all xi = 1. Both -1 gives t = -3 and hence xi = 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.
© John Scholes
jscholes@kalva.demon.co.uk
28 Sep 1998
Last corrected/updated 26 Sep 2003