2nd CaMO 1970 Problem 1 Find all triples of real numbers such that the product of any two of the numbers plus the third is 2. Solution
: (1, 1, 1) and (-2, -2, -2). Let the numbers by x, y, z, so x + yz = 2 (1), y + zx = 2 (2), z + xy = 2 (3). Subtracting (2) from (1) gives (x - y)(1 - z) = 0, so x = y or z = 1. If x = y, then x + xz = 2 and z + x 2 = 2, so x 3 - 3x + 2 = 0. Factorising, (x - 1) 2 (x + 2) = 0, so x = 1, y = 1, z = 1 or x = -2, y = -2, z = -2. If z = 1, then x + y = 2 and xy = 1. Hence x 2 - 2x + 1 = 0. Factorising (x - 1) 2 = 0, so x = 1. 2nd CaMO 1970 © John Scholes jscholes@kalva.demon.co.uk 10 June 2002