1st Iberoamerican 1985 Problem A1 Find all integer solutions to: a + b + c = 24, a 2 + b 2 + c 2 = 210, abc = 440.
ab + bc + ca = ( (a + b + c) 2 - (a 2 + b 2 + c 2 ) )/2 = 183, so a, b, c are roots of the cubic x 3 - 24x 2 + 183x - 440 = 0. But it easily factorises as (x - 5)(x - 8)(x - 11) = 0, so the only solutions are permutations of (5, 8, 11). 1st Ibero 1985 © John Scholes jscholes@kalva.demon.co.uk 1 July 2002