16th Irish 2003 Problem A1 Find all integral solutions to (m 2 + n)(m + n 2 ) = (m + n) 3 .
(m, n) = (0, n), (m, 0), (-5, 2), (-1, 1), (1, -1), (2, -5), (4, 11), (5, 7), (7, 5), (11, 4). Solution Expanding and cancelling, we get m 2 n 2 + mn = 3m 2n + 3mn 2 . So either m = 0, or n = 0, or (dividing by mn) mn + 1 = 3m + 3n. Rearranging the last case: mn - 3m - 3n + 9 = 8. Factorising: (m - 3)(n - 3) = 8. Hence m - 3 = -8, -4, -2, -1, 1, 2, 4 or 8. 1st Irish 1988 © John Scholes jscholes@kalva.demon.co.uk 14 June 2003 Last updated/corrected 14 Jun 03