10th Irish 1997 Problem A1 Find all integer solutions to 1 + 1996m + 1998n = mn.
(m,n) = (1999,1997 2 +1996), (3995,3993), (1997 2 +1998,1997) Solution We have (m-1998)(n-1996) = mn - 1996m - 1998n + 1996·1998 = 1996·1998 + 1 = 1997 2 . But 1997 is prime, so m-1998 = 1, 1997 or 1997 2 . Comment. Any olympiad competitor should factorise the year before competing! 10th Irish 1997 © John Scholes jscholes@kalva.demon.co.uk 17 Dec 2003 Last updated/corrected 17 Dec 03