24th BMO 1988 Problem 4 Find all positive integer solutions to 1/a + 2/b - 3/c = 1.
(a,b,c) = (2,1,2), (2,3,18), (1,2n,3n), (n,2,3n) Solution If a ≥ 3, then 2/b = 1 + 3/c - 1/a > 2/3, so b  1/2, so b ≤ 3. The cases b = 1, 2 we deal with below. If b = 3, then c = 18, which gives a solution. If b = 1, then 3/c = 1 + 1/a > 1, so c = 1 or 2. c = 1 does not give a solution. c = 2 gives a = 2, which is a solution. If b = 2, then 1/a = 3/c, so c = 3a. This gives a solution for any a. 24th BMO 1988 © John Scholes jscholes@kalva.demon.co.uk 9 February 2004 Last corrected/updated 9 Feb 04