25th ASU 1991 Problem 1 Find all integers a, b, c, d such that ab - 2cd = 3, ac + bd = 1.
(a,b,c,d) = (1,3,1,0), (-1,-3,-1,0), (3,1,0,1), (-3,-1,0,-1) Solution 11 = (ab - 2cd) 2 + 2(ac + bd) 2 = (a 2 + 2d 2 )(b 2 + 2c 2 ), so we must have either (1) a 2 + 2d 2 = 1, b 2 + 2c 2 = 11, or (2) a 2 + 2d 2 = 11, b 2 + 2c 2 = 1. (1) gives a = ±1, d = 0, b = ±3, c = ±1. If a = 1 and d = 0, then ac + bd = 1 implies c = 1, and ab - 2cd = 3 implies b = 3. Similarly, if a = -1, then c = -1, and b = -3. Similarly, (2) gives (a,b,c,d) = (3,1,0,1), (-3,-1,0,-1). Thanks to Suat Namli 25th ASU 1991 © John Scholes jscholes@kalva.demon.co.uk 2 March 2004 Last corrected/updated 2 Mar 04