21st APMC 1998 Problem 3 Find all real solutions to x 3 = 2 - y, y 3 = 2 - x.
x = y = 1. Solution Getting a degree 9 equation for y leads nowhere. If x = y, then x 3 + x - 2 = (x - 1)(x 2 + x + 2) = 0, so x = y = 1 (because x 2 + x + 2 = (x + ½)2 + 7/4 is always positive). So suppose x ≠ y. wlog x > y. So 0 = x 3 + y - 2  1 and hence x 3 > x. Also 0 = y 3 + x - 2 > y 3 + y - 2 = (y-1)(y 2 + y + 2), so y  x + y 3 = 2. Contradiction. So y  2, so x 3 > 8, so y = 2 - x 3 < -6. Contradiction. So there are no solutions with x ≠ y. 21st APMC 1998 © John Scholes jscholes@kalva.demon.co.uk 28 Nov 2003 Last corrected/updated 28 Nov 03