16th Australian 1995 Problem B1 Find all real k such that there is only one real solution x, y to y = x + k, x 2 + y 2 + 2x ≤ 1.
-1, 3 Solution We can write the desired inequality as (x+1) 2 + (x+k) 2 ≤ 2. This has infinitely many, 1 or 0 solutions depending on whether |k-1|  2, so we need |k-1| = 2. 16th Aus 1995 © John Scholes jscholes@kalva.demon.co.uk 7 February 2004 Last updated/corrected 7 Feb 04