16th VMO 1978

Find all values of the parameter m such that the equations x² = 2^|x| + |x| - y - m = 1 - y² have only one root.

Answer

0, 2

Solution

If x is a non-zero solution, then -x is another, so we must have x = 0. Hence 0 = 1 - y - m = 1 - y². So y = ±1, so m = 0 or 2, and each of these gives just one root.

Thanks to Suat Namli

16th VMO 1978

© John Scholes
jscholes@kalva.demon.co.uk
6 March 2004
Last corrected/updated 6 Mar 04