A. 405. The real numbers a, b, c, x, y, z satisfy abc>0 and xyz>0. Prove that
(Korean competition problem)
Deadline expired on 16 October 2006.
Solution. We will use Nesbitt's inequality: for arbitrary positive numbers A,B,C,
(This is equivalent to the AM-HM inequality for numbers B+C, C+A and A+B.)
By the rearrangement inequality, bz+cyby+cz and
It can be obtained similarly that
Applying these estiamtes and Nesbitt's inequality on the numbers a2x2, b2y2 and c2z2,