MathNet 2005 — Part 02 Question 174

Assume that positive numbers $a$, $b$, $c$, $x$, $y$, $z$ satisfy $cy + bz = a$; $az + cx = b$ and $bx + ay = c$. Find the minimum value of the function $f(x, y, z) = \frac{x^2}{1+x} + \frac{y^2}{1+y} + \frac{z^2}{1+z}$.