Question 159
Let $x$, $y$ and $z$ be positive reals such that $xyz = 1$. Find the largest possible value of the constant $C$ such that the inequality
$$
\left(\frac{x}{1+x}\right)^2 + \left(\frac{y}{1+y}\right)^2 + \left(\frac{z}{1+z}\right)^2 \ge C
$$
must hold under the described condition.
$$
\left(\frac{x}{1+x}\right)^2 + \left(\frac{y}{1+y}\right)^2 + \left(\frac{z}{1+z}\right)^2 \ge C
$$
must hold under the described condition.