MathNet 2010 — Part 05 Question 77

Let $a, b, c \in (0, 1)$ and $x, y, z \in (0, \infty)$ be such that $a = (bc)^x$, $b = (ca)^y$, $c = (ab)^z$. Show that
$$
\sum \frac{1}{x+y+2} \le 1.
$$