Question 126
Let $a$, $b$, $c$ be positive real numbers such that $ab + bc + ca = 1$. Prove that
$$
8abc \le \left(\sqrt{bc} + \frac{1}{2a + \sqrt{bc}}\right) \left(\sqrt{ca} + \frac{1}{2b + \sqrt{ca}}\right) \left(\sqrt{ab} + \frac{1}{2c + \sqrt{ab}}\right).
$$
$$
8abc \le \left(\sqrt{bc} + \frac{1}{2a + \sqrt{bc}}\right) \left(\sqrt{ca} + \frac{1}{2b + \sqrt{ca}}\right) \left(\sqrt{ab} + \frac{1}{2c + \sqrt{ab}}\right).
$$