Question 182
For positive real numbers $a$, $b$, $c$, which satisfy the condition $ab + bc + ca = 1$, prove the following inequality:
$$
\left(\sqrt{bc} + \frac{1}{2a + \sqrt{bc}}\right) \cdot \left(\sqrt{ca} + \frac{1}{2b + \sqrt{ca}}\right) \cdot \left(\sqrt{ab} + \frac{1}{2c + \sqrt{ab}}\right) \ge 8abc.
$$
$$
\left(\sqrt{bc} + \frac{1}{2a + \sqrt{bc}}\right) \cdot \left(\sqrt{ca} + \frac{1}{2b + \sqrt{ca}}\right) \cdot \left(\sqrt{ab} + \frac{1}{2c + \sqrt{ab}}\right) \ge 8abc.
$$