Question 78
Let $a$, $b$ and $c$ be positive real numbers such that $a b c = 1$. Prove the inequality
$$
\left(a b + b c + \frac{1}{c a}\right)\left(b c + c a + \frac{1}{a b}\right)\left(c a + a b + \frac{1}{b c}\right) \geq (1 + 2 a)(1 + 2 b)(1 + 2 c)
$$
$$
\left(a b + b c + \frac{1}{c a}\right)\left(b c + c a + \frac{1}{a b}\right)\left(c a + a b + \frac{1}{b c}\right) \geq (1 + 2 a)(1 + 2 b)(1 + 2 c)
$$