Question 92
The sum of positive real numbers $a$, $b$, $c$ equals $1$. Prove the inequality
$$
\sqrt{\frac{1}{a} - 1}\sqrt{\frac{1}{b} - 1} + \sqrt{\frac{1}{b} - 1}\sqrt{\frac{1}{c} - 1} + \sqrt{\frac{1}{c} - 1}\sqrt{\frac{1}{a} - 1} \ge 6.
$$
$$
\sqrt{\frac{1}{a} - 1}\sqrt{\frac{1}{b} - 1} + \sqrt{\frac{1}{b} - 1}\sqrt{\frac{1}{c} - 1} + \sqrt{\frac{1}{c} - 1}\sqrt{\frac{1}{a} - 1} \ge 6.
$$