Question 18
Let $a$, $b$ and $c$ be positive real numbers such that $ab + bc + ca = 1$. Prove that
$$
\sqrt[4]{\frac{\sqrt{3}}{a} + 6\sqrt{3}b} + \sqrt[4]{\frac{\sqrt{3}}{b} + 6\sqrt{3}c} + \sqrt[4]{\frac{\sqrt{3}}{c} + 6\sqrt{3}a} \le \frac{1}{abc}.
$$
When does the equality hold?
$$
\sqrt[4]{\frac{\sqrt{3}}{a} + 6\sqrt{3}b} + \sqrt[4]{\frac{\sqrt{3}}{b} + 6\sqrt{3}c} + \sqrt[4]{\frac{\sqrt{3}}{c} + 6\sqrt{3}a} \le \frac{1}{abc}.
$$
When does the equality hold?