Question 127
Prove that
$$
\left(a^2 + \frac{c^2}{b^2}\right) \left(b^2 + \frac{a^2}{c^2}\right) \left(c^2 + \frac{b^2}{a^2}\right) \ge \frac{abc}{\sqrt{3}} \left(a + \frac{1}{a}\right) \left(b + \frac{1}{b}\right) \left(c + \frac{1}{c}\right)
$$
for all positive real $a, b, c$.
$$
\left(a^2 + \frac{c^2}{b^2}\right) \left(b^2 + \frac{a^2}{c^2}\right) \left(c^2 + \frac{b^2}{a^2}\right) \ge \frac{abc}{\sqrt{3}} \left(a + \frac{1}{a}\right) \left(b + \frac{1}{b}\right) \left(c + \frac{1}{c}\right)
$$
for all positive real $a, b, c$.