Question 111
For positive real numbers $a$, $b$ and $c$ such that $a + b + c = abc$, prove that
$$
\sum_{\text{cyc}} \frac{a}{a^2 + 1} \le \frac{\sqrt{abc}}{3\sqrt{2}} \sum_{\text{cyc}} \frac{\sqrt{a^3 + b^3}}{ab + 1}.
$$
$$
\sum_{\text{cyc}} \frac{a}{a^2 + 1} \le \frac{\sqrt{abc}}{3\sqrt{2}} \sum_{\text{cyc}} \frac{\sqrt{a^3 + b^3}}{ab + 1}.
$$