Question 42
Positive numbers $a$, $b$, $c$ satisfy $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 3$. Prove the inequality
$$
\frac{1}{\sqrt{a^3 + b}} + \frac{1}{\sqrt{b^3 + c}} + \frac{1}{\sqrt{c^3 + a}} \le \frac{3}{\sqrt{2}}
$$
$$
\frac{1}{\sqrt{a^3 + b}} + \frac{1}{\sqrt{b^3 + c}} + \frac{1}{\sqrt{c^3 + a}} \le \frac{3}{\sqrt{2}}
$$