Question 176
Prove that for all positive real numbers $a, b, c$ such that $a b c=1$ the following inequality holds:
$$
\frac{a}{2 b+c^{2}}+\frac{b}{2 c+a^{2}}+\frac{c}{2 a+b^{2}} \leqslant \frac{a^{2}+b^{2}+c^{2}}{3}
$$
$$
\frac{a}{2 b+c^{2}}+\frac{b}{2 c+a^{2}}+\frac{c}{2 a+b^{2}} \leqslant \frac{a^{2}+b^{2}+c^{2}}{3}
$$