BALKAN-SL 2017 A1

Let $a, b, c$ be positive real numbers such that $a b c=1$. Prove that

$$\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+a^{5}+b^{2}} \leq 1 .$$