Problem 1. Let $a, b, c$ be positive real numbers such that$$a^2 + b^2 + c^2 \ge 3.$$ Prove that$$\frac{a^4}{a^2 + 2b + 2c} + \frac{b^4}{b^2 + 2c + 2a} + \frac{c^4}{c^2 + 2a + 2b} \ge \frac{3}{5}.$$