JBMO 2026 Problem 1

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