Question 172
Let $\alpha, \beta, \gamma$ be the angles of a triangle opposite to the sides $a, b, c$ respectively. Prove the inequality
$$
2\left(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma\right) \geq \frac{a^2}{b^2 + c^2} + \frac{b^2}{a^2 + c^2} + \frac{c^2}{a^2 + b^2}
$$
$$
2\left(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma\right) \geq \frac{a^2}{b^2 + c^2} + \frac{b^2}{a^2 + c^2} + \frac{c^2}{a^2 + b^2}
$$