Question 11
Let $a, b, c, d$ be positive real numbers satisfying $a + b + c + d = 4$. Prove that
$$
\sum_{\text{cyc}} \frac{3a^3}{a^2 + ab + b^2} + \sum_{\text{cyc}} \frac{2ab}{a+b} \geq 8.
$$
$$
\sum_{\text{cyc}} \frac{3a^3}{a^2 + ab + b^2} + \sum_{\text{cyc}} \frac{2ab}{a+b} \geq 8.
$$