Question 179
Let $a$, $b$, $c$ be positive real numbers. Prove that
$$
\frac{(2a + b + c)^2}{2a^2 + (b+c)^2} + \frac{(2b + c + a)^2}{2b^2 + (c+a)^2} + \frac{(2c + a + b)^2}{2c^2 + (a+b)^2} \le 8.
$$
$$
\frac{(2a + b + c)^2}{2a^2 + (b+c)^2} + \frac{(2b + c + a)^2}{2b^2 + (c+a)^2} + \frac{(2c + a + b)^2}{2c^2 + (a+b)^2} \le 8.
$$