Question 142
Let $a$, $b$, $c$ be real numbers with sum $a + b + c = 3$. Prove that
$$
\frac{a^2 + b^2 + c^2}{2} + \frac{1}{a^2 + 1} + \frac{1}{b^2 + 1} + \frac{1}{c^2 + 1} \geq 3.
$$
$$
\frac{a^2 + b^2 + c^2}{2} + \frac{1}{a^2 + 1} + \frac{1}{b^2 + 1} + \frac{1}{c^2 + 1} \geq 3.
$$