Question 116
Let $a$, $b$, $c \gt 0$ be real numbers so that $a + b + c = 1$. Prove that
$$
\frac{bc + a + 1}{a^2 + 1} + \frac{ca + b + 1}{b^2 + 1} + \frac{ab + c + 1}{c^2 + 1} \le \frac{39}{10}.
$$
$$
\frac{bc + a + 1}{a^2 + 1} + \frac{ca + b + 1}{b^2 + 1} + \frac{ab + c + 1}{c^2 + 1} \le \frac{39}{10}.
$$