Middle European Mathematical Olympiad 2016 Problem I-1

Let $n \geq 2$ be an integer and $x_1, x_2, \ldots, x_n$ be real numbers satisfying

(a) $x_j > -1$ for $j = 1, 2, \ldots, n$ and

(b) $x_1 + x_2 + \cdots + x_n = n$.

Prove the inequality $$\sum_{j=1}^n \frac{1}{1 + x_j} \geq \sum_{j=1}^n \frac{x_j}{1 + x_j^2}$$

and determine when equality holds.