Question 36
Let $n \in \mathbb{N}^*$. Find all real numbers $x_1, x_2, \dots, x_n$ such that $x_k \in [0, 2k)$, for every $k \in \{1, 2, \dots, n\}$, and
$$ \frac{1}{2-x_1} + \frac{2}{4-x_2} + \dots + \frac{n}{2n-x_n} = \frac{x_1}{1} + \frac{x_2}{2} + \dots + \frac{x_n}{n}. $$
$$ \frac{1}{2-x_1} + \frac{2}{4-x_2} + \dots + \frac{n}{2n-x_n} = \frac{x_1}{1} + \frac{x_2}{2} + \dots + \frac{x_n}{n}. $$