Question 135
Let $m$ be a positive integer and let $n = m^2 + 1$. Determine all real numbers $x_1, x_2, \dots, x_n$ satisfying
$$
x_i = 1 + \frac{2m x_i^2}{x_1^2 + x_2^2 + \dots + x_n^2}, \quad i = 1, 2, \dots, n.
$$
$$
x_i = 1 + \frac{2m x_i^2}{x_1^2 + x_2^2 + \dots + x_n^2}, \quad i = 1, 2, \dots, n.
$$