Question 96
Let $n$ be a positive integer and $x_1, x_2, \dots, x_n \gt 0$ be real numbers so that
$$
x_1 + x_2 + \dots + x_n = \frac{1}{x_1^2} + \frac{1}{x_2^2} + \dots + \frac{1}{x_n^2}.
$$
Show that for each positive integer $k \le n$, there are $k$ numbers among $x_1, x_2, \dots, x_n$ whose sum is at least $k$.
$$
x_1 + x_2 + \dots + x_n = \frac{1}{x_1^2} + \frac{1}{x_2^2} + \dots + \frac{1}{x_n^2}.
$$
Show that for each positive integer $k \le n$, there are $k$ numbers among $x_1, x_2, \dots, x_n$ whose sum is at least $k$.