Question 129
Let $x_i \ge 0$ ($i = 1, 2, \dots, m$) be real numbers, $n \ge 2$, and $\sum_{i=1}^{m} x_i = S$. Prove that
$$
\sum_{i=1}^{m} \sqrt[n]{\frac{x_i}{S - x_i}} \ge 2,
$$
with equality if and only if two of the $x_i$ are equal and nonzero and all the others are $0$.
$$
\sum_{i=1}^{m} \sqrt[n]{\frac{x_i}{S - x_i}} \ge 2,
$$
with equality if and only if two of the $x_i$ are equal and nonzero and all the others are $0$.