Problem

Source: 2017 Taiwan TST Round 3

Tags: inequalities



There are $m$ real numbers $x_i \geq 0$ ($i=1,2,...,m$), $n \geq 2$, $\sum_{i=1}^{m} x_i=S$. Prove that \[ \sum_{i=1}^{m} \sqrt[n]{\frac{x_i}{S-x_i}} \geq 2, \]The equation holds if and only if there are exactly two of $x_i$ are equal(not equal to $0$), and the rest are equal to $0$.