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Easy inequality from Taiwan TST

Source: 2017 Taiwan TST Round 3

April 13, 2018
inequalities

Problem Statement

There are mm real numbers xi≥0x_i \geq 0 (i=1,2,...,mi=1,2,...,m), n≥2n \geq 2, ∑i=1mxi=S\sum_{i=1}^{m} x_i=S. Prove that\\ ∑i=1mxiS−xin≥2, \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 xix_i are equal(not equal to 00), and the rest are equal to 00.
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