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inequality on n variables greater than -1

Source: MEMO 2016 I1

August 24, 2016
inequalitiesinequalities proposed

Problem Statement

Let n≥2n \ge 2 be an integer, and let x1,x2,…,xnx_1, x_2, \ldots, x_n be reals for which:
(a) xj>−1x_j > -1 for j=1,2,…,nj = 1, 2, \ldots, n and
(b) x1+x2+…+xn=n.x_1 + x_2 + \ldots + x_n = n.
Prove that ∑j=1n11+xj≥∑j=1nxj1+xj2 \sum_{j = 1}^{n} \frac{1}{1 + x_j} \ge \sum_{j = 1}^{n} \frac{x_j}{1 + x_j^2} and determine when does the equality occur.
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