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Equation implies inequality for real k greater than 1

Source: RMO

January 16, 2012
inequalitiesinequalities proposedBalkan

Problem Statement

Let nn be an integer number greater than 22, let x1,x2,…,xnx_{1},x_{2},\ldots ,x_{n} be nn positive real numbers such that ∑i=1n1xi+1=1\sum_{i=1}^{n}\frac{1}{x_{i}+1}=1 and let kk be a real number greater than 11. Show that: ∑i=1n1xik+1≥n(n−1)k+1\sum_{i=1}^{n}\frac{1}{x_{i}^{k}+1}\ge\frac{n}{(n-1)^{k}+1} and determine the cases of equality.
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