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There exists a k satisfying this inequality

Source: South African MO 2005 Q5

May 27, 2012
inequalitiesinductioninequalities unsolvedn-variable inequality

Problem Statement

Let x1,x2,…,xnx_1,x_2,\dots,x_n be positive numbers with product equal to 1. Prove that there exists a k∈{1,2,…,n}k\in\{1,2,\dots,n\} such that xkk+x1+x2+⋯+xk≥1−12n.\frac{x_k}{k+x_1+x_2+\cdots+x_k}\ge 1-\frac{1}{\sqrt[n]{2}}.
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