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China 2010 quiz5 Problem 1

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September 13, 2010
inequalitiesinequalities unsolved

Problem Statement

Given integer n≥2n\geq 2 and positive real number aa, find the smallest real number M=M(n,a)M=M(n,a), such that for any positive real numbers x1,x2,⋯ ,xnx_1,x_2,\cdots,x_n with x1x2⋯xn=1x_1 x_2\cdots x_n=1, the following inequality holds: ∑i=1n1a+S−xi≤M\sum_{i=1}^n \frac {1}{a+S-x_i}\leq M where S=∑i=1nxiS=\sum_{i=1}^n x_i.
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