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Inequality for the n positive integers a_i

Source: Chinese MO 2004

November 3, 2008
inequalitiesalgebrapolynomialn-variable inequality

Problem Statement

For a given positive integer n≥2n\ge 2, suppose positive integers aia_i where 1≤i≤n1\le i\le n satisfy a1<a2<…<ana_1<a_2<\ldots <a_n and ∑i=1n1ai≤1\sum_{i=1}^n \frac{1}{a_i}\le 1. Prove that, for any real number xx, the following inequality holds (∑i=1n1ai2+x2)2≤12⋅1a1(a1−1)+x2\left(\sum_{i=1}^n\frac{1}{a_i^2+x^2}\right)^2\le\frac{1}{2}\cdot\frac{1}{a_1(a_1-1)+x^2}
Li Shenghong
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