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Source: Bulgarian TST 2007 for Balkan MO and ARO, II day Problem 3

April 9, 2007
inequalities proposedinequalities

Problem Statement

Let n≥2n\geq 2 is positive integer. Find the best constant C(n)C(n) such that ∑i=1nxi≥C(n)∑1≤j<i≤n(2xixj+xixj)\sum_{i=1}^{n}x_{i}\geq C(n)\sum_{1\leq j<i\leq n}(2x_{i}x_{j}+\sqrt{x_{i}x_{j}}) is true for all real numbers xi∈(0,1),i=1,...,nx_{i}\in(0,1),i=1,...,n for which (1−xi)(1−xj)≥14,1≤j<i≤n.(1-x_{i})(1-x_{j})\geq\frac{1}{4},1\leq j<i \leq n.
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