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If product of tan x_i is 2^(n/2) then inequality holds

Source: Chinese Mathematical Olympiad 2003 Problem 3

February 18, 2012
inequalitiestrigonometryinequalities proposed

Problem Statement

Given a positive integer nn, find the least λ>0\lambda>0 such that for any x1,…xn∈(0,π2)x_1,\ldots x_n\in \left(0,\frac{\pi}{2}\right), the condition ∏i=1ntan⁡xi=2n2\prod_{i=1}^{n}\tan x_i=2^{\frac{n}{2}} implies ∑i=1ncos⁡xi≤λ\sum_{i=1}^{n}\cos x_i\le\lambda.
Huang Yumin
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