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Unusual inequality!

Source: Romania TST 2013 Day 5 Problem 1

January 21, 2015
inequalitiestrigonometryinequalities unsolved

Problem Statement

Let nn be a positive integer and let x1x_1, …\ldots, xnx_n be positive real numbers. Show that: min⁡(x1,1x1+x2,⋯ ,1xn−1+xn,1xn)≤2cos⁡πn+2≤max⁡(x1,1x1+x2,⋯ ,1xn−1+xn,1xn). \min\left ( x_1,\frac{1}{x_1}+x_2, \cdots,\frac{1}{x_{n-1}}+x_n,\frac{1}{x_n} \right )\leq 2\cos \frac{\pi}{n+2} \leq\max\left ( x_1,\frac{1}{x_1}+x_2, \cdots,\frac{1}{x_{n-1}}+x_n,\frac{1}{x_n} \right ).
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