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Regional Olympiad - FBH 2018 Grade 12 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

September 18, 2018
inequalitiesalgebralogarithms

Problem Statement

If numbers x1x_1, x2x_2,...,xnx_n are from interval (14,1)\left( \frac{1}{4},1 \right) prove the inequality: log⁡x1(x2−14)+log⁡x2(x3−14)+...+log⁡xn−1(xn−14)+log⁡xn(x1−14)≥2n\log _{x_1} {\left(x_2-\frac{1}{4} \right)} + \log _{x_2} {\left(x_3-\frac{1}{4} \right)}+ ... + \log _{x_{n-1}} {\left(x_n-\frac{1}{4} \right)} + \log _{x_n} {\left(x_1-\frac{1}{4} \right)} \geq 2n
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