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Cyclic, logarithmic inequality

Source: Romanian District Olympiad 2007, Grade X, Problem 1

October 7, 2018
inequalitieslogarithmalgebra

Problem Statement

Let be three real numbers a,b,c, a,b,c, all in the interval (0,∞) (0,\infty ) or all in the interval (0,1). (0,1). Prove the following inequality: ∑cyclog⁡abc≥4⋅∑cyclog⁡abc. \sum_{\text{cyc}}\log_a bc\ge 4\cdot\sum_{\text{cyc}} \log_{ab} c .
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