A Cyclic Ratio Comparison

★ 8/10Substitution inequality Rearrangement Inequality

Problem

Let $a_1, a_2, \ldots, a_n > 0$ be real numbers. Prove that

\sum_{i=1}^n \frac{a_{i-1}}{a_i} \geqslant \sum_{i=1}^n \frac{a_{i-1} + a_i + 1}{a_i + a_{i+1} + 1},

where indices are cyclic: $a_0 = a_n$ and $a_{n+1} = a_1$ .

Answer

Solution

Difficulty	8/10
Topics	Substitution, inequality, Rearrangement Inequality

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