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Inequality involving the terms of a fast increasing finite series

Source: Romania National Olympiad 2016, grade ix, p.2

August 25, 2019
inequalitiesn-variable inequality

Problem Statement

Let be a natural number n≥2 n\ge 2 and n n positive real numbers a1,an,…,an a_1,a_n,\ldots ,a_n that satisfy the inequalities \sum_{j=1}^i a_j\le a_{i+1} ,  \forall i\in\{ 1,2,\ldots ,n-1 \} . Prove that ∑k=1n−1akak+1≤n/2. \sum_{k=1}^{n-1} \frac{a_k}{a_{k+1}}\le n/2 .
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