Problem

Source: China Jiangxi , Jul 30, 2017

Tags: inequalities, algebra, China, BPSQ



Let $a_1,a_2,\cdots,a_{n+1}>0$. Prove that$$\sum_{i-1}^{n}a_i\sum_{i=1}^{n}a_{i+1}\geq \sum_{i=1}^{n}\frac{a_i a_{i+1}}{a_i+a_{i+1}}\cdot \sum_{i=1}^{n}(a_i+a_{i+1})$$