RO-NMO 2016 Problem 92

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