Suppose we have $n$ positive integers: $x_{1},x_{2},x_{3},..,x_{n}; \ \ n\geq 3 $ .
I'm required to show the following, for each $ 3 \leq k\leq n$:
\begin{align} \Big(\displaystyle\sum_{i=1}^{n}\frac{x_{i+1}^{k+1}x_{i-1}}{x_{i}^{k+1}x_{i}}\Big)+\displaystyle\sum_{l=2}^{k-1}\Big((k-l-1)\Big( \displaystyle\sum_{i=1}^{n} \frac{x_{i+1}^{l-1}}{x_{i}^{l-1}} \Big) \Big( \displaystyle\sum_{\substack{j_{1},j_{2},..,j_{k-l}\\ (*)}}\frac{x_{j_{1}}}{x_{j_{1}-1}}\frac{x_{j_{2}}}{x_{j_{2}-1}}...\frac{x_{j_{k-l}}}{x_{j_{k-l}-1}} \Big) \Big) \\ \geq n\Big( \displaystyle\sum_{\substack{j_{1},j_{2},..,j_{k-1}\\ (*)}}\frac{x_{j_{1}}}{x_{j_{1}-1}}\frac{x_{j_{2}}}{x_{j_{2}-1}}...\frac{x_{j_{k-1}}}{x_{j_{k-1}-1}} \Big) \end{align}
Here, (*) implies summation with permutation allowed, but where no two of the $j_{m},j_{n}$ are equal, for $m \neq n$.
I thought about using basic AM-GM inequality by combining specific terms, but could not make progress.