Show $\frac{x_1}{x_n} + \frac{x_2}{x_{n-1}} + \frac{x_3}{x_{n-2}} + \dots + \frac{x_n}{x_1} \geq n$

By the AM-GM inequality, we have $$\frac{x_1/x_n+x_2/x_{n-1}+\cdots+x_n/x_1}{n}\geq\sqrt[n]{\frac{x_1}{x_n}\frac{x_2}{x_{n-1}}\cdots\frac{x_n}{x_1}}=\sqrt[n]{1}=1.$$

HINT: Use AM-GM inequality for real positive $a_i$s (where $1\le i\le n$)

$$\frac{\sum_{1\le r\le n} a_i}n\ge \sqrt[n]{\prod_{ 1\le r\le n}a_i}$$

Dude. You shouldn't need all those fancy things. Just pair up each $\frac{x_i}{x_{n-i}} + \frac{x_{n-i}}{x_i}$ and examine each of those separately. Take into account even $n$ and odd $n$, since there'll be an unpaired fraction when $n$ is odd.

Spoiler:

When $n$ is odd, identify what the unpaired fraction is, then it's value will be obvious.