Problems

The \(n\) positive numbers \(x_{1},x_{2},\dots,x_{n}\), where \(n\ge3\), satisfy $$ x_{1}=1+\frac{1}{x_{2}}\, ,\ \ \ x_{2}=1+\frac{1}{x_{3}}\, , \ \ \ \dots\; , \ \ \ x_{n-1}=1+\frac{1}{x_{n}}\, , $$ and also $$ \ x_{n}=1+\frac{1}{x_{1}}\, . $$ Show that

1. \(x_{1},x_{2},\dots,x_{n}>1\),
2. \({\displaystyle x_{1}-x_{2}=-\frac{x_{2}-x_{3}}{x_{2}x_{3}}}\),
3. \(x_{1}=x_{2}=\cdots=x_{n}\).