Given $n$ positive numbers $x_1,\ldots,x_n$ ($n\ge 3$) such that the product $x_1x_2\cdots x_n=1$, show that
I am having trouble demonstrating this for even $n=3$, but it seems that once that case is established the method for the general case should be identical. I am sure there is some nice application of AM-GM or Cauchy that will solve it. Notice that in each term we have the numerator of degree $8$, and in the denominator there is one factor of degree $4$ and also a linear factor which is quite an annoyance. I have tried to "homogenize" the linear factor so that it is of degree 4 (thus each fraction of degree 0) by doing some substitutions, but I have been unsuccessful with this.
If anyone can provide a hint (e.g suggest a substitution that may be helpful), I would greatly appreciate it. This problem has been bothering me for a while now.