Johan Gunardi wrote: Let $ x_1,x_2,\ldots,x_n$ be real numbers greater than 1. Show that \[ \frac {x_1x_2}{x_3} + \frac {x_2x_3}{x_4} + \cdots + \frac {x_nx_1}{x_2}\ge4n \] and determine when the equality holds. try $ x_1=x_2=\cdots =x_n=2$, $ LHS=2n<4n$..

The correct problem is: Let $ x_1,x_2,\dots,x_n$ be real numbers such that $ x_i>1$ for $ i=1,2,3,\dots,n$. Prove that \[ \frac{x_1x_2}{x_3-1}+\frac{x_2x_3}{x_4-1}+\frac{x_3x_4}{x_5-1}+\dots + \frac{x_nx_1}{x_2-1} \ge 4n\] and determine when the equality occurs. I've ever posted it before...

I think n should be greater than 2

Let $ y_i=x_i-1$ for $ i=1,2,...,n$. Then the inequality is \[ \sum \frac{y_iy_{i+1}+y_i+y_{i+1}+1}{y_{i+2}}=\sum \frac{y_iy_{i+1}}{y_{i+2}}+\frac{1}{y_{i+2}}+\sum \frac{y_i}{y_{i+2}}+\frac{y_{i+1}}{y_{i+2}}\]Using the AM-GM twice, \[ \ge 2\sum 2\sqrt{ \frac{y_iy_{i+1}}{y_{i+2}^2} }\ge 4n\sqrt[n]{\prod\frac{y_iy_{i+1}}{y_{i+2}^2}}=4n\] by the AM-GM on each sum. (an example of equality is when $ y_i=1$ or $ x_i=2$ $ i=1,2,...,n$.

Altheman wrote: Let $ y_i = x_i - 1$ for $ i = 1,2,...,n$. Then the inequality is \[ \sum \frac {y_iy_{i + 1} + y_i + y_{i + 1} + 1}{y_{i + 2}} = \sum \frac {y_iy_{i + 1}}{y_{i + 2}} + \frac {1}{y_{i + 2}} + \sum \frac {y_i}{y_{i + 2}} + \frac {y_{i + 1}}{y_{i + 2}} \] Using the AM-GM twice, \[ \ge 2\sum 2\sqrt {\frac {y_iy_{i + 1}}{y_{i + 2}^2} }\ge 4n\sqrt [n]{\prod\sqrt{\frac {y_iy_{i + 1}}{y_{i + 2}^2}} }= 4n \] by the AM-GM on each sum. (an example of equality is when $ y_i = 1$ or $ x_i = 2$ $ i = 1,2,...,n$.

See also here: http://www.artofproblemsolving.com/Forum/viewtopic.php?t=240954