gobathegreat wrote: If numbers $x_1$, $x_2$,...,$x_n$ are from interval $\left( \frac{1}{4},1 \right)$ prove the inequality: $\log _{x_1} {\left(x_2-\frac{1}{4} \right)} + \log _{x_2} {\left(x_3-\frac{1}{4} \right)}+ ... + \log _{x_{n-1}} {\left(x_n-\frac{1}{4} \right)} + \log _{x_n} {\left(x_1-\frac{1}{4} \right)} \geq 2n$ Because $\log_x\left(x-\frac{1}{4}\right)\geq2$ for all $\frac{1}{4}<x<1$ and AM-GM: $$\sum_{i=1}^n\log_{x_i}\left(x_{i+1}-\frac{1}{4}\right)\geq n\sqrt[n]{\prod_{i=1}^n\frac{\ln\left(x_{i+1}-\frac{1}{4}\right)}{\ln{x_i}}}=n\sqrt[n]{\prod_{i=1}^n\frac{\ln\left(x_{i}-\frac{1}{4}\right)}{\ln{x_i}}}\geq n\sqrt[n]{\prod_{i=1}^n2}=2n.$$