# Prove or disprove this inequality involving fractions

Let $n$ be a positive integer. Assuming $x_1, x_2, x_3,...,x_{2n}$ are all positive real numbers, we need to prove or disprove: $$\frac{x_1}{x_2}+\frac{x_2}{x_3}+...+\frac{x_{2n}}{x_1} \geq n+\frac{x_1+x_{n+1}}{x_2+x_{n+2}}+\frac{x_2+x_{n+2}}{x_3+x_{n+3}}+...+\frac{x_n+x_{2n}}{x_1+x_{n+1}}$$ Another question: if this inequality fails for general situations, for which $n$ does this inequality hold?

Any help is heartily appreciated.

• from where does this inequality come? – Dr. Sonnhard Graubner Jan 21 '17 at 14:24
• From an old post in 2008. – apprenant Jan 21 '17 at 14:27