Let $n>3$ and $x_1, x_2, \ldots, x_n$ be positive real numbers with $x_1x_2\cdots x_n=1$. Prove that
$$
\frac{1}{1+x_1+x_1x_2}+\frac{1}{1+x_2+x_2x_3}+\cdots+\frac{1}{1+x_n+x_nx_1}>1.
$$
I’d like to homogenize the degree of the denominator of each term but I ran out of tricks. Any help will be appreciated. Thanks.
Best Answer
You can make the homogenization by the following way.
Let $x_1=\frac{a_2}{a_1},$ $x_2=\frac{a_3}{a_2},$..., $x_{n-1}=\frac{a_n}{a_{n-1}},$ where all $a_i>0$, $a_{n+1}=a_1$ and $a_{n+2}=a_2$.
Thus, the condition gives $x_n=\frac{a_1}{a_n}$ and $$\sum_{i=1}^n\frac{1}{1+x_1+x_2}=\sum_{i=1}^n\frac{1}{1+\frac{a_{i+1}}{a_i}+\frac{a_{i+2}}{a_i}}=\sum_{i=1}^n\frac{a_i}{a_i+a_{i+1}+a_{i+2}}>\sum_{i=1}^n\frac{a_i}{a_1+a_2+...+a_n}=1.$$