suppose $x_i>0$ for $i=1,2,\cdots,n$ and $x_1+x_2+\cdots+x_n=1$ show that $$\frac{x_1}{1+x_2+x_3+\cdots+x_n}+\frac{x_2}{1+x_1+x_3+\cdots+x_n}+\frac{x_3}{1+x_1+x_2+\cdots+x_n}+\cdots+\frac{x_n}{1+x_1+x_2+\cdots+x_{n-1}}\ge\frac{n}{2n-1}$$ attempt
from the equalitys i could rewrite $1-x_1=x_2+\cdots+x_n$ and then $$\frac{x_1}{1+x_2+x_3+\cdots+x_n}+\frac{x_2}{1+x_1+x_3+\cdots+x_n}+\frac{x_3}{1+x_1+x_2+\cdots+x_n}+\cdots+\frac{x_n}{1+x_1+x_2+\cdots+x_{n-1}}=\frac{x_1}{2-x_1}+\frac{x_2}{2-x_2}+\frac{x_3}{2-x_3}+\cdots+\frac{x_n}{2-x_n}$$ but idk how to proced from there, so how do i solve this? the fact $x_i>0$ makes me assume AM-GM inequality could be used but idk how one would use.