CN-LEAGUE 2016 Problem 10

Let $f(x)$ is an odd function on $R$ , $f(1)=1$ and $f(\frac{x}{x-1})=xf(x)$ $(\forall x<0)$ . Find the value of $f(1)f(\frac{1}{100})+f(\frac{1}{2})f(\frac{1}{99})+f(\frac{1}{3})f(\frac{1}{98})+\cdots +f(\frac{1}{50})f(\frac{1}{51}).$