Putnam 1979 A6

Problem A-6 Let $0<p_{i}<1$ for $i=1,2, \ldots, n$. Show that $$\sum_{i=1}^{n} \frac{1}{\left|x-p_{i}\right|}<8 n\left(1+\frac{1}{3}+\frac{1}{5}+\cdots+\frac{1}{2 n-1}\right)$$ for some $x$ satisfying $0<x<1$.