Show that $\sum_{i=1}^n \frac{1}{|x-p_i|}\le 8n\sum_{i=1}^n \frac{1}{2i-1}$ for some $x,0\le x \le 1$

Let $0\le p_i \le 1$ for $i = 1,2,\dots n.$ Show that $\displaystyle\sum_{i=1}^n \dfrac{1}{|x-p_i|}\le 8n\displaystyle\sum_{i=1}^n \dfrac{1}{2i-1}$ for some $x,0\le x \le 1.$

First, it seems reasonable to convert the RHS to a closed form. I know $\dfrac{\pi}{4}=\displaystyle\sum_{i=1}^\infty (-1)^{i-1}\dfrac{1}{2i-1},$ but I don't think that will be useful for this problem. Also, I don't think the LHS can be simplified using a telescoping series. I think I should pick several candidates $x_j$ and then take $\displaystyle\sum_{j=1}^n \dfrac{1}{|x_j-p_j|}.$ I should probably also consider different intervals of $[0,1].$

Any help would be appreciated.

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