Let $\{x_1,\dots,x_n\}$ be distinct positive real numbers. Define the $n\times n$ matrix $H$, where $H_{ij} = \frac{2}{x_i + x_j}$. Prove that:
$$ \mathbf{1}^\top H^{-1} \mathbf{1} = x_1+\dots+x_n $$
where $\mathbf{1}$ is the $n\times 1$ vector of all 1's.
I can show that $H\succeq 0$ (positive semidefinite) and $H\succ 0$ when the $x_i$ are distinct, but I'm not sure how to proceed further. For a reference, see Proving a symmetric Cauchy matrix is positive semidefinite.
Edit: I think this should be true even when the $x_i$ are not necessarily positive? (so long as $x_i+x_j \neq 0$ for all $i,j$).
