Let $x_1,$ $x_2,$ $x_3,$ $x_4,$ $x_5$ be distinct positive integers such that $x_1 + x_2 + x_3 + x_4 + x_5 = 100.$ Compute the maximum value of the expression \begin{align*} &\frac{(x_2 x_5 + 1)(x_3 x_5 + 1)(x_4 x_5 + 1)}{(x_2 - x_1)(x_3 - x_1)(x_4 - x_1)} + \frac{(x_1 x_5 + 1)(x_3 x_5 + 1)(x_4 x_5 + 1)}{(x_1 - x_2)(x_3 - x_2)(x_4 - x_2)} \\ &\quad + \frac{(x_1 x_5 + 1)(x_2 x_5 + 1)(x_4 x_5 + 1)}{(x_1 - x_3)(x_2 - x_3)(x_4 - x_3)} + \frac{(x_1 x_5 + 1)(x_2 x_5 + 1)(x_3 x_5 + 1)}{(x_1 - x_4)(x_2 - x_4)(x_3 - x_4)}. \end{align*}
There is probably some clever insight that I haven't seen, any ideas?
If the terms are factored out, the expression equals $x_5^3$ (see Helmut's comment, I checked it with Matlab).
Since we have that $x_1,$ $x_2,$ $x_3,$ $x_4,$ $x_5$ are distinct positive integers such that $x_1 + x_2 + x_3 + x_4 + x_5 = 100$, the maximum value of the expression will be obtained for the smallest possible distinct integers $x_1,$ $x_2,$ $x_3,$ $x_4$. Choose $(1,2,3,4)$ for these four values, then the maximum value of the expression is $90^3 = 729000$.