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Mathematics

20 March, 14:20

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*}