Solution:

Let $S = x_1 + x_2 + \cdots + x_{100}.$ The condition says $x_k = (S - x_k) - k,$ so $x_k = \frac{S - k}{2}$ for every $k.$ Summing over $k = 1, \ldots, 100,$ $$S = \frac{100S - (1 + 2 + \cdots + 100)}{2} = \frac{100S - 5050}{2},$$ so $98S = 5050$ and $S = \frac{2525}{49}.$

Then $$x_{50} = \frac{S - 50}{2} = \frac{2525 - 2450}{98} = \frac{75}{98},$$ which is in lowest terms, so $m + n = 75 + 98 = 173.$