A sequence of numbers $$x_{1},x_{2},x_{3},\ldots,x_{100}$$ has the property that, for every integer $$k$$ between $$1$$ and $$100,$$ inclusive, the number $$x_{k}$$ is $$k$$ less than the sum of the other $$99$$ numbers. Given that $$x_{50} = \frac{m}{n}$$, where $$m$$ and $$n$$ are relatively prime positive integers, find $$m + n$$.

(第十八届AIME1 2000 第10题)