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)