2000 AIME I Problem 10

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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.