What Would This Even Look Like?

Algebra Level 5

In a sequence of real numbers \(x_{0},x_{1},x_{2},x_{3},\ldots,x_{99},x_{100}\), for every pair of integers \(a,b\), such that \(0 \leq a < b \leq 100\), the sum \(x_{a}+x_{b}\) is \(a+b\) less than the sum of the other 99 numbers in the sequence.

As an explicit example, \(x_{0}+x_{1}=x_{2}+x_{3}+x_{4}+\cdots+x_{100}-(0+1)\)

If the value of \(x_{50}\) can be expressed in the form \(\frac{m}{n}\) for co-prime positive integers \(m,n\), find the value of \(m+n\).

Inspired by the AIME Exam

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