I Thought These Were Partial Fractions

Algebra Level 5

If \(x_{1},x_{2},.....,x_{71}\) satisfy

\[ \sum_{j=1}^{71}\frac{x_{j}}{i+j}=\frac{4}{2i+1} \text{ for } i=1,2,...,71. \]

Suppose \(\sum_{j=1}^{71}\frac{x_{j}}{2j+1} = \frac{p}{q},\) where \(p\) and \(q\) are coprime integers. Find \(p+q\).

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