I Thought These Were Partial Fractions

Algebra Level 5

If x1,x2,.....,x71x_{1},x_{2},.....,x_{71} satisfy

j=171xji+j=42i+1 for i=1,2,...,71. \sum_{j=1}^{71}\frac{x_{j}}{i+j}=\frac{4}{2i+1} \text{ for } i=1,2,...,71.

Suppose j=171xj2j+1=pq,\sum_{j=1}^{71}\frac{x_{j}}{2j+1} = \frac{p}{q}, where pp and qq are coprime integers. Find p+qp+q.

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