Not simply Telescopic

Algebra Level 4

\[\begin{cases} A = \dfrac{1}{1\times 2 } + \dfrac{1}{3 \times 4} + \dfrac 1{5 \times 6} + \cdots + \dfrac{1}{99 \times 100} \\ B= \dfrac{1}{51 \times 100} + \dfrac{1}{52 \times 99} + \dfrac 1{53 \times 98} + \cdots + \dfrac{1}{100 \times 51} \end{cases} \]

Given the above and that \(\dfrac{A}{B} = \dfrac{m}{n}\), where \(m\) and \(n\) are coprime positive integers. Find \(m+n\).

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