How do we approach this summation question?

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How do we find the following sum---

$\displaystyle\sum_{m=1}^\infty \displaystyle\sum_{n=1}^\infty \frac{mn^2}{2^n(n2^m+m2^n)}$

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Yes. This problem is just a slight variant of the Putnam 1999 A4 problem. You can see the solution here: