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2023 WSMO Speed Round Problems/Problem 5

Problem

There exists a rational polynomial $f(x)$ such that for all $x$ in the range $(0,1),$ $f(x)=\sum_{n=1}^{\infty}nx^n.$ If the maximum of $f(x)$ over $[6,9]$ is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

Solution

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