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

We have $\begin{align*} f(x) &= x+2x^2+3x^3+4x^4+\ldots\text{and}\\ xf(x) &= x^2+2x^3+3x^4+\ldots.\\ \end{align*}$ So, $f(x)-xf(x) = x+x^2+x^3+\ldots = \frac{x}{1-x}\implies f(x) = \frac{x}{(1-x)^2}$ for $x$ in the range $(0,1)$ . For $x>1,$ $f(x)$ is strictly decreasing, meaning $f(x)$ is maximized at $x=6.$ Thus, our answer is $\frac{6}{(1-6)^2} = \frac{6}{25}\implies6+25 = \boxed{31}.$

~pinkpig

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2023 WSMO Speed Round Problems/Problem 5

Problem

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