View source for 2023 WSMO Speed Round Problems/Problem 5

==Problem==

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

==Solution==
We have 
<cmath>\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*}</cmath>
So, <cmath>f(x)-xf(x) = x+x^2+x^3+\ldots = \frac{x}{1-x}\implies f(x) = \frac{x}{(1-x)^2}</cmath> for <math>x</math> in the range <math>(0,1)</math>. For <math>x>1,</math> <math>f(x)</math> is strictly decreasing, meaning <math>f(x)</math> is maximized at <math>x=6.</math> Thus, our answer is <cmath>\frac{6}{(1-6)^2} = \frac{6}{25}\implies6+25 = \boxed{31}.</cmath>

~pinkpig