2023 WSMO Speed Round Problems/Problem 4

Problem

A right circular cone is inscribed in a right prism as shown. If the ratio of the volume of the cone to the volume of the prism is $\frac{m}{n}\pi,$ for relatively prime positive integers $m$ and $n,$ find $m+n.$ $[asy] import three; import graph3; defaultpen(linewidth(0.8)); size(200); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle); draw((0,0,0)--(0,0,1)); draw((1,0,0)--(1,0,1)); draw((1,1,0)--(1,1,1)); draw((0,1,0)--(0,1,1)); draw(Circle((0.5,0.5,0),0.5),dashed); draw((0.5,0.5,1)--(0.5,0,0),dashed); draw((0.5,0.5,1)--(0.5,1,0),dashed); draw((0.5,0.5,1)--(1,0.5,0),dashed); draw((0.5,0.5,1)--(0,0.5,0),dashed); [/asy]$

Solution

Let $s$ and $h$ denote the sidelength and height of the right prism, respectively. The ratio of the two volumes is equal to $\frac{\frac{1}{3}\cdot\left(\frac{s}{2}\right)^2\cdot h\pi}{s^2\cdot h} = \frac{\frac{\pi}{12}\cdot s^2\cdot h}{s^2\cdot h} = \frac{\pi}{12}\implies1+12 = \boxed{13}.$

~pinkpig

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2023 WSMO Speed Round Problems/Problem 4

Problem

Solution