2023 WSMO Team Round Problems/Problem 5

Problem

A monkey is throwing darts at the dart board pictured below. The dart is equally likely to land anywhere on the board. Point values for the three regions are labeled and the radii the three circles are $1,2,3,$ respectively. If the expected value of points the monkey gets from 5 dart throws is $\frac{m\pi}{n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$ $[asy] size(6cm); fill(circle((0,0), 6), red); fill(circle((0,0), 4), green); fill(circle((0,0), 2), yellow); label("3",(0,5)); label("5",(0,3)); label("7",(0,0)); [/asy]$

Solution

The areas of the $3,5,7$ point regions are $5\pi,3\pi,\pi,$ respectively. So, the expected points of each throw is $\frac{5\pi}{9\pi}\cdot3+\frac{3\pi}{9\pi}\cdot5+\frac{\pi}{9\pi}\cdot7 = \frac{37}{9}.$ The expected points the monkeu gets from 5 dart throws is $\frac{37}{9} = \frac{185}{9}\implies185+9 = \boxed{194}.$

~pinkpig

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2023 WSMO Team Round Problems/Problem 5

Problem

Solution