Difference between revisions of "2023 WSMO Team Round Problems/Problem 5"
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| + | The areas of the <math>3,5,7</math> point regions are <math>5\pi,3\pi,\pi,</math> respectively. So, the expected points of each throw is <cmath>\frac{5\pi}{9\pi}\cdot3+\frac{3\pi}{9\pi}\cdot5+\frac{\pi}{9\pi}\cdot7 = \frac{37}{9}.</cmath> The expected points the monkeu gets from 5 dart throws is <cmath>\frac{37}{9} = \frac{185}{9}\implies185+9 = \boxed{194}.</cmath> | ||
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| + | ~pinkpig | ||
Latest revision as of 13:58, 13 September 2025
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 
respectively. If the expected value of points the monkey gets from 5 dart throws is 
for relatively prime positive integers 
and 
find 
![[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]](http://latex.artofproblemsolving.com/5/d/0/5d0f4813246775d7c34ea18d51b83596d9fd8557.png)
Solution
The areas of the 
point regions are 
respectively. So, the expected points of each throw is ![\[\frac{5\pi}{9\pi}\cdot3+\frac{3\pi}{9\pi}\cdot5+\frac{\pi}{9\pi}\cdot7 = \frac{37}{9}.\]](http://latex.artofproblemsolving.com/3/8/5/385c07da60e07c2575f7abd524dfeba02553f38b.png)
The expected points the monkeu gets from 5 dart throws is ![\[\frac{37}{9} = \frac{185}{9}\implies185+9 = \boxed{194}.\]](http://latex.artofproblemsolving.com/7/2/a/72a053c1419d24d94c6e172ba4a7d8d7642c340a.png)
~pinkpig
