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Difference between revisions of "2023 AMC 10A Problems/Problem 25"

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== Solution 2 ==
 
== Solution 2 ==
 
We can actually see that the probability that <math>d(Q, R) > d(R, S)</math> is the exact same as <math>d(Q, R) < d(R, S)</math> because QR and RS have no difference. (In other words, we can just swap Q and S, meaning that can be called the same.) Therefore, we want to find the probability that <math>d(Q, R) = d(R, S)</math>.
 
We can actually see that the probability that <math>d(Q, R) > d(R, S)</math> is the exact same as <math>d(Q, R) < d(R, S)</math> because QR and RS have no difference. (In other words, we can just swap Q and S, meaning that can be called the same.) Therefore, we want to find the probability that <math>d(Q, R) = d(R, S)</math>.
 +
 
WLOG, we can rotate the icosahedron so that R is the top of the icosahedron. Then we can divide this into 2 cases:
 
WLOG, we can rotate the icosahedron so that R is the top of the icosahedron. Then we can divide this into 2 cases:
  
 
1. They are on the second layer
 
1. They are on the second layer
 +
 
There are 5 ways to put one point, and 4 ways to put the other point such that <math>d(Q, R) = d(R, S) = 1</math>. <math>5 \cdot 4 = 20</math> ways to put them on the second layer.
 
There are 5 ways to put one point, and 4 ways to put the other point such that <math>d(Q, R) = d(R, S) = 1</math>. <math>5 \cdot 4 = 20</math> ways to put them on the second layer.
  
 
2. They are on the third layer
 
2. They are on the third layer
 +
 
There are 5 ways to put one point, and 4 ways to put the other point such that <math>d(Q, R) = d(R, S) = 2</math>. <math>5 \cdot 4 = 20</math> ways to put them on the third layer.
 
There are 5 ways to put one point, and 4 ways to put the other point such that <math>d(Q, R) = d(R, S) = 2</math>. <math>5 \cdot 4 = 20</math> ways to put them on the third layer.
  
 
The total number of ways to choose P and S are <math>11 \cdot 10 = 110</math> (because there are 12 vertices), so the probability that <math>d(Q, R) = d(R, S)</math> is <math>\frac{20+20}{110} = \frac{4}{11}</math>.
 
The total number of ways to choose P and S are <math>11 \cdot 10 = 110</math> (because there are 12 vertices), so the probability that <math>d(Q, R) = d(R, S)</math> is <math>\frac{20+20}{110} = \frac{4}{11}</math>.
 +
 
Therefore, the probability that <math>d(Q, R) > d(R, S)</math> is <math>\frac{1 - \frac{4}{11}}{2} = \boxed{\textbf{(A) }\frac{7}{22}}</math>
 
Therefore, the probability that <math>d(Q, R) > d(R, S)</math> is <math>\frac{1 - \frac{4}{11}}{2} = \boxed{\textbf{(A) }\frac{7}{22}}</math>
  
 
~Ethanzhang1001
 
~Ethanzhang1001

Revision as of 20:53, 9 November 2023

If A and B are vertices of a polyhedron, define the distance d(A, B) to be the minimum number of edges of the polyhedron one must traverse in order to connect A and B. For example, $\overline{AB}$ is an edge of the polyhedron, then d(A, B) = 1, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then d(A, B) = 2. Let Q, R, and S be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that $d(Q, R) > d(R, S)$?

$\textbf{(A) }\frac{7}{22}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{3}{8}\qquad\textbf{(D) }\frac{5}{12}\qquad\textbf{(E) }\frac{1}{2}$


Video Solution 1 by OmegaLearn

https://youtu.be/Wc6PFNq5PAM

Solution 1

We can imagine the icosahedron as having 3 layers. 1 vertex at the top, 5 vertices below connected to the top vertex, 5 vertices below that which are 2 edges away from the top vertex, and one vertex at the bottom that is 3 edges away. WLOG because the icosahedron is symmetric around all vertices, we can say that R is the vertex at the top. So now, we just need to find the probability that S is on a layer closer to the top than Q. We can do casework on the layer S is on to get \[\frac{5}{11} \cdot \frac{6}{10} + \frac{5}{11} \cdot \frac{1}{10} = \frac{35}{110} = \frac{7}{22}\] So the answer is $\boxed{A}$. -awesomeparrot

Solution 2

We can actually see that the probability that $d(Q, R) > d(R, S)$ is the exact same as $d(Q, R) < d(R, S)$ because QR and RS have no difference. (In other words, we can just swap Q and S, meaning that can be called the same.) Therefore, we want to find the probability that $d(Q, R) = d(R, S)$.

WLOG, we can rotate the icosahedron so that R is the top of the icosahedron. Then we can divide this into 2 cases:

1. They are on the second layer

There are 5 ways to put one point, and 4 ways to put the other point such that $d(Q, R) = d(R, S) = 1$. $5 \cdot 4 = 20$ ways to put them on the second layer.

2. They are on the third layer

There are 5 ways to put one point, and 4 ways to put the other point such that $d(Q, R) = d(R, S) = 2$. $5 \cdot 4 = 20$ ways to put them on the third layer.

The total number of ways to choose P and S are $11 \cdot 10 = 110$ (because there are 12 vertices), so the probability that $d(Q, R) = d(R, S)$ is $\frac{20+20}{110} = \frac{4}{11}$.

Therefore, the probability that $d(Q, R) > d(R, S)$ is $\frac{1 - \frac{4}{11}}{2} = \boxed{\textbf{(A) }\frac{7}{22}}$

~Ethanzhang1001

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