Question

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, if $\overline{AB}$ 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 $d(Q,R)\gt d(R,S)?$

$\dfrac{7}{22}$

$\dfrac{1}{3}$

$\dfrac{3}{8}$

$\dfrac{5}{12}$

$\dfrac{1}{2}$

Answer: A

Concepts:graph theory basic probability symmetry

Difficulty rating: 2170

Solution:

Fix $R.$ Among the other $11$ vertices of the icosahedron, $5$ are at distance $1,$ $5$ are at distance $2,$ and $1$ (the antipode) is at distance $3.$

Choosing ordered distinct $Q,S,$ the probability that $d(Q,R)=d(R,S)$ is $\dfrac{5\cdot 4+5\cdot 4}{11\cdot 10}=\dfrac{40}{110}=\dfrac{4}{11}.$

By the symmetry between $Q$ and $S,$ $\begin{gathered} P(d(Q,R)\gt d(R,S))\\ {}=\dfrac{1-\tfrac{4}{11}}{2}\\ {}=\dfrac{7}{22}. \end{gathered}$

Thus, the correct answer is A.

2023 AMC 12A Problem 21

Answer: A

Difficulty rating: 2170

Solution:

Problem 21 in Other Years