2023 AMC 12A Problem 21

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

Small hint:

Fix

R.

The other

11

vertices split into

5

at distance

1,

5

at distance

2,

and

1

at distance

3

Big hint:

By symmetry

P(d(Q,R)\gt d(R,S))

=\dfrac{1-P(\text{equal})}{2}

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

Small hint:

Big hint:

Solution:

Problem 21 in other years