2001 AMC 12 Problem 14

Question

Given the nine-sided regular polygon

A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9,

how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set

\{A_1, A_2, \ldots, A_9\}?

30

36

63

66

72

Each pair of vertices is a side of exactly two equilateral triangles

This counts

2\binom{9}{2}

triangles, but the three all-vertex triangles get counted three times each

Each of the

\binom{9}{2} = 36

pairs of vertices is a side of exactly two equilateral triangles, giving

72

triangles counted with multiplicity.

The triangles

A_1 A_4 A_7,

A_2 A_5 A_8,

and

A_3 A_6 A_9

have all three vertices in the set, so each is counted three times instead of once, an overcount of

2

apiece.

The number of distinct triangles is

72 - 3 \cdot 2 = 66.

Thus, the correct answer is D.

Problem 14 in other years