2001 AMC 12 Problem 14

Below is the professionally curated solution for Problem 14 of the 2001 AMC 12, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2001 AMC 12 solutions, or check the answer key.

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Concepts:counting shapes in figures regular polygon double counting

Difficulty rating: 1710

14.

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

$30$

$36$

$63$

$66$

$72$

Solution:

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.

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