Competition problem — Combinatorics

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Combinatorics Difficulty 5.0 AIME, harder Find the answer

A3. Below you see a regular nonagon with all its diagonals.

How many isosceles triangles are there whose three different vertices are also vertices of the nonagon?

(A triangle is isosceles if two or three of its sides have the same length.)
A) 27
B) 30
C) 33
D) 36
E) 39

Your answer

Multiple choice: answer with the letter of the option you want.

Solution

A3. B) 30

First, we count the number of equilateral triangles: there are 3 of them. Next, we count the isosceles triangles that are not equilateral. For such a triangle, there are 9 possibilities for the 'top'. Then there are 3 possibilities for the 'base'. This gives $9 \times 3=27$ possibilities. In total, there are thus $3+27=30$ isosceles triangles.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.