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Problem 680

AMC 10/12, early questions

Combinatorics Difficulty 3.9 Find the answer HMMO · United States · 2020

Three distinct vertices of a regular $2020$ -gon are chosen uniformly at random. The probability that the triangle they form is isosceles can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Compute $100 a + b$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Solution:
The number of isosceles triangles that share vertices with the $2020$ -gon is $2020 \cdot 1009$ , since there are $2020$ ways to choose the apex of the triangle and then $1009$ ways to choose the other two vertices. (Since $2020$ is not divisible by $3$ , there are no equilateral triangles, so no triangle is overcounted.)

Therefore, the probability is
$\frac{2020 \cdot 1009}{\binom{2020}{3}} = \frac{2020 \cdot 2018 / 2}{2020 \cdot 2019 \cdot 2018 / 6} = \frac{3}{2019} = \frac{1}{673}$

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