HMMT 十一月 2020 · 冲刺赛 · 第 19 题

HMMT November 2020 — Guts Round — Problem 19

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

解析

[11] Three distinct vertices of a regular 2020-gon are chosen uniformly at random. The probability that a the triangle they form is isosceles can be expressed as , where a and b are relatively prime positive b integers. Compute 100 a + b . Proposed by: Hahn Lheem Answer: 773 Solution: The number of isosceles triangles that share vertices with the 2020-gon is 2020 · 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 2020 · 1009 2020 · 2018 / 2 3 1 ( ) = = = . 2020 2020 · 2019 · 2018 / 6 2019 673 3