HMMT 十一月 2019 · GEN 赛 · 第 4 题

HMMT November 2019 — GEN Round — Problem 4

In 4 ABC , AB = 2019, BC = 2020, and CA = 2021. Yannick draws three regular n -gons in the plane of 4 ABC so that each n -gon shares a side with a distinct side of 4 ABC and no two of the n -gons overlap. What is the maximum possible value of n ?

解析

In 4 ABC , AB = 2019, BC = 2020, and CA = 2021. Yannick draws three regular n -gons in the plane of 4 ABC so that each n -gon shares a side with a distinct side of 4 ABC and no two of the n -gons overlap. What is the maximum possible value of n ? Proposed by: Carl Schildkraut Answer: 11 If any n -gon is drawn on the same side of one side of 4 ABC as 4 ABC itself, it will necessarily overlap with another triangle whenever n > 3. Thus either n = 3 or the triangles are all outside ABC . The ◦ n − 2 interior angle of a regular n -gon is 180 · , so we require n n − 2 ◦ ◦ 360 · + max( ∠ A, ∠ B, ∠ C ) < 360 . n ◦ As 4 ABC is almost equilateral (in fact the largest angle is less than 60 . 1 ), each angle is approximately ◦ 60 , so we require n − 2 360 · < 300 = ⇒ n < 12 . n Hence the answer is n = 11.