HMMT 十一月 2021 · 冲刺赛 · 第 17 题

HMMT November 2021 — Guts Round — Problem 17

[10] Let ABCDEF GH be an equilateral octagon with ∠ A ∠ C ∠ E ∠ G and ∠ B ∠ D ∠ F = = = = = = m ∠ H . If the area of ABCDEF GH is three times the area of ACEG , then sin B can be written as , n where m, n are positive integers and gcd( m, n ) = 1. Find 100 m + n .

解析

[10] Let ABCDEF GH be an equilateral octagon with ∠ A ∠ C ∠ E ∠ G and ∠ B ∠ D ∠ F = = = = = = m ∠ H . If the area of ABCDEF GH is three times the area of ACEG , then sin B can be written as , n where m, n are positive integers and gcd( m, n ) = 1. Find 100 m + n . Proposed by: Daniel Zhu Answer: 405 Solution: Assume AC = 1. Note that from symmetry, it can be seen that all angles in ACEG must be equal. Further, by similar logic all sides must be equal which means that ACEG is a square. Additionally, as AB = BC, ABC is an isosceles triangle, which means the octagon consists of a unit square with four isosceles triangles of area 1 / 2 attached. Now, if the side length of the octagon is s, and ∠ B = 2 θ, then we obtain that 1 1 2 2 s sin(2 θ ) = = ⇒ 2 s cos( θ ) sin( θ ) = 1 . 2 2 1 Further, since the length AC is equal to 1 , this means that s sin( θ ) = . From this, we compute 2 2 2 s sin( θ ) cos( θ ) 1 2 s cos( θ ) = = = 2 . 1 s sin( θ ) 2 s sin( θ ) 1 1 2 √ √ So tan( θ ) = = . From this, sin( θ ) = and cos( θ ) = , which means sin( B ) = sin(2 θ ) = s cos( θ ) 2 5 5 1 2 4 √ √ 2 · · = . 5 5 5