HMMT 二月 2024 · 冲刺赛 · 第 19 题

HMMT February 2024 — Guts Round — Problem 19

[11] Let A A . . . A be a regular nonadecagon. Lines A A and A A meet at X . Compute ∠ A XA . 1 2 19 1 5 3 4 7 5 √ 4 3 3 3

解析

[11] Let A A . . . A be a regular nonadecagon. Lines A A and A A meet at X . Compute ∠ A XA . 1 2 19 1 5 3 4 7 5 Proposed by: Nithid Anchaleenukoon ◦ 1170 13 π Answer: = 19 38 Solution: X A 5 A A 6 4 A A 7 3 A A 8 2 A A 9 1 Inscribing the nondecagon in a circle, note that 1 1 ̂ ̂ ̂ ∠ A XA = ( A A − A A ) = A A A = ∠ A A X. 3 5 1 3 4 5 5 3 4 5 3 2 2 Thus A X = A A = A A , so 5 5 3 5 7 1 1 ◦ ∠ A XA = 90 − ∠ XA A = ∠ A A A 7 5 5 7 1 5 7 2 2 ◦ 1 1 13 1170 ◦ ̂ = A A A = · · 360 = . 1 8 7 4 4 19 19 √ 4 3 3 3