HMMT 十一月 2024 · THM 赛 · 第 6 题

HMMT November 2024 — THM Round — Problem 6

Let F ELDSP AR be a regular octagon, and let I be a point in its interior such that ∠ F IL = ∠ LID = ∠ DIS = ∠ SIA . Compute ∠ IAR in degrees.

解析

Let F ELDSP AR be a regular octagon, and let I be a point in its interior such that ∠ F IL = ∠ LID = ∠ DIS = ∠ SIA . Compute ∠ IAR in degrees. Proposed by: Derek Liu ◦ 165 ◦ Answer: 82 . 5 = 2 Solution: D S L I P E A F R Observe that I lies on line DR due to symmetry, so ID ∥ F L . Thus ∠ F LI = ∠ LID = ∠ F IL , im- plying that triangle F IL is isosceles with F I = F L . Similarly, AI = AS . Since F LSA is a square, 1 ◦ F I = F L = AS = AI = F A . Therefore, F IA is equilateral, so ∠ AIR = ∠ F IA = 30 and 2 ◦ ◦ ◦ 1 ◦ ∠ IAR = 180 − 30 − · 135 = 82 . 5 . 2