HMMT 二月 2021 · 冲刺赛 · 第 3 题

HMMT February 2021 — Guts Round — Problem 3

[8] A semicircle with radius 2021 has diameter AB and center O . Points C and D lie on the semicircle ◦ such that ∠ AOC < ∠ AOD = 90 . A circle of radius r is inscribed in the sector bounded by OA and OC and is tangent to the semicircle at E . If CD = CE , compute b r c .

解析

[8] A semicircle with radius 2021 has diameter AB and center O . Points C and D lie on the semicircle ◦ such that ∠ AOC < ∠ AOD = 90 . A circle of radius r is inscribed in the sector bounded by OA and OC and is tangent to the semicircle at E . If CD = CE , compute b r c . Proposed by: Hahn Lheem Answer: 673 Solution: We are given m ∠ EOC = m ∠ COD and ◦ m ∠ AOC + m ∠ COD = 2 m ∠ EOC + m ∠ COD = 90 . ◦ ◦ So m ∠ EOC = 30 and m ∠ AOC = 60 . Letting the radius of the semicircle be R , we have 1 ( R − r ) sin ∠ AOC = r ⇒ r = R, 3 so ⌊ ⌋ 2021 b r c = = 673 . 3