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HMMT 十一月 2009 · 冲刺赛 · 第 12 题

HMMT November 2009 — Guts Round — Problem 12

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 8 ] Let ω be a circle of radius 1 centered at O . Let B be a point on ω , and let l be the line tangent to ◦ ω at B . Let A be on l such that ∠ AOB = 60 . Let C be the foot of the perpendicular from B to OA . Find the length of line segment OC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nd 2 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUND
解析
  1. [ 8 ] Let ω be a circle of radius 1 centered at O . Let B be a point on ω , and let l be the line tangent to ◦ ω at B . Let A be on l such that ∠ AOB = 60 . Let C be the foot of the perpendicular from B to OA . Find the length of line segment OC . 1 ◦ 1 Answer: We have OC/OB = cos(60 ). Since OB = 1, OC = . 2 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nd 2 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUND