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HMMT 十一月 2019 · 冲刺赛 · 第 29 题

HMMT November 2019 — Guts Round — Problem 29

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [15] In isosceles 4 ABC , AB = AC and P is a point on side BC . If ∠ BAP = 2 ∠ CAP , BP = 3, and CP = 1, compute AP .
解析
  1. [15] In isosceles 4 ABC , AB = AC and P is a point on side BC . If ∠ BAP = 2 ∠ CAP , BP = 3, and CP = 1, compute AP . Proposed by: Milan Haiman √ Answer: 2 √ √ 3 1 3 π Let ∠ CAP = α , By the Law of Sines, = which rearranges to cos α = ⇒ α = . This sin 2 α sin α 2 6 √ √ π 2 2 2 implies that ∠ BAC = . By the Pythagorean Theorem, 2 AB = ( 3 + 1) , so AB = 2 + 3. 2 √ √ √ √ ( 3+1)(2+ 3) 2 √ Applying Stewart’s Theorem, it follows that AP = − 3 ⇒ AP = 2. 3+1