HMMT 十一月 2024 · 冲刺赛 · 第 23 题

HMMT November 2024 — Guts Round — Problem 23

[12] Consider a quarter-circle with center O , arc AB , and radius 2 . Draw a semicircle with diameter OA lying inside the quarter-circle. Points P and Q lie on the semicircle and segment OB , respectively, such that line P Q is tangent to the semicircle. As P and Q vary, compute the maximum possible area of triangle BQP . 2

解析

[12] Consider a quarter-circle with center O , arc AB , and radius 2. Draw a semicircle with diameter OA lying inside the quarter-circle. Points P and Q lie on the semicircle and segment OB , respectively, such that line P Q is tangent to the semicircle. As P and Q vary, compute the maximum possible area of triangle BQP . Proposed by: Daeho Jacob Lee 1 Answer: = 0 . 5 2 Solution: A P B O Q Note that we can bound the area of △ BQR by 1 [ BQP ] = BQ · QP sin ∠ BQP 2 1 ≤ BQ · QP 2 1 = BQ (2 − BQ ) 2 1 ≤ . 2 The maximum occurs when Q is the midpoint of segment OB . 2