HMMT 二月 2023 · 几何 · 第 4 题

HMMT February 2023 — Geometry — Problem 4

Let ABCD be a square, and let M be the midpoint of side BC . Points P and Q lie on segment AM AQ ◦ such that ∠ BP D = ∠ BQD = 135 . Given that AP < AQ , compute . AP

解析

Let ABCD be a square, and let M be the midpoint of side BC . Points P and Q lie on segment AM AQ ◦ such that ∠ BP D = ∠ BQD = 135 . Given that AP < AQ , compute . AP Proposed by: Ankit Bisain, Luke Robitaille √ Answer: 5 ◦ ◦ ∠ BAD Solution: Notice that ∠ BP D = 135 = 180 − and P lying on the opposite side of BD as C 2 means that P lies on the circle with center C through B and D . Similarly, Q lies on the circle with center A through B and D . Let the side length of the square be 1. We have AB = AQ = AD , so AQ = 1. To compute AP , let E be the reflection of D across C . We have that E lies both on AM and the circle centered at C through B and D . Since AB is tangent to this circle, 2 AB = AP · AE √ √ 2 1 √ by power of a point. Thus, 1 = AP · 5 = ⇒ AP = . Hence, the answer is 5. 5