HMMT 十一月 2024 · 冲刺赛 · 第 18 题

HMMT November 2024 — Guts Round — Problem 18

[10] Let ABCD be a rectangle whose vertices are labeled in counterclockwise order with AB = 32 and ′ ′ ′ ◦ AD = 60 . Rectangle AB C D is constructed by rotating ABCD counterclockwise about A by 60 . Given ′ ′ that lines BB and DD intersect at point X , compute CX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2024, November 09, 2024 — GUTS ROUND Organization Team Team ID#

解析

[10] Let ABCD be a rectangle whose vertices are labeled in counterclockwise order with AB = 32 and ′ ′ ′ ◦ AD = 60. Rectangle AB C D is constructed by rotating ABCD counterclockwise about A by 60 . ′ ′ Given that lines BB and DD intersect at point X , compute CX . Proposed by: David Wei Answer: 34 Solution: ′ D ′ C X A D ′ B B C The key claim is the following. ◦ Claim 1. ∠ BXD = 90 . ′ ′ ◦ ′ ◦ ′ ◦ Proof. We see that ∠ ABB = ∠ AD D = 60 and ∠ BAD = 90 + ∠ DAD = 150 , from which we get ′ ◦ ′ ′ ∠ BXD = ∠ BXD = 90 . Note that the fact that BB and DD are perpendicular is true regardless of how much we rotate the rectangle. Now since ∠ BAD = ∠ BXD , we establish that X lies on the circumcircle of ABCD , which has di- √ ◦ ′ ◦ 2 2 ameter 32 + 60 = 68. Moreover, we have ∠ CDX = 90 + ∠ ADD = 150 , so we discover that CX = 34 by applying the extended law of sines to △ CDX .