HMMT 二月 2023 · 冲刺赛 · 第 24 题

HMMT February 2023 — Guts Round — Problem 24

[18] Let AXBY be a cyclic quadrilateral, and let line AB and line XY intersect at C . Suppose AX · AY = 2 6, BX · BY = 5, and CX · CY = 4. Compute AB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2023, February 18, 2023 — GUTS ROUND Organization Team Team ID# ∑ n − 1

解析

[18] Let AXBY be a cyclic quadrilateral, and let line AB and line XY intersect at C . Suppose 2 AX · AY = 6, BX · BY = 5, and CX · CY = 4. Compute AB . Proposed by: Luke Robitaille 242 Answer: 15 Solution: Observe that AC CY 4 ACX ∼ 4 Y CB = ⇒ = AX BY AC CX 4 ACY ∼ 4 XCB = ⇒ = AY BX Mulitplying these two equations together, we get that ( CX · CY )( AX · AY ) 24 2 AC = = . BX · BY 5 Analogously, we obtain that ( CX · CY )( BX · BY ) 10 2 BC = = . AX · AY 3 Hence, we have √ √ √ 24 10 11 30 AB = AC + BC = + = , 5 3 15 implying the answer. ∑ n − 1