HMMT 十一月 2016 · 冲刺赛 · 第 6 题

HMMT November 2016 — Guts Round — Problem 6

[ 6 ] Let ABCD be a rectangle, and let E and F be points on segment AB such that AE = EF = F B . If CE intersects the line AD at P , and P F intersects BC at Q , determine the ratio of BQ to CQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2016, November 12, 2016 — GUTS ROUND Organization Team Team ID#

解析

[ 6 ] Let ABCD be a rectangle, and let E and F be points on segment AB such that AE = EF = F B . If CE intersects the line AD at P , and P F intersects BC at Q , determine the ratio of BQ to CQ . Proposed by: Eshaan Nichani Answer: 1 / 3 Because 4 P AE ∼ 4 P DC and AE : DC = 1 : 3, we have that P A : P D = 1 : 3 = ⇒ P A : AB = P A : BC = 1 : 2. Also, by similar triangles 4 P AF ∼ 4 QBF , since AF : BF = 2 : 1, P A : BQ = 2 : 1. 1 1 1 1 1 Then BQ = P A = · BC = BC . Then BQ : CQ = . 2 2 2 4 3