HMMT 十一月 2016 · 冲刺赛 · 第 6 题
HMMT November 2016 — Guts Round — Problem 6
题目详情
- [ 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#
英文原题
[ 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
英文解析
- [ 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 11
Then BQ = P A = · BC = BC . Then BQ : CQ = .
2 2 2 4 3