HMMT 十一月 2023 · 冲刺赛 · 第 11 题

HMMT November 2023 — Guts Round — Problem 11

[8] Let ABCD and W XY Z be two squares that share the same center such that W X ∥ AB and W X < AB . Lines CX and AB intersect at P , and lines CZ and AD intersect at Q . If points P , W , and Q are collinear, compute the ratio AB/W X .

解析

[8] Let ABCD and W XY Z be two squares that share the same center such that W X ∥ AB and W X < AB . Lines CX and AB intersect at P , and lines CZ and AD intersect at Q . If points P , W , and Q are collinear, compute the ratio AB/W X . Proposed by: Edward Yu √ Answer: 2 + 1 Solution: P T A B W X Q Y Z D C Without loss of generality, let AB = 1. Let x = W X . Then, since BP W X is a parallelogram, we have 1 − x 1 − x 3 x − 1 BP = x . Moreover, if T = XY ∩ AB , then we have BT = , so P T = x − = . Then, from 2 2 2 △ P XT ∼ △ P BC , we have 3 x − 1 P T P B x 2 = = ⇒ = 1 − x XT BC 1 2 = ⇒ 3 x − 1 = x (1 − x ) √ = ⇒ x = ± 2 − 1 . √ √ 1 √ Selecting only positive solution gives x = 2 − 1. Thus, the answer is = 2 + 1. 2 − 1