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HMMT 十一月 2025 · GEN 赛 · 第 1 题

HMMT November 2025 — GEN Round — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Let ABCD be a rectangle. Let X and Y be points on segments BC and AD , respectively, such that ◦ ∠ AXY = ∠ XY C = 90 . Given that AX : XY : Y C = 1 : 2 : 1 and AB = 1, compute BC .
解析
  1. Let ABCD be a rectangle. Let X and Y be points on segments BC and AD , respectively, such that ◦ ∠ AXY = ∠ XY C = 90 . Given that AX : XY : Y C = 1 : 2 : 1 and AB = 1, compute BC . Proposed by: Derek Liu, Marin Hristov Hristov Answer: 3 Solution: Y A D B C X Z Let Z be the foot from Y to BC . Then, △ ABX ∼ △ XZY . Since XY /AX = 2, we know XZ = 2 AB = 2 and BX = ZY / 2 = 1 / 2. Similarly, CZ = 1 / 2, so BC = 1 / 2 + 2 + 1 / 2 = 3 .