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HMMT 十一月 2022 · 冲刺赛 · 第 4 题

HMMT November 2022 — Guts Round — Problem 4

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [6] Let ABCD be a square of side length 2. Let points X, Y, and Z be constructed inside ABCD such that ABX , BCY , and CDZ are equilateral triangles. Let point W be outside ABCD such that triangle √ DAW is equilateral. Let the area of quadrilateral W XY Z be a + b , where a and b are integers. Find a + b.
解析
  1. [6] Let ABCD be a square of side length 2. Let points X, Y, and Z be constructed inside ABCD such that ABX , BCY , and CDZ are equilateral triangles. Let point W be outside ABCD such that √ triangle DAW is equilateral. Let the area of quadrilateral W XY Z be a + b , where a and b are integers. Find a + b. Proposed by: Kevin Min Answer: 10 √ Solution: W XY Z is a kite with diagonals XZ and W Y , which have lengths 2 3 − 2 and 2, so the √ √ area is 2 3 − 2 = 12 − 2.