PUMaC 2023 · 几何（B 组） · 第 2 题

PUMaC 2023 — Geometry (Division B) — Problem 2

The area of the largest square that can be inscribed in a regular hexagon with sidelength 1 √ can be expressed as a − b c where c is not divisible by the square of any prime. Find a + b + c .

解析

The area of the largest square that can be inscribed in a regular hexagon with sidelength 1 √ can be expressed as a − b c where c is not divisible by the square of any prime. Find a + b + c . Proposed by Adam Huang Answer: 21 Let our regular hexagon be ABCDEF with center O . It is easy to see that the largest square must be congruent to a square W XY Z centered at O , where W, X, Y, Z lie on sides AB, CD, DE, F A such that W X ∥ F B and XY ∥ BC . Let c = AW , b = W B , and d = W X . √ Clearly b + c = 1. By drawing an altitude from A in △ ZAW , we find d = c 3. By drawing √ b b altitudes from B, C in trapezoid BCXW , we find d = +1+ = b +1. Therefore c 3 = b +1, 2 2 √ √ √ so that c ( 3 + 1) = 2, and so c = 3 − 1. Hence d = 3 − 3, which yields an area of √ √ 2 (3 − 3) = 12 − 6 3 and an answer of 12 + 6 + 3 = 21.