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HMMT 十一月 2023 · GEN 赛 · 第 2 题

HMMT November 2023 — GEN Round — Problem 2

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. A regular n -gon P P . . . P satisfies ∠ P P P = 178 . Compute n. 1 2 n 1 7 8
解析
  1. A regular n -gon P P . . . P satisfies ∠ P P P = 178 . Compute n. 1 2 n 1 7 8 Proposed by: Derek Liu Answer: 630 Solution: Let O be the center of the n -gon. Then ◦ 360 ◦ ◦ ∠ P OP = 2(180 − ∠ P P P ) = 4 = , 1 8 1 7 8 90 ̂ which means the arc P P that spans 7 sides of the n -gon also spans 1 / 90 of its circumcircle. Thus 1 8 n = 7 · 90 = 630 .