HMMT 二月 2022 · 冲刺赛 · 第 11 题

HMMT February 2022 — Guts Round — Problem 11

[7] A regular dodecagon P P · · · P is inscribed in a unit circle with center O . Let X be the intersection 1 2 12 of P P and OP , and let Y be the intersection of P P and OP . Let A be the area of the region bounded 1 5 2 1 5 4 [ by XY , XP , Y P , and minor arc P P . Compute ⌊ 120 A ⌋ . 2 4 2 4

解析

[7] A regular dodecagon P P · · · P is inscribed in a unit circle with center O . Let X be the intersection 1 2 12 of P P and OP , and let Y be the intersection of P P and OP . Let A be the area of the region 1 5 2 1 5 4 [ bounded by XY , XP , Y P , and minor arc P P . Compute ⌊ 120 A ⌋ . 2 4 2 4 Proposed by: Gabriel Wu Answer: 45 ◦ Solution: The area of sector OP P is one sixth the area of the circle because its angle is 60 . 2 4 The desired area is just that of the sector subtracted by the area of equilateral triangle OXY . 1 Note that the altitude of this triangle is the distance from O to P P , which is . Thus, the side 1 5 2 √ √ √ 3 3 π 3 length of the triangle is , implying that the area is . Thus, we find that A = − . Thus, 3 12 6 12 √ 120 A = 20 π − 10 3 ≈ 62 . 8 − 17 . 3, which has floor 45.