HMMT 十一月 2008 · 冲刺赛 · 第 27 题

HMMT November 2008 — Guts Round — Problem 27

[ 13 ] ABCDE is a regular pentagon inscribed in a circle of radius 1. What is the area of the set of points inside the circle that are farther from A than they are from any other vertex? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . st 1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUND

解析

[ 13 ] ABCDE is a regular pentagon inscribed in a circle of radius 1. What is the area of the set of points inside the circle that are farther from A than they are from any other vertex? π Answer: Draw the perpendicular bisectors of all the sides and diagonals of the pentagon with 5 one endpoint at A . These lines all intersect in the center of the circle, because they are the set of points equidistant from two points on the circle. Now, a given point is farther from A than from point X if it is on the X side of the perpendicular bisector of segment AX . So, we want to find the area of the set of all points which are separated from A by all of these perpendicular bisectors, which turns π o out to be a single 72 sector of the circle, which has area . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . st 1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 SATURDAY 2008 — GUTS ROUND