HMMT 二月 2014 · 冲刺赛 · 第 15 题

HMMT February 2014 — Guts Round — Problem 15

[ 8 ] Given a regular pentagon of area 1, a pivot line is a line not passing through any of the pentagon’s vertices such that there are 3 vertices of the pentagon on one side of the line and 2 on the other. A pivot point is a point inside the pentagon with only finitely many non-pivot lines passing through it. Find the area of the region of pivot points. 2 2

解析

[ 8 ] Given a regular pentagon of area 1, a pivot line is a line not passing through any of the pentagon’s vertices such that there are 3 vertices of the pentagon on one side of the line and 2 on the other. A pivot point is a point inside the pentagon with only finitely many non-pivot lines passing through it. Find the area of the region of pivot points. √ 1 Answer: (7 − 3 5) Let the pentagon be labeled ABCDE . First, no pivot point can be on 2 the same side of AC as vertex B . Any such point P has the infinite set of non-pivot lines within the hourglass shape formed by the acute angles between lines P A and P C . Similar logic can be applied to points on the same side of BD as C , and so on. The set of pivot points is thus a small pentagon with sides on AC, BD, CE, DA, EB . The side ratio of this small pentagon to the large pentagon is √ 3 − 5 ◦ 2 (2 cos(72 )) = , 2 so the area of the small pentagon is ( ) √ 2 √ 3 − 5 1 = (7 − 3 5) . 2 2 2 2