HMMT 二月 2014 · 冲刺赛 · 第 15 题

英文解析

15. [ 8 ] Given a regular pentagon of area 1, a pivot line is a line not passing through any of the pentagon’svertices such that there are 3 vertices of the pentagon on one side of the line and 2 on the other. Apivot 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.
    √
    Answer: (7 − 3 5) Let the pentagon be labeled ABCDE . First, no pivot point can be on 1
    the same side of AC as vertex B . Any such point P has the infinite set of non-pivot lines within the 2
    hourglass shape formed by the acute angles between lines P A and P C . Similar logic can be appliedto points on the same side of BD as C , and so on. The set of pivot points is thus a small pentagonwith 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 )) = ,
    so the area of the small pentagon is 2
    ( )
    √ 2
    √
    3 − 5 1 = (7 − 3 5) .
    2 2
    2 2