HMMT 十一月 2008 · 冲刺赛 · 第 35 题

英文解析

35. [ 25 ] Find max { Perimeter( T ) } for T a triangle contained in a regular septagon (7-sided figure) of unitedge length. Write your answer N to 2 places after the decimal. If the correct answer rounded to 2
    decimal places is A , you will receive 0 points if N < A and b max { 0 , 25 − 50 · ( N − A ) }c points otherwise.
    Answer: 5.85086 Let the septagon be A A . . . A .
    0 1 6
    If x is a point that can move along the x-axis, the distance from x to a fixed point p is a convexfunction in the x-coordinate. Therefore, the sum of the distances from x to two other points is convextoo, so if x is constrained to lie on a closed line segment, its maximum value is attained at an endpoint.
    Therefore, the triangle of maximal perimeter has vertices at the vertices of the pentagon. The trianglewith the largest such perimeter has almost evenly spaced vertices, so triangle A A A has the maximal
    0 2 4
    perimeter. It has area 5 . 85 . . . .