HMMT 十一月 2008 · 冲刺赛 · 第 35 题

HMMT November 2008 — Guts Round — Problem 35

[ 25 ] Find max { Perimeter( T ) } for T a triangle contained in a regular heptagon (7-sided figure) of unit edge 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 · ( A − N ) }c points otherwise.

解析

[ 25 ] Find max { Perimeter( T ) } for T a triangle contained in a regular septagon (7-sided figure) of unit edge 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 convex function in the x-coordinate. Therefore, the sum of the distances from x to two other points is convex too, 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 triangle with 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 . . . .