HMMT 二月 2006 · TEAM1 赛 · 第 10 题
HMMT February 2006 — TEAM1 Round — Problem 10
题目详情
英文原题
- [15] Suppose we have an n -gon such that each interior angle, measured in degrees, is a positive integer.
°
Suppose further that all angles are less than 180 , and that all angles are different sizes. What is the maximum possible value of n ? Prove your answer.
What do the following problems have in common? [170]
解析
英文解析
- [15] Suppose we have an n -gon such that each interior angle, measured in degrees, is a
°
positive integer. Suppose further that all angles are less than 180 , and that all anglesare different sizes. What is the maximum possible value of n ? Prove your answer.
Answer: 26
Solution: Let’s work with the exterior angles (each is 180 minus the interior angle).
Then the conditions on the exterior angles are identical to the conditions on the interiorangles: each is a positive integer between 1 and 179 inclusive. The sum of the exteriorangles is exactly 360. However, the sum of 1 through 27 is 27 · 28 / 2 = 378 , which istoo large. We can get 26 using angles of 1 through 25 (sum 325) and an angle of 35.
The previous problem shows that this is actually possible.
What do the following problems have in common? [170]