HMMT 二月 2006 · TEAM1 赛 · 第 10 题

HMMT February 2006 — TEAM1 Round — Problem 10

[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

[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. 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 interior angles: each is a positive integer between 1 and 179 inclusive. The sum of the exterior angles is exactly 360. However, the sum of 1 through 27 is 27 · 28 / 2 = 378 , which is too 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]