HMMT 二月 2021 · 冲刺赛 · 第 2 题

HMMT February 2021 — Guts Round — Problem 2

[8] Suppose there exists a convex n -gon such that each of its angle measures, in degrees, is an odd prime number. Compute the difference between the largest and smallest possible values of n .

解析

[8] Suppose there exists a convex n -gon such that each of its angle measures, in degrees, is an odd prime number. Compute the difference between the largest and smallest possible values of n . Proposed by: Andrew Gu Answer: 356 ◦ Solution: We can’t have n = 3 since the sum of the angles must be 180 but the sum of three odd numbers is odd. On the other hand, for n = 4 we can take a quadrilateral with angle measures ◦ ◦ ◦ ◦ 83 , 83 , 97 , 97 . The largest possible value of n is 360. For larger n we can’t even have all angles have integer measure, and 179 happens to be prime. So, the answer is 360 − 4 = 356.