HMMT 二月 2020 · 冲刺赛 · 第 6 题

HMMT February 2020 — Guts Round — Problem 6

[5] Two sides of a regular n -gon are extended to meet at a 28 angle. What is the smallest possible value for n ?

解析

[5] Two sides of a regular n -gon are extended to meet at a 28 angle. What is the smallest possible value for n ? Proposed by: James Lin Answer: 45 Solution: We note that if we inscribe the n -gon in a circle, then according to the inscribed angle 1 theorem, the angle between two sides is times some x − y , where x and y are integer multiples of the 2 1 360 arc measure of one side of the n -gon. Thus, the angle is equal to times an integer multiple of , so 2 n 1 360 · k · = 28 for some integer k . Simplifying gives 7 n = 45 k , and since all k are clearly attainable, 2 n the smallest possible value of n is 45.