Question

一个凸 $18$ 边形的各角度数组成一个递增的等差数列，并且每个角度都是整数。求最小角的度数。

The degree measures of the angles of a convex $18$ -sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.

答案

答案：143

知识点：角度和等差数列奇偶性

难度评级：1920

解答：

一个 $18$ 边形的内角和为 $180 \cdot 16 = 2880$ 度。若最小角为 $a$ ，公差为 $d$ ，则 $18a + 153d = 2880$ ，即 $2a + 17d = 320$ 。因为 $a$ 和 $d$ 都是整数， $17d$ 必须为偶数，所以 $d$ 是偶数；又因为数列递增， $d \ge 2$ 。

凸性要求最大角 $a + 17d = \frac{320 + 17d}{2}$ 小于 $180$ ，所以 $17d \lt 40$ ，从而 $d \le 2$ 。因此 $d = 2$ ，且 $a = \frac{320 - 34}{2} = 143$ 。

The interior angles of an $18$ -gon sum to $180 \cdot 16 = 2880$ degrees. If the smallest angle is $a$ and the common difference is $d,$ then $18a + 153d = 2880,$ i.e. $2a + 17d = 320.$ Since $a$ and $d$ are integers, $17d$ must be even, so $d$ is even, and $d \ge 2$ because the sequence is increasing.

Convexity requires the largest angle $a + 17d = \frac{320 + 17d}{2}$ to be less than $180,$ so $17d \lt 40$ and $d \le 2.$ Thus $d = 2$ and $a = \frac{320 - 34}{2} = 143.$

2011 AIME II 第 3 题

答案：143

难度评级：1920

解答：

其他年份的第 3 题