Question

一个圆被分成 $12$ 个扇形。每个扇形的圆心角度数都是整数，并且这十二个角的度数构成一个等差数列。最小扇形角的最小可能度数是多少？

Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?

$5$

$6$

$8$

$10$

$12$

答案：C

知识点：等差数列丢番图方程最优化

难度评级：1420

解答：

设最小扇形角为 $a$ ，等差数列的公差为 $d$ 。

这些角的总和为。 $\dfrac{12}{2}(a + a + 11d) = 12a + 66d$

总和是 $360$ ，所以 $12a + 66d = 360$ $2a + 11d = 60.$

要使 $a$ 尽可能小，就要让 $d$ 尽可能大。若 $d = 5$ ，则 $a$ 不是整数；因此 $d = 4$ ，且 $a = 8$ 。

所以正确答案是 C。

Let $a$ be the smallest possible sector angle and $d$ be the difference in the arithmetic sequence.

Then we have that $\dfrac{12}{2}(a + a + 11d) = 12a + 66d$ is the sum of the arithmetic sequence.

We have that this sums to $360,$ so $12a + 66d = 360$ $2a + 11d = 60.$

We want to minimize $a,$ so we maximize $d.$ If $d = 5,$ then $a$ is not an integer, so $d = 4$ and $a = 8.$

Thus, C is the correct answer.

2012 AMC 10A 第 10 题

答案：C

难度评级：1420

解答：

其他年份的第 10 题