Question

Mary 将一个圆分成 $12$ 个扇形。这些扇形的圆心角用度数表示时都是整数，并且构成一个等差数列。最小的扇形角可能是多少度？

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：等差数列最优化

难度评级：1240

解答：

设最小角为 $a$ ，公差为 $d \ge 0$ 。所有角的和为 $12a + 66d = 360$ ，所以 $2a + 11d = 60$ 。

要使 $a$ 小，就让 $d$ 尽量大。由于 $11d$ 必须为偶数， $d$ 为偶数，而 $d = 4$ 给出 $2a = 60 - 44 = 16$ ，所以 $a = 8$ 。更大的偶数 $d$ 会使 $a$ 非正。

因此，正确答案是 C。

Let $a$ be the smallest angle and $d \ge 0$ the common difference. The sum of the angles is $12a + 66d = 360,$ so $2a + 11d = 60.$

To make $a$ small, take $d$ large. Since $11d$ must be even, $d$ is even, and $d = 4$ gives $2a = 60 - 44 = 16,$ so $a = 8.$ A larger even $d$ makes $a$ non-positive.

Thus, the correct answer is C.

2012 AMC 12A 第 7 题

答案：C

难度评级：1240

解答：

其他年份的第 7 题