AMC12 2012 A

AMC12 2012 A · Q7

AMC12 2012 A · Q7. It mainly tests Arithmetic sequences basics, Circle theorems.

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?

玛丽将一个圆分成12个扇形。这些扇形的圆心角（以度为单位）都是整数，并且形成一个等差数列。最小的扇形角度数可能是多少？

(A) 5 5

(B) 6 6

(D) 10 10

(E) 12 12

Answer

Correct choice: (C)

正确答案：（C）

Solution

Let $a_1$ be the first term of the arithmetic progression and $a_{12}$ be the last term of the arithmetic progression. From the formula of the sum of an arithmetic progression (or arithmetic series), we have $12*\frac{a_1+a_{12}}{2}=360$, which leads us to $a_1 + a_{12} = 60$. $a_{12}$, the largest term of the progression, can also be expressed as $a_1+11d$, where $d$ is the common difference. Since each angle measure must be an integer, $d$ must also be an integer. We can isolate $d$ by subtracting $a_1$ from $a_{12}$ like so: $a_{12}-a_1=a_1+11d-a_1=11d$. Since $d$ is an integer, the difference between the first and last terms, $11d$, must be divisible by $11.$ Since the total difference must be less than $60$, we can start checking multiples of $11$ less than $60$ for the total difference between $a_1$ and $a_{12}$. We start with the largest multiple, because the maximum difference will result in the minimum value of the first term. If the difference is $55$, $a_1=\frac{60-55}{2}=2.5$, which is not an integer, nor is it one of the five options given. If the difference is $44$, $a_1=\frac{60-44}{2}$, or $\boxed{\textbf{(C)}\ 8}$

设等差数列的首项为$a_1$，末项为$a_{12}$。由等差数列（等差级数）求和公式可得 $12*\frac{a_1+a_{12}}{2}=360$，从而 $a_1 + a_{12} = 60$。等差数列的最大项 $a_{12}$ 也可表示为 $a_1+11d$，其中 $d$ 为公差。由于每个角度都必须是整数，$d$ 也必须是整数。将 $a_1$ 从 $a_{12}$ 中减去可得：$a_{12}-a_1=a_1+11d-a_1=11d$。因为 $d$ 是整数，首末两项之差 $11d$ 必须能被 $11$ 整除。由于总差必须小于 $60$，我们检查小于 $60$ 的 $11$ 的倍数作为 $a_1$ 与 $a_{12}$ 的差。我们从最大的倍数开始，因为差越大首项越小。若差为 $55$，则 $a_1=\frac{60-55}{2}=2.5$，不是整数，也不在给出的五个选项中。若差为 $44$，则 $a_1=\frac{60-44}{2}$，即 $\boxed{\textbf{(C)}\ 8}$。

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