AMC10 2012 A

AMC10 2012 A · Q10

AMC10 2012 A · Q10. It mainly tests Sequences & recursion (algebra), Arithmetic sequences basics.

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?

Mary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。可能的最小扇形角的度量是多少度？

(A) 5 5

(B) 6 6

(D) 10 10

(E) 12 12

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Let $a$ be the initial term and $d$ the common difference for the arithmetic sequence. Then the sum of the degree measures of the central angles is $$a+(a+d)+\cdots+(a+11d)=12a+66d=360,$$ so $2a+11d=60$. Letting $d=4$ yields the smallest possible positive integer value for $a$, namely $a=8$.

答案（C）：设 $a$ 为等差数列的首项，$d$ 为公差。则这些圆心角度数的总和为 $$a+(a+d)+\cdots+(a+11d)=12a+66d=360,$$ 因此 $2a+11d=60$。取 $d=4$ 可使 $a$ 取得最小的正整数值，即 $a=8$。

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