2012 AMC 12A Problem 7

Below is the professionally curated solution for Problem 7 of the 2012 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2012 AMC 12A solutions, or check the answer key.

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Concepts:arithmetic sequence optimization

Difficulty rating: 1240

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?

$5$

$6$

$8$

$10$

$12$

Solution:

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.

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