2012 AMC 10A Problem 10

Question

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

Answer: C

Concepts:arithmetic sequence Diophantine Equation optimization

Difficulty rating: 1420

Small hint:

Let the arithmetic sequence be

a,a+d,\ldots,a+11d

Big hint:

Use

12a+66d=360

and maximize

d

while keeping

a

positive

Solution:

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.