2009 AMC 10A Problem 19

Question

Circle

A

has radius

100.

Circle

B

has an integer radius

r \lt 100

and remains internally tangent to circle

A

as it rolls once around the circumference of circle

A.

The two circles have the same points of tangency at the beginning and end of circle

B

’s trip. How many possible values can

r

have?

4

8

9

50

90

Divide the circumference of circle

A

by the circumference of circle

B

to count the rolls

That number is

\dfrac{100}{r},

which must be an integer greater than

1

The circumferences are

200\pi

and

2\pi r,

so the initial point of tangency returns after

\dfrac{200\pi}{2\pi r} = \dfrac{100}{r}

rolls.

For this to be an integer greater than

1,

r

must be a divisor of

100

less than

100:

namely

1, 2, 4, 5, 10, 20, 25,

and

50.

That is

8

values.

Thus, the correct answer is B.

Problem 19 in other years