2009 AMC 10A Problem 19

Problem 19 of 25HarderGeometry Number Theory

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?

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Solution

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.