1993 AMC 12 Problem 8

Question

Let

C_1

and

C_2

be circles of radius

1

that are in the same plane and tangent to each other. How many circles of radius

3

are in this plane and tangent to both

C_1

and

C_2?

2

4

5

6

8

A radius-

3

circle can be internally or externally tangent to each unit circle

For its center, consider distance pairs

(4,4),

(2,2),

(4,2),

and

(2,4)

Let the centers of the tangent unit circles be

2

units apart. The new center must be

4

units from a unit-circle center for external tangency and

2

units away for internal tangency. The distance pair

(4,4)

gives two centers, as does

(2,2).

Each mixed pair

(4,2)

and

(2,4)

gives one center because the corresponding center-circles are internally tangent. Hence there are

2+2+1+1=6

circles. Thus the correct answer is D.

Problem 8 in other years