2025 AMC 12A Problem 24

Question

A circle of radius

r

is surrounded by

12

circles of radius

1,

externally tangent to the central circle and sequentially tangent to each other, as shown. Then

r

can be written as

\sqrt{a} + \sqrt{b} + c,

where

a,

b,

and

c

are integers. What is

a + b + c?

3

5

7

9

11

The

12

outer centers form a regular

12

-gon of circumradius

r + 1,

with adjacent centers

2

apart

So

2(r + 1)\sin 15^\circ = 2,

and

\dfrac{1}{\sin 15^\circ} = \sqrt{6} + \sqrt{2}

The centers of the

12

outer circles lie on a circle of radius

r + 1,

forming a regular

12

-gon. Adjacent centers are

2

apart (both circles have radius

1

), and the central angle between them is

30^\circ.

Thus

2(r + 1)\sin 15^\circ = 2,

so

r + 1 = \dfrac{1}{\sin 15^\circ}.

Since

\sin 15^\circ = \dfrac{\sqrt{6} - \sqrt{2}}{4},

r + 1 = \frac{4}{\sqrt{6} - \sqrt{2}} = \sqrt{6} + \sqrt{2}.

Then

r = \sqrt{6} + \sqrt{2} - 1,

so

a + b + c = 6 + 2 - 1 = 7.

Thus, the correct answer is C.

Problem 24 in other years