1994 AIME Problem 2

Question

A circle with diameter

\overline{PQ}

of length

10

is internally tangent at

P

to a circle of radius

20.

Square

ABCD

is constructed with

A

and

B

on the larger circle,

\overline{CD}

tangent at

Q

to the smaller circle, and the smaller circle outside

ABCD.

The length of

\overline{AB}

can be written in the form

m+\sqrt n,

where

m

and

n

are integers. Find

m+n.

Answer

Place the large circle at the origin and put

P

and

Q

on a diameter

If the square’s side is

s

, its chord side

AB

lies at distance

|10-s|

from the large circle’s center

Put the large circle at the origin with

P=(20,0)

and

Q=(10,0).

Let the square’s side be

s.

Because the smaller circle is outside the square,

\overline{CD}

lies on

x=10

and the parallel chord

\overline{AB}

lies on

x=10-s.

A chord of the radius-

20

circle then gives

s=2\sqrt{400-(10-s)^2}.

Squaring yields

s^2-16s-240=0,

so

s=8+4\sqrt{19}=8+\sqrt{304}.

Thus

m+n=8+304=312.

Problem 2 in other years