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

Answer: 312

Concepts:chord square (geometry)quadratic

Difficulty rating: 2170

Small Hint:

Place the large circle at the origin and put $P$ and $Q$ on a diameter

Big Hint:

If the square’s side is $s$ , its chord side $AB$ lies at distance $|10-s|$ from the large circle’s center

Solution:

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 $m+n=8+304=312.$

1994 AIME Problem 2

Answer: 312

Difficulty rating: 2170

Small Hint:

Big Hint:

Solution:

Problem 2 in Other Years