2014 AIME II Problems/Problem 8
Solution 1
Using the diagram above, let the radius of be
, and the radius of
be
. Then,
, and
, so the Pythagorean theorem in
gives
. Also,
, so
Noting that
, we can now use the Pythagorean theorem in
to get
Solving this quadratic is somewhat tedious, but the constant terms cancel, so the computation isn't terrible. Solving gives for a final answer of
.