Difference between revisions of "2014 AIME II Problems/Problem 8"
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+ | Circle <math>C</math> with radius 2 has diameter <math>\overline{AB}</math>. Circle D is internally tangent to circle <math>C</math> at <math>A</math>. Circle <math>E</math> is internally tangent to circle <math>C</math>, externally tangent to circle <math>D</math>, and tangent to <math>\overline{AB}</math>. The radius of circle <math>D</math> is three times the radius of circle <math>E</math>, and can be written in the form <math>\sqrt{m}-n</math>, where <math>m</math> and <math>n</math> are positive integers. Find <math>m+n</math>. | ||
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==Solution 1== | ==Solution 1== | ||
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Revision as of 22:52, 8 April 2014
Problem
Circle with radius 2 has diameter
. Circle D is internally tangent to circle
at
. Circle
is internally tangent to circle
, externally tangent to circle
, and tangent to
. The radius of circle
is three times the radius of circle
, and can be written in the form
, where
and
are positive integers. Find
.
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
.