Difference between revisions of "2015 AIME II Problems/Problem 15"
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==Problem== | ==Problem== | ||
− | Circles <math>\mathcal{P}</math> and <math>\mathcal{Q}</math> have radii <math>1</math> and <math>4</math>, respectively, and are externally tangent at point <math>A</math>. Point <math>B</math> is on <math>\mathcal{P}</math> and point <math>C</math> is on <math>\mathcal{Q}</math> | + | Circles <math>\mathcal{P}</math> and <math>\mathcal{Q}</math> have radii <math>1</math> and <math>4</math>, respectively, and are externally tangent at point <math>A</math>. Point <math>B</math> is on <math>\mathcal{P}</math> and point <math>C</math> is on <math>\mathcal{Q}</math> so that line <math>BC</math> is a common external tangent of the two circles. A line <math>\ell</math> through <math>A</math> intersects <math>\mathcal{P}</math> again at <math>D</math> and intersects <math>\mathcal{Q}</math> again at <math>E</math>. Points <math>B</math> and <math>C</math> lie on the same side of <math>\ell</math>, and the areas of <math>\triangle DBA</math> and <math>\triangle ACE</math> are equal. This common area is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. |
==Solution== | ==Solution== |
Revision as of 10:33, 27 March 2015
Problem
Circles and
have radii
and
, respectively, and are externally tangent at point
. Point
is on
and point
is on
so that line
is a common external tangent of the two circles. A line
through
intersects
again at
and intersects
again at
. Points
and
lie on the same side of
, and the areas of
and
are equal. This common area is
, where
and
are relatively prime positive integers. Find
.