Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 12"
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== Problem == | == Problem == | ||
− | In [[quadrilateral]] <math>\displaystyle ABCD,</math> <math>\displaystyle m \angle DAC= m\angle DBC </math> and <math>\displaystyle \frac{[ADB]}{[ABC]}=\frac12.</math> If <math>\displaystyle AD=4,</math> <math>\displaystyle BC=6</math>, <math>\displaystyle BO=1,</math> and the [[area]] of <math>\displaystyle ABCD</math> is <math>\displaystyle \frac{a\sqrt{b}}{c},</math> where <math>\displaystyle a,b,c</math> are [[relatively prime]] [[positive integer]]s, find <math>\displaystyle a+b+c.</math> | + | In [[quadrilateral]] <math>\displaystyle ABCD,</math> <math>\displaystyle m \angle DAC= m\angle DBC </math> and <math>\displaystyle \frac{[ADB]}{[ABC]}=\frac12.</math> <math>O</math> is defined to be the intersection of the diagonals of <math>ABCD</math>. If <math>\displaystyle AD=4,</math> <math>\displaystyle BC=6</math>, <math>\displaystyle BO=1,</math> and the [[area]] of <math>\displaystyle ABCD</math> is <math>\displaystyle \frac{a\sqrt{b}}{c},</math> where <math>\displaystyle a,b,c</math> are [[relatively prime]] [[positive integer]]s, find <math>\displaystyle a+b+c.</math> |
Revision as of 22:09, 16 February 2007
Problem
In quadrilateral
and
is defined to be the intersection of the diagonals of
. If
,
and the area of
is
where
are relatively prime positive integers, find
Note*: and
refer to the areas of triangles
and
Solution
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Problem Source
AoPS users 4everwise and Altheman collaborated to create this problem.