Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 12"

m (defined O)
m
Line 1: Line 1:
 
== 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> <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>
+
In [[quadrilateral]] <math>ABCD,</math> <math>m \angle DAC= m\angle DBC </math> and <math>\frac{[ADB]}{[ABC]}=\frac12.</math> <math>O</math> is defined to be the intersection of the diagonals of <math>ABCD</math>. If <math>AD=4,</math> <math>BC=6</math>, <math>BO=1,</math> and the [[area]] of <math>ABCD</math> is <math>\frac{a\sqrt{b}}{c},</math> where <math>a,b,c</math> are [[relatively prime]] [[positive integer]]s, find <math>a+b+c.</math>
  
  
−
Note*: <math>\displaystyle[ABC]</math> and <math>\displaystyle[ADB]</math> refer to the areas of [[triangle]]s <math>\displaystyle ABC</math> and <math>\displaystyle ADB.</math>
+
Note*: <math>[ABC]</math> and <math>[ADB]</math> refer to the areas of [[triangle]]s <math>ABC</math> and <math>ADB.</math>
  
 
==Solution==
 
==Solution==
 
{{solution}}
 
{{solution}}
  
−
----
+
==See also==
−
 
 
 
*[[Mock AIME 2 2006-2007/Problem 11 | Previous Problem]]
 
*[[Mock AIME 2 2006-2007/Problem 11 | Previous Problem]]
−
 
 
*[[Mock AIME 2 2006-2007/Problem 13 | Next Problem]]
 
*[[Mock AIME 2 2006-2007/Problem 13 | Next Problem]]
−
 
 
*[[Mock AIME 2 2006-2007]]
 
*[[Mock AIME 2 2006-2007]]
  

Revision as of 09:11, 16 August 2008

Problem

In quadrilateral $ABCD,$ $m \angle DAC= m\angle DBC$ and $\frac{[ADB]}{[ABC]}=\frac12.$ $O$ is defined to be the intersection of the diagonals of $ABCD$. If $AD=4,$ $BC=6$, $BO=1,$ and the area of $ABCD$ is $\frac{a\sqrt{b}}{c},$ where $a,b,c$ are relatively prime positive integers, find $a+b+c.$


Note*: $[ABC]$ and $[ADB]$ refer to the areas of triangles $ABC$ and $ADB.$

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it.

See also


Problem Source

AoPS users 4everwise and Altheman collaborated to create this problem.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Mock_AIME_2_2006-2007_Problems/Problem_12&oldid=27442"