# Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 7"

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− | *[[Mock AIME 1 2006-2007/Problem 6 | Previous Problem]] | + | *[[Mock AIME 1 2006-2007 Problems/Problem 6 | Previous Problem]] |

− | *[[Mock AIME 1 2006-2007/Problem 8 | Next Problem]] | + | *[[Mock AIME 1 2006-2007 Problems/Problem 8 | Next Problem]] |

*[[Mock AIME 1 2006-2007]] | *[[Mock AIME 1 2006-2007]] |

## Revision as of 15:52, 3 April 2012

## Problem

Let have and . Point is such that and . Construct point on segment such that . and are extended to meet at . If where and are positive integers, find (note: denotes the area of ).

## Solution

We can immediately see that quadrilateral is cyclic, since . We then have, from Power of a Point, that . In other words, . is then 2, and is 1. We can now use Menelaus on line with respect to triangle :

This shows that .

Now let , for some real . Therefore , and . Similarly, and . The desired ratio is then

Therefore .