Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 14"

 
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*[[Mock AIME 1 2006-2007 Problems/Problem 13 | Previous Problem]]
  
*[[Mock AIME 1 2006-2007/Problem 15 | Next Problem]]
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*[[Mock AIME 1 2006-2007 Problems/Problem 15 | Next Problem]]
  
 
*[[Mock AIME 1 2006-2007]]
 
*[[Mock AIME 1 2006-2007]]

Latest revision as of 14:50, 3 April 2012

Problem

Three points $A$, $B$, and $T$ are fixed such that $T$ lies on segment $AB$, closer to point $A$. Let $AT=m$ and $BT=n$ where $m$ and $n$ are positive integers. Construct circle $O$ with a variable radius that is tangent to $AB$ at $T$. Let $P$ be the point such that circle $O$ is the incircle of $\triangle APB$. Construct $M$ as the midpoint of $AB$. Let $f(m,n)$ denote the maximum value $\tan^{2}\angle AMP$ for fixed $m$ and $n$ where $n>m$. If $f(m,49)$ is an integer, find the sum of all possible values of $m$.

Solution

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