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

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


[[Mock AIME 1 2006-2007]]
==Solution==
{{solution}}
 
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*[[Mock AIME 1 2006-2007/Problem 13 | Previous Problem]]
 
*[[Mock AIME 1 2006-2007/Problem 15 | Next Problem]]
 
*[[Mock AIME 1 2006-2007]]

Revision as of 18:43, 22 August 2006

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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