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 1 2006-2007 Problems/Problem 14"
Line 7: Line 7:
 
----
 
----
  
−
*[[Mock AIME 1 2006-2007/Problem 13 | Previous Problem]]
+
*[[Mock AIME 1 2006-2007 Problems/Problem 13 | Previous Problem]]
  
−
*[[Mock AIME 1 2006-2007/Problem 15 | Next Problem]]
+
*[[Mock AIME 1 2006-2007 Problems/Problem 15 | Next Problem]]
  
 
*[[Mock AIME 1 2006-2007]]
 
*[[Mock AIME 1 2006-2007]]

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

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Mock_AIME_1_2006-2007_Problems/Problem_14&oldid=45988"