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 10"
m
Line 1: Line 1:
−
10. In <math>\triangle ABC</math>, <math>AB</math>, <math>BC</math>, and <math>CA</math> have lengths <math>3</math>, <math>4</math>, and <math>5</math>, respectively. Let the incircle, circle <math>I</math>, of <math>\triangle ABC</math> touch <math>AB</math>, <math>BC</math>, and <math>CA</math> at <math>C'</math>, <math>A'</math>, and <math>B'</math>, respectively. Construct three circles, <math>A''</math>, <math>B''</math>, and <math>C''</math>, externally tangent to the other two and circles <math>A''</math>, <math>B''</math>, and <math>C''</math> are internally tangent to the circle <math>I</math> at <math>A'</math>, <math>B'</math>, and <math>C'</math>, respectively. Let circles <math>A''</math>, <math>B''</math>, <math>C''</math>, and <math>I</math> have radii <math>a</math>, <math>b</math>, <math>c</math>, and <math>r</math>, respectively. If <math>\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are positive integers, find <math>m+n</math>.
+
==Problem==
 +
In <math>\triangle ABC</math>, <math>AB</math>, <math>BC</math>, and <math>CA</math> have lengths <math>3</math>, <math>4</math>, and <math>5</math>, respectively. Let the incircle, circle <math>I</math>, of <math>\triangle ABC</math> touch <math>AB</math>, <math>BC</math>, and <math>CA</math> at <math>C'</math>, <math>A'</math>, and <math>B'</math>, respectively. Construct three circles, <math>A''</math>, <math>B''</math>, and <math>C''</math>, externally tangent to the other two and circles <math>A''</math>, <math>B''</math>, and <math>C''</math> are internally tangent to the circle <math>I</math> at <math>A'</math>, <math>B'</math>, and <math>C'</math>, respectively. Let circles <math>A''</math>, <math>B''</math>, <math>C''</math>, and <math>I</math> have radii <math>a</math>, <math>b</math>, <math>c</math>, and <math>r</math>, respectively. If <math>\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n}</math> where <math>m</math> and <math>n</math> are positive integers, find <math>m+n</math>.
  
−
[[Mock AIME 1 2006-2007]]
+
==Solution==
 +
{{solution}}
 +
 
 +
----
 +
 
 +
*[[Mock AIME 1 2006-2007/Problem 9 | Previous Problem]]
 +
 
 +
*[[Mock AIME 1 2006-2007/Problem 11 | Next Problem]]
 +
 
 +
*[[Mock AIME 1 2006-2007]]

Revision as of 18:40, 22 August 2006

Problem

In $\triangle ABC$, $AB$, $BC$, and $CA$ have lengths $3$, $4$, and $5$, respectively. Let the incircle, circle $I$, of $\triangle ABC$ touch $AB$, $BC$, and $CA$ at $C'$, $A'$, and $B'$, respectively. Construct three circles, $A''$, $B''$, and $C''$, externally tangent to the other two and circles $A''$, $B''$, and $C''$ are internally tangent to the circle $I$ at $A'$, $B'$, and $C'$, respectively. Let circles $A''$, $B''$, $C''$, and $I$ have radii $a$, $b$, $c$, and $r$, respectively. If $\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n}$ where $m$ and $n$ are positive integers, find $m+n$.

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_10&oldid=9782"