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 "2013 Mock AIME I Problems/Problem 1"

(Created page with "== Problem 1 == Two circles <math>C_1</math> and <math>C_2</math>, each of unit radius, have centers <math>A_1</math> and <math>A_2</math> such that <math>A_1A_2=6</math>. Le...")
(No difference)

Revision as of 00:05, 5 March 2017

Problem 1

Two circles $C_1$ and $C_2$, each of unit radius, have centers $A_1$ and $A_2$ such that $A_1A_2=6$. Let $P$ be the midpoint of $A_1A_2$ and let $C_3$ be a circle externally tangent to both $C_1$ and $C_2$. $C_1$ and $C_3$ have a common tangent that passes through $P$. If this tangent is also a common tangent to $C_2$ and $C_1$, find the radius of circle $C_3$.

Solution

Let $A_3$ be the center of circle $C_3$ and $Q$ be the point of tangency between $C_3$ and $C_2$. Note that triangles $PQA_2$ and $A_3PA_2$ are similar, so $\frac{A_3A_2}{PA_2}=\frac{PA_2}{A_2Q}=3$ and $A_3A_2=r+1=9$. Thus the radius of $C_3$ is $\boxed{8}$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2013_Mock_AIME_I_Problems/Problem_1&oldid=84407"