Art of Problem Solving
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

Mock AIME II 2012 Problems/Problem 8: Difference between revisions

Created page with "==Problem== Let <math>A</math> be a point outside circle <math>\Omega</math> with center <math>O</math> and radius <math>9</math> such that the tangents from <math>A</math> to <..."
 
(No difference)

Latest revision as of 02:11, 5 April 2012

Problem

Let $A$ be a point outside circle $\Omega$ with center $O$ and radius $9$ such that the tangents from $A$ to $\Omega$, $AB$ and $AC$, form $\angle BAO=15^{\circ}$. Let $AO$ first intersect the circle at $D$, and extend the parallel to $AB$ from $D$ to meet the circle at $E$. The length $EC^2=m+k\sqrt{n}$, where $m$,$n$, and $k$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n+k$.

Solution

Let $OB$ intersect $ED$ at $J$. Note that $\angle DJO=90$, thus $\angle EOJ=\angle BOD=\angle COD=75$. Then we have $\angle EOC=360-3(75)=135$. We can use the law of cosines to find $EC$, $EC^2=9^2+9^2-2(\cos 135)\cdot 9\cdot 9=2(81)+81\sqrt{2}=162+81\sqrt{2}$. Thus, we have $162+81+2=\boxed{245}$.

Retrieved from "https://wiki.aopstest.com/wiki/index.php?title=Mock_AIME_II_2012_Problems/Problem_8&oldid=46147"