Mock AIME II 2012 Problems/Problem 8


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 $m+n+k$.


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 $162+81+2=\boxed{245}$.