Mock AIME II 2012 Problems/Problem 8

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

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Mock AIME II 2012 Problems/Problem 8

Problem

Solution