# Geometry

 1. (18 p.) Let $$K$$ and $$L$$ be the points on the sides $$AB$$ and $$BC$$ of an equilateral triangle $$ABC$$ such that $$AK=5$$ and $$CL=2$$. If $$M$$ is the point on $$AC$$ such that $$\angle KML=60^o$$, and if the area of the triangle $$KML$$ is equal to $$14\sqrt3$$ then the side of the triangle $$ABC$$ can assume two values $$\frac{a\pm \sqrt b}c$$ for some natural numbers $$a$$, $$b$$, and $$c$$. If $$b$$ is not divisible by a perfect square other than 1, find the value of $$b$$.

 2. (12 p.) Let $$A,B,C$$ be points in the plane such that $$AB=25$$, $$AC=29$$, and $$45^\circ< \angle BAC< 90^\circ$$. Semicircles with diameters $$\overline{AB}$$ and $$\overline{AC}$$ intersect at a point $$P$$ with $$AP=20$$. Find the length of line segment $$\overline{BC}$$.

 3. (6 p.) Let $$K$$ and $$M$$ be the points on the sides $$AB$$ and $$AC$$, respectively, of an equilateral triangle $$ABC$$ such that $$BK=10$$, $$MK=12$$, and $$MC=8$$. Then the side of the triangle $$ABC$$ is equal to $$p+\sqrt q$$ for some integers $$p$$ and $$q$$. Evaluate $$p+q$$.

 4. (12 p.) The area of the triangle $$ABC$$ is 70. The coordinates of $$B$$ and $$C$$ are $$(12,19)$$ and $$(23,20)$$, respectively, and the coordinates of $$A$$ are $$(p,q)$$. The line containing the median to side BC has slope -5. Find the largest possible value of p+q.

 5. (50 p.) Let $$\triangle ABC$$ be a triangle with $$AB=13$$, $$BC=14$$, and $$CA=15$$. Let $$O$$ denote its circumcenter and $$H$$ its orthocenter. The circumcircle of $$\triangle AOH$$ intersects $$AB$$ and $$AC$$ at $$D$$ and $$E$$ respectively. Suppose $$\tfrac{AD}{AE}=\tfrac mn$$ where $$m$$ and $$n$$ are positive relatively prime integers. Find $$m-n$$.

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