# Geometry

 1. (16 p.) Assume that all sides of the convex hexagon $$ABCDEF$$ are equal and the opposite sides are parallel. Assume further that $$\angle FAB = 120^o$$. The $$y$$-coordinates of $$A$$ and $$B$$ are 0 and 2 respectively, and the $$y$$-coordinates of the other vertices are 4, 6, 8, 10 in some order. The area of $$ABCDEF$$ can be written as $$a\sqrt b$$ for some integers $$a$$ and $$b$$ such that $$b$$ is not divisible by a perfect square other than 1. Find $$a+b$$.

 2. (12 p.) The right circular cone has height 4 and its base radius is 3. Its surface is painted black. The cone is cut into two parts by a plane parallel to the base, so that the volume of the top part (the small cone) divided by the volume of the bottom part equals $$k$$ and painted area of the top part divided by the painted are of the bottom part also equals $$k$$. If $$k$$ is of the form $$p/q$$ for two relatively prime numbers $$p$$ and $$q$$, calculate $$p+q$$.

 3. (11 p.) The angle $$\angle C$$ of the isosceles triangle $$ABC$$ ($$AC = BC$$) has measure of $$106^o$$. M is a point inside the triangle such that $$\angle MAC = 7^o$$ and $$\angle MCA = 23^o$$. The measure of the $$\angle CMB$$ in degrees can be written as a fraction $$p/q$$ for relatively prime integers $$p$$ and $$q$$. Calculate $$p+q$$.

 4. (11 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. (47 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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