# Geometry

 1. (12 p.) Let $$AXYZB$$ be a convex pentagon inscribed in a semicircle with diameter $$AB$$. Suppose $$AZ-AX=6$$, $$BX-BZ=9$$, $$AY=12$$, and $$BY=5$$. Find the greatest integer not exceeding the perimeter of quadrilateral $$OXYZ$$, where $$O$$ is the midpoint of $$AB$$.

 2. (48 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 $$m-n$$.

 3. (8 p.) Given a rhombus $$ABCD$$, the circumradii of the triangles $$ABD$$ and $$ACD$$ are 12.5 and 25. Find the area of $$ABCD$$.

 4. (17 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 $$a+b$$.

 5. (13 p.) Let $$ABC$$ be a rectangular triangle such that $$\angle C=90^o$$ and $$AC = 7$$, $$BC = 24$$. Let $$M$$ be the midpoint of $$AB$$ and $$D$$ point on the same side of $$AB$$ as $$C$$ such that $$DA = DB = 15$$. The area of the triangle $$CDM$$ can be expressed as $$\frac{p\sqrt q}r$$ for positive integers $$p$$, $$q$$, $$r$$ such that $$q$$ is not divisible by a perfect square and $$(p,r)=1$$. Find area $$p+q+r$$.

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