# Geometry

 1. (29 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. (20 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 $$p+q$$.

 3. (20 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 $$AB$$.

 4. (18 p.) Let $$ABC$$ be a triangle with sides 3, 4, 5 and $$DEFG$$ a $$6 \times 7$$ rectangle. A line divides $$\triangle ABC$$ into a triangle $$T_1$$ and a trapezoid $$R_1$$. Another line divides the rectangle $$DEFG$$ into a triangle $$T_2$$ and a trapezoid $$R_2$$, in such a way $$T_1\sim T_2$$ and $$R_1\sim R_2$$. The smallest possible value for the area of $$T_1$$ can be expressed as $$p/q$$ for two relatively prime positive integers $$p$$ and $$q$$. Evaluate $$p+q$$.

 5. (10 p.) Let $$ABCD$$ be a convex quadrilateral such that $$AB\perp BC$$, $$AC\perp CD$$, $$AB=18$$, $$BC=21$$, $$CD=14$$. Find the perimeter of $$ABCD$$.

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