# General Practice Test

 1. (22 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. (19 p.) Let $$ABCD$$ be a convex quadrilateral with $$AB = CD = 180$$. Assume further that the perimeter of $$ABCD$$ is 640, $$AD \neq BC$$, and $$\angle A = \angle C$$. Then $$\cos \angle A$$ can be represented as $$p/q$$ for relatively prime positive integers $$p$$ and $$q$$. Calculate $$p+q$$.

 3. (24 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 $$b$$.

 4. (14 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. (19 p.) Consider the set $$S\subseteq(0,1]^2$$ in the coordinate plane that consists of all points $$(x,y)$$ such that both $$[\log_2(1/x)]$$ and $$[\log_5(1/y)]$$ are even. The area of $$S$$ can be written in the form $$p/q$$ for two relatively prime integers $$p$$ and $$q$$. Evaluate $$p+q$$.

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