# General Practice Test

 1. (25 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. (21 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 $$p+q$$.

 3. (23 p.) A bug moves around a triangle wire. At each vertex it has 1/2 chance of moving towards each of the other two vertices. The probability that after crawling along 10 edges it reaches its starting point can be expressed as $$p/q$$ for positive relatively prime integers $$p$$ and $$q$$. Determine $$p+q$$.

 4. (7 p.) Let $$T$$ be a regular tetrahedron. Assume that $$T^{\prime}$$ is the tetrahedron whose vertices are the midpoints of the faces of $$T$$. The ratio of the volumes of $$T^{\prime}$$ and $$T$$ can be expressed as $$p/q$$ where $$p$$ and $$q$$ are relatively prime integers. Determine $$p+q$$.

 5. (23 p.) If $$ABCD$$ is a convex quadrilateral with $$AB=200$$, $$BC=153$$, $$BD=300$$, $$\angle BAC=\angle BDC<90^{\circ}$$ and $$\angle ABD=\angle BCD$$, determine $$CD.$$

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