# Geometry

 1. (9 p.) Let $$\alpha$$ be the angle between vectors $$\vec a$$ and $$\vec b$$ with $$|\vec a|=2$$ and $$|\vec b|=3$$, given that the vectors $$\vec m=2\vec a-\vec b$$ and $$\vec n=\vec a+5\vec b$$ are orthogonal. If $$\cos\alpha=\frac pq$$ with $$q>0$$ and $$\gcd(p,q)=1$$, compute $$p+q$$.

 2. (32 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. (32 p.) Let $$A,B,C$$ be points in the plane such that $$AB=25$$, $$AC=29$$, and $$45^\circ< \angle BAC< 90^\circ$$. Semicircles with diameters $$\overline{AB}$$ and $$\overline{AC}$$ intersect at a point $$P$$ with $$AP=20$$. Find the length of line segment $$\overline{BC}$$.

 4. (12 p.) A right circular cylinder has a diameter 12. Two plane cut the cylinder, the first perpendicular to the axis and the second at a $$45^o$$ angle to the first, so that the line of intersection of the two planes touches the cylinder at a single point. The two cuts remove a wedge from the cylinder. If $$V$$ is the volume of the wedge calculate $$V/\pi$$.

 5. (12 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 $$p+q$$.

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