# Geometry

 1. (14 p.) Given a rhombus $$ABCD$$, the circumradii of the triangles $$ABD$$ and $$ACD$$ are 12.5 and 25. Find the area of $$ABCD$$.

 2. (31 p.) Let $$BC$$ be a chord of length 6 of a circle with center $$O$$ and radius 5. Point $$A$$ is on the circle, closer to $$B$$ that to $$C$$, such that there is a unique chord $$AD$$ which is bisected by $$BC$$. If $$\sin\angle AOB=\frac pq$$ with $$q>0$$ and $$\gcd(p,q)=1$$, find $$p+q$$.

 3. (23 p.) The right circular cone has height 4 and its base radius is 3. Its surface is painted black. The cone is cut into two parts by a plane parallel to the base, so that the volume of the top part (the small cone) divided by the volume of the bottom part equals $$k$$ and painted area of the top part divided by the painted are of the bottom part also equals $$k$$. If $$k$$ is of the form $$p/q$$ for two relatively prime numbers $$p$$ and $$q$$, calculate $$p+q$$.

 4. (8 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. (21 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 $$p+q$$.

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