# Geometry

 1. (14 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 $$p+q$$.

 2. (20 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. (41 p.) Let $$\triangle ABC$$ have $$AB=6$$, $$BC=7$$, and $$CA=8$$, and denote by $$\omega$$ its circumcircle. Let $$N$$ be a point on $$\omega$$ such that $$AN$$ is a diameter of $$\omega$$. Furthermore, let the tangent to $$\omega$$ at $$A$$ intersect $$BC$$ at $$T$$, and let the second intersection point of $$NT$$ with $$\omega$$ be $$X$$. The length of $$\overline{AX}$$ can be written in the form $$\tfrac m{\sqrt n}$$ for positive integers $$m$$ and $$n$$, where $$n$$ is not divisible by the square of any prime. Find $$m+n$$.

 4. (13 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 $$\overline{BC}$$.

 5. (9 p.) Given a rhombus $$ABCD$$, the circumradii of the triangles $$ABD$$ and $$ACD$$ are 12.5 and 25. Find the area of $$ABCD$$.

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