# Geometry

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

 2. (37 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 $$m+n$$.

 3. (16 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 $$a+b$$.

 4. (13 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 $$p+q$$.

 5. (18 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 $$b$$.

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