# Geometry

 1. (7 p.) Let $$ABCD$$ be a convex quadrilateral such that $$AB\perp BC$$, $$AC\perp CD$$, $$AB=18$$, $$BC=21$$, $$CD=14$$. Find the perimeter of $$ABCD$$.

 2. (19 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 $$CD.$$

 3. (16 p.) Let $$ABC$$ be a rectangular triangle such that $$\angle C=90^o$$ and $$AC = 7$$, $$BC = 24$$. Let $$M$$ be the midpoint of $$AB$$ and $$D$$ point on the same side of $$AB$$ as $$C$$ such that $$DA = DB = 15$$. The area of the triangle $$CDM$$ can be expressed as $$\frac{p\sqrt q}r$$ for positive integers $$p$$, $$q$$, $$r$$ such that $$q$$ is not divisible by a perfect square and $$(p,r)=1$$. Find area $$p+q+r$$.

 4. (46 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 $$m+n$$.

 5. (9 p.) A triangle $$ABC$$ has sides 13, 14, 15. The triangle $$ABC$$ is rotated about its centroid for an angle of $$180^0$$ to form a triangle $$A^{\prime}B^{\prime}C^{\prime}$$. Find the area of the union of the two triangles.

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