# Geometry

 1. (6 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. (13 p.) Let $$AXYZB$$ be a convex pentagon inscribed in a semicircle with diameter $$AB$$. Suppose $$AZ-AX=6$$, $$BX-BZ=9$$, $$AY=12$$, and $$BY=5$$. Find the greatest integer not exceeding the perimeter of quadrilateral $$OXYZ$$, where $$O$$ is the midpoint of $$AB$$.

 3. (4 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 $$p+q$$.

 4. (54 p.) Let $$\triangle ABC$$ be a triangle with $$AB=13$$, $$BC=14$$, and $$CA=15$$. Let $$O$$ denote its circumcenter and $$H$$ its orthocenter. The circumcircle of $$\triangle AOH$$ intersects $$AB$$ and $$AC$$ at $$D$$ and $$E$$ respectively. Suppose $$\tfrac{AD}{AE}=\tfrac mn$$ where $$m$$ and $$n$$ are positive relatively prime integers. Find $$m-n$$.

 5. (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 $$p+q$$.

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