# Geometry

 1. (9 p.) Let $$X$$ be a square of side length 2. Denote by $$S$$ the set of all segments of length 2 with endpoints on adjacent sides of $$X$$. The midpoints of the segments in $$S$$ enclose a region with an area $$A$$. Find $$[100A]$$.

 2. (15 p.) The area of the triangle $$ABC$$ is 70. The coordinates of $$B$$ and $$C$$ are $$(12,19)$$ and $$(23,20)$$, respectively, and the coordinates of $$A$$ are $$(p,q)$$. The line containing the median to side BC has slope -5. Find the largest possible value of p+q.

 3. (6 p.) Let $$T$$ be a regular tetrahedron. Assume that $$T^{\prime}$$ is the tetrahedron whose vertices are the midpoints of the faces of $$T$$. The ratio of the volumes of $$T^{\prime}$$ and $$T$$ can be expressed as $$p/q$$ where $$p$$ and $$q$$ are relatively prime integers. Determine $$p+q$$.

 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. (22 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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