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Geometry
1.
(57 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 \( mn \).
2.
(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 \( ABCD \).
3.
(8 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 \( [100A] \).
4.
(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 \( p+q \).
5.
(21 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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