Geometry

1. (9 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 \( p+q \).

2. (16 p.)
Given a rhombus \( ABCD \), the circumradii of the triangles \( ABD \) and \( ACD \) are 12.5 and 25. Find the area of \( ABCD \).

3. (11 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 \( ABCD \).

4. (34 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 \( p+q \).

5. (27 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 \( p+q \).




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