IMOmath

Geometry

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

2. (32 p.)
The angle \( \angle C \) of the isosceles triangle \( ABC \) (\( AC = BC \)) has measure of \( 106^o \). M is a point inside the triangle such that \( \angle MAC = 7^o \) and \( \angle MCA = 23^o \). The measure of the \( \angle CMB \) in degrees can be written as a fraction \( p/q \) for relatively prime integers \( p \) and \( q \). Calculate \( p+q \).

3. (32 p.)
Let \( A,B,C \) be points in the plane such that \( AB=25 \), \( AC=29 \), and \( 45^\circ< \angle BAC< 90^\circ \). Semicircles with diameters \( \overline{AB} \) and \( \overline{AC} \) intersect at a point \( P \) with \( AP=20 \). Find the length of line segment \( \overline{BC} \).

4. (12 p.)
A right circular cylinder has a diameter 12. Two plane cut the cylinder, the first perpendicular to the axis and the second at a \( 45^o \) angle to the first, so that the line of intersection of the two planes touches the cylinder at a single point. The two cuts remove a wedge from the cylinder. If \( V \) is the volume of the wedge calculate \( V/\pi \).

5. (12 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 \( p+q \).





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