IMOmath

Geometry

1. (14 p.)
Given a rhombus \( ABCD \), the circumradii of the triangles \( ABD \) and \( ACD \) are 12.5 and 25. Find the area of \( ABCD \).

2. (31 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 \( p+q \).

3. (23 p.)
The right circular cone has height 4 and its base radius is 3. Its surface is painted black. The cone is cut into two parts by a plane parallel to the base, so that the volume of the top part (the small cone) divided by the volume of the bottom part equals \( k \) and painted area of the top part divided by the painted are of the bottom part also equals \( k \). If \( k \) is of the form \( p/q \) for two relatively prime numbers \( p \) and \( q \), calculate \( p+q \).

4. (8 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. (21 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 \( p+q \).





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