IMOmath

Geometry

1. (30 p.)
If \( ABCD \) is a convex quadrilateral with \( AB=200 \), \( BC=153 \), \( BD=300 \), \( \angle BAC=\angle BDC<90^{\circ} \) and \( \angle ABD=\angle BCD \), determine \( CD. \)

2. (23 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. (9 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 \( V/\pi \).

4. (23 p.)
Let \( AXYZB \) be a convex pentagon inscribed in a semicircle with diameter \( AB \). Suppose \( AZ-AX=6 \), \( BX-BZ=9 \), \( AY=12 \), and \( BY=5 \). Find the greatest integer not exceeding the perimeter of quadrilateral \( OXYZ \), where \( O \) is the midpoint of \( AB \).

5. (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 \( ABCD \).





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