IMOmath

Geometry

1. (16 p.)
Assume that all sides of the convex hexagon \( ABCDEF \) are equal and the opposite sides are parallel. Assume further that \( \angle FAB = 120^o \). The \( y \)-coordinates of \( A \) and \( B \) are 0 and 2 respectively, and the \( y \)-coordinates of the other vertices are 4, 6, 8, 10 in some order. The area of \( ABCDEF \) can be written as \( a\sqrt b \) for some integers \( a \) and \( b \) such that \( b \) is not divisible by a perfect square other than 1. Find \( a+b \).

2. (12 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 \( p+q \).

3. (11 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 \( p+q \).

4. (11 p.)
The area of the triangle \( ABC \) is 70. The coordinates of \( B \) and \( C \) are \( (12,19) \) and \( (23,20) \), respectively, and the coordinates of \( A \) are \( (p,q) \). The line containing the median to side BC has slope -5. Find the largest possible value of p+q.

5. (47 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 \( m-n \).





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