IMOmath

Geometry

1. (13 p.)
Let \( ABC \) be a triangle with sides 3, 4, 5 and \( DEFG \) a \( 6 \times 7 \) rectangle. A line divides \( \triangle ABC \) into a triangle \( T_1 \) and a trapezoid \( R_1 \). Another line divides the rectangle \( DEFG \) into a triangle \( T_2 \) and a trapezoid \( R_2 \), in such a way \( T_1\sim T_2 \) and \( R_1\sim R_2 \). The smallest possible value for the area of \( T_1 \) can be expressed as \( p/q \) for two relatively prime positive integers \( p \) and \( q \). Evaluate \( p+q \).

2. (10 p.)
Given a rhombus \( ABCD \), the circumradii of the triangles \( ABD \) and \( ACD \) are 12.5 and 25. Find the area of \( ABCD \).

3. (6 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. (22 p.)
Let \( K \) and \( L \) be the points on the sides \( AB \) and \( BC \) of an equilateral triangle \( ABC \) such that \( AK=5 \) and \( CL=2 \). If \( M \) is the point on \( AC \) such that \( \angle KML=60^o \), and if the area of the triangle \( KML \) is equal to \( 14\sqrt3 \) then the side of the triangle \( ABC \) can assume two values \( \frac{a\pm \sqrt b}c \) for some natural numbers \( a \), \( b \), and \( c \). If \( b \) is not divisible by a perfect square other than 1, find the value of \( b \).

5. (46 p.)
Let \( \triangle ABC \) have \( AB=6 \), \( BC=7 \), and \( CA=8 \), and denote by \( \omega \) its circumcircle. Let \( N \) be a point on \( \omega \) such that \( AN \) is a diameter of \( \omega \). Furthermore, let the tangent to \( \omega \) at \( A \) intersect \( BC \) at \( T \), and let the second intersection point of \( NT \) with \( \omega \) be \( X \). The length of \( \overline{AX} \) can be written in the form \( \tfrac m{\sqrt n} \) for positive integers \( m \) and \( n \), where \( n \) is not divisible by the square of any prime. Find \( m+n \).





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