IMOmath

Geometry

1. (18 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 \( b \).

2. (12 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 \( \overline{BC} \).

3. (6 p.)
Let \( K \) and \( M \) be the points on the sides \( AB \) and \( AC \), respectively, of an equilateral triangle \( ABC \) such that \( BK=10 \), \( MK=12 \), and \( MC=8 \). Then the side of the triangle \( ABC \) is equal to \( p+\sqrt q \) for some integers \( p \) and \( q \). Evaluate \( p+q \).

4. (12 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. (50 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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