IMOmath

Geometry

1. (19 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 \( p+q \).

2. (19 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 \( AB \).

3. (29 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 \( b \).

4. (19 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. (11 p.)
A triangle \( ABC \) has sides 13, 14, 15. The triangle \( ABC \) is rotated about its centroid for an angle of \( 180^0 \) to form a triangle \( A^{\prime}B^{\prime}C^{\prime} \). Find the area of the union of the two triangles.





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