IMOmath

Geometry

1. (7 p.)
Let \( \alpha \) be the angle between vectors \( \vec a \) and \( \vec b \) with \( |\vec a|=2 \) and \( |\vec b|=3 \), given that the vectors \( \vec m=2\vec a-\vec b \) and \( \vec n=\vec a+5\vec b \) are orthogonal. If \( \cos\alpha=\frac pq \) with \( q>0 \) and \( \gcd(p,q)=1 \), compute \( p+q \).

2. (35 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 \( a+b \).

3. (28 p.)
Let \( ABC \) be a rectangular triangle such that \( \angle C=90^o \) and \( AC = 7 \), \( BC = 24 \). Let \( M \) be the midpoint of \( AB \) and \( D \) point on the same side of \( AB \) as \( C \) such that \( DA = DB = 15 \). The area of the triangle \( CDM \) can be expressed as \( \frac{p\sqrt q}r \) for positive integers \( p \), \( q \), \( r \) such that \( q \) is not divisible by a perfect square and \( (p,r)=1 \). Find area \( p+q+r \).

4. (12 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 \( ABCD \).

5. (15 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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