IMOmath

Geometry

1. (7 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 \( ABCD \).

2. (19 p.)
If \( ABCD \) is a convex quadrilateral with \( AB=200 \), \( BC=153 \), \( BD=300 \), \( \angle BAC=\angle BDC<90^{\circ} \) and \( \angle ABD=\angle BCD \), determine \( CD. \)

3. (16 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. (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 \( m+n \).

5. (9 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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