IMOmath

Geometry

1. (14 p.)
The right circular cone has height 4 and its base radius is 3. Its surface is painted black. The cone is cut into two parts by a plane parallel to the base, so that the volume of the top part (the small cone) divided by the volume of the bottom part equals \( k \) and painted area of the top part divided by the painted are of the bottom part also equals \( k \). If \( k \) is of the form \( p/q \) for two relatively prime numbers \( p \) and \( q \), calculate \( p+q \).

2. (20 p.)
Let \( BC \) be a chord of length 6 of a circle with center \( O \) and radius 5. Point \( A \) is on the circle, closer to \( B \) that to \( C \), such that there is a unique chord \( AD \) which is bisected by \( BC \). If \( \sin\angle AOB=\frac pq \) with \( q>0 \) and \( \gcd(p,q)=1 \), find \( p+q \).

3. (41 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 \( m+n \).

4. (13 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 \( \overline{BC} \).

5. (9 p.)
Given a rhombus \( ABCD \), the circumradii of the triangles \( ABD \) and \( ACD \) are 12.5 and 25. Find the area of \( ABCD \).





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