IMOmath

Geometry

1. (9 p.)
Let \( X \) be a square of side length 2. Denote by \( S \) the set of all segments of length 2 with endpoints on adjacent sides of \( X \). The midpoints of the segments in \( S \) enclose a region with an area \( A \). Find \( [100A] \).

2. (15 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.

3. (6 p.)
Let \( T \) be a regular tetrahedron. Assume that \( T^{\prime} \) is the tetrahedron whose vertices are the midpoints of the faces of \( T \). The ratio of the volumes of \( T^{\prime} \) and \( T \) can be expressed as \( p/q \) where \( p \) and \( q \) are relatively prime integers. Determine \( p+q \).

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. (22 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 \( p+q \).





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