IMOmath

General Practice Test

1. (32 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 \( CD. \)

2. (22 p.)
Let \( ABC \) be a triangle with sides 3, 4, 5 and \( DEFG \) a \( 6 \times 7 \) rectangle. A line divides \( \triangle ABC \) into a triangle \( T_1 \) and a trapezoid \( R_1 \). Another line divides the rectangle \( DEFG \) into a triangle \( T_2 \) and a trapezoid \( R_2 \), in such a way \( T_1\sim T_2 \) and \( R_1\sim R_2 \). The smallest possible value for the area of \( T_1 \) can be expressed as \( p/q \) for two relatively prime positive integers \( p \) and \( q \). Evaluate \( p+q \).

3. (12 p.)
Find the least positive integer \( n \) such that when its leftmost digit is deleted, the resulting integer is equal to \( n/29 \).

4. (17 p.)
Given a rhombus \( ABCD \), the circumradii of the triangles \( ABD \) and \( ACD \) are 12.5 and 25. Find the area of \( ABCD \).

5. (15 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 \( [100A] \).





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