IMOmath

General Practice Test

1. (26 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. (8 p.)
How many pairs of integers \( (x,y) \) are there such that \( x^2-y^2=2400^2 \)?

3. (21 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 \( ABCD \).

4. (4 p.)
Let \( n \) be the largest positive integer for which there exists a positive integer \( k \) such that \[ k\cdot n! = \frac{(((3!)!)!}{3!}.\] Determine \( n \).

5. (39 p.)
Find the minimum value of \( \frac{9x^2\sin^2x+4}{x\sin x} \) for \( 0< x< \pi \).





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