# 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 $$0< x< \pi$$.

2005-2019 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax