# General Practice Test

 1. (17 p.) Find the minimum value of $$\frac{9x^2\sin^2x+4}{x\sin x}$$ for $$0< x< \pi$$.

 2. (25 p.) A bug moves around a triangle wire. At each vertex it has 1/2 chance of moving towards each of the other two vertices. The probability that after crawling along 10 edges it reaches its starting point can be expressed as $$p/q$$ for positive relatively prime integers $$p$$ and $$q$$. Determine $$p+q$$.

 3. (21 p.) Let $$ABC$$ be a rectangular triangle such that $$\angle C=90^o$$ and $$AC = 7$$, $$BC = 24$$. Let $$M$$ be the midpoint of $$AB$$ and $$D$$ point on the same side of $$AB$$ as $$C$$ such that $$DA = DB = 15$$. The area of the triangle $$CDM$$ can be expressed as $$\frac{p\sqrt q}r$$ for positive integers $$p$$, $$q$$, $$r$$ such that $$q$$ is not divisible by a perfect square and $$(p,r)=1$$. Find area $$p+q+r$$.

 4. (28 p.) It is given that $$181^2$$ can be written as the difference of the cubes of two consecutive positive integers. Find the sum of these two integers.

 5. (7 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 $$p+q$$.

2005-2017 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax