# General Practice Test

 1. (36 p.) A sequence $$x_n$$ of real numbers satisfies $$x_0=0$$ and $$|x_{n}|=|x_{n-1}+1|$$ for $$n\geq 1$$. Find the minimal value of $$|x_1+x_2+\dots+ x_{2008}|$$.

 2. (2 p.) The square $$\begin{array}{|c|c|c|} \hline x&20&151 \\\hline 38 & & \\ \hline & & \\ \hline\end{array}$$ is magic, i.e. in each cell there is a number so that the sums of each row and column and of the two main diagonals are all equal. Find $$x$$.

 3. (26 p.) The right circular cone has height 4 and its base radius is 3. Its surface is painted black. The cone is cut into two parts by a plane parallel to the base, so that the volume of the top part (the small cone) divided by the volume of the bottom part equals $$k$$ and painted area of the top part divided by the painted are of the bottom part also equals $$k$$. If $$k$$ is of the form $$p/q$$ for two relatively prime numbers $$p$$ and $$q$$, calculate $$p+q$$.

 4. (21 p.) Find the minimum value of $$\frac{9x^2\sin^2x+4}{x\sin x}$$ for $$0< x< \pi$$.

 5. (12 p.) Let $$a$$, $$b$$, and $$c$$ be non-real roots of the polynimal $$x^3+x-1$$. Find $\frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.$

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