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\begin{document}
\begin{center}
\Large University of Mississippi \\ \normalsize High School Mathematics Contest -- Team Competition \\ March 24, 2006
\end{center}
\begin{enumerate}
\testitem Customers arrive randomly at a rate of $x$ per hour at a ticket counter, and according to a model of the ticket line when one attendant is working, the average length of the line as a function of $x$ is given by $f(x)=\frac{x^2}{400-20x}$ where $0\leq x<20$. It is decided that it average line length should not exceed 9 people. For what values of $x$ will a second attendant need to be employed?
%\testitem What is the remainder when $2004^{2004}$ is divided by 18.
%\testitem Consider the sequence $\{1, 4, 18, 96, \ldots \}$ defined by $a_n=n^2(n-1)!$ Find coefficients $p$ and $q$ in terms of $n$ such that $a_{n+1}=pa_n+q$ and prove that your relationship works for all $n>0$.
\testitem For three of the four following letter sequences, find the number of ways you can arrange the letters in the sequence so that no two adjacent letters are the same. For example, MPSISISISIP would be one suitable rearrangement of the letters of Mississippi. \\ \vspace{-0.2in}\begin{center}MISSISSIPPI \qquad FLORIDA \qquad ALABAMA \qquad ALASKA \end{center}
\testitem If you collect change in a change drawer over several months, what fraction of the coins would be expected to be dimes? Assume the probability of receiving any value of change between 0 and 99 cents is the same for each purchase, and assume that the clerk always makes change in the most efficient manner possible.
\testitem How many integer values may $\frac{a+1}{a}+\frac{a+1}{ab}$ take on, where $a$ and $b$ are integers?
\testitem The rectangles in a set $S$ may be arranged to form a square. There exists a subset of $S$ that consists of three squares, no two of which are congruent to each other. What is the smallest possible number of rectangles in $S$? Explain why there may be no fewer.
\testitem {\parpic[r]{\includegraphics{images/pizza}} A circular pizza is divided using six cuts as shown in the diagram, so that the two perpendicular diameters are each divided into four equal parts by parallel cuts. How many times larger is one of the the largest pieces than one of the smallest ones? Express your answer in terms of $\pi$.\picskip{0}}
%\testitem Given that $\sin^3\theta+\cos^3\theta=\sqrt{2}/2\cos\alpha(2-\cos2\alpha)$ is true for all $\theta$, find $\alpha$ in terms of $\theta$.
\testitem Four flattened colored cubes are shown below. Each of the cubes' faces has been colored red (R), blue (B), green (G), or yellow (Y). The cubes are stacked on top of each other in numerical order with cube \#1 on bottom. The goal is to find an orientation of each cube so that on each of the four visible sides of the stack all four colors appear. Find a solution, and for each side of the stack, list the colors from bottom to top. List the sides in clockwise order.
\begin{tabular}{c|c|cc} \cline{2-2}
&R& & \\ \hline
\multicolumn{1}{|c}{R}&\multicolumn{1}{|c}{Y} & \multicolumn{1}{|c}{G} & \multicolumn{1}{|c|}{B} \\ \hline
& R & & \\ \cline{2-2}
\end{tabular} \hspace{1cm}
\begin{tabular}{c|c|cc} \cline{2-2}
& G & & \\ \hline
\multicolumn{1}{|c}{R} & \multicolumn{1}{|c}{R} & \multicolumn{1}{|c}{B} & \multicolumn{1}{|c|}{Y} \\ \hline
& Y & & \\ \cline{2-2}
\end{tabular} \hspace{1cm}
\begin{tabular}{c|c|cc} \cline{2-2}
& G & & \\ \hline
\multicolumn{1}{|c}{B} & \multicolumn{1}{|c}{B} & \multicolumn{1}{|c}{R} & \multicolumn{1}{|c|}{Y} \\ \hline
& G & & \\ \cline{2-2}
\end{tabular} \hspace{1cm}
\begin{tabular}{c|c|cc} \cline{2-2}
& B & & \\ \hline
\multicolumn{1}{|c}{G} & \multicolumn{1}{|c}{Y} & \multicolumn{1}{|c}{R} & \multicolumn{1}{|c|}{B} \\ \hline
& Y & & \\ \cline{2-2}
\end{tabular}
\testitem Note: $n!$, read $n$ \textit{factorial}, is defined by $n!=n\cdot(n-1)\cdot(n-2)\cdot\cdots\cdot2\cdot1$ \\
(a) How many primes numbers are in the following sequence? \\ \begin{center}8!+2, 8!+3, 8!+4, 8!+5, 8!+6, 8!+7, 8!+8 \end{center}
(b) Show that no matter how large a positive integer $N$ is, there
exists a string of $N$ consecutive composite numbers.\\
\end{enumerate}
\end{document}