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\begin{center}
\Large \textbf{University of Mississippi} \\ \vspace{1mm} \normalsize 2\raisebox{3.6pt}{\scriptsize{nd}} Annual High School Mathematics Contest \\ Team Competition \\ October 21, 2006
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\testitem For real numbers $r$, let $\lfloor r \rfloor$ be the greatest integer that is less than or equal to $r$. Solve the inequality \[\lfloor x \rfloor +\lfloor x+3 \rfloor \leq 17\]
\testitem Prove that there exist finitely many sets of three consecutive primes for which the first two add up to the third. (Primes are considered consecutive if all the integers between them are composite).
\testitem Rationalize the denominator and simplify: \[\frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{5}}\]
\testitem What is the maximum number of bishops that may be placed on a $5\times5$ chess board without any bishop being in a square threatened by any other bishop? (A bishop threatens any square that lies on the same diagonal as the bishop).
\testitem If both $a$ and $b$ are strictly between 0 and 1, prove that $a+b-ab$ must be strictly between 0 and 1.
\testitem A certain pattern is used to generate a sequence of numbers, the first few of which are shown below. Prove that no matter how far this sequence is continued, no 4 will ever appear as a digit.
\[1,
11,
21,
1211,
111221,
312211,
13112221,
1113213211 \]
{\parpic[r]{\includegraphics[width=4cm]{images/dominoes.eps}} \testitem Dominoes of dimension $2''\times1''$ are used to cover a $1''\times 8''$ strip of $1''\times1''$ squares. Dominoes may be placed either along the strip so as to cover two of the squares, or they may be placed so that one square inch of the domino lies above or below the strip. One such covering for a strip for which is shown. Find the number of ways there are to cover the strip. (Coverings that differ by a rotation are considered distinct.)
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\testitem Evaluate \[\frac{\displaystyle{\sum_{k=0}^{20}\cos\left(\frac{\pi(k-5)}{20}\right)}}{\displaystyle{\sum_{k=0}^{20}\sin\left(\frac{\pi k}{20}\right)}}\]
{\parpic[r]{\includegraphics[width=4cm]{images/parking.eps}}\testitem A flat, triangular parking lot has dimensions 15 car lengths by 20 car lengths by 25 car lengths, where a ``car length'' is a fixed unit of measure that is a little longer than the average car's length. A parking space must be rectangular, with dimensions one car length long by one-half car length wide, and every space must have 1.5 car lengths of clearance behind it (area within the parking lot where there is no parking space). Different parking spaces may share the same clearance space, and every spot must be accessible from some point on the perimeter of the lot, not counting the curb that bounds that parking space itself. For simplicity, assume that the long side of every space is perpendicular to the 15-car-length side of the lot. Determine the maximum number of parking spaces the lot can hold.
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\testitem Let $[\mathcal{F}]$ represent the area of figure $\mathcal{F}$. If $ABCD$ is a trapezoid with $AD\parallel BC$ and $O$ is the intersection of the trapezoid's diagonals, prove that \[\frac{[AOB]}{[ABCD]}<\frac{1}{4}\]
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