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\begin{document}
\begin{center}
\Large \textbf{University of Mississippi} \\ \vspace{1mm} \normalsize 2\raisebox{3.6pt}{\scriptsize{nd}} Annual High School Mathematics Contest \\
Individual Competition \\ October 21, 2006
\end{center}
\begin{multicols}{2}
\testitem Beginning with your feet on the ground, you ascend a ten-rung ladder by going up three rungs, then going down two rungs, going up three, then down two, and so on. Each movement, up or down one rung, takes 0.2 seconds. After how many seconds will your feet reach the top rung?
\itm 7.6 \itm 7.8 \itm 8.0 \itm 8.2 \itm \nota
\testitem One piece of gum, six ice cream cones, and four sodas cost \$10.68, whereas a piece of gum, four ice cream cones, and six sodas cost \$11.06. What is the positive difference in price between one ice cream cone and one soda?
\itm 17\textcent \itm 18\textcent \itm 19\textcent \itm 20\textcent \itm \nota
\testitem A rotating sphere has energy $E$ given by $E=\frac{1}{2}I\omega^2$, where $I$ is given in terms of the mass $M$ and radius $R$ of the sphere by $I=\frac{2}{5}MR^2$. Solve for $\omega$ in terms of $E$, $M$, and $R$.
\itm $\sqrt{\frac{2E}{I}}$
\itm $\sqrt{\frac{5I}{2M}}$
\itm $\sqrt{\frac{5E}{M}}\cdot\frac{1}{R}$
\itm $\frac{5E}{MR^2}$
\itm \nota
\testitem Which of the following is equal to $9!+8!$, where $n!$ is defined to be $n(n-1)(n-2)\cdots3\cdot2\cdot1$?
\itm $10!$
\itm $17!$
\itm $\frac{11\cdot8!}{9}$
\itm $ \frac{10!}{9}$
\itm \nota
\testitem In the United States, FM radio channels fall in the range 87.9 MHz to 107.9 MHz, with a channel spacing of 0.2 MHz. How many different channels are there? \\
\begin{minipage}{\textwidth/2}
\itmn 100
\itm 200
\itm 201
\itm 202
\itm \nota \end{minipage}
\testitem How many square feet must be in the area of Anderson's living room if his living room furniture takes up 24 ft.$^2$ and 60\% of his living room floor isn't covered by furniture?
\itm 44 \itm 50 \itm 56 \itm 60 \itm \nota
%\testitem Solve the following equation for $t$ in terms of the other variables. You may assume that the variables do not take on values that make the denominators of any of the answer choices zero.
%\[
%t^2(1-u^2)+v^2=at^2
%\]
%\itm $\displaystyle{\frac{v}{\sqrt{a-1+u^2}}}$
%\itm $\displaystyle{\sqrt{\frac{t^2(1-u^2)+v^2}{a}}}$
%\itm $\displaystyle{\frac{v}{\sqrt{1-a+u^2}}}$
%\itm $\displaystyle{\frac{|v|}{\sqrt{a-1+u^2}}}$
%\itm \nota
\testitem Let $x$ and $y$ be two real numbers that, when expressed in simplest radical form, may be written as $x=a\sqrt{b}$ and $y=c\sqrt{d}$ for natural numbers $a$, $b$, $c$, and $d$. If $xy$ is an integer, then which of the following statements, if any, must be true?
\itm $a$ and $c$ must both have divisors that are perfect squares
\itm $b=d$
\itm $a^2b=c^2d$
\itm $b$ and $d$ must be divisible by $a$ and $c$, respectively
\itm \nota
\testitem Let $k$ and $a$ be fixed constants. The function $P_A(z)=\frac{k}{z}$ is designed to approximate the function $P(z)=\frac{kz}{z^2-a^2}$ for large $z$. Which inequality describes exactly the values of $z$ for which $P_A(z)$ approximates $P(z)$ with an error of less than 1\%?
\itm $z=8ak$
\itm $z<16ak$
\itm $z>16ak$
\itm $z>11a$
\itm $z>a\sqrt{101}$
\testitem Find the greatest integer that does not exceed $(10^{12}+10)^{1/2}$.
\itm 99 \itm 100 \itm 1,000,000 \itm 1,000,001 \itm 1,000,002
\testitem If $\alpha$ is chosen at random from the set $\{3,6,9,\ldots,99\}$ and $\beta$ from the set $\{-99,-96,\ldots,-3\}$, then how many different possible values are there for $\alpha+\beta$?
\itm 59 \itm 65 \itm 334 \itm 1089 \itm \nota
%\testitem Angela has four coupons to use at a clothing store. The discount from each coupon may only be applied to the price of the item after the discount from the previous coupon. If her coupons are worth 10\%, 20\%, 30\%, and 40\%, then in what order should she apply her coupons for the maximum discount?
%\itm 10\%, 20\%, 30\%, and 40\%
%\itm 40\%, 30\%, 20\%, and 10\%
%\itm 30\%, 20\%, 10\%, and 40\%
%\itm 40\%, 10\%, 20\%, and 30\%
\pagebreak
\testitem Which answer choice is a graph of $y=\sin(2^x)$ over the range $x=5$ to $x=7$?
\itm \includegraphics[width=4cm]{images/ans1.eps}
\itm \includegraphics[width=4cm]{images/ans2.eps}
\itm \includegraphics[width=4cm]{images/ans3.eps}
\itm \includegraphics[width=4cm]{images/ans4.eps}
{\parpic[r]{\includegraphics[width=4cm]{images/orbit.eps}} \testitem Let $M(n)$ represent the number of times in year $n$ that the moon passes through the circle that is the earth's orbit around the sun. Find the mode of the set $\{ M(1900), M(1901), M(1902), \\ \ldots, M(2005), M(2006)\}$ Assume the period of the moon's orbit is 28 days, and that the orbit of the earth and the moon are circular and coplanar.
\itm 12
\itm 24
\itm 26
\itm 28
\itm \nota
\picskip{0}}
\begin{minipage}{\linewidth}\testitem Determine which answer choice (if any) may fit correctly in the following blank: If $p$ is a perfect square and $p-1$ is prime, then \blank.
\itm $p$ must be is greater than 10
\itm there are infinitely many possibilities for $p$
\itm $2p+1$ must be a perfect square
\itm there are more than 5 possibilities for $p$
\itm \nota \end{minipage}
%\testitem If event A has probability $p$ and event $B$ has probability $q$, then which of the following is equal to the probability that \textit{either} A \textit{or} B will happen?
%\itm $p(1-q)$ \itm $pq$ \itm $1-p-q$ \itm $p+q$ \itm $1-(1-p)(1-q)$
\begin{minipage}{\linewidth}\testitem How many four-digit numbers have the property that reversing the digits yields a four-digit number that is divisible by 9? (A number is not considered a four digit number if its leading digit is 0.)
\itm 899 \itm 900 \itm 999 \itm 1000 \itm \nota \end{minipage}
\testitem Define a \textit{nice} set of convex polygons to be a set that can be arranged in the plane in such a way that they fit together to form another convex polygon that has no more sides than the polygon in the set that has the most sides. (For example, a set of four equilateral triangles is nice, whereas a set of three equilateral triangles is not.) What is the smallest possible number of elements for a nice set of polygons that includes a pentagon and at least one other polygon?
\itm 3
\itm 4
\itm 5
\itm 6
\itm \nota
\testitem Pete orders a $14''$ pizza and Jonathan orders a $10''$ pizza of the same thickness. Pete remarks, ``I have 2 times as much pizza as you do!'' To the nearest percent, what is his percent error?
\itm 2\% \itm 41\% \itm 42\% \itm 43\% \itm \nota
\testitem Solve for $x$, express your answer as a decimal, and add up all the digits (e.g. if $x=8.79$ were the solution, then $8+7+9=24$ would be the answer.)
\[\sqrt{x+1}+\sqrt{x-1}=10
\] \itm 8 \itm 10 \itm 12 \itm 14 \itm \nota
\testitem How many perfect squares are factors of $14^{14}\cdot17^{17}$?
\itm 72 \itm 81 \itm 84 \itm 576 \itm 624
\pagebreak \testitem The equation $100x^{100}-1=0$ has two distinct real solutions $x=r$ and $x=s$. Which of the following is true of $rs$?
\itm $rs<-1/4$
\itm $-1/4\leq rs<0$
\itm $rs=0$
\itm $00$ \\ iii) $b^2<4ac$
\itm (i) only
\itm (ii) only
\itm (i) and (ii)
\itm (ii) and (iii)
\itm \nota
\picskip{0}}
\testitem What is the maximum value of the absolute value of the determinant of a $3\times3$ matrix, all of whose entries are 0 or 1?
\itm 2
\itm 3
\itm 4
\itm 5
\itm \nota
\pagebreak
\begin{minipage}{\linewidth}
\testitem Compute $\displaystyle{\frac{10^{18}+10^3}{1,000,010}}$
\itm 999999000900
\itm 999990000100
\itm 999900000900
\itm 999900000100
\itm \nota \\
\testitem Denote the side lengths of triangle $\bigtriangleup ABC$ by $a$, $b$, and $c$ where the side of length $a$ is across from vertex $A$, and that of length $b$ is across from $B$. Given that $\sin A\sin B=\sin(A+B)$, which of the following gives a formula for the area of $\bigtriangleup ABC$?
\itm $ab\sin C$ \itm $c^2/2$ \itm $abc/4(a+b+c)$ \itm $ab\cos C/2$ \itm \nota \\
\testitem What is the smallest value that the function $f(a,b)=\sqrt{4+a^2}+\sqrt{4+(a-b)^2}+\sqrt{1+(14-b)^2}$ takes on for real values for $a$ and $b$?
\itm 12 \itm 13 \itm $\sqrt{5}+2\sqrt{2}+\sqrt{122}$ \itm $\sqrt{219}$ \itm $\sqrt{221}$
\end{minipage}
\end{multicols}
\end{document}