\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{amssymb}
\pagestyle{empty}
\textwidth = 6.5 in
\textheight = 9 in
\oddsidemargin = 0.0 in
\evensidemargin = 0.0 in
\topmargin = 0.0 in
\headheight = 0.0 in
\headsep = 0.0 in
\parskip = 0.2in
\parindent = 0.0in
\newcounter{prob}
\setcounter{prob}{1}
\newcommand\itm{\nopagebreak \vspace{0.5mm} \newline (\alph{prob}) \stepcounter{prob}\nopagebreak}
\newcommand\testitem{\item \setcounter{prob}{1} \nopagebreak}
\begin{document}
\begin{center}
\Large University of Mississippi \\ \normalsize High School Mathematics Contest -- Individual Competition \\ March 24, 2006
\end{center}
\begin{enumerate}
\testitem How many numbers are in the set $\{747, 748, 749, \ldots 1048, 1049 \}$?
\itm 301 \itm 302 \itm 303 \itm 1049
\testitem Which of the following is equal to $\log_62+\log_63$?
\itm 1 \itm $\log_62\log_63$ \itm $\sqrt{6}+\sqrt{3}$ \itm $\log_65$
\testitem The area of a right triangle is $30$, and the difference between the lengths of the two legs is 7. What is the sum of the lengths of the two legs?
\itm 17 \itm 18 \itm 19 \itm 20
\testitem One square yard is equivalent to how many square feet?
\itm 1/27 \itm 1/3 \itm 27 \itm 9
\testitem When 1.00234152 is converted to a fraction and simplified, let the numerator of the resulting fraction be called $n$ and the denominator $d$. Which of the following is true?
\itm $nd$ \itm $n=d$ \itm $n=100d$
\testitem Which of the answer choices is the largest?
\itm $4.1\times10^{-25}$ \itm $9\times10^{-26}$ \itm $4.1\times10^{-25}-10^{-50}$ \itm $40\times10^{-24}$
\testitem Find the probability of rolling a prime number with the roll of a die (singular of \textit{dice}).
\itm 1/6 \itm 1/3 \itm 1/2 \itm 2/3
\testitem Which of the following numbers is a solution to the equation $\displaystyle{\frac{x^6+x-2}{24}=0}$? \itm 1 \itm 0 \itm -1 \itm -2
\testitem The area of a $45^{\circ}$ sector of a circle of radius 2 is \underline{nearest} to which of the following?
\itm 0 \itm 1 \itm 2 \itm 3
\testitem In right triangle $\bigtriangleup ABC$ with right angle at vertex $C$ and $m\angle B=30^{\circ}$ find the greatest integer that is not greater than $100\sin A\cos A\tan A, \csc A \sec A \cot A$, and call it $r$. Which of the following is true?
\itm $r=0$ \itm $03$. What is the smallest number $n$ such that $a_n$ is divisible by the prime number 929? (Note: $n!$ is defined by $n!=n\cdot(n-1)\cdot(n-2)\cdot\cdots\cdot2\cdot1$).
\itm 3 \itm 4 \itm 5 \itm 6
\testitem Let the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{8}=1$ be $A$ and $B$, and consider $C(2\sqrt{3},\sqrt{2})$. What is $AC+BC$?
\itm 8 \itm $\sqrt{8}$ \itm 4 \itm 16
\testitem A circle in the plane has one chord whose endpoints are (0,10) and (8,8) and another chord whose endpoints are $(-4,-2)$ and $(4,-1)$. What is the $x$-coordinate of its center?
\itm 0 \itm 7/20 \itm 9/17 \itm 11/24
\testitem How many different integer values may $\displaystyle{\frac{mn}{m^2+n^2}}$ take on, where $m$ and $n$ are nonnegative real numbers?
\itm 0 \itm 1 \itm 2 \itm infinitely many
\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 $ab\cos C/2$
\testitem What is the smallest value that the function $f(a,b)=\sqrt{2^2+a^2}+\sqrt{2^2+(a-b)^2}+\sqrt{1^2+(12-b)^2}$ takes on over real values for $a$ and $b$?
\itm 13 \itm $14$ \itm $\sqrt{200}$ \itm $\sqrt{216}$
\end{enumerate}
\end{document}