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\markboth{2008 UNIVERSITY OF MISSISSIPPI INDIVIDUAL COMPETITION}
{2008 UNIVERSITY OF MISSISSIPPI INDIVIDUAL COMPETITION}
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\begin{document}
\begin{center}
\Large \textbf{University of Mississippi} \\ \vspace{1mm} \normalsize
4\raisebox{3.6pt}{\scriptsize{th}} Annual High School Mathematics Contest \\
Individual Competition \\ \input{date.tex} \vspace{0.6cm}
\end{center}
\thispagestyle{empty}
\begin{multicols}{2}
\testitem{Two solutions of $(t+2)^2+12t=669$ are $t=19$ and $t=-35$.
Which of the following is also a solution of this equation?
\answers{22}{23}{$-14$}{$-31$}{\corr{\nota}}}
\testitem{A Latin square is a square table of symbols in which each symbol
appears exactly once in every row and column. Find the symbol that
belongs in position indicated by the question mark in the following
partial Latin square:
\begin{center}
\begin{tabular}{|c|c|c|c|} \hline
\smiley & \corrhide{\manstar} & \mancube & \phone \\ \hline
\mancube & \phone & ? & \corrhide{\manstar} \\ \hline
\manstar & \corrhide{\mancube} & \corrhide{\phone} & \corrhide{\smiley} \\ \hline
\corrhide{\phone} & \smiley & \corrhide{\manstar} & \mancube \\ \hline
\end{tabular}
\end{center}\answersn{\corr{\smiley}}{\manstar}{\phone}{\mancube}{Cannot
be determined}}
\testitem{Lagrange's four-square theorem states that every positive integer
may be written as the sum of four perfect squares. For example,
$38=6^2+1^2+1^2+0^2$. Write 88 in the form $a^2+b^2+c^2+d^2$ where
$a,b,c,$ and $d$ are integers, and calculate $|a|+|b|+|c|+|d|$.
\answers{\corr{16}}{18}{20}{24}{\nota}}
\testitem{How many prime factors does 391 have?
\answers{1}{\corr{2}}{3}{4}{More than 4}}
\testitem{In a certain country, the four states $A$, $B$, $C$, and $D$ have
populations of 11, 33, 44, and 77, respectively. Presidential elections
in this country are based on an electoral college system, and the four
states have 1, 3, 4, and 7 electoral college votes, respectively. The
candidate who receives the most individual votes in each state gets all
of that state's electoral college votes, and the candidate who gets the
most electoral college votes wins. In an election between two
candidates, what is the minimum number of individual votes that the
winning candidate could receive? \answers{\corr{45}}{54}{56}{62}{\nota}}
\testitem{A square's side length is a whole number of centimeters, its
diagonal is shorter than 20 centimeters, and its area is $A$. How many
possible values are there for $A$? \answers{9}{11}{12}{\corr{14}}{16}}
% \testitem{An integer $a$ is said to \emph{divide} an integer $b$ if $b\div a$
% is an integer. How many integers divide 80?
% \answers{\corr{20}}{24}{25}{80}{Infinitely many}}
\testitem{Josephine wants to cut a rectangular cake into 5 pieces of equal
area as shown. Find the ratio $x/h$.
\begin{center}
\includegraphics[width=0.75\linewidth]{images/cake.mps}
\end{center}
\answersn{1/5}{3/7}{1/3}{Cannot be determined}{\corr{\nota}}}
\testitem{Chester is asked to find every real number which is equal to its
cube. His solution is shown below. In which step did he make an error?
\begin{align}
x^3&=x \\
x^2&=1 \\
x^2-1&=0 \\
(x+1)(x-1)&=0 \\
x+1=0 \quad \text{or} \quad x-1=0 &\implies \fbox{$x\in\{-1,1\}$}
\end{align}\answersn{\corr{Going from (1) to (2)}}{Going from (2) to (3)}{Going
from (3) to (4)}{Going from (4) to (5)}{No error was made.}}
\testitem{The graphs of four polynomial functions $f$, $g$, $h$, and
$f+g+h$ are shown below. Determine which of the graphs represents the
function $f+g+h$.
\begin{center}
\includegraphics[width=8cm]{images/fgandh.mps}
\end{center}
\answersn{A}{\corr{B}}{C}{D}{Cannot be determined.}}
\testitem{Which of the following statements are true? \vspace*{2mm} \\
\hspace*{0.5cm}I. An acute triangle can be isosceles. \\
\hspace*{0.5cm}II. An obtuse triangle can be isosceles. \\
\hspace*{0.5cm}III. A right triangle can be isosceles.
\answers{I and II}{I only}{II only}{\corr{I, II, and III}}{None of the statements are true.}}
\testitem{Which of the following is equal to $4\times0.\overline{63}$?
\answers{$2.\overline{55}$}{\corr{$2.\overline{54}$}}{$2.\overline{53}$}
{$2.\overline{52}$}{$2.\overline{5}$}}
\testitem{Which of the following numbers is closest on the real number line
to the point which is halfway from $10^{-13}$ to $10^{-7}$?
\answers{$10^{-7}$}{\corr{$10^{-8}$}}{$10^{-9}$}{$10^{-10}$}{$10^{-13}$}}
\testitem{To the nearest whole percent, 89\% of the people that work with
Mitchell keep plants in their office. What is the sum of the digits of
the minimum possible number of people who work with Mitchell?
\answers{\corr{9}}{7}{6}{3}{\nota}}
\testitem{For how many of the following numbers $x$ is it true that
$\sqrt[4]{x^2}=\sqrt{x}$? \[4, \quad \frac{13}{2}, \quad -\sqrt{2}, \quad
\pi, \quad -3\] \answers{0}{2}{\corr{3}}{4}{\nota}}
\testitem{What is the maximum number of nonzero entries in a $2\times 2$
matrix whose square is the zero matrix?
\answers{1}{2}{3}{\corr{4}}{\nota}}
\testitem{Evaluate $16^5-12\cdot16^4-60\cdot16^3-60\cdot16^2-54\cdot16+10$.
\answers{$-16$}{90}{\corr{170}}{232}{\nota}}
\testitem{Let $r=\sqrt{111}-\sqrt{110}+\sqrt{10000^{-1}}$. Which of the
following is true?
\answers{$5<100r<5.5$}{\corr{$5.5<100r<6$}}{$6<100r<6.25$}{$6.25<100r<6.5$}{\nota}}
\testitem{Sam's grandfather was born in 1918, and Sam was born in 1986.
Therefore, in 2004, Sam's grandfather was 86 years old, having been born
in '18, while Sam was 18 years old, having been born in '86. Suppose
that the birth year of person A is chosen uniformly at random
between the years 1901 and 1950 inclusive, while the birth year of person
B is chosen uniformly at random between 1951 and 1999 inclusive. What is
the probability that there will be a year in which person A's age is
equal to the number of years between 1900 and the birth year of person B,
and vice versa? \answers{1/5}{1/4}{1/3}{1/2}{\corr{\nota}}}
\testitem{Eight numbers are chosen uniformly at random (independently of
one another) in the interval [0,1]. What is the probability that the
second largest of the numbers is less than 0.9?
\answers{$(0.9)^7$}{$(0.9)^7+(0.9)^8$}{$0.1\cdot(0.9)^8$}{\corr{$1.7\cdot(0.9)^7$}}{\nota}}
\testitem{What is the least integer $n$ such that there is exactly one
perfect square among the numbers $n$, $n+1$, $\ldots$, $n+1000$?
\answers{$562501$}{$250001$}{\corr{$62501$}}{$22501$}{\nota}}
\testitem{Find the greatest integer which is not greater than
\[\log_3\left(3^{97}+3^{96}+3^{95}+\cdots+3^{33}\right).\]
\answers{101}{99}{98}{\corr{97}}{\nota}}
\testitem{For each $x\in[-1,1]$, let the function $\cos\inv(x)$ denote the
usual inverse of the cosine function. That is, $\cos\inv(x)$ is the
unique $\theta\in[0,\pi]$ with
$\cos\theta=x$. Solve \[\cos\inv\left(\frac{3}{5}\right)+\cos\inv(x) =
\cos\inv\left(-\frac{63}{65}\right).\]
\answers{$\frac{4}{5}$}{$\frac{7}{24}$}{$-\frac{8}{17}$}{$-\frac{7}{24}$}{\corr{\nota}}}
\testitem{A list of 177 positive numbers has a unique mode. The number 3
appears exactly 45 times in the list, the number 2 appears exactly 64
times, and the sum of the 177 numbers is 885. Find the largest possible
sum of the mean, the median, and the mode of the numbers.
\answers{9}{11}{\corr{17}}{19}{\nota}}
\testitem{Define $n(a,b,c)$ to be the number of real solutions $x$ of the
equation \[\sqrt{x-a}+\sqrt{x-b}=\sqrt{c}.\] Find
$n(1,2,3)+n(4,4,0)+n(4,4,4)+n(25,15,5)$.
\answers{1}{2}{\corr{3}}{4}{\nota}}
\testitem{Find the remainder when $x^{48}-9x^{47}+20x^{46}+x$ is divided by
$x^2-4x-5$.
\answers{29}{\corr{$-4x+25$}}{$-4x+29$}{$x+29$}{\nota}}
\testitem{Find the least value of $a$ so that the graph of the function
$g(x)=-\sqrt{a^2-x^2-12x-36}+9$ intersects the graph of
$h(x)=\sqrt{4a^2-x^2}+1$.
\answers{\corr{$10/3$}}{$5$}{$5\sqrt{2}$}{$5\sqrt{3}$}{\nota}}
\testitem{There is an (nondegenerate)
equilateral triangle $\bigtriangleup ABC$ having all three vertices on
the right branch of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$
with $A=(a,0)$. Find the length of an altitude of this equilateral
triangle. \answers{\mbox{$\rule[-10pt]{0pt}{20pt}\displaystyle{\frac{a^2b+3ab^2}{a^2-3b^2}}$}}
{\corr{\mbox{$\rule[-10pt]{0pt}{30pt}\displaystyle{\frac{a^3+3a^2b}{a^2-3b^2}}$}}}
{\mbox{$\rule[-10pt]{0pt}{30pt}\displaystyle{\frac{a^2b+3ab^2}{a^2+3b^2}}$}}
{\mbox{$\rule[-10pt]{0pt}{30pt}\displaystyle{\frac{a^3+3a^2b}{a^2+3b^2}}$}}{\nota}}
\testitem{Jamye won a large number of 2\% off coupons for purchases at Blue
Bottom Jeans. She is allowed to apply as many coupons as she wishes on
each purchase, and the discounts are applied successively. After all the
discounts are applied, a tax of 8.5\% is added. What is the smallest
number of coupons she must use in order for the total cost of a pair of
blue jeans (after tax) to be less than the original price of the jeans?
(Note: It might be useful that $1.085^{-1}=0.92165...$).
\answers{2}{3}{4}{\corr{5}}{\nota}}
\testitem{The two quarter-circles shown below are externally tangent with
centers at opposite vertices of a unit square, and the larger quarter-circle
passes through two vertices of the square. A small circle is drawn
externally tangent to each of the two quarter-circles and to the side of
the square. What is the radius of the small circle?
\begin{center}
\includegraphics[width=0.65\linewidth]{images/circles.mps}
\end{center}
\answersn
{\rule[-10pt]{0pt}{20pt}$\displaystyle{\frac{5\sqrt{2}-7}{2}}$}
{\rule[-10pt]{0pt}{30pt}$\displaystyle{\frac{2\sqrt{2}-1}{32}}$}
{\corr{\rule[-10pt]{0pt}{30pt}$\displaystyle{\frac{9-4\sqrt{2}}{49}}$}}
{\rule[-10pt]{0pt}{30pt}$\displaystyle{\frac{11-6\sqrt{2}}{128}}$}
{\rule[-10pt]{0pt}{25pt}\nota}}
\testitem{Any given toss of a certain weighted coin has a probability $p$
of resulting in heads and $1-p$ of resulting in tails. The probability
that an even number of heads will result in 99 tosses of the coin is 3/4.
Find $p$.
\answers{\corr{$1/2-2^{-100/99}$}}{$1/2+2^{-99}$}{$1/2-2^{-99}$}{$1-2^{-1/99}$}{\nota}}
\end{multicols}
\end{document}