%Format: AMS TeX
\hyphenation{Mass-achusetts Central Missouri Wis-con-sin Muthuvel}
\raggedbottom
\loadmsam
\loadmsbm
\loadeufm
\def\R{\Bbb R}
\def\Z{\Bbb Z}
\def\Q{\Bbb Q}
\def\C{\Bbb C}
\def\N{\Bbb N}
\tolerance=1600
\hsize=33pc
\vsize=42pc
\baselineskip=12pt
\lineskip=12pt
\centerline{\bf 1999 Missouri MAA Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session I}
\bigskip
1.  Let $P \ne (0,0)$ be a point on the parabola $y=x^2$.  The normal 
line to the parabola at $P$ will intersect the parabola at another point, 
say $Q$.  Find the coordinates of $P$ so that the length of the arc 
of the parabola between $P$ and $Q$ is a minimum.
\bigskip
2.  Give a precise characterization of those points in the plane which do 
$\underline{\text{not}}$ lie on a tangent line to the curve $y = x^4 - 
6x^2$. 
\bigskip
3.  On a $5 \times 5$ square matrix place 13 black counters and 12 
white counters in alternating checkerboard fashion.  Remove the black 
counter in the center square.  Player A controls the white counters 
and B the black.  They take turns moving one of their counters 
orthogonally to the vacant square until a player loses by being unable 
to move.  Which player has a winning strategy?  What is the strategy? 
\bigskip
4.  If $x_0 = 5$ and $x_{n+1} = x_n + 1/x_n$, prove that for 
all $n \ge 1$
$$2n < x_n ^2 - 25 < 47n/23 .$$   
\bigskip
5.  For an $n \times n$ matrix $X$, if $\det (\lambda I - X ) = 0$ 
then we say $\lambda$ is an eigenvalue of $X$.  Let $A$ be an $m 
\times n$ matrix and let $B$ be an $n \times m$ matrix.  Prove that 
$AB$ and $BA$ have the same non-zero eigenvalues.
\vfill\eject
\centerline{\bf 1999 Missouri MAA Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session II}
\bigskip
1.  Let $SC$ be the semicircle with $y \ge 0$ centered at $(1,0)$ with 
radius 1.  Let $C_a$ be the circle with radius $a > 0$ and center $(0,0)$ 
and denote the point $(0,a)$ by $P$.  Consider the line through $P$ and 
the intersection of $SC$ and $C_a$.  What is the limiting position of the 
$x$-intercept of this line as $a \to 0$?
\bigskip
2.  Find the limit
$$\lim_{N \to \infty} \biggl( 1 - 2 \sum_{n=1}^N {1 \over 16n^2 - 1} 
\biggr) .$$
\bigskip
3.  For $n$ positive real numbers with minimum $m$ and maximum $M$, let 
$A$ and $G$ denote their arithmetic and geometric means.  Prove that
$$A - G \ge n^{-1} ( \sqrt M - \sqrt m )^2 .$$
\bigskip
4.  Find all possible continuous and differentiable curves $C$ which have 
the following properties.  The curve $C$ lies in the first quadrant and 
contains the point $(0,0)$.  Whenever $P$ is on $C$ the interior of the 
rectangle $R$ bounded by the coordinate axes and horizontal and vertical 
lines through $P$ is separated into two parts by $C$.  When the part 
adjacent to the $x$-axis is rotated about the $x$-axis and the part 
adjacent to the $y$-axis is rotated about the $y$-axis, two solids of 
equal volume are generated. 
\bigskip
5.  Let $A_n$ denote the $n \times n$ matrix whose $(i,j)$ entry is 
$\text{GCD} (i,j)$.  Compute $\det (A_n)$. 
\bye
