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\hyphenation{Mass-achusetts Central Missouri Wis-con-sin Muthuvel}
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\centerline{\bf 1999 Missouri MAA Collegiate Mathematics Competition}
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\centerline{\bf Session I}
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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 $y = x^4 -
6x^2$.
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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 $n \ge 1$
$$2n < x_n ^2 - 25 < 47n/23 .$$
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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.
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\centerline{\bf 1999 Missouri MAA Collegiate Mathematics Competition}
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\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 $a \to 0$?
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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
$$A - G \ge n^{-1} ( \sqrt M - \sqrt m )^2 .$$
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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.
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5. Let $A_n$ denote the $n \times n$ matrix whose $(i,j)$ entry is
$\text{GCD} (i,j)$. Compute $\det (A_n)$.
\bye