\input amstex
\input epsfx
\hyphenation{Mass-achusetts Central Missouri Wis-con-sin Muthuvel}
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\def\R{\Bbb R}
\def\Z{\Bbb Z}
\def\Q{\Bbb Q}
\def\C{\Bbb C}
\def\N{\Bbb N}
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\centerline{\bf 2008 Missouri Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session I}
\bigskip
1.  A straight line segment of arbitrary positive slope $m$ is drawn from the origin $O$ to the point of
intersection $A$ in quadrant I with the ellipse $x^2 + 4y^2 = 4$.  A line segment is then drawn parallel to
the $x$-axis from $A$ over to the $y$-axis, which the segment meets at point $B$.  Points $O$, $A$, $B$ are
thus the vertices of a right triangle.  Deduce the value of $m$ that maximizes the area $R$ of triangle
$OAB$, and prove that this really is the maximum area.
\bigskip
2.  Let $a$ be a positive real number.  The Lemniscate of Bernoulli is defined by
$$(x^2 + y^2)^2 = a^2 (x^2 - y^2) .$$
Find the area bounded by the Lemniscate of Bernoulli.
\bigskip
3.  The sequence of Catalan numbers, $\{ C_n \}_{n=1}^\infty$, is defined by
$$C_n = {1 \over n+1} {2n \choose n} .$$
Does there exist a member of the sequence that is not a natural
number?  Find one, or prove that there is none.
\bigskip
4.  Suppose a belt is stretched tightly over two circular pulleys with radii $r_1$ and $r_2$, whose
centers are $d$ units apart with $d > r_1 + r_2$.  If $r_1 > r_2$, find a formula for the total length
of the belt in terms of $r_1$, $r_2$, and $d$.
\bigskip
5.  Evaluate
$$\sum_{k=1}^\infty {1 \over {k+n \choose k}}$$
for $n \ge 2$.  What is this series when $n=1$?
\vfill\eject
\centerline{\bf 2008 Missouri Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session II}
\bigskip
1.
\vskip 1pt
\item{(a)} Let $f(x) = x^3 + x$.  Let $g(x)$ be the inverse function of $f(x)$.  Find $g' (10)$.
\smallskip
\item{(b)} For $x>0$, define $h(x) = 1/f(x)$.  Prove that the function $f(x) + h(x)$ has its absolute minimum
when $x=g(1)$.
\bigskip
2.  Consider the lattice $L = \{ (x,y) \ :\ x,y \in \Z \}$.  Color the lattice using an arbitrary coloring
scheme with 2008 available colors. Prove or disprove: In every coloring scheme, there must be a rectangle
whose four vertices all lie in $L$ and are colored with the same color.
\bigskip
3.  Find $b>1$ such that the graphs of $y = \log _b (x)$ and $y=b^x$ intersect in exactly one point, i.e.,
are tangent to one another.
\bigskip
4.  Suppose that
$${2x+3 \over x^2 - 2x + 2}$$
has the Taylor series
$$\sum_{k=0}^\infty a_k x^k .$$
Find the sum of the odd numbered coefficients, i.e., find
$$\sum_{k=0}^\infty a_{2k+1} = a_1 + a_3 + a_5 + \cdots .$$
\bigskip
5.  For each integer $n$, let $a_n = 8n^2 + 3n + 10$ and $b_n = 3n^2 + n + 3$.  Since $a_1 = 21$ and $b_1 = 7$,
we can write $\gcd (a_1, b_1) = 7$, where $\gcd$ denotes the greatest common divisor.  Find
$\max _{n \in \Z} \gcd (a_n, b_n)$.
\bye
