\input amstex
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\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}
\def\dps{\displaystyle}
\newsymbol\nmid 232D
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\centerline{\bf 2009 Missouri Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session I}
\bigskip
1. For the parabola having equation $y = -x^2$ let $a<0$ and $b>0$ with $P : (a,-a^2)$
and $Q : (b,-b^2)$. Let $M$ be the midpoint of $PQ$ and let $R$ be the intersection of the
vertical line through $M$ with the parabola. Finally, let $l$ be the tangent line to the
parabola at $Q$. Prove that every vertical line segment with one end on $PQ$ and the
other end on $l$ is bisected by the line through $Q$ and $R$.
\bigskip
2. Prove that for any $x$ in the half-open interval $(0, \pi/2 \rbrack$ one has
$$\left( {\sin x \over x} \right) ^3 > \cos x .$$
\bigskip
3. Let $A$ be a set with $|A|=n$, and let $k$ be a
positive integer. Determine the number of subset sequences of the
form
$\dps S_1 \subseteq S_2 \subseteq \cdots \subseteq S_k \subseteq A$.
\bigskip
4. Find the value of the infinite product $$\left( {7 \over 9}
\right) \cdot \left( {26 \over 28} \right) \cdot \left( {63 \over
65} \right) \cdot \cdots = \lim_{n \to \infty} \prod _{k=2}^n
\left( {k^3 - 1 \over k^3 + 1} \right) .$$
\bigskip
5. Evaluate
$$\int\!\!\! \intop_S\!\!\! \int \min \{ x, y, z \} \, \text{d}V ,$$
where $S = \{ (x,y,z) \in \R^3 \ \vert \ 0 \le x \le 1, \ 0 \le y \le 1, \ 0 \le z \le 1 \}$.
\vfill\eject
\centerline{\bf 2009 Missouri Collegiate Mathematics Competition} \vskip 6pt
\centerline{\bf Session II}
\bigskip
1. A piece of wire of length $x$ is cut into two pieces, one of which is formed into a
square and the other into a circle, such that the total area enclosed by the two figures is
a positive constant $A$. Find the ratio of the length of the edge of the square to the length
of the radius of the circle that makes the length of the wire a maximum. Similarly, find the
ratio that makes the length of the wire a minimum.
\bigskip
2. Determine the number of subsets $S$ of the set $\{ 1, 2, \ldots , n \}$ such that $S$
contains no two consecutive integers. Express the answer in terms of the Fibonacci
numbers ($F_1 = 1$, $F_2 = 1$, $F_n = F_{n-1} + F_{n-2}$ for $n \ge 3$) and prove your answer.
\bigskip
3. Consider a piece of paper glued to the outside of the cylinder $x^2 + y^2 = 1$. Suppose
that we open a compass to a radius $r$ ($0 < r < 2$), put the stationary point of the compass
at the point $(1,0,0)$ on the cylinder, and draw a ``circle'' on the paper (that is, we use
the pencil end of the compass to draw a curve).
If we now remove the paper from the cylinder and draw a coordinate system with the origin at
the stationary compass point, the $Y$ axis in the same direction as the original $z$ axis,
and the $X$ axis oriented appropriately, we can now consider the ``circle'' as a plane
figure. Find an equation for this figure in the $XY$-plane.
\bigskip
4. Determine
$$\lim_{n \to \infty} \sum_{k=1}^n \left( \sin \left( {\pi/2 \over k} \right) - \cos \left(
{\pi/2 \over k} \right) - \sin \left( {\pi/2 \over k+2} \right) + \cos \left( {\pi/2 \over
k+2} \right) \right) .$$
\bigskip
5. Mersenne primes continue to make news. A number $M_p = 2^p - 1$ is a Mersenne prime if
and only if $p$ is prime and $2^p-1$ is also prime. Let the operator $DS$ denote ``form the
sum of the digits''; for example, $DS(5119) = 16$. Let the operator $DR$ denote ``execute
$DS$ repeatedly until a result in the interval $\lbrack 1,9 \rbrack$ is obtained''; for
example, $DR (5119) = DS^2 (5119) = DS (16) = 7$.
\medskip
\item{(a)} Prove the following lemma.
\medskip
$\underline{\text{Lemma}}$. If $A$ and $B$ are natural numbers, then $$DR(AB) = DR(DR(A)
\cdot DR(B)) .$$
\medskip
\item{(b)} Prove that for any Mersenne prime greater than 7, $DR(M_p) = 1 \text{ or } 4$.
(You may use the Lemma in part (a) without proving part (a)).
\bye