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\begin{center}
2013 Missouri Collegiate Mathematics Competition \\ Session I
\end{center}

\noindent 1.  Consider the sets of consecutive integers $\{ 1 \}$, $\{ 2, 3 \}$, $\{ 4, 5, 6 \}$,
$\{ 7, 8, 9, 10 \}$, $\ldots$, where each set contains one more element than the preceding one,
and where the first element of each set is one more than the last element of the preceding set.
Let $S_n$ be the sum of the elements in the $n$th set.  Find $S_{32}$.
\ \\ \ \\
\noindent 2. Find all positive real solutions $x$ to the equation
$$4 \lfloor x \rfloor = 3x \{ x \} ,$$
where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{ x \} = x
- \lfloor x \rfloor$.
\ \\ \ \\
\noindent 3. Evaluate the following where $p>1$.
$$\frac{1 + \frac{1}{2^p} + \frac{1}{3^p} + \frac{1}{4^p} + \cdots}
{1 - \frac{1}{2^p} + \frac{1}{3^p} - \frac{1}{4^p} + \cdots} .$$
\ \\ \ \\
\noindent 4. Neither \textit{Mathematica} nor \textit{Maple} can find the exact value of the
following definite integral.  Can you?  We think you can.  Do it!
$$\int_0^2 (3x^2 - 3x + 1) \cos (x^3 - 3x^2 + 4x - 2) \, dx .$$
\ \\ \ \\
\noindent 5.  Define the sequence $\{ a_n \}_{n=1}^\infty$ by
$$a_1 = 1,\ \ a_2 = 1,\ \ a_3 = 2,$$
and for $n \ge 3$,
$$a_{n+1} = \frac{n a_n a_{n-2}}{a_{n-1}} .$$
\begin{itemize}
\item[(a)] Prove that $a_n$ is a positive integer for all $n \ge 1$.
\item[(b)] Define the sequence $\{ b_n \}_{n=1}^\infty$ by
$$b_n = \frac{a_n}{\sqrt {(n+1)!}} \ \ \ \ \textrm{for}\ \ n \ge 1 .$$
Prove that $\{ b_n \}$ is bounded.
\end{itemize}
\vfill\eject
\begin{center}
2013 Missouri Collegiate Mathematics Competition \\ Session II
\end{center}

\noindent 1. Let $r$ be a real number with $0 < r \le 1$.  Let $S(r)$ be the region in the
$(x,y)$-plane bounded by the curve $y = x^r$, the $x$-axis, and the line $x=1$.  Let $G(r)$
denote the centroid of $S(r)$.  Let $\ell _r$ be the line that passes through the origin and $G(r)$.
Let $T(r)$ be the region in the $(x,y)$-plane bounded by $\ell _r$, the $x$-axis, and the line
$x=1$.  Find the value of $r$ that minimizes the ratio of the area of $T(r)$ to the area of $S(r)$.
\ \\ \ \\
\noindent 2. Let
$$f(r) = \sum_{j=2}^{2013} \frac{1}{j^r} = \frac{1}{2^r} + \frac{1}{3^r} + \cdots + \frac{1}{2013^r}
.$$
Find
$$\sum_{k=2}^\infty f(k) .$$
\ \\ \ \\
\noindent 3. Define an \textit{Egyptian triangle} to be one that is similar to a 3-4-5 right
triangle.  A square sheet of paper is folded so that one corner touches the midpoint of the
opposite side.  The folded sheet will thus be composed of three triangles ``one sheet thick''
and a quadrilateral ``two sheets thick.''  Prove that all three triangles are Egyptian triangles.
\ \\ \ \\
\noindent 4. Prove that $\frac{x^2 + y^2}{4} \le e^{x+y-2}$ for $x \ge 0$ and $y \ge 0$.
\ \\ \ \\
\noindent 5. A point $(x,y)$ is chosen at random from the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, $(1,1)$ (uniform distribution on the unit square).  Find the probability that $x$ and $y$ are side lengths of an isosceles triangle of perimeter at most 1.

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