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\begin{center}
2011 Missouri Collegiate Mathematics Competition \\ Session I
\end{center}


\noindent 1.  Consider the parabola $x^2=4\,p\,y$, with vertex at the origin, focus at
$F(0,p)$, and directrix $y= -p$.  Let $P(x,y)$ be a point on the parabola and let
$R(x,-p)$ be the intersection of the vertical line through $P$ with the directrix.
Determine $P$ so that the triangle $PFR$ is equilateral.
\ \\ \ \\
\noindent 2. Prove that the inequality $\sin x + \arcsin x > 2x$ holds for all values of $x$ such that
$0 < x \le 1$.
\ \\ \ \\
\noindent 3. For what non-negative integers $n$ does there exist a
polynomial $P_n$ of degree $n$ with integer coefficients satisfying
$P_n(k)=2^k$ for all integers $k$, $0 \leq k \leq n$? Find all such
polynomials.
\ \\ \ \\
\noindent 4. Let $a$ and $b$ be real numbers with $b\neq0$ and let $C\left( \mathbb{R}\right)$
denote the set of continuous functions from the reals to the reals.  Define $T:C\left(
\mathbb{R}\right)  \rightarrow C\left(  \mathbb{R}\right)  $ by $\left(
Tf\right)  \left(  x\right)  =ax+b\int_{0}^{x}f\left(  t\right)  dt$. Find all
fixed points of $T^{2}$, where $T^{2}$ denotes the composition of $T$ with
itself.
\ \\ \ \\
\noindent 5. Let the function $F$ be given by
$$F(x) = e^{-x} - \left(1-\frac{x}{n}\right)^n.$$
Show that, for $n\ge 2$,
$$0\le F(x) \le \frac{e^{-1}}{n} \qquad \hbox{on} \qquad [0,n].$$
\vfill\eject
\begin{center}
2011 Missouri Collegiate Mathematics Competition \\ Session II
\end{center}

\noindent 1. For $0<r<1$, let $n_{r}$ denote the line that is normal to the curve
$y=x^{r}$ at the point $\left(  1,1\right)  $, and let $S_{r}$ denote the
region in the first quadrant of the $xy$-plane bounded by the $x$-axis, the
curve $y=x^{r}$, and the line $n_{r}$. Find the value of $r$ that minimizes
the area of $S_{r}$.
\ \\ \ \\
\noindent 2. A rectangular strip is
divided into $n$ bands of equal width  as in the following figure. \\ \ \\
\centerline{\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline 1&2&\phantom{2}&\phantom{2}&\phantom{2}&\phantom{2}&\phantom{2}&n \\ \hline \end{tabular}}
\ \\
Each band is colored by one of $m$
colors. Two patterns are considered identical if one is a (left-to-right) mirror reflection of the other.
Determine the number of distinct patterns with $m$ colors and $n$ bands.
\ \\ \ \\
\noindent 3. Let $M$ be an arbitrary $3 \times 3$ matrix, each of whose entries $m_{ij}$ is drawn
from the set $\{ -1,
1 \}$, and let $\det M$ be the determinant of $M$.  Prove that $\vert \det M \vert \le 4$.
\ \\ \ \\
\noindent 4. Calculate
$$\int_0^2 \left( \sqrt {1+x^3} + \root 3 \of {x^2 + 2x} \right)\, dx .$$
\ \\ \ \\
\noindent 5. Evaluate the series
$$\sum_{n=0}^\infty\frac1{2011^{2^n}-2011^{-{2^n}}}=\frac1{2011^1-2011^{-1}}
+\frac1{2011^2-2011^{-2}}+\frac1{2011^4-2011^{-4}}+\dots$$
and express it as a rational number.

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