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\begin{center}
2016 Missouri Collegiate Mathematics Competition \\ Session I
\end{center}

\noindent 1.  Let $\mathcal{R}$ be the region in the first quadrant bounded by the $x$-axis and the graph of $y=\sin x$ from $x=0$ to $x=\pi $. The graph of $y=\cos x$ divides $\mathcal{R}$ into two regions. Let $\mathcal{R}_{1}$ be
the subregion of $\mathcal{R}$ bounded on the left by $y=\cos x$, on the right by $y=\sin x$, and below by the $x$-axis. Find a positive number $b$ such that
the line $y=b$ divides $\mathcal{R}_{1}$ into two regions of equal area.
\ \\ \ \\
\noindent 2. Find, with proof, the sum of the series
$$\sum_{n=1}^\infty\sin\frac1{2^{n+1}}\cos\frac3{2^{n+1}}.$$
\ \\ \ \\
\noindent 3. Find, with proof, all positive integers $x$ satisfying $\displaystyle{3^{(2^{x!})}=2^{(3^{x!})}+1}$.
\ \\ \ \\
\noindent 4. Find a polynomial with integer coefficients that has $\displaystyle{2^{\frac{1}{5}} + 2^{-\frac{1}{5}}}$ as a root.
\ \\ \ \\
\noindent 5. For each positive integer $n$, let
\[
\mathcal{R}_{n}=\left\{ \left( x,y\right) :0\leq x\leq n\text{ and }0\leq
y\leq \sqrt{x}\right\} .
\]
Let $N\left( n\right) $ denote the number of points in $\mathcal{R}_{n}$
whose coordinates are both integers. Compute
\[
\lim\limits_{n\rightarrow \infty }\dfrac{N\left( n\right) }{n^{3/2}}.
\]
\vfill\eject
\begin{center}
2016 Missouri Collegiate Mathematics Competition \\ Session II
\end{center}

\noindent 1. Evaluate
$$\sum_{k=1}^n k 2^{k-1}$$
for $n=1, 2, 3, \ldots $.
\ \\ \ \\
\noindent 2. Evaluate
\[
\int_{1}^{\infty }\dfrac{\left\lfloor x\right\rfloor }{x^{3}}\,dx,
\]
where $\left\lfloor x\right\rfloor $ denotes the greatest integer function.
\ \\ \ \\
\noindent 3.  Find the smallest positive angle $\theta$ in degrees satisfying
$$\sin^2(2015\theta) + \cos^2 (2016\theta)=1.$$
\ \\ \ \\
\noindent 4. If $a$ and $b$ are the radii of two spheres, tangent to each other and to a
plane (on the same side of the plane), the radius $x$ of the largest sphere
which can pass between them is given by what formula?
\ \\ \ \\
\noindent 5. In a regular octagon each side is colored blue or yellow.  From such a coloring, another
coloring will be obtained ``in one step'' as follows: if the two neighbors of a side have different
colors, the ``new'' color of the side will be blue, otherwise the color will be yellow.  (Note that
the colors
are modified simultaneously.)  Show that after a finite number, say $N$, of moves, all sides will be
colored yellow.  What is the least value of $N$ that works for all possible colorings?

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