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\begin{center}
2012 Missouri Collegiate Mathematics Competition \\ Session I
\end{center}

\noindent 1.  Two circles in the first quadrant are tangent to each other and both are tangent
to the $x$-axis and the line $y=mx$, where $m$ is a positive constant.  Find the ratio
of the radius of the larger circle to the radius of the smaller circle as a function of $\theta$,
the angle between the $x$-axis and the line $y=mx$.
\ \\ \ \\
\noindent 2. Prove that $\dps AB - BA \not = I_n$ for any $n \times n$ matrices
$A$ and $B$ over the real numbers, where $I_n$ denotes the $n \times n$ identity matrix.
\ \\ \ \\
\noindent 3. Find
$$\lim_{n \to \infty} \left( \frac{1}{\sqrt n \sqrt {n+1}} + \frac{1}{\sqrt n \sqrt {n+2}} + \cdots
+ \frac{1}{\sqrt n \sqrt {2n}} \right) .$$
\ \\ \ \\
\noindent 4.
\ \\ \ \\
\noindent Part 1. Let $n$ be a positive integer and let $P_{n-1} (x)$ be the polynomial equal to
$P_{n-1} (x) = \frac{x^n - 1}{x-1}$.  Find $P_{n-1} (1)$.
\ \\ \ \\
\noindent Part 2. On the circle of radius 1, let $V_1$, $V_2$, $\ldots$, $V_n$ be the vertices of a
regular $n$-gon
inscribed in the circle.  Let $\lambda _k = dist (V_1, V_k)$, $k=1, 2, \ldots , n$.  Show that
$$\prod_{k=2}^n \lambda _k = n.$$
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\noindent Part 3. For the regular $n$-gon in Part 2, find the product of the lengths of all the line
segments joining the vertices.
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\noindent 5.  Let $S$ be a set of twelve distinct positive integers such that for distinct
$a$, $b$, $c$, and $d$ in $S$,
$a+b \ne c+d$.  Prove that the largest element in $S$ is greater than 56.
\vfill\eject
\begin{center}
2012 Missouri Collegiate Mathematics Competition \\ Session II
\end{center}

\noindent 1. Express 2012 as a sum of (two or more) consecutive integers.
\ \\ \ \\
\noindent 2. Let $f$ be a function that has a continuous derivative over the interval $[a, b]$, and
let $f(a)=f(b)=0$.  Prove that
\[\max_{a\le x \le b} |f'(x)| \ge \frac{4}{(b-a)^2} \int_a^b |f(x)| dx. \]
\ \\ \ \\
\noindent 3. For real $a>0$ define the sequence $\{x_n\}$ by
$$x_{n+1}=a(x_n^2 + 4), \qquad x_0 = 0 .$$
Determine necessary and sufficient conditions on $a$ for $\ds{\lim_{n\to \infty}x_n}$ to exist
and be finite.
\ \\ \ \\
\noindent 4. Let $P(x)$ be a polynomial in $x$ of degree $n>2$.  Suppose that $a \ne b$, 
$P(a) = n_1$, and 
$P(b) = n_2$.  What is the remainder $R(x)$ when $P(x)$ is divided by $(x-a)(x-b)$?
\ \\ \ \\
\noindent 5. Let $S$ be the set of points in $\mathbb{R}^2$ that constitute the graph of
$y=x^2$, $-2 \le x \le
2$, and let $d(p_1, p_2)$ denote the Euclidean distance between $p_1, p_2 \in S$.  Determine $p_1$
and $p_2$ that maximizes $d(p_1,p_2)$.

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