\documentclass [12pt]{article}
\usepackage{amsfonts}
\usepackage{geometry}
\usepackage{amsmath}
\setcounter{MaxMatrixCols}{30}
\usepackage{amssymb}
\usepackage{graphicx}

\setlength{\topmargin}{-0.5in} \setlength{\leftmargin}{0in}
\setlength{\oddsidemargin}{0in} \setlength{\evensidemargin}{0in}
\setlength{\textwidth}{6in} \setlength{\textheight}{8.84in}
\newcommand{\lbin}{\left(\!\begin{array}{c}}
\newcommand{\rbin}{\end{array}\!\right)}
\newcommand{\ldet}{\left|\begin{array}}
\newcommand{\rdet}{\end{array}\right|}
\newcommand{\lmat}{\left(\begin{array}}
\newcommand{\rmat}{\end{array}\right)}
\newcommand{\dspace}{\baselineskip 22pt}
\newcommand{\sspace}{\baselineskip 14pt}
\newcommand{\dps}{\displaystyle}
\newcommand{\qed}{\hfill\rule{2mm}{2mm} \\}
\newcommand{\ds}{\displaystyle}
\DeclareMathOperator{\tr}{tr}
\pagestyle{myheadings}

\begin{document}
\thispagestyle{empty}

\begin{center}
2017 Missouri Collegiate Mathematics Competition \\ Session I
\end{center}

\noindent 1.  An \textit{autobiographical number} is a natural number with ten digits or less in which the first digit of the number (reading from left to right) tells you how many zeros are in the number, the second digit tells how many 1's, the third digit tells you how many 2's, and so on.  For example,
\begin{equation*}
6,210,001,000
\end{equation*}
is an autobiographical number.  Find the smallest autobiographical number and prove that it is the smallest.
\ \\ \ \\
\noindent 2. Recall that the trace $\tr(A)$ of an $n\times n$ matrix $A$ is the sum of the entries on the main diagonal of $A$. Let $A^T$ denote the transpose of the matrix $A$. Show that if $A$ and $B$ are $n\times n$ real-valued matrices with $\tr(AA^T+BB^T)=\tr(AB+A^TB^T)$, then it must be the case that $A=B^T$.
\ \\ \ \\
\noindent 3. Let $f$ be a real-valued continuous function on the closed interval $[0,1]$. Assuming that $\int_0^1\, f(x)\, dx=\frac{\pi}{4}$, show that there is some $y$ with $0<y<1$ and $\frac{1}{1+y}<f(y)<\frac{1}{2y}$.
\ \\ \ \\
\noindent 4. A triangle $ABC$
has positive integer sides. Suppose $A=2B$, and $C>90^{\circ}$. Find the
minimum length of the perimeter of $ABC$.
\ \\ \ \\
\noindent 5. Suppose that $x$ and $y$ are chosen at random from the
uniform distribution on the square $(0,2]\times(0,2]$. Given that $\left(
x,y,1\right)  $ are the sides of a triangle, find the probability that the
triangle is acute.
\vfill\eject
\begin{center}
2017 Missouri Collegiate Mathematics Competition \\ Session II
\end{center}

\noindent 1. A smooth function $f(x)$ has $f''(x) >0$ for all $x$ in $[0,1]$.  For each point $a$ in $[0,1]$, draw the tangent line to $y=f(x)$ at the point where $x=a$.  Let $A(a)$ be the area bounded by the curve $y=f(x)$, the tangent line at $a$, and the lines $x=0$ and $x=1$.  What value(s) of $a$ minimizes the area?
\ \\ \ \\
\noindent 2. Let $n$ be a positive integer.  Evaluate
\[
\int_{0}^{\pi/2}\dfrac{\sin\left[  \left(  2n+1\right)  x\right]  }{\sin
x}\,dx.
\]
\ \\ \ \\
\noindent 3.  Let $n$ be a fixed positive integer. Let $M_{n}\left( \mathbb{Z}_{2}\right) $
denote the set of $n\times n$ matrices with entries in $\mathbb{Z}_{2}$.
Given $A\in M_{n}\left( \mathbb{Z}_{2}\right) $, what is the probability
that $A$ is invertible?  (Hint: Recall that $\mathbb{Z}_{2} = \{ 0,1 \}$ and all arithmetic is modulo 2.)
\ \\ \ \\
\noindent 4. Find, with proof, all solutions in nonnegative integers $x$, $y$,
$z$ to
\begin{equation}
2^{x}+4^{y}=6^{z}.
\end{equation}
\ \\ \ \\
\noindent 5.  Given $2017$ points in the plane, what is the largest number of line segments that can be drawn connecting these points such that distinct segments intersect only at their endpoints?

\end{document}
