Date: Mon, 17 Apr 2000 09:37:35 -0400 (EDT) From: Andreas Blass To: fac@math.lsa.umich.edu, grad@math.lsa.umich.edu Subject: Undergrad competition report REPORT OF THE UNDERGRADUATE COMPETITION COMMITTEE The 17th annual University of Michigan Undergraduate Mathematics Competition took place on the morning of Saturday, April 8, 2000. Nine students participated. Chetan Balwe won first prize and Richi Raj second. Justin Altman and Dapeng Zhu received honorable mention. As in the past few years, the competition included a lot of questions --- more than any student would be likely to solve in the three hours available, and the questions varied widely in difficulty. For your amusement, a LaTeX file of the questions is appended to this report. Also appended is a PostScript file, chord.eps, used by the LaTeX file to draw a picture. (You don't really need the picture; I think the problem is clear even without it.) The first of the two numbers in brackets after each problem is the number of contestants solving the problem essentially correctly; the second is the number of other students getting at least half credit. I thank the following people who contributed problems: Tim Callahan, Chris Connell, Razvan Gelca, Bob Griess, Mel Hochster, John Lott, Boris Mitavskiy, and Hugh Montgomery. Andreas Blass ------------------------------------------ \documentclass[12pt]{article} \usepackage{amssymb} \usepackage{amsmath} \newtheorem{prtemp$$}{$$Problem} \newenvironment{pr$$}{$$\begin{prtemp}\normalfont$$}{$$\end{prtemp}} \newenvironment{eq$$}{$$\begin{displaymath}$$}{$$\end{displaymath}} \newenvironment{eqnum$$}{$$$$}{$$} \newenvironment{eqs$$}{$$\begin{eqnarray*}$$}{$$\end{eqnarray*}} \newenvironment{eqsnum$$}{$$\begin{eqnarray}$$}{$$\end{eqnarray}} \newenvironment{ls$$}{$$\begin{itemize}$$}{$$\end{itemize}} \newenvironment{lsnum$$}{$$\begin{enumerate}$$}{$$\end{enumerate}} \newcommand{\ger}[1]{\ensuremath{\mathfrak {#1}}} \newcommand{\scr}[1]{\ensuremath{\mathcal {#1}}} \newcommand{\bld}[1]{\ensuremath{\mathbf {#1}}} \newcommand{\bbb}[1]{\ensuremath{\mathbb {#1}}} \begin{document} \begin{center} {\Large \bfseries Undergraduate Math Competition 17}\\ \end{center} Write on the front of your blue book the code letter you were given when you registered for the competition. Do not write your name anywhere on your blue book. Each problem is worth the same amount. For full credit, you must prove that your answers are correct even when the question doesn't say prove.'' There are lots of problems of \emph{widely} varying difficulty. It is \emph{not} expected that anyone will solve them all (or even nearly all); look for ones that seem easy and fun. \begin{pr} A \emph{weakly magic} square of order $n$ and sum $s$ is an $n\times n$ square array of integers such that each of the $n$ rows, each of the $n$ columns, and each of the two diagonals have sum $s$. (The entries in the square need not be consecutive integers and they need not be distinct.) For each positive integer $n$, determine the possible sums $s$ for weakly magic squares of order $n$. [0,2] \end{pr} \begin{pr} A \emph{palindromic} number is one that's the same when written backwards (base 10), like 197791. Find all the palindromic primes that have an even number of digits. [6,0] \end{pr} \begin{pr} $A$, $B$, $C$, and $D$ are points on the boundary of a circle, chosen randomly, independently, and with uniform probability. What is the probability that the line segments $\overline{AB}$ and $\overline{CD}$ intersect? [6,0] \end{pr} \begin{pr} Prove that, for all integers $n\geq2$, $$\sum_{t=n}^{2n-2}\frac1{t^2}<\frac1{2n-1}. \quad [2,0]$$ \end{pr} \begin{pr} You are blindfolded. A pile of $x$ coins is placed in front of you. You are told that $y$ of them are heads. You are not able to tell, by touch, which side of a coin is heads or tails. Your task is to arrange the $x$ coins in two piles in such a way that the number of heads in the one pile is the same as in the other pile. You get one attempt, and no feedback. [3,0] \end{pr} \begin{pr} Start with 6 points in general position in 3-dimensional space. (I'll say later what general position'' means.) Partition them arbitrarily into 3 sets of 2 points each. For each of these 3 pairs, join the two points by a line segment, and let $A$, $B$, and $C$ be the midpoints of these three line segments. General position'' means in this problem that $A$, $B$, and $C$ are not collinear, so that it makes sense to talk about the triangle $ABC$. Let $P$ be the centroid of triangle $ABC$ (the point where the three medians meet). Prove that $P$ depends only on the original 6 points, not on the way they were divided into 3 sets of 2. [5,0] \end{pr} \begin{pr} What is the largest absolute value attained by the function $f(z)=z^{2000}-z^4+1$ as $z$ ranges over the unit circle in the complex plane? [5,0] \end{pr} \begin{pr} Can the rational numbers in the interval $[0,1]$ be enumerated as a sequence $q_1,q_2,\dots$ in such a way that $\displaystyle{\sum_{n=1}^\infty \frac{q_n}n}$ converges? [2,0] \end{pr} \begin{pr} Two circles of radius 1 are tangent, as shown: \vbox{ \special{psfile=chord.eps voffset=-85 hoffset=120 vscale=100 hscale=100} \vskip1.2in} \noindent A line segment is drawn from the center of one circle and intersects the first circle at~$A$ and the second circle at~$B$ and~$C$. If the line segments $AB$ and $BC$ have the same length, find this length. %$\overline{AB}=\overline{BC}=d$, find~$d$. [4,0] \end{pr} \begin{pr} A subset $S$ of the integers from 1 to 177 contains at least one integer from each of the three element sets $\{1,\,2,\,3\}$, $\{4,\,5,\,6\}$, $\{7,\,8,\,9\}$, $\{10,\,11,\,12\}$, \dots, $\{175,\,176,\,177\}$. Prove that $S$ contains distinct integers $a$ and $b$ such that $a$ divides $b$ evenly. 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