Date: Wed, 14 Apr 1999 17:21:40 -0400 (EDT)
From: Andreas Blass <ablass@math.lsa.umich.edu>
To: fac@math.lsa.umich.edu, grad@math.lsa.umich.edu
Subject: (UM)^2 C^16 report

REPORT OF THE UNDERGRADUATE COMPETITION COMMITTEE

The 16th annual University of Michigan Undergraduate Mathematics
Competition took place on the morning of Saturday, April 3, 1999.
Nineteen students participated.  Dapeng Zhu and Chetan Balwe were tied
for first place, and Kurt Steinkraus was third.

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.
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: Mariah Birgen,
Tim Callahan, Mel Hochster, Mark Krosky, Alexander Kurganov, John
Lott, and David Winter. 

Andreas Blass
------------------------------------------
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\begin{document}

\begin{center}
{\Large \bfseries Undergraduate Math Competition 16}\\
\end{center}

\begin{pr}
Find the last six digits of $1999^{1999}$. [5,2]
\end{pr}

\begin{pr}
For any integer~$n\geq2$, factor it into primes.  Then add up all the
prime factors (including duplicates) and call this~$f(n)$.  Then
create the sequence $\{a_n:n\geq2\}$ with $a_n=1/f(n)$.  For example, a
portion of this sequence would be calculated like this:
$$\vbox{\ialign{\strut$#$\hfil&$\quad\Rightarrow\quad$#&$#$\hfil
                &$\quad\Rightarrow\quad$#&$#$\hfil\cr
        n=14=2\times7&&f(14)=2+7=9&&a_{14}=1/9\cr
        n=15=3\times5&&f(15)=3+5=8&&a_{15}=1/8\cr
        n=16=2\times2\times2\times2&&f(16)=2+2+2+2=8&&a_{16}=1/8\cr
        n=17&&f(17)=17&&a_{17}=1/17\cr
        n=18=2\times3\times3&&f(18)=2+3+3=8&&a_{18}=1/8.\cr}}$$
The sequence~$\{a_n:n\geq2\}$ begins
$$\def\\#1,{{\frac1{#1}},}
        \left\{\\2,\\3,\\4,\\5,\\5,\\7,\\6,\\6,\\7,\\11,\\7,\\13,
        \\9,\\8,\\8,\\17,\\8,\\19,\\9,\\10,\\13,
        \\23,\\9,\\10,\\15,%\\9,%\\11,%\\29,\\10,\\31,
        \ldots\,\right\}.$$
Prove that $\lim_{n\to\infty}a_n=0$.  Does the
        series~$\sum_{n=1}^\infty a_n$
converge? [2,3]
\end{pr}

\begin{pr}
How many solutions in non-negative integers $m$ and $n$ are there to
the equation $6m+11n=1999$? [9,2]
\end{pr}

\begin{pr}
(a) If each of three circles of radius 1 is (externally) tangent to the
other two, find the radius of the little circle inscribed in the space
among them.

(b) If each of 2000 spheres of radius 1 in 1999-dimensional space is
(externally) tangent to all of the others, find the radius of the
little sphere inscribed in the space in the midst of them. [0,7]
\end{pr}

\begin{pr}
Suppose you have nine coins and a balance.  Eight of the coins weigh
the same and the ninth is lighter.  How can you tell which coin is the
light one by using the balance only twice?  (One use of the balance
means choosing two disjoint subsets of the coins and finding out which
has the smaller total weight.) [18,0]
\end{pr}

\begin{pr}
A floor is tiled with squares of side 1, in the usual checkerboard
fashion.  A small square of side $L\leq1/\sqrt2$ is thrown at random on the
floor.  What is the probability that it does not land on any cracks? [2,0]
\end{pr}

\begin{pr}
Find all positive integer solutions of $a^b=b^a$.  (Be sure to prove
that you have all the solutions.) [3,0]
\end{pr}

\begin{pr}
 A set $S$ has two binary operations $\#$ and $*$ on it, and the
following axioms hold: 
\begin{lsnum}
\item There is an element $z$ in $S$  such that $z  \# s = s$ for all
$s \in S$.
\item For all $s,\,t, u \in S$  if  $s \# u = t \# u$ then $s =
t$.
\item For all $s,\,t \in S$ if  $z*s = z*t$ then $s = t$.
\item For all $s,\,t,\,u \in S$,  $(s \# t)*u =  (s*u)
\#(t*u)$.
\end{lsnum}
Prove that $S = \{z\}$. [5,0]
\end{pr}

\begin{pr}
Assume:
\begin{lsnum}
\item $f(x)$ is a real-valued function defined on the whole real line.
\item $f(x)$ is bounded on each finite interval.
\item The difference $f(x+1)-f(x)$ approaches a limit $l$ as $x\to\infty$. 
\end{lsnum}
Prove that the ratio $f(x)/x$ also approaches the same limit $l$ as
$x\to\infty$. [1,1]
\end{pr}

\begin{pr}
Take a regular icosahedron inscribed in a unit sphere with
center~$O$, and let $A$ and $B$ be two vertices connected by an edge.
Calculate~$\cos(\angle AOB)$.  You may use the fact that
$$\cos\Bigl(\frac{2\pi}5\Bigr)={\frac{\sqrt5-1}4}.$$ [0,0]
\end{pr}

\end{document}