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\begin{center}
${\bf 36\th}$ \textbf{United States of America Mathematical
Olympiad}
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\begin{center}
\textbf{Day I \hspace{6mm} 12:30 PM -- 5 PM EDT}
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\begin{center}
\textbf{April 24, 2007}
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\bigskip

\begin{enumerate}

\item %VAN1
Let $n$ be a positive integer. Define a sequence by setting
$a_1=n$ and, for each $k >1$, letting $a_k$ be the unique integer
in the range $0 \leq a_k \leq k-1$ for which $a_1+a_2+\dots+a_k$
is divisible by $k$. For instance, when $n=9$ the obtained
sequence is $9,1,2,0,3,3,3,\dots$~. Prove that for any $n$ the
sequence $a_1,a_2,a_3,\dots$ eventually becomes constant.



\vspace*{.3in}

\item %GAL2
A square grid on the Euclidean plane consists of all points
$(m,n)$, where  $m$ and $n$ are integers. Is it possible to cover
all grid points by an infinite family of discs with
non-overlapping interiors if each disc in the family has radius at
least 5?


\vspace*{.3in}

\item %ROU1
Let $S$ be a set containing $n^2 + n - 1$ elements, for some
positive integer $n$.  Suppose that the $n$-element subsets of $S$
are partitioned into two classes.  Prove that there are at least
$n$ pairwise disjoint sets in the same class.


\end{enumerate}


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{\small
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Copyright \copyright\ \ Committee on the American Mathematics
Competitions,\\
Mathematical Association of America
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}

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