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\begin{center}
${\bf 34\th}$ \textbf{United States of America Mathematical
Olympiad}
\end{center}

\begin{center}
\textbf{Day I \hspace{6mm} 12:30 PM -- 5 PM EDT}
\end{center}

\begin{center}
\textbf{April 18, 2006}
\end{center}

\bigskip


\begin{enumerate}
\item %LIU1

Find all positive integers $n$ such that, for some integer $k>1$,
there exist $k$ positive rational numbers $a_1, a_2, \dots, a_k$
whose sum and product are both $n$.

\item %HEU1
Can a square in the plane be covered by two smaller squares?

\item %AND1
For any positive integer $n$ let $p(n)$ be the greatest prime
divisor of $n$. Find all polynomials $f$ with integral coefficients
such that the sequence $p(f(n^2))-2n$ is upper-bounded.

%\item %GIBBS1
%For positive integers $k$ and $N$ let $P(k,N)$ be the following
%proposition:
%\\
%``Given $2k+1$ distinct positive integers with sum greater than $N$,
%there exists a subset of size $k$ with sum greater than
%$N/2$.''\\
%For a given $k$, find in terms of $k$, $N_t$, the maximum value of
%$N$ for which $P(k,N)$ is true, and $N_f$, the minimum value of $N$
%for which $P(k,N)$ is false.

\newpage

\begin{center}
${\bf 34\th}$ \textbf{United States of America Mathematical
Olympiad}
\end{center}

\begin{center}
\textbf{Day II \hspace{6mm} 12:30 PM -- 5 PM EDT}
\end{center}

\begin{center}
\textbf{April 19, 2006}
\end{center}

\item %KED1
For $x$ a real number, let $\{x\} = x - \lfloor x \rfloor$ denote
the fractional part of $x$. Let $p$ be a prime number and let $s$ be
an integer with $0 < s < p$. Prove that there exist integers $m,n$
with $0<m<n<p$ and
\[
\left\{ \frac{sm}{p} \right\} < \left\{ \frac{sn}{p} \right\} <
\frac{s}{p}
\]
if and only if $s$ is not a divisor of $p-1$.

\item %SUN2
A (one player) game is played as follows. Given a positive natural
number $n$ one move consist of replacing $n$ either by $n+1$ or by
$n+2^{m_n+1}$, where $2^{m_n}$ is the largest power of 2 dividing
$n$ (in particular, we can add 2 to odd numbers). The game starts
with the number 1. Show that if $k$ is positive integer greater than
1, then the minimal number of moves needed to reach the number
$2^ik$, $i \geq 0$, is strictly greater than the minimal number of
moves needed to reach $2^i$.

\item %FENG3
Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be
points on sides $AD$ and $BC$, respectively such that $AE/ED =
BF/FC$. Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$,
respectively. Prove that the circumcircles of triangles $SAE$,
$SBF$, $TCF$, and $TDE$ pass through a common point.


\end{enumerate}
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