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${\bf 34\th}$ \textbf{United States of America Mathematical
Olympiad}
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\begin{center}
\textbf{Day II \hspace{6mm} 12:30 PM -- 5 PM EDT}
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\begin{center}
\textbf{April 20, 2005}
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\bigskip

\begin{enumerate}

\item %JOHN1
Legs $L_1, L_2, L_3, L_4$ of a square table each have
 length $n$, where $n$ is a positive integer.  For how many ordered
 4-tuples $(k_1, k_2, k_3, k_4)$ of nonnegative integers can we cut
 a piece of length $k_i$ from the end of leg $L_i \; (i=1,2,3,4)$ and
still have a stable table?
 (The table is {\em stable} if it can be placed so that
 all four of the leg ends touch the floor. Note that a cut leg of length 0
is permitted.)

\vspace{5mm}

\item %KED1
Let $n$ be an integer greater than 1. Suppose $2n$ points are
given in the plane, no three of which are collinear. Suppose $n$
of the given $2n$ points are colored blue and the other $n$
colored red. A line in the plane is called a {\em balancing line}
if it passes through one blue and one red point and, for each side
of the line, the number of blue points on that side is equal to
the number of red points on the same side. Prove that there exist
at least two balancing lines.


\vspace{5mm}

\item %AND3
 For $m$ a positive integer, let $s(m)$ be the sum of the
 digits of $m$.
For $n\ge 2$, let $f(n)$ be the minimal $k$ for which there
 exists a set $S$ of $n$ positive integers such that
 $s\left(\sum_{x\in X} x\right)=k$ for any nonempty subset $X\subset S$.
 Prove that there are constants $0<C_1<C_2$ with
$$C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.$$


\end{enumerate}

\vfill
{\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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