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The 35th International Mathematical Olympiad (July 13-14, 1994, Hong Kong)
{\noindent \bf 1.}
Let $m$ and $n$ be positive integers. Let $a_1, a_2, \dots, a_m$
be distinct elements of $\{1, 2, \dots, n\}$ such that whenever $a_i +
a_j \leq n$ for some $i, j$, $1 \leq i \leq j \leq m$, there exists $k$,
$1 \leq k \leq m$, with $a_i + a_j = a_k$. Prove that
\[
\frac{a_1 + a_2 + \cdots + a_m}{m} \geq \frac{n+1}{2}.
\]
{\noindent \bf 2.}
$ABC$ is an isosceles triangle with $AB = AC$. Suppose that
\begin{enumerate}
\item $M$ is the midpoint of $BC$ and $O$ is the point on the line
$AM$ such that $OB$ is perpendicular to $AB$;
\item $Q$ is an arbitrary point on the segment $BC$ different from
$B$ and $C$;
\item $E$ lies on the line $AB$ and $F$ lies on the line $AC$
such that $E$, $Q$, $F$ are distinct and collinear.
\end{enumerate}
Prove that $OQ$ is perpendicular to $EF$ if and only if $QE = QF$.
{\noindent \bf 3.}
For any positive integer $k$, let $f(k)$ be the number of
elements in the set $\{k+1, k+2, \dots, 2k\}$ whose base 2 representation
has precisely three 1s.
\begin{itemize}
\item (a) Prove that, for each positive integer $m$, there exists at
least one positive integer $k$ such that $f(k) = m$.
\item (b) Determine all positive integers $m$ for which there exists
exactly one $k$ with $f(k) = m$.
\end{itemize}
{\noindent \bf 4.}
Determine all ordered pairs $(m, n)$ of positive integers such that
\[
\frac{n^3 + 1}{mn - 1}
\]
is an integer.
{\noindent \bf 5.}
Let $S$ be the set of real numbers strictly greater than $-1$.
Find all functions $f: S \to S$ satisfying the two conditions:
\begin{enumerate}
\item $f(x + f(y) + xf(y)) = y + f(x) + yf(x)$ for all $x$ and $y$ in $S$;
\item $\frac{f(x)}{x}$ is strictly increasing on each of the intervals
$-1 < x < 0$ and $0 < x$.
\end{enumerate}
{\noindent \bf 6.}
Show that there exists a set $A$ of positive integers with the
following property: For any infinite set $S$ of primes there exist two
positive integers $m \in A$ and $n \notin A$ each of which is a product
of $k$ distinct elements of $S$ for some $k \geq 2$.
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