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{\footnotesize December 2005}

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\begin{document}

\begin{centre}
  \bf \Large
 Functional Equations (S)
\end{centre}

%\subsection*{Theory}
%
%specific values, injective, surjective, bijective, eliminate
%trivial solutions, cauchy's equation and methods to extend to
%$\R$, multiplicative functions, transformation, substitute in
%things to make things cancel, keep substituting, substitute some
%more...

\subsection*{Problems}

\begin{enumerate}

\item (Russia 2000) Find all functions $f\map{\R}{\R}$ such that
the following equation holds for all real $x$, $y$ and $z$: \[
f(x+y)+f(y+z)+f(z+x)\geq 3f(x+2y+3z).
\]

\item Suppose $f$ is a real-valued function defined on the real
numbers satisfying the following properties:
\begin{enumerate}
\item $f(x+y)+f(x-y)=2f(x)f(y)$ for all $x$, $y$, \item There
exists $x_0\in\R$ with $f(x_0)=-1$.
\end{enumerate}
Prove that $f$ is periodic.

\item (Balkan 1987) Consider a function $f\map{\R}{\R}$ such that
$f(0)=1/2$ and there exists $a$ such that for all $x$ and $y$,
\[
f(x+y)=f(x)f(a-y)+f(y)f(a-x).
\]
Prove $f$ is constant.

\item Find all surjective functions from the set of positive
integers to itself with the property that $m|n$ if and only if
$f(m)|f(n)$.

\item (IMO 1992) Find all functions $f\map{\R}{\R}$ such that
$f(x^2+f(y))=y+f(x)^2$.

\item Find all increasing functions $f\map{\N}{\N}$ with the
property that $f(f(n))=3n$ for all $n$.

\item $f\map{\N}{\N}$ (0$\in\N$) $$xf(y)+yf(x)=(x+y)f(x^2+y^2)$$

\item $f\map{\Q}{\Q}$, $$f(x+f(y))=f(x)+y$$

\item $f\map{\R}{\R}$,
$$(f(x)+f(z))(f(y)+f(t))=f(xy-zt)+f(xt+yz)$$


\item (IMO 1999) Find all functions $f\map{\R}{\R}$ such that for
all real $x$ and $y$ we have
$$f(x-f(y))=f(f(y))+xf(y)+f(x)-1.$$

\end{enumerate}


\end{document}
