\documentclass {article}
\begin {document}
\begin {center}
The 2nd examination for IMO
\end {center}
\begin {itemize}
\item {Problem \# 1}
Consider $A_1,A_2...A_n$ points in the same plane $\alpha$ and V outside
this plane where $n\geq 4$. A plane $\pi$ cuts the edges
$VA_1, VA_2\ldots VA_n$ in $B_1, B_2\ldots B_n$ respectively. Prove
that if $A_1...A_n$ and $B_1...B_n$ are similar n-gons then $\pi$ is
paralallel with $\alpha$
\item {Problem \#2}
Consider $A$ the set of integers that can be written as $a^2+2\cdot b^2$
where $a,b\in \Z$ and $b\neq0$. If $p$ is prime and $p^2\in A$ then
$p\in A$.
\item {Problem \#3}
Consider $p\geq5$ a prime number, $k \in \{0,1\ldots p-1\}$ and $A$ the
set of all positive integers that do not contain $k$ in their base $p$
expansion. Find the greatest number of terms that a non-constant
arithmetic progression consisting of elements of $A$ can have.
\item {Problem \# 4}
Consider $p,q,r$ pairwise distinct prime numbers $>$ 2 and $A= \{p^a \cdot
q^b \cdot r^c \mid 0\leq a,b,c\leq 5, a,b,c\in Z\}$ Find the lowest number
$n$ such that any subset of $A$ with $n$ elements contains two distinct
elements $x$ and $y$ such that $x\mid y$.
\end {itemize}
\end {document}