RE: [obm-l] IMO!?!?

[Ralph Teixeira <RALPH@xxxxxx>]

Segue aqui em TeX a prova (veja no final). Depois a gente ve onde na Web ela
esta/estaria/estarah.

Acho que o pessoal foi bem, mas prefiro esperar um pouco mais para dar a
noticia oficial. Jah terminamos de corrigir duas questoes dos Brasileiros.
Uma foi um arraso (Questao 6 -- 1/0/0/0/0/0), mas a outra foi razoavel
(Questao 1 -- 0/7/7/6/6/6).

Internet aqui estah dificil... Se der, eu ou o Ed mandamos mais noticias aa
medida que tudo se desenrolar.

Obrigado,
        Ralph

{\bf Problem 1}\par\nobreak

Let $n$ be a positive integer. \ Let $T$ be the set of
points $(x,y)$ in the plane where $x$ and $y$ are non-negative
integers and $x+y<n$. \ Each point of $T$ is coloured red or
blue. \ If a point $(x,y)$ is red, then so are all points $(x',y')$
of $T$ with both $ $$x'\leq x$ and $y'\leq y$. \ Define an $X$-set
to be a set of $n$ blue points having distinct $x$-coordinates,
and a $Y$-set to be a set of $n$ blue points having distinct
$y$-coordinates. \ Prove that the number of $X$-sets is equal
to the number of $Y$-sets.

{\bf Problem 2}\par\nobreak

Let \ $BC$ be a diameter of the circle ${\Gamma}$ with
centre $O$. \ Let $A$ be a point on $\Gamma$ such that $0{{}^\circ
}<\angle AOB<120{{}^\circ}$. \ Let $D$ be the midpoint of the
arc $AB$ not containing $C$. \ The line through $O$ parallel
to $DA$ meets the line $AC$ at $J$. \ The perpendicular bisector
of $OA$ meets $\Gamma$ at $E$ and at $F$. \ Prove that $J$ is
the incentre of the triangle $CEF$.

{\bf Problem 3}\par\nobreak

Find all pairs of integers $m,n\geq3$ such that there
exist infinitely many positive integers $a$ for which
$$a^m+a-1\over a^n+a^2-1$$

is an integer.

{\bf Problem 4}\par\nobreak

Let $n$ be an integer greater than 1. \ The positive
divisors of $n$ are $d_1,d_2,\ldots,d_k$ where

\vskip 6pt\leftskip=0pt plus1fil\parfillskip=0pt
$1=d_1<d_2<\cdot\,\cdot\,\cdot<d_k=n$.

\vskip 6pt\leftskip=0pt\parfillskip=0pt plus1fil
\relax Define $D=d_1d_2+d_2d_3+\,\cdot\,\cdot\,\cdot\,+d_{k-1}d_k$.

\vskip 6pt\hangindent=27pt\leftskip=27pt
\noindent\setbox0=\hbox{(a)}\tabnone\setbox0=\hbox{Prove that
$D<n^2$.}\dimen1=-27pt\tableft

\vskip 6pt\hangindent=27pt
\noindent\setbox0=\hbox{(b)}\tabnone\setbox0=\hbox{Determine
all $n$ for which $D$ is a divisor of $n^2$.}\dimen1=-27pt\tableft

{\bf Problem 5}\par\nobreak

Find all functions $f$ from the set $\rm r$ of real numbers
to itself such that

\vskip 6pt\linedif=\baselineskip\advprev$$\left(f(x)+f(z)\right
)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz)$$

for all $x,y,z,t$ in $\rm r$.

{\bf Problem 6}\par\nobreak

Let $\Gamma_1,\Gamma_2,\ldots,\Gamma_n$ be circles of
radius 1 in the plane, where $n\geq3$. \ Denote their centres
by $O_1,O_2,\ldots,O_n$ \ respectively. \ Suppose that no line
meets more than two of the circles. \ Prove that
$$\sum^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over
4}.$$


-----Original Message-----
From: Rodrigo Villard Milet
To: Obm
Sent: 24/07/02 11:17
Subject: [obm-l] IMO!?!?

Onde eu acho a prova da imo de hj ?!? Se alguém já tiver, por favor
mande para a lista.
Obrigado !
 
Villard
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