Re: New try 3

["Luis Lopes" <llopes@xxxxxxxxxxxxx>]

Saudac,o~es,

Envio a mensagem do prof. Rousseau que acabei de
receber. Vou escrever pra ele dando as instruc,o~es
pra se tornar membro da nossa lista.

Ale'm da matema'tica iremos aprender ingle^s tambe'm.

[ ]'s
Lui's

-----Mensagem Original-----
De: <ccrousse@memphis.edu>
Para: Luis Lopes <llopes@ensrbr.com.br>
Enviada em: Sexta-feira, 4 de Agosto de 2000 12:01
Assunto: Re: New try 3


Dear Luis:

    This morning, I had 4 messages - "another try", "new try 1", "new try
2",
and
"new try 3".  I hope that this means that all of your messages got through.
I would be happy to be on your list.  Please keep in mind that often the
time that I have to devote to working on the problems will be quite limited,
due to other responsibilities, but I will be glad to help insofar as I am
able.  This coming semester, I shall be working with a group of
undergraduates
here as they prepare to take the W. L. Putnam exam.  Perhaps you and
your fellow problem enthusiasts would be interested in having the
materials that I prepare for these students.  If so, I can send them to
you (as LaTeX files) as they are ready,

All the best,

Cecil

Luis Lopes wrote:

> The last one.
>
> Regards,
> Luís
>
> -----Mensagem Original-----
> De: Luis Lopes <llopes@ensrbr.com.br>
> Para: <ccrousse@memphis.edu>
> Enviada em: Quinta-feira, 3 de Agosto de 2000 16:44
> Assunto: Re: Inequalities
>
> > Dear Cecil,
> >
> > Thank you for your solution. One of the members of the list had already
> sent
> > in one solution but yours is much simpler. I wonder whether I can
forward
> > your message (with your email address)  to the list. I think its members
> will
> > appreciate and be happy with your participation.
> >
> > We have good problem-solvers on it and solutions to some of the last IMO
> > problems have been sent in.
> > I will wait for your answer before proposing your inequality. We have a
> > problem in writing maths on the emails. Everyone has his own way to
> > represent the math language. I intend to collect some problems/solutions
> and
> > create a pdf file with TeX with them. When it is large enough I would
put
> it
> > on the list.
> >
> > In the meantime I will forward you two problems that came from the list.
> > Source: Moldavia Republic www.
> > These nobody gave an answer. But please, don't misunderstand me. I do
not
> > expect you to solve (answer me) these problems or do not want to have
the
> > feeling that I am disturbing or bothering you. I lack experience and
time
> in
> > trying to solve olympic problems and like sharing experience.
> >
> > Thank you for all your attention and kindly replies.
> >
> > CHEERS,
> > Luís Lopes
> >
> > ===== My translation
> > 1)
> > Encontre todas as funcoes f:R->R que verifiquem a relacao
> >    x*f(x)=[x]*f({x})+{x}*f([x]), para todo x pertencente a R, onde [x]
> > representa a parte inteira de x e {x} representa a parte fracionaria de
x.
> >
> > Find out all the functions f:R->R that satisfy the relation
> >    x*f(x)=[x]*f({x})+{x}*f([x]), for all $x\in R$, where [x]  represents
> the
> > integer part of x and {x} represents  its  fractional part.
> >
> > 2)
> > Encontre todos os valores inteiros de m para os quais a equacao
> >    [((m^2)*x-13/1999)]=(x-12)/2000 tem 1999 solucoes reais distintas,
onde
> > [k] representa a parte inteira de k
> >
> > Find out all integer values of $m$ for which the equation
> > [((m^2)*x-13/1999)]=(x-12)/2000 has 1999 different real solutions, where
> [k]
> > represents the integer part of k
> >
> > I understand the following: the equation is
> >
> > $$\left\lfloor (m^2)x - \frac{13}{1999} \right\rfloor =
> \frac{x-12}{2000}$$
> >
> > Source (I think):  http://www.olsedim.com/olympiad/problems.html
> >
> >  ========