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Re: [obm-l] Questões da USAMO



Claudio,
Obrigado pela indicação do site (aliás, muito
interessante e útil)!!

Onde eu posso encontrar os problemas da olimpiadas
polonesa de 83?

Daniel S. Braz

========================

 --- Claudio Buffara <claudio.buffara@terra.com.br>
escreveu: > on 26.04.04 15:38, Daniel Silva Braz at
> dsbraz@yahoo.com.br wrote:
> 
> > Pessoal,
> > Problemas da olimpíadas americana...
> >  
> > 1. Let S be a set with 2002 elements, and let N be
> an
> > integer with 0 · N · 22002. Prove
> > that it is possible to color every subset of S
> either
> > black or white so that the following
> > conditions hold:
> > (a) the union of any two white subsets is white;
> > (b) the union of any two black subsets is black;
> > (c) there are exactly N white subsets.
> > 
> > 2. Let ABC be a triangle such that
> > 
> > (cot A/2)^2 + (2cot A/2)^2 + (3cot A/2)^2 =
> (6s/7r)^2
> > 
> > where s and r denote its semiperimeter and its
> > inradius, respectively. Prove that
> > triangle ABC is similar to a triangle T whose side
> > lengths are all positive integers with
> > no common divisor and determine these integers.
> > 
> > 3. Prove that any monic polynomial (a polynomial
> with
> > leading coefficient 1) of degree n
> > with real coefficients is the average of two monic
> > polynomials of degree n with n real
> > roots.
> > 
> > Daniel Silva Braz
> > 
> Oi, Daniel:
> 
> As solucoes para estes problemas da USAMO estao
> aqui:
> http://www.kalva.demon.co.uk/usa/usa02.html
> 
> Obviamente, se alguem tiver alguma solucao diferente
> das apresentadas no
> site, deve envia-la para a lista.
> 
> Por outro lado, talvez seja mais interessante
> concentrar os esforcos em
> problemas olimpicos cujas solucoes nao estejam
> disponiveis - por exemplo, a
> olimpiada polonesa de 1983.
> 
> []s,
> Claudio. 
> 
> 
>
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