Re: [obm-l] Questões da USAMO

[Claudio Buffara <claudio.buffara@xxxxxxxxxxxx>]

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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