# Re: [obm-l] Questões da USAMO

```Acho q não hein...

USAMO = United States of America Mathematical Olympiad```
```
Daniel S. Braz

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

--- Johann Peter Gustav Lejeune Dirichlet
<[EMAIL PROTECTED]> escreveu: > Nossa,
tem uma olimpiada na qual participam todas as
> Americas???!!!!
>
> Daniel Silva Braz <[EMAIL PROTECTED]>
> wrote:Pessoal,
>
> 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
> 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
>
>
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