Re: [obm-l] Problemas da IMO

[edmilson motta <edeale@xxxxxxxxx>]

Parabéns, Gugu.

Isso só confirma que você é um dos melhores criadores
de problemas (no bom sentido) do mundo. 
Os últimos bancos já indicavam que era só uma questão
de tempo (para quem não sabe, o Gugu já colocou vários
problemas nas short lists).

O Brasil confirma que está evoluindo em todos os
sentidos!!

Abraços, Ed.




--- gugu@impa.br wrote:
> 
> 
> Prova da IMO retirada do Site
> http://www.mathlinks.go.ro/
> 
> O Problema 1 é nois que mandou...
> 
> 
> First Day - 44th IMO 2003 Japan 
> 
> 1. Let A be a 101-element subset of the set
> S={1,2,3,...,1000000}. Prove that 
> there exist numbers t_1, t_2, ..., t_{100} in S such
> that the sets 
> 
> Aj = { x + tj | x is in A } for each j = 1, 2, ...,
> 100 
> 
> are pairwise disjoint. 
> 
> 
> 2. Find all pairs of positive integers (a,b) such
> that the number 
> 
> a^2 / ( 2ab^2-b^3+1) is also a positive integer. 
> 
> 3. Given is a convex hexagon with the property that
> the segment connecting the 
> middle points of each pair of opposite sides in the
> hexagon is  sqrt(3) / 2 
> times the sum of those sides' sum. 
> 
> Prove that the hexagon has all its angles equal to
> 120. 
> 
> 
> Second Day - 44th IMO 2003 Japan 
> 
> 4. Given is a cyclic quadrilateral ABCD and let P,
> Q, R be feet of the 
> altitudes from D to AB, BC and CA respectively.
> Prove that if PR = RQ then the 
> interior angle bisectors of the angles < ABC and <
> ADC are concurrent on AC. 
> 
> 5. Let x1 <= x2 <= ... <= xn be real numbers, n>2. 
> 
> a) Prove the following inequality: 
> 
> (sum  ni,j=1 | xi - xj | ) 2 <= 2/3 ( n^2 - 1 )sum
> ni,j=1 ( xi - xj)^2 
> 
> b) Prove that the equality in the inequality above
> is obtained if and only if 
> the sequence (xk) is an arithemetical progression. 
> 
> 6. Prove that for each given prime p there exists a
> prime q such that n^p - p 
> is not divisible by q for each positive integer n. 
> 
> 
> 
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