Caros Ed et al,
Eu queria agradecer (com algum atraso; eu estava na SBPC em Recife, com o
Paulo Jose', e nao estava facil conseguir computador) as mensagens (um tanto
exageradas, como a sua e a do Wagner) sobre ter entrado um probleminha nosso
na IMO. E' claro que eu tambem fiquei contente...
Essa prova da IMO deu trabalho. Acho (e espero) que as notas de corte
devam estar baixas esse ano... E' para mandar solucoes (mesmo feias) ?
Abracos,
Gugu
>
>Parab�ns, Gugu.
>
>Isso s� confirma que voc� � um dos melhores criadores
>de problemas (no bom sentido) do mundo.
>Os �ltimos bancos j� indicavam que era s� uma quest�o
>de tempo (para quem n�o sabe, o Gugu j� colocou v�rios
>problemas nas short lists).
>
>O Brasil confirma que est� evoluindo em todos os
>sentidos!!
>
>Abra�os, Ed.
>
>
>
>
>--- [EMAIL PROTECTED] wrote:
>>
>>
>> Prova da IMO retirada do Site
>> http://www.mathlinks.go.ro/
>>
>> O Problema 1 � 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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