Re: [obm-l] Problemas da IMO

["Eduardo Wagner" <wagner@xxxxxxx>]

Parabens Gugu!!!!!

Voce eh realmente f (no bom sentido).
Eu ja carimbei alguns na OIM, mas na IMO,
sequer tentei.

Um grande abraco,

E. Wagner.

----------
>From: gugu@impa.br
>To: obm-l@mat.puc-rio.br
>Subject: [obm-l] Problemas da IMO
>Date: Mon, Jul 14, 2003, 3:38 PM
>

>
>
> 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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