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. > > > > ------------------------------------------------- > This mail sent through IMP: http://horde.org/imp/ > ========================================================================= > Instru��es para entrar na lista, sair da lista e > usar a lista em > http://www.mat.puc-rio.br/~nicolau/olimp/obm-l.html > ========================================================================= __________________________________ Do you Yahoo!? SBC Yahoo! DSL - Now only $29.95 per month! http://sbc.yahoo.com ========================================================================= Instru��es para entrar na lista, sair da lista e usar a lista em http://www.mat.puc-rio.br/~nicolau/olimp/obm-l.html =========================================================================
