on 26.04.04 15:38, Daniel Silva Braz at [EMAIL PROTECTED] wrote: > Pessoal, > Problemas da olimp�adas americana... > � > 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 > inradius, respectively. Prove that > 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 > Oi, Daniel:
As solucoes para estes problemas da USAMO estao aqui: http://www.kalva.demon.co.uk/usa/usa02.html Obviamente, se alguem tiver alguma solucao diferente das apresentadas no site, deve envia-la para a lista. Por outro lado, talvez seja mais interessante concentrar os esforcos em problemas olimpicos cujas solucoes nao estejam disponiveis - por exemplo, a olimpiada polonesa de 1983. []s, Claudio. ========================================================================= 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 =========================================================================
