Pessoal, Problemas da olimpíadas 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 ______________________________________________________________________ Yahoo! Messenger - Fale com seus amigos online. Instale agora! http://br.download.yahoo.com/messenger/ ========================================================================= Instruções para entrar na lista, sair da lista e usar a lista em http://www.mat.puc-rio.br/~nicolau/olimp/obm-l.html =========================================================================