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
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