[obm-l] Prova da IMC - 1o. dia (correcao)

[yurigomes@xxxxxxxxxxxxxx]

 Bem, num sei se a correcao que fiz no ultimo email vai chegar, entaum estou
novamente corrigindo o enunciado do problema 1.

 
>1) Let S be an infinite set of real numbers such that
>|s_1 + s_2 + ... + s_k| < 1 for every finite subset
>{s_1,s_2,...,s_k} of S. Show that S is countable.
>
>2)Let P(x) = x^2 - 1. How many distinct real solutions
>does the following equation have:
>P(P(...(P(x))...)) = 0? [com P sendo aplicado 2004
>vezes]
>
>3) Let S_n be the set of all sum x_1+x_2+...x_n, where
>n>=2, 0<=x_1,...,x_n<="pi"/2 and
>sin(x_1) + sin(x_2) + ... + sin(x_n) = 1
>a) Show that S_n is an interval.
>b)Let l_n be the length of S_n. Find lim(n->infinito)(l_n). 
>
>4)Suppose n>=4 and let M be a finite set of n points in
>R^3, no four of which lie in a plane. Assume that the
>points can be coloured black or white so that any of
>the sphere which intersect M in at least four points have
>the property that exactly half of the points in the
>intersection of M and the sphere are white. Prove that
>all of the points in M lie on one sphere.
>
>5) Let X be a set of binomial(2k-4, k-2) + 1 real numbers,
>k>=2. Prove that there exists a monotone sequence x_1, x_2, ..., x_k in
X such that |x_{i+1} - x_1| >= 2|x_i - x_1|
>for all i = 2,...,k-1.
>
>6) For every complex number z != 0,1 define
> f(z) := sum((log z)^(-4)),
>where the sum is over all branches of the complex
>logarithm.
>a) Show that there are two polynomials P and Q such
>that f(z) = P(z)/Q(z) for all z in C\{0,1}
>b) Show that for all z in C\{0,1}
>f(z)=z(z^2 + 4z + 1)/6(z-1)^4. 


Até mais, 

Yuri



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