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Re: Fw: Putnam 2001




   Oi,
O que é Putnam? É tipo uma imo?







>From: "Marcio" <mcohen@iis.com.br>
>Reply-To: obm-l@mat.puc-rio.br
>To: <obm-l@mat.puc-rio.br>
>Subject: Fw: Putnam 2001
>Date: Sun, 2 Dec 2001 15:26:30 -0200
>
>Aqui estao as questoes do Putnam 2001. Ainda nao tive tempo para pensar em
>todas. O autor desse email parece ser muito bom, mas mesmo assim ele disse
>que ainda nao conseguiu fazer 3 questoes, como vcs podem ver ai em baixo.
>Eu ja tentei fazer as questoes desde a A1 ateh A5. A A5 ainda me restam 6
>valores de x para considerar, nao estou conseguindo elimina-los. As 
>solucoes
>para os problemas A eu achei na internet em algum lugar, nao me lembro
>exatamente onde ("putnam 2001 problems" no google deve mostrar o site).
>Descobri que eu deixei de considerar um caso importante na solucao do A3. A
>minha solucao do A4 ficou meio grande, e deu resposta diferente da desse
>site. Com certeza a minha esta errada, mas ainda nao achei aonde (eu fiz 
>por
>vetores, deu uma conta grande, mas diferente da que ta la).
>O A5 eu ainda nao li a solucao. O A6 eu vou tentar agora, mas eh improvavel
>eu obter algum avanco, haja vista que o autor do email ainda nao conseguiu
>fazer. Se eu descobrir algo mando pra lista! Os 3 primeiros sao bem mais
>faceis..
>Tentem fazer tmb!
>Os problemas B's eu tento outro dia :)
>
>Eu nao estou cronometrando, mas acho que um dos principais obstaculos dessa
>prova eh o tempo. Vc tem que pensar em 6 questoes no curto periodo de 3
>horas.
>
>Como curiosidade, um dos americanos que fecharam a IMO esse ano, ja foi
>(antes de entrar para a universidade!) um fellow putnam, o q significa que
>ele conseguiu ficar entre as 5 melhores pontuacoes individuais do putnam
>(nao sei em q ano foi isso).
>
>Abracos,
>Marcio
>
>----- Original Message -----
>From: <rusin@math.niu.edu>
>To: <mcohen@iis.com.br>
>Sent: Sunday, December 02, 2001 1:19 PM
>Subject: Re: Putnam 2001
>
>
> > Here are the questions to the 62nd annual Putnam exam, held today
> > (Dec 1 2001).  You have 6 hours; good luck :-)
> >
> > I will post the answers I have as soon as I can type them up.
> > (Right now I lack answers to  A6, B5, B6 .)
> >
> > There are links to Putnam problems and solutions at
> > http://www.math.niu.edu/~rusin/problems-math/
> >
> >
> >
> > A1. Consider a set  S  and a binary operation  *  on  S  (that is, for
> > each  a, b  in  S,  a*b  is in  S).  Assume that  (a*b)*a = b  for all
> > a, b  in  S.  Prove that  a*(b*a) =b  for all a, b  in  S.
> >
> > A2. You have coins  C1, C2, ..., C_n.  For each  k,  coin  C_k  is 
>biased
> > so that, when tossed, it has probability  1/(2k+1)  of falling heads.
> > If the  n  coins are tossed, what is the probability that the
> > number of heads is odd? Express the answer as a rational function of  n.
> >
> > A3. For each integer  m,  consider the polynomial
> > P_m(x) = x^4 - (2m+4) x^2 + (m-2)^2
> > For what values of  m  is  P_m(x)  the product of two nonconstant
> > polynomials with integer coefficients?
> >
> > A4. Triangle  ABC  has area  1.  Points  E,F,G  lie, respectively, on
> > sides  BC, CA, AB  such that  AE  bisects  BF  at point  R,
> > BF bisects  CG  at point  S,  and  CG  bisects  AE  at point  T.
> > Find the area of triangle  RST.
> > [Illustration deleted.]
> >
> > A5. Prove that there are unique positive integers  a, n  such that
> > a^(n+1) - (a+1)^n = 2001.
> >
> > A6. Can an arc of a parabola inside a circle of radius  1 have length
> > greater than  4 ?
> >
> > B1. Let  n  be an even positive integer. Write the numbers  1, 2, ..., 
>n^2
> > in the squares of an  n x n  grid so that the  k-th row, from left to
> > right, is
> > (k-1)n + 1,  (k-1)n + 2, ..., (k-1)n + n.
> > Color the squares of the grid so that half of the squares in each row
> > and in each column are red and the other half are black (a checkerboard
> > coloring is one possibility). Prove that for each such coloring, the
> > sum of the numbers on the red squares is equal to the sum of the numbers
> > on the black squares.
> >
> > B2. Find all pairs of real numbers  (x,y)  satisfying the system of
>equations
> >
> > 1/x + 1/(2y) = (x^2 + 3 y^2) ( 3 x^2 + y^2 )
> > 1/x - 1/(2y) = 2(y^4 - x^4)
> >
> > B3. For any positive integer  n  let  <n>  denote the closest integer
> > to  sqrt(n).  Evaluate
> > \sum_{n=1}^{\infty}  ( 2^{<n>} + 2^{-<n>} ) / 2^n
> >
> > B4. Let  S  denote the set of rational numbers different from -1, 0, and
>1.
> > Define  f : S --> S  by  f(x) = x - 1/x . Prove or disprove that
> > \intersect_{n=1}^{\infty}  f^(n) (S) = \emptyset,
> > where  f^(n) = f o f o  ... o f   (n  times).
> >
> > (Note:  f(S)  denotes the set of all values  f(s)  for  s \in  S. )
> >
> > B5. Let  a  and  b  be real numbers in the interval  (0, 1/2)  and
> > let  g  be a continuous real-valued function such that
> > g(g(x)) = a g(x) + b x  for all real  x.  Prove that  g(x) = c x  for
> > some constant  c.
> >
> > B6. Assume that  (a_n)_{n >= 1}  is an increasing sequence of positive
> > real numbers such that  lim_{n->\infty}  a_n / n = 0.  Must there
> > exist infinitely many positive integers  n  such that
> > a_{n-i} + a_{n+i} < 2 a_n  for  i = 1, 2, ..., n-1 ?
> >
>


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