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Re: Fw: Putnam 2001
E uma competiçao para alunos do ciclo basico de universidades dos EUA e
Canada. Para vergonha dos matematicos, em geral os vencedores se tornam
medicos ou advogados.
Fernanda Medeiros wrote:
>
> Oi,
> O que é Putnam? É tipo uma imo?
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>> 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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