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USAMO 1982, problem #4

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  • jbrennen
    Phil s posting of an IMO problem from this year prompted me to go back and look up the USAMO (USA Math Olympiad) problems from the days when I competed on that
    Message 1 of 6 , Jul 31, 2002
      Phil's posting of an IMO problem from this year prompted me
      to go back and look up the USAMO (USA Math Olympiad) problems
      from the days when I competed on that test.

      I found the following gem from 1982. I competed on the USAMO
      in 1982, and I was amazed to see this question, because I
      honestly don't remember it from 20 years ago, despite my
      rather intimate knowledge of the question nowadays:

      (4) Prove that there exists a positive integer k such that
      k*2^n+1 is composite for every positive integer n.

      I wish I could go back and see how I answered this one
      as a 16-year old kid. :-)
    • Jon Perry
      (4) Prove that there exists a positive integer k such that k*2^n+1 is composite for every positive integer n. Is 0 a positive integer? ( coz then we can
      Message 2 of 6 , Jul 31, 2002
        (4) Prove that there exists a positive integer k such that
        k*2^n+1 is composite for every positive integer n.

        Is 0 a positive integer? ('coz then we can immediately abandon k=p-1)

        Jon Perry
        perry@...
        http://www.users.globalnet.co.uk/~perry/maths
        BrainBench MVP for HTML and JavaScript
        http://www.brainbench.com


        -----Original Message-----
        From: jbrennen [mailto:jack@...]
        Sent: 31 July 2002 16:58
        To: primenumbers@yahoogroups.com
        Subject: [PrimeNumbers] USAMO 1982, problem #4


        Phil's posting of an IMO problem from this year prompted me
        to go back and look up the USAMO (USA Math Olympiad) problems
        from the days when I competed on that test.

        I found the following gem from 1982. I competed on the USAMO
        in 1982, and I was amazed to see this question, because I
        honestly don't remember it from 20 years ago, despite my
        rather intimate knowledge of the question nowadays:

        (4) Prove that there exists a positive integer k such that
        k*2^n+1 is composite for every positive integer n.

        I wish I could go back and see how I answered this one
        as a 16-year old kid. :-)





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      • Norman Luhn
        ... Prove that there exists a positive integer k ... My hint, the solve is k=16, because we don t find a prime of form 2^2^k+1 where k 4 or k=2^16- k*2^n+1 is
        Message 3 of 6 , Jul 31, 2002
          --- Jon Perry <perry@...> schrieb: > (4)
          Prove that there exists a positive integer k
          > such that
          > k*2^n+1 is composite for every positive integer
          > n.
          >
          > Is 0 a positive integer? ('coz then we can
          > immediately abandon k=p-1)
          >
          > Jon Perry

          My hint, the solve is k=16, because we don't find a
          prime of form 2^2^k+1 where k>4 or k=2^16-> k*2^n+1 is
          prime ?! 0 isn't a positiv integer, it is a neutral
          element.



          Norman

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        • jbrennen
          ... Before you folks spend any more time on this, read: http://primes.utm.edu/glossary/page.php/SierpinskiNumber.html
          Message 4 of 6 , Jul 31, 2002
            --- In primenumbers@y..., Norman Luhn <nluhn@y...> wrote:
            > --- Jon Perry <perry@g...> schrieb: > (4)
            > > Is 0 a positive integer? ('coz then we can
            > > immediately abandon k=p-1)
            > > Jon Perry
            > My hint, the solve is k=16, because we don't find a
            > prime of form 2^2^k+1 where k>4 or k=2^16-> k*2^n+1 is
            > prime ?! 0 isn't a positiv integer, it is a neutral
            > element.
            > Norman

            Before you folks spend any more time on this, read:

            http://primes.utm.edu/glossary/page.php/SierpinskiNumber.html
          • Nathan Russell
            ... *nod* I took the first and second tests - the ones that aren t the olympiad, I forget what they re called - and became the first kid in my district to be
            Message 5 of 6 , Aug 3, 2002
              At 03:57 PM 7/31/2002 +0000, Jack Brennen wrote:
              >Phil's posting of an IMO problem from this year prompted me
              >to go back and look up the USAMO (USA Math Olympiad) problems
              >from the days when I competed on that test.
              >
              >I found the following gem from 1982. I competed on the USAMO
              >in 1982, and I was amazed to see this question, because I
              >honestly don't remember it from 20 years ago, despite my
              >rather intimate knowledge of the question nowadays:
              >
              >(4) Prove that there exists a positive integer k such that
              > k*2^n+1 is composite for every positive integer n.
              >
              >I wish I could go back and see how I answered this one
              >as a 16-year old kid. :-)

              *nod*

              I took the first and second tests - the ones that aren't the olympiad, I
              forget what they're called - and became the first kid in my district to be
              allowed to take the second.

              I don't think there was anything about primes, though, sadly - I might have
              gotten the other ten points and been allowed to take the USAMO if there had
              been.

              This is drifting OT, but are kids supposed to be able to take the Olympiad
              tests throughout high school? I've heard things that seem to imply that,
              here and elsewhere, but my school only allowed it for seniors who were in
              accelerated math (perhaps 10-20 people a year). Perhaps that's why I was
              the first able to make it to the invitational (level 2) test?

              Nathan
            • Jack Brennen
              ... The AHSME (Annual High School Math Exam) and the AIME (Annual Invitational Math Exam). I think that approximately 1% of the AHSME contestants get invited
              Message 6 of 6 , Aug 3, 2002
                Nathan Russell wrote:
                > I took the first and second tests - the ones that aren't the olympiad, I
                > forget what they're called - and became the first kid in my district to be
                > allowed to take the second.

                The AHSME (Annual High School Math Exam) and the AIME (Annual Invitational
                Math Exam). I think that approximately 1% of the AHSME contestants get
                invited to take the AIME. That top 1% is also not evenly distributed --
                some magnet schools and specialized math/science schools routinely get
                30 to 50 students into the AIME every year. Outside of these top-rung
                high schools, probably 1 in 250 students advances to the AIME. The USAMO
                contestants are chosen based on the AHSME-AIME combined score, but it's not
                a simple "make the cut" threshold -- non-seniors have it easier, and I
                believe that every state of the US must be represented.

                > This is drifting OT, but are kids supposed to be able to take the Olympiad
                > tests throughout high school? I've heard things that seem to imply that,
                > here and elsewhere, but my school only allowed it for seniors who were in
                > accelerated math (perhaps 10-20 people a year). Perhaps that's why I was
                > the first able to make it to the invitational (level 2) test?

                The USA Olympiad is certainly open to students as young as 8th grade, perhaps
                even younger. I know that in many US high schools, only the "top-level" math
                teacher knows anything about the AHSME-AIME-USAMO trilogy of tests. Unless
                that teacher seeks out precocious students in younger grades, they may never
                be aware that they are eligible for the AHSME. I took the AHSME for the
                first time in 7th grade (12 years old), and I had to take the exam at a
                different school, since my school knew nothing about the test. I only knew
                about it because I had been "discovered" by the county math team coach, who
                insisted that I find a way to take the exam. In 9th grade, I took the USAMO
                for the first time -- then again in 10th and 12th grades, including a top-12
                finish my senior year.
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