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Re: [PrimeNumbers] USAMO 1982, problem #4

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  • 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 1 of 6 , Aug 3, 2002
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      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 2 of 6 , Aug 3, 2002
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        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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