Re: USAMO 1982, problem #4
- --- In primenumbers@y..., Norman Luhn <nluhn@y...> wrote:
> --- Jon Perry <perry@g...> schrieb: > (4)Before you folks spend any more time on this, read:
> > 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
- At 03:57 PM 7/31/2002 +0000, Jack Brennen wrote:
>Phil's posting of an IMO problem from this year prompted me*nod*
>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. :-)
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
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 Russell wrote:
> I took the first and second tests - the ones that aren't the olympiad, IThe AHSME (Annual High School Math Exam) and the AIME (Annual Invitational
> forget what they're called - and became the first kid in my district to be
> allowed to take the second.
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 OlympiadThe USA Olympiad is certainly open to students as young as 8th grade, perhaps
> 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?
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.