- 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. :-) - Nathan Russell wrote:
> I took the first and second tests - the ones that aren't the olympiad, I

The 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 Olympiad

The 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.