Also, there are a bunch of contest problems on John Scholes's website: http://www.kalva.demon.co.uk/ . It seems to be down at the moment, but I'm sure it'll be back soon. Of the contests I'm familiar with:
AIME is for fairly advanced high school students (say, on average, one person per high school might take it, and most won't get any of them right)
USAMO is a step up from AIME -- the top ~150 high school students in the US take it each year
IMO is a step up from USAMO -- basically, don't even bother
Putnam is an exam for US university students -- it's generally considered difficult, since the median score is usually 0, even among those willing to take it in the first place, but A1 and B1 are usually pretty easy on every test; I would say it's between AIME and USAMO