The AMC (American Math Competition) is a series of tests which lead to the selection of the U.S. Math Olympiad Team which competes in the International Math Olympiad.  Along with MathCounts, the AMC series of competitions is the premier math contest for middle school students.

The AMC 8 is for students in grade 8 and below.  It is held in November.  Students are allowed 40 minutes to complete the 25-question multiple-choice exam.  Middle-school students who score well on the AMC 8 often take the AMC 10 in February.  A high score on the AMC 10 or 12 qualifies a student to take the AIME, which leads to the USAMO and eventually the MOSP and the IMO.  More information can be found at the AMC website.

A very well thought out mock AMC10 from AoPS user dnkywin.

A very hard (AMC12+/AIME level) Mock AMC exam created by AoPS users Caelestor and gfour84.  View the discussion posts linked here.

Submit your own mock competition!

I have pulled most of the sample exams I had posted.  You can purchase past exams from the MAA here.  $25 buys you every exam and solution for the past 10 years (buy the High School Math Club Package, one of the last few items on the list).

I am leaving the most recent exams only.

Harold Reiter has the exams for the years he chaired the AMC on his web site here.  (Scroll down to the AMC section ... there is a ton of great stuff linked from his site.)

Sample AMC8 Exams

Sample AMC10 Exams

Sample AMC12/AHSME Exams

Sample AIME Exams

Number Theory:  Primes, Divisibility, Factoring, Factoring 'Tricks', Factorials, Different Bases, Units Digits.  (52 pages)  Last Updated February 6, 2009.

Draft Counting unit.  Mostly completed now.  (42 pages).  Updated 2/26/09.

Draft Probability unit.  Mostly complete, but could use some revisions.  NEW!  Updated 4/26/09.