If you want to "Exercise your mind daily with a problem from the AMC-8, AMC-10, or AMC-12, provided by MAA's American Mathematics Competitions," visit the MAA Minute Math website.
Note that the official solutions provided are correct, but are not always the most efficient, so it's very worthwhile to think about more creative approaches of your own.
Minute math also provides interesting data on the percentage of test-takers who got each question correct. If you get the question wrong, you can take comfort in the fact that you generally have lots of company! (And bear in mind that the sample of AMC contest-takers is not a random sample of the population. The pool of students who take AMC exams includes a disproportionate number of strong and enthusiastic math students.) Whether you got the question right or wrong, it's interesting to look at the "distractor" wrong answers, which often give insights into common misconceptions.
It's important not only to understand your own blind spots, but also to recognize the blind spots of other people. This is true both in math competitions (which often have team-based components, so it's important to anticipate areas in which teammates may make mistakes), and also in the real world (perhaps the financial bubble which led to the recent meltdown would have been "pricked" sooner if more financial decision makers had considered the blind spots of other financial decision makers.)
If you want to look at the entire database of problems used on the AMC8/10/12 contests over the past decade, they are available here in a format that allows you to select particular types of problems, for example, discrete math, geometry, etc.