As I prepare for the AMC 10 and 12 and AIME this year, I find it very hard to see similarities in preparation for these challenging exams.
For starters, AMC exams have 25 problems that you need to solve in 75 minutes with multiple choices. This means that you have on average 3 minutes per problem. If you qualify for AIME by scoring high on AMCs then the time constraints are very different. AIME exams have 15 problems and you get three hours to solve them which means 12 minutes per problem and there are no multiple choices. Guessing on AIME does not work as the answer could be anywhere from 000 to 999. The average score among these AIMEs is typically between 3 to 6. For example, 2016 AIME II had an average score of 3.52 among 975 students who took it in USA. There are also very few students who score more than 10 in AIME. For USAMO qualification, one has to score typically in the range of 7-9 assuming high scores on AMCs.
Here is what I am doing this year. I practice for both AMCs and AIMEs using past exams. For AIME, I have enrolled into AoPS class in addition. The class goes over the concept and then there are 10 problems that you solve within a week which reinforces the concept. These are not easy problems and some assignments have taken 3-5 hours for me to solve. I find AIME solutions to be a lot more tedious and involve multiple steps where you can make silly calculation mistakes as compared to AMCs. I don’t mix the days where I practice both of the tests. For example, during the holidays, there are days when I take just AMCs and there are days when I just took an AIME per day. My scores have improved as I practice more but I still have a long ways to go in being an expert. The AIMEs are a lot more humbling exams than AMCs.
Good luck to everyone as they prepare for the tests in 2017! The AoPS website is your best source for past exams and solutions.