How to Prepare for the AIME Math Contest
What is the AIME Contest
The AIME (American Invitational Math Examination) is a math contest organized by the Mathematical Association of America (MAA). The AIME contest contains 15 questions that must be completed within 3 hours. Each answer is a 3 digit integer from 000 - 999. There is no penalty for guessing but there is only a 1 in 1000 chance of guessing something correctly.
Who can take the AIME
To take the AIME, you must qualify based on your score on AMC 10 or AMC 12 contest. The top 10 - 15% on the AMC 12 and the top 5 - 10% on the AMC 10 are invited to take the American Invitational Mathematics Examination (AIME).
You can also qualify for AIME from USAMTS (USA Mathematical Talent Search), which is a proof based competition with 3 rounds over 3 months. However, this is much harder than making it through the AMCs. Even if you qualify for AIME through USAMTS, it will be much harder to qualify for USA(J)MO as the index for USAMTS is "100 times usamts score divided by 75" so your maximum index will be 100 (compared to AMC max score of 150), and you would need to score very high on AIME to have any chance at USA(J)MO qualification.
Where can you take the AIME
AIME must be taken at the same center where you took the AMC 10/12 which helped you qualify for the AIME.
How to prepare for the AIME contest
Step I: Books
These books are comprehensive and cover all the important concepts in depth with many practice problems.
Volume 2 book by AoPS: This book contains a good overview of the important concepts for the AIME although is not very comprehensive.
Intermediate Counting & Probability: Advanced book for learning Combinatorics at the AIME level
UKMT Number Theory: Since AoPS does not have an Intermediate Number Theory Book, this book is the best alternative. It's perfect for AIME with some early olympiad level concepts. The book covers modular arithmetic, sequences, Diophantine Equations, and many other important concepts.
UKMT Geometry: Since AoPS does not have an Intermediate Geometry Book, this book is the best alternative. It's perfect for AIME with some early olympiad level concepts. The book includes Pythagoras, trigonometry, circle theorems, Ceva and Menelaus, geometrical inequalities, and coordinate geometry.
Euclidean Geometry in Mathematical Olympiads: This is an advanced, olympiad-level book in Euclidean geometry. It is best for those wanting to tackle 11 - 15 geometry on the AIME.
Step II: Handouts
These are great for specifically practicing AIME-style problems. However, they are much less comprehensive than the above books
Youth Conway provides high-quality, free, virtual instruction as well as engaging mathematical immersion activities in a time when many similar events have been canceled due to this global crisis.
*Highly Recommended Resource
David is an MAA problem writer and has created a document with many great problems for practice
Math Divulged has some good handouts and YouTube videos for AIME level.
Math Divulged is composed of dedicated competitive math veterans, including MathCounts National Champion, MOPpers, IMO Gold Medal winner, etc
Euclid's Orchard has high quality handouts for Polynomials, Trigonometry, Recursion, and Modular Arithmetic.
Some good handouts from the Bergen County Academies Math Team
Practice Contests & Mocks
In order to practice speed and accuracy, it's important to take full AIME tests for timed practice.
List of Mock Contests
1. Official AIME from previous years with solutions: Use these tests with a 75 minute timer to simulate test taking situations. The AMC's are increasing in difficulty every year, so only AMC 12 contests from 2021 Fall and after are representative of the difficulty of recent AMC 12.
2. AIME problems (till 2014) Sorted By Topic: This is a great resource to practice AIME problems topic by topic. For each topic, the problems are sorted roughly in increasing order of difficulty. A strategy you can consider is use this handout to practice topic by topic problems and do the AIME's 2015 onwards as practice tests.
3. CIME Mock Contests: Nearly all of the tests created by CMC (even the ones from 2017) are extremely high quality and close in difficulty to AMC's nowadays.
4. AIME Practice Tests Book by AlphaStar Academy: This book contains four AIME practice exams with new problems not used in any past competitions and with insightful solutions.
5. AoPS Forum with Many Mocks: This forum has many mocks created by AoPS users. Most of them are great quality and good to practice with but they can often be hard. Some ones I recommend are DIME, EIME, and XYZIME but there are probably many other great mocks you can find here
6. More mocks from AoPS forum: Another list of mocks from AoPS.
For even more practice, there are some useful websites which help you practice specific topics and improve problem solving speed.
1. Alcumus: Free resource by AoPS to practice problems from specific topics.
2. AMC Trivial: This site generates problems from past contests. You can even make custom mocks from older questions and it can give problems from specific topics or difficulty levels.
3. MAATester: This site makes it easy to search for problems related to any topic from various contests.