Everything you need to know to ace every level of the AMC competition.
The American Mathematics Competitions (AMC) are the first step toward becoming one of America’s top math students and advancing to elite contests such as the Mathematical Olympiad. But the AMC is much more than just a pathway to national recognition. For many students, the AMC math competition provides a rare opportunity to engage in pure problem-solving in a challenging and fun setting.
Whether you’re aiming for the highest levels of competition or simply want to test your skills and see how you compare nationally, this complete guide will help you prepare for the AMC-8, AMC-10, and AMC-12.
Love math? Great! Keep reading. We’ll explain everything you need to know — from how AMC scoring works to the problem-solving skills and strategies that lead to success.
Who takes the AMC? Is it only for math geniuses?
Is the AMC math competition just for prodigies who were contemplating right angles in their cribs and estimating pi from the circumference of their baby Binky? Definitely not.
The AMC is for any student who enjoys problem-solving. Notice we didn’t say it’s only for students who love traditional school math. While a love of math certainly helps, many students find that standard classroom math can feel repetitive or uninspiring—and that’s not what the AMC is about.
The AMC challenges students to think creatively, spot patterns, and approach problems in new ways. If you enjoy puzzles, logic, and “aha!” moments more than repetitive drills, the AMC may be a great fit for you.
Try this!
Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?
That’s an actual problem on the AMC-8 from 2020. If you like puzzles and games, the AMC is for you. Do you want to check if you got the right answer? At the end of our article, we’ll tell you who was in the middle car and give you the solutions to all the interesting problems scattered throughout this Ultimate Guide.
Can you improve your problem-solving skills?
Even if the AMC problems you’ve seen so far feel overwhelming, you can still excel at the AMC math competition. Research on performance in logic puzzles, strategy games, and mathematical problem solving consistently shows that students improve most when they enjoy learning and approach challenges with a growth mindset.
So what does it take to succeed on AMC-style problems?
One key factor is deliberate practice—solving many problems in a focused, thoughtful way, ideally with guidance and feedback. Whether students work with a problem-solving teacher, take a structured class, or study independently using high-quality books, deliberate practice helps them build the reasoning skills the AMC rewards.
Through this kind of preparation, students learn to apply core AMC problem-solving strategies, including:
- Working backward
- Working systematically
- Looking for patterns
- Visualizing relationships
- Iterating on an initial best guess
- Reasoning logically
As these strategies become second nature, students gain confidence in their mathematical reasoning. They learn how to combine creativity with careful thinking—an essential skill for earning strong AMC scores.
For more than 30 years, IMACS has focused on teaching these core STEM and problem-solving skills. Whether your goal is simply to explore the AMC or to pursue top scores, a free IMACS class can help your child develop strong reasoning skills that extend far beyond any single competition. Families can also learn more about why IMACS works and read IMACS success stories from students who have gone on to excel in advanced math and beyond.
Why all the numbers? What do the 8,10, and 12 mean?
The AMC math competitions are offered at three levels: AMC-8, AMC-10, and AMC-12. The number refers to the intended grade level for the contest—not a strict requirement.
For example, the Mathematical Association of America (MAA) designs the AMC-8 primarily for students in 8th grade or below. However, students are not required to take the AMC that matches their exact grade level, and many choose to compete at a higher level.
It’s common for:
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6th graders (and occasionally 5th graders) to take the AMC-8
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7th graders to choose between the AMC-8 and AMC-10
As long as a student has not exceeded the grade level or age limit for a contest, they are eligible to participate. Students from around the world—including those studying outside the United States—are welcome to compete.
Here are the official AMC age and grade limits for each level:
- AMC-8
- Currently 8th grade or lower
- 14.5 years old or younger
- AMC-10
- Currently 10th grade or lower
- 17.5 years old or younger
- AMC-12
- Currently 12th grade or lower
- 19.5 years old or younger
How do I get to the Math Olympiad from the AMC?
You don’t have to aspire to be an elite math competitor to take the AMC, but for students who do dream of reaching the highest levels, the AMC-10 and AMC-12 are the first steps on the path to the International Mathematical Olympiad (IMO).
Here’s how the progression works.
While the AMC-8 is an excellent introduction to competition math, it does not factor into Olympiad qualification. The Olympiad pathway begins with strong performance on the AMC-10 or AMC-12.
Each year, top scorers on the AMC-10 and AMC-12—based on annual score cutoffs—are invited to take the American Invitational Mathematics Examination (AIME).
Students who perform exceptionally well on the AIME may then qualify for one of two national Olympiads:
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USAJMO (USA Junior Mathematical Olympiad), typically for younger students qualifying via the AMC-10
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USAMO (United States of America Mathematical Olympiad), typically for students qualifying via the AMC-12
While exact numbers vary from year to year, only a few hundred students nationwide reach this level.
From there, the strongest performers are invited to the Mathematical Olympiad Program (MOP), an intensive summer training program. Finally, six students are selected from this group to represent the United States on the International Mathematical Olympiad (IMO), competing against teams from more than 100 countries worldwide.
What kind of problems does the Olympiad team solve?
Even if your immediate goal is simply to do well on the AMC, it’s never too early to understand what sets Olympiad-level problems apart.
Like AMC and AIME problems, Olympiad questions emphasize deep reasoning and creativity. Although calculus is permitted at the Olympiad level, it is never required for the AMC, AIME, or Olympiad competitions.
A core principle of these contests is that success should reward thinking, not just years of advanced coursework. Olympiad problems draw from areas such as:
- Combinatorics
- Projective Geometry
- Complex Geometry
- Functional Equations
- Number Theory
The essential facts you need to know about the AMC.
How many problems are on the AMC-8, and how much time will I get?
There is some time pressure on every AMC exam, so students must balance speed with careful reasoning. Here’s a breakdown of how many questions appear on each contest and how much time students are given:
- AMC-8
- 25 Questions
- 40 Minutes
- About 90 seconds per question
- AMC-10
- 25 Questions
- 75 Minutes
- About 3 Minutes per question
- AMC-12
- 25 Questions
- 75 Minutes
- About 3 Minutes per question
How is the AMC-8 scored? Is it better to guess or leave a problem blank?
Good news for students taking the AMC-8: there is no penalty for wrong answers.
Each correct answer is worth 1 point, and both incorrect and blank answers are worth 0 points. This means there is no scoring advantage to leaving a question blank.
A smart strategy is to skip problems that seem time-consuming and return to them after completing easier questions. If time is running short, it is always better to make an educated guess than to leave an answer blank.
Time yourself on this practice problem from the 2019 competition. How long does it take you?
Ike and Mike go into a sandwich shop with a total of $30.00 to spend. Sandwiches cost $4.50 each, and soft drinks cost $1.00 each. Ike and Mike plan to buy as many sandwiches as they can and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how many items will they buy?
Keep in mind AMC problems are multiple-choice. Does it help you solve the problem faster if you know the right answer is one of the following: (a) 6, (b) 7, (c) 8, (d) 9, or (e) 10?
Think you got it? You can find the answer at the end of our guide.
Even if this took a long time, don’t worry about it. Analytical thinking is something you can improve with practice and instruction. And if you figured it out quickly? Great! We’ll have more sample problems for you in the next section.
How are the AMC-10 and the AMC-12 Scored? Guess or skip hard problems?
The AMC-10 and AMC-12 use a simple scoring system that rewards correct answers without penalizing mistakes.
Each exam consists of 25 multiple-choice questions, and:
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Each correct answer earns 1 point
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Incorrect and blank answers earn 0 points
There is no penalty for guessing on the AMC-10 or AMC-12, and there is no advantage to leaving a question blank.
A common and effective strategy is to:
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Skip problems that seem especially time-consuming
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Solve as many easier problems as possible first
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Return to harder questions if time allows
If time is running short and you’re unsure of an answer, it is always better to make an educated guess than to leave a question unanswered.
While thoughtful problem selection and time management matter, modern AMC scoring no longer requires complex guessing strategies. Success depends far more on strong problem-solving skills, logical reasoning, and efficient use of time than on scoring tricks.
Can you compete at the same AMC level multiple times in one year?
The AMC-8 is offered once per year, so students can only take it a single time during a competition cycle. However, the AMC-10 and AMC-12 are each offered twice per year, which gives students more flexibility.
Both the AMC-10 and AMC-12 have two versions:
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Version A
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Version B
That means there are five total AMC contests each year:
AMC-8, AMC-10A, AMC-10B, AMC-12A, and AMC-12B.
Why are there “A” and “B” versions of the AMC-10 and 12?
The Mathematical Association of America (MAA) offers two versions of the AMC-10 and AMC-12 so students have multiple testing opportunities in a single year.
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AMC-10A and AMC-12A are administered on the same day
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AMC-10B and AMC-12B are administered on a later date, also on the same day
The A and B versions are completely different exams. However, because AMC-10 and AMC-12 are given on the same day, a small number of problems may overlap between the 10 and the 12 on that test date.
This overlap is intentional. Both exams begin at a similar difficulty level, but the AMC-12 increases in difficulty more quickly, covering more advanced material later in the test.
Students are not allowed to take the AMC-10 and AMC-12 on the same day, but it is permitted to take both exams in the same year by alternating versions. For example:
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AMC-10A and AMC-12B
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AMC-12A and AMC-10B
One benefit of this structure is that students aiming for Olympiad qualification can attempt to qualify separately for the USAJMO (via AMC-10) and the USAMO (via AMC-12) in the same year.
When and where do the AMC competitions take place?
For the 2025–2026 AMC competition year, the contests were offered on the following dates:
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AMC 10 A and AMC 12 A: November 5, 2025
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AMC 10 B and AMC 12 B: November 13, 2025
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AMC 8: January 22–28, 2026
Because AMC competition dates and registration deadlines are set annually, students and families should always confirm current information through the official Mathematical Association of America (MAA) website.
In general, regular registration closes about two months before the competition date, with late registration ending roughly one month prior. Since host sites may fill quickly, early registration is strongly recommended.
AMC exams are hosted at many middle schools, high schools, universities, and learning centers. Start by asking your school whether they plan to host the AMC this year. If not, you can consult the MAA’s list of approved AMC testing sites. Many of these locations host AMC exams year after year and may be available again for the upcoming competition cycle.
When do you get your scores?
What’s harder than being in the American Mathematics Contest? Waiting for your scores to come back!
In most cases, AMC scores are released about 3–4 weeks after the competition date. Scores are typically distributed through the school or organization that hosted the exam, though some students may also receive score notifications electronically depending on how they registered.
If it has been more than 30 days since you took the exam and you haven’t received your results, you can contact the Mathematical Association of America (MAA) for assistance.
Can you use a calculator on the AMC-8?
Try this problem out. It’s from the 2017 AMC-8.
Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn’t know her house number. She tells him, “My house number has two digits, and exactly three of the following four statements about it are true.”
- It is prime.
- It is even
- It is divisible by 7.
- One of its digits is 9.
This information allows Malcolm to determine Isabella’s house number. What is its ones-place digit?
No calculators are allowed on any AMC exam, including the AMC-8, AMC-10, and AMC-12.
The AMC is designed to test problem-solving ability and mathematical reasoning, not computational speed. That’s why calculators were removed from AMC competitions entirely in 2008. Every AMC problem can be solved using logical thinking, insight, and careful reasoning—without the need for technology.
Because of this emphasis, strong AMC performance depends less on memorized procedures and more on understanding why math works. While many school math programs focus on applying standard methods, AMC problems often require flexible thinking and creative approaches.
For students preparing for the AMC, enrichment programs that build deep mathematical reasoning skills—rather than narrow test-specific tricks—tend to be the most effective. These skills carry over not only to AMC competitions, but also to broader STEM fields such as robotics, engineering, medicine, and data science.
IMACS is one such program, emphasizing reasoning-based learning and placing students by skill level rather than age or grade. IMACS offers a free placement class, which helps families understand a student’s strengths and areas for growth as they prepare for AMC-style problem solving.
What about taking the same AMC level in multiple years?
As long as a student has not exceeded the age or grade limit, they may take the same AMC competition in multiple years. Many students use this opportunity to build experience, improve problem-solving speed, and steadily raise their scores over time.
Younger students often begin with the AMC-8 and return to the contest year after year as their skills develop. Starting early allows students to focus on building strong mathematical reasoning skills rather than rushing to cover advanced content.
Developing these skills early can be especially helpful for students who plan to continue with AMC competitions through middle and high school. Programs like IMACS emphasize systematic problem-solving and logical thinking, helping students build a strong foundation for long-term success.
IMACS offers a free placement class, allowing families to better understand a student’s current skill level and the best path forward for AMC preparation.
What is a good score on the AMC?
A “good” AMC score depends on the difficulty of the test and how other students perform that year. Because the AMC is intentionally challenging, even strong students often answer many questions incorrectly.
On the AMC-8, the average score in a typical year is well below a perfect 25, which surprises many first-time competitors. This is good news for students: you don’t need to get every question right to perform well.
Each year, the Mathematical Association of America recognizes top performers with:
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Honor Roll (typically awarded to the top-performing students nationwide)
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Distinguished Honor Roll (reserved for the highest scorers)
The exact score cutoffs for these honors vary from year to year, depending on the difficulty of the exam. As a result, AMC success is best measured relative to peers, not against a fixed score target.
The takeaway for students and families is encouraging: consistent improvement, strong reasoning skills, and smart problem-solving strategies matter far more than perfection.
Do you think you can earn Distinguished Honor Roll? Even top scorers don’t need a perfect paper to stand out. Try this AMC-8 problem from 2016 and see how you do.
Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is 132.
(You’ll find the answer to this problem at the end of the article.)
What specific math topics does the AMC test?
What math topics do you need to know for the AMC-8 competition?
The AMC-8 math competition tests far more than routine middle school math. While the contest draws on topics commonly taught in advanced or gifted programs, many questions require creative reasoning and multi-step problem-solving.
Common AMC-8 math topics include:
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Exponents and number properties
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Counting and probability
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Estimation and number sense
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Proportional reasoning
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Basic geometry (including the Pythagorean Theorem)
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Spatial visualization
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Introductory statistics
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Interpreting graphs and tables
According to the Mathematical Association of America (MAA), AMC-8 problems may also involve early algebra concepts such as linear or quadratic relationships, coordinate geometry, and more advanced geometric reasoning.
Although a strong foundation in school mathematics is essential, high AMC-8 scores depend even more on logical thinking, pattern recognition, and mathematical creativity. Many problems are designed to reward insight rather than memorization, and students often encounter approaches they have not seen in a standard classroom setting.
Students who perform well on the AMC-8 typically prepare by:
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Practicing with past AMC-8 problems
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Learning multiple solution strategies
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Strengthening reasoning skills across number theory, geometry, and combinatorics
For families seeking a structured path to AMC-8 success, enrichment programs like IMACS focus on the reasoning-based skills that AMC problems emphasize. IMACS classes begin as early as 1st and 2nd grade, and students who start by 3rd or 4th grade can develop the depth of thinking needed to pursue distinguished honor roll recognition and, for some, a perfect AMC-8 score.
Let’s try another one. You got this!
Peter, Emma, and Kyler played chess with each other. Peter won 4 games and lost 2 games. Emma won 3 games and lost 3 games. If Kyler lost 3 games, how many games did he win?
What math do you need to know for the AMC-10 and 12?
he AMC-10 and AMC-12 draw on material commonly taught in middle school and high school math programs, including topics through pre-calculus. However, simply knowing the formulas and procedures isn’t enough to succeed.
AMC-10 and AMC-12 problems often require students to:
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Apply familiar concepts in unfamiliar ways
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Combine ideas from multiple areas of math
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Reason logically through multi-step problems
While many questions test core algebra, geometry, counting, probability, and number theory, the biggest challenge is not advanced content—it’s how the problems are framed.
Some problems are relatively straightforward for well-prepared students, while others are intentionally designed to stretch even the strongest competitors. This wide range of difficulty is what makes the AMC such a powerful measure of problem-solving ability.
Here are some problems that have appeared on the AMC-10 and the AMC-12:
The numbers 3, 5, 7, a, and b have an average (arithmetic mean) of 15. What is the average of a and b? (From the 2020 AMC-10)
Here is one of the most straightforward problems from the 2020 AMC-12. About 85% of the competitors got this one right:
Carlos took 70% of a whole pie. Maria took one-third of the remainder. What portion of the whole pie was left?
What about the hardest problems? On the 2020 AMC-12A, this was one of the three most difficult, as measured by how many participants got it right.
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to re-roll. After re-rolling, he wins if and only if the sum of the numbers face up on the three dice is exactly 7. Jason always plays to optimize his chances of winning. What is the probability that he chooses to re-roll exactly two of the dice?
Only 6% of the competitors got this one correct.
If some of these AMC-10 or AMC-12 problems feel intimidating, that’s normal. With consistent practice and the right kind of training, students can learn to break complex problems into manageable pieces and steadily improve their performance over time.
What are the benefits of participating in the AMC?
Beyond the challenge itself, participating in the American Mathematics Competitions helps students develop problem-solving and reasoning skills that extend far beyond any single test. These skills are valuable not only for future math competitions, but also for advanced STEM coursework and careers.
High-performing students may also earn recognition through AMC honor rolls and related distinctions, and many students choose to include AMC participation as part of their broader academic profile. Just as importantly, the AMC helps students build confidence in tackling unfamiliar and complex problems.
When should AMC preparation begin?
The AMC is not something students can cram for. Success depends on long-term development of mathematical reasoning, not memorization or routine practice.
Many students begin building these skills in elementary school, especially if they plan to compete in the AMC-8 over multiple years. Starting early allows students to focus on thinking strategies rather than rushing to learn advanced content.
That said, it is never too late to start. Students who begin focused preparation in upper elementary or middle school can still make significant gains with consistent practice and the right kind of instruction.
Programs like IMACS emphasize multi-year development of problem-solving skills and place students by skill level rather than grade, helping families find an appropriate starting point at any age.
How can students build AMC problem-solving skills?
Most students who become high-scorers on the AMC start preparing in early elementary school. Some prepare by competing in events like Math Kangaroo, while others take specialized courses like IMACS Mathematics Enrichment. Many kids improve their mathematical creativity and reasoning skills in a family culture that promotes puzzles, games, and the joy of problem-solving. You can also join a math club at school. Definitely include these in your preparation plans, but nothing will keep you focused and get you better, faster, than engaging in a multi-year mathematical reasoning curriculum.
The best time to start this type of curriculum is 1st or 2nd grade. Students who build problem-solving skills over a few years are the most successful across all of the AMC levels. Another reason to start in 1st or 2nd grade is that competing as a 5th or 6th grader in a competition designed for 8th graders is a huge confidence booster.
What if you are already in 4th, 5th, or 6th grade? The earlier you start, the better, but it’s never too late to build these skills. It is absolutely possible to begin a program like IMACS in 5th or 6th grade and get the practice you need to get a top score by 8th grade.
Classes are best, but there are other great ways to prep!
While structured classes are one effective approach, there are many ways students can strengthen the reasoning skills the AMC rewards. A balanced preparation plan may include:
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Solving past AMC problems
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Working through high-quality problem-solving books such as Try How To Solve It by George Polya, Problem-Solving Strategies by Arthur Engel, and Challenging Problems in Algebra by Alfred Posamentier and Charles Salkind
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Playing logic games and puzzles, such as the classic Think Fun games, Rush Hour, and Chocolate Fix (created by an IMACS alumnus Mark Engelberg. Here is his story, from IMACS to puzzle inventor!)
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Participating in math clubs or competitions
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Engaging regularly with challenging, non-routine problems
Unexpected pro-tips for competition day
The most important preparation for the AMC happens before competition day, but a few simple habits can help students perform at their best.
The night before the test, focus on rest and a calm mindset. Getting a good night’s sleep—and maintaining a consistent sleep schedule in the days leading up to the competition—can make a real difference in focus and stamina.
On test day, keep things simple:
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Eat a balanced breakfast
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Arrive early and relaxed
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Sit comfortably and stay focused on one problem at a time
If nerves set in, taking a few minutes before the test to write down worries or mentally reset can help clear your mind and free up mental energy for problem-solving.
Ultimately, AMC success isn’t about last-minute tricks. A rested mind, steady focus, and confidence built through preparation go a long way on competition day.
Want to try more AMC problems?
Here is sample problem 11 from the 2019 AMC 8 competition…
The eighth-grade class at Lincoln Middle School has 93 students. Each student takes a science class or a foreign language class or both. There are 70 eighth-graders taking a science class, and there are 54 eighth-graders taking a foreign language class. How many eighth-graders take only a science class and not a foreign language class?
Only 20% of the mathletes got this problem (#16) from the 2019 AMC-8 correct:
Qiang drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to average 50 miles per hour for the entire trip?
The answer options are: 45, 62, 90, 110, and 135.
What did you get?
Ready for more?
The next step in AMC preparation
Competing in the AMC math competitions at any level is a great way to build problem-solving skills for future STEM challenges—and just as importantly, it’s fun.
If you know a student in 1st through 6th grade who enjoys puzzles and challenges, share this AMC guide and invite them to try a free IMACS assessment. During this session, an IMACS instructor meets one-on-one with each student over Zoom and works through engaging problems to identify existing strengths and areas for growth.
This personalized approach helps families understand what skills a young problem-solver already has—and what they can develop next to prepare for AMC-style challenges.
Thanks for reading our guide. And here’s our final piece of advice for the AMC:
Answers to AMC Problems in the Ultimate Guide (with video explanations when available).
Who sat where on the train? Problem # 6, 2020, AMC-8. The answer is Aaron. Video explanation.
Ike and Mike get lunch. Problem #1, 2019, AMC-8. The answer is 9.
Malcom visits Isabella. Problem #8, 2017, AMC-8. The answer is 8.
Reverse the numbers and get 132. Problem #11, 2016, AMC-8. The answer is 7.
Chess Wins and Losses. Problem #13, 2017, AMC-8. Answer = 1. Video explanation.
Average of 15. Problem #2. 2020, AMC-10. The answer is 30. Video explanation.
Carlos has some pie. Problem #1, 2020 AMC-12. Answer = 20%. Video explanation.
Jason rolls some dice. Problem #25, 2020, AMC-10. The answer is (7/36).
Lincoln Middle School. Problem #11. 2019, AMC-8. The answer is 39.
Qiang drives. Problem #16, 2019, AMC-8. The answer is 110.
The problems shared here are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.
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