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# 2021 AMC Math Competition Guide. Love Math? Score Better!

## Everything you need to know to ace every level of the AMC competition.

The American Mathematics Competition, better known as the AMC, is the first step to becoming America’s best mathlete and going all the way to the Math Olympiad. But the AMC is so much more than just the pathway to winning the title of America’s greatest high school mathematician. Even for academic mortals, the AMC’s 75 minutes of pure problem-solving fun is likely one of the most enjoyable tests you’ll ever take.

So, whether you’re aiming for the lofty heights of the Olympiad, or you just want to test your chops and see if you can get a perfect score, this complete guide will help you prepare for the AMC-8, the AMC-10, and the AMC 12.

Love math? Great! Then keep reading. We’ll explain everything you need to know, from the AMC scoring system to the skills you need to succeed.

## Who takes the AMC? Is it only for math geniuses?

Is the AMC just for prodigies who were contemplating the right angles on the bars of their crib and estimating pi from the circumference of their baby Binky? Definitely not! The AMC is for anyone who loves problem-solving. Notice, we didn’t say, “it’s for anyone who loves math.” At least, there is no requirement you love traditional school math. Sure, loving math is great, but standard math classes may have sometimes left you bored, and it isn’t really what the AMC is all about. (“OK, class, tell me what seven times eight is.” Again? Really?)

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 problem above melted your brain, you can still excel at the competition. Many researchers have looked closely at what skills lead to dominating at board games, logic puzzles (like Sudoku), or these types of AMC problems. The science is unambiguous: if you enjoy learning and have a growth mindset  you can definitely become a master problem-solver.

What precisely does it take to shine at these kinds of problems? One key is deliberate practice; that’s just a way of saying: do lots of problems in a focused way, preferably while getting the right instruction.

Whether you work with a problem-solving teacher, take a class, or teach yourself with books (and we have book recommendations below), you’ll use your deliberate practice sessions to master these core skills:

• Working backward
• Working systematically
• Looking for patterns
• Visualizing
• Iterating on an initial best guess
• Reasoning logically

Once you have learned these strategies, you’ll be a pro at mathematical reasoning. You’ll be able to summon all of your mathematical knowledge and creativity and be on your way to a top score!

## Why all the numbers? What does the 8,10, and 12 Mean?

The AMC has three levels: AMC-8, AMC-10, and AMC-12. The number represents the target grade. For example, the Mathematical Association of America calibrates the AMC-8 for 8th graders. But that doesn’t mean you can’t take a higher level than your current grade.

Some 6th graders (even sometimes some 5th graders) take the AMC-8. Some 7th graders take the AMC-8, but others decide to take the AMC-10. As long as you haven’t graduated from the grade the competition is for, and you haven’t aged out, you are eligible to compete. Whether you are an American student in elementary school or doing secondary school mathematics in Australia, there is an AMC competition right for you. Here are the 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 mathlete to take the AMC, but it is the first step to getting on the US International Mathematical Olympiad (IMO) team. How many steps are there between a great AMC score and the IMO team? Here’s how it works.

Being a top scorer on the AMC-8 gives you bragging rights. A high score on either the AMC-10 (top 2.5%) or AMC-12 (top 5%) gives you the chance to compete again on a test known as the AIME, the American Invitational Mathematics Examination.

Crushed the AIME? Yeah! Depending on whether you made the selection cut-off with an AMC-10 score or an AMC-12 score, you’ll be invited to either the USA Junior Mathematical Olympiad (USAJMO) or the United States of America Mathematical Olympiad (USAMO). And yes, you can qualify for both! While it can vary a bit year-to-year, in general, about 500 kids reach these lofty heights.

From there, you’ll get invited to the Mathematical Olympiad Summer program. Finally, the top six students who have beaten these levels will win one of the six coveted spots on the US Olympiad team—called the USIMO—that competes each year against 100 other countries. It’s pretty amazing to think that all these different teams of six mathletes at the world competition represent 90% of the planet’s population.

### What Kind of Problems does the Olympiad Team Do?

Even though you’re here to learn all about how to ace the AMC, it’s never too early to dream of becoming one of the most elite mathletes in the world. The Olympiad problems (like the AMC and the AIME problems) require deep problem-solving skills. While it’s OK to use calculus in the Olympiad (and might even be useful), it’s never needed to solve any AMC, AIME, or Olympiad question.

A founding principle of these competitions is that winning should reward mathematical reasoning and creativity, not the number of years you’ve been grinding through common core classes. The Olympiad level will hit you with advanced Algebra and Precalculus concepts and problems, but the competition explores a lot of other interesting topics too, like:

• 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 a bit of time pressure, and you must perform with the clock ticking. Here is the lowdown on how many questions there are and exactly how fast you must be:

• AMC-8
• 25 Questions
• 40 Minutes
• 90 Seconds per question
• AMC-10
• 25 Questions
• 75 Minutes
• 3 Minutes per question
• AMC-12
• 25 Questions
• 75 Minutes
• 3 Minutes per question

### How is the AMC-8 Scored? Is it better to guess or leave a problem blank?

Good news if you are taking the AMC-8: there is no penalty for wrong answers. Every question is worth the same number of points, so if a problem has you stumped, or you know it will take you a long time, skip it and come back after you’ve tackled easier problems. (If you think you won’t have time to go back, you should definitely guess! )

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 do 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?

When you get to the AMC-10 and AMC-12, the scoring system changes. There are still 25 questions, and all correct problems are worth an equal number of points, six per answer. But once you get to these two competitions, the AMC punishes you for non-strategic guessing.

Like the AMC-8, an incorrect answer will still get zero points, but a blank question is worth a point and a half. Like the AMC-8, there are five multiple-choice answers. If you leave every answer blank, congrats, you just got a score of 37.5 (1.5 * 25). That leaves us with the big question: when should you guess? This question could be an AMC problem itself!

How many answers (of the five offered) do you need to eliminate before guessing is a good idea if each blank question is worth 1.5 points, and each right answer is worth 6 points?

A purely random guess (with five options) would give you an expected value of 1.2 points per guess. That’s worse than the 1.5 points you’d get for leaving it blank. If you can’t eliminate anything, it’s better to leave it empty. Once you can cross off one of the five options, it’s a wash—guessing among the remaining four possibilities has an expected value of 1.5, exactly what you’d score for skipping.

As soon as you can cross off two of those five multiple-choice options: guess!

With two choices eliminated, guessing is worth (on average) two points per problem. If you want to read more about great guessing strategies for the AMC-10 and the AMC-12, check out the MIT Scholar and Olympiad Gold medalist Tanya Khovanova’s blog post on the topic.

### Can you compete at the same AMC level multiple times in one year?

The AMC-8 competition occurs only once per year, so it would be impossible to take it multiple times in a single year, but it is possible to take either the 10 twice in the same year or take the 12 twice in the same year.

The 10 and 12 each have two versions, the “A” and then the “B.” That makes five total competition events per year: 8, 10A, 10B, 12A, and 12B.

### Why is there is an “A” and “B” version of the AMC-10 and 12?

The AMC-10 and the AMC-12 are each offered twice in a single year. The MAA designates the first test date of the year “A” and the second, “B.” The “A” version of the test and the “B” version of the tests are entirely different.

The 10A and 12A always take place on the same day. Later in the year (also on the same day), the 10B and 12B happen. Notably, the 10A and 12A share several of the same problems. Likewise, the 10B and 12B share a few problems. It might seem odd that the 10th-grade competition and the 12th-grade one have a few questions that are the same, but it is true. A consensus among AMC competitors is that both the AMC-10 and the AMC-12 start at about the same level of difficulty, and each test gets progressively more demanding. The difference is that the 12 ramps up to more challenging questions faster. So while it’s true that a few questions overlap, the 12 is still quite a bit harder than the 10!

It is strictly forbidden for a student to take the 10 and 12 on the same day, but it is fine to take them both in the same year. How? The mathlete could take the 10A and 12B or flip the sequence and take the 12A and the 10B.

Enabling students who want to try both (the 10 and the 12) in the same year is one reason the MAA has these two distinct versions of the test. If the Olympiad is your goal, one more interesting fact is that taking both AMC 10 and AMC 12 gives you a shot at qualifying for the USAMO and USAJMO separately.

### When and where do the AMC competitions take place?

The AMC-8 contest happens in November, while the AMC-10 and AMC-12 occur in January or February. They are always around the same time, although the specific date may change each year.

In 2021, the 10 and 12 were on Thursday, February 4th and Wednesday, February 10th. The MAA (the competition organizers) haven’t yet announced the 2022 dates yet, but will soon.

Keep in mind, the regular registration deadline is usually about two months before the competition date, and the late entry is a month before. Sign up early, so you don’t miss your shot!

The easiest way to find out where you can take any of the AMC tests is to use the zip-code search tool on the Mathematical Association of America’s website.

### When do you get your scores?

What’s harder than being in the American Mathematics Contest? Waiting for your scores to come back!

The scoring process takes about 3 to 4 weeks from your competition date. You’ll get your score by email. The AMC office is happy to help you out if you haven’t received your scores anytime 30 days after you’ve competed. You can call them at (800) 331-1622 or email maaservice@maa.org.

### 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?

You don’t need a calculator for that! Is it tricky? Easy? Either way, the important takeaway is that the work here is the puzzle, not computing big numbers.

The AMC looks for creativity and problem-solving chops, which is why no MAA AMC program ever needs a calculator. You could bring one in with you many years ago, but they banned them entirely in 2008.

### What about taking the same level many years in a row?

Calling all high school mathematics champs! As long as you haven’t aged out, you are welcome to try any AMC competition as many times as you’d like. And good news for all of you elementary and middle school students: the Mathematical Association of America says that kids as young as eight have competed.

Over the last few years, about 6% of all the AMC-8 competitors are in 5th grade or below. Another 19% of the competitors are in 6th grade. If you want to compete, early elementary school is the ideal for learning the mathematical reasoning and problem-solving skills you’ll need to succeed. If you are in elementary school (which is first to fifth grade in the USA), IMACS is a superb choice for early training.

Whether your goal is to compete while you are still in the elementary grades (as 25% of the AMC-8 competitors do) or in middle and high school, building the analytic and mathematical reasoning skills you’ll need for the AMC should start as soon as possible. The IMACS program teaches problem-solving and mathematical reasoning in a systematic and rigorous way, laying the foundational skills middle and high school competitors need to win. IMACS offers a free assessment for future AMC competitors—you can sign up here.

### What is a good score on the AMC?

In the most recent year, the average AMC-8 score was ten questions right out of 25. Since a perfect score on the AMC-8 would be 25, this may seem strikingly low. However, it’s a hopeful sign to aspiring AMC champions: the test is so tough that students can get many problems wrong and still do very well! Even honor roll scores are often far lower than perfect. Here are the average and honors scores for each AMC this past year.

• On the AMC-8, out of the 25 questions on the test:
• The average score was 10 right
• Honor Roll (top 5%) was 18 right
• Distinguished Honor Roll (top 1%) was 21 right

If you’d like to look at more statistics about the distribution of scores, the MAA AMC math program website has a neat tool for seeing how well the almost 100,000 competitors fared each year.

Do you think you can get distinguished honor roll? You can miss four problems and still be in the top 1%. Give this one, from 2016, a try.

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 standards does the AMC test?

### What math topics do you need to know for the AMC-8 competition?

The AMC-8 includes, but is not limited to, topics you’d see in a gifted middle school mathematics program. For example: exponents, counting and probability, estimation, proportional reasoning, basic geometry (including the Pythagorean Theorem), spatial visualization, simple statistics, and interpreting graphs and tables. The MAA, the organization that writes the competition questions, says that the problems can also include linear or quadratic functions and equations, coordinate geometry, and advanced geometry.

Although having a solid understanding of classic academic math concepts is essential, your creativity and mathematical reasoning skills will play an even larger role in your success. In fact, on the MAA website, they show which common core standards match with each problem. Last year, 11 of the 25 questions at AMC-8 did not map to US math standards. (Look at the “Item Difficulty” report to see which ones).

If you want to be a contender for a top score, a rigorous extracurricular program like IMACS, emphasizing logic and mathematical reasoning, is the best way to succeed. IMACS is an excellent program for 1st and 2nd graders, but even starting IMACS by 3rd grade or 4th will prepare you for the distinguished honor roll and even a perfect score.

If you want to learn more about the IMACS program and how it gives kids the skills they need to be an elite mathlete, sign-up for a free zoom assessment.

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?

The AMC-10 and 12 will require an understanding of a typical high school curriculum up to pre-calculus concepts. However, merely having these skills isn’t enough. 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 last year. If this seems impossible, keep in mind that this was one of the three hardest questions on the competition’s most challenging level. With the right training and practice, you’ll be able to do these problems!

## Here are a few other benefits of participating

Beyond being eligible to take the AIME, high-scoring students on the AMC are also up for other prizes and honors. If you’re a top scorer, you might receive a book prize or a plaque, and you’ll likely have your name recommended to colleges and STEM programs.

There are also scholarships and special prizes for young women. The AMC-10 and AMC-12 have recently begun offering the top twenty scoring women a \$1000 scholarship and there are certificates of excellence for the 580 highest scoring women. So, if you are a high school student, keep the AMC on your radar, and you might end up with some money for college!

While the love of math is its own reward, it’s notable that many top universities have spaces on their applications for sharing AMC scores. How much do AMC and AIME scores matter to top schools like MIT? Discussions on college application chat boards claim that a great score can significantly impact admissions, but no one outside the admission offices truly knows. Notably, competing in the AMC is a graduation requirement for all students at a few famous STEM-focused high schools, like Stuyvesant in New York City. When you take the AMC, you’re in good company!

## The best time to start AMC training

### How long does AMC prep take?

You can’t cram for the AMC. It is not even something that you will get great at by doing all of your homework (of course, keep doing all of your homework!). Instead, it relies on subtle skills that grow by solving similar practice problems with the right type of pedagogical support. Because each AMC problem is so different from all the others, success isn’t a matter of memorizing formulas. The very best preparation requires taking mathematical reasoning classes.

Good news: you can learn mathematical reasoning skills; they aren’t innate. Better news: the skills you develop for the AMC are what you’ll need for future careers. Whether you want to be an engineer, physicist, research scientist, or computer programmer, you are simultaneously preparing for interesting, rewarding careers as you prepare for the AMC.

Given the importance of these skills in both very cool careers and also in the world’s most prestigious universities (like Stanford), it is surprising that only a handful of extracurricular programs in the United States focus intensively on developing these critical mathematical abilities. While there are several great AMC prep options, the IMACS curriculum starts earlier (1st grade) than the others and has helped mathletes realize their greatest potential since 1993.

### At what age should AMC Prep begin?

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!

There are so many ways to building mathematical reasoning skills.

This AMC guide has covered a lot of the reasons taking a specialized class is essential. But there are also many other ways to get better. Here are some suggestions for our favorite books, puzzles, and games to help build the mental muscles mathletes need.

Puzzles and online resources for building mathematical reasoning skills

One of the best ways to develop the thinking skills for the AMC is to have fun! Logic games (available on Amazon or at local toy stores) make great birthday and holiday gifts, and online puzzles can entertain for hours. These resources are all great for exploring the fun of logic and problem solving.

### Books for Improving Problem-Solving and Mathematical Capabilities

Improving problem-solving abilities through a targeted course like IMACS is the most efficient way to build AMC skills. Still, problem-solving strategy books should also be on your prep list if you are shooting for honor roll or better.

What books to get? PrepScholar, one of the best SAT, GRE, and GMAT prep and tutor providers (featured in articles by the Wall Street Journal and Time Magazine), has a great AMC info page. They recommend three top books for aspiring mathletes:

PrepScholar’s recommendation of Arthur Engel’s book, “Problem Solving Strategies,” is a nice endorsement of IMACS, as Mr. Engel helped develop part of IMACS’ curriculum!

## Unexpected pro-tips for competition day.

### The night before the big day

The night before you take the AMC, relax and get into a healthy mindset. As everyone will tell you—go to bed early and get a good night’s sleep. This obvious advice is too often ignored. But what fewer people talk about is how important it is to get into the right sleep patterns up a week before the competition. Check out what this article in the Harvard Crimson (their student newspaper) has to say about it:

Biorhythmic gymnastics: Try to get on a sleeping pattern that has you waking up around test time every day for a week or so in advance so that your biological clock can acclimate to the test day schedule for optimal academic performance.

### Common test-taking advice is old news. What does science say?

Who doesn’t know that you should eat a nutritious breakfast the morning of the big day? Instead of looking at all the commonplace advice, let’s look at recent science and learn a few surprising ways you can boost your score.

Two intriguing pieces of research present a few unconventional but winning tactics.

The first, from the University of San Francisco:

“…good posture can mean better scores. Sitting up straight [during a test or competition] aids performance, researchers find.”

And the second, from scientists at the University of Chicago::

Writing about worries eases anxiety and improves test performance.

Researchers found that students who were prone to test anxiety improved their high-stakes test scores by nearly one grade point after they were given 10 minutes to write about what was causing them fear, according to a study in science. […] Pressure-filled situations can deplete a part of the brain’s processing power known as working memory, which is critical to many everyday activities. Working memory is a sort of mental scratch pad that allows people to retrieve and use information relevant to the task at hand. But it is a limited resource, and when worries creep up, the working memory people normally use to succeed becomes overburdened.

[…] The writing exercise allowed students to unload their anxieties before taking the test and accordingly freed up brainpower needed to complete the test successfully — brainpower that is normally occupied by worries about the test, explained the study’s senior author, Sian Beilock, an associate professor in psychology at the University.

## 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?

### The next step to get ready for the AMC

Competing in the AMC at any level is a great way to get ready for any STEM challenge in the future. More importantly, it’s fun!

If you know a child in 1st through 6th grade, send them this AMC guide and have them try the IMACS free assessment. An IMACS teacher will meet with any budding mathlete over Zoom. Through a series of fun puzzles and problems, IMACS will determine what skills the young problem-solver already has and what skills he or she still needs to develop to compete.

Thanks for reading our guide, and here is our last, best 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.

Author

## IMACS Staff

IMACS Live! is designed to give children a competitive edge by teaching them how to think critically using logic and reasoning. With over 25 years of experience, IMACS offers online after-school, weekend, homeschool, and summer classes for gifted and talented 1st-12th grade students in math enrichment, computer programming, logic puzzles, university-level mathematics, and university-level computer science. Highly experienced IMACS Live! instructors teach all courses via Zoom sessions. Secondary school students unable to attend using Zoom may be eligible for online self-paced courses through Elements of Mathematics: Foundations (EMF) or university-level courses through eIMACS.