The AMC 10/12 exams are coming up in less than a month, so today, I thought I'd share some of my tried-and-true AMC tips and tricks!
For a little background, the AMC (American Math Competition) is the national math competition that qualifies any mathlete for the next round of the prestigious MAA competition series, the AIME (American Invitational Math Exam). If you're new to math competitions, AMC is one of the best places to start: it's the most standardized and well-known competition in the math world! You can learn more about it here.
This is a follow-up to the last article, where I compiled the 40 most important AMC problems for you to solve during your preparation process, so make sure you check that out! The problems are grouped by category and aimed at the intersection of AMC 10 and AMC 12, and I carefully chose them to cover the widest range of topics possible, so they will help you understand what content you understand and what you might want to learn or review before the AMC. Go here to visit it!
Today's article will provide some tips for the short-term as well as the long-term. I'll start by giving you some advice on what you should focus on in the next few weeks, and I will also provide some tips and strategies you might want to incorporate into a longer, in-depth AMC preparation process if you want to thoroughly prepare over the course of this year for next year's AMCs. Finally, we'll end with some tips you can use while solving problems and taking the exam.
Also, if you are a middle schooler and want AMC 8 tips, here is an article from last year that should help you out! This is actually GLeaM's most popular article to date.
Let's get to the article!
Short-Term AMC Tips
Here are some steps you can take over the next few weeks that would be extremely effective!
Solve some problems! You can find lots here: https://artofproblemsolving.com/wiki/index.php/AMC_10_Problems_and_Solutions or https://artofproblemsolving.com/wiki/index.php/AMC_12_Problems_and_Solutions. Remember that a very good score does not require solving that many problems on the exam, and practicing is super important to help you hone your skills to get the score you want! Here are some other places to find practice problems: https://artofproblemsolving.com/wiki/index.php/Mock_AMC and https://ziml.areteem.org/ziml/practicecontests.php?p=MockAMC. You can also, of course, check out last week's article for a systematic topic-themed problem set to review more effectively!
Take a few timed (75-minute) real-scenario AMC exams, using the resources for practice problems listed above. Make sure you bring all the materials you're allowed to have with you during the AMC exam: pencils, erasers, blank scratch paper, rulers, compasses, highlighters, and colored pens. Calculators are NOT allowed.
Check out this list of topics and formulas: Eashan Gandotra's Formulas for Pre-Olympiad Math. Do not feel like you need to know all of them (you don't!) but review and remember at least the beginning ones in each section and remind yourself of what you do know before the competition. It's a great resource to see all the facts at once, and it'll help you find a few tidbits to learn before you take the AMC!
Long-Term AMC Tips
Solve lots and lots of problems! Competition math is unique in that, for the most part, there really isn't a curriculum of things you need to understand per se (it's a lot more about the reasoning and critical thinking skills you develop over time), and all you need to know in terms of content can be found in the books I mention below. The best way to get a grasp on the kinds of questions that are asked in the AMC and to get better is to simply practice a lot! Like I mentioned earlier, check out the AMC 10 problem database (https://artofproblemsolving.com/wiki/index.php/AMC_10_Problems_and_Solutions) and the AMC 12 database (https://artofproblemsolving.com/wiki/index.php/AMC_12_Problems_and_Solutions), as well as these two resources: https://artofproblemsolving.com/wiki/index.php/Mock_AMC and https://ziml.areteem.org/ziml/practicecontests.php?p=MockAMC for practice problems. I also know that the MATHCOUNTS Trainer (at https://artofproblemsolving.com/mathcounts_trainer) and For the Win! (at https://artofproblemsolving.com/ftw) are amazing game-show-style ways to try your hand at some problems. These are geared more towards the MATHCOUNTS and AMC 10 in terms of difficulty level.
Read math books. If you're willing to invest in books, the absolute best books are the Art of Problem Solving book series: Art of Problem Solving Volume 1: The Basics and Art of Problem Solving Volume 2: and Beyond. You'll need to get the solution manuals as well. I would probably start with the first volume and work your way up to the second, but that depends on how confident you feel with competition math already! You could start at Volume 2 if you feel relatively confident with the competition math in middle school and on MATHCOUNTS (if that is something you've done) as that's mostly what Volume 1 covers. Another book I've found to be really helpful is the Three-Year MATHCOUNTS Marathon by Karen Ge. I know that this is geared towards the MATHCOUNTS competition, but the difficulty level of the problems in the book are way above middle school. All the concepts explored in the book are very relevant for varsity (and JV) competitions, which relate so well to the AMC, and you can get so much out of the concise explanations throughout the book. Also, this book focuses mainly on AIME problems than the AMC, so it is a bit further down the road from AMC, but A Gentle Introduction to the American Invitational Mathematics Exam by Scott A. Annin is also a fantastic resource. Feel free to ask in the comments if you want even more book recommendations!
Take an online course. If you're super invested in this, the Art of Problem Solving has a lot of online courses specifically tailored to the AMC and AIME series, including a Special AMC Seminar series that lasts for one weekend a few weeks before the AMC, where you're given a comprehensive crash course. There are also some free videos on the AoPS website that are helpful.
Attend other math competitions! This is pretty self-explanatory; finding all the competitions in your area and competing is the best way to practice for the AMC. If you live in Georgia, feel free to reach out because I can definitely discuss local opportunities with you.
Check out cheat sheets! One way I personally really like to study for math competitions is by finding "cheat sheets," essentially documents that compile formulas for you to learn. This is by no means something meant for you to memorize or for you to feel like you have to know entirely (because honestly, you can do really well on the AMC without using any formulas!). However, seeing lists of formulas and trying to figure out how they were derived and how they would be used helps you see the swath of potential concepts in math tournaments and learn how they all fit together. My favorite at the moment is Eashan Gandotra's Formulas for Pre-Olympiad Math, but I also like Coach Monk's High School Playbook and the outline All of Math in Three Pages. You can also check out Jim Sukha's ARML Handbook and Tom Davis's Contest Geometry Handbook. (All of those descriptions are linked, so click on them!) Again, these are not for you to memorize but to give you a reference sheet as you pick up concepts in contest math and to give you something to review before the AMC or any competition. It really helps me get in the mindset before taking a math competition to review one of these sheets, so that's something unique I've picked up over the years.
Tips for Taking the AMC Competition!
Manage your time. You only have 75 minutes, so keep an eye on the clock (bring a watch!), and leave the last couple of minutes to check and review and make sure you bubbled everything correctly. I don't want to lay out elaborate timing rules, because the AMC experience is different for everybody depending on your goals, but a general rule of thumb if you want to advance to the AIME is: answer around the first 10 in 25 minutes, around the first 15 in 50 minutes, and after that, try to answer at least 19 questions in total. (You only need to get about 16-18 right to make the AIME.) Time yourself and pick a target score to figure out where your own timing agenda might fall, but remember to focus more on accuracy than timing; Do not rush! (Another rule of thumb is to make sure you are not stuck on one problem for 2-3 minutes. If you can't get anywhere in that amount of time, keep moving.)
Avoid careless errors. This is easier said than done, but keeping your work organized on your scratch paper and slowing down are two great ways to avoid accidentally multiplying incorrectly or blundering something similar. Stay careful and attentive and your score could go way up! Here's a great article by the Art of Problem Solving founder, Richard Rusczyk, about learning to avoid 'stupid mistakes.' It is better to answer fewer questions and be entirely certain about them than to answer every single question and rush through the test.
Read, read, and read! This goes right along with avoiding careless errors. Make sure you're interpreting the problem correctly. I recommend reading the problem once to get the gist of it and then immediately reading it a second time, paying more attention to details. Also, read the answer choices to see if you can rule anything out immediately. Then, after you've solved the problem, read it one more time to make sure you're actually answering what they're asking.
Annotate! Underline key words, like not, integers, or consecutive, and highlight or circle the specific question they are asking you (what quantity they want you to find). Since the 2021 exams are online, you might not be able to write directly on the test, so if you cannot, keep a piece of paper beside you where you can write down ideas and thoughts, as well as keep track of answer choices you might have eliminated.
Organize your scratch paper into sections. Divide your paper up into four boxes on each side. Label each box with a question number and keep your work for that question in there, so if you want to review something, it's easy to find! You might want to leave a half-page or even a full-page for the harder questions at the end. It also helps to only work on the front of the paper, so you don't accidentally lose something that might be on the back. Write neatly so you can check your work at the end!
Draw large, precise, to-scale diagrams for geometry questions! When you have a very large diagram, it is easy to see when two lines might be parallel or when a triangle might be a right triangle. You still have to go through the math to fully convince yourself these facts are true, but having an accurate diagram is a major tip-off to what the answer might be. Also, draw your diagrams in PEN. Often, you have to draw other lines and figures on top of your diagram that you'll be erasing and retrying, so if you have your main diagram in pen, you won't have to worry about removing it and having to continually redraw it.
Guess if you can eliminate the answer choices down to 3. The AMC penalizes you for guessing wildly, but if you can get rid of two answer choices, go ahead and guess because the odds are in your favor! You can also often even use the answer choices themselves to solve the problem. Try plugging them in as potential values and/or just using them to get an idea of what type of number you're looking for as an answer.
Dive in! Some problems are easier than they seem, so just go ahead and try it out. Often, if there is a long equation that looks pretty complicated, there is a pretty easy way to go around it if you just think about it for a bit. Test writers often put intimidating problems later in the test because they look hard, rather than because they are actually hard, so they tend to be easier than you think they'll be. Most of all, don't just sit and stare. Gravitating towards problems you know you can do is a great way to get a good score, but cycle back around if you have time, and at least try as many as you can! For geometry, a good way to keep moving is drawing in extra lines, like a perpendicular from a point to a line or a line through a point that is parallel to another line.
Keep your eye on the ball! Looking at what the question is asking can save you a lot of time. For example, if the question asks for the sum x+y, there is no need to individually solve for x and y. Make sure you coordinate your solution process so it gets to the answer you need as simply and quickly as possible. Also, remove anything overly ugly first: square expressions to get rid of square roots, exponentiate to get rid of logs, and clear denominators. If you're ever stuck, go back to the question and ask yourself if there are any pieces of information you haven't used yet. Often, those facts are the key to the problem.
Start small. If the question involves variables, plug in a few simple values and see what happens, especially with sequences and series. Look for patterns. If the problem seems too complicated, see if you can find a simpler version to try.
Check your work! Set aside 5-10 minutes to go over your computations at the end of the competition, and also check as you go. If you have extended computations or a lot of manipulation, double-checking that each individual step is going properly will save you a lot of time if something goes awry all the way at the end of your solution process. "Sanity check" your final answers to make sure they make sense, in terms of scale and type of answer. For example, if the question asks you for the sum of the factors of 36, the answer cannot be a fraction, and you may be able to catch an error! If it is feasible, even try to plug your answer back into the question!
The standard tips: I might as well tell you to get a good night's sleep and eat a healthy breakfast. Cliché, but it honestly helps so much!
I hope this was helpful! Feel free to reach out (either through the comments or by emailing me) if you want even more specific AMC-related advice. I will also leave two resources here if you'd like to read other articles and discover other strategies for the AMC:
Have a great weekend, everybody!