Thoughts on the AMC 10+12

Fall 2022 AMC 10+12 Thoughts with Dr. Wang and Mr. John

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With the Fall 2022 AMC 10A, 12A, 10B, and 12B complete, Areteem’s Academic Director Dr. Wang and Director of Curriculum & Instruction Mr. John share their thoughts on the exams and answer five questions about the problems.

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Five Questions with Dr. Wang and Mr. John

1) What are your overall thoughts on the 2022 AMC 10+12 exams?

Dr. Wang: The overall trend in the AMC 10 and 12 during recently years is that the problems are getting harder by the year. The problems this year can be described as follows: the first part of each test paper, roughly #1 to #17, are generally similar in difficulty to previous years, perhaps a little easier; the second part, roughly #18 to #25, are generally harder. The most common knowledge points and techniques are all being tested, such as Vieta’s Theorem, complex numbers (especially roots of unity), trigonometry identities, place values, telescoping sums, etc. There are also a few advanced ideas or methods that are involved, such as Method of Undetermined Coefficients (10B #21/12B #20), Lipschitz conditions (10B #24), and linear algebra (10B #18); some have ideas taken from old math olympiad problems (but simplified), such as 10B #14 and 10B #25/12B #23.

Mr. John: The questions this year were definitely challenging, as they have been in recent years. A strong foundation is more important than ever, as the more difficult questions rely on multiple key ideas or theorems. I would also say some difficulty is added by longer questions, with multiple steps or cases to consider. Thus, students must answer questions (especially the beginning questions, which tend to be easier) quickly, so they have time for the harder ones. I do worry a little that the tests are becoming less approachable to the general high school math student. Of course I don’t expect a student taking the AMC 10 or 12 for the first time to be able to compete with students who have spent the year preparing, but I hope that a first time student is still able to solve some of the problems, plus be able to understand the solutions for many more. Hopefully they are inspired to learn more and try again the next year!

2) What is your favorite problem from this year?

Mr. John: I covered two B exam problems I liked, AMC 10B #22 (12B #21) and 10B #24, in my Thursday Tidbit’s video last week. One of my other favorite problems was AMC 10A #22 (12A #19) about arrangements of cards. To me, it’s a great example of a “harder but approachable” problem. There’s some good careful thinking / exploration to notice you’re really just picking subsets, and the problem can definitely be used to help teach or reinforce some basic counting skills and identities. I also want to give a quick shoutout to AMC 10B #5, which pretty easy to do directly, but also has some fun factoring patterns that can speed it up.

Dr. Wang: One problem I like is #17 from the AMC 12A. I always like an interesting application of the trigonometry identities. This question involves the double-angle and triple-angle formulas; alternatively it can be solved with sum-to-product formulas. It is also a fresh angle to ask the question—asking about the range of a parameter, which can be expressed in terms of intervals. Another problem I like is AMC 12B #17. This one appears to be adapted from an old math olympiad question too, and the key idea gives a pretty clever method.

3) Were there any problems you disliked? Why?

Dr. Wang: The one problem I dislike is AMC 10A #25. The question involves high order Diophantine equations, and there are no clever ideas except for some standard divisibility analysis, and the whole process is quite tedious.

Mr. John: I personally am not a big fan of AMC 10B #18 about systems of equations and when they have non-trivial solutions. As a general question, it’s definitely interesting, and could be a good challenge question to use for students learning about systems of equations to further explore the ideas. For the contest, however, it can be a little messy to consider all the different cases and students who have familiarity with linear algebra (which is something with less emphasis both in school and on previous exams) have a large advantage over other students.

4) Which one problem do you recommend students review and fully understand?

Mr. John: AMC 12A #25 is my recommended problem for students to revisit and make sure they understand. It is a more difficult problems with various steps, but even students preparing for AMC 10 would benefit from reviewing it. The problem is also a nice mix of geometry (tangent circles and lines) and algebra / number theory (Pythagorean triples, completing the rectangle, etc.), so it’s a great way to study multiple subjects at once. Even if it’s not required for this problem, be sure to review how to generate Pythagorean triples! For students preparing for AMC 12 who want a good problem to use for reviewing roots of unity, #21 from the AMC 12A is a good problem to review as well.

Dr. Wang: I would vote for 10A #24/12A #24. Although the case analysis is also a little tedious, it is a nice problem to practice careful case work. In fact if one lists the cases clearly, the whole problem only requires a few minutes to finish. If one use complementary counting, the case work is much simpler. Also, the related famous problem, “Parking Functions”, has a very nice bijection solution, which provides a lot of insights if one wants to learn how to convert one problem to an equivalent one that is in fact easier to solve. Another problem worth studying is 12A #23. At first glance, there seems to be no easier way than doing brute force calculations, but as one starts to explore the denominators of the sums, thinking about why some the denominators can be different from the least common multiples, the question becomes rather interesting, and the solution is pretty clever.

5) What advice do you have for students preparing for next year?

Dr. Wang: The learning of math should not be limited to preparation for math competitions. The concepts and problem-solving techniques that are frequently involved in the math contests such as AMC and ZIML are all very important in future STEM study and real-life applications. Areteem courses are designed with these goals in mind—not only help you get ready for math competitions, but also prepare you for future studying in school and solving real world problems in your field of endeavor. This is also aligned with the true purpose of the math competitions like the AMC series. Therefore, our courses systematically help students learn the concepts and techniques needed in the contests, and encourage them to see beyond the contests themselves. Interestingly, when they study with the bigger picture in mind, the students actually get better results in the math contests.

Mr. John: Practice, practice, practice! The best way to keep improving is to be doing math as often as possible, every day is best! The ZIML Daily Magic Spells and Monthly Contests are perfect for tracking your progress over time. Courses that focus on building a strong foundation, such as Areteem’s Math Challenge courses are great as well. That being said, any math is better than no math. Math riddles, puzzles, and games (such as Sudoku and KenKen) help build logical thinking skills and exposure new mathematical ideas, even in contexts outside of math contests, only helps! Be sure you are having fun while learning!

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