Getting Started with the USAMTS

If you have decided to participate in the USA Mathematical Talent Search (USAMTS), congratulations! The USAMTS is one of the most prestigious and challenging math contests, so you will be competing against some of the best and brightest middle and high schoolers in the nation. Don’t be intimidated, though! Just try your best—math is supposed to be fun!

The USAMTS is different from math exams you take in school and other math contests. Most notably, we place very little emphasis on speed. We are much more interested in developing your ability to think deeply to solve challenging and interesting math problems. This is why we give students at least a month to solve each round of problems.

Each school year there are three USAMTS rounds. Each round consists of five problems—one puzzle and four proof-based problems. For each proof-based problem, you need to write a full solution explaining how you solved the problem. We encourage students to do all three rounds, but students are welcome to participate in Rounds 2 and 3 even if they don’t participate in Round 1.

We’ve written this article to help both new and experienced competitors get the most out of their USAMTS experience. See the Key Steps below (recommended reading for everyone). See Getting Started with Competitive Math for a guide on problem solving and writing proofs (recommended for those who are new to proof-based math contests).

Key Steps

1. If you are new to the USAMTS, register. If you are a returning competitor, use your existing account. At the beginning of every contest year, you will be prompted to update your information.
2. Read the contest rules.
3. Read the current problems.
4. Solve the problems. The problems are supposed to be challenging, so read some of our tips or our Getting Started with Competitive Math guide if you get stuck on a USAMTS problem.
5. Write solutions. We encourage you to start writing your solutions early. As with any other writing, creating a draft and returning to it later with a fresh set of eyes is a great way to recognize steps that need to be explained more thoroughly. This can also help you discover possible gaps or errors in your solution, and give you time to correct them. Also see our LaTeX guide, which include template files for solutions.
6. Submit your solutions. Once you’ve written your solutions, submit them online or by mail prior to the submission deadline.
7. Review grader feedback. Our team will read your solutions and provide you with written feedback to help you develop your math, problem-solving, and technical writing skills. If you believe you should have received more points on a problem, you can submit a protest.

Most importantly, enjoy the problems and have fun!

Competing in the USAMTS

First off, this guide is not a substitute for reading the rules, so make sure to read those as well.

Each year, there are 3 USAMTS rounds. Each round will consist of 5 problems—1 puzzle and 4 proof-based problems—for which you will be given at least one month to solve.

When you are done solving the problems, submit your solutions, remembering our rules for submitting. If you are submitting online, remember that you must submit a single PDF file. No other file format will be accepted. If you have handwritten your solutions and don’t have access to a scanner, you can use a scanner app on your phone, such as Google Drive’s scanner or Genius Scan, to scan your work.

If you wish to make changes to your online submission prior to the submission deadline, you may do so by resubmitting online. The new submission will overwrite your previous submission.

After each round, you will be notified when your submission has been graded. You can check your scores and feedback on our My Scores page. If you feel that the graders have made a mistake grading your work, you may submit a protest on our Protest page.

After we finish grading Round 3, we will announce the cutoffs for prizes, and send prize-winners instructions on how to claim their prizes.

Check out the USAMTS forum on AoPS to discuss the contest with fellow students. Just remember not to discuss the problems until after the round is over.

Using LaTeX for USAMTS

LaTeX is a document typesetting system widely used in academia. Unlike “what you see is what you get” typesetting systems like Microsoft Word and Google Docs, LaTeX is written as a .tex file, which is just a plain-old text file, and then compiled into a .pdf file. As such, LaTeX has a bit of a learning curve. However, learning LaTeX is well worth it as it allows you to create very high quality documents, and if you plan to major in a STEM field, you most likely will have to learn it at some point in your life anyway.

We strongly encourage you to learn LaTeX and write your solutions using it. AoPS user greenturtle3141 has kindly provided a beginner’s guide to LaTeX for USAMTS participants. Even if you decide not to typeset your entire document with LaTeX, we still recommend using a tool like TeXeR to format your equations and formulas. You are not required to use LaTeX, though, so feel free to handwrite your solutions or use a word processor like Microsoft Word if you prefer.

The easiest tool to use to write LaTeX is Overleaf, which is an online editor. The steps to install LaTeX for offline use are a little more involved since it varies so much by operating system, but it’s still an option if you prefer. Also, there is a page on the AoPS wiki about LaTeX, if you want to read more.

If you need to include drawings or figures in your solution, we recommend starting out by hand-drawing or using a tool like Geogebra. If you are using LaTeX, you can then include a scan or screenshot in your LaTeX document. Another option is using a vector graphics language like Asymptote directly in your LaTeX document, but doing so is more difficult and time-consuming than using Geogebra, so we don’t recommend this for beginners.

LaTeX Solutions Template

If you do choose to use LaTeX, we recommend that you use the official USAMTS template: solutions.tex. Instructions for how to use the template are in the file. You will also need to download the USAMTS include file: usamts.tex. Put this file in the same folder that you put your solutions.

Handwritten Solutions Template

If you choose to handwrite your solutions, we recommend printing out and using the USAMTS solutions form. Remember to write your name, username, and user ID on every page, along with the round and problem numbers. Also, if you choose to handwrite, please write neatly.

Using Reference Material

You are allowed to reference books or the internet when solving the problems, as long as you do not ask for help on sites like the AoPS forums or Stack Overflow. Remember that you must cite or prove every non-trivial theorem that you use, unless it is “well known.”

Every problem on the USAMTS is possible to solve using only “well known” results.

Using Computers

You are allowed to use computers and calculators when solving problems. This includes things like writing programs in Python or using WolframAlpha. However, if you do write a program that is used as part of your solution, you must include the input, output, and source code of your program, along with enough explanation for a grader to understand your code. If you use an answer engine like WolframAlpha, you must include the exact query that you used and the exact output.

If you used a computer to help you find a pattern or make a conjecture that you then prove rigorously without the use of a computer, then you don’t need to include the data, code, or explanations mentioned above.

Using computers can be helpful for some USAMTS problems, though every problem on the USAMTS is possible to solve without the use of a computer.

Tips

• Start early.
• Take breaks. Sometimes not thinking about a problem for a few hours or a few days can help you come up with new ideas for how to solve it.
• Look at past years’ solutions to get an idea of how your solutions should be written.
• Check your work. If your solution includes a lot of algebra, substitute some values into your equations to make sure that they are correct. If you find a number-theoretic or combinatorial formula, check a few cases to make sure that the formula gives the right answer. If you write a program, print some debug output to verify that the program doesn’t have any bugs.
• Draw accurate diagrams. If hand-drawing, use a straightedge and compass. If you don’t want to hand-draw, you can use drawing software such as GeoGebra or Asymptote.

Getting Started with Competitive Math

Competitive math is a topic that many others have covered in far greater depth than what we could ever cover on single web page. Here is a list of resources to help you get started on your competitive math journey:

• The AoPS Wiki has a wide selection of resources, tutorials, and practice problems.
• “How to Write a Math Solution” by Richard Rusczyk is an article that gives a good overview of how to write well for proof-based math contests.
• The Art of Problem Solving, by Richard Rusczyk and The Art and Craft of Problem Solving, by Paul Zeitz are both great books that introduce many of the strategies, techniques, and theorems needed to succeed in competitive math. The former book is directed more towards middle schoolers, while the latter is directed more towards high schoolers.
• Taking part in other contests, such as MATHCOUNTS and the AMC, can be good practice, although these contests have a vastly different format from the USAMTS.

In general, competitive math is like most other skills—the best way to get better is to practice regularly with tasks that are challenging for you, and to routinely get feedback on your work. Also, smart people rarely become smart on their own. Finding a community of peers, whether in-person or online, who are also doing competitive math, will give you a good source of insights, support, and motivation.