We’ve put together a list of our favorite tips to help you take your math competition game to the next level. They are ordered chronologically starting at test preparation and leading up through the end of an exam. Some of these reference question numbers from our 2019-2020 Benchmark exam. If you’d like to share your own, please let us know at mathclubchair@rmsptsa.org.

1) Practice past test problems.

The absolute best way to prepare for a given competition is to master the questions from previous versions of that competition. Most competitions make some or all of their previous exams available, so check out the competitions page for links. For past RMS exams, please see our exams page.

Patterns and concepts repeat themselves, and you will get better at identifying questions with tricks and gotchas through experience. Sometimes there is no trick to a given question and you need to grind out the answer through casework, so figuring that out early saves you a lot of time.

Art of Problem Solving is a great resource with a lot of past competitions that include both answers and detailed solutions.

2) Memorize the Mathcounts Toolkit and other standard facts.

The Mathcounts Handbook provides a core set of common formulas and shortcuts condensed into a few sheets (Toolkit) that is really easy to digest. A lot of competition questions are just applications of these formulas, so having them memorized will save you a lot of time. Question #3 (area of a rhombus given both diagonals) is a simple calculation if you have the formula memorized. We’ve extracted the toolkit into a single file you can download through the link on the schedule page.

There are many other facts that are considered common knowledge for math tests, especially around measurement. Be sure you know the metric system (milli, centi, kilo, mega, etc.), imperial system (feet in a yard, cups in a gallon, etc.), and more. Question #2 (feet in 57 miles) is easy to answer if you know the number of feet in a mile before going in, but impossible to guess otherwise.

3) Master your math fundamentals.

A lot of questions will come down to your ability to quickly and correctly perform fundamental operations with big numbers. Many questions, such as #11 (height of smallest tower of five), #12 (conversion rates), and #25 (centimeters in an inch), were just a series of 2+ digit multiplication and division problems sometimes involving decimals.

At the same time, understand that there are some reasonable limitations to what exams can expect of you. If you find yourself needing to multiply two 10-digit numbers by hand, then consider that you may be missing something in the form of a more elegant solution. This comes up a lot with exponents and factorials, so expect fractions to have a lot of cancelling out.

4) Know what you can bring to the competition and don’t forget it.

Pretty much every competition requires that you bring pencils. Prefer mechanical pencils that do not need to be sharpened and bring a spare just in case.

Some competitions will provide scratch paper and will not allow you to bring your own. If you are allowed to bring your own, prefer graph paper (if allowed). This will make it easier to draw geometric diagrams to scale.

Some competitions will allow you to bring a calculator for some or all rounds. Be sure to bring a calculator that meets the requirements of the competition. They will almost surely disallow the use of phones or anything that can connect to the internet.

5) Manage your exam time so that you can at least read every question.

This might sound obvious, but it’s really important and can score you critical points. There is a pattern across exams where questions get harder as you get deeper in. As a result, some competitors will grind through questions one by one from the beginning and often don’t even read them all by the time the exam is over. That’s why devious test designers (like us) will often slip an easier question near the end of the exam. For example, #25 (calculating centimeters per inch) is a straightforward set of multiplication and division operations. However, multiple top 10 ties were broken because one competitor solved this relatively easy question while the other left it blank, presumably because they had not gotten to it by the time the exam ended.

We recommend the following time management strategy for math competition exams:

  1. Read the question. If you are confident that you know exactly how to answer it right now, then do so. It’s okay if it has a series of operations that might take you a while to complete. Always prefer the questions you’re highly confident in to those you’re not completely sure about.
  2. Otherwise, if you think you can figure out how to solve the question, but don’t have a solution right now, circle the question number on the problem sheet to indicate that you will come back to it.
  3. If you have no idea how to answer the question, put an X through the number on your problem sheet. Don’t burn time on these until you’ve resolved all the other questions.
  4. After you’ve gone through the entire exam, return to the beginning and review any circled questions using the same process. If a solution isn’t clear after re-reading the question, move on to the next one and solve the ones you can.
  5. After your second pass, review all of the answers you’ve put down. At this point you have a better chance of catching mistakes in your existing answers than grinding away at problems you don’t have solutions to.
  6. If you are confident in all of your answers, take another pass through the circled questions.
  7. When there are only X-ed out questions remaining, use the remaining time to craft educated guesses for them (if there is no penalty).

6) Don’t be intimidated by long questions.

These questions are usually a series of steps or parameters that are much easier to solve after you make a diagram, table, and/or system of equations. #14 (RMS student count) looks like a hassle but is fairly easy if you just make a table and fill it in based on the provided facts.

7) Draw out geometry questions if not given a diagram.

When a geometry question is given without a diagram, it’s usually easy to answer after you draw it out. In fact, that’s probably the reason they’re not giving you a diagram in the first place. #9 (find angle within a triangle) would have been way too easy if we provided a diagram.

On a related note, don’t automatically trust diagrams provided on the exam. They’re usually accurate, but sometimes will be intentionally misleading.

8) Be sure to give the answer the question is asking for.

This is one of the most common mistakes competitors can make if they’re not careful. Sometimes questions have a lot of data or complexity that can be distracting, such as question #23 (horse betting). A lot of answers where given as the horse (“D”) when the question was asking for an absolute return on investment in dollars (“-.50”).

Unfortunately, some devious test makers (like us) love to try to trick competitors. They’ll set up a question where the hard part is figuring out the area of a circle using obscure details about other geometric shapes, but then quietly slip in that the answer should be “the radius squared” and not the area itself. Question #4 (base-3 math) is a good example where many answers were given in base-10 when it was required they be in base-3.

9) Format answers as expected.

Similar to given the answer being requested, be sure to give the answer in the format requested. It’s important to know what the default form of answers is for your exam and to then pay close attention to questions that ask you to use something else. These formats will vary from competition to competition, but there are some general rules that usually apply.

  • Give the least amount of information required to meet competition standards. Answers like “x = 5” may be marked incorrect if the expected answer is “5”.
  • Fractional answers must be given as reduced, rationalized fractions. We had many answers for #19 (clock angles) that were correct in decimal form, but incorrect because they were required to be fractions per the rules on the front page.
  • Simplify radicals. √8 should be 2√2.
  • Leave mathematical constants like pi (π) as-is unless specifically instructed to calculate and round.
  • Money answers should be rounded to the nearest cent unless it’s a whole-dollar amount. Do not put a currency sign.
  • Don’t put units.
  • For time, don’t forget AM/PM. We had wrong answers for #10 (broken clock) because they left off “PM”.
  • Don’t put commas in numbers to avoid getting them confused as decimal points.

Some specific form of answers docs are available at:

10) Don’t leave blanks if there is no penalty.

The title says it all here. If you’re at the end of the exam and have blank answers on a test where this is no penalty for guessing incorrectly, put something (anything!) down. If you have time, try to come up with educated guesses based on the information you have. If you have no idea, 0’s and/or 1’s are a good choice. Devious test designers (like us) will often put questions with these answers on very challenging problems late in the exam. For example, the answer to #22 (randomly translated triangles) was 0 and the answer to #26 (point translation) was 1.

When selecting an educated guess, consider the type of answer you’re giving. If it’s a probability, go with a reasonable fraction. If it’s a percentage you know is under 100%, guess within that range.