Art of Problem Solving (2024)

Many AoPS Community members and online school students have been participants at National MATHCOUNTS, including many Nationals Countdown Round participants in the past decade. MATHCOUNTS is a large national mathematics competition and mathematics coaching program that has served millions of middle school students since 1984. Sponsored by the CNA Foundation, National Society of Professional Engineers, the National Council of Teachers of Mathematics, and others including Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.

Region: USA
Type: Free Response
Difficulty: 0.5 - 2.5
Difficulty Breakdown:

Countdown: 0.5 (School/Chapter), 1 (State/National)
Sprint: 1-1.5 (School/Chapter), 2-2.5 (State/National)
Target: 1.5 (School), 2 (Chapter), 2-2.5 (State/National)


  • 1 MATHCOUNTS Curriculum
  • 2 Past State Team Winners
  • 3 MATHCOUNTS Competition Structure
    • 3.1 Sprint Round
    • 3.2 Target Round
    • 3.3 Team Round
    • 3.4 Countdown Round
    • 3.5 Chapter and State Competitions
    • 3.6 National Competition
    • 3.7 Ciphering Round
    • 3.8 Masters Round
    • 3.9 Scoring and Ranking
  • 4 MATHCOUNTS Competition Levels
    • 4.1 School Competition
    • 4.2 Chapter Competition
    • 4.3 State Competition
    • 4.4 National Competition
      • 4.4.1 National Competition Sites
  • 5 What comes after MATHCOUNTS?
  • 6 See also...


MATHCOUNTS curriculum includes arithmetic, algebra, counting, geometry, number theory, probability, and statistics. The focus of MATHCOUNTS curriculum is in developing mathematical problem solving skills.

Before 1990, MATHCOUNTS chose particular areas of mathematics to highlight each year before changing the focus of the competition more broadly to problem solving.

Past State Team Winners

  • 1984: Virginia
  • 1985: Florida
  • 1986: California
  • 1987: New York
  • 1988: New York
  • 1989: North Carolina
  • 1990: Ohio
  • 1991: Alabama
  • 1992: California
  • 1993: Kansas
  • 1994: Pennsylvania
  • 1995: Indiana
  • 1996: Wisconsin
  • 1997: Massachusetts
  • 1998: Idaho
  • 1999: Massachusetts
  • 2000: California
  • 2001: Virginia
  • 2002: California
  • 2003: California
  • 2004: Illinois
  • 2005: Texas
  • 2006: Virginia
  • 2007: Texas
  • 2008: Texas
  • 2009: Texas
  • 2010: California
  • 2011: California
  • 2012: Massachusetts
  • 2013: Massachusetts
  • 2014: California
  • 2015: Indiana
  • 2016: Texas
  • 2017: Texas
  • 2018: Texas
  • 2019: Massachusetts
  • 2020: CANCELLED
  • 2021: New Jersey
  • 2022: New Jersey
  • 2023: Texas
  • 2024: Texas

MATHCOUNTS Competition Structure

Sprint Round

30 problems are given all at once. Students have 40 minutes to complete the Sprint Round. This round is very fast-paced and requires speed and accuracy as well. The first 20 problems are usually the easiest problems in the competition, and the last 10 problems can be as hard as some of the Team Round questions. No calculators are allowed during this round.


Target Round

8 problems are given 2 at a time. Students have 6 minutes to complete each set of two problems. Students may not go back to previous rounds (or forwards to future rounds) even if they finish before time is called. Calculators are allowed for the Target Round. Usually comprised of one "confidence booster" and one hard problem.

Team Round

10 problems in 20 minutes for a team of 4 students. These problems typically include some of the most difficult problems of the competition. Use of a calculator is allowed (and required for some questions).

Countdown Round

High scoring individuals compete head-to-head until a champion is crowned. People compete from off a screen taking 45 seconds or less to finish the problem. The Countdown round is run differently in various different chapter, state, and national competitions. In the national competitions, it is the round that determines the champion. Calculators are not allowed, but scratch paper will be provided.

Chapter and State Competitions

In the chapter and state competitions, the countdown round is not mandatory. However, if it is deemed official by the chapter or state, the following format must be used:

  • The 10th place written finisher competes against the 9th place written finisher. A problem is displayed, and both competitors have 45 seconds to answer the question, and the first competitor to correctly answer the question receives one point. The person who gets the most correct out of three questions (not necessarily two out of three) is the winner.
  • The winner of the first round goes up against the 8th place finisher.
  • The winner of the second round goes up against the 7th place finisher.

This process is continued until the countdown round reaches the top four written competitors. Starting then, the first person to get three questions correct wins (as opposed to the best-out-of-three rule).

If the countdown round is unofficial, any format may be used, and if it is unofficial it will not determine your placement towards the final results and may be determined by another format. Single-elimination bracket-style tournaments are common.

National Competition

At the national competition, there are some structural changes to the countdown round. The top 12 (not the top 10) written finishers make it to the countdown round, and the format is changed from a ladder competition to a single-elimination tournament where the top four written competitors get a bye. This setup makes it far more likely for a 12th place finisher to become champion, and it makes it less likely for a first-place written finisher to become champion, equalizing the field. But even then, a 12th place written competitor will have less of a chance to become champion than the top 4, because the top 4 get a bye. Until the semi-finals, the scoring is best out of five advances.

In the first round and the second round, the person to correctly answer the most out of 5 questions wins. However, at the semifinals, the rules slightly change— the first person to correctly answer four questions wins.

Ciphering Round

In some states (most notably Florida), there is an optional ciphering round. Very similar to countdown (in both difficulty and layout), a team sends up a representative to go against all representatives from the other teams. A problem is shown on a screen and students work fast to answer the problem. The students give their answer and after 45 seconds the answer is shown and the answers are checked to see if they are right. The fastest correct answer gets five points, the next fastest gets 4, etc. There are 4 questions per individual and teams send up 4 people. A perfect score is then 80. Often times the questions take clever reading skills. For example, one question was "How much dirt is in a 3 ft by 3 ft by 4 ft hole?" The answer was 0 because there is no dirt in a hole.

Masters Round

Top students give in-depth explanations to challenging problems. This round was optional at the state level competition and mandatory at the national competition (up to 2011). At nationals the top two on the written and countdown participate. In 2012, it was replaced by the Reel Math Challenge (now called the Math Video Challenge).

Scoring and Ranking

An individual's score is their total number of correct sprint round answers plus 2 times their total number of correct target round answers. This total is out of a maximum of Art of Problem Solving (1) points.

A team's score is the individual scores of its members divided by 4 plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a relatively low team score and a phenomenal individual score, as the team score is only roughly 30% of the total team score. Note that when there are less than four members the score will become less.

MATHCOUNTS Competition Levels

School Competition

Students vie for the chance to make their school teams. Problems at this level are generally the easiest and most basic in curriculum.

Chapter Competition

Chapter competitions serve as a selection filter for state competitions. A few states don't need to host chapter competitions due to a small population size.

State Competition

The top 4 students in each state form the state team for the national competition. The coach of the top school team at the state level is invited to coach the state team at the national competition. Interestingly, the coach of a state team is not necessarily the coach of any of the state's team members. State competition competitors may be determined from the Chapter competition, based on the population of that particular state.

National Competition

National Competition Sites

For many years, the National MATHCOUNTS competition was held in Washington, D.C. More recently, the competition has changed venues often.

  • The 2024 competition was held in Washington, D.C.
  • The 2023 competition was held in Orlando, Florida.
  • The 2022 competition was held in Washington, D.C.
  • The 2021 competition was an online event.
  • The 2020 competition was canceled due to the COVID-19 pandemic, but was due to be held in Orlando, Florida.
  • The 2019 competition was held in Orlando, Florida.
  • The 2018 competition was held in Washington, D.C.
  • The 2017 competition was held in Orlando, Florida.
  • The 2016 competition was held in Washington, D.C.
  • The 2015 competition was held in Boston, Massachusetts.
  • The 2014 competition was held in Orlando, Florida.
  • The 2013 competition was held in Washington, D.C.
  • The 2012 competition was held in Orlando, Florida.
  • The 2011 competition was held in Washington, D.C.
  • The 2009 and 2010 competitions were held in Orlando, Florida.
  • The 2008 competition was held in Denver, Colorado.
  • The 2007 competition was held in Fort Worth, Texas.
  • The 2006 competition was held in Arlington, Virginia.
  • The 2005 competition was held in Detroit, Michigan.
  • The 2004 competition was held in Washington, D.C.
  • The 2002 and 2003 competitions were held in Chicago, Illinois.

What comes after MATHCOUNTS?

Give the following competitions a try and take a look at the List of United States high school mathematics competitions.

  • American Mathematics Competitions
  • American Regions Math League
  • Mandelbrot Competition
  • Mu Alpha Theta

See also...

  • List of national MATHCOUNTS teams
  • Mathematics competition resources
  • Math contest books
  • Math books
  • List of United States middle school mathematics competitions
  • List of United States high school mathematics competitions
  • 2006 MATHCOUNTS Countdown Video
Art of Problem Solving (2024)
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