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Mathcounts: The Most Exciting Math Competition for U.S. Middle Schoolers

A road leads to a snow-capped mountain in the distance and "Mathcounts" is written on the road.

Expand the limits of your math skills with Mathcounts!

Whether you count down the days until your next math class or count down the minutes until it ends, Mathcounts is the challenge for you. Mathcounts offers the opportunity to pursue math competitively, which is exhilarating for both those who want to push themselves to the limits of their abilities and those who find traditional academic math classes too slow.

Mathcounts, founded by the National Society of Professional Engineers and the National Council of Teachers of Mathematics, has organized competitions and programs for U.S. middle school students since 1983. You can compete no matter how good (or bad) you might think you are at math. The goal of the Mathcounts competitions is to improve students’ feelings about math and to improve their problem-solving abilities.

More than 250,000 students each year are involved with Mathcounts programs and offerings. If you want to learn how to get involved, keep reading!

 

A road runs up the center of a canyon. The word "challenge yourself: Mathcounts" are superimposed over the road and horizon.

 

Mathcounts runs three separate programs. Which ones are right for you?

The Mathcounts organization runs three math programs. The first is an enrichment program that provides advice and resources to educators trying to start math clubs.

The second is a contest that asks students to film and produce a video solving a math problem and demonstrating its real-world relevance.

The third, the most popular and best-known, is the Competition Series, a national middle school competition for aspiring mathletes. The Competition Series is the third and the oldest Mathcounts program, running every year since 1983. It’s open to all 6th, 7th, and 8th graders in America. Homeschoolers must complete an additional registration form confirming their grade level to participate.

The Competition Series has been held in person most years, with students competing in four levels across several months, with each level consisting of four rounds over two days. The Competition Series is one of the biggest and most prestigious middle school math competitions in America.

 

Are you ready for Mathcounts?

What are the topics covered, and are they appropriate for middle schoolers?

While Mathcounts problems will generally be more challenging than a typical middle school math homework assignment, they will cover the same topics. But, even though Mathcounts covers these standard concepts found in U.S. 6th, 7th, and 8th-grade curriculums, there are also occasional problems that touch on algebra, probability, approximation, geometry, and statistics–topics more commonly found in high school math programs.

Mathcounts problems are challenging yet also fun. They aim to test your problem-solving skills, not just your knowledge and memorization of math rules. Take, for example, this sample problem:

“Carla is mixing cherry, grape, and lime candies in a bowl. Since her favorite flavor is cherry, she wants 2/5 of the candies to be cherry. Since her least favorite flavor is lime, she wants 1/4 of the candies to be lime. What fraction of the candies will be grape? Express your answer as a common fraction.”

Can you get the answer? Give it a try before you keep reading!

The solution to this problem is 7/20. First, we label the fraction of grape candies as x. We then know that 1/4 + 2/5 + x = 1 since only three types of candy are in the bowl. Now, we will convert the non-variable numbers to different forms to easily add and subtract them. The expression will become 5/20 + 8/20 + x = 20/20. We can then simplify the expression, so it becomes 13/20 + x = 20/20, and then solve for x algebraically to get x = 7/20. So the fraction of the candies that will be grape is 7/20.

This question tests two simple middle school math skills: adding and subtracting fractions and working with unknown variables. However, by combining these two skills in an engaging word problem, this question becomes a challenge of understanding how to combine these two concepts in a real-world situation. 

 

Image of bridge of greenscape with superimposed words about Mathcounts on the bridge.

 

What is the Mathcounts Trainer Contest?

The Mathcounts Trainer Contest is another opportunity for those interested in competing to gain experience before the official competition. The Mathcounts Trainer Contest is a practice competition held online and runs concurrently with the actual competition. It begins August 15th, shortly before registration for the official Mathcounts competition opens, and ends April 5th, a month before the National Championships.

Students who sign up for the Trainer Contest will get a daily practice problem on the Trainer Contest website. Each day a competitor plays, they will gain points for every question answered correctly. When the competition closes, the student who has accrued the most points will be declared the winner over the seven months. 

Students work on these problems and compete in the Trainer Contest individually, building their skills to compete in the real Mathcounts competition. While the Trainer Contest does declare a winner, there are no prizes offered, and winning does not affect your standing in the actual contest. Trainer Contest participants have the advantage of developing their skills and are likely better prepared when they move forward into the official Mathcounts Competition Series. But more importantly, the Trainer problems are fun!

 

How the Mathcounts competition works: there are four rounds in each level, and four levels, from your local school to the National stage!

Each of the four official levels of competition–Chapter, Chapter Invitational, State, and National–typically has four rounds. The rounds are known as Sprint, Target, Team, and Countdown. (The Mathcounts organizers have made some temporary changes to the structure because of COVID, but the contest will likely revert to the old “four levels, four rounds” when in-person competition resumes.)

The rounds occur in the following order, and the questions get more complicated each new round:

 

Sprint Round

The Sprint Round is the first round in every level of competition. Competitors must focus on combining speed with accuracy. In the Sprint, a mathlete needs to try to complete 30 written problems in 40 minutes. You may not use a calculator, and you must work individually.

Sprint Round Basics

  • Type: Individual competition
  • # of Problems: 30 problems
  • Time: 40 minutes
  • Focus: Speed and accuracy
  • Calculator Rules: No calculator

Since the Sprint Round focuses more on the challenge of a time constraint than the challenge of the math problems, the questions offered here are generally the easiest of all four rounds. Here is a question from the 2020 State Competition Sprint Round:

A school wishes to use square tiles of artificial turf to cover an outdoor play area that measures 40 feet by 72 feet. Only whole tiles that are congruent squares of the largest possible size will be used. How many such square tiles are needed?

On a problem like this, it can really help to draw a picture to help visualize what the question is asking.

Answer: 45. We want to partition the 40 ft × 72 ft play area into squares of size x ft by x ft. To avoid gaps and overlap without cutting tiles, both 40 and 72 must be divisible by x; since we want the largest possible tiles, we need x to be the greatest common divisor of 40 and 72, which is 8. We will need 40/8 = 5 rows each with 72/8 = 9 tiles for a total tile count of 5 × 9 = 45 square tiles.

 

Target Round

The Target Round makes you utilize your problem-solving and mathematical logic abilities. It consists of four pairs of written problems. The two problems in each pair are not related to each other, but the difficulty is roughly equivalent. You will have six minutes to solve each pair, and the problems get more difficult throughout the round. You are still required to work as an individual, and you may use a calculator.

Target Round Basics

  • Type: Individual competition
  • # of Problems: 4 pairs (8 problems)
  • Time: 6 minutes per pair (24 minutes total)
  • Focus: Problem-solving and mathematical logic
  • Calculator Rules: Calculator allowed

Here is a question from the 2020 State Competition Target Round:

Both truth-tellers and liars populate a remote island. Truth-tellers always tell the truth, and liars always lie. To conduct a census of the island, Erin gathers all 72 of its inhabitants and asks each person how many truth-tellers and liars there are. She receives 61 responses saying “there are 32 truth-tellers and 40 liars” and 11 responses saying “there are 11 truth-tellers and 61 liars”. How many liars are there?

Answer: The truth-tellers will correctly state how many truth-tellers there are, whereas the liars will state a wrong count of truth-tellers. The group of truth-tellers will be that group whose answer regarding the number of truth-tellers matches the number of people in the group. The only such match is 11, so the group claiming 11 truth-tellers are truth-tellers, and they told the truth when they said there are 61 liars.

If you like these types of problems, IMACS has logic puzzle classes  you’ll love! And check out our logic puzzle article which has a lot of great practice problems.

 

Team Round

This round tests the strength of your collaboration with your Mathcounts team members. Working in teams, with calculators permitted, you must complete ten written problems in 20 minutes. One of the challenges of this round is determining how to divide the work–teams can solve the problems however they want. In addition, these questions are usually the most difficult of the competition; since you have more help to solve them, they are more challenging than the questions in the individual rounds. This round will determine the winning team, but it will have no effect on individual standings in the competition.

Team Round Basics

  • Type: Team competition
  • # of Problems: 10 problems
  • Time: 20 minutes
  • Focus: Difficult problems and team strategy
  • Calculator Rules: Calculator allowed

 

Countdown Round

The fourth round is the most intense and is designed to test your ability to perform with speed and accuracy under pressure. The top 10 competitors are selected from the first two rounds, and only they are allowed to compete in the Countdown Round. The Countdown Round is the only spoken round. It is named for its ladder-style elimination: the round begins with the 10th place competitor competing against the 9th place competitor. Both competitors have 45 seconds to solve the same problem, without calculators, and the first competitor to answer the question right gets the point. The competitor who gets the most correct out of the three questions is the winner. The winner then goes up against the competitor in 8th place, then that winner goes up against the competitor in 7th place.  If you want to read more details on how the Countdown Round works, the official Mathcounts rules and procedures are on their website.

Countdown Round Basics

  • Type: Individual competition
  • # of Problems: Variable
  • Time: 45 seconds per problem
  • Focus: Difficult problems, performance under pressure and verbalization
  • Calculator Rules: No calculator

 

A flat plain with a road that cuts through the center has the words "journey starts here: Mathcounts" superimposed over the landscape.

 

What is the timeline of the Mathcounts Competition?

The Mathcounts Competition Series takes place over several months, consisting of four levels of competition spread out by approximately four weeks each. At each level, competitors will either be eliminated or will advance to the next level. The problems increase in difficulty at each level, with perfect scores on the written portions being relatively common in the Chapter Competition and very rare at the National Competition. At every level, there is both a winning team and a winning individual. Here are the levels and what you can expect in each one:

Chapter Competition

The first level of competition is the Chapter Competition. Schools are allowed to register up to fifteen individuals, including the four students designated as the school team. School coaches will select the students who get to compete. Students who register but who do not attend an official Mathcounts school (like homeschoolers or students from smaller schools) are guaranteed one of the spots to participate in this round.

Chapter Invitational Competition 

The top 20% of competitors from the Chapter Competition advance to the next level: the Chapter Invitational Competition. At the Chapter Invitational Competition, the top five students from each chapter and the next ten highest-scoring students in the Sprint and Target Rounds will move forward to the State Competition. 

State Competition

Once again, the top competitors and the top team from each Chapter Invitational Competition will advance to the State Competition. After the State Competition, the top four individuals will be selected in each state, even if they come from different schools and different teams. In each state this group of four becomes the official state team. The coach of the winning team at the state level becomes the coach of this new team.

National Competition

The top four mathletes from each state will be the official team of that state. Every team–one from each state–gets an all-expense-paid trip to Washington D.C. and advances to the exciting National Competition: the Raytheon Technologies National Mathcounts Competition.

At the National Competition, a winning team and a winning individual are declared. The individual winner will receive the prize and is the official Mathcounts National Champion for that year. Do you have what it takes? It’s never too early to start growing the skills you’ll need to be the Mathcounts National Champion!

IMACS understands what it takes to be a Mathcounts National Champion. IMACS’ founder, Burt Kaufman, coached the winning Mathcounts national team in 1985, brought his team to nationals again in 1986, and mentored Brian Ewald, the 1986 National Champion. There are a lot of exciting IMACS success stories. Are you ready to create yours?

 

A winding road curves around a rocky mountain with the words, "Ready to Compete: Mathcounts" superimposed over the road.

 

What happens when you win? Mathcounts scholarships and prizes.

Mathcounts Competition Prize

Everyone on the National winning team and the top individual competitor at Nationals wins scholarship prizes. Every person on the winning team receives a $2,000 scholarship for college, and the individual champion receives the $20,000 Donald G. Weinert College Scholarship.

Mathcounts Alumni Scholarship

If you give Mathcounts everything you’ve got but didn’t end up winning the National Championship title, you are still eligible for a scholarship– you just have to wait a bit! Mathcounts offers anyone who competed at any level the opportunity to win the $3000 Alumni College Scholarship.

The Alumni Scholarship is open to U.S. high school seniors who participated in any level of Mathcounts during middle school and who plan to begin college immediately after their high school graduation. To apply, you must fill out an application that includes your academic and extracurricular information, as well as several written questions about your experience competing in Mathcounts. 

The Mathcounts organization generally chooses 4-5 winners, along with 10-20 finalists and semifinalists for the Alumni College Scholarship. Don’t miss the deadline: you can only apply during the fall of your senior year of high school!

 

Image of a snow-capped mountain with a road leading to the mountain. The words, "Let the race begin: Mathcounts" are superimposed over the road.

 

How to prepare to become a Mathcounts champion

Mathcounts Preparation Books

Books are always a great place to start to improve your math skills, whether you are preparing for competitions like Mathcounts or the AMC Math Competition, or even if you just want to have fun solving cool logic puzzles.

Here are some of our favorite books of practice problems, advice, and information about Mathcounts:

All-Time Greatest Mathcounts Problems – Patrick Vennebush

Mathcounts Tips for Beginners – Yongcheng Chen

The Three Year Mathcounts Marathon – Karen Ge

 

Mathcounts Preparation Courses

Because students are eligible for Mathcounts beginning in 6th grade, it is best to start preparing during elementary school (first to fifth grade in the U.S.) If you are in elementary school and want to develop the problem-solving skills a Mathcounts champion needs, IMACS is an excellent choice. If you want to learn more about IMACS and what it can offer you, try a free class.

 

Mathcounts Sample Questions and Answers

Along with the books and courses listed above, there are also other helpful practice resources available online. Here are some of our favorite websites that offer free Mathcounts practice problems:

The Mathcounts Problem Archive

Problems from Past Mathcounts Competitions

2021 Mathcounts Practice Handbook

 

Love math and want more challenges than you are getting in school?

IMACS graduates rave about the IMACS experience. Check out some success stories here!

 

Author

IMACS Staff

IMACS Live! is designed to give children a competitive edge by teaching them how to think critically using logic and reasoning. With over 25 years of experience, IMACS offers online after-school, weekend, homeschool, and summer classes for gifted and talented 1st-12th grade students in math enrichment, computer programming, logic puzzles, university-level mathematics, and university-level computer science. Highly experienced IMACS Live! instructors teach all courses via Zoom sessions. Secondary school students unable to attend using Zoom may be eligible for online self-paced courses through Elements of Mathematics: Foundations (EMF) or university-level courses through eIMACS.