Each year, students have access to a variety of math competitions. Beginning in elementary school, students can participate in UIL or Texas-based TMSCA, which focus on academic curriculum. In middle school, options expand to AMC 8 and MATHCOUNTS. High school students can take part in AMC 10/12, AIME, and Math Olympiad, among others. Additionally, private organizations such as Mathleague and Kangaroo Math offer further opportunities.

 

While there are many more competitions available, becoming familiar with these will make it easier to explore others. The links below provide past exam samples, which are an excellent starting point. I have also included recommended study materials under the Books section to help with preparation.

 

• TMSCA (Texas Math and Science Coaches Association):

• UIL (The University Interscholastic League)

• ​AMC 8 (American Mathematics Competitions 8) 

• AMC 10 A/B (American Mathematics Competitions 10 A/B)

• AMC 12 A/B (American Mathematics Competitions 12 A/B)

• MATHCOUNTS: (https://www.mathcounts.org/resources/past-competitions)

• AIME (American Invitational Mathematics Examination)

• USA(J)MO (United States of America Junior Mathematical Olympiad)

• USAMO (United States of America Mathematical Olympiad)

• ARML (American Regions Mathematics League)

• Mathleague:(https://mathleague.org/freetests.php)

• Kangaroo Math: (https://mathkangaroo.org/mks/practice/free-question-samples/)

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There are several more math competitions available both at national and international levels. Here is a link where you can explore more: https://artofproblemsolving.com/wiki/index.php/List_of_United_States_high_school_mathematics_competitions?srsltid=AfmBOoraY3hmO3q-vCRl3mw_LfydyjTKV1NvSMXDCICpp49nhb_YbO-8

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You can also find something specific to you state in the above link. Ex: Texas. 

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Here’s a breakdown of some math contests, including their formats and how to register. Once you get familiar with just a few, it becomes much easier to navigate and find information about other competitions.

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I believe TMSCA and UIL are among the best math and science competitions for schools and students to participate in. These contests focus on academic topics, making them both engaging and accessible for students. Both competitions are divided into three levels:
 

• Elementary

• Middle School

• High School – Other than the awards, students also receive scholarships at this level.
   

Each division includes questions from all grade levels within that category. For example, the high school competition covers math topics from 9th to 12th grade. However, awards are given at grade level, meaning a 3rd grader and a 5th grader may take the same test but are judged separately.

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Additionally, UIL offers competitions beyond math and science, including music, art, speech & debate, and computer science. Visit UIL website for more detailed information.

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Contest Format: TMSCA and UIL competitions include four main events:

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1. Number Sense (NS)– A fast-paced mental math test where students solve problems without calculators or scratch paper.

2. Calculator (Calc)– Tests students on their ability to use calculators efficiently for complex math problems.

3. General Math (GM)– Covers a wide range of math topics, including algebra, geometry, and word problems.

4. General Science (GS) – Focuses on biology, chemistry, physics, and earth science concepts.

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These contests challenge students' problem-solving skills and help them prepare for higher-level math and science competitions.

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Contest Date & Location: This information can be downloaded each year from the organization website.

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• TMSCA: https://www.tmsca.org/index.php?option=com_content&view=article&id=161&Itemid=580

• UIL: https://www.uiltexas.org/academics/invitational-meets

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Contest Types: Do not confuse a contest type with the contest format. The format remains the same no matter which type you pick. There is always a NS, Calc, GM or GS available no matter which of the following contest types you go to. Contest type is just a different levels of contest. In the links given above, you should be able to clearly determine which type of contest it is. 

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• Invitationals – These are practice rounds held multiple times throughout the year, allowing students to get familiar with the contest format and improve their skills. Awards are typically given to the top 6–10 students in each grade level. Students can participate in as many invitationals as they want, depending on which schools host them and how accessible they are.
   

• Qualifiers – Like invitationals, there are multiple qualifier rounds, and students can compete in as many as they choose. The key difference is that a strong score in a qualifier can earn a student a spot in the state championship.
   

• State – This is the highest level of these competitions. To qualify, a student only needs to achieve the required score in one qualifier round. While participating in multiple invitationals and qualifiers is encouraged for practice, only one qualifying score is needed to advance. In high school, the state competition carries even more significance, as top-scoring students may receive scholarship awards.

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How to Register Your School: Both UIL & TMSCA require approval from the school, so the first step is to talk to your math teacher or math club coach if your school has one. If your school isn’t registered due to resource or time constraints, consider volunteering to handle the process yourself.

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If you're a new parent or student volunteer, this might seem overwhelming at first, but I’ve been through it with the help of my parents, and I can assure you—it’s straightforward!

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Here’s how to get started:

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1. Fill out the registration forms at the following links:
    

   • TMSCA Registration

   • UIL Registration
      

2. Email the contact person listed on the website with your completed forms—they’ll guide you through the next steps.

3. There is a small registration fee (~$50), which can be shared among all interested students.

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Once registered, your school can start competing in these exciting contests!

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How to Register for an Event (Invitationals, Qualifiers, etc.): Once your school is registered, the coach or parent volunteer will receive a test registration link and login details. Using these credentials, they can log in and register students for individual events.

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Test Fees: Each test typically comes with a small fee. For example, TMSCA charges around $7–$8 per test (NS, Calc, GM, GS). This fee is usually collected from student parents and paid at the contest location during the morning check-in process.

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AMC: The Path to Math Olympiads: The AMC series serves as a stepping stone toward math Olympiads, with AMC 8 being the starting point for middle school students. Since Olympiads are extremely challenging, AMC 8 helps build a strong foundation in problem-solving and mathematical thinking.

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While AMC 8 is designed for middle schoolers, elementary students can take it as soon as they feel ready—starting in 3rd grade is ideal. This gives students plenty of time to develop their foundational skills before moving on to higher-level contests.

 

Similarly, while competing in AMC 8 during middle school, students can also attempt AMC 10 A/B to get an early start. This might seem ambitious, but for those aiming to qualify for AIME and eventually reach Math Olympiads, this is the best approach. By starting early, students can identify their weak areas and take advantage of the many resources available to improve over time.

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Different Versions ( 8 / 10 A-B / 12 A-B): As the name suggests, AMC 8 is designed for students up to 8th grade, but students can start at any age when they feel ready and continue taking it until they reach 8th grade. The same rule applies to AMC 10 and AMC 12, where students can participate until they reach 10th or 12th grade, respectively.

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For AMC 10 and AMC 12, there are two versions: A & B, which serve as two opportunities to qualify for the next level, AIME (American Invitational Mathematics Examination). Both versions have the same difficulty level and are scheduled one week apart, typically in November each year. Students can choose to take both versions or just one if they feel confident about qualifying for AIME. ​Each year, cutoff scores for qualification vary, based on test difficulty and overall student performance nationwide.​CUT OFF SCORES.

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Contest Format: Each version of AMC has a slightly different format. The breakdown is give below:

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Contest Date & Location:

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• AMC 10/12 are usually held in Nov with A & B versions scheduled a week apart. 

• AMC 8 is held in January of every school year.

• Location is usually a local school, math center or a university.

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Contest Type:

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• AMC 8: Just a single test. There is no advancement path after the test as this helps students prepare for AMC 10.Students are recognized by the organization based on their scores. Higher scores can get students 'Honors' or 'Honors with distinction' awards.

• AMC 10/12: 2 opportunities (A & B) to get a high score to qualify for AIME, based on a cut-off score set each year.

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How to Register?

Individuals cannot register directly, but AMC contest registration is quite flexible. Typically, a middle/High school math club or teacher can assist with registration. If you are an elementary school student, you can reach out to your zoned middle school for assistance. If that is not an option, private math learning centers, local universities, or colleges can also help with registration. Since AMCs are widely recognized as the pathway to Math Olympiads, most organizations are familiar with the process and should be able to register you at no additional cost beyond the test fee. A regular google search can find test center locations near you or you can look at the following link: https://maa.org/amcreg/ 

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​​The AIME (American Invitational Mathematics Examination) is an intermediate-level math competition in the Mathematical Olympiad pathway. It is an invitation-only contest for students who score high enough on the AMC 10 or AMC 12.

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There are 2 tests available on AIME — AIME I and AIME II. Both versions have different sets of questions, but their structure and difficulty level are similar. A student may take either AIME I or AIME II, but not both. Schools typically choose which version to administer.

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Contest Format

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• Number of Questions: 15

• Time Limit: 3 hours

• Format: Open-ended numerical answers (not multiple choice)

• Scoring:

  • +10 points for each correct answer

  • No penalty for incorrect or unanswered questions

  • Maximum score: 150

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The AIME is significantly more challenging than AMC 10/12 and is designed to test creative problem-solving skills. High scorers may advance to USAJMO (for AMC 10 qualifiers) or USAMO (for AMC 12 qualifiers).

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Contest Dates & Location 
 

• AIME I & II are usually held in Febuary with I & II versions scheduled a week apart. 

• Location is usually a local school, math center or a university.

• Students can take either AIME I OR AIME II, but not both.

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Topics Covered in AIME

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AIME covers advanced problem-solving topics that extend beyond AMC 10/12. The questions require creative approaches and deep mathematical thinking. Topics include: Algebra, Geometry, Number Theory, Combinatorics & Probability.  Calculus is not required for AIME, but problems often require creative problem-solving techniques such as substitution, symmetry, and clever casework and having an understanding of calculus can help.

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How to Register?

AIME is an invitation only competition which means that students who have qualifying scores in AMC 10/12 will be invited to participate. The school will generally help the student to register for the same.

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The USAJMO (USA Junior Mathematical Olympiad) and USAMO (USA Mathematical Olympiad) are proof-based math competitions for top-performing students in the AIME. These exams serve as the final round before selection for the Mathematical Olympiad Program (MOP) and the USA International Math Olympiad (IMO) team.

 

Contest Format

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• Days: 2 days

• Number of Problems: 6 (3 per day)

• Time Limit: 4.5 hours per day (total of 9 hours)

• Format:

  • Proof-based questions (students must write full solutions, not just final answers)

  • Problems require deep reasoning and clear mathematical proofs

• Scoring:

  • Each problem is worth 7 points, making the maximum score 42

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Contest Dates & Location:

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• Both USAJMO & USAMO are held in March.

• The test is held over two days, with each day containing 3 problems.

• Location is usually a local school, math center or a university.

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Topics Covered: USAJMO and USAMO problems come from a broad range of mathematical fields, including:

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1. Algebra

• Polynomial identities

• Functional equations

• Inequalities (AM-GM, Cauchy-Schwarz, etc.)

• Sequences and series

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2. Geometry

• Euclidean geometry (triangles, circles, and transformations)

• Coordinate and vector geometry

• Advanced constructions

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3. Number Theory

• Modular arithmetic and congruences

• Diophantine equations

• Prime numbers and divisibility

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4. Combinatorics

• Graph theory

• Counting arguments and bijections

• Recursion and generating functions

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Key Difference Between USAJMO & USAMO

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• USAJMO: Intended for AMC 10 qualifiers (easier problems, but still proof-based).

• USAMO: Intended for AMC 12 qualifiers (higher difficulty level).

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Students who excel in these contests may be invited to MOP (Mathematical Olympiad Program) for further training and possible selection for the USA IMO team.

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How to Register?

USA(J)MO & USAMO is an invitation only competition which means that students who have qualifying scores in AIME will be invited to participate. The school will generally help the student to register for the same.

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MATHCOUNTS is a nationwide middle school math competition in the U.S. Unlike AMC’s, this competition is only for Middle school students from 6~8th grade.  Students must go through multiple levels of competition to reach the top level – School, Chapter, State and National.

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Contest Format:

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The format of the test remains the same no matter the level (School or Chapter etc.,). The contest has four rounds:

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1. Sprint Round:

   • 30 questions, 40 minutes

   • No calculators

   • Focuses on speed and accuracy

2. Target Round:

   • 8 questions (given in pairs), 24 minutes

   • Calculators allowed

   • Requires deep problem-solving skills

3. Team Round:

   • 10 questions, 20 minutes

   • Teams of four work together

   • Calculators allowed

4. Countdown Round: (Optional at chapter/state, official at nationals)

   • Head-to-head fast-paced competition

   • No calculators

   • Tests speed and mental math skills

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Contest Levels:

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School Competition (October – January) : Students participate in the school-level competition to qualify for the next round. The math coach determines the test date, usually between October and January. Selection for the Chapter Round depends on the chapter’s policy.

• Official School Team: A group of 4 students is selected to compete as a team.

• Individual Competitors: At least 10 students (or more, depending on chapter rules) are chosen to compete individually.

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Chapter Competition (February): Each state is divided into multiple region-based chapters, where schools compete against one another.

• All students participate in every competition round (Sprint, Target, Team, etc.).

• Individual Rankings: Every participant is ranked based on their scores.

• State Qualifiers: The top 12 individuals advance to the State Round.

• Countdown Round: The top 8 individuals compete in a fast-paced head-to-head contest to determine the chapter champion.

• Team Advancement: Although all students complete the Team Round, only the official school teams are considered for team rankings. Depending on the chapter size, the top 3–7 teams qualify for the State Round.

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State Competition (March) : Winners from the Chapter Round compete at the state level to earn a spot in the National Round.

• Team and Individual Rankings:

  • The top teams and individuals from across the state compete.

  • Awards are given to the top 10 individuals and top 4 teams.

• National Qualifiers: Only the highest-ranked individuals advance to the National Competition.

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National Competition (May)

The top 4 students from each state qualify for Nationals, held in Washington, D.C.

• These students represent their state and compete for national titles.

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How to Register?

MATHCOUNTS is handled by the school's math club or a Volunteer at school. The students are selected based on their performance in the school round. Many competitive schools hold multiple tests to finalize on the Team of 4 and the individuals that will go on to represent the school at the chapter, state etc., levels. Speaking to your math teacher will help in finidng more details on how your school picks the students.

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Mathleague.org is an organization that runs mathematics competitions for students at the elementary, middle, and high school levels across the United States and internationally. These contests prepare students for future competitions such as AMC 8/10/12, AIME, and MathCounts, building problem-solving skills and mathematical intuition.

 

I highly recommend this competition for elementary school students who are interested in higher levels of math competitions in later years. You do not need to be a part of any math club or selected through any test process to qualify for this competition. However, registration of school is required. Following are the details:

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School Registration

Registering the school is quite simple and free. You math club teacher or student volunteer can help fill up a simple online form and submit to the Mathleague team. https://mathleague.org/membership.php.

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 The team at Mathleague is very responsive and they will respond to your email within a day. Email for any queries at: customercare@mathleague.org

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Student Registration:

If the school is registered the schools name should appear in the list of student registration form. Simply fill in the student registration for the contest to participate. The list of all events and location is released ahead of time, so keep checking the 'EVENTS' link on their website to find a contest taking place closer to you and register to participate.

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NOTE: You can register just as a single student using the events links given below. You do not need a whole team, if unavailable. Start of your journey and others will join in!

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Contest Dates & Locations:

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• Elementary: https://mathleague.org/esevents.php 

• Middle: https://mathleague.org/msevents.php 

• High: https://mathleague.org/hsevents.php

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Contest Format:

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• Elementary School Contests (Grades 3-6)

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1. Number Sense (10 minutes, 80 questions)

   • A mental math test where students solve problems without calculators or scratch paper.

   • No erasures or changes allowed once an answer is written.

2. Sprint Round (40 minutes, 30 questions)

   • Focuses on speed and accuracy without a calculator.

   • Questions increase in difficulty, covering various mathematical topics.

3. Target Round (24 minutes, 8 questions in pairs of 2)

   • Each pair of questions is given 6 minutes to complete.

   • Calculators are allowed.

   • Emphasizes problem-solving and critical thinking.

4. Team Round (20 minutes, 10 questions)

   • Teams of up to four students collaborate.

   • Calculators are allowed.

   • Tests communication and teamwork skills.

5. Countdown Round (Unofficial at some contests)

   • A head-to-head speed round for the top individuals based on Sprint and Target scores to declare a champion.

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Advancement:

• Top-performing students qualify for the State Championship, and from there, top scorers advance to the National Championship.

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• Middle School Contests (Grades 6-8)

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1. Sprint Round (40 minutes, 30 questions)

   • No calculators.

   • Tests speed and accuracy across a broad range of math topics.

2. Target Round (24 minutes, 8 questions in sets of 2)

   • Each set has a 6-minute time limit.

   • Calculators are permitted.

   • Encourages deep thinking and multi-step problem solving.

3. Team Round (20 minutes, 10 questions)

   • Teams of up to four students.

   • Calculators allowed.

   • Rewards collaboration and shared strategies.

4. Countdown Round (Optional at most contests, official at Nationals)

   • Fast-paced, head-to-head buzzer competition.

   • Used at higher levels to determine final rankings.

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Advancement:

• Students advance from local to state competitions.

• Top performers at state level qualify for the National Championship.

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• High School Contests (Grades 9-12)

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1. Sprint Round (60 minutes, 30 questions)

   • A longer format than middle school.

   • No calculators allowed.

2. Target Round (24 minutes, 8 questions in pairs)

   • Each pair has 6 minutes.

   • Calculators are allowed.

3. Team Round (20 minutes, 10 questions)

   • Four-member teams work together.

   • Calculators are permitted.

4. Power Round (Optional, 60 minutes)

   • A proof-based section requiring deeper mathematical reasoning.

   • Students must justify their solutions in written format.

5. Relay Round (Optional at some contests)

   • Similar to the middle school relay format.

6. Countdown Round (Official at National Level)

   • A head-to-head buzzer competition among top-scoring students.

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Advancement:

• Top students from local meets qualify for State Championships.

• The highest scorers at state level advance to National Championships.​