US Math Competitions
# Exploring the World of Math Competitions in the United States

This document introduces each Math Competition conducted in the United States. It describes the purpose of each test, the various levels, and the question types.

Please refer to the Competition Schedules document for the 2024 competition schedules.

Contrary to its name, the Kangaroo Math Contest (mathkangaroo.org) was not born in Australia. It was created in France by André Deledicq, a professor at the University of Paris, and his colleagues Jean-Pierre Boudine and Claude Deschamps, a French mathematician 1991. The contest was initially intended for French students but later became an international event. The Kangaroo Math Contest is usually held every year, and more and more schools and educational institutions in the United States have joined the event, increasing its popularity.

The contest has demonstrated that children enjoy solving fun and exciting problems within their reach. This bridges the gap between the standard and often dull examples and difficulties in school textbooks and the challenging and demanding problems of more advanced Math Olympiads.

The contest's primary purpose is to engage as many children as possible in solving mathematical problems and show each student that problem-solving can be lively, exciting, and entertaining. This purpose has been achieved successfully: in 2023, over 6 million children from more than 70 countries participated in the contest.

All students from grades 1-12 of general educational institutions who have paid the registration fee can participate in the contest without any prior selection. (Like in other countries, the contest is not free, but the cost has been meager in recent years - about $21. This money covers the expenses of the competition and rewards many participants with small but varied prizes).

The distribution of levels in the Kangaroo Competition is as follows:

The contest occurs in all schools on the same day, the third Thursday in March. But during the pandemic years, some schools had the test on a different day. Please check the Math Kangaroo website for the exact schedule.

When choosing problems, two principles are followed: firstly, solving problems should be fun, and secondly, "Kangaroo" is, although not very hard, still a contest, so the most skilled and prepared should win.

All problems in the Olympiad are split into three categories, with ten issues in each (for the younger ones, the last, most challenging section has only six issues). The first section has easy, often funny problems, each worth 3 points. These problems are chosen so contestants can solve at least one and have fun. They are within the reach of anyone who reads the condition carefully and does not need special training. But even in them, there are surprising questions and even cunning "traps," so it cannot be said that they are accessible to the contestants. The problems worth 4 points are meant for high school students and "good students" to solve independently. These problems are much more complex than the 3-point ones and, usually, are closer to the school curriculum. The last section has complex, non-standard problems, each worth 5 points. They are made so that even the most prepared students have something to think about. One must show cleverness, the ability to reason independently, and observation to solve them.

"Kangaroo" is a huge contest, and the rewards based on its results are enormous. Accordingly, the main form of rewards are diplomas and small prizes with the symbols of the contest, but these prizes go to almost every school that participates in the Olympiad.

The Noetic Learning Math Contest (NLMC) is a semiannual problem-solving competition for elementary and middle school students. It aims to enhance students' mathematical skills and foster their interest in math by presenting them with engaging problems to solve independently within a given time frame.

During the contest, participants are challenged to solve 20 creative problems without a calculator, with 45 minutes for the paper-pencil format and 50 minutes for the online version.

For the 2023-2024 school year, they will continue offering both paper-pencil and online contest versions.

Their Approach:

- Grade-level Tailored Problems: Unique problems are designed for each grade level, ensuring that students face challenges appropriate to their skill level.

- Broad Recognition: Comprehensive recognition is provided to participants, with the top 10% receiving National Honor Roll medals, the top 50% receiving Honorable Mention ribbons, and each team's top scorer earning a Team Winner medal.

Recognition and Awards:

- Team Winner Medal: Awarded to the highest scorer of each team.

- National Honor Roll Medal: Given to the top 10% of all participants nationwide.

- National Honorable Mention Ribbon: Awarded to the top 50% of all participants nationwide.

- Team Achievement Plaque: Presented to the top 10% of teams (excluding after-school institution teams).

The Mathematical Olympiads for Elementary and Middle Schools (moems.org) is a math competition program for grades 4 through 8 students. It is designed to stimulate enthusiasm and a love for mathematics, encourage mathematical creativity, and develop problem-solving skills.

Here are some key points about the MOEMS contest:

- Format: MOEMS consists of five monthly contests from November to March. Each contest consists of five non-routine problems.
- Problem Types: The problems presented in MOEMS are designed to be challenging and require creative problem-solving. They cover various mathematical topics and may involve logical reasoning and critical thinking.
- Scoring: Students earn points for correct answers, and their scores contribute to both individual and team rankings. Recognition is given at the local, regional, and national levels.
- Division E and Division M: MOEMS has two divisions for elementary students (grades 4 and 5) and middle school students (grades 6, 7, and 8).
- Math Club Participation: Many schools participate in MOEMS through math clubs, where students work collaboratively to solve problems. Math clubs can enhance students' problem-solving skills and foster a sense of camaraderie.
- Certificates and Awards: Participants receive certificates based on their performance, and high achievers may be eligible for additional awards and recognition.
- Purpose: MOEMS aims to allow students to solve problems creatively beyond the regular classroom curriculum. It helps nurture a positive attitude toward mathematics and fosters a sense of achievement.

Overall, MOEMS is a popular math contest that encourages students to explore the beauty and depth of mathematics while honing their problem-solving abilities.

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Mathcounts (mathcounts.org) is a prestigious mathematics competition program in the United States aimed at middle school students. Here are key points about the Mathcounts contest:

- Participants: Mathcounts is designed for students in grades 6-8, typically covering ages 11-14. It is a middle school-focused competition.
- Format: The Mathcounts competition consists of several rounds, including the School, Chapter, State, and National rounds. The contest format involves both individual and team competitions.
- Individual Round: Participants solve math problems individually, testing their problem-solving skills and mathematical knowledge.
- Team Round: Teams of up to four students collaborate to solve more complex problems. This round encourages teamwork and effective communication.
- Countdown Round: The top-scoring students from the written rounds engage in a "Countdown Round," a fast-paced oral competition where questions are presented to individuals who must respond quickly.
- Sprint and Target Rounds: These written rounds challenge participants with various mathematical problems. The Sprint Round consists of 30 rapid-fire questions, while the Target Round features more in-depth issues solved in pairs.
- Competition Levels: Mathcounts start at the school level, progressing to Chapter, State, and finally, the National Competition, where the most successful students from each state compete.
- Math Club Involvement: Many schools have Mathcounts clubs or teams that prepare for the competition throughout the academic year. Dedicated coaches often guide students in honing their problem-solving skills.
- National Math Club: Besides the competition, Mathcounts provides resources for a National Math Club, encouraging students to engage in mathematical activities and challenges beyond the official contests.
- Purpose: Mathcounts promotes math excellence, teamwork, and enthusiasm among middle school students. It provides an opportunity for talented students to showcase their mathematical abilities.

Overall, Mathcounts is a well-regarded math competition program that has contributed to the development of countless students interested in mathematics across the United States.

The USA Mathematical Talent Search (USAMTS) is a yearly, free, three-round math competition nationwide for middle and high school students. Unlike other contests, it emphasizes problem-solving over speed, giving participants at least a month to solve five problems for each round. Justifications are required for all but the first problem, which is a puzzle. The USAMTS aims to develop problem-solving and writing skills while providing a pathway to the International Mathematical Olympiad. Students who score 68 or above (out of 75) on the USAMTS are eligible to participate in the AIME, the second stage in the selection process for the team representing the USA at the IMO.

AMC (maa.org) - American Mathematics Contest - is an examination series that builds problem-solving skills and mathematical knowledge in middle and high school students. The AMC is organized by the Mathematical Association of America (MAA).

The AMC series consists of several different exams based on grade level:

- AMC 8: For students in grade 8 and below. It's a 25-question, 40-minute multiple-choice examination.
- AMC 10: For students in grade 10 and below. It's a 25-question, 75-minute multiple-choice examination.
- AMC 12: For students in grade 12 and below. Like the AMC 10, it's a 25-question, 75-minute multiple-choice examination.

The primary purpose of these competitions is to awaken interest in mathematics and develop talent. It includes all the significant topics from the curriculum: Algebra, Number theory, Geometry, and Combinatorics. Participation in the AMC can lead to more opportunities, such as being invited to the American Invitational Mathematics Examination (AIME) and the USA Mathematical Olympiad (USAMO), depending on the scores achieved. Moreover, it helps develop competitiveness and test-taking skills, allowing students to learn throughout their careers.

The American Invitational Mathematics Examination (maa.org), often AIME, is an advanced mathematics competition primarily for high school students. It represents a critical step in the progression of mathematically talented students in the United States, bridging the gap between the American Mathematics Contest (AMC) and the most elite level of mathematical challenges, such as the USA Mathematical Olympiad and USA Junior Mathematical Olympiad(USAMO/USAJMO).

AIME is a distinctive contest, primarily because it targets students with top performance in either AMC 10 or AMC 12. The examination format is also different: It consists of 15 questions, with each answer being an integer in the range of 0 to 999. Candidates are given 3 hours to tackle these problems, reflecting the complexity and depth of the mathematical thinking required.

What truly sets AIME apart is its focus on original and innovative problem-solving skills and applying sophisticated mathematical concepts. The problems are designed to encourage out-of-the-box thinking and often require a synthesis of multiple areas of mathematics. This approach tests and nurtures a deep and comprehensive understanding of mathematical principles.

Excelling in this examination can lead to opportunities such as an invitation to the prestigious USAMO/USAJMO, and it is often considered a marker of high potential and achievement in mathematics.

The United States of America Mathematical Olympiad (maa.org), commonly known as the USAMO/USAJMO, is a highly prestigious mathematics competition in the United States, primarily aimed at high school students. This Olympiad marks the pinnacle of high school mathematics competitions in the U.S. and serves as a gateway to international mathematical arenas, such as the International Mathematical Olympiad (IMO).

The USAMO/USAJMO is an invitation-only competition for those who achieve outstanding scores in the American Mathematics Competitions (AMC) and the American Invitational Mathematics Examination (AIME). This rigorous selection process ensures that only the most talented and skilled young mathematicians participate in the Olympiad. Those who perform exceptionally well in the AMC12 are invited to the USAMO, while top scorers in the AMC10 are invited to the USAJMO.

The format of the USAMO/USAJMO is uniquely challenging: it consists of a two-day examination comprising three extensive, proof-based mathematical problems. These problems require mathematical knowledge and deep creativity, ingenuity, and advanced problem-solving skills. Each day's exam lasts four and a half hours, highlighting the complexity and depth of the problems posed.

Success in the USAMO/USAJMO is a significant achievement, often regarded as an indicator of exceptional mathematical talent and potential. High performers in the USAMO/USAJMO are usually candidates for the U.S. team in the International Mathematical Olympiad, where they represent the country on a global stage.

The highest achievers in the USAMO and USAJMO are offered invitations to join the Mathematical Olympiad Program (MOP) during the summer immediately following the competitions. Those participating in MOP become eligible for selection to the six-member team representing the United States of America at the International Mathematical Olympiad the following summer.

The International Mathematical Olympiad (IMO-official.org), commonly called the IMO, is the world's most prestigious mathematics competition for high school students. Held annually in a different country, it represents the pinnacle of mathematical problem-solving at the pre-university level. It brings together the most brilliant young minds from around the globe.

The IMO is an invitation-only competition, with each participating country selecting and training a team of up to six students through national competitions and rigorous preparation. This ensures that only the most exceptional young mathematicians, capable of representing their country on the international stage, participate.

The format of IMO is practically the same as USAMO, but the problems themselves differ, of course, in complexity. Each participant has a chance to receive a medal according to their performance:

- Core Thresholds: Each participant's score is calculated out of the maximum possible total, which is determined by the points available from all the problems in the competition. The maximum score is usually 42 points (each of the six issues is worth 7 points).
- Medal Allocation: The allocation of gold, silver, and bronze medals is based on the distribution of scores among all participants. The basic guideline is:

- Gold Medals: Awarded to approximately the top 1/12 of the contestants.
- Silver Medals: These are given to approximately the next 1/6 of the contestants.
- Bronze Medals: Allocated to approximately the next 1/4 of the contestants.

4. Honorable Mentions: Participants who do not win a medal but solve at least one problem completely (earning 7 points for a problem) typically receive an honorable mention.

5. Team Performance vs. Individual Awards: It's important to note that medals are awarded based on individual performance, not team performance. Each country's team comprises up to six students, but their scores are considered independently when determining medal eligibility.

Participation in the IMO is more than a competition; it's an unparalleled opportunity for young mathematicians to engage with complex mathematical challenges, showcase their talents on an international stage, and be part of a global community that shares a deep passion for mathematics. The experience of competing in the IMO often has a profound and lasting impact on the participants, fostering a lifelong love for mathematics and problem-solving.

The William Lowell Putnam Mathematical Competition® is the top math contest for undergraduates in the U.S. and Canada.(WLPMC) It happens yearly on the first Saturday of December, and it features two 3-hour sessions, each with six challenging math problems for individual participants to solve.

Past year papers for this competition can be found here.