AMC 8 Problems (Mathcounts Problems / Math Olympiad Problems) and Solutions

Tyler York
AMC 8 Problems

Given the growing interest in elite math competitions over the last couple of years — possibly due to the US Team’s surprising four wins since 2015 at the International Math Olympiad — it’s a good time to discuss the differences between the most popular math competitions in the country for elementary and middle schoolers (let’s start them young!). As well as to highlight some example AMC 8 problems and solutions for those who want to practice or are interested in getting into the competitions.

What elementary and middle-school math competitions are available?

The many available competitions include MOEMS, Mathcounts, and the AMC series, including the AMC 8. Of these, the AMC series is the oldest and the most highly respected. 

The 2019 Math Olympiad winners. Photo credit:

What is the Math Olympiad?

The Mathematical Olympiads is a worldwide math competition for elementary and middle school students, organized by the Mathematical Olympiads for Elementary and Middle School (MOEMS). The objective of the competition is to stimulate enthusiasm, foster creativity, and strengthen the student’s intuition in mathematical problem-solving. 

This annual competition is held from November to March, where a test is given every month to a team of up to 35 students. Each test lasts 30 minutes and is administered via the participating school’s preferred method, on paper or online. The tests are then sent to MOEMS for evaluation and scoring.

About two dozen nations participate in the Math Olympiad, which is separated into two divisions: Elementary and Middle School. Math Olympiad problems in the elementary level come from 4th to 6th-grade math concepts, and Middle School level questions are from grades 7th and 8th-grade math concepts. If they wish to, 4th to 6th graders may participate in the Middle School level.

A thing to note here, and to avoid confusion, is that this Math Olympiad is different from the USA Mathematical Olympiad, which is for high school students who have qualified through their AMC 10, AMC 12, or USAMTS scores.

AMC 8 problems
Photo by Jeswin Thomas on Unsplash

What is Mathcounts?

Mathcounts, meanwhile, is a nationwide middle school mathematics competition held in different venues in the United States. Its sponsors include the National Society of Professional Engineers, the CNA Foundation, and the National Council of Teachers of Mathematics. Just like the Olympiads, Mathcounts’ objective is to build problem-solving skills through multiple levels of fun and engaging, “bee” style contests.

Unlike the Math Olympiad, Mathcounts is taken in-person and is considered to be much more difficult. Mathcounts problems are taken from subjects that include algebra, geometry, number theory, counting, and probability.

The competition is classified into four stages: school, chapter, state, and national. In each stage, the questions increase in difficulty. In fact, one can expect to see perfect scores at the school level, while at the national level, a perfect score is very rare.

The top individuals and teams from the chapter levels advance to the state competitions. At the state level of Mathcounts, only the top 4 individuals will compete at the Nationals. The trainer or coach of the team in first place at the state level becomes that Mathcounts trainer for the National competition. At the Nationals, all the participants compete individually and go through the Sprint and Target Rounds. The top 4 individuals from each state will compete in the Team Round to represent their state.

Photo by Anoushka P on Unsplash

What is the AMC 8?

The AMC 8 is part of the American Mathematics Competitions (AMC) series, which includes competitions at both the middle- and high-school levels. It is administered by the Mathematical Association of America. The AMC 8 is the hardest of the three math competitions on this list. Many consider Math Olympiad and Mathcounts to be good preparation for the AMC 8. 

This 40-minute competition is for middle schoolers in the 8th grade or below who are under 14.5 years old on competition day. It is held at the school’s choice of online or in person at the host school’s facilities, and is proctored by a teacher or school administrator. The test itself consists of 25 multiple-choice questions on topics that include elementary geometry, counting and probability, spatial visualization, proportional reasoning, and interpreting graphs and tables. Expect questions about coordinate geometry, linear or quadratic functions and equations, and basic algebra as well.

It is also considered an excellent test of general problem-solving skills, as it is structured in a way that permits multiple (creative) solution paths for most questions.

Students who excel in the AMC 8 may also choose to participate in the more advanced  AMC 10. Doing so will give them the opportunity to compete in the more prestigious invitational competitions such as the American Invitational Mathematics Examination (AIME), and the United States of America Mathematical Olympiad (USAMO), or the United States of America Junior Mathematical Olympiad (USAJMO). These, in turn, are the basis for the selection of the US’s team at the International Mathematical Olympiad (IMO).

AMC 8 Math Problem Samples

Curious to see if you can take on the challenge of the AMC 8? Below are five sample AMC 8 problems that you can try to solve. The complete answers can also be found below. Don’t peek!

Sample problems can be found here: PDF Link

The AMC-8 is the introduction to this world, but it’s meant to be elite math. Don’t be surprised if you don’t get them all right on your first try. The AMC series of exams will likely be the most difficult you’ve faced to date, but that’s also part of the fun.

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