AMC 8 vs AMC 10 vs AMC 12

The American Mathematics Competition (AMC) is a distinguished, exam-based math tournament for students in grades K–12, organized into the AMC 8, AMC 10, and AMC 12 for different grade levels. In short: the AMC 8 is a 40-minute, 25-question middle-school contest; the AMC 10 and AMC 12 are 75-minute, 25-question high-school contests that serve as the qualifying round for the American Invitational Mathematics Examination (AIME) and, eventually, the USA and International Mathematical Olympiads.

Each competition presents its own challenges and opportunities for students with strong quantitative skills, enabling them to advance to higher-level national contests and strengthen their applications to selective high schools and colleges.

AMC 8

The AMC 8 is a 25-question math competition for students in 8th grade and below, with a 40-minute time limit. The exam covers material commonly taught in middle school mathematics, including counting and probability, estimation, proportional reasoning, geometry (including the Pythagorean Theorem), spatial reasoning, real-world applications, and interpreting graphs and tables. Some of the more advanced problems may also introduce concepts from introductory algebra, including linear and quadratic equations, coordinate geometry, and related topics.

Each correct answer is worth one point, and there is no penalty for guessing.

Why take the AMC 8:

• Build familiarity with advanced mathematics competitions, preparing you for further exams down the road
• Explore mathematical concepts in-depth, honing your skills before entering high school and higher-level classes
• Make yourself a more competitive applicant for selective high school programs, especially those that emphasize quantitative ability

AMC 8 problems

AMC 8 problems are structured in the following ways:

• Multiple-choice questions with five answer choices (only one is correct)
• Easier questions are at the start of the test and become progressively more difficult
• No calculators are allowed on the exam
• About 1.5 minutes are allowed per problem

AMC 8 problems are usually less straightforward than questions you may have seen on a math test in school. The test requires students to think logically and use multiple steps to solve problems.

Here’s an example problem from a past AMC 8 exam that asks students to visualize shapes that can be made from a piece of wire:

Haruki has a piece of wire that is 24 centimeters long. He wants to bend it to form each of the following shapes, one at a time:

• A regular hexagon with side length 5 cm.
• A square with area 36 cm².
• A right triangle whose legs are 6 and 8 cm long.

Which of the shapes can Haruki make?

A. Triangle only

B. Hexagon and square only

C. Hexagon and triangle only

D. Square and triangle only

E. Hexagon, triangle, and square

To find the solution, a student can calculate the perimeter of each shape to see if it equals 24 cm:

• A regular hexagon with a side length of 5 cm has a perimeter of 30 cm (too long to make).
• A square with area 36 cm² has a side length of 6 cm (since 6 × 6 = 36), giving a perimeter of 24 cm: this works.
• A right triangle with legs 6 and 8 cm has a hypotenuse of 10 cm (by the Pythagorean theorem), for a total perimeter of 24 cm: this also works.
• Answer D, “Square and triangle only,” is correct.

Remember: On the AMC exams, there may be multiple ways to solve the same problem. For example, a student could use the process of elimination to remove all options that include “Hexagon” and exclude “Square,” which means they don’t need to calculate the missing side of the triangle.

Time-saving strategies can make a huge difference on the AMC.

AMC 10

The AMC 10 is a 25-question mathematics competition intended for students in 10th grade or below, with a 75-minute time limit. The extra time (compared to the AMC 8) accounts for the competition’s more challenging content. Its syllabus includes elementary algebra, basic geometry, area and volume formulas, introductory number theory, and elementary probability. Unlike the AMC 12, it does not cover more advanced subjects such as trigonometry, higher-level algebra, or advanced geometry. Overall, the material closely aligns with concepts typically taught in middle school and early high school math courses.

Students who place in approximately the top 2.5% on the AMC 10 qualify for the American Invitational Mathematics Examination (AIME). Strong performance on the AIME can lead to qualification for the U.S. Mathematical Olympiad and, ultimately, international mathematical competitions.

The AMC 10 rewards leaving blanks over guessing incorrectly: each correct answer earns 6 points, each blank earns 1.5 points, and each incorrect answer earns 0 points, for a maximum possible score of 150.

Why take the AMC 10?

• Complete the first step to qualifying for the U.S. Mathematical Olympiad (USAMO) team and competing in the International Mathematics Olympiad (IMO)
• Qualify for the AIME with a high enough score
• Strengthen your college application, especially for students entering STEM programs

AMC 10 problems

AMC 10 problems are structured in the following ways:

• Multiple-choice questions with five answer choices (only one is correct)
• No proofs or written portions are included
• Easier questions are at the start of the test and become progressively more difficult
• No calculators are allowed on the exam
• Three minutes are allowed per problem on average

Like the AMC 8, the AMC 10 features problems that ask students to think “outside of the box” while solving.

Here’s an example AMC 10-style problem:

A box contains 10 pounds of a nut mix that is 50 percent peanuts, 20 percent cashews, and 30 percent almonds. A second nut mix contains 20 percent peanuts, 40 percent cashews, and 40 percent almonds. The two mixes are combined in one box, resulting in a new nut mix that is 40 percent peanuts. How many pounds of cashews are now in the box?

A. 3.5

B. 4

C. 4.5

D. 5

E. 6

To solve, a student can:

• Let x be the number of pounds in the second mix. The first box contributes 10 × 0.5 = 5 lb of peanuts; the second contributes 0.2x lb.
• Set up the equation 5 + 0.2x = 0.4(10 + x) and solve to get x = 5 lb of the second mix.
• Total cashews = (10 × 0.2) + (5 × 0.4) = 2 + 2 = 4 lb, or answer choice B.

AMC 12

The AMC 12 is a 25-question mathematics competition intended for students in 12th grade or below. Participants have 75 minutes to complete the exam, which covers the full range of high school math topics, including advanced algebra, geometry, and trigonometry, but does not include calculus. Students who are under 19.5 years old on the day of the contest are eligible to participate.

The competition expects students to be familiar with elementary algebra, basic geometry concepts such as the Pythagorean Theorem and area/volume formulas, introductory number theory, and basic probability.

Students who score in approximately the top 5% on the AMC 12 qualify for the AIME. Every year, the MAA sets a specific score cutoff designed to invite a consistent number of students, usually between 6,000 and 7,000.

The AMC 12 uses the same scoring system as the AMC 10: each correct answer earns 6 points, each blank earns 1.5 points, and each incorrect answer earns 0 points, for a maximum possible score of 150.

Why take the AMC 12?

• Take the first step toward qualifying for the USAMO and potentially the IMO
• Earn a qualification for the AIME with a high enough score
• Enhance college applications, particularly for students entering quantitative-heavy fields

AMC 12 problems

AMC 12 problems are designed with the following format and rules:

• Each question is multiple choice with five possible answers, only one of which is correct
• The exam begins with more accessible problems and gradually increases in difficulty
• Calculators are not permitted during the test
• Students have an average of three minutes per question
• Like the AMC 8 and AMC 10, the AMC 12 emphasizes creative problem-solving and rewards students who can think beyond standard classroom methods

Here is an example problem from a previous AMC 12 competition:

In the figure shown below, major arc AD and minor arc BC have the same center, O. Also, A lies between O and B, and D lies between O and C. Major arc AD, minor arc BC, and each of the two segments AB and CD has length 2π. What is the distance from O to A?

A. 1

B. 1 - π + √π² + 1

C. π / 2

D. √π² + 1 / 2

E. 2

Students must be able to determine that:

• The ratio between the radius and arc length is constant
• For the inner circle, the corresponding arc is 2πr - 2π, or the circumference minus the major arc length
• The outer circle radius is r + 2π, with a corresponding arc of 2π. This gives us:

r / 2πr - 2π = r + 2π / 2π

• Simplyfying the equation, we get r = (r + 2π)(r - 1), or r² + (2π - 2)r - 2π = 0, which is answer B when solving using the quadratic formula

AMC 12 problems often combine multiple high-school topics (algebra, geometry, trigonometry, and number theory) into a single multi-step question. A typical AMC 12 geometry problem might ask you to set up an equation relating the arc lengths and radii of two concentric circles, then solve the resulting quadratic equation. Working through past AMC 12 papers on the official MAA archive is the best way to see the full range of styles.

How many students take the AMC each year?

Over 300,000 students participate in the AMC annually. The AMC 10 and 12 act as a gateway to the AIME, which hosts between 6,000 and 7,000 students a year. From there, roughly the top 500 students move on to either the USAMO or the USA Junior Mathematical Olympiad (USAJMO). Out of the roughly 270 who participate in the USAMO, only the very top performers are selected for the Mathematical Olympiad Summer Program (MOP). From that group, the top 6 students are selected to form the six-member IMO team and compete internationally.

How do I sign up for the AMC, and where does the competition take place?

To sign up for the AMC, students and families can:

1. Register on the Mathematical Association of America (MAA) website
2. Pay the registration fee (those who register before the early-bird deadline receive a discount)
3. Decide their competition date (AMC 10 and 12 offer two exam dates, denoted A or B, while the AMC 8 is generally offered once a year)

The AMC 8 usually takes place in early winter (January), while the AMC 10 and 12 are held in the fall (November).

The AMC is administered at an official, authorized testing site, which is likely to be your middle or high school. You can find a participating school nearby if yours does not offer it, but you must be a full-time student enrolled in a U.S. or Canadian school to participate.

Wrapping up

The AMC practice problems above are designed to give you a feel for the types of questions you’ll encounter on the official exam. Rather than testing straightforward calculations, the AMC challenges you to apply critical thinking and problem-solving skills to multi-step questions. The most effective way to build both speed and comprehension is through consistent practice with AMC 8, 10, and 12 problems and full-length exams.

Developing a strong foundation in geometry and logical reasoning at the middle and high school levels is essential, but success on the AMC also depends on solving problems quickly, often in under a few minutes per question. With focused practice and the right guidance, you can strengthen the strategies and intuition needed to approach each problem efficiently and confidently.