What are the main differences between the AMC 8 and the AMC 10/12? How can you plan a learning path from the AMC 8 to higher-level competitions? With typical high-frequency exam questions as examples.

As one of the most globally recognized mathematics competitions, the AMC (American Mathematics Competitions) provides a progressive challenge platform for students of different ages and mathematical levels through its tiered competition system. The AMC 8, AMC 10, and AMC 12 are designed for students in different grade levels, sharing close connections while also featuring significant differences.

I. The Three‑Level Leap from AMC 8 to AMC 10/12: Understanding the Fundamental Differences

The AMC competition system was founded in 1950 by the Mathematical Association of America (MAA) and has become one of the most globally trusted mathematics assessments, with its scores widely recognized in the United States, Canada, the United Kingdom, and many other countries. Although they belong to the same AMC series, the AMC 8, AMC 10, and AMC 12 differ significantly across several dimensions. In terms of participants and exam format, these three levels are designed for students of different age groups, with key differences summarized in the following table:

Comparison of Eligibility Requirements and Exam Formats for AMC 8, AMC 10, and AMC 12

Here’s a clear, natural English translation of your text:In terms of knowledge scope and difficulty, the three competition levels also show a distinct progression:

AMC8 covers elementary and middle school mathematics, including integers, fractions, decimals, percentages, ratios, number theory, everyday geometry, area, volume, probability and statistics, logical reasoning, and more. The questions are relatively easy, similar to the level of domestic primary school Olympiad competitions.

AMC10 covers the ninth- and tenth-grade mathematics curriculum, including elementary algebra, basic geometry, elementary number theory, and probability, but does not include trigonometry, advanced algebra, or advanced geometry. In recent years, the exam has also added new types of questions such as coloring problems and chessboard problems, making the content broader and the difficulty gradually increasing.

AMC12 is the most difficult of the three levels, covering the entire high school mathematics curriculum, including advanced algebra, advanced geometry, trigonometry, complex numbers, logarithms, and logarithmic functions. Both the problem-solving difficulty and the required techniques are significantly higher, placing greater demands on students’ mathematical thinking abilities. It is important to note that there is no direct advancement relationship between AMC8, AMC10, and AMC12. Students can choose the competition level that suits their age and mathematical ability without having to participate in each one in sequence.

II. From Beginner to Master: A Study Path Plan for the AMC Competitions

A successful AMC competition journey requires systematic planning based on a student’s grade level and mathematical foundation. Below is a reference study path plan for students at different grade levels:

Grade‑by‑Grade AMC Competition Study Planning Recommendations

For students in lower grades (Grades 3–5), the AMC8 is an ideal starting point. The goal at this stage is not to pursue a high score, but to spark interest and build a strong foundation. A recommended target is around 15 points, and earning the Global Honor Roll can be considered a success. When preparing, students should focus on consolidating elementary school math content, gradually learn middle school basics, and develop fundamental competition‑level thinking and skills.

For students in middle grades (Grades 6–8), they can aim for higher awards in the AMC8, such as the Honor Roll (Top 5% globally) or the Distinguished Honor Roll (Top 1% globally). This stage requires systematic study of the four core modules—algebra, geometry, number theory, and combinatorics—to build a solid foundation for transitioning to the AMC10/12.

A practical criterion for determining readiness to move up to the AMC10 is consistently scoring above 20 points in the AMC8. For early high‑scorers, further preparation for the AMC10 is recommended, as it is the core competition in the AMC series.

Preparing for the AMC10 requires completing at least the in‑school math curriculum up to Grade 10 and setting aside more than six months for focused, topic‑specific preparation.

For upper‑grade students (Grades 8–12), directly preparing for the AMC12 is a more efficient choice. The AMC12 covers a broader and deeper range of topics than the AMC10, adding nearly mandatory content such as logarithms, trigonometry, and complex numbers.

A good indicator of readiness for the AMC12 is the ability to correctly answer around 18 questions in the AMC10.

Achieving Top 5% or Top 2.5% in the AMC10/12 not only earns prestigious honors but also qualifies students for the more advanced AIME competition, which holds significant weight in university applications.

III. Detailed Explanation of High‑Frequency Exam Questions: Mastering Core AMC Problem‑Solving Skills

To achieve excellent results in the AMC competitions, it is not enough to merely master the knowledge points; students must also become familiar with typical question types and problem‑solving techniques. Below are the common question types and solution strategies for both the AMC8 and AMC10/12.

Typical AMC8 Question Types and Solutions

The difficulty level of AMC8 questions increases gradually, and they are generally divided into basic questions (1–10), intermediate questions (11–20), and challenging questions (21–25). Different strategies are needed for each difficulty level.

Example of an Algebra Problem: The product of two positive integers is 24, and their sum is 11. Find the larger number.

Solution Strategy: Let the two numbers be x and y, with x > y. Set up the equations xy = 24 and x + y = 11. Solving these gives x = 8 and y = 3 (or x = 3 and y = 8), so the larger number is 8. This question tests integer factorization and the concepts behind quadratic equations.

Example of a Geometry Problem: Calculate the area of a lattice polygon.

Solution Strategy: The area of a lattice polygon can be found using the dissection method or the complementary shape method. For example, in a problem asking for the area of a small kite, the result is 21 square inches. This type of question focuses on calculating areas of lattice figures, often using Pick’s Theorem or dissection/complementary techniques.

Example of a Number Theory and Combinatorics Problem: Remove one number from the set {15, 16, 17, 18, 19} so that the sum of the remaining four numbers is a multiple of 4. Find the number that was removed.

Solution Strategy: The total sum of the five numbers is 15 + 16 + 17 + 18 + 19 = 85. When 85 is divided by 4, the remainder is 1. To make the sum of the remaining four numbers a multiple of 4, we need to remove a number that also leaves a remainder of 1 when divided by 4. Checking each number: 15 ÷ 4 leaves remainder 3, 16 leaves 0, 17 leaves 1, 18 leaves 2, and 19 leaves 3. Therefore, the number to remove is 17. This question tests properties of remainders and divisibility.

Typical AMC10/12 Question Types and Solutions

The difficulty of AMC10/12 questions increases significantly, especially the last five questions (21–25), which are the key to achieving high scores.

Example of an Algebra and Geometry Combined Problem: A rectangle has an area of 24 and a perimeter of 20. Find the value of the sum of the squares of its length and width.

Solution Strategy: Given length × width = 24 and length + width = 10 (derived from the perimeter 20), we need to find length² + width². Recall the perfect square formula: (length + width)² = length² + 2×length×width + width². Therefore, length² + width² = (length + width)² – 2×length×width = 10² – 2×24 = 100 – 48 = 52. The core of such problems is recognizing the algebraic relationships between geometric quantities.

Example of a Number Theory and Combinatorics Combined Problem: Find the largest positive integer N such that there exist N distinct positive integers where the sum of the exponents of 2 and 3 in the prime factorization of each number is at most 2.

Solution Strategy: This question combines number theory and combinatorics. The key is to divide the problem into three cases: sums of exponents equal to 0, 1, or 2. Then systematically count the numbers that satisfy the conditions in each case. Such problems require precise understanding of the problem statement and systematic analysis through classification.Efficient Problem‑Solving Techniques

Mastering efficient problem‑solving techniques can significantly improve scores in AMC competitions:

Specific Value Method: When a geometric figure is not uniquely determined, assume special conditions and then calculate; when finding maximum or minimum values, consider extreme cases; in ratio problems, assume a total value for easier calculation.

Measurement Method: For some geometry problems, draw the figure under special conditions and directly measure side lengths or angles to obtain the answer (though recent examiners have tried to avoid making this possible).

Pattern Recognition: For sequence problems, start with simple cases; for remainder problems, look for cyclic patterns.

Elimination Method: Eliminate options based on parity; exclude options by considering possible value ranges; test options one by one in logical reasoning problems.

Time Management Strategy: For the 40‑minute AMC8 exam, it is recommended to spend about 15–20 minutes on easy questions (1–15), reserve 15–20 minutes for difficult questions (21–25), and leave 5 minutes for checking.

For the 75‑minute AMC10/12 exams, answer the first 15 questions quickly while ensuring accuracy to leave ample time for the later, more difficult questions.

The journey from AMC8 to AMC10/12 is full of challenges. The AMC8 tests elementary school math and some middle school knowledge, while the AMC10/12 cover a wider range of high school topics such as trigonometry, complex numbers, and advanced algebra.

Scan the QR code to join the exclusive AMC8 study community, where you’ll find a wealth of preparation materials and trial courses waiting for you to explore!

Students preparing for the competition can download for free:

• 2000–2025 AMC8 real exam questions in both Chinese and English

• Detailed explanations

• Formula sheets

• Vocabulary lists

• Recommended preparation books

• Lecture notes and more

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