AMC Syllabus: AMC 8, AMC 10 and AMC 12 Topics Explained

The AMC syllabus covers four areas at every level: algebra, geometry, number theory, and counting and probability.

AMC 8 tests them at middle school depth, AMC 10 adds quadratics, modular arithmetic and casework, and AMC 12 adds trigonometry, logarithms and complex numbers. No AMC level includes calculus, proofs, matrices or formal statistics. All three papers are 25 multiple-choice questions with no calculator.

If you have searched for the AMC maths competition syllabus and come back with a vague list, “algebra, geometry, number theory”, you are not alone.

Most articles online list topics without explaining what they mean at each level, how hard they are, or whether your child actually needs calculus.

This guide breaks down the American Mathematics Competitions curriculum level by level, AMC 8, AMC 10, and AMC 12, with specific subtopics, a side-by-side comparison, topic weightings, the difficulty curve across a paper, AIME score benchmarks, and a clear progression ladder.

What Is the AMC? A Quick Overview

The American Mathematics Competitions (AMC) is a national problem-solving competition run by the Mathematical Association of America (MAA).

Over 300,000 students across 50+ countries participate each year. There are three exam levels:

Exam	Grade Level	Format	When
AMC 8	Grade 8 and below (under 15.5 on contest day)	25 questions / 40 min	January
AMC 10	Grade 10 and below (under 17.5)	25 questions / 75 min	November
AMC 12	Grade 12 and below (under 19.5)	25 questions / 75 min	November

All three are multiple-choice with no calculators allowed. High scorers on AMC 10 and AMC 12 advance to the AIME, the next step toward the USA Mathematical Olympiad (USAMO).

Two sittings of AMC 10 and AMC 12 are offered each year, the A and the B, on different dates. You may take one A paper and one B paper, and your better result counts for qualification. For eligibility details see AMC age limits, and for this cycle’s dates see AMC test dates.

Is There an Official AMC Syllabus PDF?

No. The MAA does not publish a formal syllabus document for any AMC level. It states a content boundary (the AMC 10 covers the high school curriculum through grade 10, the AMC 12 the full curriculum excluding calculus) and leaves the rest to be inferred from past papers.

That is why every “AMC syllabus PDF” you find online is someone’s reconstruction rather than an official list, this page included. The reconstruction is reliable because the topic mix has been stable for years, but it is worth knowing that no official document exists to check it against. The closest thing to an authoritative source is the papers themselves, which is why working through AMC past papers tells you more than any topic list.

AMC 8 Syllabus Breakdown

The AMC 8 tests middle-school mathematics, but the problem-solving style is harder than school exams. Here is what is covered:

The Five Core Knowledge Areas on the AMC 8

Every AMC 8 question falls into one of five areas. There is nothing else on the paper.

Arithmetic and Number Sense, fractions, decimals, percentages, ratios, and proportional reasoning, applied through multi-step word problems rather than simple calculations.

Introductory Algebra, setting up and solving linear equations, basic coordinate geometry, and simple quadratic expressions in context.

Geometry Basics, area and perimeter of rectangles, triangles, and circles, the Pythagorean theorem, spatial reasoning, and composite figure problems.

Counting and Probability (Combinatorics), basic permutations (ordered arrangements) and combinations (unordered selections), and straightforward probability. For example: “How many ways can 4 students stand in a line?”

Number Theory Basics, prime numbers, prime factorization, divisibility rules, greatest common divisors (GCD), and least common multiples (LCM).

How Much of the AMC 8 Is Algebra Versus Number Theory Versus Combinatorics?

Roughly: geometry and arithmetic take five to seven questions each, counting and probability three to five, algebra three to five, and number theory two to four.

Area	Questions per paper	Share of the paper
Geometry and spatial reasoning	5 to 7	About 25%
Arithmetic and word problems	5 to 7	About 25%
Counting and probability	3 to 5	About 16%
Algebra	3 to 5	About 16%
Number theory	2 to 4	About 12%

The split moves a little each year, but no area has disappeared and none has taken over. If you are allocating study time, the honest answer is that geometry earns the most marks and combinatorics costs the most marks, because schools teach the first and skip the second.

How AMC 8 Difficulty Is Distributed Across the 25 Questions

The AMC 8 is deliberately laid out from easiest to hardest, and knowing the shape of the curve changes how you pace a paper.

Questions	Difficulty	What they test
1 to 10	Accessible to most participants	Single-step arithmetic, reading a diagram, direct percentage or ratio work
11 to 18	Moderate	Two or three steps combined, often algebra set up from a word problem
19 to 25	Hard	Multiple ideas combined, casework, non-obvious geometry configurations

Most students should be finishing 1 to 10 quickly and accurately, because those marks are worth exactly as much as question 25. A student who scores 15 is usually losing marks to careless errors in the first half rather than failing the last five questions.

Which Geometry Topics Appear Most Often on the AMC 8?

Area and perimeter lead, followed by the Pythagorean theorem and angle chasing.

• Area and perimeter of composite figures, the single most frequent geometry type. See area and perimeter for the foundations.
• The Pythagorean theorem, usually hidden inside a larger figure rather than asked directly.
• Angle relationships, parallel lines, triangle angle sums, exterior angles. See types of angles.
• Triangle properties, similarity and special triangles. See types of triangles.
• Circles, area, circumference, and sectors, less frequent than the above but reliably present.
• Spatial and 3D reasoning, nets, counting cubes, and folding problems.

Essential Number Theory Topics for the AMC 8

Number theory is only two to four questions, but they cluster in the hard half of the paper, so they decide whether a student reaches the Honor Roll.

• Prime numbers and prime factorization. Start with prime and composite numbers.
• Divisibility rules, which turn several questions a year into a ten-second check. See divisibility rules.
• Factors, multiples, GCD and LCM.
• Remainders and simple modular thinking, without the formal notation.
• Digit and place value problems.

What Algebra Skills Does AMC 8 Need Beyond School?

School algebra teaches you to solve an equation you are given. The AMC 8 mostly asks you to build the equation first, then solve it, which is a different skill.

Specifically, beyond the standard curriculum:

• Translating a word problem into variables quickly, including choosing which quantity to call x.
• Working with two unknowns through substitution, before formal systems are taught.
• Recognising when algebra is not the fastest route. Many AMC 8 questions that look algebraic are faster by testing the answer choices.
• Handling expressions rather than just equations, simplifying and comparing without solving.

How it compares to school: AMC 8 topics overlap with Grades 6 to 8 school math, but the problems demand more creative thinking. A school test asks you to find the area of a triangle. An AMC 8 question shows you a triangle inside a pentagon and asks you to find the shaded region, using the same formula, but in a less obvious way.

Did the AMC 8 Syllabus Change for 2026-27?

No. The AMC 8 syllabus has not changed, and no new topics were added. The five knowledge areas above have been stable for years.

What has changed is emphasis rather than content, covered in the trends section below, and the cutoff scores, which rise with the field rather than the syllabus. If you see a claim that new topics were added for a particular year, it is almost always describing a shift in how often an existing topic appears.

AMC 10 Syllabus Breakdown

The AMC 10 is a significant step up. It is designed for Grade 10 and below, but most students benefit from dedicated preparation starting in Grade 7 or 8. For a full strategy, see how to prepare for the AMC.

Intermediate Algebra, quadratic equations and functions, factoring techniques, systems of equations, arithmetic and geometric sequences, inequalities, and polynomial manipulation.

Advanced Geometry, triangle similarity and congruence, circle theorems (inscribed angles, tangent lines, arc lengths), coordinate geometry with complex configurations, and 3D shapes (prisms, pyramids, cylinders).

Combinatorics, this expands significantly from AMC 8. Expect permutations with restrictions, the multiplication principle, casework (splitting problems into cases), complementary counting, and probability with combinatorial reasoning. This is the area students most underestimate.

Number Theory, modular arithmetic (clock-like arithmetic with remainders), properties of primes, the Euclidean algorithm, and Diophantine equations (whole-number solutions to equations).

What Does the AMC 10 Cover That the AMC 8 Does Not?

Four things carry across from AMC 8, but every one of them gets deeper:

Area	New on the AMC 10
Algebra	Quadratics, systems, sequences, inequalities, polynomial manipulation
Geometry	Circle theorems, similarity proofs, coordinate geometry, 3D solids
Combinatorics	Casework, complementary counting, restricted permutations
Number theory	Modular arithmetic, the Euclidean algorithm, Diophantine equations

Nothing is removed. A student moving up from the AMC 8 keeps everything they learned and adds a layer, which is why the transition is manageable within a year of focused work.

Does AMC 10 Include Calculus?

No. AMC 10 does not include calculus. The MAA officially excludes it. Derivatives, integrals, and limits do not appear anywhere on the AMC 10. This is the single most common misconception about the AMC maths competition syllabus.

How it compares to school: AMC 10 overlaps with parts of Algebra I, Geometry, and Algebra II, but combinatorics and number theory are rarely taught in school, making specific competition prep essential.

AMC 10A vs AMC 10B: What Is Actually Different?

Nothing in the syllabus. This is the most common misunderstanding about the two sittings.

	AMC 10A	AMC 10B
Syllabus	Identical	Identical
Format	25 questions, 75 minutes	25 questions, 75 minutes
Date	Earlier November date	Later November date
Difficulty	Set to the same target	Set to the same target
Can you take both?	Yes, one A and one B	Yes, and your better score counts

The MAA writes both papers to the same specification, so neither is officially the easier one. In practice the two papers do land slightly differently each year, which is why historical cutoffs for A and B differ, sometimes by 10 points or more. But that variation is unpredictable in advance, so there is no strategy in choosing one over the other.

The one genuine strategic point: sit both. Two attempts at the same syllabus, with the better score counting, is the cheapest score improvement available to any AMC student.

The same applies to AMC 12A and 12B.

AMC 12 Syllabus Breakdown

The AMC 12 covers the full high school mathematics curriculum. It is the hardest of the three levels and is open to students in Grade 12 and below.

Advanced Algebra, everything in AMC 10, plus polynomial division, Vieta’s formulas, complex numbers, logarithms, exponential functions, and function composition and inverses.

Trigonometry (Limited), the six trig functions, the unit circle, law of sines, and law of cosines. This is not a full trigonometry course, it appears primarily in geometry-based problems.

Advanced Combinatorics, inclusion-exclusion, conditional probability, expected value, and combinatorial identities.

Functions, domain and range, piecewise functions, absolute value functions, and transformations.

Precalculus (Limited), deeper sequences and series, informal limits (conceptual, not computational), and isolated polar coordinate problems.

Does AMC 12 Include Calculus?

No. AMC 12 does not include calculus. The MAA is explicit: calculus is excluded from the AMC 12. Your child does not need it.

How it differs from AMC 10: AMC 12 adds trigonometry, complex numbers, logarithms, and deeper function theory. About 10 to 15 questions overlap with the AMC 10, but the remaining problems require a significantly broader mathematical toolkit. AMC 10 vs AMC 12 covers which to choose if you are eligible for both.

AMC 8 vs AMC 10 vs AMC 12: Side-by-Side Comparison

Feature	AMC 8	AMC 10	AMC 12
Algebra	Basic (linear, word problems)	Intermediate (quadratics, sequences)	Advanced (polynomials, logs, complex numbers)
Geometry	Basics (area, Pythagorean)	Intermediate (circles, 3D, similarity)	Advanced (trig, harder configurations)
Number Theory	Basic (primes, GCD, LCM)	Intermediate (modular arithmetic)	Advanced
Combinatorics	Basic (permutations, probability)	Intermediate (casework, complementary)	Advanced (inclusion-exclusion, expected value)
Trigonometry	No	No	Limited
Calculus	No	No	No
Questions and time	25 in 40 min	25 in 75 min	25 in 75 min
Scoring	1 point per correct answer	6 correct, 1.5 blank, 0 wrong	6 correct, 1.5 blank, 0 wrong
Typical average score	12.42 in 2026, out of 25	About 57 to 62 out of 150	About 57 to 66 out of 150
Leads to	AMC 10 preparation	AIME at 100	AIME at 85

Note the scoring row, because it is widely misreported. On AMC 10 and 12 a blank is worth 1.5 points and a wrong answer is worth 0.

There is no negative marking. A blind guess across five options averages 1.2 points, slightly less than leaving it, so guessing only pays once you can eliminate two options.

Topic Weight Distribution

Knowing which topics appear most often helps you prepare smarter, not harder.

AMC 8 (25 questions)

• Geometry and spatial reasoning: 5 to 7 questions, consistently the largest category
• Arithmetic and word problems: 5 to 7 questions, appears at every difficulty level
• Counting and probability: 3 to 5 questions, growing in recent exams
• Algebra: 3 to 5 questions, increasingly word-problem driven
• Number theory: 2 to 4 questions, essential for harder questions

AMC 10 (25 questions)

• Algebra: 7 to 9 questions, the largest single category
• Geometry: 6 to 8 questions, consistent throughout
• Combinatorics and probability: 4 to 6 questions, the biggest differentiator for high scorers
• Number theory: 3 to 5 questions, common in mid-difficulty problems

AMC 12 (25 questions)

• Algebra (including functions and complex numbers): 8 to 10 questions, dominant
• Geometry (including trigonometry): 5 to 7 questions
• Combinatorics and probability: 4 to 6 questions, often the hardest problems
• Number theory: 3 to 4 questions

Key insight: combinatorics is the most underprepared topic for students coming from school math, and it is heavily represented across all three levels. Prioritize it.

How the Topic Mix Has Shifted Recently

The official syllabus has not changed in years. What has changed is emphasis, and three trends are worth knowing when you choose practice material.

Counting and probability is growing. Recent AMC 8 papers lean on it more than papers from a decade ago, and it now regularly appears in the first ten questions rather than only in the hard half.

Algebra is increasingly disguised as word problems. Fewer questions ask you to solve a stated equation. More ask you to build the equation from a described situation, then solve it. Practising bare algebraic manipulation prepares you for the second step but not the first.

Geometry has moved toward configuration over formula. The formulas tested are the same ones, but recent problems hide them inside composite or unfamiliar figures, so recognising which figure you are looking at matters more than recalling the formula.

None of this changes what to study. It changes how to practise, which is by working real past papers rather than topic drills alone.

AIME Qualification and Score Benchmarks

The AMC syllabus question most students really want answered is what score they need. Here are the benchmarks.

The AIME Floor Is Now a Fixed Number

This is the most important recent change to the AMC. Historically, AIME qualification was a percentage cut, roughly the top 2.5% of AMC 10 takers and the top 5% of AMC 12 takers, with the actual score published only after the exam.

From the 2026 to 2027 season the MAA has replaced that with fixed thresholds: 100 on the AMC 10 and 85 on the AMC 12.

That is a meaningful change for how you plan. Under the old system you could not know your target until results came out, and the number moved a lot year to year. Now it is published in advance, so AIME qualification becomes a concrete goal.

For context, here are the cutoffs from the final years of the percentage system. The year shown is the AIME year, so the 2026 row reflects the AMC sat in November 2025.

AIME year	AMC 10A	AMC 10B	AMC 12A	AMC 12B
2026	105	99	96	100.5
2025	94.5	105	76.5	88.5
2027 onward	100 (fixed)	100 (fixed)	85 (fixed)	85 (fixed)

Look at the AMC 12A column. A 76.5 qualified for the 2025 AIME and a 96 was needed for 2026, a swing of nearly 20 points on the same paper in consecutive years. That volatility is exactly what the fixed floor removes.

Award Benchmarks at a Glance

Award	AMC 8	AMC 10 and AMC 12
Top 5%	Honor Roll, 21 in 2026	Distinction
Top 1%	Distinguished Honor Roll, 24 in 2026	Honor Roll of Distinction
Achievement Roll	15+ for grade 6 and below	90+ for grade 8 and below on the AMC 10
AIME invitation	Not applicable	100 on AMC 10, 85 on AMC 12 from 2027

On the AMC 10 and AMC 12 the top-5% Distinction has recently landed between 105 and 127.5, and the top-1% Honor Roll of Distinction between 132 and 139.5. Both are percentile awards rather than fixed scores, so they move each year with the difficulty of the paper.

What is a good AMC score breaks these down in full, and average AMC math scores shows how the distribution actually looks. For the awards themselves, see AMC math competition awards.

The Progression Ladder: AMC 8 to AMC 10 to AMC 12 to AIME

Think of the AMC pathway as a ladder, with each step requiring stronger foundations and new mathematical tools:

AMC 8 (Foundation), arithmetic, basic algebra, geometry basics, counting basics.

AMC 10 (Intermediate), adds quadratics, combinatorics, number theory, advanced geometry.

AMC 12 (Advanced), adds trigonometry, logarithms, complex numbers, advanced functions.

AIME (Elite), integer-answer format, 15 questions. From 2027 the paper is sat in two 90-minute parts rather than one three-hour block. No multiple choice.

How to Move From AMC 8 to AMC 10

The syllabus gap is smaller than it looks, and most of the work is in two areas rather than all four.

1. Keep everything you have. Nothing from the AMC 8 is dropped. The arithmetic, basic geometry and counting all reappear, just inside harder problems.
2. Learn quadratics properly. Factoring, the quadratic formula, Vieta’s formulas and the discriminant between them account for a large share of the new algebra.
3. Learn casework and complementary counting. This is the biggest genuine gap, because school does not teach it at all.
4. Add modular arithmetic. It converts a class of AMC 10 number theory problems from impossible to routine.
5. Rebuild your pacing. You move from 1.6 minutes per question to 3, but the questions take far longer than twice as long, so the skip strategy matters more.

A reasonable timeline is one year: sit the AMC 8 in January, then work on the four points above through spring and summer for the following November’s AMC 10. AMC 8 vs AMC 10 covers when a student is ready.

What Is the AIME Syllabus?

There is no separate AIME syllabus. The AIME draws on the same four areas as the AMC 12, tested more deeply and with problems that combine two or three areas in a single question.

What changes is the format and the tolerance for error. Answers are integers from 000 to 999, there is no multiple choice to work backwards from, and there is no partial credit. A student who is comfortable across the full AMC 12 syllabus has the content they need for the AIME. What they usually lack is the stamina for problems that take fifteen minutes rather than three. AIME problems shows what that step up looks like.

What Is NOT on the AMC Syllabus

This is where most confusion lives. Here is a clear list of what the AMC does not test:

• Calculus, not on AMC 8, AMC 10, or AMC 12. No derivatives, integrals, or limits.
• Trigonometry, not on AMC 8 or AMC 10. Limited on AMC 12 only.
• Proof-based mathematics, the AMC is multiple choice. No written proofs are required, that is USAMO and IMO territory.
• Statistics, formal methods like standard deviation or regression are not tested.
• Matrices and linear algebra, not included at any AMC level.
• Calculators, not permitted at any level, which means no problem requires one.

Where to Start Based on Your Grade

Current Grade	Recommended Starting Point
Grade 4 to 5	AMC 8 (exploration, low pressure)
Grade 6	AMC 8 with focused preparation
Grade 7	AMC 8 this year, then begin AMC 10 prep
Grade 8	AMC 8 (if new) or AMC 10 (if AMC 8 experienced)
Grade 9	AMC 10 with solid preparation
Grade 10	AMC 10 or AMC 12 (based on math level)
Grade 11 to 12	AMC 12

If your child has already taken the AMC 10, review their score against what is a good AMC score and identify which topic areas cost the most points.

That analysis should drive the next preparation phase. For younger students just starting, free math olympiad training online builds problem-solving foundations early, and AMC 8 vs AMC 10 covers when to make the jump.

How to Turn This Syllabus Into a Study Plan

A syllabus is a checklist, not a plan. Here is the order that works for most students.

1. Confirm your level and the dates. Check AMC age limits and eligibility, then AMC test dates and how to register for the AMC.
2. Audit yourself against the topic list above. Mark each subtopic as confident, shaky, or never seen. Most students find combinatorics and number theory in the third column.
3. Fix the never-seen topics first. These cost the most marks per hour of study, because they are rarely taught in school.
4. Work past papers by level, under time. Start with AMC past papers, then AMC 8 practice tests, AMC 10 practice tests or AMC 12 practice tests, and use free AMC practice resources for volume.
5. Log every miss by topic, not by paper. After five papers the pattern in your error log tells you exactly where the next month should go.

A 24-Week AMC 8 Study Plan Across the Five Areas

If you want a concrete structure rather than a principle, this is the shape that covers all five knowledge areas with time left for timed practice.

Weeks	Focus	What you should be doing
1 to 4	Number theory	Primes, divisibility, GCD and LCM. The smallest area, done first because it is the least familiar.
5 to 9	Counting and probability	Permutations, combinations, casework. The highest-value gap for most students.
10 to 14	Geometry	Area and perimeter of composite figures, Pythagoras, angle chasing, 3D reasoning.
15 to 18	Algebra and word problems	Translating situations into equations, then solving.
19 to 22	Full past papers, untimed	One paper a week, reviewing every miss in detail.
23 to 24	Full past papers, timed	40 minutes, no calculator, exactly as on the day. Pacing only.

The review is the part people skip and the part that works. A paper you have analysed properly is worth five you have simply completed. How to get better at solving math olympiad questions covers the review method, and math olympiad questions shows the question styles to expect.

For the full version with weekly structure across all levels, see how to prepare for the AMC.

Key Takeaways

• The AMC syllabus is four areas at every level: algebra, geometry, number theory, and counting and probability.
• No AMC level includes calculus, proofs, matrices, formal statistics or calculators.
• The MAA publishes no official syllabus PDF, so every topic list online is reconstructed from past papers.
• On the AMC 8, geometry and arithmetic take about half the paper between them, and the questions run easiest to hardest across 1 to 25.
• The AMC 8 syllabus did not change for 2026-27. Emphasis shifts, content does not.
• Combinatorics is the biggest gap between school maths and the AMC at every level, so it earns the most marks per hour of study.
• AIME qualification is now a fixed 100 on the AMC 10 and 85 on the AMC 12, replacing a cutoff that swung nearly 20 points year to year.

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