How to Prep for the AMC 10 and AMC 12 Math Competitions in 2026

The American Mathematics Competitions (AMC) 10 and AMC 12 are the front door of the U.S. math-olympiad pipeline. They are administered by the Mathematical Association of America (MAA) each November, and a strong score is one of the most powerful credentials a high schooler can put on a college application — especially for selective STEM programs at MIT, Caltech, Princeton, Stanford, and CMU. This 2026 guide explains the format, who should sit for which test, and how to train your way to AIME qualification.

What Are the AMC 10 and AMC 12?

The AMC is a 75-minute, 25-question, multiple-choice math contest. There are two versions:

• AMC 10 — open to students in grade 10 or below, age 17.5 or younger on contest day.
• AMC 12 — open to students in grade 12 or below, age 19.5 or younger on contest day.

Both have two annual administrations: the “A” test in early November and the “B” test about a week later. Students may sit for both the A and B versions of either contest.

The content gap between the two:

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• AMC 10 stops at the end of high school Algebra 2 and Geometry. No trig, no logs, no advanced algebra.
• AMC 12 includes trigonometry, logarithms, complex numbers, and basic statistics, plus the full AMC 10 content.

Format and Scoring

That scoring structure is the most important fact about the AMC. Blanks are worth 1.5 points; wrong answers are worth 0. So random guessing destroys your score. Even a 1-in-5 random guess has expected value 6 × 0.2 = 1.2 points, which is less than the 1.5 you get for leaving it blank. You must eliminate at least one answer before guessing.

What Score Earns AIME Qualification?

To qualify for the American Invitational Mathematics Examination (AIME) — the next level above the AMC — you need either:

• AMC 10: a score of about 100+, or top 2.5% of all takers in your version (A or B).
• AMC 12: a score of about 85+, or top 5% of all takers.

The exact cutoffs are set after the test and vary by year. As a rule of thumb, target 120 on AMC 10 or 100 on AMC 12 to confidently qualify.

The 5 Topic Clusters That Drive the AMC

If you pull apart 10 years of AMC 10 and AMC 12 papers, the questions cluster into five buckets:

1. Algebraic manipulation — equations, inequalities, polynomial roots, function composition. ~25–30% of the test.
2. Number theory — divisibility, prime factorization, modular arithmetic, base systems. ~15–20%.
3. Counting and probability — combinations, permutations, casework, complementary counting, expected value. ~15–20%.
4. Geometry — triangles, circles, coordinate geometry, areas, similar triangles, the Power of a Point theorem. ~25%.
5. Sequences, logs, complex numbers, trig (AMC 12 only) — ~10–15%.

Training in those buckets — rather than in random problem sets — is how serious AMC students gain 30–50 points over a school year.

The “Difficulty Curve” — and the Slot Strategy

The 25 questions on an AMC are roughly arranged by difficulty:

• Problems 1–10: warm-ups. A strong Algebra 2 student should reliably solve these. Aim for 10/10 correct = 60 points.
• Problems 11–18: middle. Real problem-solving begins. Most AIME qualifiers get 6–8 of these correct.
• Problems 19–25: hard. Many problems require multiple insights. AIME qualifiers usually get 1–3 of these correct.

A clean “slot strategy” for an AIME-target student:

That’s a 118.5 — comfortably over the AMC 10 AIME cutoff. You can leave 7 problems blank and still qualify. Internalizing that takes the panic out of the back half of the test.

The Power of “Casework”

The single most valuable problem-solving habit on the AMC is casework: breaking a hard counting or geometry problem into a small number of clean sub-cases and solving each one.

Example: “How many positive integers less than 1000 contain exactly one ‘7’ in their decimal representation?”

Rather than a clever formula, split by digit count:
– 1-digit: just the number 7 → 1 case
– 2-digit: AB where exactly one digit is 7. Two sub-cases (7B with B ≠ 7, A7 with A ≠ 0 and A ≠ 7) → 9 + 8 = 17.
– 3-digit: ABC, exactly one digit is 7. Three sub-cases → 9·9 + 8·9 + 8·9 = 225.

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Total: 1 + 17 + 225 = 243.

Casework isn’t elegant, but it’s bulletproof. Every AIME qualifier I know lives in casework.

An 8-Week AMC Training Plan

Week 1 — Diagnose. Take a full past AMC 10 or AMC 12 (free at MAA’s “Art of Problem Solving / AMC archive”). Score it. Read every problem you missed.

Week 2 — Algebra cluster. Work 30 AMC-style algebra problems. Drill polynomial roots, function composition, Vieta’s formulas (sum and product of roots).

Week 3 — Number theory cluster. Modular arithmetic, prime factorization patterns, divisibility tricks, last-digit problems. Source: any AoPS Intro to Number Theory chapter.

Week 4 — Counting and probability cluster. Permutations, combinations, the multiplication principle, complementary counting, expected value, casework drills.

Week 5 — Geometry cluster. Triangle properties, similar triangles, circle theorems (Inscribed Angle, Power of a Point), coordinate geometry.

Week 6 — Advanced topics (AMC 12 only). Logarithms, complex numbers, trig identities, sequences and series.

Week 7 — Full timed mocks. Take two complete AMCs (one A-form, one B-form) under real timing. Score honestly. Identify your “slot 19–25” patterns.

Week 8 — Final review. Make a personal “trap sheet” of repeated errors. Review your slot strategy. Sleep schedule shift. No new material.

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Five Strategies That Move AMC Scores

1. Eliminate before guessing. Never random-guess. The scoring penalizes it. If you can’t eliminate at least one answer, leave it blank for the +1.5.

2. Use the answer choices as data. AMC writers craft distractors carefully. If your answer doesn’t match any choice, you made an arithmetic error — recompute, don’t restart.

3. Plug small cases. When a problem has a parameter (“for all positive integers n”), test n = 1, 2, 3 by hand. Pattern-spotting is faster than algebra on roughly a quarter of AMC problems.

4. Skip aggressively in the first pass. Take a 25-question test in 3 passes: easy ones first, mid-tier on the second pass, leave the brutal back-five for last. Most AIME qualifiers run this exact pattern.

5. Track your time per slot. 1–10 should take ~20 minutes total. 11–18 should take ~30 minutes. That leaves ~25 minutes for problems 19–25 (or for review). If you’re at 30 minutes when you finish problem 10, you’re going too slow.

What Parents Can Do

• Register through the student’s school. Most U.S. high schools host the AMC; if yours doesn’t, find a nearby school that does — MAA maintains a registration locator.
• Buy the “Art of Problem Solving” textbooks. Introduction to Counting and Probability and Introduction to Number Theory are the two highest-ROI books a serious AMC student can own.
• Subscribe to AoPS Online classes. The “AMC 10/12 Problem Series” courses are the standard for serious competitors.
• Don’t push for AMC 12 before the student is ready. Smart students who could clear 100 on AMC 10 sometimes score 60 on AMC 12 because trig and logs aren’t fluent yet. Pick the right test for your skill set.

Free AMC Resources Worth Using

• Art of Problem Solving (AoPS) AMC Archive — every past AMC problem, with student-written solutions.
• MAA AMC Site — official registration and past papers.
• EffortlessMath AMC Practice — topic-by-topic problem sets aligned to the five clusters.

Frequently Asked Questions

Should I take AMC 10 or AMC 12?
If you’re in 10th grade or below, take AMC 10 — you’re eligible, and the content cap matches your coursework. Once you’ve covered trig and logs, switch to AMC 12. Strong sophomores often take both: AMC 10 for an AIME shot, AMC 12 for the experience.

Are calculators allowed?
No. No calculators on AMC, AIME, or USAMO.

What’s the next level after AMC?
A qualifying score advances you to the AIME (American Invitational Mathematics Examination). Top AIME scorers advance to the USAMO/USAJMO. Top USAMO scorers may be invited to the Math Olympiad Summer Program (MOSP) and the U.S. International Math Olympiad team.

How much do AMC results matter for college?
A qualifying AMC score (and especially an AIME score) is one of the most weighted credentials in selective STEM admissions at schools like MIT, Caltech, Princeton, Stanford, and CMU.

What’s the most common AMC mistake?
Trying to power-solve problem 19 in 5 minutes. The slot strategy beats heroics. Bank the easy points, don’t bleed the hard ones.

The Bottom Line

The AMC 10 and AMC 12 are competitive but coachable. Five topic clusters, a slot strategy that targets AIME at 118 points, eight weeks of structured training, and the discipline to leave problems blank rather than guess randomly — that’s the path. Walk it on purpose and AIME qualification stops being a lottery and starts being an expected outcome.

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