The American Mathematics Competition 12 (AMC 12) is a 30-question, 75-minute multiple-choice exam administered by the Mathematical Association of America (MAA) to students in grades 10 through 12 and anyone under 19.5 years of age. It is the first step in a pathway that can lead to the American Invitational Mathematics Examination (AIME) and ultimately to the USA Mathematical Olympiad (USAMO) and international competition.
This free AMC 12 practice test PDF gives you printable problems that reflect the style, difficulty distribution, and topic coverage of the actual exam. Work through them offline, time yourself, and use your results to guide focused practice in the areas where you need the most improvement.
AMC 12 preparation requires consistent problem-solving practice across algebra, geometry, number theory, and combinatorics. The exam does not reward rote memorization — it rewards pattern recognition, creative problem formulation, and efficient solution strategies developed through repeated exposure to competition-style problems.
The AMC 12 is designed so that brute-force calculation is rarely the intended path to a solution. Problems at the harder end of the exam often have elegant solutions that become apparent once the right approach is identified. Developing a toolkit of strategies — and knowing when to apply each — is the single most important skill for improving your score.
Working backwards from answer choices (back-solving) is effective when the answer choices are simple numbers and the forward calculation would be complex. Plugging in specific numbers works well for problems involving general variables or percentages. Drawing accurate diagrams for geometry problems prevents assumption errors and often reveals relationships not obvious from the text alone.
Symmetry exploitation is a powerful shortcut. When a problem has symmetric structure — equal quantities, balanced expressions, or figures with obvious symmetry — the answer often has a simple form. Recognizing symmetry early can eliminate entire classes of cases and simplify calculation dramatically.
Casework and complementary counting are essential for combinatorics and probability. For casework, identify the smallest number of disjoint cases that cover all possibilities and handle each systematically. For complementary counting, subtract the count of unwanted outcomes from the total — this is often faster when the complementary set is simpler than the target set.
Parity arguments — reasoning about whether quantities are even or odd — quickly eliminate answer choices and narrow the solution space in number theory and algebra problems. Modular arithmetic, particularly working mod 2, mod 3, mod 4, and mod 9, catches patterns that are invisible in ordinary arithmetic. Practicing mod operations until they are instinctive is a high-return investment for AMC 12 preparation.
Algebra is the largest content domain on the AMC 12. Problems range from systems of equations and polynomial factoring to sequences and series, function composition, logarithms, complex numbers, and inequalities. The exam regularly features problems where recognizing a disguised algebraic identity — difference of squares, Sophie Germain identity, AM-GM inequality — is the key insight.
Sequences and series appear frequently. For arithmetic sequences, know the general term formula and sum formula. For geometric sequences, know the general term, finite sum, and infinite sum (for |r| < 1). The AMC 12 often combines sequences with other topics — for example, geometric sequences with integer constraints that require number theory arguments to resolve.
Number theory problems on the AMC 12 test divisibility rules, prime factorization, greatest common divisor (GCD), least common multiple (LCM), Euler's totient function, and the properties of perfect squares and perfect cubes. Modular arithmetic underlies many number theory problems — practice finding patterns in residues, applying Fermat's Little Theorem, and recognizing when Chinese Remainder Theorem arguments simplify a problem.
Complex numbers appear in AMC 12 problems more often than AMC 10. Know the polar form, De Moivre's theorem for powers and roots, and the geometric interpretation of complex multiplication as rotation and scaling. Problems involving roots of unity are a recurring AMC 12 topic that combines complex numbers with polynomials and symmetry arguments.
Geometry on the AMC 12 covers plane geometry (triangles, circles, polygons, coordinate geometry) and extends to three-dimensional geometry and trigonometry. Triangle problems frequently involve area calculations using multiple methods — base-height formula, Heron's formula, and the formula Area = (1/2)ab sin C. Knowing several area methods and recognizing which applies to the given information is a core geometry skill.
Circle theorems are tested heavily: inscribed angle theorem, central angle theorem, power of a point, intersecting chords, tangent-secant relationships, and the relationship between arc length, chord length, and central angle. Problems combining circles with triangles — particularly cyclic quadrilaterals and inscribed/circumscribed circle configurations — require knowing the extended law of sines and Ptolemy's theorem.
Trigonometry on the AMC 12 goes beyond right triangle ratios. Know the law of sines, law of cosines, product-to-sum and sum-to-product identities, double and half angle formulas, and the graphs and properties of sine, cosine, and tangent. Inverse trigonometric functions and their compositions appear in challenging problems, as do problems requiring you to find exact values at non-standard angles using angle addition identities.
Coordinate geometry problems on the AMC 12 typically involve finding intersections, distances, slopes, and areas in the coordinate plane. Know the distance formula, midpoint formula, section formula, area of a triangle given coordinates (shoelace formula), and the conditions for perpendicular and parallel lines. Locus problems — finding the set of all points satisfying a given condition — often have elegant solutions when translated into the coordinate plane.
Combinatorics and probability form a significant portion of the AMC 12 and are among the topics most students find challenging without targeted preparation. Counting problems require fluency with permutations, combinations, the multiplication principle, complementary counting, inclusion-exclusion, and stars-and-bars for distributing identical objects into distinct bins.
The inclusion-exclusion principle extends naturally to probability: P(A or B) = P(A) + P(B) - P(A and B). Problems involving at least one occurrence of an event almost always benefit from complementary probability — compute 1 minus the probability that none of the events occur, which is typically a product of independent probabilities or a simpler geometric probability argument.
Expected value problems appear regularly. Know that expected value is linear even when events are not independent — this makes many seemingly complex expected value calculations tractable by decomposing the random variable into simpler indicator variables. Geometric probability — probability proportional to length, area, or volume — tests integration of geometry and probability concepts.
Counting paths in grids, arranging objects with constraints, and distributing items into groups with restrictions are standard AMC 12 combinatorics problem types. Practice identifying which counting framework applies — does the problem require ordered or unordered selection? Are items identical or distinct? Are the bins labeled or unlabeled? — before attempting to count, because selecting the wrong framework leads to systematic errors that are difficult to catch without re-reading the problem.
Set a 75-minute timer and work through all 30 problems without consulting notes, formulas, or a calculator — the actual AMC 12 prohibits calculators, so practicing without one builds the mental arithmetic fluency the exam requires. At the 75-minute mark, stop and score your answers using the AMC 12 scoring formula: +6 for correct, -1.5 for blank, 0 for wrong.
After scoring, sort your missed questions by topic area and difficulty. AMC 12 problems are generally arranged from easier (problems 1–10) to harder (problems 21–30), though this is not absolute. Review every problem you missed regardless of difficulty, and also review problems you solved by guessing or inefficient methods — finding a cleaner solution for problems you already got right is valuable practice.
For each missed problem, work through the solution step by step before checking an official solution. If your method was correct but you made an arithmetic error, note the error type and practice targeted arithmetic exercises to eliminate that error pattern. If your approach was fundamentally wrong, identify which concept or strategy the problem required and add it to your targeted study list.
Consistent practice over months produces far better AMC 12 results than intensive cramming in the week before the exam. Aim to work 10–15 competition problems per day across all topic areas, alternating between timed sets (simulating real exam conditions) and untimed deep-dives where you work a hard problem until you fully understand the solution without hints.
Official AMC 12 past exams from the MAA are the best practice resource. Work through exams from recent years first, since they reflect the current difficulty distribution and problem style. For additional problem sources, AIME past exams provide harder problems in the same topic areas, and state competition exams (ARML, NYSML, HMMT) offer variety.
Track your scores across practice sessions. Plot your score on each timed practice exam and look for upward trend with decreasing variance. If your scores plateau, analyze whether the plateau is across all topics or concentrated in one area. Targeted drill on a specific weak topic for two to three weeks typically breaks a score plateau faster than general problem sets.
This PDF is one part of a complete AMC 12 preparation plan. For interactive timed practice questions organized by topic and difficulty, with detailed solution explanations and step-by-step answer rationales, visit the full amc 12 practice test page. Pairing printable problem sets with online practice gives you the flexibility to study offline combined with performance tracking to measure your progress toward qualifying for the AIME.