This AMC 10/12 strategy and reference sheet covers the contest format, pacing, common problem types, and formulas students use most often. It helps students choose efficient methods instead of doing long computations under time pressure. The focus is on recognizing patterns, estimating wisely, and using core high school math facts quickly.
Students in grades 9-12 can use it for review before practice tests and competitions.
The most important tools include algebraic identities, modular arithmetic, prime factorization, area and angle formulas, counting rules, and probability. Many AMC problems can be simplified by trying small cases, using symmetry, drawing a diagram, or working backward from the answer choices. Strong strategy means skipping strategically, checking units and constraints, and avoiding overcomplicated methods.
A good solution is usually accurate, short, and easy to verify.
Key Facts
- The AMC 10 and AMC 12 each have multiple-choice questions in minutes, so average pacing is minutes per problem.
- The scoring rule is points for a correct answer, points for a blank answer, and points for an incorrect answer.
- Use the identity to factor differences of squares quickly.
- For a quadratic , the roots are and their sum is .
- If integers are counted modulo , then means divides .
- The area of a triangle can be found with or .
- For independent events, , and for complementary events, .
- When order matters, use permutations such as , and when order does not matter, use combinations such as .
Vocabulary
- Modular arithmetic
- A system for comparing integers by their remainders after division by a fixed positive integer.
- Invariant
- A quantity or property that stays the same while other parts of a problem change.
- Complement
- The set of outcomes in a probability problem that are not in the event being considered.
- Combination
- A selection of objects where order does not matter, often counted by .
- Similar triangles
- Triangles with equal corresponding angles and proportional corresponding side lengths.
- Vieta's formulas
- Rules that connect the roots of a polynomial to its coefficients, such as the root sum for .
Common Mistakes to Avoid
- Guessing too early, because AMC scoring rewards blanks more than wrong answers when you cannot eliminate choices confidently.
- Using a long algebraic expansion when factoring is available, because identities like often reveal a faster path.
- Forgetting that combinations ignore order, because using instead of can overcount arrangements by a factor of .
- Applying a geometry formula without checking the needed height or angle, because requires a perpendicular height and not just any side length.
- Ignoring answer choices and constraints, because AMC problems often become much easier after estimating, bounding, or testing possible values.
Practice Questions
- 1 On the AMC, a student answers questions correctly, leaves blank, and gets the rest wrong. What is the score?
- 2 Find the remainder when is divided by .
- 3 How many ways are there to choose students from a group of students?
- 4 A problem can be solved by direct computation, testing answer choices, or drawing a diagram. Explain how you would decide which method to try first on an AMC problem.
Understanding AMC 10/12 Strategy & Reference
AMC work is partly about deciding what not to do. A difficult-looking problem may contain one small condition that controls everything. Read for limits, such as positive integers, distinct choices, or a value between zero and one.
These words remove many possible cases. Before calculating, state what the problem is asking for and what each given fact actually tells you. A common error is solving for an intermediate value when the answer requires a related quantity, such as a square, a remainder, or a total count.
Keep exact values when possible. Early rounding can change a close answer choice.
Multiple-choice answers provide useful information, but they must be used carefully. Plugging in answer choices works best when the choices are numerical and the original condition is easy to test. Start with a middle choice if the result changes in one direction as the input grows.
This can eliminate half the choices at once. Plugging in simple legal values is powerful for expressions that claim to work for every value. Choose values that expose issues, including zero, one, negative one, or equal variables when allowed.
Estimation is another filter. Compare size, sign, parity, and last digits before doing detailed arithmetic. If an answer must be odd or must be less than one, several choices may disappear immediately.
Number theory problems reward organized bookkeeping. Prime factorization turns questions about divisors, greatest common factors, and least common multiples into exponent counting. For a positive integer, each prime exponent in a divisor can range from zero up to the exponent in the number.
Remainders often follow repeating cycles. Powers of a number can have a short pattern in their last digit or in their remainder after division by a fixed number. Write several terms, identify the cycle length, then reduce the exponent using that length.
Be alert to a difference between divisibility and equality. A statement that one number divides another does not tell you their exact values.
In geometry, a clean drawing is part of the solution. Mark equal lengths, parallel lines, right angles, and known angle measures directly on the diagram. Do not trust the picture to be drawn to scale.
Similar triangles are especially important because they connect angle facts to ratios of lengths and areas. If every length is multiplied by a scale factor, area is multiplied by the scale factor squared. Coordinate geometry can simplify a figure when slopes, distances, or midpoints matter.
In counting and probability, define the outcome process first. Ask whether choices are made with replacement, whether order creates different outcomes, and whether cases overlap. Listing small cases can reveal missing restrictions.
Practice should include reviewing wrong answers by cause. Label each mistake as a reading error, a concept gap, a counting overlap, or a time decision. That habit makes later practice far more useful than simply checking the correct letter.