Check List: Math Competition

🟦 TIER 1 — MathCounts Core (Middle School Foundation)

Algebra

• Linear equations

• Proportions & ratios

• Percent problems

• Simple systems (substitution)

• Basic inequalities

• Absolute value basics

• Simple functional relationships

Number Theory

• Divisibility rules

• Prime factorization

• GCD/LCM

• Basic modular arithmetic (mod 2, mod 5, mod 10 patterns)

• Counting divisors (simple cases)

• Base conversions

Geometry

• Area & perimeter

• Pythagorean theorem

• Special right triangles

• Similarity & scale factors

• Basic angle chasing

• Coordinate geometry basics

Counting & Probability

• Set Theory (Basic)

• Multiplication/addition principles

• Simple permutations/combinations

• Complementary counting

• Basic probability

• Casework (small cases)

Skills to Master Before AMC 8

• Fast arithmetic

• Pattern recognition

• Efficient casework

• Translating word problems into equations

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🟩 TIER 2 — AMC 8 (Upper Middle School Competition)

Algebra

• Systems of equations (elimination)

• Quadratic recognition (factoring simple quadratics)

• Functional notation

• Inequality reasoning

• Rate problems (work, distance)

• Mixture problems

Number Theory

• Modular arithmetic (mod 7, mod 9, mod 11)

• Divisor counting (general)

• Prime factorization puzzles

• Diophantine equations (simple)

• Digit problems

Geometry

• Circle basics (arc, sector)

• Coordinate geometry (distance, midpoint)

• Similar triangles (AMC-style)

• 3D geometry basics (nets, volume)

Counting & Probability

• More advanced casework

• Permutations with restrictions

• Combinations (conceptual)

• Probability with geometric shapes

Skills to Master Before AMC 10

• Multi-step reasoning

• Algebraic manipulation speed

• Geometry intuition

• Counting logic

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🟧 TIER 3 — AMC 10 (High School Entry Competition)

Algebra

• Quadratic equations (full mastery)

• Completing the square

• Quadratic inequalities

• Rational expressions

• Functional transformations

• Vieta’s formulas (intro)

Number Theory

• Modular arithmetic (full operations)

• Euclidean algorithm

• Diophantine equations (general)

• Counting divisors/sum of divisors

• Base arithmetic

• Congruence solving

Geometry

• Power of a point

• Cyclic quadrilaterals

• Coordinate geometry with algebra

• Similarity + ratios (advanced)

• 3D geometry with cross-sections

Counting & Probability

• Combinations/permutations (full)

• Binomial coefficients

• Pascal’s triangle identities

• Complementary probability

• Expected value (intro)

Skills to Master Before AMC 12

• Algebra fluency

• Geometry theorems

• Counting strategy selection

• Modular arithmetic comfort

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🟪 TIER 4 — AMC 12 (Upper High School Competition)

Algebra

• Polynomial division

• Remainder theorem

• Factor theorem

• Rational root theorem

• Functional inverses

• Inequality techniques (AM-GM intro)

Number Theory

• Modular arithmetic with large numbers

• Chinese Remainder Theorem (conceptual)

• Fermat’s Little Theorem

• Euler’s Totient (basic)

• Order modulo n

Geometry

• Stewart’s theorem

• Ptolemy’s theorem

• Trigonometric geometry

• Coordinate geometry with conics

• Advanced similarity

• Area ratios

Counting & Probability

• Inclusion-exclusion

• Pigeonhole principle

• Stars and bars

• Recurrence relations (simple)

• Expected value (full)

Skills to Master Before AIME

• Multi-theorem geometry

• Polynomial reasoning

• Modular arithmetic fluency

• Counting with structure

• Strategic problem decomposition

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🟥 TIER 5 — AIME (Elite Problem-Solving Tier)

Algebra

• Functional equations

• Symmetric sums

• Vieta’s formulas (full power)

• Inequality techniques (AM-GM, Cauchy-Schwarz)

• Clever substitutions

• Telescoping series

• Recurrence relations (nontrivial)

Number Theory

• Chinese Remainder Theorem

• Euler’s Totient Theorem

• Quadratic residues

• Order of elements

• Diophantine equations (hard)

• Frobenius coin problem

• Pythagorean triple generation

Geometry

• Ceva’s theorem

• Menelaus’s theorem

• Ptolemy’s theorem (full use)

• Power of a point (advanced)

• Radical axis

• Coordinate geometry with transformations

• Shoelace theorem

• Barycentric intuition (optional)

Counting & Probability

• Advanced combinatorics

• Inclusion-exclusion (multi-layer)

• Generating functions (light exposure)

• Expected value with states

• Probability with geometry + algebra

Trigonometry

• Full identity

• Law of Sines/Cosines in proofs

• De Moivre’s theorem

• Roots of unity

• Trig equations with structure

Skills Required to Succeed at AIME

• Deep pattern recognition

• Multi-step reasoning chains

• Ability to combine topics

• Comfort with nonstandard approaches

• Persistence and structured scratchwork