📚 AMC 10/12 — Exam Syllabus

A streamlined, what-you-actually-see syllabus for AMC 10/12. It lists AMC 10 core topics and the few key additions for AMC 12 (mainly logarithms, trigonometry, complex numbers) plus light extensions of AMC 10 tools.

✅ Use this as your planning backbone. If a topic isn’t listed here, it’s likely AIME-level or too rare for AMC 10/12.

🗺️ Quick Topic Map

DomainAMC 10 — Core CoverageAMC 12 — Adds/Extends
🧮 Algebra & FunctionsLinear/quadratic; factoring & identities; remainder/factor; basic Vieta; rational expressions (domains, extraneous roots); inequalities (sign charts, absolute value, AM–GM lite); systems (linear + simple nonlinear); exponents/radicals; sequences (AP/GP, telescoping); basic function ops/graphsLogarithms (laws, change-of-base, solving); a bit more discriminant/parameter sweeps; slightly richer functional-equation plugs
📐 GeometrySimilarity & congruence; circle theorems; Power of a Point; cyclic quads basics; coordinate line/circle, slope/distance/midpoint; shoelace; simple transformations; 3D formulas & netsPlane trigonometry (unit circle values, identities, basic trig equations); law of sines/cosines; some deeper cyclic/similarity setups
🎲 Counting & ProbabilitySum/product rules; complement; P&C (with/without repetition, indistinguishable); Stars & Bars (nonneg/positive + simple bounds); PIE ≤ 3 sets; Pigeonhole; grid paths, anagrams, seatings; probability (conditional, independence); EV via linearity/indicators; Binomial/Geometric recognition; simple HypergeometricSlightly richer PIE/bounded distributions; derangements; a bit more geometric probability
🔢 Number TheoryDivisibility, gcd/lcm, Euclid; prime factorization; modular arithmetic (residues, inverses when $\gcd=1$); last-digit cycles; simple FLT/Euler reductions; CRT with small coprime moduli; linear Diophantine $ax+by=c$; digits/bases & divisibility testsMore modular chains/orders, light valuation use (e.g., trailing zeros)
📘 “Precalculus”—Logs, Trig, Complex (algebra, modulus/argument, simple De Moivre/roots of unity geometry)

🧮 Algebra & Functions (Core for AMC 10)

1) Linear & Quadratic

  • Factor/expand, complete the square; vertex/intercepts/axis; discriminant sense.
  • Viète (sum/product of roots), quick parameter checks.

2) Polynomials

  • Remainder/Factor theorems; basic Vieta relations; synthetic division when natural.
  • Classic identities: $(a\pm b)^2$, $a^2-b^2$, $a^3\pm b^3$, $(a+b)^3$.

3) Rational Expressions & Equations

  • Domain first; cancellation rules; avoid/spot extraneous solutions.

4) Inequalities & Absolute Value

  • Sign charts; piecewise splitting; AM–GM for quick bounds (light use).

5) Systems

  • Substitution/elimination for linear; simple nonlinear pairs (e.g., line–circle).

6) Exponents & Radicals

  • Laws of exponents; rational exponents; conjugate/rationalize tactics.

7) Sequences & Series

  • AP/GP formulas; finite/infinite GP; simple telescoping sums.

8) Functions

  • Composition/inverses; graph transformations; piecewise; light functional-equation plugging for symmetry/fixed points.

AMC 12 Adds in Algebra

  • Logarithms: $\log$ laws, change-of-base, solving equations/inequalities, growth comparisons.
  • More discriminant/parameter sweeps (counting roots, tangency conditions).

📐 Geometry (Core for AMC 10)

1) Triangles

  • Similarity & congruence (SSS/SAS/ASA/AAS); area: $A=\tfrac12 ab\sin C$, Heron.
  • Special points: centroid, incenter, circumcenter, orthocenter (basic properties).

2) Circles & Cyclic Figures

  • Central/inscribed angles; chord–tangent–secant relations; Power of a Point.
  • Cyclic quadrilaterals: opposite angles supplementary; equal angles $\leftrightarrow$ equal arcs.

3) Coordinate Geometry

  • Slope, distance, midpoint; line & circle equations; collinearity/concurrency via algebra.
  • Shoelace for polygon area.

4) Transformations & 3D Basics

  • Reflections/rotations/translations; symmetry arguments.
  • 3D volumes/surfaces (prisms, pyramids, cylinders, cones, spheres), nets & cross-sections; space diagonal.

AMC 12 Adds in Geometry

  • Trig toolkit: unit circle values, identities (Pythagorean, angle addition/double/half), basic trig equations.
  • Law of sines/cosines, $A=\tfrac12 ab\sin C$ as an algebraic lever.
  • Slightly deeper cyclic/similarity chains (still AMC-level).

🎲 Counting & Probability (Core for AMC 10)

1) Counting Principles & P&C

  • Sum/product rules; complement; with/without repetition; indistinguishable objects; circular basics.

2) Stars & Bars / PIE / Pigeonhole

  • Nonnegative/positive solutions; simple bounds/caps.
  • PIE up to 3 sets; Pigeonhole classic bounds and constructions.

3) Classic Patterns

  • Grid paths; anagrams; seatings with restrictions (gaps/adjacency); committees with conditions.

4) Probability & Expected Value

  • Sample space models; conditional probability; independence checks.
  • Linearity of expectation; indicator variables (expected counts).
  • Binomial/Geometric recognition; simple Hypergeometric selections.

AMC 12 Adds in C&P

  • Slightly richer PIE/bounds; derangements; more geometric probability setups.

🔢 Number Theory (Core for AMC 10)

1) Divisibility & GCD/LCM

  • Euclidean algorithm; $\gcd!\cdot!\operatorname{lcm}=\text{product}$; prime factorization.

2) Modular Arithmetic

  • Residues; modular inverses when $\gcd=1$; last-digit cycles; reduce big exponents.
  • Quick use of Fermat/Euler to tame powers.

3) CRT & Simple Diophantine

  • CRT with small pairwise coprime moduli; solve two-mod systems.
  • Linear forms $ax+by=c$, coin problems (two-coin Frobenius by recognition).

4) Digits & Bases

  • Divisibility tests (3/9/11); digital sums; base-$b$ reasoning.

AMC 12 Adds in NT

  • Longer modular chains, order/cycle reasoning; light valuations (e.g., trailing zeros).

🚫 What to De-Emphasize for AMC 10/12

  • Heavy inequality machinery (Jensen, full Cauchy/Schwarz proofs, rearrangement as theory).
  • Generating functions/Burnside/Polya; Catalan/Stirling as formal topics.
  • Inversion, spiral similarity; advanced triangle-center geometry.
  • Conics beyond circles; vectors/matrices/eigenvalues beyond basics.
  • Quadratic reciprocity, Pell’s equation, continued fractions; complex analysis.

🎯 Study Flow (Practical)

  1. Master AMC 10 Core in all four domains (Algebra, Geometry, C&P, Number Theory).
  2. Layer AMC 12 Adds: Logs, Trig, Complex (plus light extensions in each domain).
  3. Drill pattern recognition: grid paths, stars & bars, PIE ≤ 3 sets, power-of-a-point, discriminant sweeps, modular cycles.
  4. After each practice set: error review → log the miss reason (concept vs. slip vs. strategy).
  5. Weekly checkpoint: 1 short mixed section (25–35 min) focusing on timing + accuracy.

🧩 Micro-Checklist Before Test Day

  • Domains covered: Algebra ✔︎ Geometry ✔︎ Counting/Prob ✔︎ Number Theory ✔︎ (+ Logs/Trig/Complex for AMC 12).
  • Tools ready: factor/identity bank, PoP & cyclic cues, stars-bars/PIE/pigeonhole, modular inverse/CRT.
  • Habits: domain first (for rational/log/radical), verify after squaring/cross-multiplying, check units/signs/extremes, use complement/linearity.

Keep it simple: nail the core, add logs/trig/complex, practice patterns, and review mistakes with discipline.

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