🗺️ AMC Scope Map — Algebra Coverage Matrix

Essential reference for understanding what algebra topics are tested at each AMC level.

📊 Coverage Legend

SymbolMeaning
✅Core — Essential for this level
🔶Often — Frequently tested
🔸Stretch — AMC12 extension
❌Rare — Minimal coverage

📋 Complete Topic Matrix

TopicAMC10AMC12Notes
Basic Operations✅✅Order of operations, fractions
Factoring Patterns✅✅$(a\pm b)^2$, $a^2-b^2$, $a^3\pm b^3$
Linear Equations✅✅One-variable, slope-intercept
Linear Inequalities✅✅Interval notation, graphing
Absolute Value🔶✅Piecewise, distance interpretation
Quadratic Equations✅✅Factoring, quadratic formula
Quadratic Inequalities🔶✅Sign charts, graphing
Completing the Square🔶✅Vertex form, optimization
Discriminant🔶✅Nature of roots, parameter analysis
Polynomial Division❌🔶Long division, synthetic division
Remainder/Factor Theorems❌🔶$f(a)$ remainder, $(x-a)$ factors
Vieta’s Formulas🔶✅Sum/product of roots
Rational Expressions🔶✅Simplifying, domain restrictions
Rational Equations🔶✅Cross-multiplying, extraneous roots
Radical Equations🔶✅Isolating, squaring, verification
Rational Exponents🔶✅$a^{m/n} = \sqrt[n]{a^m}$
Systems (2×2 Linear)✅✅Substitution, elimination
Systems (3×3 Linear)❌🔶Gaussian elimination
Nonlinear Systems🔶✅Line-circle, parabola-line
AM-GM Inequality❌🔶Arithmetic-geometric mean
Cauchy-Schwarz❌🔸Vector form, contest applications
Arithmetic Sequences✅✅$a_n = a_1 + (n-1)d$
Geometric Sequences✅✅$a_n = a_1 \cdot r^{n-1}$
Series Sums🔶✅Finite AP/GP, infinite GP
Telescoping❌🔶Partial fractions, cancellation
Functional Equations❌🔶$f(x+a)$, $f(-x)$, injectivity
Exponential Equations🔶✅$a^x = b$, change of base
Logarithmic Equations❌🔶$\log_a x = b$, properties
Complex Numbers❌🔶$i^2 = -1$, conjugates, modulus
Parameter Analysis🔶✅Discriminant, degeneracy cases

🎯 AMC10 Focus Areas

Essential (Must Master)

  • Basic operations and factoring
  • Linear equations and inequalities
  • Quadratic equations and factoring
  • Systems of 2×2 linear equations
  • Arithmetic and geometric sequences

Important (Frequently Tested)

  • Absolute value equations
  • Rational expressions and equations
  • Radical equations
  • Vieta’s formulas
  • Parameter analysis

🎯 AMC12 Extensions

Core Additions

  • Polynomial theory (division, remainder/factor theorems)
  • Advanced inequalities (AM-GM, Cauchy-Schwarz)
  • Functional equations
  • Complex numbers
  • Logarithmic equations

Advanced Techniques

  • Telescoping series
  • 3×3 systems
  • Advanced parameter analysis
  • Symmetry substitutions

📈 Study Priority

Phase 1: AMC10 Essentials

  1. Master all ✅ topics for AMC10
  2. Practice 🔶 topics for AMC10
  3. Build problem-solving speed

Phase 2: AMC12 Preparation

  1. Add AMC12 core topics
  2. Master advanced techniques
  3. Practice mixed problem types

Phase 3: Contest Readiness

  1. Review scope map regularly
  2. Focus on weak areas
  3. Time management practice

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