🧩 Problem Types — Pattern Recognition & Templates
Learn to recognize and solve common AMC algebra problem patterns efficiently.
🎯 Pattern Categories
🔢 Factoring & Manipulation
- Factoring Templates — Classic patterns, Sophie Germain, grouping
- Symmetry Substitutions — $u=x+\frac{1}{x}$, $t=x^2$, cyclic sums
🎯 Equation Solving
- Discriminant & Parameter Sweeps — Count solutions, range of parameters
- Rational Equations — Cross-multiplying safely, domain first
- Radical Equations — Isolate, square, verify, conjugate tricks
- Absolute Value Casework — Breakpoints, symmetry, sum of distances
🎲 Systems & Advanced
- Systems Linear & Nonlinear — Line-circle, parabola-line, substitution flows
- Sequence Closed Forms — Finite GP/AP sums, telescoping
- Functional Equation Templates — $f(x+a)/f(-x)$ invariances, injectivity tests
- Exponential & Log Tricks — Compare growth, base alignment, logs of both sides
📝 Word Problems
- Word Problems — Work/rate/mixture, ratio/proportion, units
🎯 How to Use This Section
Pattern Recognition
- Read the problem — Look for key phrases and structure
- Identify the pattern — Match to known problem types
- Apply the template — Use the standard solution approach
- Verify the answer — Check for extraneous solutions
Study Strategy
- Master each pattern — Understand recognition cues
- Practice templates — Learn solution steps by heart
- Work examples — See patterns in action
- Time yourself — Build speed and accuracy
📊 Pattern Recognition Guide
| Problem Type | Recognition Cues | Key Template |
|---|---|---|
| Factoring | “Factor completely”, “Simplify” | Look for common patterns, factor by grouping |
| Discriminant | “How many solutions”, “For what values” | Use $\Delta = b^2 - 4ac$ |
| Rational | Fractions with variables, “Solve” | Cross-multiply, check domain |
| Radical | Square roots, “Solve for $x$” | Isolate, square, verify |
| Absolute Value | $ | x |
| Systems | Multiple equations, “Find all solutions” | Substitution or elimination |
| Sequences | “Find the sum”, “What is the $n$th term” | Use AP/GP formulas |
| Functional | $f(x+a)$, $f(-x)$, “Find $f(x)$” | Substitution, symmetry |
| Word Problems | “How many”, “What is the ratio” | Set up equations, solve |
⚡ Quick Tips
- Start with the end — What are you trying to find?
- Look for patterns — Don’t reinvent the wheel
- Use templates — Follow proven solution methods
- Check your work — Verify answers make sense
- Practice speed — Time yourself on each pattern
🎯 Contest Strategy
Time Management
- Easy patterns: 1-2 minutes
- Medium patterns: 3-5 minutes
- Hard patterns: 5-8 minutes
- Skip if stuck: Move on after 3 minutes
Pattern Priority
- Factoring — Most common, quick wins
- Discriminant — Parameter problems
- Systems — Word problems
- Sequences — Sum problems
- Advanced — Time permitting
Start with Factoring Templates to master the most common pattern!
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