🧠 Problem-Solving Tips — Strategies & Heuristics
Actionable strategies and checklists for efficient AMC algebra problem solving.
🎯 How to Use This Section
Before Contests
- Review checklists — Ensure you know all strategies
- Practice decision trees — Learn when to use each technique
- Time yourself — Build speed and efficiency
During Contests
- Follow checklists — Don’t skip steps
- Use decision trees — Choose the right approach
- Manage time — Don’t get stuck on one problem
Study Strategy
- Master basic strategies — Domain first, extraneous solutions
- Practice advanced techniques — Symmetry substitutions, discriminant playbook
- Build speed — Time yourself on each strategy
- Review regularly — Keep strategies fresh
📚 Strategy Categories
🔍 Problem Analysis
- Domain first — Check restrictions before manipulating
- Extraneous solutions — Always verify after squaring/cross-multiplying
- Pattern recognition — Look for common problem types
🎯 Solution Techniques
- Symmetry substitutions — Try $u=x+\frac{1}{x}$, $t=x^2$
- Discriminant playbook — Count roots, enforce conditions
- Factor before expand — Look for common patterns
⚡ Optimization
- Sign and bounds — Use AM-GM for lower bounds
- Graph sense — Visualize intersections and monotonicity
- Timing strategies — When to skip and move on
🎯 Quick Reference
| Strategy | When to Use | Key Points |
|---|---|---|
| Domain First | Any equation with restrictions | Check denominators, radicands, log arguments |
| Extraneous Check | After squaring or cross-multiplying | Always substitute back into original |
| Symmetry Sub | Cyclic expressions, $x + \frac{1}{x}$ | Look for patterns that simplify |
| Discriminant | Parameter problems, root counting | Use $\Delta = b^2 - 4ac$ |
| Factor First | Complex expressions | Look for common factors before expanding |
| AM-GM | Optimization problems | Use for lower bounds on positive expressions |
| Graph Sense | Intersection problems | Visualize curves and their relationships |
| Time Management | All problems | Don’t spend more than 3 minutes on one problem |
⚡ Contest Tips
Time Management
- Easy problems: 1-2 minutes
- Medium problems: 3-5 minutes
- Hard problems: 5-8 minutes
- Skip threshold: 3 minutes without progress
Problem Priority
- Factoring problems — Usually quick wins
- Quadratic problems — Well-practiced techniques
- System problems — Good if you see the approach
- Advanced problems — Time permitting
Common Pitfalls
- Domain errors — Always check restrictions
- Sign errors — Watch negative signs carefully
- Extraneous solutions — Verify all answers
- Time traps — Don’t get stuck on one problem
See Problem-Solving Tips for detailed strategies and checklists!
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