💡 Problem-Solving Tips

Strategic approaches, checklists, and timing advice to maximize your AMC performance.

🎯 How to Use This Section

1. Study the strategies - Learn different approaches to problem-solving
2. Practice the checklists - Use them during practice problems
3. Master the timing - Know when to move on and when to persist
4. Apply the tips - Use them in actual AMC competitions

📚 Tip Categories

🧠 Complete Problem-Solving Guide

Comprehensive collection of strategies, checklists, and timing advice.

🎯 Quick Strategy Overview

When Stuck

1. Graph it - Visualize the problem
2. Try symmetry - Look for patterns
3. Test special cases - Use simple examples
4. Check domains - Verify constraints

Algebra Sanity Checks

1. Factor first - Look for common factors
2. Compare growth rates - For limits and asymptotes
3. Check for extraneous roots - Especially with log/abs/rational
4. Verify solutions - Substitute back

Trig Sanity Checks

1. Check angle ranges - Principal values for inverse trig
2. Use periodicity - Don’t forget all solutions
3. Verify identities - Make sure they’re applicable

Complex Number Playbook

1. Rotate/scale via multiplication - Use polar form
2. Use argument addition - For rotations
3. Polygon roots-of-unity geometry - Regular polygons

Sequences/Series Strategy

1. Express $a_{n+1}-a_n$ - For telescoping
2. Telescope - Look for cancellation
3. Partial sums before limits - For convergence

⏰ Timing Strategy

AMC 10 (75 minutes, 25 problems)

• Easy problems (1-10): 1-2 minutes each
• Medium problems (11-20): 3-4 minutes each
• Hard problems (21-25): 5-7 minutes each
• Total time: ~60 minutes, 15 minutes for review

AMC 12 (75 minutes, 25 problems)

• Easy problems (1-10): 1-2 minutes each
• Medium problems (11-20): 3-5 minutes each
• Hard problems (21-25): 6-8 minutes each
• Total time: ~65 minutes, 10 minutes for review

When to Move On

• Spent 3+ minutes on easy problem
• Spent 5+ minutes on medium problem
• Spent 8+ minutes on hard problem
• No clear path forward after 2-3 attempts

📋 Quick Decision Tree

Problem Type Recognition

1. Functions → Check domain, use transformations
2. Polynomials → Try Vieta’s, factor, remainder theorem
3. Trig → Use identities, check quadrants
4. Logs/Exp → Check domain, use laws
5. Complex → Convert to polar, use De Moivre’s
6. Sequences → Look for patterns, use formulas
7. Geometry → Draw diagram, use coordinate geometry
8. Inequalities → Try AM-GM, Cauchy-Schwarz

Solution Strategy

1. Direct approach - Straightforward calculation
2. Substitution - Change variables to simplify
3. Symmetry - Use geometric or algebraic symmetry
4. Special cases - Test simple examples first
5. Contradiction - Assume opposite and find contradiction

🎯 AMC Success Checklist

• Read carefully - Understand what’s being asked
• Identify type - Which pattern does this match?
• Check constraints - Domain, range, special cases
• Try standard approach - Use known techniques
• Simplify - Look for shortcuts and identities
• Verify answer - Check it makes sense
• Consider alternatives - Is there a simpler way?

Next: Problem-Solving Tips | Back: Precalculus Mastery Guide