💡 Complete Problem-Solving Guide

Master the art of AMC problem-solving with comprehensive strategies, checklists, and timing advice.

🎯 The AMC Mindset

Core Principles

Efficiency over perfection - Find the fastest correct solution
Pattern recognition - Most problems follow known patterns
Strategic guessing - Eliminate wrong answers systematically
Time management - Know when to move on

Problem-Solving Philosophy

Start simple - Try the most direct approach first
Look for patterns - AMC problems often have elegant solutions
Use symmetry - Geometric and algebraic symmetry are powerful
Check your work - Verify answers make sense

📋 The 5-Step Process

Step 1: Read & Understand (30 seconds)

What’s being asked? - Identify the goal
What’s given? - List all information
What’s the context? - Which topic area?
Any constraints? - Domain, range, special cases

Step 2: Identify the Pattern (30 seconds)

Function problems - Domain, range, transformations
Polynomial problems - Vieta’s, factoring, remainder theorem
Trig problems - Identities, unit circle, laws of sines/cosines
Log/Exp problems - Laws, domain restrictions, change of base
Complex problems - Polar form, De Moivre’s, roots of unity
Sequence problems - Arithmetic, geometric, telescoping
Geometry problems - Coordinate geometry, distance, area
Inequality problems - AM-GM, Cauchy-Schwarz, optimization

Step 3: Choose Strategy (1 minute)

Direct calculation - Straightforward approach
Substitution - Change variables to simplify
Symmetry - Use geometric or algebraic symmetry
Special cases - Test simple examples first
Contradiction - Assume opposite and find contradiction
Induction - For sequence or pattern problems

Step 4: Execute & Simplify (2-4 minutes)

Apply the strategy - Work through the solution
Look for shortcuts - Identities, cancellations, patterns
Simplify continuously - Don’t carry complex expressions
Check intermediate steps - Verify calculations

Step 5: Verify & Move On (30 seconds)

Check the answer - Does it make sense?
Verify constraints - Domain, range, special cases
Consider alternatives - Is there a simpler way?
Move on - Don’t overthink

🔍 Pattern Recognition Guide

Function Problems

Look for: $f(x)$, domains, ranges, inverses, composition Common patterns:

Domain restrictions (square roots, logarithms, fractions)
Function composition $(f \circ g)(x) = f(g(x))$
Inverse functions $f^{-1}(x)$ vs $\frac{1}{f(x)}$
Transformations (shifts, scales, reflections)

Example: “Find domain of $\sqrt{\log_2(x-1)}$” Pattern: Nested domain restrictions Strategy: Work from inside out

Polynomial Problems

Look for: Roots, coefficients, Vieta’s relationships Common patterns:

Vieta’s formulas for root relationships
Remainder theorem for polynomial division
Factoring techniques (difference of squares, sum/difference of cubes)
Symmetric sum identities

Example: “If $x^2 - 5x + 6 = 0$ has roots $r$ and $s$, find $r^2 + s^2$” Pattern: Vieta’s + symmetric sum identity Strategy: Use $r^2 + s^2 = (r+s)^2 - 2rs$

Trigonometric Problems

Look for: $\sin$, $\cos$, $\tan$, angles, identities Common patterns:

Unit circle values for special angles
Identity manipulation (Pythagorean, addition, double angle)
Law of sines and cosines for triangles
Inverse trigonometric functions

Example: “Solve $\sin(2x) = \cos(x)$ for $0 \leq x < 2\pi$” Pattern: Multiple trig functions Strategy: Use double angle formula, then factor

Logarithm/Exponential Problems

Look for: $\log$, $\ln$, $e^x$, growth/decay Common patterns:

Logarithm laws (product, quotient, power)
Change of base formula
Domain restrictions (arguments must be positive)
Growth and decay applications

Example: “Solve $\log_2(x-1) + \log_2(x+1) = 3$” Pattern: Multiple logarithms Strategy: Combine using product law, check domain

Complex Number Problems

Look for: $i$, polar form, rotations, roots of unity Common patterns:

Polar form $z = re^{i\theta}$
De Moivre’s theorem for powers
Roots of unity forming regular polygons
Geometric interpretations in complex plane

Example: “Find all complex numbers $z$ such that $z^3 = -8$” Pattern: Roots of complex numbers Strategy: Convert to polar form, use De Moivre’s

Sequence/Series Problems

Look for: $a_n$, $S_n$, arithmetic, geometric, telescoping Common patterns:

Arithmetic sequences: $a_n = a_1 + (n-1)d$
Geometric sequences: $a_n = a_1 r^{n-1}$
Telescoping series with cancellation
Binomial theorem for expansions

Example: “Find sum of first 20 terms of arithmetic sequence with $a_1 = 3$ and $d = 4$” Pattern: Arithmetic sequence sum Strategy: Use $S_n = \frac{n}{2}(a_1 + a_n)$

Coordinate Geometry Problems

Look for: Points, distances, areas, conic sections Common patterns:

Distance formula and midpoint formula
Line equations (point-slope, slope-intercept)
Circle equations and properties
Area formulas (triangle, polygon)

Example: “Find area of triangle with vertices $(0,0)$, $(3,4)$, $(6,0)$” Pattern: Triangle area calculation Strategy: Use coordinate geometry area formula

Inequality Problems

Look for: Maximum/minimum, optimization, AM-GM, Cauchy-Schwarz Common patterns:

AM-GM inequality for optimization
Cauchy-Schwarz inequality for dot products
Constrained optimization problems
Equality conditions

Example: “Find minimum of $x + \frac{1}{x}$ for $x > 0$” Pattern: AM-GM optimization Strategy: Apply AM-GM to $x$ and $\frac{1}{x}$

⏰ Timing Strategy

AMC 10 Timing (75 minutes, 25 problems)

Problems 1-10: 1-2 minutes each (15 minutes total)
Problems 11-20: 3-4 minutes each (35 minutes total)
Problems 21-25: 5-7 minutes each (30 minutes total)
Review time: 5 minutes
Total: 75 minutes

AMC 12 Timing (75 minutes, 25 problems)

Problems 1-10: 1-2 minutes each (15 minutes total)
Problems 11-20: 3-5 minutes each (40 minutes total)
Problems 21-25: 6-8 minutes each (30 minutes total)
Review time: 5 minutes
Total: 75 minutes

When to Move On

Easy problem (1-10): After 3 minutes
Medium problem (11-20): After 5 minutes
Hard problem (21-25): After 8 minutes
No clear path: After 2-3 different approaches

🚨 Common Traps & How to Avoid Them

Domain/Range Traps

Trap: Forgetting domain restrictions Fix: Always check domains for square roots, logarithms, fractions Example: $\sqrt{x-2} + \log(x+1)$ requires $x \geq 2$ and $x > -1$ → $x \geq 2$

Sign Traps

Trap: Wrong signs in formulas Fix: Double-check signs in Vieta’s formulas, trig identities Example: Sum of roots = $-\frac{b}{a}$, not $\frac{b}{a}$

Extraneous Solution Traps

Trap: Solutions that don’t work in original equation Fix: Always check solutions, especially after squaring Example: Squaring both sides of $\sqrt{x} = x-2$ can introduce extraneous solutions

Unit Traps

Trap: Confusing degrees and radians Fix: Be consistent with units throughout the problem Example: $\sin 30° = \frac{1}{2}$, but $\sin 30$ (radians) ≈ $-0.988$

Notation Traps

Trap: Confusing similar notations Fix: Be careful with $f^{-1}(x)$ vs $\frac{1}{f(x)}$, $\log_a(xy)$ vs $\log_a x \cdot \log_a y$ Example: $f^{-1}(x)$ is inverse function, $\frac{1}{f(x)}$ is reciprocal

📊 Strategic Guessing

Elimination Strategy

Read all choices - Don’t just pick the first reasonable answer
Eliminate obviously wrong - Use domain restrictions, common sense
Test boundary cases - Check extreme values
Use symmetry - Look for patterns in the choices

Estimation Strategy

Round numbers - Use approximations for complex calculations
Check orders of magnitude - Make sure your answer is reasonable
Use known values - Leverage special angles, common log values
Compare to similar problems - Use previous experience

Pattern Recognition in Choices

Look for relationships - Do choices follow a pattern?
Check for symmetry - Are there pairs of similar answers?
Use the problem structure - Does the answer format match the question?
Consider special cases - What happens at boundaries?

🎯 Final Checklist

Before Starting

Read the entire problem - Don’t jump to conclusions
Identify the topic - Which area of precalculus?
Check for constraints - Domain, range, special cases
Look for patterns - Similar to problems you’ve seen?

During Solving

Use the right approach - Don’t overcomplicate
Simplify continuously - Don’t carry complex expressions
Check intermediate steps - Verify calculations
Look for shortcuts - Identities, cancellations, patterns

Before Moving On

Verify your answer - Does it make sense?
Check constraints - Domain, range, special cases
Consider alternatives - Is there a simpler way?
Time check - Should you move on?

If Stuck

Try a different approach - Don’t get stuck in one method
Look for symmetry - Geometric or algebraic patterns
Test special cases - Use simple examples
Check your work - Did you make an error?

🏆 AMC Success Mindset

Confidence

You know the material - Trust your preparation
Patterns are learnable - Most problems follow known types
Practice makes perfect - The more you solve, the better you get

Efficiency

Time is precious - Don’t waste time on dead ends
Move on when stuck - Come back later if time permits
Use shortcuts - Look for elegant solutions

Accuracy

Check your work - Verify answers make sense
Read carefully - Don’t miss important details
Stay focused - Don’t let one problem derail you

Back to: Problem-Solving Tips

💡 Complete Problem-Solving Guide#

🎯 The AMC Mindset#

Core Principles#

Problem-Solving Philosophy#

📋 The 5-Step Process#

Step 1: Read & Understand (30 seconds)#

Step 2: Identify the Pattern (30 seconds)#

Step 3: Choose Strategy (1 minute)#

Step 4: Execute & Simplify (2-4 minutes)#

Step 5: Verify & Move On (30 seconds)#

🔍 Pattern Recognition Guide#

Function Problems#

Polynomial Problems#

Trigonometric Problems#

Logarithm/Exponential Problems#

Complex Number Problems#

Sequence/Series Problems#

Coordinate Geometry Problems#

Inequality Problems#

⏰ Timing Strategy#

AMC 10 Timing (75 minutes, 25 problems)#

AMC 12 Timing (75 minutes, 25 problems)#

When to Move On#

🚨 Common Traps & How to Avoid Them#

Domain/Range Traps#

Sign Traps#

Extraneous Solution Traps#

Unit Traps#

Notation Traps#

📊 Strategic Guessing#

Elimination Strategy#

Estimation Strategy#

Pattern Recognition in Choices#

🎯 Final Checklist#

Before Starting#

During Solving#

Before Moving On#

If Stuck#

🏆 AMC Success Mindset#

Confidence#

Efficiency#

Accuracy#