🧠 Problem-Solving Tips — Detailed Strategies

Comprehensive strategies for efficient AMC algebra problem solving.

🔍 Problem Analysis Checklist

Before Starting

  • Read the problem twice — Understand what’s being asked
  • Identify the type — Factoring, equation solving, system, etc.
  • Check for restrictions — Domain, range, parameter conditions
  • Look for patterns — Common problem types or techniques

Domain First Strategy

When to use: Any equation with fractions, radicals, or logarithms

Checklist:

  • Denominators — Set equal to zero, exclude those values
  • Radicands — Must be non-negative for real solutions
  • Log arguments — Must be positive
  • Write restrictions — Note all excluded values

Example: Solve $\frac{x+1}{x-2} = \frac{x-3}{x+4}$

  • Domain: $x \neq 2, -4$
  • Cross-multiply: $(x+1)(x+4) = (x-3)(x-2)$
  • Expand: $x^2 + 5x + 4 = x^2 - 5x + 6$
  • Solve: $10x = 2$, so $x = \frac{1}{5}$
  • Check: $\frac{1}{5} \neq 2, -4$ ✓

🎯 Solution Techniques

Extraneous Solutions Strategy

When to use: After squaring both sides or cross-multiplying

Checklist:

  • Identify the operation — Squaring, cross-multiplying, etc.
  • Solve the equation — Find all potential solutions
  • Substitute back — Check each solution in original equation
  • Eliminate extraneous — Remove solutions that don’t work

Example: Solve $\sqrt{x+1} = x-1$

  • Square both sides: $x+1 = (x-1)^2 = x^2 - 2x + 1$
  • Rearrange: $x^2 - 3x = 0$, so $x = 0, 3$
  • Check $x = 0$: $\sqrt{1} = -1$ ✗ (extraneous)
  • Check $x = 3$: $\sqrt{4} = 2$ ✓
  • Answer: $x = 3$

Symmetry Substitutions

When to use: Cyclic expressions, $x + \frac{1}{x}$, $x^2$ patterns

Common substitutions:

  • $u = x + \frac{1}{x}$ — For expressions involving $x$ and $\frac{1}{x}$
  • $t = x^2$ — For even powers of $x$
  • $u = x + k$ — To complete the square
  • $u = \sqrt{x}$ — For radical expressions

Example: Solve $x^2 + \frac{1}{x^2} = 7$

  • Let $u = x + \frac{1}{x}$
  • Then $u^2 = x^2 + 2 + \frac{1}{x^2}$, so $x^2 + \frac{1}{x^2} = u^2 - 2$
  • Equation becomes: $u^2 - 2 = 7$, so $u^2 = 9$, so $u = \pm 3$
  • Solve $x + \frac{1}{x} = 3$: $x^2 - 3x + 1 = 0$, so $x = \frac{3 \pm \sqrt{5}}{2}$
  • Solve $x + \frac{1}{x} = -3$: $x^2 + 3x + 1 = 0$, so $x = \frac{-3 \pm \sqrt{5}}{2}$

Discriminant Playbook

When to use: Parameter problems, counting solutions

Checklist:

  • Identify the equation — Usually quadratic in form
  • Calculate discriminant — $\Delta = b^2 - 4ac$
  • Analyze cases — $\Delta > 0$, $\Delta = 0$, $\Delta < 0$
  • Apply conditions — What does the problem require?

Example: For what values of $k$ does $x^2 + kx + 1 = 0$ have no real solutions?

  • Discriminant: $\Delta = k^2 - 4(1)(1) = k^2 - 4$
  • No real solutions when $\Delta < 0$: $k^2 - 4 < 0$
  • Solve: $k^2 < 4$, so $-2 < k < 2$

⚡ Optimization Strategies

Factor Before Expand

When to use: Complex expressions, looking for patterns

Checklist:

  • Look for common factors — Factor out GCF first
  • Check for patterns — Difference of squares, perfect squares, etc.
  • Group terms — Look for grouping opportunities
  • Use identities — Apply known factoring patterns

Example: Simplify $(x^2 - 1)(x^2 + 1) - (x^2 - 1)^2$

  • Factor out common factor: $(x^2 - 1)[(x^2 + 1) - (x^2 - 1)]$
  • Simplify: $(x^2 - 1)(2) = 2(x^2 - 1) = 2(x-1)(x+1)$

Sign and Bounds

When to use: Optimization problems, finding extrema

Checklist:

  • Identify the expression — What needs to be optimized?
  • Check for symmetry — Look for even/odd patterns
  • Use AM-GM — For positive expressions
  • Consider boundaries — Check extreme values

Example: Find minimum value of $x^2 + \frac{1}{x^2}$ for $x > 0$

  • Use AM-GM: $\frac{x^2 + \frac{1}{x^2}}{2} \geq \sqrt{x^2 \cdot \frac{1}{x^2}} = 1$
  • So $x^2 + \frac{1}{x^2} \geq 2$
  • Equality when $x^2 = \frac{1}{x^2}$, so $x = 1$
  • Minimum value is $2$

🎯 Timing Strategies

Time Allocation

  • Easy problems (1-2 minutes): Basic factoring, simple equations
  • Medium problems (3-5 minutes): Systems, quadratics, rational equations
  • Hard problems (5-8 minutes): Advanced techniques, parameter analysis
  • Skip threshold: 3 minutes without progress

Decision Tree

  1. Can I solve this quickly? → Yes: Solve it
  2. Do I see a clear approach? → Yes: Try it for 2-3 minutes
  3. Am I making progress? → Yes: Continue for 1 more minute
  4. Still stuck? → Skip and come back later

Common Time Traps

  • Over-complicating — Look for simple approaches first
  • Getting stuck — Don’t spend more than 3 minutes on one problem
  • Perfectionism — Good enough is better than perfect
  • Second-guessing — Trust your first approach if it’s reasonable

📝 Practice Checklist

Before Contests

  • Review all strategies — Ensure you know them by heart
  • Practice decision trees — Learn when to use each technique
  • Time yourself — Build speed and efficiency
  • Review common pitfalls — Avoid known mistakes

During Contests

  • Follow checklists — Don’t skip steps
  • Use decision trees — Choose the right approach
  • Manage time — Don’t get stuck on one problem
  • Stay calm — Don’t panic if you don’t know something

After Contests

  • Review mistakes — Learn from errors
  • Identify weak areas — Focus study on these
  • Practice more — Build confidence and speed
  • Update strategies — Refine your approach

Back: Tips Overview | Next: Algebra Basics