Mastering Efficient Problem Solving in Competitive Math
A Guide to Strategic Thinking, Case Reduction, and Symmetry Exploitation

1. Diagnose the Problem Type

Cues: Keywords like “how many,” “maximize,” or “prove for all $n$.”
Goal: Identify the domain (algebra, combinatorics, etc.) and complexity.
Example:

  • “Find all integer solutions” → Number theory (modular arithmetic, Bézout’s identity).
  • “How many ways…” → Combinatorics (permutations, inclusion-exclusion).

2. Exploit Symmetry

Cues: Interchangeable variables (e.g., $x, y, z$), homogeneous equations, geometric figures, combinatorics scenarios.
Strategies:

  • Consider the symmetry: Think of group theory: given a problem environment, try to find a set and a group action that preserves some invariants.
  • Group identical cases: For repeated objects, use multinomial coefficients instead of enumerating permutations.
  • Assume equality first: Test $x = y$ in symmetric equations.

Example:

  • Problem: Solve $x + y + z = 6$, $xy + yz + zx = 11$.
    • Assume $x = y$: $2x + z = 6$ and $x^2 + 2xz = 11$. Solve for $x$, then $z$.

3. Leverage Bounds and Constraints

Cues: Inequalities, divisibility conditions, or fixed totals (e.g., $a + b + c = 1$).
Strategies:

  • Bound variables: Restrict values using constraints (e.g., $x > 0$ → $x \geq 1$).
  • Optimize with AM-GM: For problems like “maximize $xy$ given $x + y = 12$,” use $xy \leq \left(\frac{x+y}{2}\right)^2$.

Example:

  • Problem: “Find all positive integers $x, y$ with $x + y = 10$.”
    • Bounds: $1 \leq x \leq 9$, $y = 10 - x$. Only 9 cases needed.

4. Use Mathematical Identities and Theorems

Cues: Repeated terms, known theorem keywords (e.g., “collinear,” “cyclic quadrilateral”).
Strategies:

  • Factor or expand: Rewrite expressions using identities (e.g., $a^2 - b^2 = (a-b)(a+b)$).
  • Apply theorems: Ceva’s Theorem (collinearity), Chinese Remainder Theorem (modular systems).

Example:

  • Problem: Solve $17x \equiv 5 \mod 23$.
    • Use the Extended Euclidean Algorithm to find $x \equiv 17^{-1} \cdot 5 \mod 23$.

5. Avoid Brute Force with Patterns

Cues: Large numbers, recursive definitions, or “prove existence.”
Strategies:

  • Modular cycles: For $7^{123} \mod 10$, note $7^1=7$, $7^2=9$, $7^3=3$, $7^4=1$ (cycle of 4).
  • Pigeonhole Principle: Prove existence without enumeration (e.g., 367 people → 2 share a birthday).

Example:

  • Problem: “Prove in any 5-card hand, two cards share a suit.”
    • 5 cards (pigeons) and 4 suits (holes) → At least two in one suit.

6. Divide and Conquer

Cues: Recursive structures, multiple variables, or “for all $n$.”
Strategies:

  • Case analysis: Split by parity (even/odd) or residue classes.
  • Iterative computation: Precompute small cases for sequences (e.g., Fibonacci).

Example:

  • Problem: “Prove $n^2 - n$ is even for all integers $n$.”
    • Split into cases: If $n$ is even, $n^2$ and $n$ are even. If $n$ is odd, $n^2$ and $n$ are odd.

7. Verify and Refine

Cues: “Find all solutions,” edge cases, or bounds.
Strategies:

  • Back-substitute: Plug answers into original equations.
  • Test extremes: Check $n = 0$, $x = 1$, or degenerate shapes.

Example:

  • Problem: Solve $\sqrt{x + 3} = x - 1$.
    • Squaring gives $x + 3 = x^2 - 2x + 1$ → $x^2 - 3x - 2 = 0$. Check solutions $x = 2$ (valid) and $x = -1$ (invalid).

Quick-Reference Summary

  1. Diagnose the Problem: Identify domain and keywords.
  2. Exploit Symmetry: Assume equality, group identical cases.
  3. Use Bounds: Restrict variables with constraints.
  4. Apply Theorems/Identities: Factor, expand, or invoke theorems.
  5. Avoid Brute Force: Modular cycles, Pigeonhole Principle.
  6. Divide and Conquer: Split into cases or precompute.
  7. Verify: Back-substitute and test edge cases.