10 Tips
1. Always hunt for structure before computing
AoPS problems are designed so brute force explodes. Look for invariants, symmetry, parity, or monotonicity first, especially in NT and algebra sections.
2. Rephrase the problem in your own math language
Translate wordy statements into equations, inequalities, or set relations. Vol. 1 drills this heavily in algebra and number theory.
3. Exploit small cases strategically
Try n = 1, 2, 3 only to discover patterns or eliminate cases, not to guess answers. Vol. 2 combinatorics rewards this a lot.
4. Use bounds aggressively
Many AoPS inequalities and NT problems are solved by tight upper or lower bounds rather than exact values.
5. Check parity and mod early
Before deep work, test mod 2, mod 3, mod 4, or mod 9. Vol. 1 NT problems often collapse instantly with this.
6. Convert geometry to algebra when stuck
Coordinate bash, vectors, or complex numbers are often intended even if the problem looks synthetic.
7. Reverse-engineer the answer choices
For AMC-style problems, plug answers back in or analyze constraints implied by choices. AoPS explicitly trains this.
8. Look for bijections in counting
Vol. 2 combinatorics problems often become trivial once you re-interpret the count as something else.
9. Write one clean solution path, not many partial ones
AoPS problems punish indecision. Commit to a strategy after 3 to 5 minutes or reset completely.
10. Assume the problem is elegant
If your solution is messy, it is almost certainly wrong or incomplete. AoPS problems nearly always have a clean core idea.
The AoPS Volume 1 and AoPS Volume 2 books are great for contest mathematics.
AoPS Volume 1 Breakdown (Foundations)
Tier S
Number Theory
- Divisibility, modular arithmetic, remainders
- Backbone of AMC 10–AIME
- Many problems reduce to “just mod something”
Algebra
- Manipulating expressions, inequalities, functional thinking
- Almost every contest problem touches algebra somewhere
Counting & Probability
- Casework, complementary counting, basic bijections
- Huge AMC payoff, especially mid-to-late problems
Tier A (very high value)
Geometry
- Angle chasing, similar triangles, area ratios
- Less advanced than Vol. 2 geometry but still vital
Sequences
- Patterns, recursive definitions, telescoping
- Common AMC 10/12 and Euclid Part A
Tier B (useful but situational)
Polynomials
- Roots, Vieta, plugging in smart values
- Appears often, but usually in contained ways
Coordinate Geometry
- Distance, slopes, midpoints
- High yield if you are efficient; otherwise time-sink
AoPS Volume 2 (Advanced)
Tier S (non-negotiable for AIME / Euclid)
Advanced Number Theory
- Orders, LTE ideas, Diophantine flavor
- Dominates AIME and Euclid Part B
Advanced Counting
- Inclusion–exclusion, recursion, invariants
- Separates strong solvers from average ones
Tier A (very strong payoff)
- Functional Equations
- Plug-in tactics, symmetry, injectivity
- Extremely common in harder contests
Advanced Algebra
- Inequalities, transformations, bounding
- Especially relevant for Euclid and Olympiad-style problems
Tier B (important but specialized)
- Advanced Geometry
- Power of a point, cyclic quadrilaterals
- High payoff, but requires practice density
Polynomials (Advanced)
- Factor chasing, symmetric polynomials
- Appears less frequently but can be decisivesivedvanced)
- Factor chasing, symmetric polynomials
- Appears less frequently but can be decisive