PS / Synthesis
Problem-Solving Strategies
Summit Challenge
Choose representations and arguments that turn difficult-looking problems into tractable ones.
Preparation: Algebra, geometry, counting and the logarithm definition. Later stages combine subjects.
Current coverage: 12 strategy stages: reversal, invariants, extremal reasoning, telescoping, induction and connections.
LEVEL 1 · DIFFICULTY 1/5 within this course
Useful First Moves
Reverse processes, track invariants and force repetition.
Explore 3 knowledge islands
Choose an island to read its lesson.
3 stages · 24 library problems · 108 checked variations · 2 × 6 assessment questions
Read the lessons →LEVEL 2 · DIFFICULTY 2/5 within this course
Exploit Structure
Balance extremes, cancel terms and justify patterns.
Explore 3 knowledge islands
Choose an island to read its lesson.
3 stages · 24 library problems · 108 checked variations · 2 × 6 assessment questions
Read the lessons →LEVEL 3 · DIFFICULTY 3/5 within this course
Change the Representation
Use residues, bijections and shared-altitude geometry.
Explore 3 knowledge islands
Choose an island to read its lesson.
3 stages · 24 library problems · 108 checked variations · 2 × 6 assessment questions
Read the lessons →LEVEL 4 · DIFFICULTY 4/5 within this course
Combine Ideas
Recover totals, telescope logarithms and transform recurrences.
Explore 3 knowledge islands
Choose an island to read its lesson.
3 stages · 24 library problems · 108 checked variations · 2 × 6 assessment questions
Read the lessons →Progress earned through practice
Earn two independent practice stars and one application star, then complete the library skill check for that stage. The check covers the stage’s library prompts, so changing numbers alone is not enough. Hints help you learn; your existing stars and earned rewards stay yours. Clear every stage to clear a level.
Every stage includes an explanation, two worked examples and eight library practice problems, plus checked variations with changing numbers and applications. Each level also has a separate unit test and a later review form. Parents see actual attempts, hint use and results.
These are independently authored core-skills courses, with the current scope listed for each course. They do not reproduce complete textbook contents. Difficulty describes the teaching sequence within each course; course preparation matters when comparing subjects. This site is not affiliated with or endorsed by AoPS. Results reflect these exercises and are not standardized scores. Content release math-courses-2026.09.27-university-guides-v2.