Dynamic Programming
Turn a huge search into small states by recording what the past can no longer change.
View deck and teaching arcSerious problems, every other Saturday.
COSMO STEM Circles meet biweekly to leave room for the kind of follow-up hard problems deserve: a first attempt, a second look, and a reasoned explanation.
What is coming next
Each topic connects to a public deck, so students can preview the ideas and return to them between meetings.
-
Geometry moves: symmetry, transformations, and invariants
AMC-style geometry begins with what a transformation preserves, then turns those observations into proof.
Open the Transformations and Symmetry deck -
Number theory: modular arithmetic and Chinese remainders
Use cycles, congruences, and constructive thinking to make huge-number questions manageable.
Open the Modular Arithmetic deck -
Counting without double-counting
Build reliable cases, complements, and bijections for AMC and AIME counting problems.
Open the Combinatorics deck -
Algebraic structure: inequalities and polynomial roots
Look for equality cases, factors, and transformations before reaching for a calculation.
Open the Inequalities deck -
Probability: expected value, cases, and conditional information
Translate a situation into outcomes, weights, and a conclusion that can be checked.
Open the Expected Value deck -
Problem-solving tools: invariants, coloring, and proof
Track what cannot change to turn an impossible-looking search into a short argument.
Open the Invariants deck
Past study topics
This archive collects COSMO's published AMC/AIME-style mathematics materials. Publication dates identify the resource release, not individual attendance or a record of a live session.
42 problem-solving decks Search every COSMO deck
Mathematical Modeling
Turn a situation into variables, assumptions, relationships, and a testable conclusion.
View deck and teaching arcError-Correcting Codes
Add structured redundancy so corrupted messages can be detected or corrected.
View deck and teaching arcExponents and Radicals
Factor common powers, respect domains, and let exponent laws expose the simplest equation.
View deck and teaching arcPower of a Point
Translate a circle diagram into one invariant product: secants, tangents, and the lengths that keep multiplying.
View deck and teaching arcInvariants
Parity, coloring, preserved quantities, and the strategy of proving impossibility without tracing every move.
View deck and teaching arcCyclic Geometry
Let right angles, supplementary angles, and intersecting chords turn a circle diagram into equations.
View deck and teaching arcTelescoping Products
Factor each term before multiplying; a long chain can collapse to two endpoints.
View deck and teaching arcApplied Optimization
Decision variables, constraints, objectives, feasible choices, and why a locally good move can fail globally.
View deck and teaching arcFunctional Equations
Use substitutions, normalization, and a hidden additive rule to reconstruct an unknown function.
View deck and teaching arcMarkov Chains
Model systems where the next state depends on the present state, not the full path.
View deck and teaching arcGenerating Functions
Store an entire sequence as coefficients of one formal power series.
View deck and teaching arcRandom Walks
Repeated random steps produce patterns in position, spread, and hitting probability.
View deck and teaching arcRecurrence Relations
Define a sequence by how each new term depends on earlier ones.
View deck and teaching arcBayes' Theorem
Reverse a conditional probability by combining evidence with prior prevalence.
View deck and teaching arcMatchings
Pair vertices without reuse and use neighborhood conditions to test whether everyone can be matched.
View deck and teaching arcConditional Probability
Update the sample space after learning that an event occurred.
View deck and teaching arcExpected Value
Outcome tables, net payoff, fair games, linearity, and what a long-run average can and cannot predict.
View deck and teaching arcGraph Coloring
Assign limited labels so adjacent objects never conflict.
View deck and teaching arcTransformations and Symmetry
Classify motions by what they preserve and use composition to reveal symmetry structure.
View deck and teaching arcCatalan Numbers
One sequence counts balanced parentheses, triangulations, and paths that avoid crossing a boundary.
View deck and teaching arcPlanar Graphs
Use faces and Euler's formula to prove that some networks cannot be drawn without crossings.
View deck and teaching arcIntroductory Cryptography
Use modular inverses and one-way arithmetic to separate encryption from decryption.
View deck and teaching arcVectors and Matrices
Use coordinates and matrices to represent directions and transformations.
View deck and teaching arcBase Representations
Place value is a polynomial in the base, so changing base changes notation, not quantity.
View deck and teaching arcBinomial Coefficients
Combinations count choices, paths, and coefficients through the same recurrence.
View deck and teaching arcFermat and Euler Theorems
Use multiplicative cycles to reduce enormous exponents modulo n.
View deck and teaching arcComplex Numbers
Treat multiplication as scaling and rotation in the plane.
View deck and teaching arcTrees and Spanning Trees
A tree is the minimal connected network: every edge is necessary and every pair has one path.
View deck and teaching arcChinese Remainder Theorem
Reconstruct one number from compatible remainder clues.
View deck and teaching arcInclusion-Exclusion
Add overlapping counts, then repair the overcount systematically.
View deck and teaching arcLinear Diophantine Equations
Decide when ax + by = c has integer solutions and describe every solution.
View deck and teaching arcPolynomial Roots
Connect factors, roots, coefficients, and graphs without treating them as separate tricks.
View deck and teaching arcGraph Theory
Vertices, edges, degree, Euler routes, trees, and the local counts that decide global network questions.
View deck and teaching arcPrime Factorization
Break a number into prime building blocks and use exponents to compare divisibility.
View deck and teaching arcPigeonhole Principle
More objects than containers force at least one collision.
View deck and teaching arcSequences and Series
Recognize arithmetic and geometric structure, then justify finite and infinite sums.
View deck and teaching arcEuclidean Algorithm
Replace a large gcd problem with smaller remainder problems without changing the answer.
View deck and teaching arcCombinatorics
Cases, complements, bijections, and the bookkeeping habits that prevent hidden overcounting.
View deck and teaching arcModular Arithmetic
Remainders, cycles, congruences, and why huge-number questions often need only a small repeating pattern.
View deck and teaching arcInequalities
Compare quantities using structure, equality cases, and transformations that preserve order.
View deck and teaching arcProof and Induction
Evidence versus proof, the precise role of an inductive hypothesis, and how a chain reaches every case.
View deck and teaching arcA usable plan for teachers
A 75-minute circle can open with one problem, move into small-group attempts, pause for a targeted explanation, test the idea on an extension, and end with one written reflection. The deck supplies the content; the room supplies the thinking.