Patterns, Logic, and Enrichment
Patterns, Functions, and GeneralizationIdentify, extend, generate, compare, and explain visual, numeric, growing, and input-output rules.Logic, Constraint, Deduction, and CombinatoricsUse visible and hidden clues, eliminations, impossibility, truth, parity, invariants, systematic lists, tree diagrams, product-rule readiness, guarantees, counterexamples, contradiction, and nonblocking olympiad-method adapters while routing arithmetic, data, geometry, spatial, symbolic, fraction, measurement, ratio, and pattern-primary work to owning families.Limits, Series, and CalculusEstablish equality between infinite series or products using finite truncations, absolute convergence and explicit limiting arguments. Compute Fourier coefficients and use a justified convergence theorem to evaluate an exact series sum, including endpoint behavior. Derive a limiting proportion of integer structures by comparing finite counts and controlling truncation or boundary errors. Determine the limit of a recursively defined numerical sequence and prove convergence using monotonicity, bounds, contraction or an equivalent exact argument. Estimate a definite or improper integral and prove an explicit relative-error guarantee by valid comparison, transformation or remainder bounds.Complex Numbers and Trigonometric StructuresDerive multiple-angle polynomial identities from a recurrence and prove their validity for all stated indices and input values. Represent all nth roots of unity and derive integer-power sums with complete divisibility and negative-exponent cases.