# Question batches 36,000 questions have bilingual step-by-step derivations. Browse original batches or search and filter in the question explorer. [Search all questions & derivations →](/en/explore/) 1 [2](/en/practice/?page=2#results) … [15](/en/practice/?page=15#results)[Next →](/en/practice/?page=2#results) 360 results · Page 1 of 15 Question batch · 100 questions ## [Complete Finite Neighborhoods, Improvement Witnesses and Attained Bounds](/en/practice/batch-20261015-advance-20261119-r07-optimization-problem-foundations/?from=%2Fen%2Fpractice%2F) Optimization problem batch-20261015-advance-20261119-r07-optimization-problem-foundations Question batch · 100 questions ## [Ordered Gradient Linearity, Negative Directions and Metric Identification](/en/practice/batch-20261015-advance-20261119-r06-gradient-vector-foundations/?from=%2Fen%2Fpractice%2F) Gradient vector batch-20261015-advance-20261119-r06-gradient-vector-foundations Question batch · 100 questions ## [Affine Residual Scaling, Forward Error and Tolerance Conventions](/en/practice/batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations/?from=%2Fen%2Fpractice%2F) Numerical approximation and error batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations Question batch · 100 questions ## [Regular Equality Constraints, Multiplier Residuals and Necessity Boundaries](/en/practice/batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations/?from=%2Fen%2Fpractice%2F) Necessary Lagrange multiplier condition for equality constraints batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations Question batch · 100 questions ## [Exact Diagonal-Quadratic Gradient Steps and Strict Safe-Step Traces](/en/practice/batch-20261015-advance-20261119-r03-gradient-descent-foundations/?from=%2Fen%2Fpractice%2F) Gradient descent batch-20261015-advance-20261119-r03-gradient-descent-foundations Question batch · 100 questions ## [Nonuniform Composite Trapezoids, Affine Exactness and Signed Reference Error](/en/practice/batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations/?from=%2Fen%2Fpractice%2F) Composite trapezoidal quadrature batch-20261015-advance-20261119-r02-trapezoidal-quadrature-foundations Question batch · 100 questions ## [Root-First Newton Updates, Exact Finite Orbits and Convergence Boundaries](/en/practice/batch-20261015-advance-20261119-r01-newton-method-foundations/?from=%2Fen%2Fpractice%2F) Newton method for root finding batch-20261015-advance-20261119-r01-newton-method-foundations Question batch · 100 questions ## [DFA Loop Invariants, Pumping Quantifiers and Every Legal Adversarial Split](/en/practice/batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations/?from=%2Fen%2Fpractice%2F) Pumping lemma for regular languages batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations Question batch · 100 questions ## [Typed Regular Expressions, Complete Bounded Languages and Membership](/en/practice/batch-20261015-advance-20261118-r06-regular-expression-foundations/?from=%2Fen%2Fpractice%2F) Regular expression batch-20261015-advance-20261118-r06-regular-expression-foundations Question batch · 100 questions ## [Prime-Power Extraction, Product Additivity and Sum Cancellation](/en/practice/batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations/?from=%2Fen%2Fpractice%2F) p-adic valuation of integers batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations Question batch · 100 questions ## [Factorial Congruence, Complete Inverse Pairing and the Modulus-Two Exception](/en/practice/batch-20261015-advance-20261118-r04-wilson-theorem-foundations/?from=%2Fen%2Fpractice%2F) Wilson theorem batch-20261015-advance-20261118-r04-wilson-theorem-foundations Question batch · 100 questions ## [Precommitted Finite Intervals, Pointwise Coverage and Selection Boundaries](/en/practice/batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations/?from=%2Fen%2Fpractice%2F) Confidence-interval coverage in a finite model batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations Question batch · 100 questions ## [Finite-Valued IID Strong-Law Scope and Fixed-n Audits](/en/practice/batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations/?from=%2Fen%2Fpractice%2F) Strong law of large numbers for finite-valued IID sequences batch-20261015-advance-20261118-r02-strong-law-large-numbers-finite-valued-iid-foundations Question batch · 100 questions ## [Declared IID CLT Premises, Exact Sum Moments and Finite-Lattice Boundaries](/en/practice/batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations/?from=%2Fen%2Fpractice%2F) Central limit theorem for finite-variance IID sequences batch-20261015-advance-20261118-r01-central-limit-theorem-finite-variance-iid-foundations Question batch · 100 questions ## [Inclusion-Maximal Connected Subspaces and Finite Clopen Partitions](/en/practice/batch-20261015-advance-20261117-r07-topological-connected-component-foundations/?from=%2Fen%2Fpractice%2F) Connected component of a topological space batch-20261015-advance-20261117-r07-topological-connected-component-foundations Question batch · 100 questions ## [Explicit Interval and Box Paths with Full Continuity Obligations](/en/practice/batch-20261015-advance-20261117-r06-path-connected-space-foundations/?from=%2Fen%2Fpractice%2F) Path-connected space batch-20261015-advance-20261117-r06-path-connected-space-foundations Question batch · 100 questions ## [Surjective Fibers, Saturated Opens and Final-Topology Scope](/en/practice/batch-20261015-advance-20261117-r05-quotient-topology-foundations/?from=%2Fen%2Fpractice%2F) Quotient topology batch-20261015-advance-20261117-r05-quotient-topology-foundations Question batch · 100 questions ## [Complete Rectangle-Union Topologies and Coordinate Continuity](/en/practice/batch-20261015-advance-20261117-r04-finite-product-topology-foundations/?from=%2Fen%2Fpractice%2F) Finite product topology batch-20261015-advance-20261117-r04-finite-product-topology-foundations Question batch · 100 questions ## [Whole-Domain Error Bounds and a Fixed-Approximant Continuity Transfer](/en/practice/batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations/?from=%2Fen%2Fpractice%2F) A uniform limit of continuous functions is continuous batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations Question batch · 100 questions ## [Absolute Ratio Limits, Domination and Limit-One Inconclusiveness](/en/practice/batch-20261015-advance-20261117-r02-ratio-test-series-foundations/?from=%2Fen%2Fpractice%2F) Ratio test for series batch-20261015-advance-20261117-r02-ratio-test-series-foundations Question batch · 100 questions ## [Nonnegative Tail Comparisons and Correct Convergence Directions](/en/practice/batch-20261015-advance-20261117-r01-series-comparison-test-foundations/?from=%2Fen%2Fpractice%2F) Comparison tests for series batch-20261015-advance-20261117-r01-series-comparison-test-foundations Question batch · 100 questions ## [Qualified Units, First Positive Returns and Phi Divisibility](/en/practice/batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations/?from=%2Fen%2Fpractice%2F) Multiplicative order modulo n batch-20261015-advance-20261116-r07-multiplicative-order-mod-n-foundations Question batch · 100 questions ## [All Topological Orders, Source-Removal Traces and Explicit Cycles](/en/practice/batch-20261015-advance-20261116-r06-topological-ordering-dag-foundations/?from=%2Fen%2Fpractice%2F) Topological ordering of a directed acyclic graph batch-20261015-advance-20261116-r06-topological-ordering-dag-foundations Question batch · 100 questions ## [Complete Fibers, Positive-Mass Conditional Laws and Sufficiency](/en/practice/batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations/?from=%2Fen%2Fpractice%2F) Sufficient statistic in a finite model batch-20261015-advance-20261116-r05-finite-sufficient-statistic-foundations 1 [2](/en/practice/?page=2#results) … [15](/en/practice/?page=15#results)[Next →](/en/practice/?page=2#results) 360 results · Page 1 of 15