# 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/) [← Previous](/en/practice/?page=11#results)[1](/en/practice/?page=1#results) … [11](/en/practice/?page=11#results) 12 [13](/en/practice/?page=13#results) … [15](/en/practice/?page=15#results)[Next →](/en/practice/?page=13#results) 360 results · Page 12 of 15 Question batch · 100 questions ## [Euclidean Division and Signed Integer Protocols](/en/practice/batch-20261009-advance-20261013-r02-integer-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Integer batch-20261009-advance-20261013-r02-integer-foundations Question batch · 100 questions ## [Positive Right-Triangle Sides and Square Residuals](/en/practice/batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Pythagorean theorem batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations Question batch · 100 questions ## [Complete Typed Maps, Ranges, and Query Guards](/en/practice/batch-20261008-advance-20261012-r07-function-mapping-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Function mapping batch-20261008-advance-20261012-r07-function-mapping-foundations Question batch · 100 questions ## [Directed Components, Principal Lengths, and Midpoints](/en/practice/batch-20261008-advance-20261012-r06-euclidean-length-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Euclidean length batch-20261008-advance-20261012-r06-euclidean-length-foundations Question batch · 100 questions ## [Exact Angle Qualification and Labeled Right Vertices](/en/practice/batch-20261008-advance-20261012-r05-right-triangle-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Right triangle batch-20261008-advance-20261012-r05-right-triangle-foundations Question batch · 100 questions ## [Typed Membership and Finite Predicate Extensions](/en/practice/batch-20261008-advance-20261012-r04-set-membership-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Set membership batch-20261008-advance-20261012-r04-set-membership-foundations Question batch · 100 questions ## [Closed Truths, Open Bindings, and Candidate Reports](/en/practice/batch-20261008-advance-20261012-r03-proposition-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Proposition batch-20261008-advance-20261012-r03-proposition-foundations Question batch · 100 questions ## [Tagged Choice Counts and Actual Branch Widths](/en/practice/batch-20261008-advance-20261012-r02-counting-principle-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Basic counting principles batch-20261008-advance-20261012-r02-counting-principle-foundations Question batch · 100 questions ## [Finite Induction Scope and Separate Base/Step Defects](/en/practice/batch-20261008-advance-20261012-r01-mathematical-induction-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Mathematical induction batch-20261008-advance-20261012-r01-mathematical-induction-foundations Question batch · 100 questions ## [Finite Principal-Filter Images and Direction Guards](/en/practice/batch-20261008-advance-20261011-r07-tends-to-relation-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Tends-to relation batch-20261008-advance-20261011-r07-tends-to-relation-foundations Question batch · 100 questions ## [Possible Outcomes, Recording Fibers, and Completeness](/en/practice/batch-20261008-advance-20261011-r06-sample-space-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Sample space batch-20261008-advance-20261011-r06-sample-space-foundations Question batch · 100 questions ## [Inclusion Obstructions and Strictness Witnesses](/en/practice/batch-20261008-advance-20261011-r05-subset-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Subset batch-20261008-advance-20261011-r05-subset-foundations Question batch · 100 questions ## [Finite Set Extension, Equality, and Powersets](/en/practice/batch-20261008-advance-20261011-r04-set-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Set batch-20261008-advance-20261011-r04-set-foundations Question batch · 100 questions ## [Environments, Simultaneous Substitution, and Original Domains](/en/practice/batch-20261008-advance-20261011-r03-variable-expression-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Variables and algebraic expressions batch-20261008-advance-20261011-r03-variable-expression-foundations Question batch · 100 questions ## [Exact Rational Operations and Natural Eligibility](/en/practice/batch-20261008-advance-20261011-r02-arithmetic-operations-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Arithmetic operations batch-20261008-advance-20261011-r02-arithmetic-operations-foundations Question batch · 100 questions ## [Zero-Inclusive Naturals and Partial Predecessors](/en/practice/batch-20261008-advance-20261011-r01-natural-number-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Natural number batch-20261008-advance-20261011-r01-natural-number-foundations Question batch · 100 questions ## [Finite Predicate Partitions and Ordered Quantifier Counts](/en/practice/batch-20261008-advance-20261010-r07-quantifier-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Quantifiers and explicit domains batch-20261008-advance-20261010-r07-quantifier-foundations Question batch · 100 questions ## [Centered Probability Contributions and Positive Support](/en/practice/batch-20261008-advance-20261010-r06-variance-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Variance in finite probability models batch-20261008-advance-20261010-r06-variance-foundations Question batch · 100 questions ## [Shared-Model Expectations and Positive-Mass Recovery](/en/practice/batch-20261008-advance-20261010-r05-expectation-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Expectation in finite models batch-20261008-advance-20261010-r05-expectation-foundations Question batch · 100 questions ## [Finite Conditional Laws and Candidate Defects](/en/practice/batch-20261008-advance-20261010-r04-conditional-probability-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Conditional probability in finite models batch-20261008-advance-20261010-r04-conditional-probability-foundations Question batch · 100 questions ## [Exact Squared Distances, Signs, and Tie Groups](/en/practice/batch-20261008-advance-20261010-r03-distance-formula-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Euclidean distance in an equal-unit orthonormal frame batch-20261008-advance-20261010-r03-distance-formula-foundations Question batch · 100 questions ## [Finite Set Cardinalities and Ordered Partitions](/en/practice/batch-20261008-advance-20261010-r02-set-operations-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Set operations batch-20261008-advance-20261010-r02-set-operations-foundations Question batch · 100 questions ## [Complete Polynomial Coefficients and Ordered Values](/en/practice/batch-20261008-advance-20261010-r01-polynomial-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Univariate polynomials over the rationals batch-20261008-advance-20261010-r01-polynomial-foundations Question batch · 100 questions ## [Real 2×2 Characteristic Polynomial and Residual Depth](/en/practice/batch-20261007-advance-20261009-r07-eigenvalue-foundations/?from=%2Fen%2Fpractice%2F%3Fpage%3D12) Real eigenvalues and nonzero eigenvectors of 2×2 matrices batch-20261007-advance-20261009-r07-eigenvalue-foundations [← Previous](/en/practice/?page=11#results)[1](/en/practice/?page=1#results) … [11](/en/practice/?page=11#results) 12 [13](/en/practice/?page=13#results) … [15](/en/practice/?page=15#results)[Next →](/en/practice/?page=13#results) 360 results · Page 12 of 15