OCME · Bilingual mathematics & derivations

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 →

Question batch · 100 questions

Euclidean Division and Signed Integer Protocols

Integer

batch-20261009-advance-20261013-r02-integer-foundations

Question batch · 100 questions

Positive Right-Triangle Sides and Square Residuals

Pythagorean theorem

batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations

Question batch · 100 questions

Complete Typed Maps, Ranges, and Query Guards

Function mapping

batch-20261008-advance-20261012-r07-function-mapping-foundations

Question batch · 100 questions

Directed Components, Principal Lengths, and Midpoints

Euclidean length

batch-20261008-advance-20261012-r06-euclidean-length-foundations

Question batch · 100 questions

Exact Angle Qualification and Labeled Right Vertices

Right triangle

batch-20261008-advance-20261012-r05-right-triangle-foundations

Question batch · 100 questions

Typed Membership and Finite Predicate Extensions

Set membership

batch-20261008-advance-20261012-r04-set-membership-foundations

Question batch · 100 questions

Closed Truths, Open Bindings, and Candidate Reports

Proposition

batch-20261008-advance-20261012-r03-proposition-foundations

Question batch · 100 questions

Tagged Choice Counts and Actual Branch Widths

Basic counting principles

batch-20261008-advance-20261012-r02-counting-principle-foundations

Question batch · 100 questions

Finite Induction Scope and Separate Base/Step Defects

Mathematical induction

batch-20261008-advance-20261012-r01-mathematical-induction-foundations

Question batch · 100 questions

Finite Principal-Filter Images and Direction Guards

Tends-to relation

batch-20261008-advance-20261011-r07-tends-to-relation-foundations

Question batch · 100 questions

Possible Outcomes, Recording Fibers, and Completeness

Sample space

batch-20261008-advance-20261011-r06-sample-space-foundations

Question batch · 100 questions

Inclusion Obstructions and Strictness Witnesses

Subset

batch-20261008-advance-20261011-r05-subset-foundations

Question batch · 100 questions

Finite Set Extension, Equality, and Powersets

Set

batch-20261008-advance-20261011-r04-set-foundations

Question batch · 100 questions

Environments, Simultaneous Substitution, and Original Domains

Variables and algebraic expressions

batch-20261008-advance-20261011-r03-variable-expression-foundations

Question batch · 100 questions

Exact Rational Operations and Natural Eligibility

Arithmetic operations

batch-20261008-advance-20261011-r02-arithmetic-operations-foundations

Question batch · 100 questions

Zero-Inclusive Naturals and Partial Predecessors

Natural number

batch-20261008-advance-20261011-r01-natural-number-foundations

Question batch · 100 questions

Finite Predicate Partitions and Ordered Quantifier Counts

Quantifiers and explicit domains

batch-20261008-advance-20261010-r07-quantifier-foundations

Question batch · 100 questions

Centered Probability Contributions and Positive Support

Variance in finite probability models

batch-20261008-advance-20261010-r06-variance-foundations

Question batch · 100 questions

Shared-Model Expectations and Positive-Mass Recovery

Expectation in finite models

batch-20261008-advance-20261010-r05-expectation-foundations

Question batch · 100 questions

Finite Conditional Laws and Candidate Defects

Conditional probability in finite models

batch-20261008-advance-20261010-r04-conditional-probability-foundations

Question batch · 100 questions

Exact Squared Distances, Signs, and Tie Groups

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

Set operations

batch-20261008-advance-20261010-r02-set-operations-foundations

Question batch · 100 questions

Complete Polynomial Coefficients and Ordered Values

Univariate polynomials over the rationals

batch-20261008-advance-20261010-r01-polynomial-foundations

Question batch · 100 questions

Real 2×2 Characteristic Polynomial and Residual Depth

Real eigenvalues and nonzero eigenvectors of 2×2 matrices

batch-20261007-advance-20261009-r07-eigenvalue-foundations