Mathematics

Apply the division algorithm to quotient, remainder, and divisibility problems.

Prove the gcd-preserving step used by the Euclidean algorithm.

Use the extended Euclidean algorithm to express a gcd as a linear combination.

Determine and describe all integer solutions of a linear Diophantine equation.

Prove the existence of prime factorizations by strong induction.

Prove Euclid’s lemma using Bézout’s identity.

Prove the uniqueness of prime factorization using Euclid’s lemma.

Prove that there are infinitely many primes.

Prove that congruence modulo n is an equivalence relation compatible with addition and multiplication.

Simplify sums, products, and powers using modular arithmetic.

Derive the gcd criterion for solvability of a linear congruence.

Compute a modular inverse using the extended Euclidean algorithm.

Prove Fermat’s little theorem using a permutation of nonzero residue classes.

Prove Euler’s theorem using a reduced residue system.

Solve simultaneous congruences using the elementary Chinese remainder theorem.

Use quadratic residues modulo small integers to prove that a specified Diophantine equation has no solution.

Prove a triangle theorem using an appropriate congruence criterion.

Prove that the perpendicular bisectors of a triangle meet at its circumcenter.

Prove the Pythagorean theorem using similar triangles.

Prove the inscribed-angle theorem for a circle.

Prove a collinearity or concurrency result using coordinates or vectors.

Represent translations, rotations, and reflections of the plane with coordinate formulas.

Derive the geometric effects of composing two reflections in lines.

Derive the rational parametrization of a circle and use it to classify primitive Pythagorean triples.

Model a finite situation as a simple graph and identify its vertices, edges, and adjacencies.

Prove the handshaking lemma and derive a consequence about odd-degree vertices.

Distinguish walks, trails, paths, and cycles, and extract a path from a walk.

Execute breadth-first search to obtain shortest-path distances and a spanning tree.

Prove equivalent characterizations of a tree using connectivity, cycles, and edge count.

Prove the degree criterion for the existence of an Euler trail or Euler circuit.

Either construct a bipartition of a graph or exhibit an odd cycle obstructing one.

Prove that a graph is bipartite if and only if it has no odd cycle.

Use degree, distance, and cycle invariants to prove that two graphs are not isomorphic.

Identify bridges and cut vertices and explain how their removal changes connectivity.

Prove the deletion-contraction recurrence for the number of spanning trees.

Use deletion–contraction to count spanning trees in a cycle and in a specified small graph.

Determine the chromatic number of a specified graph by proving matching upper and lower bounds.

Prove Euler’s formula for connected plane graphs by reducing to a spanning tree.

Derive edge bounds for simple planar graphs from Euler’s formula and face lengths.

Certify the nonplanarity of K₅ and K₃,₃ using planar edge bounds.

Verify whether a specified distance function satisfies the metric axioms.

Describe open balls and determine whether specified subsets of a metric space are open.

Prove that arbitrary unions and finite intersections of open sets are open.

Determine the interior, closure, and boundary of a specified subset of a metric space.

Prove or disprove convergence of a sequence using the metric definition.

Prove continuity of a map between metric spaces using an epsilon–delta argument.

Prove the open-set inverse-image criterion for continuity between metric spaces.

Use the inverse-image criterion to prove that a specified map is continuous or discontinuous.

Verify whether a specified collection of subsets defines a topology.

Construct the topology generated by a basis.

Prove continuity of a map using inverse images of basis elements.

Determine the subspace topology inherited by a specified subset.

Determine basic open sets in a product topology.

Verify that a relation is an equivalence relation and describe its quotient set.

Construct a quotient topology induced by a specified identification.

Prove that two spaces are homeomorphic by checking an explicit map and its inverse.

Determine whether a specified space is connected by finding or ruling out a separation.

Prove that intervals in ℝ are connected.

Prove that continuous images of connected spaces are connected.

Prove that every path-connected space is connected.

Determine compactness of a specified space directly from the open-cover definition.

Prove compactness of a closed interval directly from the open-cover definition.

Prove that continuous images of compact spaces are compact.

Use connectedness or compactness to prove that two specified spaces are not homeomorphic.

Construct the circle, cylinder, and Möbius strip as quotient spaces of intervals or rectangles.

Construct the torus and Klein bottle from edge identifications of a rectangle.

Determine the boundary components and orientability of the cylinder and Möbius strip.

Compute cellular Euler characteristic for standard cell structures on the sphere, torus, and real projective plane.

Prove that Euler characteristic is unchanged by subdivision of a finite cell structure.

Construct the planar dual of a connected plane graph and identify how bridges and loops correspond.

Translate a plane-map coloring problem into a vertex-coloring problem on its dual graph.

Distinguish the cylinder, Möbius strip, torus, and Klein bottle using boundary, orientability, and Euler characteristic.