graph theory
number of vertices with odd degrees must be even
number of edges of a tree is one less than the number of vertices
Euler's formula for planar graphs
there are exactly five regular polyhedra (incomplete)
a graph has an Euler circuit if and only if the degree of every vertex is even (incomplete)
a graph has an Euler trail if and only if there are at most two vertices with odd degree (incomplete)
the four color theorem (incomplete)
Brook's theorem (incomplete)
Vizing's theorem (incomplete)
Hall's marriage theorem (incomplete)
factorials, permutations and combinations
using combinations to find a number in the Pascal's triangle
number of ways of arranging n objects with k identical objects
sequence and series
finding a term in an arithmetic sequence
finding a term in a geometric sequence
sum of a finite geometric series
sum to infinity of a geometric series
sum of the first n positive integers
sum of the squares of the first n positive integers
sum of the cubes of the first n positive integers
showing that the harmonic series diverges
sum of the series iri for i ∈ [1,n]
sum to infinity of the series iri where r ∈ (-1,1)
sum of an arithmetico-geometric series
The closed formula for a sequence will be a degree k polynomial if and only if the sequence is Δk-constant (incomplete)
the characteristic root technique (incomplete)