Combinatorics problems
382 problems · Counting arguments, binomial coefficients, graphs and bijections.
Ex.185Four-digit numbers with distinct digits
Counting principles
High school
Ex.215Definition and basic values
Binomial coefficients
High school
Ex.216Pascal's triangle
Binomial coefficients
High school
Ex.217Symmetry of binomial coefficients
Binomial coefficients
High school
Ex.226Counting functions
Counting principles
High school
Ex.227Handshakes at a party
Counting principles
High school
Ex.228The pigeonhole principle
Counting principles
High school
Ex.229The product rule
Counting principles
High school
Ex.230The sum rule
Counting principles
High school
Ex.237Basic graph terminology
Graph theory
High school
Ex.238The complete graph $K_n$
Graph theory
High school
Ex.239Parity of odd-degree vertices
Graph theory
High school
Ex.240The handshake lemma
Graph theory
High school
Ex.243The two-set inclusion-exclusion formula
Inclusion-exclusion
High school
Ex.244Disjoint unions
Inclusion-exclusion
High school
Ex.245Proof of the two-set formula
Inclusion-exclusion
High school
Ex.259The pigeonhole principle on birthdays
Pigeonhole principle
High school
Ex.260Socks in the dark
Pigeonhole principle
High school
Ex.315Sum of binomial coefficients
Binomial coefficients
L1
Ex.322Arrangements of MISSISSIPPI
Counting principles
High school
Ex.330Pascal's rule
Binomial coefficients
L1
Ex.331Pigeonhole modulo 4
Pigeonhole principle
L1
Ex.335Stars and bars
Counting principles
L1
Ex.340The handshake lemma
Graph theory
L1
Ex.400Alternating sum of binomial coefficients
Binomial coefficients
High school
Ex.401The binomial theorem
Binomial coefficients
High school
Ex.402A specific expansion
Binomial coefficients
High school
Ex.403Key binomial identities
Binomial coefficients
High school
Ex.404Sum of binomial coefficients
Binomial coefficients
High school
Ex.405Monotonicity of binomial row
Binomial coefficients
High school
Ex.406Even vs odd subsets
Binomial coefficients
High school
Ex.407Pascal's identity
Binomial coefficients
High school
Ex.408Simplifying binomial quotients
Binomial coefficients
High school
Ex.409Recurrence for coefficients
Binomial coefficients
High school
Ex.410Binomial as a quotient of factorials
Binomial coefficients
High school
Ex.411Sum over odd $k$
Binomial coefficients
High school
Ex.412Sum over odd and even sizes
Binomial coefficients
High school
Ex.413Sum over even $k$
Binomial coefficients
High school
Ex.414Sum of powers of 2
Binomial coefficients
High school
Ex.415Sum with $3^k$
Binomial coefficients
High school
Ex.437Arrangements
Counting principles
High school
Ex.438The binomial theorem
Counting principles
High school
Ex.439Pascal's identity
Counting principles
High school
Ex.440Expanding $(x-y)^n$
Counting principles
High school
Ex.441Choosing with repetition
Counting principles
High school
Ex.442Combinations
Counting principles
High school
Ex.443Choosing a mixed committee
Counting principles
High school
Ex.444Counting injections
Counting principles
High school
Ex.445Permutations
Counting principles
High school
Ex.446Pigeonhole with balls
Counting principles
High school
Ex.447Subsets and their complements
Counting principles
High school
Ex.448Round-robin tournament
Counting principles
High school
Ex.469The complement of a graph
Graph theory
High school
Ex.470Connected components
Graph theory
High school
Ex.471Paths and cycles
Graph theory
High school
Ex.472Summary of graph theory
Graph theory
High school
Ex.473Trees
Graph theory
High school
Ex.474Leaves of a tree
Graph theory
High school
Ex.479The three-set inclusion-exclusion formula
Inclusion-exclusion
High school
Ex.480Counting complement via IE
Inclusion-exclusion
High school
Ex.481Derangements of $3$ elements
Inclusion-exclusion
L1
Ex.482Derangements of $4$ elements
Inclusion-exclusion
L1
Ex.483Even vs odd subsets
Inclusion-exclusion
High school
Ex.484Exact intersection count
Inclusion-exclusion
High school
Ex.485Summary of inclusion-exclusion
Inclusion-exclusion
High school
Ex.486Non-constant functions to $\{0,1\}$
Inclusion-exclusion
High school
Ex.487Nested sets
Inclusion-exclusion
L1
Ex.488Surjections to $\{0,1\}$
Inclusion-exclusion
High school
Ex.525Among $n+1$ from $2n$, two consecutive
Pigeonhole principle
High school
Ex.526Consecutive integers
Pigeonhole principle
L1
Ex.527Difference divisible by $n$
Pigeonhole principle
L1
Ex.528Pigeonhole with numbers
Pigeonhole principle
L1
Ex.529More socks
Pigeonhole principle
High school
Ex.530Summary of pigeonhole techniques
Pigeonhole principle
High school
Ex.531Generalized pigeonhole with $100$ people
Pigeonhole principle
High school
Ex.532Balls in boxes
Pigeonhole principle
High school
Ex.533Proof of the generalized pigeonhole principle
Pigeonhole principle
L1
Ex.534A pair with difference of $n$
Pigeonhole principle
L1
Ex.535Pairs summing to $2n+1$
Pigeonhole principle
L1
Ex.601A block of consecutive terms divisible by n
Pigeonhole principle
M1
Ex.602Dominoes on a mutilated chessboard
Counting principles
M1
Ex.603Replacing two numbers by their difference
Counting principles
M1
Ex.604Regions cut by n lines
Counting principles
M1
Ex.605A combinatorial proof that Σ C(n,k)² = C(2n,n)
Binomial coefficients
M1
Ex.606Two people with the same number of friends
Pigeonhole principle
M1
Ex.607Two take-away games
Counting principles
M1
Ex.608The expected number of fixed points of a random permutation
Counting principles
M1
Ex.635A subtraction game with moves 1, 3 and 4
Counting principles
M1
Ex.636How many colours does a graph with m edges need?
Graph theory
M1
Ex.637Six people: three friends or three strangers
Pigeonhole principle
M1
Ex.638Tiling a 2 × n board with dominoes and squares
Counting principles
M1
Ex.701Derangements of four objects
Inclusion-exclusion
L2
Ex.799Parity of binomial coefficients
Binomial coefficients
L1
Ex.800A combinatorial proof
Binomial coefficients
Olympiad
Ex.801Combinatorial proof
Binomial coefficients
High school
Ex.802Compositions via binomial coefficients
Binomial coefficients
Olympiad
Ex.803Expansion of a large binomial
Binomial coefficients
High school
Ex.804Telescoping with binomials
Binomial coefficients
Olympiad
Ex.805The hockey-stick identity
Binomial coefficients
L1
Ex.806An inequality with binomials
Binomial coefficients
L1
Ex.807Counting with binomials
Binomial coefficients
L1
Ex.808The largest binomial coefficient
Binomial coefficients
L1
Ex.809Multinomial identity
Binomial coefficients
Olympiad
Ex.810Generalized binomial coefficients
Binomial coefficients
L1
Ex.811Polynomial expansion with binomials
Binomial coefficients
L1
Ex.812Coefficients of a power
Binomial coefficients
L1
Ex.813Application to sums
Binomial coefficients
L1
Ex.814Sum with two powers
Binomial coefficients
L1
Ex.815Vandermonde's identity
Binomial coefficients
L1
Ex.816Weighted sum of binomial coefficients
Binomial coefficients
L1
Ex.871Circular arrangements
Counting principles
L1
Ex.872Arrangements with repetition
Counting principles
High school
Ex.873A bijective proof
Counting principles
Olympiad
Ex.874Ordering books on a shelf
Counting principles
High school
Ex.875Counting Boolean functions
Counting principles
L1
Ex.876Counting cliques
Counting principles
Olympiad
Ex.877Coloring problems
Counting principles
L1
Ex.878Combinations with repetition
Counting principles
High school
Ex.879Complement counting
Counting principles
High school
Ex.880Degrees in a graph
Counting principles
L1
Ex.881Handshake lemma
Counting principles
L1
Ex.882Counting pairs within subsets
Counting principles
Olympiad
Ex.883Counting with at least one
Counting principles
L1
Ex.884At least one property
Counting principles
Olympiad
Ex.885Counting unions
Counting principles
L1
Ex.886Counting configurations
Counting principles
L1
Ex.887Sets of even size
Counting principles
Olympiad
Ex.888Exact intersection count
Counting principles
Olympiad
Ex.889Counting onto functions
Counting principles
Olympiad
Ex.890Monotone lattice paths
Counting principles
Olympiad
Ex.891Pigeonhole with integers
Counting principles
Olympiad
Ex.892Generalized pigeonhole
Counting principles
L1
Ex.893Pigeonhole proof
Counting principles
L1
Ex.894Subset sums
Counting principles
Olympiad
Ex.895Scheduling tournaments
Counting principles
L1
Ex.957BFS and DFS
Graph theory
L1
Ex.958Bipartite graphs
Graph theory
L1
Ex.959Cliques and independent sets
Graph theory
L1
Ex.960A graph with a unique degree sequence
Graph theory
L1
Ex.961Hamiltonian cycles (example)
Graph theory
L1
Ex.962Graph isomorphism
Graph theory
L1
Ex.963Matchings
Graph theory
L1
Ex.964Euler's formula application
Graph theory
L1
Ex.965Spanning trees
Graph theory
L1
Ex.966Characterizations of trees
Graph theory
L1
Ex.967Diameter of a tree
Graph theory
L1
Ex.968Trees as minors
Graph theory
L1
Ex.986Permutations with at least one fixed point
Inclusion-exclusion
L1
Ex.987Binomial identity via IE
Inclusion-exclusion
Olympiad
Ex.988Coprime to $100$
Inclusion-exclusion
L1
Ex.989Numbers coprime to $30$ up to $100$
Inclusion-exclusion
L1
Ex.990Counting numbers with at least one property
Inclusion-exclusion
L1
Ex.991Derangements of $5$ elements
Inclusion-exclusion
L1
Ex.992Derangements of $6$ elements
Inclusion-exclusion
L1
Ex.993Derangement recurrence
Inclusion-exclusion
L1
Ex.994Derangements
Inclusion-exclusion
L1
Ex.995Euler's totient via inclusion-exclusion
Inclusion-exclusion
L1
Ex.996Exactly one property
Inclusion-exclusion
L1
Ex.997The general inclusion-exclusion formula
Inclusion-exclusion
L1
Ex.998Items with no defect
Inclusion-exclusion
L1
Ex.999Letters and envelopes
Inclusion-exclusion
L1
Ex.1000Elements with none of the properties
Inclusion-exclusion
L1
Ex.1001Surjections from $5$ to $3$
Inclusion-exclusion
L1
Ex.1002Inclusion-exclusion in a survey
Inclusion-exclusion
High school
Ex.1003Counting with divisibility
Inclusion-exclusion
L1
Ex.1004The sieve of Eratosthenes as IE
Inclusion-exclusion
L1
Ex.1005Surjections from $4$ to $3$
Inclusion-exclusion
L1
Ex.1006Counting surjections via inclusion-exclusion
Inclusion-exclusion
L1
Ex.1007Partial three-set data
Inclusion-exclusion
L1
Ex.1008Exactly one of two properties
Inclusion-exclusion
L1
Ex.1009Two triples
Inclusion-exclusion
L1
Ex.1010IE with complements
Inclusion-exclusion
L1
Ex.1088$\{1,2,3,\ldots\}$ and pairwise differences
Pigeonhole principle
Olympiad
Ex.1089Eratosthenes via pigeonhole
Pigeonhole principle
L1
Ex.1090Sum of digits
Pigeonhole principle
Olympiad
Ex.1091Binomial coefficient bound
Pigeonhole principle
Olympiad
Ex.1092Consecutive integers whose sum is divisible
Pigeonhole principle
L1
Ex.1093Divisibility via pigeonhole
Pigeonhole principle
L1
Ex.1094Three same-color integers
Pigeonhole principle
Olympiad
Ex.1095Points in an equilateral triangle
Pigeonhole principle
Olympiad
Ex.1096An even number of odd vertices
Pigeonhole principle
Olympiad
Ex.1097The handshake lemma by pigeonhole
Pigeonhole principle
L1
Ex.1098Integer grid points
Pigeonhole principle
Olympiad
Ex.1099Pigeonhole in a Latin square
Pigeonhole principle
Olympiad
Ex.1100Many odd numbers
Pigeonhole principle
Olympiad
Ex.1101Modular pigeonhole
Pigeonhole principle
L1
Ex.1102A multiple of $n$ from $0$s and $1$s
Pigeonhole principle
Olympiad
Ex.1103An odd number in a triple
Pigeonhole principle
Olympiad
Ex.1104Points in a unit square
Pigeonhole principle
L1
Ex.1105Subset sums
Pigeonhole principle
L1
Ex.1106Remainders of consecutive integers
Pigeonhole principle
Olympiad
Ex.1107Pigeonhole in a table
Pigeonhole principle
L1
Ex.1108A sequence of $0$s and $1$s
Pigeonhole principle
Olympiad
Ex.1234The game of coins on a round table
Counting principles
Olympiad
Ex.1235Derangements and the number 1/e
Inclusion-exclusion
Olympiad
Ex.1236Every tournament has a Hamiltonian path
Graph theory
Olympiad
Ex.1237Finding the heavy coin with a balance
Counting principles
Olympiad
Ex.1238Flipping rows and columns of a board
Counting principles
Olympiad
Ex.1239A tree has at least as many leaves as its maximum degree
Graph theory
Olympiad
Ex.1240Monotone subsequences (Erdős–Szekeres)
Pigeonhole principle
Olympiad
Ex.1241A finite set in which every point is a midpoint
Counting principles
Olympiad
Ex.1242Choosing n + 1 numbers among 1, …, 2n
Pigeonhole principle
Olympiad
Ex.1243Subsets without two consecutive integers
Counting principles
Olympiad
Ex.1244Tiling a 10 × 10 board with 1 × 4 tiles
Counting principles
Olympiad
Ex.1245Splitting a parliament into two houses
Graph theory
Olympiad
Ex.1246Tiling a 2ⁿ × 2ⁿ board with L-trominoes
Counting principles
Olympiad
Ex.1247Binary strings with no three equal consecutive digits
Counting principles
M1
Ex.1248A graph is bipartite exactly when it has no odd cycle
Graph theory
Olympiad
Ex.1249When can a graph be drawn without lifting the pen?
Graph theory
Olympiad
Ex.1250Gray codes: a cyclic list of all bit strings with one change at a time
Counting principles
Olympiad
Ex.1251The fewest kings controlling the board
Pigeonhole principle
M1
Ex.1252Knights and liars around a table
Pigeonhole principle
M1
Ex.1253The expected number of records
Counting principles
M1
Ex.1254Sorting by adjacent swaps needs exactly the number of inversions
Counting principles
M1
Ex.1255Stars and bars with upper bounds
Inclusion-exclusion
M1
Ex.1256Surjections and splitting a set into three groups
Inclusion-exclusion
Olympiad
Ex.1257Gas stations on a circular road
Counting principles
Olympiad
Ex.1306A cut containing at least half of the edges
Graph theory
M1
Ex.1307A 7 × 7 grid with no rectangle of marked cells
Counting principles
Olympiad
Ex.1308Vandermonde's identity and a weighted sum of squares
Binomial coefficients
M1
Ex.1309Partitions into distinct parts and into odd parts
Counting principles
Olympiad
Ex.1310Drawing K₈ without lifting the pen
Graph theory
Olympiad
Ex.1311Five points contain a convex quadrilateral
Pigeonhole principle
Olympiad
Ex.1312Nim and the XOR rule
Counting principles
M1
Ex.1313The 15-puzzle with two tiles swapped
Counting principles
Olympiad
Ex.1526Binomial coefficient inverses
Binomial coefficients
Olympiad
Ex.1527An identity with inverses
Binomial coefficients
L1
Ex.1528Catalan via binomial
Binomial coefficients
L1
Ex.1529Central binomial coefficients
Binomial coefficients
Olympiad
Ex.1530Choosing subsets of subsets
Binomial coefficients
L1
Ex.1531Binomial coefficients and primes
Binomial coefficients
Olympiad
Ex.1532A combinatorial sum
Binomial coefficients
Olympiad
Ex.1533A combinatorial sum with choice
Binomial coefficients
Olympiad
Ex.1534Cyclic identity with binomials
Binomial coefficients
Olympiad
Ex.1535Approximation via binomial theorem
Binomial coefficients
Olympiad
Ex.1536Fibonacci and binomials
Binomial coefficients
Olympiad
Ex.1537Generating function of binomial row
Binomial coefficients
L1
Ex.1538A complex binomial identity
Binomial coefficients
Olympiad
Ex.1539Sum of sines
Binomial coefficients
Olympiad
Ex.1540An identity with squares
Binomial coefficients
Olympiad
Ex.1541Application to integration
Binomial coefficients
L1
Ex.1542An integral identity
Binomial coefficients
L1
Ex.1543The multinomial theorem
Binomial coefficients
L1
Ex.1544Multinomial coefficients
Binomial coefficients
L1
Ex.1545Partial sums of binomials
Binomial coefficients
Olympiad
Ex.1546Hexagon identity
Binomial coefficients
Olympiad
Ex.1547Pentagon identity
Binomial coefficients
Olympiad
Ex.1548Weighted power sum
Binomial coefficients
Olympiad
Ex.1549Quadratic coefficient sum
Binomial coefficients
Olympiad
Ex.1550Moments of the binomial row
Binomial coefficients
L1
Ex.1551Sum of squares of binomial coefficients
Binomial coefficients
L1
Ex.1552Stirling numbers and binomials
Binomial coefficients
M1
Ex.1553Convolution of binomial rows
Binomial coefficients
Olympiad
Ex.1554Reciprocals of binomial coefficients
Binomial coefficients
Olympiad
Ex.1555Sum with $k^2\binom{n}{k}$
Binomial coefficients
L1
Ex.1556A generalized Vandermonde
Binomial coefficients
L1
Ex.1603Generating function for binomial coefficients
Counting principles
Olympiad
Ex.1604Catalan applications
Counting principles
Olympiad
Ex.1605A combinatorial identity
Counting principles
L1
Ex.1606Derangements
Counting principles
L1
Ex.1607Double counting
Counting principles
Olympiad
Ex.1608Another double count
Counting principles
Olympiad
Ex.1609Alternating sum identity
Counting principles
Olympiad
Ex.1610Counting coprime integers
Counting principles
L1
Ex.1611Derangements via inclusion-exclusion
Counting principles
Olympiad
Ex.1612Derangement formula
Counting principles
Olympiad
Ex.1613A combinatorial identity via inclusion-exclusion
Counting principles
Olympiad
Ex.1614Surjection formula proof
Counting principles
Olympiad
Ex.1615Sieve formula
Counting principles
Olympiad
Ex.1616Surjections via inclusion-exclusion
Counting principles
Olympiad
Ex.1617Surjections onto $3$ elements
Counting principles
Olympiad
Ex.1618Three-set inclusion-exclusion
Counting principles
L1
Ex.1619Three-set counts
Counting principles
L1
Ex.1620Counting with matrices
Counting principles
Olympiad
Ex.1621The multinomial theorem
Counting principles
L1
Ex.1622Lattice paths
Counting principles
Olympiad
Ex.1623Permutations with fixed points
Counting principles
L1
Ex.1624Pigeonhole with 100 numbers
Counting principles
Olympiad
Ex.1625Multiples via pigeonhole
Counting principles
Olympiad
Ex.1626Pigeonhole in a graph
Counting principles
Olympiad
Ex.1627Pigeonhole with points
Counting principles
Olympiad
Ex.1628Sum via pigeonhole
Counting principles
Olympiad
Ex.1629Counting surjections
Counting principles
L1
Ex.1630Vandermonde's identity
Counting principles
Olympiad
Ex.1681Planar bipartite bound
Graph theory
L1
Ex.1682Chromatic number
Graph theory
L1
Ex.1683Graphical degree sequences
Graph theory
L1
Ex.1684Dijkstra's algorithm
Graph theory
L1
Ex.1685Eulerian circuits
Graph theory
L1
Ex.1686The four-color theorem (statement)
Graph theory
L1
Ex.1687Hall's marriage theorem (statement)
Graph theory
L1
Ex.1688$K_{3,3}$ is not planar
Graph theory
L1
Ex.1689Kőnig's theorem
Graph theory
M1
Ex.1690Kruskal's algorithm
Graph theory
L1
Ex.1691Minimum degree and connectivity
Graph theory
L1
Ex.1692Moore bound
Graph theory
M1
Ex.1693NP-hard problems in graphs
Graph theory
M1
Ex.1694Planar graphs and Euler's formula
Graph theory
L1
Ex.1695Prim's algorithm
Graph theory
L1
Ex.1696Prüfer sequences
Graph theory
M1
Ex.1697Ramsey numbers
Graph theory
L1
Ex.1698Cayley's formula
Graph theory
M1
Ex.1712Four-set inclusion-exclusion
Inclusion-exclusion
L1
Ex.1713Application to probability
Inclusion-exclusion
L1
Ex.1714At least two properties
Inclusion-exclusion
L1
Ex.1715Inclusion-exclusion in ballot problems
Inclusion-exclusion
Olympiad
Ex.1716Colorings avoiding patterns
Inclusion-exclusion
Olympiad
Ex.1717Numbers coprime to $210$
Inclusion-exclusion
L1
Ex.1718Path counting via IE
Inclusion-exclusion
Olympiad
Ex.1719Derangements via double counting
Inclusion-exclusion
Olympiad
Ex.1720Pairwise but not triple
Inclusion-exclusion
L1
Ex.1721A multi-property application
Inclusion-exclusion
L1
Ex.1722Committees by inclusion-exclusion
Inclusion-exclusion
L1
Ex.1723Combining multiple IE applications
Inclusion-exclusion
Olympiad
Ex.1724Probability a permutation has a fixed point
Inclusion-exclusion
L1
Ex.1725Proof of inclusion-exclusion
Inclusion-exclusion
L1
Ex.1726Sieve formula
Inclusion-exclusion
L1
Ex.1727Colorings with constraints
Inclusion-exclusion
L1
Ex.1728Avoiding two letters
Inclusion-exclusion
L1
Ex.1729Two obstacles via IE
Inclusion-exclusion
Olympiad
Ex.1730Words avoiding letters
Inclusion-exclusion
L1
Ex.1803Difference exactly $10$
Pigeonhole principle
Olympiad
Ex.1804Difference of at most $10$
Pigeonhole principle
Olympiad
Ex.1805Circles and pigeonhole
Pigeonhole principle
Olympiad
Ex.1806Coloring integers
Pigeonhole principle
Olympiad
Ex.1807Friends and enemies
Pigeonhole principle
Olympiad
Ex.1808Graph theory via pigeonhole
Pigeonhole principle
Olympiad
Ex.1809Increasing or decreasing subsequence
Pigeonhole principle
Olympiad
Ex.1810Monochromatic triangle
Pigeonhole principle
Olympiad
Ex.1811Rectangle with lattice points
Pigeonhole principle
Olympiad
Ex.1812Separated integers
Pigeonhole principle
Olympiad
Ex.1813Subsets of fixed size
Pigeonhole principle
Olympiad
Ex.1814A subset with sum divisible by $n$
Pigeonhole principle
Olympiad
Ex.1815Sum of two distinct elements
Pigeonhole principle
Olympiad
Ex.1816A triple summing to a multiple
Pigeonhole principle
Olympiad
Ex.1817Triangle area via pigeonhole
Pigeonhole principle
Olympiad
Ex.1938Triangle-free graphs have at most n²/4 edges (Mantel)
Graph theory
Olympiad
Ex.1939Paths that never go below the axis: Catalan numbers
Binomial coefficients
Olympiad
Ex.1940Graphs with no 4-cycle
Graph theory
Olympiad
Ex.1941Hall's marriage theorem and regular bipartite graphs
Graph theory
Olympiad
Ex.1942The ballot problem
Counting principles
Olympiad
Ex.1972Chomp and the strategy-stealing argument
Counting principles
Olympiad
Ex.1973Erdős's lower bound for Ramsey numbers
Graph theory
Olympiad
Ex.1974Hall's marriage theorem and a deck of cards
Graph theory
Olympiad
Ex.1975Oddtown
Counting principles
Olympiad
Ex.1976Sperner's theorem and the LYM inequality
Binomial coefficients
Olympiad
Ex.1977The Sylvester–Gallai theorem
Pigeonhole principle
Olympiad
Ex.1978Turán's theorem
Graph theory
Olympiad
Ex.2084Chu–Vandermonde for negative upper index
Binomial coefficients
M1
Ex.2085A hard combinatorial identity
Binomial coefficients
Olympiad
Ex.2086A correct challenge identity
Binomial coefficients
Olympiad
Ex.2087Gauss's binomial theorem
Binomial coefficients
M1
Ex.2088Generating function of central binomials
Binomial coefficients
M1
Ex.2089A challenging binomial sum
Binomial coefficients
Olympiad
Ex.2090A challenge identity
Binomial coefficients
Olympiad
Ex.2091Lucas's theorem
Binomial coefficients
M1
Ex.2092Mahonian numbers
Binomial coefficients
M1
Ex.2093Partial fractions with binomials
Binomial coefficients
M1
Ex.2094Riordan arrays (statement)
Binomial coefficients
M1
Ex.2095Zeilberger's algorithm (statement)
Binomial coefficients
M1
Ex.2116Bertrand's ballot problem
Counting principles
Olympiad
Ex.2117Catalan numbers
Counting principles
Olympiad
Ex.2136Edge coloring (Vizing)
Graph theory
M1
Ex.2137Erdős–Gallai theorem (statement)
Graph theory
M1
Ex.2138Erdős–Gyárfás conjecture (statement)
Graph theory
M1
Ex.2139Expander graphs (statement)
Graph theory
M1
Ex.2140The four-color theorem proof (history)
Graph theory
M1
Ex.2141Kuratowski's theorem (statement)
Graph theory
L1
Ex.2142Graph minors
Graph theory
M1
Ex.2143Perfect graphs
Graph theory
M1
Ex.2144Spectral graph theory (statement)
Graph theory
M1
Ex.2145Szemerédi's regularity lemma (statement)
Graph theory
M1
Ex.2146Turán's theorem
Graph theory
M1
Ex.2155Both rows and columns
Inclusion-exclusion
Olympiad
Ex.2156Exactly $k$ properties
Inclusion-exclusion
Olympiad
Ex.2157The ménage problem (statement)
Inclusion-exclusion
Olympiad
Ex.2187Ramsey-type pigeonhole
Pigeonhole principle
Olympiad
Ex.2226Subsets whose sum is divisible by p
Counting principles
Olympiad
Ex.2227The Sylvester-Gallai theorem
Pigeonhole principle
Olympiad
Ex.2235Every set of n integers contains a sum-free subset of more than n/3 elements
Counting principles
Olympiad
Ex.2236Conway's soldiers cannot reach the fifth row
Counting principles
Olympiad
Ex.2237The Erdős–Ko–Rado theorem
Counting principles
Olympiad
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