aops_id int64 2 3.62M | tags listlengths 1 74 |
|---|---|
100,015 | [
"Alphabetical order",
"Combination",
"Binomial coefficient",
"License plate"
] |
100,019 | [
"Permutation",
"Inclusion–exclusion principle",
"Factorial"
] |
1,000,044 | [
"Probability",
"Binomial distribution",
"Exponentiation",
"Binomial coefficient"
] |
100,023 | [
"Geometric series",
"Mathematical induction",
"Infinite product",
"Power series"
] |
100,010 | [
"Snail",
"Day of the week",
"Date and time",
"Calendar",
"Word problem",
"Arithmetic progression",
"Modulo operation",
"Rate of change",
"Problem solving"
] |
1,000,141 | [
"Nickel (United States coin)",
"Floor function",
"Dime",
"Casework (mathematics)",
"Penny (United States coin)",
"Quarter (United States coin)",
"Linear equation",
"Coin problem",
"Generating function"
] |
100,009 | [
"Differentiation",
"Trigonometric functions",
"Square root",
"Cosine",
"Definite integral",
"Sine",
"Fundamental theorem of calculus",
"Trigonometric identities",
"Integral"
] |
1,000,136 | [
"Square root",
"Quadratic equation",
"Quadratic formula",
"Algebra",
"Completing the square",
"Polynomial"
] |
100,026 | [
"Vieta's formulas",
"Quadratic equation",
"Algebra",
"Root of a polynomial",
"Polynomial"
] |
1,000,249 | [
"Decision theory",
"Non-cooperative game",
"Dominant strategy",
"Nash equilibrium",
"Game theory",
"Prisoner's dilemma",
"Expected value"
] |
100,029 | [
"Inequality",
"Factorial",
"Arithmetic mean–geometric mean inequality",
"AM–GM inequality",
"Mathematical induction"
] |
100,036 | [
"Infinite product",
"Convergence (mathematics)",
"Divergence (mathematics)",
"Infinite series",
"Sequence (mathematics)",
"Absolute value"
] |
1,000,379 | [
"Power series",
"Generating function",
"Fibonacci number",
"Infinite series",
"Radius of convergence",
"Induction (mathematics)"
] |
1,000,306 | [
"Remainder",
"Winning strategy",
"Sum of squares",
"Divisor function",
"Modular arithmetic",
"Perfect square",
"Locker problem",
"Number theory",
"Factorization",
"Combinatorial game theory"
] |
100,035 | [
"Euler's sum of powers conjecture",
"Exponents",
"Modular arithmetic",
"Diophantine equation",
"Inequality (mathematics)",
"Fermat's Last Theorem",
"Number theory"
] |
1,000,440 | [
"Vieta's formulas",
"Imaginary unit",
"Polynomial division",
"Polynomial",
"Root of a polynomial",
"Complex number"
] |
100,046 | [
"Probability",
"Random permutation",
"Event (probability theory)",
"Permutation",
"Deck of cards",
"Playing card",
"Combinatorics",
"Conditional probability"
] |
100,047 | [
"Polynomial equation",
"SAT Subject Tests",
"Algebra",
"Fundamental theorem of algebra",
"TI-83",
"Real number",
"Root of a function",
"Polynomial"
] |
100,052 | [
"Sequence",
"Functional equation",
"Integer sequence",
"Mathematical induction",
"Square number",
"Recurrence relation"
] |
100,067 | [
"Algebraic conjugate",
"Rational number",
"Real number",
"Minimal polynomial",
"Sequence (mathematics)",
"Closed set",
"Pisot number"
] |
100,071 | [
"Algebraic identity",
"Jensen's inequality",
"Mathematical proof",
"Symmetric function",
"Inequality (mathematics)",
"Convex function",
"Lagrange multipliers",
"Second derivative test"
] |
100,061 | [
"Interval (mathematics)",
"Trigonometric functions",
"Cosine",
"Derivative",
"Inverse function",
"Equation",
"System of equations",
"Function (mathematics)",
"Monotonic function",
"Real number",
"Sine",
"Odd function"
] |
100,068 | [
"Proof (mathematics)",
"Integer sequence",
"Sequence (mathematics)",
"Mathematical induction",
"Recurrence relation",
"Polynomial"
] |
100,073 | [
"Binomial coefficient",
"Wolstenholme's theorem",
"Elementary symmetric polynomial",
"Modular arithmetic",
"Prime number",
"Finite field",
"Fermat's little theorem",
"Polynomial"
] |
100,039 | [
"Probability",
"Binomial coefficient",
"Probability theory",
"Combinatorics",
"Combination"
] |
100,076 | [
"Trigonometric functions",
"Cosine",
"Right triangle",
"Geometry",
"Angle bisector",
"Half-angle formula",
"Triangle",
"Law of cosines"
] |
100,079 | [
"Probability",
"Binomial coefficient",
"Fair coin",
"Probability theory",
"Combinatorics",
"Binomial distribution"
] |
100,075 | [
"Positive-semidefinite matrix",
"Real symmetric matrix",
"Projection (linear algebra)",
"Non-negative definite matrix",
"Hilbert space",
"Orthogonal projection",
"Eigenvalues and eigenvectors",
"Matrix norm",
"Frobenius norm"
] |
100,074 | [
"Almost everywhere",
"Dominated convergence theorem",
"Lp space",
"Fatou's lemma",
"Convergence in norm",
"Functional analysis",
"Measure theory",
"Real part and imaginary part",
"Norm (mathematics)",
"Pointwise convergence",
"Positive and negative parts of a function"
] |
100,090 | [
"Interior (topology)",
"Countable set",
"Measure zero",
"Cantor set",
"Rational number",
"Closed set",
"Open interval",
"Real number line"
] |
100,101 | [
"Integer",
"Divisibility rule",
"Arithmetic progression",
"Multiple (mathematics)",
"Natural number",
"Repdigit",
"Number theory"
] |
100,114 | [
"Jensen's inequality",
"Mathematical proof",
"Real number",
"Cyclic sum",
"Inequality (mathematics)",
"Concave function"
] |
100,130 | [
"Exponents",
"Inverse function",
"Algebra",
"Power law",
"Inequality (mathematics)",
"Mathematical problem solving"
] |
100,133 | [
"Pythagorean triple",
"Arithmetic progression",
"Quartic function",
"Pythagorean theorem",
"Rational root theorem",
"Geometric progression"
] |
100,119 | [
"Subset sum problem",
"Inequality",
"Algebra",
"Partial sum",
"Abel's identity",
"Positive integer",
"Sumset",
"Combinatorics",
"Injective function"
] |
100,160 | [
"Diameter (geometry)",
"Right triangle",
"Power of a point",
"Triangle inequality",
"Hypotenuse",
"Circumcircle"
] |
1,001,481 | [
"Residue theorem",
"Complex plane",
"Polynomial interpolation",
"Lagrange polynomial",
"Complex analysis",
"Root of a polynomial",
"Contour integral",
"Polynomial",
"Degree of a polynomial"
] |
1,001,633 | [
"Polynomial map",
"Jacobian conjecture",
"Surjective function",
"Polynomial function",
"Algebraic geometry",
"Finite field",
"Complex Euclidean space",
"Injective function",
"Complex number"
] |
100,202 | [
"Unit circle",
"Homogeneous function",
"Limit (mathematics)",
"Multivariable calculus",
"Continuity",
"Polynomial"
] |
100,172 | [
"Rectangle",
"Domino tiling",
"Tiling (mathematics)",
"Modulo operation",
"Combinatorics",
"Parity (mathematics)",
"Integer partition",
"Recurrence relation"
] |
100,173 | [
"Non-negative integer",
"Modular arithmetic",
"Integer solution",
"Diophantine equation",
"Congruence (number theory)",
"Modular arithmetic properties",
"Number theory",
"Fourth power"
] |
100,226 | [
"Sum of squares",
"Integer solution",
"Quadratic form",
"Diophantine equation",
"Number theory",
"Representation of numbers as sums of squares"
] |
1,001,604 | [
"Polynomial (mathematics)",
"Divisibility",
"Primitive root of unity",
"Combinatorics",
"Cyclotomic polynomial",
"Prime power"
] |
1,002,173 | [
"Algebraic number",
"Exponents",
"Equation solving",
"Polynomial equation",
"Surd",
"Radical (mathematics)",
"Chebyshev polynomials",
"Root of unity",
"Binomial theorem",
"Symmetric polynomial",
"Complex number"
] |
100,193 | [
"Trigonometric functions",
"Algebraic identity",
"Mathematical proof",
"Cosine",
"Inequality (mathematics)",
"Triangle",
"Sine",
"Law of cosines",
"Cyclic sum",
"Inequalities in mathematics",
"Symmetric polynomial"
] |
100,233 | [
"Pigeonhole principle"
] |
100,235 | [
"Proof by contradiction",
"Exponentiation",
"Modular arithmetic",
"Division algorithm",
"Divisibility",
"Number theory"
] |
100,255 | [
"Pigeonhole principle",
"Modular arithmetic",
"Combinatorics",
"Number theory",
"Congruence (mathematics)"
] |
100,250 | [
"Half-life",
"Equation",
"Logarithm",
"Algebra",
"Exponential function",
"Exponential decay"
] |
100,241 | [
"Hexomino",
"Triangular number",
"Tessellation",
"Polyiamond",
"Combinatorial geometry",
"Equilateral triangle",
"Mathematical tiling",
"Geometric dissection"
] |
100,229 | [
"Residue class",
"Sum of squares function",
"Divisibility rule",
"Digit sum",
"Modular arithmetic",
"Integer square",
"Quadratic residue",
"Sum of two squares theorem",
"Number theory"
] |
100,257 | [
"Euclidean algorithm",
"Irrational number",
"Pi",
"Rational number",
"Real number",
"Limit of a sequence",
"Sequence (mathematics)",
"Golden ratio",
"Continued fraction"
] |
100,274 | [
"Divisor",
"Mersenne prime",
"Prime number",
"Euclid–Euler theorem",
"Number theory",
"Perfect number"
] |
100,259 | [
"Remainder (polynomial)",
"Polynomial (mathematics)",
"Polynomial division",
"Lagrange interpolation",
"Polynomial remainder theorem"
] |
1,002,802 | [
"Power set",
"Binomial coefficient",
"Set (mathematics)",
"Strong induction",
"Union (set theory)",
"Expected value"
] |
100,277 | [
"Polynomial root",
"Triangle inequality",
"Complex analysis",
"Inequality (mathematics)",
"Polynomial",
"Absolute value",
"Complex number"
] |
1,002,828 | [
"Non-negative integer",
"Factorial",
"Geometric series",
"Generating function",
"Natural number",
"Infinite series",
"Multinomial theorem"
] |
1,002,871 | [
"Binomial coefficient",
"Summation",
"Arithmetic mean",
"Exponentiation",
"Binomial theorem",
"Inequality (mathematics)"
] |
100,287 | [
"Jensen's inequality",
"Logarithm",
"Real number",
"Inequality (mathematics)",
"Convex function"
] |
1,002,900 | [
"Sequence",
"Floor function",
"Inequality",
"Mathematical analysis",
"Proof by contradiction",
"Functional equation",
"Riemann zeta function",
"Natural number",
"Mathematical induction"
] |
100,294 | [
"Mathematical proof",
"Summation",
"Harmonic series",
"Infinite series",
"Inequality (mathematics)",
"Sequence (mathematics)"
] |
100,297 | [
"Fundamental theorem of arithmetic",
"Function (mathematics)",
"Multiplicative function",
"Prime number",
"Unique factorization domain",
"Natural number"
] |
100,299 | [
"Complex polynomial",
"Inequality",
"Mathematical proof",
"Unit circle",
"Complex plane",
"Root of unity",
"Polynomial (mathematics)",
"Triangle inequality",
"Limit of a sequence",
"Rotation (mathematics)",
"Absolute value",
"Complex number"
] |
1,002,990 | [
"Integer",
"Binomial coefficient",
"Permutation",
"Divisibility (mathematics)",
"Cyclic group",
"Necklace (combinatorics)",
"Combinatorics",
"Greatest common divisor"
] |
1,002,911 | [
"Tuple",
"Integer",
"Pigeonhole principle",
"Proof",
"Summation",
"Integer sequence",
"International Mathematical Olympiad",
"Mathematical induction",
"Greatest common divisor",
"Difference (mathematics)"
] |
1,002,943 | [
"Square (geometry)",
"Point (geometry)",
"Rectangle",
"Obtuse angle",
"Pigeonhole principle",
"Cyclic quadrilateral",
"Plane (geometry)",
"Rhombus",
"Distance",
"Angle",
"Triangle similarity",
"Equilateral triangle",
"Quadrilateral",
"Circle (geometry)",
"Perpendicular bisector",
"Para... |
100,300 | [
"Square (geometry)",
"Similar triangles",
"Power of a point",
"Triangle",
"Chord (geometry)",
"Circumcircle"
] |
1,002,994 | [
"Chinese remainder theorem",
"Integer",
"Dirichlet's theorem on arithmetic progressions",
"Quadratic nonresidue",
"Legendre symbol",
"Modular arithmetic",
"Arithmetic progression",
"Prime number",
"Perfect square",
"Squarefree integer",
"Natural number",
"Quadratic residue",
"Law of quadrati... |
1,003,005 | [
"Square (geometry)",
"Similar triangles",
"Straightedge and compass construction",
"Perpendicular",
"Cyclic quadrilateral",
"Angle",
"Triangle",
"Altitude (triangle)",
"Orthocenter"
] |
1,003,043 | [
"Functional equation",
"Function (mathematics)",
"Contradiction",
"Real number",
"Cubic function"
] |
1,003,111 | [
"Transposition (mathematics)",
"Permutation",
"Triangle",
"Area",
"Inequality (mathematics)",
"Combinatorics",
"Quadrilateral"
] |
10,032 | [
"Probability",
"Binomial coefficient",
"Playoff (sports)",
"Combinatorics",
"Mathematical induction",
"Binomial distribution",
"Geometric distribution"
] |
1,003,040 | [
"Muirhead inequality",
"Area of a triangle",
"Schur's inequality",
"Circumradius",
"Feuerbach's theorem",
"Triangle inequality",
"Feuerbach point",
"Arithmetic mean–geometric mean inequality",
"Triangle",
"Nine-point circle",
"Inequality (mathematics)",
"Substitution (mathematics)",
"Inradiu... |
100,332 | [
"Square root",
"Integer",
"Floor function",
"Real number",
"Inequality (mathematics)",
"Piecewise function"
] |
100,336 | [
"Modular arithmetic",
"Divisibility",
"Prime number",
"Diophantine equation",
"Natural number",
"Number theory",
"Greatest common divisor"
] |
100,318 | [
"Concyclic points",
"Radical axis (geometry)",
"Euclidean geometry",
"Angle bisector",
"Incenter",
"Central angle",
"Triangle",
"Inscribed angle",
"Tangent line",
"Circumcircle",
"Orthocenter",
"Circumcenter",
"Isosceles triangle"
] |
10,035 | [
"Divide and conquer algorithm",
"Binary search algorithm",
"Information theory",
"Combinatorial search",
"Decision tree",
"Weighing scale",
"Coin weighing problem",
"Balance scale"
] |
100,350 | [
"Pedal triangle",
"Perpendicular",
"Concurrency (geometry)",
"Orthogonal projection",
"Triangle",
"Isogonal conjugate",
"Circle (geometry)",
"Circumcircle"
] |
100,340 | [
"Determinant",
"Complex plane",
"Perpendicular (geometry)",
"Excenter",
"Concurrent lines",
"Symmetry (geometry)",
"Foot of perpendicular",
"Ceva's theorem",
"Triangle",
"Geometric proof",
"Excircle",
"Altitude (triangle)",
"Circumcircle",
"Orthic triangle",
"Complex number"
] |
1,003,750 | [
"Definite integral",
"Algebra",
"Binomial theorem",
"Power rule for integration",
"Integral",
"Polynomial"
] |
100,370 | [
"Complex plane",
"Absolute value",
"Pythagorean theorem",
"Complex number"
] |
1,003,681 | [
"Integer",
"Exponentiation",
"Function (mathematics)",
"Divisibility",
"Prime number",
"Diophantine equation",
"Natural number",
"Number theory"
] |
1,003,591 | [
"Dilation (mathematics)",
"Central symmetry",
"Polygon",
"Center of mass",
"Area",
"Parallel (geometry)",
"Continuity",
"Line (geometry)",
"Reflection (mathematics)"
] |
1,003,557 | [
"Integer",
"Combinatorial proof",
"Binomial coefficient",
"Summation",
"Combinatorial enumeration",
"Mathematical induction",
"Binomial identity",
"Binomial theorem",
"Combinatorial number system",
"Binomial coefficient#Identities",
"Combinatorics",
"Counting",
"Committee (disambiguation)",
... |
100,382 | [
"Line integral",
"Complex plane",
"Entire function",
"Morera's theorem",
"Sine",
"Complex analysis",
"Fubini's theorem",
"Integral"
] |
100,378 | [
"Algebraic number",
"Recurrence relation",
"Integer solution",
"Fundamental solution (Pell's equation)",
"Diophantine equation",
"Positive integer",
"Square number",
"Pell's equation"
] |
100,383 | [
"Sequence",
"Derivative",
"Borel's theorem",
"Real number",
"Infinitely differentiable function"
] |
1,003,940 | [
"Minute hand",
"Clock angle problem",
"Degree (angle)",
"Hour hand",
"Angle"
] |
100,386 | [
"Liouville's theorem (complex analysis)",
"Complex plane",
"Entire function",
"Maximum modulus principle",
"Compact space"
] |
100,388 | [
"Cauchy–Schwarz inequality",
"Algebraic inequality",
"Positive real numbers",
"AM–GM inequality",
"Inequality (mathematics)",
"Symmetric polynomial"
] |
100,394 | [
"Square root",
"Jensen's inequality",
"Inequality",
"Mathematical proof",
"Logarithm",
"Positive real numbers",
"Arithmetic mean–geometric mean inequality",
"Convex function"
] |
1,004,076 | [
"Transitive property",
"Unit conversion",
"Ratio",
"Equation"
] |
1,004,096 | [
"Inclusion–exclusion principle",
"Buffet",
"Set theory",
"Venn diagram"
] |
1,004,058 | [
"Arithmetic operations",
"Sum of a geometric series",
"Geometric series",
"Common ratio",
"Infinite series"
] |
100,413 | [
"Interval (mathematics)",
"Riemann integral",
"Identically zero function",
"Definite integral",
"Darboux integral",
"Continuous function",
"Real analysis",
"Fundamental theorem of calculus",
"Integral",
"Non-negative function"
] |
100,419 | [
"Square root",
"Mathematical proof",
"Algebra",
"Arithmetic mean–geometric mean inequality",
"Real number",
"Inequality (mathematics)"
] |
100,415 | [
"Identity element",
"Dihedral group",
"Order of an element",
"Group order",
"Conjugation (group theory)",
"Direct product of groups",
"Cyclic group",
"Group (mathematics)"
] |
100,428 | [
"Infinite product",
"Convergence (mathematics)",
"Exponentiation",
"Decimal representation",
"Real number",
"Limit of a sequence",
"Mathematical induction"
] |
1,004,329 | [
"Natural number",
"Irreducible fraction",
"Greatest common divisor",
"Euclidean algorithm"
] |
100,434 | [
"Mathematical proof",
"Exponential function",
"Real number",
"Inequality (mathematics)",
"Bernoulli's inequality"
] |
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