ksuinbragimova/final_tags · Datasets at Hugging Face

aops_id int64 2 3.62M	tags listlengths 3 32
100,015	[ "lexicographic_order", "combination", "binomial_coefficient" ]
100,019	[ "permutation", "inclusionexclusion_principle", "factorial" ]
1,000,044	[ "binomial_distribution", "exponentiation", "binomial_coefficient" ]
100,023	[ "geometric_series", "mathematical_induction", "infinite_product", "power_series" ]
100,010	[ "word_problem_mathematics", "arithmetic_progression", "modulo", "relative_change" ]
1,000,141	[ "floor_and_ceiling_functions", "linear_equation", "coin_problem", "generating_function" ]
100,009	[ "differentiation_rules", "square_root", "cosine", "sine", "fundamental_theorem_of_calculus", "list_of_trigonometric_identities", "integral" ]
1,000,136	[ "square_root", "quadratic_equation", "quadratic_formula", "completing_the_square", "polynomial" ]
100,026	[ "vietas_formulas", "quadratic_equation", "polynomial_root_finding", "polynomial" ]
1,000,249	[ "nash_equilibrium", "prisoners_dilemma", "expected_value" ]
100,029	[ "inequality_mathematics", "factorial", "mathematical_induction" ]
100,036	[ "infinite_product", "convergence_mathematics", "divergence", "series_mathematics", "sequence", "absolute_value" ]
1,000,379	[ "power_series", "generating_function", "fibonacci_numbers", "radius_of_convergence", "mathematical_induction" ]
1,000,306	[ "sum_of_squares_function", "divisor_function", "modular_arithmetic", "square", "combinatorial_game_theory" ]
100,035	[ "eulers_sum_of_powers_conjecture", "exponentiation", "modular_arithmetic", "diophantine_equation", "inequality_mathematics", "fermats_last_theorem" ]
1,000,440	[ "vietas_formulas", "imaginary_unit", "polynomial_long_division", "polynomial", "polynomial_root_finding", "complex_number" ]
100,046	[ "random_permutation", "event_probability_theory", "permutation", "conditional_probability" ]
100,047	[ "fundamental_theorem_of_algebra", "real_number", "root_of_a_function" ]
100,052	[ "sequence", "functional_equation", "integer_sequence", "mathematical_induction", "square_number", "recurrence_relation" ]
100,067	[ "complex_conjugate", "rational_number", "real_number", "minimal_polynomial_linear_algebra", "sequence", "closed_set", "pisotvijayaraghavan_number" ]
100,071	[ "algebraic_identities", "jensens_inequality", "symmetric_function", "inequality_mathematics", "convex_function", "lagrange_multiplier", "second_partial_derivative_test" ]
100,061	[ "interval_mathematics", "cosine", "derivative", "inverse_function", "equation", "system_of_equations", "function_mathematics", "monotonic_function", "real_number", "sine", "even_and_odd_functions" ]
100,068	[ "integer_sequence", "sequence", "mathematical_induction", "recurrence_relation", "polynomial" ]
100,073	[ "binomial_coefficient", "wolstenholmes_theorem", "elementary_symmetric_polynomial", "prime_number", "finite_field", "fermats_little_theorem", "polynomial" ]
100,039	[ "binomial_coefficient", "probability_theory", "combination" ]
100,076	[ "cosine", "right_triangle", "angle_bisector_theorem", "tangent_half_angle_formula", "triangle", "law_of_cosines" ]
100,079	[ "binomial_coefficient", "fair_coin", "binomial_distribution" ]
100,075	[ "symmetric_matrix", "projection_linear_algebra", "hilbert_space", "orthogonal_transformation", "eigenvalues_and_eigenvectors", "matrix_norm" ]
100,074	[ "dominated_convergence_theorem", "lp_space", "fatous_lemma", "norm_mathematics", "pointwise_convergence", "positive_and_negative_parts" ]
100,090	[ "interior_topology", "countable_set", "cantor_set", "rational_number", "closed_set", "open_set", "number_line" ]
100,101	[ "integer", "divisibility_rule", "arithmetic_progression", "multiple_mathematics", "natural_number", "repdigit" ]
100,114	[ "jensens_inequality", "real_number", "inequality_mathematics", "concave_function" ]
100,130	[ "exponentiation", "inverse_function", "power_law", "inequality_mathematics" ]
100,133	[ "pythagorean_triple", "arithmetic_progression", "quartic_function", "rational_root_theorem", "geometric_progression" ]
100,119	[ "subset_sum_problem", "inequality_mathematics", "series_mathematics", "abels_identity", "integer", "sumset", "injective_function" ]
100,160	[ "diameter", "right_triangle", "power_of_a_point", "triangle_inequality", "hypotenuse", "circumcircle" ]
1,001,481	[ "residue_theorem", "complex_plane", "polynomial_interpolation", "lagrange_polynomial", "polynomial_root_finding", "contour_integration", "polynomial", "degree_of_a_polynomial" ]
1,001,633	[ "polynomial_mapping", "jacobian_conjecture", "surjective_function", "polynomial_function", "finite_field", "complex_coordinate_space", "injective_function", "complex_number" ]
100,202	[ "unit_circle", "homogeneous_function", "limit_mathematics", "polynomial" ]
100,172	[ "rectangle", "domino_tiling", "modulo", "parity_mathematics", "integer_partition", "recurrence_relation" ]
100,173	[ "diophantine_equation", "congruence_relation", "fourth_power" ]
100,226	[ "sum_of_squares_function", "integer", "quadratic_form", "diophantine_equation" ]
1,001,604	[ "polynomial", "divisibility_rule", "root_of_unity", "cyclotomic_polynomial", "prime_power" ]
1,002,173	[ "algebraic_number", "exponentiation", "equation_solving", "polynomial", "nth_root", "radical_of_an_integer", "chebyshev_polynomials", "root_of_unity", "binomial_theorem", "symmetric_polynomial", "complex_number" ]
100,193	[ "algebraic_identities", "cosine", "inequality_mathematics", "triangle", "sine", "law_of_cosines", "symmetric_polynomial" ]
100,235	[ "proof_by_contradiction", "exponentiation", "modular_arithmetic", "division_algorithm", "divisibility_rule" ]
100,250	[ "equation", "logarithm", "exponential_function", "exponential_decay" ]
100,241	[ "hexomino", "triangular_number", "polyiamond", "combinatorial_geometry_in_the_plane", "equilateral_triangle", "geometric_dissection" ]
100,229	[ "residue_number_system", "sum_of_squares_function", "divisibility_rule", "digit_sum", "quadratic_residue", "sum_of_two_squares_theorem" ]
100,257	[ "euclidean_algorithm", "irrational_number", "pi", "rational_number", "real_number", "limit_of_a_sequence", "sequence", "golden_ratio", "continued_fraction" ]
100,274	[ "divisor", "mersenne_prime", "prime_number", "euclideuler_theorem", "perfect_number" ]
100,259	[ "polynomial", "polynomial_long_division", "interpolation", "polynomial_remainder_theorem" ]
1,002,802	[ "power_set", "binomial_coefficient", "set_mathematics", "mathematical_induction", "union_set_theory", "expected_value" ]
100,277	[ "polynomial_root_finding", "triangle_inequality", "inequality_mathematics", "polynomial", "absolute_value", "complex_number" ]
1,002,828	[ "integer", "factorial", "geometric_series", "generating_function", "natural_number", "multinomial_theorem" ]
1,002,871	[ "binomial_coefficient", "summation", "arithmetic_mean", "exponentiation", "binomial_theorem", "inequality_mathematics" ]
100,287	[ "jensens_inequality", "logarithm", "real_number", "inequality_mathematics", "convex_function" ]
1,002,900	[ "sequence", "floor_and_ceiling_functions", "inequality_mathematics", "proof_by_contradiction", "functional_equation", "riemann_zeta_function", "natural_number", "mathematical_induction" ]
100,294	[ "summation", "series_mathematics", "inequality_mathematics", "sequence" ]
100,297	[ "fundamental_theorem_of_arithmetic", "function_mathematics", "multiplicative_function", "prime_number", "unique_factorization_domain", "natural_number" ]
100,299	[ "inequality_mathematics", "unit_circle", "complex_plane", "root_of_unity", "triangle_inequality", "limit_of_a_sequence", "rotation", "absolute_value", "complex_number" ]
1,002,990	[ "integer", "binomial_coefficient", "permutation", "divisibility_rule", "cyclic_group", "necklace_combinatorics", "greatest_common_divisor" ]
1,002,911	[ "tuple", "integer", "pigeonhole_principle", "proof_truth", "summation", "integer_sequence", "mathematical_induction", "greatest_common_divisor", "difference_of_two_squares" ]
1,002,943	[ "square", "rectangle", "acute_and_obtuse_triangles", "pigeonhole_principle", "cyclic_quadrilateral", "rhombus", "angle", "similarity_system_of_triangles", "equilateral_triangle", "quadrilateral", "circle", "perpendicular_bisector", "parallelogram", "circumcenter", "isosceles_trapezoid" ]
100,300	[ "square", "similarity_system_of_triangles", "power_of_a_point", "triangle", "chord_geometry", "circumcircle" ]
1,002,994	[ "chinese_remainder_theorem", "integer", "dirichlets_theorem_on_arithmetic_progressions", "legendre_symbol", "arithmetic_progression", "prime_number", "square", "square_free_integer", "natural_number", "quadratic_reciprocity", "congruence_relation" ]
1,003,005	[ "square", "similarity_system_of_triangles", "straightedge_and_compass_construction", "perpendicular", "cyclic_quadrilateral", "angle", "triangle", "orthocenter" ]
1,003,043	[ "functional_equation", "function_mathematics", "real_number", "cubic_function" ]
1,003,111	[ "transposition_mathematics", "permutation", "triangle", "area", "inequality_mathematics", "quadrilateral" ]
10,032	[ "binomial_coefficient", "mathematical_induction", "binomial_distribution", "geometric_distribution" ]
1,003,040	[ "muirheads_inequality", "area_of_a_triangle", "schurs_inequality", "circumradius", "triangle_inequality", "feuerbach_point", "amgm_inequality", "triangle", "nine_point_circle", "inequality_mathematics", "inradius" ]
100,332	[ "square_root", "integer", "floor_and_ceiling_functions", "real_number", "inequality_mathematics", "piecewise_function" ]
100,336	[ "modular_arithmetic", "divisibility_rule", "prime_number", "diophantine_equation", "natural_number", "greatest_common_divisor" ]
100,318	[ "concyclic_points", "radical_axis", "angle_bisector_theorem", "incenter", "central_angle", "triangle", "inscribed_angle", "tangent", "circumcircle", "orthocenter", "circumcenter", "isosceles_triangle" ]
10,035	[ "divide_and_conquer_algorithm", "binary_search", "information_theory", "combinatorial_search", "decision_tree" ]
100,350	[ "pedal_triangle", "perpendicular", "concurrent_lines", "orthogonal_transformation", "triangle", "isogonal_conjugate", "circle", "circumcircle" ]
100,340	[ "determinant", "complex_plane", "perpendicular", "excenter", "concurrent_lines", "symmetry_geometry", "foot_of_a_perpendicular", "cevas_theorem", "triangle", "incircle_and_excircles", "circumcircle", "orthocenter", "complex_number" ]
1,003,750	[ "binomial_theorem", "power_rule", "integral", "polynomial" ]
100,370	[ "complex_plane", "absolute_value", "pythagorean_theorem", "complex_number" ]
1,003,681	[ "integer", "exponentiation", "function_mathematics", "divisibility_rule", "prime_number", "diophantine_equation", "natural_number" ]
1,003,591	[ "dilation_operator_theory", "centrosymmetry", "polygon", "center_of_mass", "area", "parallel_geometry", "line_geometry", "reflection_mathematics" ]
1,003,557	[ "integer", "combinatorial_proof", "summation", "mathematical_induction", "binomial_theorem", "combinatorial_number_system", "counting", "a_group", "binomial_distribution" ]
100,382	[ "line_integral", "complex_plane", "entire_function", "moreras_theorem", "sine", "fubinis_theorem", "integral" ]
100,378	[ "algebraic_number", "recurrence_relation", "fundamental_solution", "diophantine_equation", "square_number", "pells_equation" ]
100,383	[ "sequence", "derivative", "borels_theorem", "real_number", "differentiable_function" ]
100,386	[ "liouvilles_theorem_complex_analysis", "complex_plane", "entire_function", "maximum_modulus_principle", "compact_space" ]
100,388	[ "cauchyschwarz_inequality", "positive_real_numbers", "amgm_inequality", "symmetric_polynomial" ]
100,394	[ "square_root", "jensens_inequality", "inequality_mathematics", "logarithm", "positive_real_numbers", "amgm_inequality", "convex_function" ]
1,004,076	[ "transitive_relation", "conversion_of_units", "ratio", "equation" ]
1,004,096	[ "inclusionexclusion_principle", "indian_buffet_process", "venn_diagram" ]
100,413	[ "interval_mathematics", "riemann_integral", "zero_function", "darboux_integral", "continuous_function", "fundamental_theorem_of_calculus", "integral" ]
100,419	[ "square_root", "amgm_inequality", "real_number", "inequality_mathematics" ]
100,415	[ "identity_element", "dihedral_group", "order_group_theory", "direct_product_of_groups", "cyclic_group", "group_mathematics" ]
100,428	[ "infinite_product", "exponentiation", "decimal_floating_point", "real_number", "limit_of_a_sequence", "mathematical_induction" ]
1,004,329	[ "natural_number", "irreducible_fraction", "greatest_common_divisor", "euclidean_algorithm" ]
100,434	[ "exponential_function", "real_number", "inequality_mathematics", "bernoullis_inequality" ]
100,436	[ "factorial", "binomial_theorem", "natural_number", "amgm_inequality", "inequality_mathematics" ]
100,414	[ "integer", "inequality_mathematics", "pigeonhole_principle", "divisibility_rule", "prime_number", "natural_number", "congruence_relation", "square_number" ]
100,437	[ "integer", "divisibility_rule", "quadratic_form", "modular_arithmetic", "diophantine_equation", "pells_equation" ]
1,004,497	[ "partial_fraction_decomposition", "convergent_series", "telescoping_series", "summation", "factorization_of_polynomials" ]

100,015

[ "lexicographic_order", "combination", "binomial_coefficient" ]

100,019

[ "permutation", "inclusionexclusion_principle", "factorial" ]

1,000,044

[ "binomial_distribution", "exponentiation", "binomial_coefficient" ]

100,023

[ "geometric_series", "mathematical_induction", "infinite_product", "power_series" ]

100,010

[ "word_problem_mathematics", "arithmetic_progression", "modulo", "relative_change" ]

1,000,141

[ "floor_and_ceiling_functions", "linear_equation", "coin_problem", "generating_function" ]

100,009

[ "differentiation_rules", "square_root", "cosine", "sine", "fundamental_theorem_of_calculus", "list_of_trigonometric_identities", "integral" ]

1,000,136

[ "square_root", "quadratic_equation", "quadratic_formula", "completing_the_square", "polynomial" ]

100,026

[ "vietas_formulas", "quadratic_equation", "polynomial_root_finding", "polynomial" ]

1,000,249

[ "nash_equilibrium", "prisoners_dilemma", "expected_value" ]

100,029

[ "inequality_mathematics", "factorial", "mathematical_induction" ]

100,036

[ "infinite_product", "convergence_mathematics", "divergence", "series_mathematics", "sequence", "absolute_value" ]

1,000,379

[ "power_series", "generating_function", "fibonacci_numbers", "radius_of_convergence", "mathematical_induction" ]

1,000,306

[ "sum_of_squares_function", "divisor_function", "modular_arithmetic", "square", "combinatorial_game_theory" ]

100,035

[ "eulers_sum_of_powers_conjecture", "exponentiation", "modular_arithmetic", "diophantine_equation", "inequality_mathematics", "fermats_last_theorem" ]

1,000,440

[ "vietas_formulas", "imaginary_unit", "polynomial_long_division", "polynomial", "polynomial_root_finding", "complex_number" ]

100,046

[ "random_permutation", "event_probability_theory", "permutation", "conditional_probability" ]

100,047

[ "fundamental_theorem_of_algebra", "real_number", "root_of_a_function" ]

100,052

[ "sequence", "functional_equation", "integer_sequence", "mathematical_induction", "square_number", "recurrence_relation" ]

100,067

[ "complex_conjugate", "rational_number", "real_number", "minimal_polynomial_linear_algebra", "sequence", "closed_set", "pisotvijayaraghavan_number" ]

100,071

[ "algebraic_identities", "jensens_inequality", "symmetric_function", "inequality_mathematics", "convex_function", "lagrange_multiplier", "second_partial_derivative_test" ]

100,061

[ "interval_mathematics", "cosine", "derivative", "inverse_function", "equation", "system_of_equations", "function_mathematics", "monotonic_function", "real_number", "sine", "even_and_odd_functions" ]

100,068

[ "integer_sequence", "sequence", "mathematical_induction", "recurrence_relation", "polynomial" ]

100,073

[ "binomial_coefficient", "wolstenholmes_theorem", "elementary_symmetric_polynomial", "prime_number", "finite_field", "fermats_little_theorem", "polynomial" ]

100,039

[ "binomial_coefficient", "probability_theory", "combination" ]

100,076

[ "cosine", "right_triangle", "angle_bisector_theorem", "tangent_half_angle_formula", "triangle", "law_of_cosines" ]

100,079

[ "binomial_coefficient", "fair_coin", "binomial_distribution" ]

100,075

[ "symmetric_matrix", "projection_linear_algebra", "hilbert_space", "orthogonal_transformation", "eigenvalues_and_eigenvectors", "matrix_norm" ]

100,074

[ "dominated_convergence_theorem", "lp_space", "fatous_lemma", "norm_mathematics", "pointwise_convergence", "positive_and_negative_parts" ]

100,090

[ "interior_topology", "countable_set", "cantor_set", "rational_number", "closed_set", "open_set", "number_line" ]

100,101

[ "integer", "divisibility_rule", "arithmetic_progression", "multiple_mathematics", "natural_number", "repdigit" ]

100,114

[ "jensens_inequality", "real_number", "inequality_mathematics", "concave_function" ]

100,130

[ "exponentiation", "inverse_function", "power_law", "inequality_mathematics" ]

100,133

[ "pythagorean_triple", "arithmetic_progression", "quartic_function", "rational_root_theorem", "geometric_progression" ]

100,119

[ "subset_sum_problem", "inequality_mathematics", "series_mathematics", "abels_identity", "integer", "sumset", "injective_function" ]

100,160

[ "diameter", "right_triangle", "power_of_a_point", "triangle_inequality", "hypotenuse", "circumcircle" ]

1,001,481

[ "residue_theorem", "complex_plane", "polynomial_interpolation", "lagrange_polynomial", "polynomial_root_finding", "contour_integration", "polynomial", "degree_of_a_polynomial" ]

1,001,633

[ "polynomial_mapping", "jacobian_conjecture", "surjective_function", "polynomial_function", "finite_field", "complex_coordinate_space", "injective_function", "complex_number" ]

100,202

[ "unit_circle", "homogeneous_function", "limit_mathematics", "polynomial" ]

100,172

[ "rectangle", "domino_tiling", "modulo", "parity_mathematics", "integer_partition", "recurrence_relation" ]

100,173

[ "diophantine_equation", "congruence_relation", "fourth_power" ]

100,226

[ "sum_of_squares_function", "integer", "quadratic_form", "diophantine_equation" ]

1,001,604

[ "polynomial", "divisibility_rule", "root_of_unity", "cyclotomic_polynomial", "prime_power" ]

1,002,173

[ "algebraic_number", "exponentiation", "equation_solving", "polynomial", "nth_root", "radical_of_an_integer", "chebyshev_polynomials", "root_of_unity", "binomial_theorem", "symmetric_polynomial", "complex_number" ]

100,193

[ "algebraic_identities", "cosine", "inequality_mathematics", "triangle", "sine", "law_of_cosines", "symmetric_polynomial" ]

100,235

[ "proof_by_contradiction", "exponentiation", "modular_arithmetic", "division_algorithm", "divisibility_rule" ]

100,250

[ "equation", "logarithm", "exponential_function", "exponential_decay" ]

100,241

[ "hexomino", "triangular_number", "polyiamond", "combinatorial_geometry_in_the_plane", "equilateral_triangle", "geometric_dissection" ]

100,229

[ "residue_number_system", "sum_of_squares_function", "divisibility_rule", "digit_sum", "quadratic_residue", "sum_of_two_squares_theorem" ]

100,257

[ "euclidean_algorithm", "irrational_number", "pi", "rational_number", "real_number", "limit_of_a_sequence", "sequence", "golden_ratio", "continued_fraction" ]

100,274

[ "divisor", "mersenne_prime", "prime_number", "euclideuler_theorem", "perfect_number" ]

100,259

[ "polynomial", "polynomial_long_division", "interpolation", "polynomial_remainder_theorem" ]

1,002,802

[ "power_set", "binomial_coefficient", "set_mathematics", "mathematical_induction", "union_set_theory", "expected_value" ]

100,277

[ "polynomial_root_finding", "triangle_inequality", "inequality_mathematics", "polynomial", "absolute_value", "complex_number" ]

1,002,828

[ "integer", "factorial", "geometric_series", "generating_function", "natural_number", "multinomial_theorem" ]

1,002,871

[ "binomial_coefficient", "summation", "arithmetic_mean", "exponentiation", "binomial_theorem", "inequality_mathematics" ]

100,287

[ "jensens_inequality", "logarithm", "real_number", "inequality_mathematics", "convex_function" ]

1,002,900

[ "sequence", "floor_and_ceiling_functions", "inequality_mathematics", "proof_by_contradiction", "functional_equation", "riemann_zeta_function", "natural_number", "mathematical_induction" ]

100,294

[ "summation", "series_mathematics", "inequality_mathematics", "sequence" ]

100,297

[ "fundamental_theorem_of_arithmetic", "function_mathematics", "multiplicative_function", "prime_number", "unique_factorization_domain", "natural_number" ]

100,299

[ "inequality_mathematics", "unit_circle", "complex_plane", "root_of_unity", "triangle_inequality", "limit_of_a_sequence", "rotation", "absolute_value", "complex_number" ]

1,002,990

[ "integer", "binomial_coefficient", "permutation", "divisibility_rule", "cyclic_group", "necklace_combinatorics", "greatest_common_divisor" ]

1,002,911

[ "tuple", "integer", "pigeonhole_principle", "proof_truth", "summation", "integer_sequence", "mathematical_induction", "greatest_common_divisor", "difference_of_two_squares" ]

1,002,943

[ "square", "rectangle", "acute_and_obtuse_triangles", "pigeonhole_principle", "cyclic_quadrilateral", "rhombus", "angle", "similarity_system_of_triangles", "equilateral_triangle", "quadrilateral", "circle", "perpendicular_bisector", "parallelogram", "circumcenter", "isosceles_trapezoid" ]

100,300

[ "square", "similarity_system_of_triangles", "power_of_a_point", "triangle", "chord_geometry", "circumcircle" ]

1,002,994

[ "chinese_remainder_theorem", "integer", "dirichlets_theorem_on_arithmetic_progressions", "legendre_symbol", "arithmetic_progression", "prime_number", "square", "square_free_integer", "natural_number", "quadratic_reciprocity", "congruence_relation" ]

1,003,005

[ "square", "similarity_system_of_triangles", "straightedge_and_compass_construction", "perpendicular", "cyclic_quadrilateral", "angle", "triangle", "orthocenter" ]

1,003,043

[ "functional_equation", "function_mathematics", "real_number", "cubic_function" ]

1,003,111

[ "transposition_mathematics", "permutation", "triangle", "area", "inequality_mathematics", "quadrilateral" ]

10,032

[ "binomial_coefficient", "mathematical_induction", "binomial_distribution", "geometric_distribution" ]

1,003,040

[ "muirheads_inequality", "area_of_a_triangle", "schurs_inequality", "circumradius", "triangle_inequality", "feuerbach_point", "amgm_inequality", "triangle", "nine_point_circle", "inequality_mathematics", "inradius" ]

100,332

[ "square_root", "integer", "floor_and_ceiling_functions", "real_number", "inequality_mathematics", "piecewise_function" ]

100,336

[ "modular_arithmetic", "divisibility_rule", "prime_number", "diophantine_equation", "natural_number", "greatest_common_divisor" ]

100,318

[ "concyclic_points", "radical_axis", "angle_bisector_theorem", "incenter", "central_angle", "triangle", "inscribed_angle", "tangent", "circumcircle", "orthocenter", "circumcenter", "isosceles_triangle" ]

10,035

[ "divide_and_conquer_algorithm", "binary_search", "information_theory", "combinatorial_search", "decision_tree" ]

100,350

[ "pedal_triangle", "perpendicular", "concurrent_lines", "orthogonal_transformation", "triangle", "isogonal_conjugate", "circle", "circumcircle" ]

100,340

[ "determinant", "complex_plane", "perpendicular", "excenter", "concurrent_lines", "symmetry_geometry", "foot_of_a_perpendicular", "cevas_theorem", "triangle", "incircle_and_excircles", "circumcircle", "orthocenter", "complex_number" ]

1,003,750

[ "binomial_theorem", "power_rule", "integral", "polynomial" ]

100,370

[ "complex_plane", "absolute_value", "pythagorean_theorem", "complex_number" ]

1,003,681

[ "integer", "exponentiation", "function_mathematics", "divisibility_rule", "prime_number", "diophantine_equation", "natural_number" ]

1,003,591

[ "dilation_operator_theory", "centrosymmetry", "polygon", "center_of_mass", "area", "parallel_geometry", "line_geometry", "reflection_mathematics" ]

1,003,557

[ "integer", "combinatorial_proof", "summation", "mathematical_induction", "binomial_theorem", "combinatorial_number_system", "counting", "a_group", "binomial_distribution" ]

100,382

[ "line_integral", "complex_plane", "entire_function", "moreras_theorem", "sine", "fubinis_theorem", "integral" ]

100,378

[ "algebraic_number", "recurrence_relation", "fundamental_solution", "diophantine_equation", "square_number", "pells_equation" ]

100,383

[ "sequence", "derivative", "borels_theorem", "real_number", "differentiable_function" ]

100,386

[ "liouvilles_theorem_complex_analysis", "complex_plane", "entire_function", "maximum_modulus_principle", "compact_space" ]

100,388

[ "cauchyschwarz_inequality", "positive_real_numbers", "amgm_inequality", "symmetric_polynomial" ]

100,394

[ "square_root", "jensens_inequality", "inequality_mathematics", "logarithm", "positive_real_numbers", "amgm_inequality", "convex_function" ]

1,004,076

[ "transitive_relation", "conversion_of_units", "ratio", "equation" ]

1,004,096

[ "inclusionexclusion_principle", "indian_buffet_process", "venn_diagram" ]

100,413

[ "interval_mathematics", "riemann_integral", "zero_function", "darboux_integral", "continuous_function", "fundamental_theorem_of_calculus", "integral" ]

100,419

[ "square_root", "amgm_inequality", "real_number", "inequality_mathematics" ]

100,415

[ "identity_element", "dihedral_group", "order_group_theory", "direct_product_of_groups", "cyclic_group", "group_mathematics" ]

100,428

[ "infinite_product", "exponentiation", "decimal_floating_point", "real_number", "limit_of_a_sequence", "mathematical_induction" ]

1,004,329

[ "natural_number", "irreducible_fraction", "greatest_common_divisor", "euclidean_algorithm" ]

100,434

[ "exponential_function", "real_number", "inequality_mathematics", "bernoullis_inequality" ]

100,436

[ "factorial", "binomial_theorem", "natural_number", "amgm_inequality", "inequality_mathematics" ]

100,414

[ "integer", "inequality_mathematics", "pigeonhole_principle", "divisibility_rule", "prime_number", "natural_number", "congruence_relation", "square_number" ]

100,437

[ "integer", "divisibility_rule", "quadratic_form", "modular_arithmetic", "diophantine_equation", "pells_equation" ]

1,004,497

[ "partial_fraction_decomposition", "convergent_series", "telescoping_series", "summation", "factorization_of_polynomials" ]