query string | positive_passages list | negative_passages list | candidate_id string | source_doc_id string | difficulty string | length string | domains list | split_candidates list | positive_splits list | used_cluster_ids list | used_path_ids list | batch_custom_id string | sample_set_id string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Decompose this three-tone sound using a time-to-frequency analysis and report each component’s frequency, magnitude, and phase. | [
{
"doc_id": "camel_44852",
"title": "",
"text": "A sound signal is given by the equation f(t) = sin(2πt) + sin(4πt) + sin(6πt) where t is time in seconds. Use Fourier transform to find the frequencies, amplitudes, and phases of the individual sinusoidal components in the signal.\nTo find the frequencies... | [
{
"doc_id": "camel_44538",
"title": "",
"text": "A signal has a frequency content of 5 Hz, 10 Hz, and 15 Hz. Using the Fourier transform, find the amplitude and phase angle of each frequency component.\nTo find the amplitude and phase angle of each frequency component using the Fourier transform, we nee... | cbf3675cf60ae6c4336a | camel_44852 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_02",
"cluster_05",
"cluster_06"
] | [
"path_eef1cda66f5e5a01",
"path_e54f0045602caf06",
"path_7836a3572f136711"
] | qgen-000000001-00-1eec2ea4e844 | set_00 |
How can a sum of three pure tones be decomposed into frequencies, amplitudes, and phase offsets using linearity? | [
{
"doc_id": "camel_44852",
"title": "",
"text": "A sound signal is given by the equation f(t) = sin(2πt) + sin(4πt) + sin(6πt) where t is time in seconds. Use Fourier transform to find the frequencies, amplitudes, and phases of the individual sinusoidal components in the signal.\nTo find the frequencies... | [
{
"doc_id": "camel_44807",
"title": "",
"text": "A voltage signal has a frequency content of 50 Hz, 100 Hz, and 150 Hz with amplitudes of 10 V, 5 V, and 3 V respectively. Using Fourier transform, determine the frequency spectrum of the signal.\nTo determine the frequency spectrum of the given voltage si... | 77e98dad5413227866d2 | camel_44852 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_07"
] | [
"supplemental_13aba398001c7cbcf8b0"
] | qgen-000000001-01-069237394f78 | set_01 |
For an acoustic signal containing sinusoids at integer multiples of a base rate, how should its spectral peaks and phase information be identified? | [
{
"doc_id": "camel_44852",
"title": "",
"text": "A sound signal is given by the equation f(t) = sin(2πt) + sin(4πt) + sin(6πt) where t is time in seconds. Use Fourier transform to find the frequencies, amplitudes, and phases of the individual sinusoidal components in the signal.\nTo find the frequencies... | [
{
"doc_id": "camel_44854",
"title": "",
"text": "The question for the math student is:\n\nA system operates with an input signal given by x(t), where x(t) = cos(2π(50)t) for t ≥ 0. The system's output signal is y(t) = cos(2π(50)t + θ), where θ is random phase angle with uniform distribution in [0, 2π). ... | cb35da058a50cd81054b | camel_44852 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_05",
"cluster_06"
] | [
"path_5cd8d35c9706d4ef",
"path_7836a3572f136711"
] | qgen-000000001-01-069237394f78 | set_01 |
How does linearity of the time-to-frequency representation reveal the separate components of a three-tone signal, and what role does inverse analysis play in interpreting them? | [
{
"doc_id": "camel_44852",
"title": "",
"text": "A sound signal is given by the equation f(t) = sin(2πt) + sin(4πt) + sin(6πt) where t is time in seconds. Use Fourier transform to find the frequencies, amplitudes, and phases of the individual sinusoidal components in the signal.\nTo find the frequencies... | [
{
"doc_id": "camel_44521",
"title": "",
"text": "A signal has a frequency domain representation shown below:\n\n| Frequency (Hz) | Amplitude |\n|----------------|-----------|\n| 0 | 20 |\n| 1 | 10 |\n| 2 | 5 |\n| 3 | 2 |\n| 4 | 1 |\n\nCalculate the time-domain representation of the signal by performing ... | c8c7b9c6b387de53bb6b | camel_44852 | expert | long | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_02",
"cluster_03",
"cluster_07"
] | [
"path_bc61a8944c6a9c16",
"path_4de058cf15ae6986",
"supplemental_13aba398001c7cbcf8b0"
] | qgen-000000001-01-069237394f78 | set_01 |
How can frequency-domain analysis of a three-term sound signal determine each component’s frequency, amplitude, and phase? | [
{
"doc_id": "camel_44852",
"title": "",
"text": "A sound signal is given by the equation f(t) = sin(2πt) + sin(4πt) + sin(6πt) where t is time in seconds. Use Fourier transform to find the frequencies, amplitudes, and phases of the individual sinusoidal components in the signal.\nTo find the frequencies... | [
{
"doc_id": "camel_45811",
"title": "",
"text": "Consider a communication network where the signal is given by f(t) = cos(2πt) + sin(4πt) + cos(8πt). Use Fourier analysis to determine the frequency components of the signal and write the Fourier series of the signal. Specifically, find the amplitude and ... | 03f8b11859428cda7f21 | camel_44852 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_06",
"cluster_07"
] | [
"path_adaf148055424a22",
"supplemental_348495464d531ba91aa5"
] | qgen-000000001-02-a14ec3dc7314 | set_02 |
How do you find the total exposed area of a solid with a circular base, given base radius 6 cm and slant length 10 cm? | [
{
"doc_id": "camel_3452",
"title": "",
"text": "A right circular cone has a radius of 6cm and a slant height of 10cm. Determine the surface area of the cone.\nTo find the surface area of a right circular cone, we need to calculate the area of the base and the lateral surface area, and then add them toge... | [
{
"doc_id": "camel_3449",
"title": "",
"text": "Sure! Here's a precise numerical problem for the student to solve:\n\nA right circular cone has a base radius of 5cm and a slant height of 10cm. Find its total surface area.\nTo find the total surface area of a right circular cone, we need to find the sum ... | eb1220d3007048f370df | camel_3452 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_04"
] | [
"supplemental_6ef69734e7e5df586a8e"
] | qgen-000000002-00-2168f5cd45e4 | set_00 |
For a solid with a circular base, central perpendicular axis, radius 6 cm, and slant length 10 cm, how is total surface area calculated? | [
{
"doc_id": "camel_3452",
"title": "",
"text": "A right circular cone has a radius of 6cm and a slant height of 10cm. Determine the surface area of the cone.\nTo find the surface area of a right circular cone, we need to calculate the area of the base and the lateral surface area, and then add them toge... | [
{
"doc_id": "camel_3441",
"title": "",
"text": "A right circular cone has a radius of 5 cm and a slant height of 13 cm. Find its surface area correct to two decimal places.\nTo find the surface area of a right circular cone, we need to calculate the area of the base and the lateral surface area, and the... | 0606102460bab4648da5 | camel_3452 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_04"
] | [
"supplemental_8091e14a18ed0d265c02"
] | qgen-000000002-01-b115aa1e5d6e | set_01 |
How is the total surface area found for a solid with a circular base and slant height? | [
{
"doc_id": "camel_3452",
"title": "",
"text": "A right circular cone has a radius of 6cm and a slant height of 10cm. Determine the surface area of the cone.\nTo find the surface area of a right circular cone, we need to calculate the area of the base and the lateral surface area, and then add them toge... | [
{
"doc_id": "camel_3449",
"title": "",
"text": "Sure! Here's a precise numerical problem for the student to solve:\n\nA right circular cone has a base radius of 5cm and a slant height of 10cm. Find its total surface area.\nTo find the total surface area of a right circular cone, we need to find the sum ... | 611e8ef4b13b19649572 | camel_3452 | medium | short | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_04"
] | [
"supplemental_6ef69734e7e5df586a8e"
] | qgen-000000002-02-48ab94744071 | set_02 |
How can analytic geometry locate the point on a segment whose distance from one endpoint is twice its distance from the other? | [
{
"doc_id": "aqua_rat_78473",
"title": "",
"text": "On the xy co-ordinate plane, point C is (5,-2) and point D is (-1,1.5). The point on line segment CD that is twice as far from C as from D is:\nAnswer Choices: (A) (1,-1) (B) (1,1) (C) (2,0.25) (D) (3,0.5) (E) (3,1) \nLet's think about the mul... | [
{
"doc_id": "aqua_rat_35190",
"title": "",
"text": "On the xy co-ordinate plane, point C is (4,-2) and point D is (-1,2.5). The point on line segment CD that is twice as far from C as from D is:\nAnswer Choices: (A) (1,-1) (B) (1,1) (C) (2,0.25) (D) (3,0.5) (E) (3,1) \nLet's reason about the mu... | 56ff388125bcfb2d3b54 | aqua_rat_78473 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_02"
] | [
"path_c11d15b0fb3f4c24"
] | qgen-000000003-00-28611b3038a0 | set_00 |
Given two endpoints in a plane, which internal division formula finds the point where the first-to-second distance ratio equals two to one? | [
{
"doc_id": "aqua_rat_78473",
"title": "",
"text": "On the xy co-ordinate plane, point C is (5,-2) and point D is (-1,1.5). The point on line segment CD that is twice as far from C as from D is:\nAnswer Choices: (A) (1,-1) (B) (1,1) (C) (2,0.25) (D) (3,0.5) (E) (3,1) \nLet's think about the mul... | [
{
"doc_id": "camel_19739",
"title": "",
"text": "Suppose that you have a sphere of radius 5 centered at the origin in 3-dimensional space, and a plane defined by the equation 2x + y - z = 6. What is the shortest distance between a point on the sphere and a point on the plane?\nTo find the shortest dista... | b229594887d00abd52ee | aqua_rat_78473 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_02",
"cluster_07"
] | [
"path_c11d15b0fb3f4c24",
"supplemental_620280760fd906c2b024"
] | qgen-000000003-00-28611b3038a0 | set_00 |
On a coordinate plane, how can the point dividing a segment in a 2:1 distance ratio be found? | [
{
"doc_id": "aqua_rat_78473",
"title": "",
"text": "On the xy co-ordinate plane, point C is (5,-2) and point D is (-1,1.5). The point on line segment CD that is twice as far from C as from D is:\nAnswer Choices: (A) (1,-1) (B) (1,1) (C) (2,0.25) (D) (3,0.5) (E) (3,1) \nLet's think about the mul... | [
{
"doc_id": "aqua_rat_35190",
"title": "",
"text": "On the xy co-ordinate plane, point C is (4,-2) and point D is (-1,2.5). The point on line segment CD that is twice as far from C as from D is:\nAnswer Choices: (A) (1,-1) (B) (1,1) (C) (2,0.25) (D) (3,0.5) (E) (3,1) \nLet's reason about the mu... | d2665d5633fdbc0d4cdf | aqua_rat_78473 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_07"
] | [
"supplemental_cceb4ad19252e541ac68"
] | qgen-000000003-01-1335e4993314 | set_01 |
How can coordinate geometry locate the point dividing a segment in a two-to-one distance ratio? | [
{
"doc_id": "aqua_rat_78473",
"title": "",
"text": "On the xy co-ordinate plane, point C is (5,-2) and point D is (-1,1.5). The point on line segment CD that is twice as far from C as from D is:\nAnswer Choices: (A) (1,-1) (B) (1,1) (C) (2,0.25) (D) (3,0.5) (E) (3,1) \nLet's think about the mul... | [
{
"doc_id": "aqua_rat_35190",
"title": "",
"text": "On the xy co-ordinate plane, point C is (4,-2) and point D is (-1,2.5). The point on line segment CD that is twice as far from C as from D is:\nAnswer Choices: (A) (1,-1) (B) (1,1) (C) (2,0.25) (D) (3,0.5) (E) (3,1) \nLet's reason about the mu... | 5fe2c914d4f205542e18 | aqua_rat_78473 | medium | short | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_07"
] | [
"supplemental_bc9241e34ee553a31c41"
] | qgen-000000003-02-0460d9739bd2 | set_02 |
How can trigonometric relationships in a right triangle determine the missing leg when one leg is 5 and the hypotenuse is 13? | [
{
"doc_id": "camel_4040",
"title": "",
"text": "A triangle has a leg of length 5 and a hypotenuse of length 13. Find the length of the other leg using trigonometry.\nWe can use the Pythagorean theorem to find the length of the other leg. The Pythagorean theorem states that in a right-angled triangle, th... | [
{
"doc_id": "camel_2846",
"title": "",
"text": "A right triangle has one leg with a length of 5 cm, and the other leg with a length of 12 cm. Determine the length of the hypotenuse using the Pythagorean theorem.\nThe Pythagorean theorem states that in a right triangle, the square of the length of the hy... | 7658496952b16e814c72 | camel_4040 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_00"
] | [
"path_60a563f82d7b6899"
] | qgen-000000004-00-4005d6e29bbf | set_00 |
In Euclidean geometry, how is the unknown perpendicular side calculated from side lengths 5 and 13 in a right triangle? | [
{
"doc_id": "camel_4040",
"title": "",
"text": "A triangle has a leg of length 5 and a hypotenuse of length 13. Find the length of the other leg using trigonometry.\nWe can use the Pythagorean theorem to find the length of the other leg. The Pythagorean theorem states that in a right-angled triangle, th... | [
{
"doc_id": "camel_4641",
"title": "",
"text": "A right-angled triangle has a base of 10cm and a hypotenuse of 15cm. Find the length of the perpendicular side using trigonometry to determine the perimeter of the triangle.\nTo find the length of the perpendicular side, we can use the Pythagorean theorem,... | 4bf0377d0b621d607d1a | camel_4040 | hard | short | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_02",
"cluster_06"
] | [
"path_476164e976ece978",
"supplemental_43307c4f6091fb277c10"
] | qgen-000000004-00-4005d6e29bbf | set_00 |
Given a right triangle with one leg 5 and hypotenuse 13, how is the remaining leg found? | [
{
"doc_id": "camel_4040",
"title": "",
"text": "A triangle has a leg of length 5 and a hypotenuse of length 13. Find the length of the other leg using trigonometry.\nWe can use the Pythagorean theorem to find the length of the other leg. The Pythagorean theorem states that in a right-angled triangle, th... | [
{
"doc_id": "camel_2803",
"title": "",
"text": "A right triangle has one leg measuring 12 cm and the other leg measuring 16 cm. What is the length of the hypotenuse of the triangle using the Pythagorean theorem?\nThe Pythagorean theorem states that in a right triangle, the square of the length of the hy... | 7ba723b0aa552878ba96 | camel_4040 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_02",
"cluster_06"
] | [
"path_b49b55f690f60b7c",
"supplemental_8f6d90b9b8aa0119e4de"
] | qgen-000000004-01-661c624c0e02 | set_01 |
How can Euclidean geometry determine the missing leg when a right triangle has sides 5 and 13? | [
{
"doc_id": "camel_4040",
"title": "",
"text": "A triangle has a leg of length 5 and a hypotenuse of length 13. Find the length of the other leg using trigonometry.\nWe can use the Pythagorean theorem to find the length of the other leg. The Pythagorean theorem states that in a right-angled triangle, th... | [
{
"doc_id": "camel_2846",
"title": "",
"text": "A right triangle has one leg with a length of 5 cm, and the other leg with a length of 12 cm. Determine the length of the hypotenuse using the Pythagorean theorem.\nThe Pythagorean theorem states that in a right triangle, the square of the length of the hy... | d24b04e5ed908bd4ad8b | camel_4040 | medium | short | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_06",
"cluster_02"
] | [
"supplemental_8f6d90b9b8aa0119e4de",
"path_476164e976ece978"
] | qgen-000000004-02-1b8a73021252 | set_02 |
Given one leg and the hypotenuse of a right triangle, what calculation finds the remaining leg? | [
{
"doc_id": "camel_4040",
"title": "",
"text": "A triangle has a leg of length 5 and a hypotenuse of length 13. Find the length of the other leg using trigonometry.\nWe can use the Pythagorean theorem to find the length of the other leg. The Pythagorean theorem states that in a right-angled triangle, th... | [
{
"doc_id": "camel_2803",
"title": "",
"text": "A right triangle has one leg measuring 12 cm and the other leg measuring 16 cm. What is the length of the hypotenuse of the triangle using the Pythagorean theorem?\nThe Pythagorean theorem states that in a right triangle, the square of the length of the hy... | 726ebc7a85e730a7cd18 | camel_4040 | medium | short | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_02"
] | [
"path_476164e976ece978"
] | qgen-000000004-02-1b8a73021252 | set_02 |
Why does a right triangle with side lengths 5 and 13 invite an integer-side calculation for its third side? | [
{
"doc_id": "camel_4040",
"title": "",
"text": "A triangle has a leg of length 5 and a hypotenuse of length 13. Find the length of the other leg using trigonometry.\nWe can use the Pythagorean theorem to find the length of the other leg. The Pythagorean theorem states that in a right-angled triangle, th... | [
{
"doc_id": "camel_4003",
"title": "",
"text": "A right triangle has a hypotenuse of length 10 cm and one other side of length 6 cm. Find the length of the third side.\nTo find the length of the third side, we can use the Pythagorean theorem, which states that in a right triangle, the square of the leng... | cbe13a84ed8969171c2b | camel_4040 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_03",
"cluster_06"
] | [
"path_d96b33aed8f6fa49",
"supplemental_8f6d90b9b8aa0119e4de"
] | qgen-000000004-02-1b8a73021252 | set_02 |
In Euclidean geometry, how can orthogonality and an integer right-triangle relation be combined to determine the unknown side opposite the right angle when the other sides are 5 and 13? | [
{
"doc_id": "camel_4040",
"title": "",
"text": "A triangle has a leg of length 5 and a hypotenuse of length 13. Find the length of the other leg using trigonometry.\nWe can use the Pythagorean theorem to find the length of the other leg. The Pythagorean theorem states that in a right-angled triangle, th... | [
{
"doc_id": "camel_4047",
"title": "",
"text": "A triangle has a right angle at one vertex and the length of the adjacent side is 6 cm while the length of the opposite side is 8 cm. Determine the length of the hypotenuse of the triangle using trigonometric ratios.\nTo determine the length of the hypoten... | 0450b5dab781ac862c4b | camel_4040 | hard | long | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_02",
"cluster_03",
"cluster_06"
] | [
"path_476164e976ece978",
"path_d96b33aed8f6fa49",
"supplemental_8f6d90b9b8aa0119e4de"
] | qgen-000000004-02-1b8a73021252 | set_02 |
For a simple graph with seven vertices and nine edges, what minimum edge-color count is required? | [
{
"doc_id": "camel_23489",
"title": "",
"text": "Consider a simple graph G with 7 vertices and 9 edges. Find the minimum number of colors needed to color the edges of the graph such that no adjacent edges have the same color.\nTo find the minimum number of colors needed to color the edges of the graph G... | [
{
"doc_id": "camel_23458",
"title": "",
"text": "A graph has 10 vertices and 15 edges. What is the minimum number of colors required to color the edges such that no two adjacent edges have the same color?\nTo find the minimum number of colors required to color the edges such that no two adjacent edges ... | 617986ad3451afa26fde | camel_23489 | medium | medium | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_00"
] | [
"path_abfe9aedb3f0bb94"
] | qgen-000000005-01-bf95d60efd4e | set_01 |
How does a graph-coloring proof use maximum vertex degree to bound the colors needed for nine edges? | [
{
"doc_id": "camel_23489",
"title": "",
"text": "Consider a simple graph G with 7 vertices and 9 edges. Find the minimum number of colors needed to color the edges of the graph such that no adjacent edges have the same color.\nTo find the minimum number of colors needed to color the edges of the graph G... | [
{
"doc_id": "camel_22158",
"title": "",
"text": "Consider a graph G that has eight vertices and twelve edges. Determine the minimum number of colors needed to color the vertices of G such that no two adjacent vertices have the same color.\nTo determine the minimum number of colors needed to color the ve... | ea0bf90451ca29245307 | camel_23489 | medium | medium | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_04"
] | [
"path_ee375337dad7bb88"
] | qgen-000000005-01-bf95d60efd4e | set_01 |
For a simple graph with seven vertices and nine edges, how many edge colors are required using a maximum-degree bound and contradiction-based reasoning? | [
{
"doc_id": "camel_23489",
"title": "",
"text": "Consider a simple graph G with 7 vertices and 9 edges. Find the minimum number of colors needed to color the edges of the graph such that no adjacent edges have the same color.\nTo find the minimum number of colors needed to color the edges of the graph G... | [
{
"doc_id": "camel_23458",
"title": "",
"text": "A graph has 10 vertices and 15 edges. What is the minimum number of colors required to color the edges such that no two adjacent edges have the same color?\nTo find the minimum number of colors required to color the edges such that no two adjacent edges ... | b8b5d4465bf0fa9099f5 | camel_23489 | medium | medium | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
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"cluster_04"
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"path_f3d9638edeb57158"
] | qgen-000000005-02-7a56ac81d8f8 | set_02 |
How can one determine the minimum proper vertex-color count for a 12-vertex graph whose every vertex has degree four? | [
{
"doc_id": "camel_22124",
"title": "",
"text": "A student needs to determine the chromatic number of a graph G, where G has 12 vertices and the degree of each vertex is 4. What is the minimum number of colors needed to color the vertices of G such that no two adjacent vertices have the same color?\nTo ... | [
{
"doc_id": "camel_21779",
"title": "",
"text": "Consider a graph with 6 vertices. What is the minimum number of colors needed to color the vertices of this graph if no two adjacent vertices can have the same color?\nTo determine the minimum number of colors needed to color the vertices of a graph with ... | 8d4abfaab25f614acb7a | camel_22124 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_07"
] | [
"supplemental_5098502956bf1c813508"
] | qgen-000000006-00-88a251718ce8 | set_00 |
Which maximum-degree coloring bound applies to a connected degree-four graph, and which structural exceptions must be checked? | [
{
"doc_id": "camel_22124",
"title": "",
"text": "A student needs to determine the chromatic number of a graph G, where G has 12 vertices and the degree of each vertex is 4. What is the minimum number of colors needed to color the vertices of G such that no two adjacent vertices have the same color?\nTo ... | [
{
"doc_id": "camel_21796",
"title": "",
"text": "Consider a simple undirected graph G with 9 vertices (labeled 1, 2, 3, ..., 9) and 20 edges. Determine the minimum number of distinct colors required to color each vertex of G such that no two adjacent vertices have the same color.\nTo determine the minim... | 9d292215ea5ef380c8b0 | camel_22124 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_02",
"cluster_07"
] | [
"path_264595a0fad10ec7",
"supplemental_5098502956bf1c813508"
] | qgen-000000006-00-88a251718ce8 | set_00 |
For a 12-vertex graph whose every vertex has degree four, how can the minimum proper coloring requirement be bounded? | [
{
"doc_id": "camel_22124",
"title": "",
"text": "A student needs to determine the chromatic number of a graph G, where G has 12 vertices and the degree of each vertex is 4. What is the minimum number of colors needed to color the vertices of G such that no two adjacent vertices have the same color?\nTo ... | [
{
"doc_id": "camel_21779",
"title": "",
"text": "Consider a graph with 6 vertices. What is the minimum number of colors needed to color the vertices of this graph if no two adjacent vertices can have the same color?\nTo determine the minimum number of colors needed to color the vertices of a graph with ... | 4bb24c02001e8001f7eb | camel_22124 | hard | short | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_04",
"cluster_07"
] | [
"path_a34a4830705bee77",
"supplemental_9e9a027e6bb8dcfadf5d"
] | qgen-000000006-02-6b8b04e2ea0b | set_02 |
What integral identity relates Gaussian curvature, boundary geodesic curvature, and the topology of a compact surface? | [
{
"doc_id": "camel_47441",
"title": "",
"text": "A surface S is a portion of a sphere with radius r=5 centered at the origin. Find the Gauss-Bonnet theorem for S.\nThe Gauss-Bonnet theorem relates the total Gaussian curvature of a surface S to its Euler characteristic χ(S) and can be expressed as:\n\n∫∫... | [
{
"doc_id": "camel_47505",
"title": "",
"text": "What is the Gauss-Bonnet theorem for a surface with constant positive curvature 2 and a circular boundary of radius 3?\nThe Gauss-Bonnet theorem relates the total curvature of a surface to its topology. For a compact surface with boundary, the theorem sta... | 27eddc16414e37e901f4 | camel_47441 | medium | short | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
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"aops",
"theoremqa_questions"
] | [
"cluster_04"
] | [
"supplemental_2e30152b573067e0715f"
] | qgen-000000009-00-79f6e7185379 | set_00 |
For a spherical patch of radius five, how should constant Gaussian curvature enter the curvature-topology formula when the patch has boundary? | [
{
"doc_id": "camel_47441",
"title": "",
"text": "A surface S is a portion of a sphere with radius r=5 centered at the origin. Find the Gauss-Bonnet theorem for S.\nThe Gauss-Bonnet theorem relates the total Gaussian curvature of a surface S to its Euler characteristic χ(S) and can be expressed as:\n\n∫∫... | [
{
"doc_id": "camel_47505",
"title": "",
"text": "What is the Gauss-Bonnet theorem for a surface with constant positive curvature 2 and a circular boundary of radius 3?\nThe Gauss-Bonnet theorem relates the total curvature of a surface to its topology. For a compact surface with boundary, the theorem sta... | cf42240fd427fe3d910e | camel_47441 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_04",
"cluster_06"
] | [
"supplemental_2e30152b573067e0715f",
"supplemental_4ce74839ee35e266a1f6"
] | qgen-000000009-00-79f6e7185379 | set_00 |
How does the topology of a compact two-dimensional surface, including handles or removed disks, determine the topological term in its curvature integral identity? | [
{
"doc_id": "camel_47441",
"title": "",
"text": "A surface S is a portion of a sphere with radius r=5 centered at the origin. Find the Gauss-Bonnet theorem for S.\nThe Gauss-Bonnet theorem relates the total Gaussian curvature of a surface S to its Euler characteristic χ(S) and can be expressed as:\n\n∫∫... | [
{
"doc_id": "camel_46599",
"title": "",
"text": "Let us consider a 2-dimensional sphere with radius 4 units. Determine the Ricci curvature of the sphere at a point (2, 2) on the surface of the sphere.\nTo determine the Ricci curvature of a 2-dimensional sphere, we first need to understand that the Ricci... | a010e2c05624e1ef39d8 | camel_47441 | expert | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
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"cluster_03",
"cluster_04",
"cluster_06"
] | [
"supplemental_421a6b224e5efafe1875",
"supplemental_2e30152b573067e0715f",
"supplemental_4ce74839ee35e266a1f6"
] | qgen-000000009-00-79f6e7185379 | set_00 |
What integral relation connects Gaussian curvature, boundary geodesic curvature, and a surface’s topological invariant? | [
{
"doc_id": "camel_47441",
"title": "",
"text": "A surface S is a portion of a sphere with radius r=5 centered at the origin. Find the Gauss-Bonnet theorem for S.\nThe Gauss-Bonnet theorem relates the total Gaussian curvature of a surface S to its Euler characteristic χ(S) and can be expressed as:\n\n∫∫... | [
{
"doc_id": "camel_47519",
"title": "",
"text": "What is the Euler characteristic of a hemisphere with a radius of 5 cm, and how does it relate to the total sum of angles of its geodesic triangles? Use the Gauss-Bonnet theorem to answer.\nThe Euler characteristic (χ) of a surface is a topological invari... | c83e57177a8ceef6b39e | camel_47441 | medium | short | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_04"
] | [
"supplemental_2e30152b573067e0715f"
] | qgen-000000009-01-ba549a17b145 | set_01 |
For a spherical portion of radius five, how does constant Gaussian curvature combine with topology to determine its curvature integral? | [
{
"doc_id": "camel_47441",
"title": "",
"text": "A surface S is a portion of a sphere with radius r=5 centered at the origin. Find the Gauss-Bonnet theorem for S.\nThe Gauss-Bonnet theorem relates the total Gaussian curvature of a surface S to its Euler characteristic χ(S) and can be expressed as:\n\n∫∫... | [
{
"doc_id": "camel_47494",
"title": "",
"text": "A spherical surface of radius 6 has a 2 cm hole drilled through it. Determine the total Gaussian curvature of the resulting surface.\nTo determine the total Gaussian curvature of the resulting surface, we first need to find the Gaussian curvature of the o... | ba328479c91b48dbfd05 | camel_47441 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_03",
"cluster_04"
] | [
"supplemental_3b1d73230ea22b5e1b2c",
"supplemental_2e30152b573067e0715f"
] | qgen-000000009-01-ba549a17b145 | set_01 |
How should the curvature formula be modified when a compact two-dimensional surface has a boundary, and what topological quantity appears on the right-hand side? | [
{
"doc_id": "camel_47441",
"title": "",
"text": "A surface S is a portion of a sphere with radius r=5 centered at the origin. Find the Gauss-Bonnet theorem for S.\nThe Gauss-Bonnet theorem relates the total Gaussian curvature of a surface S to its Euler characteristic χ(S) and can be expressed as:\n\n∫∫... | [
{
"doc_id": "camel_47445",
"title": "",
"text": "A student needs to find the Gauss-Bonnet theorem for a surface with the following characteristics:\n- The surface is a sphere with radius 2 units.\n- The surface is oriented counterclockwise.\n- The surface has a constant curvature of 1 unit.\n\nWhat is t... | 4d58214c712e1c2d6343 | camel_47441 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_04",
"cluster_05"
] | [
"supplemental_2e30152b573067e0715f",
"supplemental_bc26a34b49eab6f4a355"
] | qgen-000000009-01-ba549a17b145 | set_01 |
For a spherical patch of radius five, what geometric and topological data are needed to evaluate its curvature integral? | [
{
"doc_id": "camel_47441",
"title": "",
"text": "A surface S is a portion of a sphere with radius r=5 centered at the origin. Find the Gauss-Bonnet theorem for S.\nThe Gauss-Bonnet theorem relates the total Gaussian curvature of a surface S to its Euler characteristic χ(S) and can be expressed as:\n\n∫∫... | [
{
"doc_id": "camel_47510",
"title": "",
"text": "A spherical surface has a radius of 10 units. Using the Gauss-Bonnet theorem, calculate the total Gaussian curvature of the surface.\nThe Gauss-Bonnet theorem states that the total Gaussian curvature of a closed surface is equal to 2π times its Euler char... | ebad7b447e1a9204f20d | camel_47441 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_03",
"cluster_04"
] | [
"supplemental_421a6b224e5efafe1875",
"supplemental_2e30152b573067e0715f"
] | qgen-000000009-02-41cd13c70f01 | set_02 |
How does classifying a compact two-dimensional surface with possible boundary determine the invariant appearing in its curvature integral, and what additional boundary quantity may be required? | [
{
"doc_id": "camel_47441",
"title": "",
"text": "A surface S is a portion of a sphere with radius r=5 centered at the origin. Find the Gauss-Bonnet theorem for S.\nThe Gauss-Bonnet theorem relates the total Gaussian curvature of a surface S to its Euler characteristic χ(S) and can be expressed as:\n\n∫∫... | [
{
"doc_id": "camel_47445",
"title": "",
"text": "A student needs to find the Gauss-Bonnet theorem for a surface with the following characteristics:\n- The surface is a sphere with radius 2 units.\n- The surface is oriented counterclockwise.\n- The surface has a constant curvature of 1 unit.\n\nWhat is t... | 61a2df5bb0b17a246942 | camel_47441 | expert | long | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_03",
"cluster_04",
"cluster_06"
] | [
"supplemental_421a6b224e5efafe1875",
"supplemental_2e30152b573067e0715f",
"supplemental_83aaa10e99a9bfa97d46"
] | qgen-000000009-02-41cd13c70f01 | set_02 |
How can a weighted graph structure and greedy distance updates determine the minimum-distance route from A to F in this matrix? | [
{
"doc_id": "camel_22003",
"title": "",
"text": "A student needs to find the shortest path between two nodes \"A\" and \"F\" in a graph. The graph is represented by the following adjacency matrix:\n\n```\n A B C D E F\nA 0 3 5 2 6 1\nB 3 0 7 3 9 5\nC 5 7 0 4 8 6\nD 2 3 4... | [
{
"doc_id": "camel_21825",
"title": "",
"text": "In a graph G with 8 vertices, what is the minimum number of colors required to color the vertices such that no two adjacent vertices have the same color?\nTo determine the minimum number of colors required to color the vertices of a graph G with 8 vertice... | 13f1a7b2b0680792e788 | camel_22003 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_01",
"cluster_04"
] | [
"path_ea0b51a889c3f607",
"path_98c10d58629427cc"
] | qgen-000000010-00-743a2b2e56b0 | set_00 |
Using an algorithm for optimum routes through a network, what distance does the given weighted adjacency matrix yield from A to F? | [
{
"doc_id": "camel_22003",
"title": "",
"text": "A student needs to find the shortest path between two nodes \"A\" and \"F\" in a graph. The graph is represented by the following adjacency matrix:\n\n```\n A B C D E F\nA 0 3 5 2 6 1\nB 3 0 7 3 9 5\nC 5 7 0 4 8 6\nD 2 3 4... | [
{
"doc_id": "camel_22343",
"title": "",
"text": "Consider a graph with 8 vertices and the following adjacency matrix:\n\n 0 1 1 0 0 0 0 0 \n 1 0 1 1 1 0 0 0 \n 1 1 0 0 1 1 1 0 \n 0 1 0 0 1 1 0 0 \n 0 1 1 1 0 1 1 1 \n 0 0 1 1 1 0 1 0 \n 0 0 1 0 1 1 0 1 \n 0 0 0 0 1 0 1 0 \n\nWhat is the diameter of this ... | 0d0aa51828657ac05223 | camel_22003 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_06"
] | [
"supplemental_d9b98b9a7988db42df76"
] | qgen-000000010-00-743a2b2e56b0 | set_00 |
Using a weighted adjacency matrix, how can one determine the minimum route cost from A to F? | [
{
"doc_id": "camel_22003",
"title": "",
"text": "A student needs to find the shortest path between two nodes \"A\" and \"F\" in a graph. The graph is represented by the following adjacency matrix:\n\n```\n A B C D E F\nA 0 3 5 2 6 1\nB 3 0 7 3 9 5\nC 5 7 0 4 8 6\nD 2 3 4... | [
{
"doc_id": "camel_22390",
"title": "",
"text": "Consider a graph G with 9 vertices and the following adjacency matrix: \n\n```\n0 1 1 1 0 0 0 0 0 \n1 0 0 0 1 1 0 0 0 \n1 0 0 0 0 1 0 0 0 \n1 0 0 0 0 1 1 0 0 \n0 1 0 0 0 0 1 1 0 \n0 1 1 1 0 0 0 1 1 \n0 0 0 1 1 0 0 0 1 \n0 0 0 0 1 1 0 0 1\n0 0 0 0 0 1 1 1 ... | 4a27d6d1abb18066b651 | camel_22003 | medium | short | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_04"
] | [
"path_8e6bac585ebf7c6d"
] | qgen-000000010-01-6d6ef5f0b0b3 | set_01 |
Given this weighted graph matrix, what minimum total edge weight connects node A to node F, and which route achieves it? | [
{
"doc_id": "camel_22003",
"title": "",
"text": "A student needs to find the shortest path between two nodes \"A\" and \"F\" in a graph. The graph is represented by the following adjacency matrix:\n\n```\n A B C D E F\nA 0 3 5 2 6 1\nB 3 0 7 3 9 5\nC 5 7 0 4 8 6\nD 2 3 4... | [
{
"doc_id": "camel_22295",
"title": "",
"text": "Consider the following undirected weighted graph with 6 vertices and 9 edges:\n\n\n\nFind the minimum spanning tree of the given graph using Prim’s algorithm. Report the total weight of the minimum spanning t... | d92cf834f2f116c76ac2 | camel_22003 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_04",
"cluster_06"
] | [
"path_8e6bac585ebf7c6d",
"supplemental_d9b98b9a7988db42df76"
] | qgen-000000010-01-6d6ef5f0b0b3 | set_01 |
Using the standard nonnegative-weight network procedure, determine the minimum route length from A to F in this adjacency matrix. | [
{
"doc_id": "camel_22003",
"title": "",
"text": "A student needs to find the shortest path between two nodes \"A\" and \"F\" in a graph. The graph is represented by the following adjacency matrix:\n\n```\n A B C D E F\nA 0 3 5 2 6 1\nB 3 0 7 3 9 5\nC 5 7 0 4 8 6\nD 2 3 4... | [
{
"doc_id": "camel_22911",
"title": "",
"text": "Find the girth of the graph G, where G has 12 vertices and 18 edges.\nThe girth of a graph is the length of the shortest cycle in the graph. To find the girth, we need to know more information about the graph G, such as the adjacency matrix or the list of... | 59cb87d4ad43a62a4180 | camel_22003 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_06"
] | [
"supplemental_8300e1de6f2b4a7d3bf0"
] | qgen-000000010-02-81c32cf3f2e5 | set_02 |
How can a finite graph-search procedure systematically update tentative distances to determine the minimum route from A to F in the given weighted matrix? | [
{
"doc_id": "camel_22003",
"title": "",
"text": "A student needs to find the shortest path between two nodes \"A\" and \"F\" in a graph. The graph is represented by the following adjacency matrix:\n\n```\n A B C D E F\nA 0 3 5 2 6 1\nB 3 0 7 3 9 5\nC 5 7 0 4 8 6\nD 2 3 4... | [
{
"doc_id": "camel_22012",
"title": "",
"text": "Consider the following weighted graph,\n\n\n\nWhat is the shortest path and the corresponding distance between vertex A and vertex D?\nTo find the shortest path between vertex A and vertex D, we can use Dijkstra's ... | 585f60a838cfa4c16f0d | camel_22003 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_07"
] | [
"supplemental_7518ec02f8995d764a76"
] | qgen-000000010-02-81c32cf3f2e5 | set_02 |
How do you calculate the volume of a metal ball from its diameter using the standard geometric formula and the constant approximately 3.14159? | [
{
"doc_id": "camel_2091",
"title": "",
"text": "A metal ball has a diameter of 10 cm. Find the volume of the sphere.\nTo find the volume of a sphere, we use the formula:\n\nVolume (V) = (4/3) * π * r^3\n\nwhere r is the radius of the sphere.\n\nGiven that the diameter of the metal ball is 10 cm, its rad... | [
{
"doc_id": "camel_2102",
"title": "",
"text": "A metal ball has a diameter of 10 cm. What is the volume of the metal used to make the ball?\nTo find the volume of the metal ball, we can use the formula for the volume of a sphere:\n\nVolume = (4/3)πr^3\n\nwhere r is the radius of the sphere.\n\nSince th... | fe56d2bf71c476695265 | camel_2091 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_06",
"cluster_07"
] | [
"supplemental_51bb266e6b3c7dddf14d",
"supplemental_2029361fa1b62cebcc1a"
] | qgen-000000011-00-f21899206aea | set_00 |
What volume results for a metal ball measuring 10 centimeters across? | [
{
"doc_id": "camel_2091",
"title": "",
"text": "A metal ball has a diameter of 10 cm. Find the volume of the sphere.\nTo find the volume of a sphere, we use the formula:\n\nVolume (V) = (4/3) * π * r^3\n\nwhere r is the radius of the sphere.\n\nGiven that the diameter of the metal ball is 10 cm, its rad... | [
{
"doc_id": "camel_2102",
"title": "",
"text": "A metal ball has a diameter of 10 cm. What is the volume of the metal used to make the ball?\nTo find the volume of the metal ball, we can use the formula for the volume of a sphere:\n\nVolume = (4/3)πr^3\n\nwhere r is the radius of the sphere.\n\nSince th... | 3af256d9e60a9f5b99f6 | camel_2091 | medium | short | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_06"
] | [
"supplemental_51bb266e6b3c7dddf14d"
] | qgen-000000011-00-f21899206aea | set_00 |
A metal ball measures 10 cm across; what exact and decimal cubic-centimeter volume does it occupy? | [
{
"doc_id": "camel_2091",
"title": "",
"text": "A metal ball has a diameter of 10 cm. Find the volume of the sphere.\nTo find the volume of a sphere, we use the formula:\n\nVolume (V) = (4/3) * π * r^3\n\nwhere r is the radius of the sphere.\n\nGiven that the diameter of the metal ball is 10 cm, its rad... | [
{
"doc_id": "camel_2102",
"title": "",
"text": "A metal ball has a diameter of 10 cm. What is the volume of the metal used to make the ball?\nTo find the volume of the metal ball, we can use the formula for the volume of a sphere:\n\nVolume = (4/3)πr^3\n\nwhere r is the radius of the sphere.\n\nSince th... | 41f659a0b1e9831e62cc | camel_2091 | hard | short | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_06",
"cluster_07"
] | [
"supplemental_51bb266e6b3c7dddf14d",
"supplemental_c657955b48f618289584"
] | qgen-000000011-01-9e609d528ae9 | set_01 |
How can the volume of a round solid be computed from its diameter, including an exact expression and decimal approximation? | [
{
"doc_id": "camel_2091",
"title": "",
"text": "A metal ball has a diameter of 10 cm. Find the volume of the sphere.\nTo find the volume of a sphere, we use the formula:\n\nVolume (V) = (4/3) * π * r^3\n\nwhere r is the radius of the sphere.\n\nGiven that the diameter of the metal ball is 10 cm, its rad... | [
{
"doc_id": "camel_3370",
"title": "",
"text": "A cube has an edge length of 4 cm. Find the volume of the sphere inscribed inside the cube. (Round your answer to two decimal places.)\nTo find the volume of the inscribed sphere, we first need to find its radius. Since the sphere is inscribed inside the c... | b93dff2ee430a7d08c49 | camel_2091 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_06",
"cluster_07"
] | [
"supplemental_51bb266e6b3c7dddf14d",
"supplemental_c657955b48f618289584"
] | qgen-000000011-01-9e609d528ae9 | set_01 |
How can the volume of a round object be found from its diameter using an irrational constant? | [
{
"doc_id": "camel_2091",
"title": "",
"text": "A metal ball has a diameter of 10 cm. Find the volume of the sphere.\nTo find the volume of a sphere, we use the formula:\n\nVolume (V) = (4/3) * π * r^3\n\nwhere r is the radius of the sphere.\n\nGiven that the diameter of the metal ball is 10 cm, its rad... | [
{
"doc_id": "camel_3400",
"title": "",
"text": "A cube has a volume of 512 cubic units. What is the volume of the sphere inscribed in the cube? Round your answer to the nearest hundredth.\nTo find the volume of the inscribed sphere, we first need to find the side length of the cube. Since the volume of ... | b669094a68eded1faf84 | camel_2091 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_06",
"cluster_07"
] | [
"supplemental_19cf950182800409fdd5",
"supplemental_c657955b48f618289584"
] | qgen-000000011-02-08b3f373eae7 | set_02 |
What volume results for a round metal object measuring 10 centimeters across? | [
{
"doc_id": "camel_2091",
"title": "",
"text": "A metal ball has a diameter of 10 cm. Find the volume of the sphere.\nTo find the volume of a sphere, we use the formula:\n\nVolume (V) = (4/3) * π * r^3\n\nwhere r is the radius of the sphere.\n\nGiven that the diameter of the metal ball is 10 cm, its rad... | [
{
"doc_id": "camel_2084",
"title": "",
"text": "A metal sphere has a radius of 6 cm. Find its volume (in cubic centimeters) using the formula for the volume of a sphere.\nThe formula for the volume of a sphere is V = (4/3)πr^3, where V is the volume and r is the radius of the sphere.\n\nGiven that the r... | ad14a9e55cae0e5619f5 | camel_2091 | medium | short | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_06"
] | [
"supplemental_19cf950182800409fdd5"
] | qgen-000000011-02-08b3f373eae7 | set_02 |
How can the rotational inertia of a uniform solid circular rod be calculated from its radius, length, and material density? | [
{
"doc_id": "camel_45953",
"title": "",
"text": "A cylindrical rod of diameter 4 cm and length 20 cm is made of a material with density 5 g/cm³. If the rod is rotated about its central axis with an angular speed of 500 rpm, find the moment of inertia of the rod about its axis. Use Fourier analysis to ex... | [
{
"doc_id": "camel_44680",
"title": "",
"text": "A thin uniform rod of length 2 meters is suspended vertically from one end. The rod is then twisted by applying torque to its lower end such that the angle of twist at the upper end is 45 degrees. Use Fourier transform to calculate the torsion constant of... | 5ec04b4b491643f65bb0 | camel_45953 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_07",
"cluster_06"
] | [
"supplemental_c6ca933dbc6c94ec56b6",
"supplemental_058f3a87c588e20169eb"
] | qgen-000000012-00-63a20b9a8414 | set_00 |
How can the constant longitudinal mass distribution of a uniform rotating rod be represented using a spectral integral? | [
{
"doc_id": "camel_45953",
"title": "",
"text": "A cylindrical rod of diameter 4 cm and length 20 cm is made of a material with density 5 g/cm³. If the rod is rotated about its central axis with an angular speed of 500 rpm, find the moment of inertia of the rod about its axis. Use Fourier analysis to ex... | [
{
"doc_id": "camel_44680",
"title": "",
"text": "A thin uniform rod of length 2 meters is suspended vertically from one end. The rod is then twisted by applying torque to its lower end such that the angle of twist at the upper end is 45 degrees. Use Fourier transform to calculate the torsion constant of... | dde55023bec0d5161142 | camel_45953 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_00"
] | [
"path_814ea49454e02fbe"
] | qgen-000000012-00-63a20b9a8414 | set_00 |
How can the rotational resistance of a uniform solid body be calculated from its radius, length, and density about its symmetry axis? | [
{
"doc_id": "camel_45953",
"title": "",
"text": "A cylindrical rod of diameter 4 cm and length 20 cm is made of a material with density 5 g/cm³. If the rod is rotated about its central axis with an angular speed of 500 rpm, find the moment of inertia of the rod about its axis. Use Fourier analysis to ex... | [
{
"doc_id": "camel_47815",
"title": "",
"text": "Given a torus with radius of the small circle equal to 2 and radius of the large circle equal to 4, determine the symmetries of the manifold.\nA torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis copl... | 03a11ab720ebcefbfaa4 | camel_45953 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_06",
"cluster_07"
] | [
"supplemental_6a3a00d5bf709d711003",
"supplemental_29d5fc961f1a9c7575c7"
] | qgen-000000012-01-5a6d5d26cc50 | set_01 |
How can a uniform axial mass distribution for a rotating solid be represented with a frequency-domain integral, and what physical quantity does it determine? | [
{
"doc_id": "camel_45953",
"title": "",
"text": "A cylindrical rod of diameter 4 cm and length 20 cm is made of a material with density 5 g/cm³. If the rod is rotated about its central axis with an angular speed of 500 rpm, find the moment of inertia of the rod about its axis. Use Fourier analysis to ex... | [
{
"doc_id": "camel_45942",
"title": "",
"text": "A piece of metal has a thermal diffusivity, α, of 9.7 x 10^-5 m^2/s. The temperature at any point of the object can be described as a function of time, t, and distance from the surface, x, such that T(x,t) = 100e^-αx^2t. Using Fourier transform, find the ... | ae9e33edcf51d399bf9c | camel_45953 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_00",
"cluster_07"
] | [
"path_13eba1f731cc7682",
"supplemental_29d5fc961f1a9c7575c7"
] | qgen-000000012-01-5a6d5d26cc50 | set_01 |
How is the rotational inertia of a uniform solid rod computed from density, radius, and length? | [
{
"doc_id": "camel_45953",
"title": "",
"text": "A cylindrical rod of diameter 4 cm and length 20 cm is made of a material with density 5 g/cm³. If the rod is rotated about its central axis with an angular speed of 500 rpm, find the moment of inertia of the rod about its axis. Use Fourier analysis to ex... | [
{
"doc_id": "camel_44680",
"title": "",
"text": "A thin uniform rod of length 2 meters is suspended vertically from one end. The rod is then twisted by applying torque to its lower end such that the angle of twist at the upper end is 45 degrees. Use Fourier transform to calculate the torsion constant of... | a646beb3a5645db8db62 | camel_45953 | medium | short | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_06",
"cluster_07"
] | [
"supplemental_76fb1fac441d4341871a",
"supplemental_7f3c25ef4550122c517c"
] | qgen-000000012-02-05f598bfdde8 | set_02 |
How can a uniform rod’s mass distribution be represented through an integral transform when evaluating its rotational inertia? | [
{
"doc_id": "camel_45953",
"title": "",
"text": "A cylindrical rod of diameter 4 cm and length 20 cm is made of a material with density 5 g/cm³. If the rod is rotated about its central axis with an angular speed of 500 rpm, find the moment of inertia of the rod about its axis. Use Fourier analysis to ex... | [
{
"doc_id": "camel_44680",
"title": "",
"text": "A thin uniform rod of length 2 meters is suspended vertically from one end. The rod is then twisted by applying torque to its lower end such that the angle of twist at the upper end is 45 degrees. Use Fourier transform to calculate the torsion constant of... | 89848ce0db4b1cf6536d | camel_45953 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_00",
"cluster_06",
"cluster_07"
] | [
"path_72a290237777e438",
"supplemental_76fb1fac441d4341871a",
"supplemental_7f3c25ef4550122c517c"
] | qgen-000000012-02-05f598bfdde8 | set_02 |
For scores 78, 82, 85, 73, 92, 68, 79, 90, 88, and 80, find the median and middle-half spread. | [
{
"doc_id": "camel_8669",
"title": "",
"text": "A group of ten students completed a math test, and their scores are as follows: 78, 82, 85, 73, 92, 68, 79, 90, 88, and 80. Calculate the median score and the interquartile range.\nFirst, we need to arrange the scores in ascending order: 68, 73, 78, 79, 80... | [
{
"doc_id": "camel_8702",
"title": "",
"text": "A group of 10 students took a test where the maximum score was 100. The scores are as follows:\n\n60, 75, 85, 50, 40, 65, 80, 70, 65, 75\n\nCalculate the median, range, and interquartile range of the scores.\nFirst, we need to arrange the scores in ascendi... | 896c305d84601c7856fa | camel_8669 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_05"
] | [
"supplemental_121ac7135114cad4e41f"
] | qgen-000000013-00-fcf5293a6a24 | set_00 |
Given these ten test scores, determine the middle value and the spread between the lower and upper quartiles, showing each calculation. | [
{
"doc_id": "camel_8669",
"title": "",
"text": "A group of ten students completed a math test, and their scores are as follows: 78, 82, 85, 73, 92, 68, 79, 90, 88, and 80. Calculate the median score and the interquartile range.\nFirst, we need to arrange the scores in ascending order: 68, 73, 78, 79, 80... | [
{
"doc_id": "camel_9819",
"title": "",
"text": "A math class consists of 25 students with test scores ranging from 55 to 98. Determine if there are any outliers in the data using the Z-score and Tukey's method. Use a significance level of 0.05.\nTo determine if there are any outliers in the data using t... | aae5ff6e18bba6350bed | camel_8669 | hard | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_00",
"cluster_05"
] | [
"path_5d65f92a3aec64af",
"supplemental_25429806b04a5acd4c3a"
] | qgen-000000013-01-dd378897b6e1 | set_01 |
Given ten scores, sort them and compute the middle value plus the difference between upper and lower quartiles. | [
{
"doc_id": "camel_8669",
"title": "",
"text": "A group of ten students completed a math test, and their scores are as follows: 78, 82, 85, 73, 92, 68, 79, 90, 88, and 80. Calculate the median score and the interquartile range.\nFirst, we need to arrange the scores in ascending order: 68, 73, 78, 79, 80... | [
{
"doc_id": "camel_8702",
"title": "",
"text": "A group of 10 students took a test where the maximum score was 100. The scores are as follows:\n\n60, 75, 85, 50, 40, 65, 80, 70, 65, 75\n\nCalculate the median, range, and interquartile range of the scores.\nFirst, we need to arrange the scores in ascendi... | 6237b1409fcffa291318 | camel_8669 | medium | medium | [
"mathematics",
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"aops",
"theoremqa_questions"
] | [
"cluster_05"
] | [
"supplemental_37d6d853e1b048fc893d"
] | qgen-000000013-02-39194c6ec6f0 | set_02 |
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