Dataset Viewer
id
stringlengths 22
32
| setting_key
stringclasses 7
values | messages
listlengths 1
1
| ground_truth
stringlengths 1
79
| dataset
stringclasses 1
value | omega_config
stringclasses 7
values | omega_split
stringclasses 1
value |
|---|---|---|---|---|---|---|
trans_matrix_rank_test_000
|
trans_matrix_rank
|
[
{
"content": "\\text{Let } A_n \\text{ be an } n \\times n \\text{ matrix where } a_{ij} = 1 \\text{ if } i \\leq j, \\text{ else } 0. \\text{ Find } \\mathrm{rank}(A_n).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
n
|
math
|
trans_matrix_rank
|
test
|
trans_matrix_rank_test_001
|
trans_matrix_rank
|
[
{
"content": "\\text{Let } B_n \\text{ be } n \\times n, \\ b_{ij} = 1 \\text{ if } i + j \\leq n + 1, \\text{ else } 0. \\text{ Find } \\mathrm{rank}(B_n).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
n
|
math
|
trans_matrix_rank
|
test
|
trans_matrix_rank_test_002
|
trans_matrix_rank
|
[
{
"content": "\\text{Let } C_n \\text{ be } n \\times n, \\ c_{ij} = i \\times j. \\text{ Find } \\mathrm{rank}(C_n).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1
|
math
|
trans_matrix_rank
|
test
|
trans_matrix_rank_test_003
|
trans_matrix_rank
|
[
{
"content": "\\text{Let } D_n \\text{ be } n \\times n, \\ d_{ij} = \\begin{cases} 1 & \\text{if } i = j \\\\ 2 & \\text{if } |i - j| = 1 \\\\ 0 & \\text{otherwise} \\end{cases}. \\text{ Find } \\mathrm{rank}(D_n).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
n
|
math
|
trans_matrix_rank
|
test
|
trans_matrix_rank_test_004
|
trans_matrix_rank
|
[
{
"content": "\\text{Let } E_n \\text{ be } n \\times n, \\ e_{ij} = \\begin{cases} 1 & \\text{if } i + j \\text{ is even} \\\\ 0 & \\text{if } i + j \\text{ is odd} \\end{cases}. \\text{ Find } \\mathrm{rank}(E_n).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
math
|
trans_matrix_rank
|
test
|
trans_matrix_rank_test_005
|
trans_matrix_rank
|
[
{
"content": "\\text{Let } F_n \\text{ be } n \\times n, \\ f_{ij} = 2^{i + j}. \\text{ Find } \\mathrm{rank}(F_n).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1
|
math
|
trans_matrix_rank
|
test
|
trans_matrix_rank_test_006
|
trans_matrix_rank
|
[
{
"content": "\\text{Let } G_n \\text{ be } n \\times n, \\ g_{ij} = \\begin{cases} 1 & \\text{if } i = 1 \\text{ or } j = 1 \\\\ 0 & \\text{otherwise} \\end{cases}. \\text{ Find } \\mathrm{rank}(G_n).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
math
|
trans_matrix_rank
|
test
|
trans_matrix_rank_test_007
|
trans_matrix_rank
|
[
{
"content": "\\text{Let } H_n \\text{ be } n \\times n, \\ h_{ij} = \\begin{cases} 1 & \\text{if } i \\leq n/2 \\text{ and } j \\leq n/2 \\\\ 0 & \\text{otherwise} \\end{cases} \\ (\\text{n even}). \\text{ Find } \\mathrm{rank}(H_n).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1
|
math
|
trans_matrix_rank
|
test
|
trans_matrix_rank_test_008
|
trans_matrix_rank
|
[
{
"content": "\\text{Let } I_n \\text{ be } n \\times n, \\ i_{ij} = \\begin{cases} i & \\text{if } i = j \\\\ 0 & \\text{otherwise} \\end{cases}. \\text{ Find } \\mathrm{rank}(I_n).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
n
|
math
|
trans_matrix_rank
|
test
|
trans_matrix_rank_test_009
|
trans_matrix_rank
|
[
{
"content": "\\text{Let } J_n \\text{ be } n \\times n, \\ j_{ij} = \\begin{cases} 1 & \\text{if } i + j = n + 1 \\\\ 0 & \\text{otherwise} \\end{cases}. \\text{ Find } \\mathrm{rank}(J_n).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
n
|
math
|
trans_matrix_rank
|
test
|
trans_func_intersection_test_000
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 3g(f(sin(0.5 PI x))) and x = 5g(f(cos(1.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
23
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_001
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 3g(f(sin(1.5 PI x))) and x = 3g(f(sin(1.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
53
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_002
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 3g(f(sin(1.5 PI x))) and x = 4g(f(sin(1.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
45
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_003
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(cos(1.5 PI x))) and x = 4g(f(sin(2.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
109
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_004
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(sin(0.5 PI x))) and x = 4g(f(sin(2.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
40
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_005
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(cos(1.0 PI x))) and x = 4g(f(sin(2.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
104
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_006
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(sin(2.0 PI x))) and x = 5g(f(sin(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
144
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_007
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 3g(f(cos(1.5 PI x))) and x = 4g(f(cos(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
72
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_008
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(cos(0.5 PI x))) and x = 3g(f(sin(2.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
36
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_009
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 3g(f(cos(0.5 PI x))) and x = 4g(f(cos(2.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
36
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_010
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(sin(1.5 PI x))) and x = 3g(f(sin(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
81
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_011
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 5g(f(cos(0.5 PI x))) and x = 5g(f(cos(0.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
16
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_012
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 5g(f(cos(2.0 PI x))) and x = 5g(f(cos(0.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
71
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_013
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 5g(f(sin(0.5 PI x))) and x = 5g(f(cos(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
56
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_014
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(cos(1.0 PI x))) and x = 4g(f(cos(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
72
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_015
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 5g(f(cos(1.5 PI x))) and x = 5g(f(sin(2.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
157
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_016
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 5g(f(sin(0.5 PI x))) and x = 5g(f(cos(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
48
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_017
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(sin(2.0 PI x))) and x = 4g(f(sin(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
158
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_018
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(cos(0.5 PI x))) and x = 5g(f(cos(0.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
14
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_019
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(cos(0.5 PI x))) and x = 5g(f(cos(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
28
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_020
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(cos(1.0 PI x))) and x = 4g(f(cos(0.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
25
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_021
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(sin(1.0 PI x))) and x = 5g(f(sin(1.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
31
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_022
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(cos(0.5 PI x))) and x = 4g(f(cos(1.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
25
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_023
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(cos(0.5 PI x))) and x = 5g(f(cos(0.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
12
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_024
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(sin(1.5 PI x))) and x = 5g(f(cos(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
121
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_025
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(cos(2.0 PI x))) and x = 4g(f(cos(1.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
72
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_026
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(sin(1.0 PI x))) and x = 4g(f(cos(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
61
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_027
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(sin(2.0 PI x))) and x = 5g(f(cos(1.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
83
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_028
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 5g(f(sin(1.5 PI x))) and x = 5g(f(cos(1.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
91
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_029
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(cos(1.5 PI x))) and x = 5g(f(cos(2.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
146
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_030
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 5g(f(cos(1.5 PI x))) and x = 5g(f(sin(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
127
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_031
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(cos(1.5 PI x))) and x = 5g(f(sin(2.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
122
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_032
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 3g(f(sin(2.0 PI x))) and x = 5g(f(cos(0.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
38
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_033
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(cos(1.0 PI x))) and x = 4g(f(sin(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
73
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_034
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(cos(1.0 PI x))) and x = 3g(f(cos(0.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
19
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_035
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(sin(2.0 PI x))) and x = 4g(f(cos(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
144
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_036
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(cos(0.5 PI x))) and x = 5g(f(sin(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
24
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_037
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(cos(1.5 PI x))) and x = 3g(f(cos(1.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
51
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_038
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(sin(1.0 PI x))) and x = 5g(f(sin(2.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
60
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_039
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 5g(f(cos(1.0 PI x))) and x = 5g(f(cos(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
78
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_040
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(sin(2.0 PI x))) and x = 5g(f(cos(2.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
217
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_041
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 3g(f(cos(2.0 PI x))) and x = 3g(f(cos(0.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
31
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_042
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 5g(f(sin(2.0 PI x))) and x = 5g(f(cos(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
192
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_043
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(sin(1.0 PI x))) and x = 5g(f(sin(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
54
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_044
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(sin(1.0 PI x))) and x = 5g(f(sin(1.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
36
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_045
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(sin(1.5 PI x))) and x = 3g(f(sin(1.0 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
49
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_046
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(sin(2.0 PI x))) and x = 4g(f(sin(0.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
38
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_047
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(sin(2.0 PI x))) and x = 5g(f(cos(0.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
50
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_048
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(sin(1.5 PI x))) and x = 5g(f(cos(0.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
40
|
math
|
trans_func_intersection
|
test
|
trans_func_intersection_test_049
|
trans_func_intersection
|
[
{
"content": "Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(cos(1.0 PI x))) and x = 5g(f(sin(1.5 PI y))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
73
|
math
|
trans_func_intersection
|
test
|
trans_de_moivre_test_000
|
trans_de_moivre
|
[
{
"content": "x^5 + 10x^3 + 20x - 4 = 0. Find a closed form expression for the real roots of x in the simplest form.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2^{\frac{3}{5}} - 2^{\frac{2}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_001
|
trans_de_moivre
|
[
{
"content": "x^5 + 5x^3 + 5x - 2 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(1 + \sqrt{2})^{\frac{1}{5}} + -1(1 + \sqrt{2})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_002
|
trans_de_moivre
|
[
{
"content": "x^5 + 10x^3 + 20x + 28 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(-14 + 2\sqrt{57})^{\frac{1}{5}} + -2(-14 + 2\sqrt{57})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_003
|
trans_de_moivre
|
[
{
"content": "x^5 + 15x^3 + 45x - 26 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(13 + 2\sqrt{103})^{\frac{1}{5}} + -3(13 + 2\sqrt{103})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_004
|
trans_de_moivre
|
[
{
"content": "x^5 + 20x^3 + 80x - 66 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(33 + \sqrt{2113})^{\frac{1}{5}} + -4(33 + \sqrt{2113})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_005
|
trans_de_moivre
|
[
{
"content": "x^5 + 30x^3 + 180x - 122 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(61 + \sqrt{11497})^{\frac{1}{5}} + -6(61 + \sqrt{11497})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_006
|
trans_de_moivre
|
[
{
"content": "x^5 - 10x^3 + 20x + 36 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(-18 + 2\sqrt{73})^{\frac{1}{5}} + 2(-18 + 2\sqrt{73})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_007
|
trans_de_moivre
|
[
{
"content": "x^5 + 5x^3 + 5x - 30 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(15 + \sqrt{226})^{\frac{1}{5}} + -1(15 + \sqrt{226})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_008
|
trans_de_moivre
|
[
{
"content": "x^5 + 40x^3 + 320x - 258 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(129 + \sqrt{49409})^{\frac{1}{5}} + -8(129 + \sqrt{49409})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_009
|
trans_de_moivre
|
[
{
"content": "x^5 + 15x^3 + 45x + 28 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(-14 + \sqrt{439})^{\frac{1}{5}} + -3(-14 + \sqrt{439})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_010
|
trans_de_moivre
|
[
{
"content": "x^5 - 5x^3 + 5x - 6 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(3 + 2\sqrt{2})^{\frac{1}{5}} + -1(3 + 2\sqrt{2})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_011
|
trans_de_moivre
|
[
{
"content": "x^5 + 25x^3 + 125x - 126 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(63 + \sqrt{7094})^{\frac{1}{5}} + -5(63 + \sqrt{7094})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_012
|
trans_de_moivre
|
[
{
"content": "x^5 + 10x^3 + 20x + 56 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(-28 + 4\sqrt{51})^{\frac{1}{5}} + -2(-28 + 4\sqrt{51})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_013
|
trans_de_moivre
|
[
{
"content": "x^5 + 5x^3 + 5x + 2 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(-1 + \sqrt{2})^{\frac{1}{5}} + -1(-1 + \sqrt{2})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_014
|
trans_de_moivre
|
[
{
"content": "x^5 + 30x^3 + 180x - 344 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(172 + 4\sqrt{2335})^{\frac{1}{5}} + -6(172 + 4\sqrt{2335})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_015
|
trans_de_moivre
|
[
{
"content": "x^5 - 20x^3 + 80x + 66 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(-33 + \sqrt{65})^{\frac{1}{5}} + -4(-33 + \sqrt{65})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_016
|
trans_de_moivre
|
[
{
"content": "x^5 + 5x^3 + 5x - 122 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(61 + \sqrt{3722})^{\frac{1}{5}} + -1(61 + \sqrt{3722})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_de_moivre_test_017
|
trans_de_moivre
|
[
{
"content": "x^5 + 50x^3 + 500x - 1252 = 0. Find a closed form expression for the real roots of x in the simplest radical form of (a + b\\sqrt{c})^{\\frac{1}{5}} + d(a + b\\sqrt{c})^{-\\frac{1}{5}} for integers a, b, c, d, as necessary, or 'None' if no real roots exist.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
(626 + 2\sqrt{122969})^{\frac{1}{5}} + -10(626 + 2\sqrt{122969})^{-\frac{1}{5}}
|
math
|
trans_de_moivre
|
test
|
trans_prob_letter_test_000
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 8 can be formed from {\"x\": 7,\"y\": 7,\"z\": 6}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
6542
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_001
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 12 can be formed from {\"p\": 12,\"q\": 11,\"r\": 11}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
531439
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_002
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 9 can be formed from {\"m\": 8,\"n\": 9,\"o\": 7}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
19663
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_003
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 12 can be formed from {\"d\": 11,\"e\": 10,\"f\": 11}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
531414
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_004
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 10 can be formed from {\"c\": 9,\"a\": 9,\"b\": 8}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
59026
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_005
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 6 can be formed from {\"f\": 5,\"g\": 6,\"h\": 4}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
715
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_006
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 11 can be formed from {\"l\": 10,\"m\": 11,\"n\": 9}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
177123
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_007
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 7 can be formed from {\"o\": 6,\"p\": 7,\"q\": 5}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2171
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_008
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 5 can be formed from {\"r\": 4,\"s\": 5,\"t\": 3}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
231
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_009
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 4 can be formed from {\"u\": 3,\"v\": 4,\"w\": 2}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
71
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_010
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 8 can be formed from {\"i\": 7,\"j\": 6,\"k\": 8}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
6543
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_011
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 9 can be formed from {\"x\": 8,\"y\": 7,\"z\": 9}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
19663
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_012
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 10 can be formed from {\"a\": 9,\"b\": 8,\"c\": 7,\"d\": 6}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1044432
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_013
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 12 can be formed from {\"e\": 11,\"f\": 10,\"g\": 9,\"h\": 8}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
16769976
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_014
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 8 can be formed from {\"i\": 7,\"j\": 6,\"k\": 5,\"l\": 4}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
63444
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_015
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 11 can be formed from {\"m\": 10,\"n\": 9,\"o\": 8,\"p\": 7}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4188756
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_016
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 9 can be formed from {\"q\": 8,\"r\": 7,\"s\": 6,\"t\": 5}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
259143
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_017
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 7 can be formed from {\"u\": 6,\"v\": 5,\"w\": 4,\"x\": 3}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
14994
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_018
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 6 can be formed from {\"y\": 5,\"z\": 4,\"a\": 3,\"b\": 2}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3228
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_019
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 5 can be formed from {\"c\": 4,\"d\": 3,\"e\": 2,\"f\": 1}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
535
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_020
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 10 can be formed from {\"g\": 9,\"h\": 8,\"i\": 7,\"j\": 6}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1044432
|
math
|
trans_prob_letter
|
test
|
trans_prob_letter_test_021
|
trans_prob_letter
|
[
{
"content": "How many distinct words of length 12 can be formed from {\"k\": 11,\"l\": 10,\"m\": 9,\"n\": 8}? (value indicates the number of letters)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
16769976
|
math
|
trans_prob_letter
|
test
|
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