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": "us... | 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... | (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: \\bo... | (-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: \\bo... | (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: \\bo... | (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: \\... | (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: \\bo... | (-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: \\boxe... | (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: \\... | (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: \\bo... | (-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... | (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: \\... | (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: \\bo... | (-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... | (-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: \\... | (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: \\bo... | (-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: \\box... | (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: \... | (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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