id
stringlengths 19
37
| setting_key
stringclasses 7
values | setting_name
stringclasses 7
values | messages
listlengths 1
1
| ground_truth
stringlengths 1
6
| dataset
stringclasses 1
value | omega_config
stringclasses 7
values | omega_split
stringclasses 1
value |
|---|---|---|---|---|---|---|---|
comp_parametric_intersection_test_015
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Considering the functions f(x) = ax^2 + bx and g(x) = px^2 + qx, where a, b, p, q can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 1 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
404
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_016
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Pick values for a, b, p from 1 to 5, how many combinations can we have so that the function f(x) = ax^2 + bx and g(x) = px have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
78
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_017
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Considering the functions f(x) = acos(bπx) and g(x) = pcos(qπx) - 2, where a, b, p, q can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 5 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
501
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_018
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Considering the functions f(x) = ax^2 + bx and g(x) = px^2 + qx, where a, b, p, q can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 1 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
414
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_019
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Pick values for a, p, q from 1 to 5, how many combinations can we have so that the function f(x) = ax + 1 and g(x) = px^2 + qx have at least 3 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_020
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Pick values for a, b, p, q from 1 to 5, how many combinations can we have so that the function f(x) = acos(bπx) and g(x) = pcos(qπx) + 2 have at least 7 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
510
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_021
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Considering the functions f(x) = ax^2 + bx + 3 and g(x) = px^2 + qx, where a, b, p, q can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 1 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
397
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_022
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Pick values for a, b, p, q from 1 to 5, how many combinations can we have so that the function f(x) = ax^2 + bx and g(x) = px^2 + qx - 5 have at least 2 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
277
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_023
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "If a, b, p, q are integers between 1 and 5 inclusive, how many different combinations of these values make the functions f(x) = ax^2 + bx and g(x) = px^2 + qx intersect at least 3 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_024
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "If a, p are integers between 1 and 5 inclusive, how many different combinations of these values make the functions f(x) = ax and g(x) = px intersect at least 2 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_025
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Pick values for a, b, p, q from 1 to 5, how many combinations can we have so that the function f(x) = acos(bπx) - 1 and g(x) = psin(qπx) - 1 have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
549
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_026
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Pick values for a, p, q from 1 to 5, how many combinations can we have so that the function f(x) = ax - 5 and g(x) = px^2 + qx - 1 have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
72
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_027
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Pick values for a, p from 1 to 5, how many combinations can we have so that the function f(x) = ax and g(x) = px have at least 3 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_028
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "If a, b, p, q are integers between 1 and 5 inclusive, how many different combinations of these values make the functions f(x) = ax^2 + bx and g(x) = px^2 + qx intersect at least 1 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
392
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_029
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Considering the functions f(x) = ax^2 + bx - 3 and g(x) = px^2 + qx, where a, b, p, q can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 1 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
395
|
math
|
comp_parametric_intersection
|
test
|
comp_matrix_rank_test_000
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Consider the matrix M = \\begin{bmatrix} a & b & c \\\\ 1 & a & b \\\\ 2 & 1 & a \\end{bmatrix} where a, b, and c are integers between 3 and 10, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
511
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_001
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Let M = \\begin{bmatrix} a & b & c \\\\ b & a & b \\\\ c & b & a \\end{bmatrix}, where a, b, and c are integers from 1 to 10 inclusive. Find the number of different combinations of values for a, b, and c that make the rank of M equal to 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
897
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_002
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Let M = \\begin{bmatrix} a & b & c & d \\\\ b & a & b & c \\\\ c & b & a & b \\\\ d & c & b & a \\end{bmatrix}, where a, b, c, and d are integers from 2 to 5 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
212
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_003
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, and c \\in [3, 9], count the number of distinct ordered triples (a, b, c) such that the matrix \\begin{bmatrix} a & 0 & 0 \\\\ 0 & b & 0 \\\\ 0 & 0 & c \\end{bmatrix} has rank 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
343
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_004
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Consider the matrix M = \\begin{bmatrix} a & b & c \\\\ b & a & b \\\\ c & b & a \\end{bmatrix} where a, b, and c are integers between 2 and 10, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 2?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
73
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_005
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Consider the matrix M = \\begin{bmatrix} a & b & c & d \\\\ b & a & b & c \\\\ c & b & a & b \\\\ d & c & b & a \\end{bmatrix} where a, b, c, and d are integers between 1 and 6, inclusive. How many different combinations of (a, b, c, d) result in a matrix with rank exactly 4?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1177
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_006
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, and c \\in [2, 10], count the number of distinct ordered triples (a, b, c) such that the matrix \\begin{bmatrix} a & b & c \\\\ 0 & b & a \\\\ 0 & 0 & c \\end{bmatrix} has rank 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
729
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_007
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Consider the matrix M = \\begin{bmatrix} a & b & c \\\\ 0 & b & a \\\\ 0 & 0 & c \\end{bmatrix} where a, b, and c are integers between 1 and 9, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
729
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_008
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Consider the matrix M = \\begin{bmatrix} a & b & c \\\\ 1 & a & b \\\\ 2 & 1 & a \\end{bmatrix} where a, b, and c are integers between 1 and 10, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
994
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_009
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, c, and d \\in [1, 5], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \\begin{bmatrix} a & 0 & 0 & 0 \\\\ 0 & b & 0 & 0 \\\\ 0 & 0 & c & 0 \\\\ 0 & 0 & 0 & d \\end{bmatrix} has rank 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
625
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_010
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Consider the matrix M = \\begin{bmatrix} a & b & c & d \\\\ b & a & 1 & 2 \\\\ c & 1 & b & 3 \\\\ d & 2 & 3 & a \\end{bmatrix} where a, b, c, and d are integers between 2 and 6, inclusive. How many different combinations of (a, b, c, d) result in a matrix with rank exactly 4?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
610
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_011
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Let M = \\begin{bmatrix} a & b & c \\\\ 0 & b & a \\\\ 0 & 0 & c \\end{bmatrix}, where a, b, and c are integers from 1 to 5 inclusive. Find the number of different combinations of values for a, b, and c that make the rank of M equal to 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
125
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_012
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, and c \\in [1, 5], count the number of distinct ordered triples (a, b, c) such that the matrix \\begin{bmatrix} a & b & c \\\\ 0 & b & a \\\\ 0 & 0 & c \\end{bmatrix} has rank 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
125
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_013
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Consider the matrix M = \\begin{bmatrix} a & 0 & 0 \\\\ 0 & b & 0 \\\\ 0 & 0 & c \\end{bmatrix} where a, b, and c are integers between 2 and 5, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
64
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_014
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Let M = \\begin{bmatrix} a & b & c \\\\ b & a & b \\\\ c & b & a \\end{bmatrix}, where a, b, and c are integers from 1 to 8 inclusive. Find the number of different combinations of values for a, b, and c that make the rank of M equal to 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
445
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_015
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, c, and d \\in [1, 5], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \\begin{bmatrix} a & b & c & d \\\\ 0 & b & a & c \\\\ 0 & 0 & c & b \\\\ 0 & 0 & 0 & d \\end{bmatrix} has rank 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
625
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_016
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, and c \\in [2, 6], count the number of distinct ordered triples (a, b, c) such that the matrix \\begin{bmatrix} a & b & c \\\\ 1 & a & b \\\\ 2 & 1 & a \\end{bmatrix} has rank 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
125
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_017
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, and c \\in [1, 6], count the number of distinct ordered triples (a, b, c) such that the matrix \\begin{bmatrix} a & b & c \\\\ b & a & 1 \\\\ c & 1 & b \\end{bmatrix} has rank 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
210
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_018
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Let M = \\begin{bmatrix} a & b & c & d \\\\ 0 & b & a & c \\\\ 0 & 0 & c & b \\\\ 0 & 0 & 0 & d \\end{bmatrix}, where a, b, c, and d are integers from 3 to 6 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
256
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_019
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, and c \\in [3, 7], count the number of distinct ordered triples (a, b, c) such that the matrix \\begin{bmatrix} a & b & c \\\\ 0 & b & a \\\\ 0 & 0 & c \\end{bmatrix} has rank 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
125
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_020
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, and c \\in [3, 9], count the number of distinct ordered triples (a, b, c) such that the matrix \\begin{bmatrix} a & b & c \\\\ b & a & 1 \\\\ c & 1 & b \\end{bmatrix} has rank 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
343
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_021
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Consider the matrix M = \\begin{bmatrix} a & b & c \\\\ b & a & 1 \\\\ c & 1 & b \\end{bmatrix} where a, b, and c are integers between 2 and 10, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
728
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_022
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Let M = \\begin{bmatrix} a & b & c \\\\ b & a & 1 \\\\ c & 1 & b \\end{bmatrix}, where a, b, and c are integers from 3 to 9 inclusive. Find the number of different combinations of values for a, b, and c that make the rank of M equal to 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
343
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_023
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Let M = \\begin{bmatrix} a & b & c & d \\\\ b & a & 1 & 2 \\\\ c & 1 & b & 3 \\\\ d & 2 & 3 & a \\end{bmatrix}, where a, b, c, and d are integers from 3 to 7 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
620
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_024
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, c, and d \\in [3, 6], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \\begin{bmatrix} a & 0 & 0 & 0 \\\\ 0 & b & 0 & 0 \\\\ 0 & 0 & c & 0 \\\\ 0 & 0 & 0 & d \\end{bmatrix} has rank 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
256
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_025
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Consider the matrix M = \\begin{bmatrix} a & b & c \\\\ 0 & b & a \\\\ 0 & 0 & c \\end{bmatrix} where a, b, and c are integers between 3 and 8, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
216
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_026
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, c, and d \\in [1, 5], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \\begin{bmatrix} a & b & c & d \\\\ b & a & 1 & 2 \\\\ c & 1 & b & 3 \\\\ d & 2 & 3 & a \\end{bmatrix} has rank 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
602
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_027
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Consider the matrix M = \\begin{bmatrix} a & b & c & d \\\\ b & a & 1 & 2 \\\\ c & 1 & b & 3 \\\\ d & 2 & 3 & a \\end{bmatrix} where a, b, c, and d are integers between 3 and 5, inclusive. How many different combinations of (a, b, c, d) result in a matrix with rank exactly 4?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
79
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_028
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Consider the matrix M = \\begin{bmatrix} a & b & c \\\\ b & a & 1 \\\\ c & 1 & b \\end{bmatrix} where a, b, and c are integers between 3 and 9, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
343
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_029
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, c, and d \\in [2, 5], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \\begin{bmatrix} a & b & c & d \\\\ b & a & 1 & 2 \\\\ c & 1 & b & 3 \\\\ d & 2 & 3 & a \\end{bmatrix} has rank 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
245
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_030
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Let M = \\begin{bmatrix} a & b & c & d \\\\ 1 & a & b & c \\\\ 2 & 1 & a & b \\\\ 3 & 2 & 1 & a \\end{bmatrix}, where a, b, c, and d are integers from 3 to 5 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
81
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_031
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Let M = \\begin{bmatrix} a & b & c \\\\ 1 & a & b \\\\ 2 & 1 & a \\end{bmatrix}, where a, b, and c are integers from 3 to 7 inclusive. Find the number of different combinations of values for a, b, and c that make the rank of M equal to 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
124
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_032
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, and c \\in [1, 7], count the number of distinct ordered triples (a, b, c) such that the matrix \\begin{bmatrix} a & b & c \\\\ b & a & b \\\\ c & b & a \\end{bmatrix} has rank 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
292
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_033
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, c, and d \\in [2, 6], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \\begin{bmatrix} a & b & c & d \\\\ b & a & 1 & 2 \\\\ c & 1 & b & 3 \\\\ d & 2 & 3 & a \\end{bmatrix} has rank 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
610
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_034
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Consider the matrix M = \\begin{bmatrix} a & b & c \\\\ 0 & b & a \\\\ 0 & 0 & c \\end{bmatrix} where a, b, and c are integers between 3 and 9, inclusive. How many different combinations of (a, b, c) result in a matrix with rank exactly 3?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
343
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_035
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, c, and d \\in [3, 7], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \\begin{bmatrix} a & b & c & d \\\\ 0 & b & a & c \\\\ 0 & 0 & c & b \\\\ 0 & 0 & 0 & d \\end{bmatrix} has rank 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
625
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_036
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, and c \\in [1, 8], count the number of distinct ordered triples (a, b, c) such that the matrix \\begin{bmatrix} a & b & c \\\\ b & a & b \\\\ c & b & a \\end{bmatrix} has rank 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
445
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_037
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Let M = \\begin{bmatrix} a & b & c & d \\\\ b & a & b & c \\\\ c & b & a & b \\\\ d & c & b & a \\end{bmatrix}, where a, b, c, and d are integers from 1 to 5 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
548
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_038
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Let M = \\begin{bmatrix} a & b & c & d \\\\ 0 & b & a & c \\\\ 0 & 0 & c & b \\\\ 0 & 0 & 0 & d \\end{bmatrix}, where a, b, c, and d are integers from 1 to 7 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2401
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_039
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, c, and d \\in [2, 6], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \\begin{bmatrix} a & b & c & d \\\\ 1 & a & b & c \\\\ 2 & 1 & a & b \\\\ 3 & 2 & 1 & a \\end{bmatrix} has rank 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
623
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_040
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Let M = \\begin{bmatrix} a & b & c & d \\\\ 1 & a & b & c \\\\ 2 & 1 & a & b \\\\ 3 & 2 & 1 & a \\end{bmatrix}, where a, b, c, and d are integers from 1 to 6 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1287
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_041
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Consider the matrix M = \\begin{bmatrix} a & b & c & d \\\\ b & a & b & c \\\\ c & b & a & b \\\\ d & c & b & a \\end{bmatrix} where a, b, c, and d are integers between 2 and 7, inclusive. How many different combinations of (a, b, c, d) result in a matrix with rank exactly 4?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1177
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_042
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Let M = \\begin{bmatrix} a & b & c \\\\ b & a & 1 \\\\ c & 1 & b \\end{bmatrix}, where a, b, and c are integers from 3 to 7 inclusive. Find the number of different combinations of values for a, b, and c that make the rank of M equal to 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
125
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_043
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Let M = \\begin{bmatrix} a & b & c \\\\ b & a & 1 \\\\ c & 1 & b \\end{bmatrix}, where a, b, and c are integers from 1 to 9 inclusive. Find the number of different combinations of values for a, b, and c that make the rank of M equal to 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
719
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_044
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, c, and d \\in [2, 7], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \\begin{bmatrix} a & 0 & 0 & 0 \\\\ 0 & b & 0 & 0 \\\\ 0 & 0 & c & 0 \\\\ 0 & 0 & 0 & d \\end{bmatrix} has rank 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1296
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_045
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, c, and d \\in [3, 6], count the number of distinct ordered quadruples (a, b, c, d) such that the matrix \\begin{bmatrix} a & b & c & d \\\\ b & a & b & c \\\\ c & b & a & b \\\\ d & c & b & a \\end{bmatrix} has rank 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
212
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_046
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "For integers a, b, and c \\in [3, 8], count the number of distinct ordered triples (a, b, c) such that the matrix \\begin{bmatrix} a & b & c \\\\ 0 & b & a \\\\ 0 & 0 & c \\end{bmatrix} has rank 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
216
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_047
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Consider the matrix M = \\begin{bmatrix} a & b & c & d \\\\ 1 & a & b & c \\\\ 2 & 1 & a & b \\\\ 3 & 2 & 1 & a \\end{bmatrix} where a, b, c, and d are integers between 2 and 5, inclusive. How many different combinations of (a, b, c, d) result in a matrix with rank exactly 4?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
255
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_048
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Let M = \\begin{bmatrix} a & b & c & d \\\\ 0 & b & a & c \\\\ 0 & 0 & c & b \\\\ 0 & 0 & 0 & d \\end{bmatrix}, where a, b, c, and d are integers from 3 to 7 inclusive. Find the number of different combinations of values for a, b, c, and d that make the rank of M equal to 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
625
|
math
|
comp_matrix_rank
|
test
|
comp_matrix_rank_test_049
|
comp_matrix_rank
|
Arithmetic (Matrix Rank) + Combinatorics (Probability)
|
[
{
"content": "Consider the matrix M = \\begin{bmatrix} a & 0 & 0 & 0 \\\\ 0 & b & 0 & 0 \\\\ 0 & 0 & c & 0 \\\\ 0 & 0 & 0 & d \\end{bmatrix} where a, b, c, and d are integers between 3 and 5, inclusive. How many different combinations of (a, b, c, d) result in a matrix with rank exactly 4?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
81
|
math
|
comp_matrix_rank
|
test
|
comp_vertex_color_test_000
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular rectangle is independently colored either red or blue with equal probability. The probability that the rectangle can then be rotated so that all of the blue vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
27
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_001
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular rectangle is independently colored with either red, blue, green, or magenta with equal probability. The probability that the rectangle can then be rotated so that all of the magenta vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
419
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_002
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular hexagon is independently colored with either red, green, yellow, or magenta with equal probability. The probability that the hexagon can then be rotated so that all of the magenta vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3201
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_003
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular square is independently colored with either blue, yellow, or magenta with equal probability. The probability that the square can then be rotated so that all of the magenta vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
131
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_004
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular square is colored either yellow with probability 0.25 or blue with probability 0.75. The probability that the square can then be rotated so that all of the blue vertices end up at positions where there had been yellow vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
323
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_005
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular octagon is independently colored with either blue, yellow, or magenta with equal probability. The probability that the octagon can then be rotated so that all of the magenta vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
9403
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_006
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular pentagon is colored either yellow with probability 0.25 or blue with probability 0.75. The probability that the pentagon can then be rotated so that all of the blue vertices end up at positions where there had been yellow vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
565
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_007
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular triangle is independently colored with either blue, green, or magenta with equal probability. The probability that the triangle can then be rotated so that all of the magenta vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
44
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_008
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular heptagon is independently colored with either red, blue, or magenta with equal probability. The probability that the heptagon can then be rotated so that all of the magenta vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3218
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_009
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular hexagon is colored either yellow with probability 0 or blue with probability 1. The probability that the hexagon can then be rotated so that all of the blue vertices end up at positions where there had been yellow vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_010
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular heptagon is independently colored with either blue, yellow, or magenta with equal probability. The probability that the heptagon can then be rotated so that all of the magenta vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3218
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_011
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular rhombus is independently colored with either red, blue, green, or yellow with equal probability. The probability that the rhombus can then be rotated so that all of the yellow vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
419
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_012
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular rhombus is independently colored with either blue, green, or yellow with equal probability. The probability that the rhombus can then be rotated so that all of the yellow vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
131
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_013
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular rhombus is independently colored with either blue, green, yellow, or magenta with equal probability. The probability that the rhombus can then be rotated so that all of the magenta vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
419
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_014
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular nonagon is colored either yellow with probability 1 or blue with probability 0. The probability that the nonagon can then be rotated so that all of the blue vertices end up at positions where there had been yellow vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_015
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular triangle is independently colored with either blue, green, or yellow with equal probability. The probability that the triangle can then be rotated so that all of the yellow vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
44
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_016
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular rhombus is independently colored with either red, green, or magenta with equal probability. The probability that the rhombus can then be rotated so that all of the magenta vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
131
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_017
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular octagon is independently colored with either red, blue, or magenta with equal probability. The probability that the octagon can then be rotated so that all of the magenta vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
9403
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_018
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular rhombus is independently colored with either blue, green, or magenta with equal probability. The probability that the rhombus can then be rotated so that all of the magenta vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
131
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_019
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular rectangle is colored either yellow with probability 0.75 or blue with probability 0.25. The probability that the rectangle can then be rotated so that all of the blue vertices end up at positions where there had been yellow vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
499
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_020
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular rectangle is independently colored with either green, yellow, or magenta with equal probability. The probability that the rectangle can then be rotated so that all of the magenta vertices end up at positions where there had been green vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
131
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_021
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular rectangle is independently colored with either red, green, or yellow with equal probability. The probability that the rectangle can then be rotated so that all of the yellow vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
131
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_022
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular pentagon is independently colored with either green, yellow, or magenta with equal probability. The probability that the pentagon can then be rotated so that all of the magenta vertices end up at positions where there had been green vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
380
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_023
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular triangle is independently colored with either red, blue, green, or yellow with equal probability. The probability that the triangle can then be rotated so that all of the yellow vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
53
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_024
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular pentagon is independently colored with either blue, green, or yellow with equal probability. The probability that the pentagon can then be rotated so that all of the yellow vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
380
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_025
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular septagon is independently colored with either green, yellow, or magenta with equal probability. The probability that the septagon can then be rotated so that all of the magenta vertices end up at positions where there had been green vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3218
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_026
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular triangle is independently colored with either red, green, yellow, or magenta with equal probability. The probability that the triangle can then be rotated so that all of the magenta vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
53
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_027
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular rectangle is colored either yellow with probability 0 or blue with probability 1. The probability that the rectangle can then be rotated so that all of the blue vertices end up at positions where there had been yellow vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_028
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular square is independently colored with either red, blue, green, yellow, or magenta with equal probability. The probability that the square can then be rotated so that all of the magenta vertices end up at positions where there had been green vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
207
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_029
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular hexagon is independently colored with either blue, yellow, or magenta with equal probability. The probability that the hexagon can then be rotated so that all of the magenta vertices end up at positions where there had been blue vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
368
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_030
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular pentagon is independently colored with either red, blue, or green with equal probability. The probability that the pentagon can then be rotated so that all of the green vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
380
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_031
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular diamond is independently colored with either red, blue, green, or magenta with equal probability. The probability that the diamond can then be rotated so that all of the magenta vertices end up at positions where there had been red vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
419
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_032
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular rhombus is colored either yellow with probability 0.25 or blue with probability 0.75. The probability that the rhombus can then be rotated so that all of the blue vertices end up at positions where there had been yellow vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
323
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_033
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular triangle is colored either yellow with probability 0.75 or blue with probability 0.25. The probability that the triangle can then be rotated so that all of the blue vertices end up at positions where there had been yellow vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
59
|
math
|
comp_vertex_color
|
test
|
comp_vertex_color_test_034
|
comp_vertex_color
|
Geometry (Polygon Color) + Combinatorics (Probability)
|
[
{
"content": "Each vertex of a regular rhombus is colored either yellow with probability 0.5 or blue with probability 0.5. The probability that the rhombus can then be rotated so that all of the blue vertices end up at positions where there had been yellow vertices is M/N, where M and N are relatively prime positive integers. Find M+N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
27
|
math
|
comp_vertex_color
|
test
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.