messages listlengths 1 1 | ground_truth stringlengths 2 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"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 answe... | 511 | MATH |
[
{
"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... | 897 | MATH |
[
{
"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 ... | 212 | MATH |
[
{
"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 |
[
{
"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 answe... | 73 | MATH |
[
{
"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... | 1177 | MATH |
[
{
"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 |
[
{
"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... | 729 | MATH |
[
{
"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 answe... | 994 | MATH |
[
{
"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}"... | 625 | MATH |
[
{
"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... | 610 | MATH |
[
{
"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}... | 125 | MATH |
[
{
"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 |
[
{
"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... | 64 | MATH |
[
{
"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}... | 445 | MATH |
[
{
"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}"... | 625 | MATH |
[
{
"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 |
[
{
"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 |
[
{
"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 ... | 256 | MATH |
[
{
"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 |
[
{
"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 |
[
{
"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 answe... | 728 | MATH |
[
{
"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}... | 343 | MATH |
[
{
"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 ... | 620 | MATH |
[
{
"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}"... | 256 | MATH |
[
{
"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... | 216 | MATH |
[
{
"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}"... | 602 | MATH |
[
{
"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... | 79 | MATH |
[
{
"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... | 343 | MATH |
[
{
"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}"... | 245 | MATH |
[
{
"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 ... | 81 | MATH |
[
{
"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}... | 124 | MATH |
[
{
"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 |
[
{
"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}"... | 610 | MATH |
[
{
"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... | 343 | MATH |
[
{
"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}"... | 625 | MATH |
[
{
"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 |
[
{
"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 ... | 548 | MATH |
[
{
"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 ... | 2401 | MATH |
[
{
"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}"... | 623 | MATH |
[
{
"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 ... | 1287 | MATH |
[
{
"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... | 1177 | MATH |
[
{
"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}... | 125 | MATH |
[
{
"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}... | 719 | MATH |
[
{
"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}"... | 1296 | MATH |
[
{
"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}"... | 212 | MATH |
[
{
"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 |
[
{
"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... | 255 | MATH |
[
{
"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 ... | 625 | MATH |
[
{
"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... | 81 | MATH |
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