data.jsonl

{"messages": [{"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"}], "ground_truth": "511", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "897", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "212", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "343", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "73", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "1177", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "729", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "729", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "994", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "625", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "610", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "125", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "125", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "64", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "445", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "625", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "125", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "210", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "256", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "125", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "343", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "728", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "343", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "620", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "256", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "216", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "602", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "343", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "245", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "124", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "292", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "610", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "343", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "625", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "445", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "548", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "2401", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "623", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "1287", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "1177", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "125", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "719", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "1296", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "212", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "216", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "255", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "625", "dataset": "MATH"}
{"messages": [{"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"}], "ground_truth": "81", "dataset": "MATH"}