id string | original_id string | family string | difficulty_level int64 | source_family string | source_level string | messages list | ground_truth string | dataset string |
|---|---|---|---|---|---|---|---|---|
omega_500_100 | logic_puzzles_grid_chip_level_3_0076 | logic_puzzles_grid_chip | 3 | logic_puzzles_grid_chip | 3 | [
{
"role": "user",
"content": "Chips, colored either black or white, are placed in the cells of a 6x6 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more... | 18 | OMEGA_500_SAMPLE |
omega_500_101 | arithmetic_matrix_rank_level_2_0022 | arithmetic_matrix_rank | 2 | arithmetic_matrix_rank | 2 | [
{
"role": "user",
"content": "What is the rank of the matrix:\n[[10, 30, -30, 0, 10],\n [2, 6, -6, 0, 2],\n [-4, -12, 12, 0, -4],\n [-2, -6, 6, 0, -2],\n [-12, -36, 36, 0, -12]]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | OMEGA_500_SAMPLE |
omega_500_102 | combinatory_probability_at_least_n_specific_fixed_level_1_0076 | combinatory_probability_at_least_n_specific_fixed | 1 | combinatory_probability_at_least_n_specific_fixed | 1 | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {e: 2, g: 1, t: 1}, shuffle the letters in the word, what is the probability of at least 1 letter 'e' remains in the same position?\nIf the probability can be written a... | 3 | OMEGA_500_SAMPLE |
omega_500_103 | combinatory_distribution_level_1_0061 | combinatory_distribution | 1 | combinatory_distribution | 1 | [
{
"role": "user",
"content": "Arrange the letters {'i': 2, 'v': 2} into 2 uniquely identified boxes that can hold [1, 3] letters respectively. Find the number of possible arrangements.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | OMEGA_500_SAMPLE |
omega_500_104 | logic_puzzles_grid_rook_level_1_0068 | logic_puzzles_grid_rook | 1 | logic_puzzles_grid_rook | 1 | [
{
"role": "user",
"content": "In a 8x9 grid, what is the fewest number of steps it takes to get from the bottom left (0, 0) to the top right (7, 8), provided you can move right or up by any number of cells (like a chess rook) at each step, subject to the constraint that you cannot move through the following... | 4 | OMEGA_500_SAMPLE |
omega_500_105 | algebra_func_intersection_level_5_0033 | algebra_func_intersection | 5 | algebra_func_intersection | 5 | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -3((2((1(-2((2((3x + 3)/(2x - 3)) - 2)/(-1((3x + 3)/(2x - 3)) - 3)) + 3) + 3)/(1(-2((2((3x + 3)/(2x - 3)) - 2)/(-1((3x + 3)/(2x - 3)) - 3)) + 3) + 1)) + 2)/(2((1(-2((2((3x + 3)/(2x - 3)) - 2)/(-1((3x + 3)/... | 8 | OMEGA_500_SAMPLE |
omega_500_106 | arithmetic_matrix_power_level_6_0082 | arithmetic_matrix_power | 6 | arithmetic_matrix_power | 6 | [
{
"role": "user",
"content": "Find the sum of all entries of the 7th power of the matrix:\n[[-12, -14, -11, 10, -2, 2, 13, -4, -13, 10],\n [1, -13, 14, 4, 8, 4, 10, 1, 7, 4],\n [-6, -14, -5, 4, -14, -13, 1, -3, -7, 8],\n [3, -13, 10, 14, -2, -9, 6, 11, -9, -6],\n [1, -4, -8, -5, -5, 10, 5, 3, 13, 9],\n [5, ... | -27517536322 | OMEGA_500_SAMPLE |
omega_500_107 | arithmetic_matrix_rank_level_3_0098 | arithmetic_matrix_rank | 3 | arithmetic_matrix_rank | 3 | [
{
"role": "user",
"content": "Compute the rank of the given 6x6 matrix:\n[[-45, 38, 31, -13, -15, -84],\n [17, 41, -7, 32, 41, 113],\n [-18, 34, 14, 4, 6, -6],\n [-10, 45, 10, 15, 20, 35],\n [-69, 146, 55, 23, 33, 0],\n [18, -34, -14, -4, -6, 6]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
... | 2 | OMEGA_500_SAMPLE |
omega_500_108 | arithmetic_matrix_determinant_level_2_0032 | arithmetic_matrix_determinant | 2 | arithmetic_matrix_determinant | 2 | [
{
"role": "user",
"content": "Calculate the determinant of the matrix, round to the nearest integer, then find the result modulo 1000:\n[[-10, -12, -12, 6],\n [-2, -8, -1, -10],\n [-12, -12, -5, -3],\n [-2, -1, 0, -1]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 510 | OMEGA_500_SAMPLE |
omega_500_109 | arithmetic_list_prime_factors_level_2_0023 | arithmetic_list_prime_factors | 2 | arithmetic_list_prime_factors | 2 | [
{
"role": "user",
"content": "What is the second-largest prime factor of 142997224?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2351 | OMEGA_500_SAMPLE |
omega_500_110 | arithmetic_matrix_eigenvalues_level_5_0071 | arithmetic_matrix_eigenvalues | 5 | arithmetic_matrix_eigenvalues | 5 | [
{
"role": "user",
"content": "What is the sum of absolute values of all eigenvalues for the following matrix?\n[[19, 4, 16, 0, -18, -12, 4, 2, 10],\n [4, 23, -6, -3, -13, -3, 0, 2, 5],\n [16, -6, 6, 14, 10, 14, 12, 4, -5],\n [0, -3, 14, -5, 14, -12, 1, 9, -8],\n [-18, -13, 10, 14, -3, 6, -6, -16, 8],\n [-12... | 238 | OMEGA_500_SAMPLE |
omega_500_111 | arithmetic_matrix_svd_level_3_0098 | arithmetic_matrix_svd | 3 | arithmetic_matrix_svd | 3 | [
{
"role": "user",
"content": "Calculate the rounded difference between the largest and smallest singular values from the SVD of:\n[[10, -10, -4, 9, 1, 5],\n [5, 0, 10, -7, 2, -4],\n [-7, 4, 8, -2, 0, -3],\n [-10, -3, 2, -7, -3, -8],\n [-2, -9, -9, 2, 4, 1],\n [-7, -3, -2, -6, 9, 7]]\n\nPresent the answer in... | 23 | OMEGA_500_SAMPLE |
omega_500_112 | arithmetic_matrix_svd_level_7_0094 | arithmetic_matrix_svd | 7 | arithmetic_matrix_svd | 7 | [
{
"role": "user",
"content": "Calculate the rounded largest singular value from the SVD of:\n[[-22, -23, -5, -9, 8, -10, 23, 23, -3, -25, -14, -12],\n [-22, -24, 10, 10, 10, -24, 8, 25, -4, 1, 3, -9],\n [25, 9, 6, -1, 21, 13, -10, 13, -20, -19, 26, 4],\n [-15, 21, 4, 17, -8, -17, 12, -15, -7, 10, 21, 14],\n... | 94 | OMEGA_500_SAMPLE |
omega_500_113 | arithmetic_matrix_inverse_level_4_0027 | arithmetic_matrix_inverse | 4 | arithmetic_matrix_inverse | 4 | [
{
"role": "user",
"content": "Find the value of the entry with the largest absolute value of the inverse of matrix 1/252 * A where A =\n[[15, 6, 1, 3, -3, -4, -2],\n [6, 8, -2, 1, 8, 2, 0],\n [1, -2, -12, -5, 0, 2, -2],\n [3, 1, -5, 11, -1, -12, -5],\n [-3, 8, 0, -1, -17, 4, -2],\n [-4, 2, 2, -12, 4, -1, 12... | -29 | OMEGA_500_SAMPLE |
omega_500_114 | algebra_linear_equation_level_4_0018 | algebra_linear_equation | 4 | algebra_linear_equation | 4 | [
{
"role": "user",
"content": "Let l be (268/14)/(4/(-7)*118/(-236)). Suppose 17558 - 17893 = -l*s. Solve -2*r - v = 2*p - 5*d - 14, 2*v - 4*g + 3*d = 4*r - 17, -5*v + g + d + 11 = 2*r + p - 4*d, 4*r + 4*v - 5*p - 4*g - 5 = 2*d, 3*p = -4*r - s*v + 2*g - 2*d - 36 for r.\n\nPresent the answer in LaTex format: ... | -2 | OMEGA_500_SAMPLE |
omega_500_115 | algebra_func_derivative_sign_level_4_0028 | algebra_func_derivative_sign | 4 | algebra_func_derivative_sign | 4 | [
{
"role": "user",
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3(2|(1sin(1π((-3(2x^2 - 3x - 1) - 3)/(-1(2x^2 - 3x - 1) + 2)) - 1) - 1) + 2| + 3)^2 + 2(2|(1sin(1π((-3(2x^2 - 3x - 1) - 3)/(-1(2x^2 - 3x - 1) + 2)) - 1) - 1) + 2| + 3) - 1 positive?\n\nPresent ... | 6 | OMEGA_500_SAMPLE |
omega_500_116 | number_theory_prime_mod_level_1_0008 | number_theory_prime_mod | 1 | number_theory_prime_mod | 1 | [
{
"role": "user",
"content": "let p=13. How many solutions m in [1, 2, ..., 13^1] satisfy m^3+1=0 (mod 13^1)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | OMEGA_500_SAMPLE |
omega_500_117 | geometry_rotation_level_3_0066 | geometry_rotation | 3 | geometry_rotation | 3 | [
{
"role": "user",
"content": "Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 8, from vertex 6 to vertex 1, from vertex 6 to vertex 9, and from vertex 3 to vertex 6. Then, rotate the entire setup, including the constructed ... | 4 | OMEGA_500_SAMPLE |
omega_500_118 | arithmetic_matrix_power_level_3_0043 | arithmetic_matrix_power | 3 | arithmetic_matrix_power | 3 | [
{
"role": "user",
"content": "For the matrix power A^4 where A =\n[[4, 2, -4, 3, -5],\n [2, 5, -2, -2, -1],\n [2, -1, 2, 4, -5],\n [4, 3, 0, -3, 2],\n [3, 0, 5, 1, -5]], determine the value at position (4, 3).\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | -53 | OMEGA_500_SAMPLE |
omega_500_119 | logic_puzzles_blocked_grid_level_2_0075 | logic_puzzles_blocked_grid | 2 | logic_puzzles_blocked_grid | 2 | [
{
"role": "user",
"content": "In a 18x15 grid, how many different paths are there from the bottom left (0, 0) to the top right (17, 14), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (3, 0), (10, 10), (4, 4), (13, 3), (15, 7), (2, ... | 915333 | OMEGA_500_SAMPLE |
omega_500_120 | arithmetic_matrix_multiplication_level_7_0033 | arithmetic_matrix_multiplication | 7 | arithmetic_matrix_multiplication | 7 | [
{
"role": "user",
"content": "Compute the value at position (8, 9) of the product matrix AB given:\nA =\n[[16, -15, -8, -24, -5, 1, -19, -23, -26, 11, -3, 22, -26],\n [18, -5, -26, -9, -16, -18, -3, 2, 12, -8, 2, 12, -13],\n [19, 22, -1, -5, -15, -18, 8, 2, -5, 17, -30, 18, 7],\n [-29, 18, -25, 15, 19, 7, 1... | 35 | OMEGA_500_SAMPLE |
omega_500_121 | algebra_func_intersection_coords_level_2_0036 | algebra_func_intersection_coords | 2 | algebra_func_intersection_coords | 2 | [
{
"role": "user",
"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-1(-1(-2cos(2πx + 1) + 2) + 4) + 1)/(3(-1(-2cos(2πx + 1) + 2) + 4) + 1) and g(x) = -1(2x^2 - 5x) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaT... | 3 | OMEGA_500_SAMPLE |
omega_500_122 | arithmetic_matrix_power_level_7_0045 | arithmetic_matrix_power | 7 | arithmetic_matrix_power | 7 | [
{
"role": "user",
"content": "Find the value at position (10, 8) of the 9th power of the matrix:\n[[-4, -16, 3, -9, -7, -8, -14, -6, -5, -7, 8, 8],\n [-6, 9, -3, -16, -13, 13, -5, -4, 14, 6, -13, -1],\n [-10, 5, 12, -6, 16, 12, -5, -10, 9, -8, -4, 8],\n [7, 0, -15, -4, 14, 12, 11, 10, -2, 14, -2, 16],\n [3,... | -4146051490366 | OMEGA_500_SAMPLE |
omega_500_123 | geometry_polygon_color_level_2_0007 | geometry_polygon_color | 2 | geometry_polygon_color | 2 | [
{
"role": "user",
"content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue. What is the maximum number of blue v... | 4 | OMEGA_500_SAMPLE |
omega_500_124 | geometry_polygon_chords_level_5_0036 | geometry_polygon_chords | 5 | geometry_polygon_chords | 5 | [
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 12 to vertex 8, from vertex 7 to vertex 1, from vertex 3 to vertex 7, from vertex 7 to vertex 4, from vertex 2 to vertex 11, from vertex 5 to vertex 1, from v... | 6 | OMEGA_500_SAMPLE |
omega_500_125 | arithmetic_matrix_determinant_level_5_0072 | arithmetic_matrix_determinant | 5 | arithmetic_matrix_determinant | 5 | [
{
"role": "user",
"content": "Compute |A| for the matrix A =\n[[-7, -14, 2, -2, -12, 1, -1, 19, -13],\n [-14, -11, 14, 22, -9, -3, 10, 4, 6],\n [2, 14, 14, -4, 6, 2, -4, 5, -5],\n [-2, 22, -4, -20, -5, -12, 8, -10, -2],\n [-12, -9, 6, -5, 0, -8, 18, 20, 6],\n [1, -3, 2, -12, -8, -25, 10, 2, -20],\n [-1, 10,... | 242 | OMEGA_500_SAMPLE |
omega_500_126 | arithmetic_matrix_inverse_level_2_0057 | arithmetic_matrix_inverse | 2 | arithmetic_matrix_inverse | 2 | [
{
"role": "user",
"content": "Find the sum of all entries of the inverse of matrix 1/96 * A where A =\n[[-4, 10, -10, 6],\n [-5, -8, -9, 11],\n [-7, -11, 7, -2],\n [-9, 2, -7, 9]]\nRound your answer to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | -82 | OMEGA_500_SAMPLE |
omega_500_127 | geometry_rotation_level_2_0006 | geometry_rotation | 2 | geometry_rotation | 2 | [
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 2, from vertex 2 to vertex 4, and from vertex 4 to vertex 7. Then, rotate the entire setup, including the constructed diagonals, 3 vertices counter... | 9 | OMEGA_500_SAMPLE |
omega_500_128 | arithmetic_matrix_multiplication_level_6_0020 | arithmetic_matrix_multiplication | 6 | arithmetic_matrix_multiplication | 6 | [
{
"role": "user",
"content": "Find the value at position (11, 9) of the product of the following matrices:\nA =\n[[6, -6, 24, -15, -11, 6, 2, -4, 15, -9],\n [0, 18, -19, 16, 0, -1, -16, 19, -7, -15],\n [-13, -18, 14, -22, 21, 5, 2, -15, 17, -25],\n [23, 21, 12, -2, 21, 5, 15, 20, 24, 0],\n [23, 24, -21, -6,... | 722 | OMEGA_500_SAMPLE |
omega_500_129 | combinatory_distribution_level_4_0083 | combinatory_distribution | 4 | combinatory_distribution | 4 | [
{
"role": "user",
"content": "Given the letters {'x': 2, 'z': 4, 'r': 2, 'm': 3}, place them into 4 indistinguishable boxes with sizes [2, 4, 2, 3]. In how many ways can you do this?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 404 | OMEGA_500_SAMPLE |
omega_500_130 | number_theory_digit_sum_level_5_0044 | number_theory_digit_sum | 5 | number_theory_digit_sum | 5 | [
{
"role": "user",
"content": "let N be the least 7-digit positive integer with the property that both the number and the number formed by reversing its digits are divisible by 8. Let Q and R be the quotient and remainder, respectively, when N is divided by 1000000. Find Q + R.\n\nPresent the answer in LaTex... | 2110000 | OMEGA_500_SAMPLE |
omega_500_131 | combinatory_probability_at_least_n_specific_fixed_level_1_0015 | combinatory_probability_at_least_n_specific_fixed | 1 | combinatory_probability_at_least_n_specific_fixed | 1 | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {r: 1, b: 1, i: 2}, shuffle the letters in the word, what is the probability of at least 1 letter 'i' remains in the same position?\nIf the probability can be written a... | 3 | OMEGA_500_SAMPLE |
omega_500_132 | arithmetic_matrix_power_level_1_0076 | arithmetic_matrix_power | 1 | arithmetic_matrix_power | 1 | [
{
"role": "user",
"content": "What is the value at position (2, 1) of the matrix A^2 where A is:\n[[-1, -3, 1],\n [-3, -3, 0],\n [1, 0, 2]]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 12 | OMEGA_500_SAMPLE |
omega_500_133 | geometry_polygon_color_level_1_0069 | geometry_polygon_color | 1 | geometry_polygon_color | 1 | [
{
"role": "user",
"content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is red. The 7-gon is then rotated clockwise by 3 positions, so that the vertex ... | 3 | OMEGA_500_SAMPLE |
omega_500_134 | arithmetic_mixed_level_1_0007 | arithmetic_mixed | 1 | arithmetic_mixed | 1 | [
{
"role": "user",
"content": "What is (-8)/(-22) + (-108)/(-66)? Express your answer as an integer or in the form p/q where p and q are coprime.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 2 | OMEGA_500_SAMPLE |
omega_500_135 | arithmetic_matrix_svd_level_1_0042 | arithmetic_matrix_svd | 1 | arithmetic_matrix_svd | 1 | [
{
"role": "user",
"content": "What is the rounded difference between the largest and smallest singular values in the SVD of the following matrix?\n[[1, 0, 1, -4],\n [0, 4, -3, -4],\n [4, 4, 4, 1],\n [3, -4, -1, -4]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 6 | OMEGA_500_SAMPLE |
omega_500_136 | arithmetic_matrix_determinant_level_2_0083 | arithmetic_matrix_determinant | 2 | arithmetic_matrix_determinant | 2 | [
{
"role": "user",
"content": "Determine the value of det(A) where A =\n[[-11, 10, -8, -2],\n [-7, 5, 8, -3],\n [-9, -4, -7, -11],\n [4, -8, 12, -7]]\nRound to the nearest integer, then give your answer as det(A) mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 859 | OMEGA_500_SAMPLE |
omega_500_137 | logic_puzzles_blocked_grid_level_1_0054 | logic_puzzles_blocked_grid | 1 | logic_puzzles_blocked_grid | 1 | [
{
"role": "user",
"content": "In a 6x8 grid, how many different paths are there from the bottom left (0, 0) to the top right (5, 7), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (2, 5), (3, 5), (0, 5), (1, 4), (3, 4), (5, 2), (0, ... | 175 | OMEGA_500_SAMPLE |
omega_500_138 | algebra_linear_equation_level_4_0058 | algebra_linear_equation | 4 | algebra_linear_equation | 4 | [
{
"role": "user",
"content": "Suppose -4*q + 3*n = -90, 5*q = -4*n + 44 + 53. Suppose 4*i - 5*m = 200 - 23, 39 - 27 = -29*i - 3*m. Solve -f = 4*z - 4*a - i*d - 2*j - 9, -2*z - 4*d + 5*j - 19 = -f, -f + 5*j = -3*z + a - d + q + 9, 3*f - 2*a + 3 = -d - j, 2*d + 5*j = f - 4*z - 3*a + 36 for f.\n\nPresent the a... | -1 | OMEGA_500_SAMPLE |
omega_500_139 | geometry_rotation_level_5_0018 | geometry_rotation | 5 | geometry_rotation | 5 | [
{
"role": "user",
"content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 2, from vertex 2 to vertex 6, from vertex 2 to vertex 4, from vertex 4 to vertex 6, from vertex 1 to vertex 4, and from vertex 1 to vertex 5. Then,... | 20 | OMEGA_500_SAMPLE |
omega_500_140 | arithmetic_gcd_level_2_0011 | arithmetic_gcd | 2 | arithmetic_gcd | 2 | [
{
"role": "user",
"content": "Calculate the highest common divisor of 1449 and 77300286.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 483 | OMEGA_500_SAMPLE |
omega_500_141 | combinatory_distribution_level_2_0000 | combinatory_distribution | 2 | combinatory_distribution | 2 | [
{
"role": "user",
"content": "Distribute the letters {'g': 1, 'y': 3, 's': 2} among 3 identical containers that can hold [2, 3, 1] letters respectively. How many different arrangements exist?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 12 | OMEGA_500_SAMPLE |
omega_500_142 | combinatory_probability_exactly_n_specific_fixed_level_2_0028 | combinatory_probability_exactly_n_specific_fixed | 2 | combinatory_probability_exactly_n_specific_fixed | 2 | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {d: 1, t: 3, v: 2}, shuffle the letters in the word, what is the probability of exact 1 letter 'v' remains in the same position?\nIf the probability can be written as t... | 4 | OMEGA_500_SAMPLE |
omega_500_143 | arithmetic_matrix_power_level_7_0055 | arithmetic_matrix_power | 7 | arithmetic_matrix_power | 7 | [
{
"role": "user",
"content": "Find the sum of absolute values of all entries of the 8th power of the matrix:\n[[8, 2, -6, 3, -2, -5, 2, -8, 5, -2, 10, -10],\n [2, 9, -4, 0, 11, 2, 0, -2, 0, -2, -1, -12],\n [-6, -4, -14, 5, 5, 8, -10, -1, -2, -3, 8, 11],\n [3, 0, 5, 16, -2, 2, 5, -4, 2, 8, -14, -6],\n [-2, 1... | 105157555449503 | OMEGA_500_SAMPLE |
omega_500_144 | algebra_func_zeros_level_3_0084 | algebra_func_zeros | 3 | algebra_func_zeros | 3 | [
{
"role": "user",
"content": "Find the number of x-intercepts of the function f(x) = 3(-3|(1(-2x^2 - 3x)^2 + 5(-2x^2 - 3x) - 3) - 2|) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 8 | OMEGA_500_SAMPLE |
omega_500_145 | combinatory_pattern_matching_level_5_0066 | combinatory_pattern_matching | 5 | combinatory_pattern_matching | 5 | [
{
"role": "user",
"content": "If you form a word of length 6 by randomly selecting letters from the multiset {m: 1, s: 5, c: 2, i: 2, r: 2}, what is the expected number of times the pattern 'r+c' appears? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 0.16 | OMEGA_500_SAMPLE |
omega_500_146 | algebra_func_derivative_sign_level_5_0036 | algebra_func_derivative_sign | 5 | algebra_func_derivative_sign | 5 | [
{
"role": "user",
"content": "For the function f(x) = -1|(-3|(2|(2((3(-1cos(3πx - 3) + 3) + 3)/(2(-1cos(3πx - 3) + 3) - 3))^2 + 5((3(-1cos(3πx - 3) + 3) + 3)/(2(-1cos(3πx - 3) + 3) - 3)) + 5) + 1| + 3) - 3| + 1)| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent th... | 30 | OMEGA_500_SAMPLE |
omega_500_147 | algebra_func_intersection_level_3_0088 | algebra_func_intersection | 3 | algebra_func_intersection | 3 | [
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -4(-1|(-3(-1x - 2)^2 - 4(-1x - 2) + 4) - 3| + 2) + 1 and g(x) = (2(2sin(3π(-5x^2 - 3x + 5)) + 3) + 2)/(1(2sin(3π(-5x^2 - 3x + 5)) + 3) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxe... | 8 | OMEGA_500_SAMPLE |
omega_500_148 | algebra_func_derivative_sign_level_5_0082 | algebra_func_derivative_sign | 5 | algebra_func_derivative_sign | 5 | [
{
"role": "user",
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = (-3(1|(4(2|(2(-2sin(3πx + 3) + 2) - 1) - 1| + 3) + 1) - 2| + 2) + 1)/(-2(1|(4(2|(2(-2sin(3πx + 3) + 2) - 1) - 1| + 3) + 1) - 2| + 2) - 1) negative?\n\nPresent the answer in LaTex format: \\boxe... | 60 | OMEGA_500_SAMPLE |
omega_500_149 | logic_puzzles_zebralogic_level_2_0071 | logic_puzzles_zebralogic | 2 | logic_puzzles_zebralogic | 2 | [
{
"role": "user",
"content": "There are 2 houses, numbered 1 to 2 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Eric`, `Arnold`\n - People have... | 1 | OMEGA_500_SAMPLE |
omega_500_150 | arithmetic_matrix_eigenvalues_level_3_0070 | arithmetic_matrix_eigenvalues | 3 | arithmetic_matrix_eigenvalues | 3 | [
{
"role": "user",
"content": "Find the eigenvalues of the matrix and calculate the largest eigenvalue by absolute value:\n[[2, -4, -8, -8, 2, 12],\n [0, 4, 14, -1, 11, 15],\n [0, 0, 4, 0, -12, -10],\n [0, 0, 0, 15, -13, 15],\n [0, 0, 0, 0, -5, 2],\n [0, 0, 0, 0, 0, 9]]\n\nPresent the answer in LaTex format:... | 15 | OMEGA_500_SAMPLE |
omega_500_151 | logic_puzzles_grid_rook_level_5_0096 | logic_puzzles_grid_rook | 5 | logic_puzzles_grid_rook | 5 | [
{
"role": "user",
"content": "In a 41x42 grid, what is the fewest number of steps it takes to get from the bottom left (0, 0) to the top right (40, 41), provided you can move right or up by any number of cells (like a chess rook) at each step, subject to the constraint that you cannot move through the follo... | 6 | OMEGA_500_SAMPLE |
omega_500_152 | combinatory_distribution_level_5_0014 | combinatory_distribution | 5 | combinatory_distribution | 5 | [
{
"role": "user",
"content": "You have {'t': 1, 'l': 4, 'z': 4, 'y': 3} and need to distribute them into 4 identical boxes with capacities [3, 4, 3, 2]. How many distinct distributions are possible?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 466 | OMEGA_500_SAMPLE |
omega_500_153 | arithmetic_matrix_eigenvalues_level_1_0057 | arithmetic_matrix_eigenvalues | 1 | arithmetic_matrix_eigenvalues | 1 | [
{
"role": "user",
"content": "Find the eigenvalues of the matrix and calculate the sum of absolute values of all eigenvalues:\n[[10, 5, -4],\n [0, 10, -1],\n [0, 0, -6]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 26 | OMEGA_500_SAMPLE |
omega_500_154 | logic_puzzles_blocked_grid_level_5_0035 | logic_puzzles_blocked_grid | 5 | logic_puzzles_blocked_grid | 5 | [
{
"role": "user",
"content": "In a 41x42 grid, how many different paths are there from the bottom left (0, 0) to the top right (40, 41), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (18, 24), (15, 5), (35, 23), (13, 32), (3, 41), ... | 457920 | OMEGA_500_SAMPLE |
omega_500_155 | algebra_func_extrema_coords_level_1_0010 | algebra_func_extrema_coords | 1 | algebra_func_extrema_coords | 1 | [
{
"role": "user",
"content": "How many local extrema (both minima and maxima) does the function f(x) = 1cos(1π(4x^2 + 3x - 2) - 3) - 3 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 573 | OMEGA_500_SAMPLE |
omega_500_156 | algebra_func_extrema_coords_level_1_0097 | algebra_func_extrema_coords | 1 | algebra_func_extrema_coords | 1 | [
{
"role": "user",
"content": "Find the x-coordinate of the local extremum with the smallest y-value of the function f(x) = -4((-2x)/(-3x)) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | -4 | OMEGA_500_SAMPLE |
omega_500_157 | algebra_func_extrema_level_2_0064 | algebra_func_extrema | 2 | algebra_func_extrema | 2 | [
{
"role": "user",
"content": "Determine the total count of local extrema points for the function f(x) = 2cos(1π(5(-2|x - 1|) - 1)) - 1 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 200 | OMEGA_500_SAMPLE |
omega_500_158 | number_theory_prime_mod_level_5_0095 | number_theory_prime_mod | 5 | number_theory_prime_mod | 5 | [
{
"role": "user",
"content": "Let p be the minimum prime where n^3+-2=0 (mod p^4) has a solution for some integer n, then find the minimum n for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 303 | OMEGA_500_SAMPLE |
omega_500_159 | arithmetic_matrix_determinant_level_6_0000 | arithmetic_matrix_determinant | 6 | arithmetic_matrix_determinant | 6 | [
{
"role": "user",
"content": "Calculate the determinant of the matrix, round to the nearest integer, then find the result modulo 1000:\n[[26, -1, 16, 4, -12, 19, -23, -5, 18, 9],\n [0, -1, -24, -28, -30, 5, -6, 6, -30, -16],\n [0, 0, 9, -23, -26, 29, -22, -30, 5, 4],\n [0, 0, 0, -22, 3, 15, -17, -27, -1, 0]... | 168 | OMEGA_500_SAMPLE |
omega_500_160 | arithmetic_matrix_multiplication_level_6_0075 | arithmetic_matrix_multiplication | 6 | arithmetic_matrix_multiplication | 6 | [
{
"role": "user",
"content": "Compute the sum of absolute values of all entries of the product matrix AB given:\nA =\n[[-25, 2, 22, -11, -9, 0, -6, -1, -13, 7, -1],\n [-18, -21, -20, -3, 4, -12, -3, -15, 5, -20, -15],\n [-13, -14, -24, 9, 10, -9, 2, 2, -4, 12, -19],\n [-11, 6, -15, -2, -23, 4, -20, -24, -12... | 50096 | OMEGA_500_SAMPLE |
omega_500_161 | algebra_linear_equation_level_3_0070 | algebra_linear_equation | 3 | algebra_linear_equation | 3 | [
{
"role": "user",
"content": "Suppose -3*l = 2*q - 16, 582*q + 60 = 586*q - l. Let y be (-38)/q + 1425/525. Solve 2*d - 4*m + 12 = -3*z, -b - 3*z + 48 = -4*d + 5*m - 0*m, 4*d - 5*m = 3*b - 36, y = -2*d + 3*m + 3*z - 32 for d.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | -4 | OMEGA_500_SAMPLE |
omega_500_162 | arithmetic_list_prime_factors_level_2_0055 | arithmetic_list_prime_factors | 2 | arithmetic_list_prime_factors | 2 | [
{
"role": "user",
"content": "What is the second-largest prime factor of 13520959865?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 103 | OMEGA_500_SAMPLE |
omega_500_163 | logic_puzzles_grid_chip_level_2_0030 | logic_puzzles_grid_chip | 2 | logic_puzzles_grid_chip | 2 | [
{
"role": "user",
"content": "Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell ... | 6 | OMEGA_500_SAMPLE |
omega_500_164 | logic_puzzles_blocked_grid_level_1_0051 | logic_puzzles_blocked_grid | 1 | logic_puzzles_blocked_grid | 1 | [
{
"role": "user",
"content": "In a 8x8 grid, how many different paths are there from the bottom left (0, 0) to the top right (7, 7), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (0, 5), (7, 2), (6, 5), (5, 7), (0, 3), (2, 6), (5, ... | 461 | OMEGA_500_SAMPLE |
omega_500_165 | arithmetic_mixed_level_1_0081 | arithmetic_mixed | 1 | arithmetic_mixed | 1 | [
{
"role": "user",
"content": "Calculate (-4197)/10 - (-48571883)/89190. Express your answer as an integer or in the form p/q where p and q are coprime.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1124/9 | OMEGA_500_SAMPLE |
omega_500_166 | combinatory_pattern_matching_level_1_0004 | combinatory_pattern_matching | 1 | combinatory_pattern_matching | 1 | [
{
"role": "user",
"content": "Form a word by randomly choosing 2 letters from the multiset {m: 1, x: 1, z: 1, d: 1}. What is the expected number of occurrences of the pattern 'x+d' in each word? Round your answer to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 0.08 | OMEGA_500_SAMPLE |
omega_500_167 | combinatory_probability_no_specific_letter_fixed_level_3_0064 | combinatory_probability_no_specific_letter_fixed | 3 | combinatory_probability_no_specific_letter_fixed | 3 | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {j: 1, q: 2, u: 5}, shuffle the letters in the word, what is the probability of no letter 'q' occupy any of their original positions?\nIf the probability can be written... | 47 | OMEGA_500_SAMPLE |
omega_500_168 | arithmetic_matrix_svd_level_1_0026 | arithmetic_matrix_svd | 1 | arithmetic_matrix_svd | 1 | [
{
"role": "user",
"content": "Calculate the rounded difference between the largest and smallest singular values from the SVD of:\n[[-2, 4, 2, -1],\n [0, -3, 3, -1],\n [4, 0, 4, -1],\n [0, 4, 3, -2]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | OMEGA_500_SAMPLE |
omega_500_169 | geometry_polygon_chords_level_2_0045 | geometry_polygon_chords | 2 | geometry_polygon_chords | 2 | [
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 2, from vertex 1 to vertex 8, from vertex 7 to vertex 3, from vertex 7 to vertex 2, from vertex 2 to vertex 6, from vertex 3 to vertex 1, from ver... | 3 | OMEGA_500_SAMPLE |
omega_500_170 | algebra_linear_equation_level_5_0039 | algebra_linear_equation | 5 | algebra_linear_equation | 5 | [
{
"role": "user",
"content": "Let v be (921/(4088319/(-1930)))/((725/(-1334))/5). Let z = 11 - 9. Solve -4*b - 5*y - 2*j - 8 = -4*q - 3*x + 2*u, -b - 3*q + 3*y - 5*x - 4*u - 2*j + 3 = 0, 4*b + 2*q + 2*j + 39 = b - 4*x + 3*u, 5*b = 4*q + 3*y - u - j + 4, 3*q - 4*u + 13 = 2*b + 2*y - x, 2*q - 3*u + z*j = v*b ... | -5 | OMEGA_500_SAMPLE |
omega_500_171 | arithmetic_matrix_inverse_level_5_0084 | arithmetic_matrix_inverse | 5 | arithmetic_matrix_inverse | 5 | [
{
"role": "user",
"content": "For the inverse of 1/450 times the matrix below, what is the value at position (1, 2) rounded to the nearest integer?\n[[9, 22, -21, 22, 6, -23, 21, 9, -12],\n [0, 5, 5, -9, -2, -3, -8, 24, -12],\n [0, 0, -11, -20, 1, -11, 24, 10, -18],\n [0, 0, 0, -13, 15, 8, -2, 14, 6],\n [0,... | -220 | OMEGA_500_SAMPLE |
omega_500_172 | algebra_func_extrema_coords_level_5_0055 | algebra_func_extrema_coords | 5 | algebra_func_extrema_coords | 5 | [
{
"role": "user",
"content": "Find the average of all x-coordinates of local extrema of the function f(x) = (3(-2e^(-1(-1(-3(-1((2x - 3)/(1x + 1)) - 1)^2 + 5(-1((2x - 3)/(1x + 1)) - 1) - 3) - 4)) - 2))/(-3(-2e^(-1(-1(-3(-1((2x - 3)/(1x + 1)) - 1)^2 + 5(-1((2x - 3)/(1x + 1)) - 1) - 3) - 4)) - 2) - 1) in the ... | 6 | OMEGA_500_SAMPLE |
omega_500_173 | algebra_func_derivative_sign_level_3_0013 | algebra_func_derivative_sign | 3 | algebra_func_derivative_sign | 3 | [
{
"role": "user",
"content": "For the function f(x) = -1|(1sin(2π(1(-3x^2 - 5x - 1)^2 - 2) + 3)) - 3| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 18 | OMEGA_500_SAMPLE |
omega_500_174 | algebra_func_area_level_5_0052 | algebra_func_area | 5 | algebra_func_area | 5 | [
{
"role": "user",
"content": "Determine the area of the region enclosed by the graphs of f(x) = (-3((-2((2(-5(-5(-5x^2 - 3x + 5)^2 + 4(-5x^2 - 3x + 5) + 2) - 5) - 2)/(2(-5(-5(-5x^2 - 3x + 5)^2 + 4(-5x^2 - 3x + 5) + 2) - 5) + 3)) + 3)/(2((2(-5(-5(-5x^2 - 3x + 5)^2 + 4(-5x^2 - 3x + 5) + 2) - 5) - 2)/(2(-5(-5(... | 6.0 | OMEGA_500_SAMPLE |
omega_500_175 | arithmetic_matrix_eigenvalues_level_5_0088 | arithmetic_matrix_eigenvalues | 5 | arithmetic_matrix_eigenvalues | 5 | [
{
"role": "user",
"content": "Calculate the largest eigenvalue by absolute value for the matrix:\n[[-15, 12, -20, 19, -15, -12, -24, 14, -23],\n [0, 7, -2, -7, -8, 23, -2, 19, 9],\n [0, 0, -1, 19, 23, 17, 8, -1, -10],\n [0, 0, 0, -3, -7, -1, -10, 25, -17],\n [0, 0, 0, 0, 8, -10, 18, 14, -8],\n [0, 0, 0, 0, ... | 20 | OMEGA_500_SAMPLE |
omega_500_176 | algebra_func_intersection_coords_level_5_0068 | algebra_func_intersection_coords | 5 | algebra_func_intersection_coords | 5 | [
{
"role": "user",
"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1(1cos(3π((-1(-3|(-5(1x^2 - 4x + 1)^2 + 3(1x^2 - 4x + 1) + 3) - 3| + 1) - 2)/(-2(-3|(-5(1x^2 - 4x + 1)^2 + 3(1x^2 - 4x + 1) + 3) - 3| + 1) + 3)) + 2))^2 + 2(1cos(3π((-1(-3|(-5(1x^2 - 4x + 1)^2 + 3(1x^... | 0 | OMEGA_500_SAMPLE |
omega_500_177 | combinatory_probability_exactly_n_specific_fixed_level_2_0079 | combinatory_probability_exactly_n_specific_fixed | 2 | combinatory_probability_exactly_n_specific_fixed | 2 | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {o: 1, i: 3, y: 2}, shuffle the letters in the word, what is the probability of exact 1 letter 'o' remains in the same position?\nIf the probability can be written as t... | 7 | OMEGA_500_SAMPLE |
omega_500_178 | number_theory_prime_mod_level_1_0030 | number_theory_prime_mod | 1 | number_theory_prime_mod | 1 | [
{
"role": "user",
"content": "let p=5. How many solutions m in [1, 2, ..., 5^4] satisfy m^3+1=0 (mod 5^4)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 1 | OMEGA_500_SAMPLE |
omega_500_179 | algebra_func_area_level_1_0030 | algebra_func_area | 1 | algebra_func_area | 1 | [
{
"role": "user",
"content": "Determine the area of the region enclosed by the graphs of f(x) = 3e^(2(-3x - 3)) - 1 and g(x) = (-2x - 3)/(1x + 3) between the vertical lines x = -0.6 and x = -0.0. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 0.0 | OMEGA_500_SAMPLE |
omega_500_180 | logic_puzzles_grid_chip_level_4_0006 | logic_puzzles_grid_chip | 4 | logic_puzzles_grid_chip | 4 | [
{
"role": "user",
"content": "Chips, colored either black or white, are placed in a 7x7 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell ... | 11 | OMEGA_500_SAMPLE |
omega_500_181 | geometry_rotation_level_1_0035 | geometry_rotation | 1 | geometry_rotation | 1 | [
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 9 and from vertex 1 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex ... | 1 | OMEGA_500_SAMPLE |
omega_500_182 | arithmetic_matrix_eigenvalues_level_6_0016 | arithmetic_matrix_eigenvalues | 6 | arithmetic_matrix_eigenvalues | 6 | [
{
"role": "user",
"content": "Find the eigenvalues of the matrix and calculate the largest eigenvalue by absolute value:\n[[-8, 20, 28, 25, -12, 1, -1, -2, 18, 14],\n [0, -1, -15, 9, -12, -13, -30, -17, 20, 16],\n [0, 0, 23, -3, 18, 19, -1, 7, 20, -27],\n [0, 0, 0, 24, 8, -28, 1, -21, -21, -12],\n [0, 0, 0,... | 30 | OMEGA_500_SAMPLE |
omega_500_183 | algebra_func_intersection_level_4_0019 | algebra_func_intersection | 4 | algebra_func_intersection | 4 | [
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -3|(2|(3(-4(3e^(1x))^2 + 5(3e^(1x)) + 5) - 3) - 3| - 1) + 2| + 1 and g(x) = -2(1|((2(-5x^2 - 5x - 1) + 1)/(-1(-5x^2 - 5x - 1) - 2)) + 2| - 1) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format:... | 5 | OMEGA_500_SAMPLE |
omega_500_184 | algebra_func_area_level_2_0091 | algebra_func_area | 2 | algebra_func_area | 2 | [
{
"role": "user",
"content": "Find the area of the region bounded by the curves f(x) = (-3(4(-4x^2 + 2x + 2) + 2) + 3)/(2(4(-4x^2 + 2x + 2) + 2) - 1), g(x) = -2cos(3π(-4x + 1) + 1) - 3, x = 3.7, and x = 3.8. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
... | 0.2 | OMEGA_500_SAMPLE |
omega_500_185 | arithmetic_matrix_eigenvalues_level_2_0098 | arithmetic_matrix_eigenvalues | 2 | arithmetic_matrix_eigenvalues | 2 | [
{
"role": "user",
"content": "What is the largest eigenvalue by absolute value for the following matrix?\n[[-5, 5, 10, 5],\n [0, -6, -10, 5],\n [0, 0, 4, 1],\n [0, 0, 0, 0]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 6 | OMEGA_500_SAMPLE |
omega_500_186 | combinatory_probability_no_fixed_points_level_5_0019 | combinatory_probability_no_fixed_points | 5 | combinatory_probability_no_fixed_points | 5 | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 5 letters from the multiset {m: 3, a: 4, d: 1, z: 2, c: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be ... | 2633 | OMEGA_500_SAMPLE |
omega_500_187 | arithmetic_gcd_level_3_0017 | arithmetic_gcd | 3 | arithmetic_gcd | 3 | [
{
"role": "user",
"content": "Calculate the greatest common factor of 51063271 and 26529276380.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 9017 | OMEGA_500_SAMPLE |
omega_500_188 | arithmetic_matrix_eigenvalues_level_6_0011 | arithmetic_matrix_eigenvalues | 6 | arithmetic_matrix_eigenvalues | 6 | [
{
"role": "user",
"content": "Find the eigenvalues of the matrix and calculate the sum of absolute values of all eigenvalues:\n[[-8, -2, 8, -4, -1, 7, -22, -20, 0, 19],\n [-2, 25, 10, -19, -8, -6, -14, -4, 13, 16],\n [8, 10, 20, -18, -8, 0, 2, -3, 16, 8],\n [-4, -19, -18, -30, -6, 4, -8, -6, -4, 15],\n [-1,... | 309 | OMEGA_500_SAMPLE |
omega_500_189 | combinatory_probability_at_least_n_specific_fixed_level_4_0027 | combinatory_probability_at_least_n_specific_fixed | 4 | combinatory_probability_at_least_n_specific_fixed | 4 | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {w: 2, e: 2, o: 4, q: 2}, shuffle the letters in the word, what is the probability of at least 1 letter 'q' remains in the same position?\nIf the probability can be wri... | 56 | OMEGA_500_SAMPLE |
omega_500_190 | algebra_func_area_level_4_0088 | algebra_func_area | 4 | algebra_func_area | 4 | [
{
"role": "user",
"content": "Determine the area of the region enclosed by the graphs of f(x) = -1|(-1((3(-3|(-1x^2 + 4x + 2) - 2| + 1) + 3)/(1(-3|(-1x^2 + 4x + 2) - 2| + 1) - 2))^2 - 4((3(-3|(-1x^2 + 4x + 2) - 2| + 1) + 3)/(1(-3|(-1x^2 + 4x + 2) - 2| + 1) - 2)) + 4) - 2| + 2 and g(x) = (1(4((-2(5x^2 + 5) -... | 0.1 | OMEGA_500_SAMPLE |
omega_500_191 | logic_puzzles_grid_knight_level_2_0015 | logic_puzzles_grid_knight | 2 | logic_puzzles_grid_knight | 2 | [
{
"role": "user",
"content": "Supposing you start in the bottom left (0, 0) cell of a 15x16 grid, what is minimum number of steps you can take to get to (12, 12), if your only valid moves are to move over 1 squares in one direction and over 4 squares in a perpendicular direction (at once)?\n\nPresent the an... | 8 | OMEGA_500_SAMPLE |
omega_500_192 | algebra_func_extrema_level_4_0096 | algebra_func_extrema | 4 | algebra_func_extrema | 4 | [
{
"role": "user",
"content": "Determine the count of local minima for the function f(x) = 3((2(-2|(-1|(3sin(3πx - 1) + 1) - 2| + 1) - 1| + 1) + 1)/(2(-2|(-1|(3sin(3πx - 1) + 1) - 2| + 1) - 1| + 1) + 2))^2 - 1((2(-2|(-1|(3sin(3πx - 1) + 1) - 2| + 1) - 1| + 1) + 1)/(2(-2|(-1|(3sin(3πx - 1) + 1) - 2| + 1) - 1|... | 189 | OMEGA_500_SAMPLE |
omega_500_193 | logic_puzzles_grid_chip_level_1_0010 | logic_puzzles_grid_chip | 1 | logic_puzzles_grid_chip | 1 | [
{
"role": "user",
"content": "Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more... | 8 | OMEGA_500_SAMPLE |
omega_500_194 | logic_puzzles_blocked_grid_level_1_0049 | logic_puzzles_blocked_grid | 1 | logic_puzzles_blocked_grid | 1 | [
{
"role": "user",
"content": "In a 6x5 grid, how many different paths are there from the bottom left (0, 0) to the top right (5, 4), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (0, 3), (2, 0), (3, 3), (5, 1), (3, 2), (1, 0), (4, ... | 2 | OMEGA_500_SAMPLE |
omega_500_195 | algebra_func_extrema_coords_level_4_0025 | algebra_func_extrema_coords | 4 | algebra_func_extrema_coords | 4 | [
{
"role": "user",
"content": "Find the x-coordinate of the leftmost local extremum of the function f(x) = 5(-3sin(3π(4((2(-1x^2 + 4x + 5) + 2)/(-2(-1x^2 + 4x + 5) - 3))^2 + 5((2(-1x^2 + 4x + 5) + 2)/(-2(-1x^2 + 4x + 5) - 3)) - 2) + 3) - 1)^2 + 4(-3sin(3π(4((2(-1x^2 + 4x + 5) + 2)/(-2(-1x^2 + 4x + 5) - 3))^2... | -2 | OMEGA_500_SAMPLE |
omega_500_196 | logic_puzzles_zebralogic_level_3_0026 | logic_puzzles_zebralogic | 3 | logic_puzzles_zebralogic | 3 | [
{
"role": "user",
"content": "There are 2 houses, numbered 1 to 2 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Arnold`, `Eric`\n - The people ... | 1 | OMEGA_500_SAMPLE |
omega_500_197 | logic_puzzles_zebralogic_level_4_0096 | logic_puzzles_zebralogic | 4 | logic_puzzles_zebralogic | 4 | [
{
"role": "user",
"content": "There are 3 houses, numbered 1 to 3 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Arnold`, `Peter`, `Eric`\n - Ev... | 1 | OMEGA_500_SAMPLE |
omega_500_198 | algebra_func_extrema_level_5_0078 | algebra_func_extrema | 5 | algebra_func_extrema | 5 | [
{
"role": "user",
"content": "How many local maxima does the function f(x) = 5((-2(3sin(1π(5(-5(5x - 5))^2 - 3(-5(5x - 5)) + 3) - 2)) - 1)/(1(3sin(1π(5(-5(5x - 5))^2 - 3(-5(5x - 5)) + 3) - 2)) + 3))^2 + 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 45 | OMEGA_500_SAMPLE |
omega_500_199 | algebra_func_derivative_sign_level_2_0052 | algebra_func_derivative_sign | 2 | algebra_func_derivative_sign | 2 | [
{
"role": "user",
"content": "For the function f(x) = 1|(-2(-2cos(2πx + 3) - 1)^2 + 2(-2cos(2πx + 3) - 1)) - 1| + 3, find the number of maximal connected intervals in [-10, 10] where f'(x) < 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 40 | OMEGA_500_SAMPLE |
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