Dataset Viewer
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_000
|
arithmetic_matrix_svd_level_4_0013
|
arithmetic_matrix_svd
| 4 |
arithmetic_matrix_svd
|
4
|
[
{
"role": "user",
"content": "Calculate the rounded largest singular value from the SVD of:\n[[-1, -4, -3, -1, 4, -7, -6],\n [1, 0, -5, -3, -4, 5, -11],\n [-8, -8, -7, 6, -5, 3, 0],\n [-12, 9, 7, 4, -6, 12, 0],\n [-9, -9, -7, 6, -1, 9, -6],\n [-3, 6, -6, -10, 0, 0, 5],\n [7, -5, -4, -6, -12, -10, 0]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
28
|
OMEGA_500_SAMPLE
|
omega_500_001
|
arithmetic_list_prime_factors_level_5_0071
|
arithmetic_list_prime_factors
| 5 |
arithmetic_list_prime_factors
|
5
|
[
{
"role": "user",
"content": "Find the second-largest prime factor of 502550486709360515258.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
48859
|
OMEGA_500_SAMPLE
|
omega_500_002
|
combinatory_probability_no_fixed_points_level_3_0016
|
combinatory_probability_no_fixed_points
| 3 |
combinatory_probability_no_fixed_points
|
3
|
[
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {g: 2, t: 4, x: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
473
|
OMEGA_500_SAMPLE
|
omega_500_003
|
arithmetic_matrix_multiplication_level_2_0074
|
arithmetic_matrix_multiplication
| 2 |
arithmetic_matrix_multiplication
|
2
|
[
{
"role": "user",
"content": "What is the sum of all entries when matrix A is multiplied by matrix B?\nA =\n[[0, 10, 5, -9, -9],\n [-3, 2, -1, -3, -7],\n [4, -10, 0, -4, 6],\n [9, 7, 1, -2, -6],\n [-1, -7, -6, -1, 0],\n [10, -1, -3, -9, -3]]\nB =\n[[5, -6, -8, 1, 2],\n [2, -1, -7, -2, -9],\n [8, -10, -4, 6, -8],\n [9, 9, 10, -8, -8],\n [6, -4, 3, 6, 7]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-777
|
OMEGA_500_SAMPLE
|
omega_500_004
|
combinatory_probability_at_least_n_specific_fixed_level_3_0033
|
combinatory_probability_at_least_n_specific_fixed
| 3 |
combinatory_probability_at_least_n_specific_fixed
|
3
|
[
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {t: 3, s: 2, o: 3}, shuffle the letters in the word, what is the probability of at least 2 letter 's' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
29
|
OMEGA_500_SAMPLE
|
omega_500_005
|
combinatory_distribution_level_5_0012
|
combinatory_distribution
| 5 |
combinatory_distribution
|
5
|
[
{
"role": "user",
"content": "Given the letters {'j': 2, 'r': 4, 'i': 2, 'c': 4}, place them into 4 indistinguishable boxes with sizes [4, 1, 2, 5]. In how many ways can you do this?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
586
|
OMEGA_500_SAMPLE
|
omega_500_006
|
number_theory_triple_count_level_5_0058
|
number_theory_triple_count
| 5 |
number_theory_triple_count
|
5
|
[
{
"role": "user",
"content": "Define N to be the number of ordered quadruples of integers (a, b, c, d) such that a, b, c, d <= 3^2 and (1)a^3 + (-4)b^3 + (-4)c^3 + (2)d^3 is divisible by 3^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
567
|
OMEGA_500_SAMPLE
|
omega_500_007
|
geometry_polygon_color_level_4_0028
|
geometry_polygon_color
| 4 |
geometry_polygon_color
|
4
|
[
{
"role": "user",
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, 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, vertex 9 is red, vertex 10 is red, vertex 11 is blue. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
OMEGA_500_SAMPLE
|
omega_500_008
|
algebra_func_extrema_level_1_0007
|
algebra_func_extrema
| 1 |
algebra_func_extrema
|
1
|
[
{
"role": "user",
"content": "Find the number of local minimum points for f(x) = 1|(3sin(1πx + 1) + 3) + 3| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
10
|
OMEGA_500_SAMPLE
|
omega_500_009
|
algebra_func_zeros_level_1_0022
|
algebra_func_zeros
| 1 |
algebra_func_zeros
|
1
|
[
{
"role": "user",
"content": "How many zeros does the function f(x) = -2cos(2π(3x^2 - 2x + 5) + 3) - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
506
|
OMEGA_500_SAMPLE
|
omega_500_010
|
logic_puzzles_grid_chip_level_3_0047
|
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 of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 0) is white, cell (1, 1) is black, cell (4, 1) is black, cell (3, 4) is white, cell (0, 1) is black, cell (2, 5) is black, cell (5, 5) is black, cell (3, 2) is white, cell (3, 3) is white. How many chips are placed on the grid?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
14
|
OMEGA_500_SAMPLE
|
omega_500_011
|
combinatory_probability_at_least_n_specific_fixed_level_2_0017
|
combinatory_probability_at_least_n_specific_fixed
| 2 |
combinatory_probability_at_least_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 {l: 4, s: 1, p: 1}, shuffle the letters in the word, what is the probability of at least 3 letter 'l' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
6
|
OMEGA_500_SAMPLE
|
omega_500_012
|
arithmetic_matrix_rank_level_2_0053
|
arithmetic_matrix_rank
| 2 |
arithmetic_matrix_rank
|
2
|
[
{
"role": "user",
"content": "Find the rank of the matrix:\n[[3, 26, 18, 30, -20],\n [15, -5, 10, -13, 2],\n [-28, -10, -12, 3, 13],\n [-11, 30, -14, 24, 14],\n [16, 35, -36, 16, -9]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
OMEGA_500_SAMPLE
|
omega_500_013
|
logic_puzzles_grid_knight_level_1_0025
|
logic_puzzles_grid_knight
| 1 |
logic_puzzles_grid_knight
|
1
|
[
{
"role": "user",
"content": "Supposing you start in the bottom left (0, 0) cell of a 7x8 grid, what is minimum number of steps you can take to get to (4, 5), if your only valid moves are like a chess knight, i.e, over two squares in one direction and over one square in a perpendicular direction in one move?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
3
|
OMEGA_500_SAMPLE
|
omega_500_014
|
arithmetic_matrix_rank_level_2_0069
|
arithmetic_matrix_rank
| 2 |
arithmetic_matrix_rank
|
2
|
[
{
"role": "user",
"content": "Determine the rank of the following matrix:\n[[18, -10, 16, 12, -16],\n [48, -24, 30, 30, -42],\n [-28, 4, 30, -10, 22],\n [28, -8, -11, 13, -23],\n [36, -16, 13, 21, -31]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
2
|
OMEGA_500_SAMPLE
|
omega_500_015
|
logic_puzzles_grid_rook_level_3_0097
|
logic_puzzles_grid_rook
| 3 |
logic_puzzles_grid_rook
|
3
|
[
{
"role": "user",
"content": "In a 25x27 grid, what is the fewest number of steps it takes to get from the bottom left (0, 0) to the top right (24, 26), 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 cells: (20, 11), (19, 0), (11, 2), (8, 18), (12, 21), (23, 1), (20, 22), (6, 2), (20, 9), (13, 20), (9, 12), (3, 23), (11, 22), (13, 18), (10, 20), (15, 19), (23, 12), (20, 13), (24, 6), (12, 26), (21, 7), (7, 22), (24, 24), (2, 7), (14, 14), (5, 20), (22, 9), (8, 8), (1, 6), (14, 26), (21, 2), (17, 22), (0, 11), (11, 24), (15, 22), (7, 25), (3, 19), (21, 21), (2, 0), (2, 13), (17, 6), (11, 9), (1, 3), (15, 15), (15, 6), (24, 2), (4, 15), (11, 5), (2, 16), (16, 16), (12, 14), (24, 19), (1, 7), (12, 12), (10, 26), (4, 24), (8, 26), (9, 4), (14, 10), (20, 1), (2, 8), (16, 23), (16, 19), (18, 12), (21, 0), (4, 8), (1, 18), (24, 11), (15, 21), (18, 22), (21, 23), (3, 1), (17, 16), (21, 6), (7, 15), (1, 23), (1, 14), (19, 25), (5, 2), (1, 17), (19, 11), (11, 8), (0, 21), (21, 9), (15, 5), (20, 19), (15, 9), (0, 8), (5, 14), (4, 21), (20, 16), (11, 4), (1, 12), (10, 21), (11, 21), (2, 12), (10, 25), (11, 6), (6, 26), (1, 2), (0, 2), (5, 13), (9, 20), (0, 19), (10, 8)? If it's impossible, answer with -1.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
8
|
OMEGA_500_SAMPLE
|
omega_500_016
|
algebra_func_derivative_sign_level_3_0098
|
algebra_func_derivative_sign
| 3 |
algebra_func_derivative_sign
|
3
|
[
{
"role": "user",
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = 3cos(2π(-3(3(-1x^2 + 1x - 4) + 2) - 2) + 3) positive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
19
|
OMEGA_500_SAMPLE
|
omega_500_017
|
logic_puzzles_blocked_grid_level_5_0031
|
logic_puzzles_blocked_grid
| 5 |
logic_puzzles_blocked_grid
|
5
|
[
{
"role": "user",
"content": "In a 47x42 grid, how many different paths are there from the bottom left (0, 0) to the top right (46, 41), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (13, 32), (14, 6), (41, 18), (37, 22), (16, 0), (6, 21), (4, 10), (42, 18), (26, 15), (35, 32), (12, 20), (6, 10), (13, 36), (34, 24), (8, 7), (37, 35), (25, 28), (8, 30), (21, 37), (38, 15), (22, 25), (25, 23), (37, 12), (1, 41), (21, 41), (5, 1), (7, 20), (1, 29), (15, 23), (27, 23), (38, 37), (25, 38), (19, 8), (25, 33), (12, 37), (37, 34), (40, 0), (43, 28), (6, 3), (18, 22), (37, 15), (13, 12), (21, 11), (26, 17), (36, 25), (15, 3), (37, 20), (8, 23), (0, 24), (39, 10), (0, 13), (33, 26), (22, 6), (17, 15), (28, 29), (11, 39), (24, 10), (29, 10), (7, 7), (18, 33), (0, 10), (44, 18), (0, 2), (7, 21), (17, 30), (32, 10), (46, 10), (10, 40), (21, 39), (32, 6), (17, 22), (41, 22), (4, 9), (32, 0), (0, 29), (18, 35), (23, 5), (13, 38), (46, 26), (21, 18), (18, 36), (5, 21), (24, 13), (44, 41), (17, 7), (41, 30), (46, 27), (14, 3), (36, 22), (33, 10), (24, 22), (19, 23), (34, 7), (17, 14), (19, 24), (8, 25), (7, 12), (3, 20), (46, 15), (16, 41), (38, 40), (28, 37), (1, 18), (25, 26), (7, 29), (9, 12), (29, 17), (44, 4), (15, 41), (30, 25), (14, 0), (3, 8), (37, 1), (30, 29), (7, 30), (33, 7), (4, 12), (3, 34), (16, 9), (39, 23), (5, 4), (41, 40), (45, 32), (31, 40), (15, 28), (41, 14), (35, 40), (10, 12), (25, 12), (30, 18), (0, 41), (18, 32), (21, 16), (43, 15), (15, 39), (2, 13), (23, 31), (12, 1), (19, 4), (42, 28), (28, 19), (20, 2), (30, 11), (34, 21), (26, 26), (6, 14), (46, 20), (5, 0), (28, 31), (11, 14), (18, 8), (45, 21), (31, 38), (20, 32), (7, 37), (32, 1), (16, 26), (8, 37), (36, 4), (45, 7), (9, 40), (17, 35), (28, 28), (9, 21), (39, 40), (31, 26), (37, 39), (27, 13), (34, 33), (6, 36), (46, 38), (9, 29), (7, 34), (7, 31), (24, 23), (19, 0), (0, 1), (33, 40), (43, 5), (44, 32), (2, 30), (5, 38), (11, 19), (32, 19), (19, 6), (14, 37), (13, 41), (0, 5), (31, 15), (9, 17), (35, 3), (24, 4), (44, 35), (44, 11), (20, 0), (31, 29), (29, 39), (35, 12), (15, 17), (13, 11), (31, 2), (43, 12), (42, 24), (5, 25), (28, 21), (29, 36), (36, 34), (17, 12), (18, 13), (6, 33), (13, 28), (11, 24), (8, 32), (2, 40), (6, 9), (41, 10), (7, 18), (28, 9), (34, 41), (37, 38), (9, 3), (39, 1), (2, 21), (4, 37), (16, 37), (13, 35), (13, 9), (11, 3), (26, 40), (37, 27), (9, 14), (41, 32), (37, 30), (45, 1), (28, 27), (10, 0), (7, 9), (25, 4), (4, 31), (14, 40), (43, 14), (37, 3), (16, 18), (43, 29), (21, 19), (11, 23), (17, 25), (11, 8), (33, 22), (20, 12), (8, 14), (33, 18), (14, 21), (8, 18), (6, 39), (21, 38), (26, 30), (14, 33), (39, 18), (4, 2), (1, 36), (8, 20), (2, 27), (21, 35), (9, 7), (10, 33), (25, 1), (23, 39), (26, 14), (14, 35), (35, 9), (14, 4), (42, 13), (24, 36), (3, 31), (4, 14), (45, 19), (44, 14), (20, 7), (38, 14), (20, 36), (8, 24), (26, 32), (3, 25), (0, 25), (10, 4), (1, 26), (28, 20), (13, 39), (9, 38), (5, 37), (23, 34), (29, 14), (21, 33), (4, 36), (21, 9), (37, 21), (23, 30), (34, 34), (23, 38), (24, 1), (15, 35), (13, 8), (29, 40), (18, 24), (12, 8), (21, 40), (22, 9), (13, 17), (38, 31), (3, 14), (10, 6), (15, 20), (29, 41), (4, 21), (34, 0), (19, 26), (2, 32), (10, 11), (35, 10), (29, 32), (17, 41), (20, 40), (12, 26), (42, 33), (46, 14), (21, 5), (1, 35), (28, 33), (28, 12), (35, 25), (12, 29), (30, 8), (16, 34), (37, 41), (18, 5), (25, 29), (39, 21), (38, 25), (21, 24), (22, 5), (37, 32), (42, 0), (3, 28), (4, 15), (15, 37), (38, 6), (29, 16), (10, 25), (41, 2), (29, 22), (7, 33), (36, 37), (16, 35), (24, 5), (25, 3), (17, 23), (17, 2), (25, 24), (37, 4), (30, 20), (20, 35), (9, 31), (7, 3), (27, 18), (13, 6), (20, 20), (2, 39), (41, 0), (12, 35), (12, 27), (42, 20), (24, 28), (12, 2), (41, 12), (23, 18), (10, 37), (29, 19), (38, 35), (19, 39), (32, 29), (14, 18), (11, 10), (28, 35), (11, 16), (33, 39), (30, 35), (34, 11), (17, 16), (26, 9), (25, 37), (46, 25), (0, 21), (5, 12), (38, 17), (2, 34), (29, 34), (41, 8), (7, 27), (32, 23), (8, 5), (13, 10), (24, 34), (35, 18), (43, 35), (16, 30), (26, 21), (8, 2), (39, 27), (26, 8), (45, 29), (43, 37), (46, 29), (12, 3), (43, 39), (27, 9), (30, 13), (1, 10), (35, 4), (46, 40), (5, 16), (46, 33), (2, 22), (43, 11), (27, 27), (19, 36), (17, 6), (24, 16), (15, 26), (45, 30), (16, 39), (40, 9), (12, 36), (34, 19), (29, 25), (45, 2), (30, 9), (16, 27), (1, 21), (1, 16), (12, 24), (10, 29), (12, 41), (15, 22), (25, 8), (18, 12), (41, 35), (44, 19), (19, 3), (43, 0), (35, 21), (8, 0), (38, 8), (19, 19), (25, 10), (7, 13), (7, 24), (39, 5), (1, 25), (4, 24), (11, 9), (40, 3), (29, 3), (45, 3), (46, 12), (33, 35), (3, 3), (40, 24), (38, 11), (9, 22), (14, 12), (30, 27), (26, 6), (31, 34), (31, 3), (29, 4), (12, 31), (43, 40), (27, 16), (18, 15), (18, 41), (11, 29), (2, 18), (29, 5), (38, 3), (44, 3), (6, 30), (43, 24), (11, 34), (35, 15), (6, 12), (43, 7), (39, 8), (17, 3), (42, 16), (10, 13), (10, 10), (1, 33), (21, 1), (22, 4), (30, 6), (9, 37), (33, 9), (10, 41), (43, 30), (10, 7), (44, 36), (5, 32), (17, 26), (16, 10), (26, 28), (1, 20), (5, 27), (13, 0), (25, 2), (0, 9), (39, 35), (35, 13), (19, 21), (24, 18), (33, 8), (43, 36), (42, 25), (33, 38), (13, 37), (2, 41), (11, 27), (18, 39), (14, 41), (32, 24), (41, 38), (28, 0), (34, 8), (19, 12), (32, 13), (17, 21), (46, 3), (32, 20), (16, 22), (22, 23), (39, 17), (12, 39), (27, 19), (46, 39), (30, 14), (9, 36), (14, 36), (19, 1), (25, 39), (18, 40), (18, 18), (42, 11), (7, 41), (15, 6), (38, 27), (6, 2), (24, 0), (12, 21), (27, 8), (6, 17), (35, 11), (21, 30), (0, 35), (10, 35), (38, 29), (37, 8), (31, 28), (41, 15), (22, 16), (26, 22), (16, 2), (25, 32), (35, 34), (19, 38), (28, 34), (24, 20), (3, 11), (45, 6), (3, 7), (40, 33), (21, 15), (5, 19), (28, 8), (17, 4), (28, 22)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
320000
|
OMEGA_500_SAMPLE
|
omega_500_018
|
arithmetic_matrix_rank_level_7_0005
|
arithmetic_matrix_rank
| 7 |
arithmetic_matrix_rank
|
7
|
[
{
"role": "user",
"content": "Compute the rank of the given 12x12 matrix:\n[[502, -77, -605, -8, -282, -163, -517, 555, -463, -477, -196, -234],\n [374, -45, -506, 18, -190, -191, -510, 473, -389, -406, -141, -298],\n [452, -110, -363, -86, -320, 82, -55, 304, -272, -263, -193, 196],\n [-168, 12, 264, -24, 72, 132, 312, -252, 204, 216, 60, 216],\n [218, -11, -363, 40, -86, -197, -451, 349, -281, -299, -76, -326],\n [352, -80, -308, -56, -240, 32, -100, 264, -232, -228, -148, 96],\n [696, -132, -726, -60, -432, -84, -462, 648, -552, -558, -282, -72],\n [656, -152, -561, -110, -452, 76, -157, 478, -422, -413, -277, 208],\n [-154, 43, 99, 40, 118, -59, -37, -77, 73, 67, 68, -122],\n [-394, 115, 231, 112, 310, -179, -145, -173, 169, 151, 176, -362],\n [44, -58, 176, -100, -108, 274, 472, -198, 142, 168, -38, 492],\n [-116, 118, -308, 196, 228, -526, -892, 354, -250, -300, 86, -948]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
2
|
OMEGA_500_SAMPLE
|
omega_500_019
|
combinatory_pattern_matching_level_2_0052
|
combinatory_pattern_matching
| 2 |
combinatory_pattern_matching
|
2
|
[
{
"role": "user",
"content": "If you form a word of length 3 by randomly selecting letters from the multiset {a: 3, y: 1, z: 1, r: 1}, what is the expected number of times the pattern 'a.*a' appears? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0.6
|
OMEGA_500_SAMPLE
|
omega_500_020
|
arithmetic_matrix_power_level_5_0026
|
arithmetic_matrix_power
| 5 |
arithmetic_matrix_power
|
5
|
[
{
"role": "user",
"content": "Calculate the matrix power A^7 and find the value at position (5, 6) of the result:\nA =\n[[-2, 0, -11, 12, -6, 7, -9, -5],\n [7, 9, 8, 6, 0, 1, -3, -2],\n [-11, 12, -4, 4, -10, 11, -7, -8],\n [-8, 9, 2, 10, 4, -4, -10, 7],\n [4, -3, -10, 5, 5, 4, 11, -10],\n [-10, -7, 0, 11, 2, 12, -4, -9],\n [1, -8, -11, -2, -4, 11, -10, -4],\n [-10, 12, -11, -1, -3, -4, 2, -5]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
468194046
|
OMEGA_500_SAMPLE
|
omega_500_021
|
arithmetic_matrix_eigenvalues_level_7_0006
|
arithmetic_matrix_eigenvalues
| 7 |
arithmetic_matrix_eigenvalues
|
7
|
[
{
"role": "user",
"content": "Determine the largest eigenvalue by absolute value of the matrix:\n[[-7, -28, -31, -7, 11, 32, 9, -34, -9, 0, 0, -10],\n [0, -9, 33, -16, -25, 2, -30, -13, 11, 10, -24, -23],\n [0, 0, 7, 32, -31, 1, -5, -27, 15, -7, 4, 5],\n [0, 0, 0, 10, -15, 0, 18, 21, -35, 27, 18, 19],\n [0, 0, 0, 0, 17, -27, 16, 21, -10, 5, -1, 27],\n [0, 0, 0, 0, 0, -19, 7, 23, 15, 18, -12, -11],\n [0, 0, 0, 0, 0, 0, 27, -14, -10, -8, 13, 35],\n [0, 0, 0, 0, 0, 0, 0, -23, 15, 20, 26, -4],\n [0, 0, 0, 0, 0, 0, 0, 0, -18, -35, 11, 30],\n [0, 0, 0, 0, 0, 0, 0, 0, 0, -4, -26, -2],\n [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 31, 15],\n [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
31
|
OMEGA_500_SAMPLE
|
omega_500_022
|
arithmetic_gcd_level_3_0009
|
arithmetic_gcd
| 3 |
arithmetic_gcd
|
3
|
[
{
"role": "user",
"content": "What is the greatest common divisor of 334156129367352385 and 3370551835?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
25342495
|
OMEGA_500_SAMPLE
|
omega_500_023
|
algebra_func_extrema_level_5_0061
|
algebra_func_extrema
| 5 |
algebra_func_extrema
|
5
|
[
{
"role": "user",
"content": "Determine the count of local maxima for the function f(x) = -1(-1((-2(5((-1(-2sin(3πx) + 3) + 1)/(-3(-2sin(3πx) + 3) + 4))^2 - 3((-1(-2sin(3πx) + 3) + 1)/(-3(-2sin(3πx) + 3) + 4)) + 2))/(-2(5((-1(-2sin(3πx) + 3) + 1)/(-3(-2sin(3πx) + 3) + 4))^2 - 3((-1(-2sin(3πx) + 3) + 1)/(-3(-2sin(3πx) + 3) + 4)) + 2) - 3)) + 5) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
36
|
OMEGA_500_SAMPLE
|
omega_500_024
|
number_theory_prime_mod_level_4_0023
|
number_theory_prime_mod
| 4 |
number_theory_prime_mod
|
4
|
[
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 6). For the least p where, for some integer n, n^5=-1 (mod p^3) has a solution, find the minimal n for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
342
|
OMEGA_500_SAMPLE
|
omega_500_025
|
arithmetic_matrix_power_level_5_0089
|
arithmetic_matrix_power
| 5 |
arithmetic_matrix_power
|
5
|
[
{
"role": "user",
"content": "Find the value at position (2, 7) of the 6th power of the matrix:\n[[-5, -4, 4, -3, 4, 0, -6, 3],\n [-4, -6, 4, 6, -4, 0, 7, 1],\n [4, 4, -7, 2, 6, 0, 0, 2],\n [-3, 6, 2, -4, 5, 0, -2, -3],\n [4, -4, 6, 5, -11, -6, 8, -8],\n [0, 0, 0, 0, -6, -11, -2, 10],\n [-6, 7, 0, -2, 8, -2, -2, -4],\n [3, 1, 2, -3, -8, 10, -4, 6]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-40007303
|
OMEGA_500_SAMPLE
|
omega_500_026
|
algebra_func_zeros_level_5_0083
|
algebra_func_zeros
| 5 |
algebra_func_zeros
|
5
|
[
{
"role": "user",
"content": "Find the number of x-intercepts of the function f(x) = -1(-2(1((3(-3|(-2cos(3πx + 3) - 1)| + 2))/(-2(-3|(-2cos(3πx + 3) - 1)| + 2) - 2))^2 - 5((3(-3|(-2cos(3πx + 3) - 1)| + 2))/(-2(-3|(-2cos(3πx + 3) - 1)| + 2) - 2)))^2 - 4(1((3(-3|(-2cos(3πx + 3) - 1)| + 2))/(-2(-3|(-2cos(3πx + 3) - 1)| + 2) - 2))^2 - 5((3(-3|(-2cos(3πx + 3) - 1)| + 2))/(-2(-3|(-2cos(3πx + 3) - 1)| + 2) - 2))) - 2) - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
259
|
OMEGA_500_SAMPLE
|
omega_500_027
|
geometry_polygon_chords_level_5_0016
|
geometry_polygon_chords
| 5 |
geometry_polygon_chords
|
5
|
[
{
"role": "user",
"content": "Suppose you have a 14-gon, with vertices numbered 1 through 14 in counterclockwise order. Draw the diagonal from vertex 9 to vertex 4, from vertex 3 to vertex 6, from vertex 7 to vertex 5, from vertex 6 to vertex 8, from vertex 10 to vertex 1, from vertex 4 to vertex 6, from vertex 9 to vertex 1, from vertex 13 to vertex 3, from vertex 2 to vertex 11, from vertex 14 to vertex 6, from vertex 2 to vertex 10, from vertex 4 to vertex 1, from vertex 4 to vertex 14, and from vertex 14 to vertex 12. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
OMEGA_500_SAMPLE
|
omega_500_028
|
combinatory_pattern_matching_level_3_0051
|
combinatory_pattern_matching
| 3 |
combinatory_pattern_matching
|
3
|
[
{
"role": "user",
"content": "If you form a word of length 4 by randomly selecting letters from the multiset {l: 1, a: 1, s: 2, n: 3, m: 2}, what is the expected number of times the pattern 'ln+' appears? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0.15
|
OMEGA_500_SAMPLE
|
omega_500_029
|
arithmetic_mixed_level_3_0090
|
arithmetic_mixed
| 3 |
arithmetic_mixed
|
3
|
[
{
"role": "user",
"content": "Evaluate (-49791)/(-11) + (-182077143670368)/40221406819. 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}"
}
] |
-93/223
|
OMEGA_500_SAMPLE
|
omega_500_030
|
number_theory_prime_mod_level_2_0085
|
number_theory_prime_mod
| 2 |
number_theory_prime_mod
|
2
|
[
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 8). For the least p where, for some integer n, n^2=-1 (mod p^2) has a solution, find the minimal n for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
38
|
OMEGA_500_SAMPLE
|
omega_500_031
|
algebra_func_extrema_level_1_0020
|
algebra_func_extrema
| 1 |
algebra_func_extrema
|
1
|
[
{
"role": "user",
"content": "Find the number of local minimum points for f(x) = 1cos(1π(2x^2 + 1x + 5) + 2) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
201
|
OMEGA_500_SAMPLE
|
omega_500_032
|
geometry_polygon_chords_level_1_0096
|
geometry_polygon_chords
| 1 |
geometry_polygon_chords
|
1
|
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 5, from vertex 6 to vertex 3, from vertex 1 to vertex 4, from vertex 6 to vertex 1, from vertex 4 to vertex 2, and from vertex 2 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
1
|
OMEGA_500_SAMPLE
|
omega_500_033
|
algebra_linear_equation_level_2_0063
|
algebra_linear_equation
| 2 |
algebra_linear_equation
|
2
|
[
{
"role": "user",
"content": "Suppose z - 1893 = 6*z - 1893. Solve 3*g - 5*k + 8 = 4*h, -4*g + 3*k - h - 7 = z, 8*g + 4*k - 1 = 5*g for g.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-1
|
OMEGA_500_SAMPLE
|
omega_500_034
|
arithmetic_matrix_multiplication_level_1_0025
|
arithmetic_matrix_multiplication
| 1 |
arithmetic_matrix_multiplication
|
1
|
[
{
"role": "user",
"content": "Find the trace (sum of diagonal elements) of the product of the following matrices:\nA =\n[[-5, -7, 1, -4],\n [1, -8, -3, 4],\n [6, -5, 4, 6],\n [-1, -4, 6, -5],\n [3, 4, -7, -5]]\nB =\n[[-5, 2, -4, 2, 1],\n [-1, -3, -3, -1, 6],\n [5, 2, 3, 4, -1],\n [-8, 6, -1, 2, 3]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
145
|
OMEGA_500_SAMPLE
|
omega_500_035
|
arithmetic_matrix_eigenvalues_level_5_0030
|
arithmetic_matrix_eigenvalues
| 5 |
arithmetic_matrix_eigenvalues
|
5
|
[
{
"role": "user",
"content": "Calculate the sum of absolute values of all eigenvalues for the matrix:\n[[4, -8, -24, 23, -13, -22, 0, -1, 4],\n [0, -19, 5, 18, -16, 13, -13, 5, -19],\n [0, 0, 16, 10, -6, -22, -19, -25, 18],\n [0, 0, 0, -13, -4, -5, -4, 2, -17],\n [0, 0, 0, 0, -2, 16, -18, -2, -10],\n [0, 0, 0, 0, 0, 24, -2, -7, 10],\n [0, 0, 0, 0, 0, 0, -16, 7, 24],\n [0, 0, 0, 0, 0, 0, 0, -5, -9],\n [0, 0, 0, 0, 0, 0, 0, 0, -15]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
114
|
OMEGA_500_SAMPLE
|
omega_500_036
|
number_theory_prime_mod_level_5_0059
|
number_theory_prime_mod
| 5 |
number_theory_prime_mod
|
5
|
[
{
"role": "user",
"content": "Let p be the least prime number for which there exists a positive integer n such that n^4+(1) is divisible by p^2. Find the least positive integer m such that m^4+(1) is divisible by p^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
110
|
OMEGA_500_SAMPLE
|
omega_500_037
|
geometry_rotation_level_4_0050
|
geometry_rotation
| 4 |
geometry_rotation
|
4
|
[
{
"role": "user",
"content": "Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 7, from vertex 9 to vertex 7, from vertex 4 to vertex 1, from vertex 8 to vertex 4, and from vertex 3 to vertex 6. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the \"smallest\" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
26
|
OMEGA_500_SAMPLE
|
omega_500_038
|
logic_puzzles_grid_rook_level_3_0046
|
logic_puzzles_grid_rook
| 3 |
logic_puzzles_grid_rook
|
3
|
[
{
"role": "user",
"content": "In a 21x26 grid, what is the fewest number of steps it takes to get from the bottom left (0, 0) to the top right (20, 25), 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 cells: (10, 22), (18, 19), (20, 22), (18, 7), (15, 17), (16, 24), (5, 23), (4, 9), (20, 7), (7, 16), (6, 22), (4, 18), (19, 16), (0, 10), (16, 16), (14, 7), (16, 21), (9, 0), (1, 7), (16, 17), (2, 21), (10, 17), (11, 17), (10, 8), (15, 19), (9, 18), (15, 0), (12, 25), (16, 7), (12, 12), (3, 5), (11, 5), (15, 4), (17, 0), (14, 21), (3, 15), (0, 5), (0, 12), (14, 18), (13, 13), (16, 0), (5, 0), (3, 22), (12, 14), (2, 6), (1, 5), (17, 2), (0, 1), (2, 0), (19, 10), (5, 22), (2, 18), (11, 18), (1, 4), (12, 4), (13, 18), (19, 18), (11, 6), (2, 13), (13, 19), (8, 3), (12, 20)? If it's impossible, answer with -1.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
OMEGA_500_SAMPLE
|
omega_500_039
|
combinatory_probability_no_specific_letter_fixed_level_3_0058
|
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 {x: 4, v: 3, p: 1}, shuffle the letters in the word, what is the probability of no letter 'v' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
41
|
OMEGA_500_SAMPLE
|
omega_500_040
|
geometry_rotation_level_5_0037
|
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 3 to vertex 6, from vertex 2 to vertex 6, from vertex 4 to vertex 6, from vertex 5 to vertex 2, from vertex 5 to vertex 3, and from vertex 1 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the \"smallest\" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
24
|
OMEGA_500_SAMPLE
|
omega_500_041
|
combinatory_distribution_level_1_0063
|
combinatory_distribution
| 1 |
combinatory_distribution
|
1
|
[
{
"role": "user",
"content": "Given the letters {'c': 1, 'b': 3}, place them into 2 labeled boxes with sizes [1, 3]. How many distinct arrangements are possible?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
2
|
OMEGA_500_SAMPLE
|
omega_500_042
|
arithmetic_gcd_level_4_0006
|
arithmetic_gcd
| 4 |
arithmetic_gcd
|
4
|
[
{
"role": "user",
"content": "Calculate the highest common factor of 11448511508945476803 and 38530971.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
1427073
|
OMEGA_500_SAMPLE
|
omega_500_043
|
algebra_func_derivative_sign_level_2_0082
|
algebra_func_derivative_sign
| 2 |
algebra_func_derivative_sign
|
2
|
[
{
"role": "user",
"content": "On how many maximal connected intervals within [-10, 10] is the derivative of f(x) = -2(1(2cos(3πx - 2))^2 - 2(2cos(3πx - 2)) + 3)^2 - 3(1(2cos(3πx - 2))^2 - 2(2cos(3πx - 2)) + 3) - 5 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
49
|
OMEGA_500_SAMPLE
|
omega_500_044
|
arithmetic_gcd_level_2_0046
|
arithmetic_gcd
| 2 |
arithmetic_gcd
|
2
|
[
{
"role": "user",
"content": "Calculate the highest common divisor of 270578734 and 866951.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
1957
|
OMEGA_500_SAMPLE
|
omega_500_045
|
arithmetic_matrix_eigenvalues_level_5_0053
|
arithmetic_matrix_eigenvalues
| 5 |
arithmetic_matrix_eigenvalues
|
5
|
[
{
"role": "user",
"content": "Calculate the sum of absolute values of all eigenvalues for the matrix:\n[[2, -14, 1, 16, 21, 4, 15, 8, 11],\n [0, -19, 6, 6, -1, 7, -19, -14, -10],\n [0, 0, 10, -13, -19, 10, 21, 25, -17],\n [0, 0, 0, 7, -3, -1, -18, 19, -4],\n [0, 0, 0, 0, 15, 3, 8, 8, 2],\n [0, 0, 0, 0, 0, -22, 0, 14, -17],\n [0, 0, 0, 0, 0, 0, 6, 2, -4],\n [0, 0, 0, 0, 0, 0, 0, -1, -16],\n [0, 0, 0, 0, 0, 0, 0, 0, 5]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
87
|
OMEGA_500_SAMPLE
|
omega_500_046
|
number_theory_digit_sum_level_4_0050
|
number_theory_digit_sum
| 4 |
number_theory_digit_sum
|
4
|
[
{
"role": "user",
"content": "let N be the third smallest 5-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 4. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
10008
|
OMEGA_500_SAMPLE
|
omega_500_047
|
logic_puzzles_zebralogic_level_2_0076
|
logic_puzzles_zebralogic
| 2 |
logic_puzzles_zebralogic
|
2
|
[
{
"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 - People have unique favorite music genres: `classical`, `rock`, `pop`\n - Everyone has something unique for lunch: `grilled cheese`, `pizza`, `spaghetti`\n\n## Clues:\n1. The person who loves classical music is somewhere to the right of the person who loves rock music.\n2. The person who loves pop music is the person who loves the spaghetti eater.\n3. Arnold is not in the third house.\n4. Arnold is the person who loves eating grilled cheese.\n5. Peter and the person who loves pop music are next to each other.\n6. Peter is not in the third house.\n\n\nChoices:\n0. pizza\n1. grilled cheese\n2. spaghetti\n\nWhen answering, please specify the number corresponding to the correct choice, instead of the choice itself.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0
|
OMEGA_500_SAMPLE
|
omega_500_048
|
logic_puzzles_grid_chip_level_4_0054
|
logic_puzzles_grid_chip
| 4 |
logic_puzzles_grid_chip
|
4
|
[
{
"role": "user",
"content": "Chips, colored either black or white, are placed in the cells of 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, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 2) is black, cell (3, 3) is black, cell (1, 5) is white, cell (1, 6) is white, cell (2, 4) is black, cell (4, 1) is black, cell (5, 1) is black, cell (6, 1) is black, cell (1, 0) is white. How many chips are placed on the grid?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
27
|
OMEGA_500_SAMPLE
|
omega_500_049
|
algebra_func_derivative_sign_level_5_0008
|
algebra_func_derivative_sign
| 5 |
algebra_func_derivative_sign
|
5
|
[
{
"role": "user",
"content": "For the function f(x) = -3(-1(-5((3(-2((1x - 1)/(-3x - 3)) + 5) + 3)/(1(-2((1x - 1)/(-3x - 3)) + 5) + 1)) + 5)^2 + 4(-5((3(-2((1x - 1)/(-3x - 3)) + 5) + 3)/(1(-2((1x - 1)/(-3x - 3)) + 5) + 1)) + 5) - 5)^2 + 2(-1(-5((3(-2((1x - 1)/(-3x - 3)) + 5) + 3)/(1(-2((1x - 1)/(-3x - 3)) + 5) + 1)) + 5)^2 + 4(-5((3(-2((1x - 1)/(-3x - 3)) + 5) + 3)/(1(-2((1x - 1)/(-3x - 3)) + 5) + 1)) + 5) - 5) + 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_050
|
arithmetic_matrix_rank_level_1_0059
|
arithmetic_matrix_rank
| 1 |
arithmetic_matrix_rank
|
1
|
[
{
"role": "user",
"content": "Determine the rank of the following matrix:\n[[2, -6, -2, -2],\n [1, -3, -1, -1],\n [1, -3, -1, -1],\n [0, 0, 0, 0]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
1
|
OMEGA_500_SAMPLE
|
omega_500_051
|
logic_puzzles_blocked_grid_level_3_0072
|
logic_puzzles_blocked_grid
| 3 |
logic_puzzles_blocked_grid
|
3
|
[
{
"role": "user",
"content": "In a 23x21 grid, how many different paths are there from the bottom left (0, 0) to the top right (22, 20), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (22, 15), (16, 3), (10, 0), (9, 10), (11, 1), (1, 1), (15, 17), (16, 10), (11, 14), (1, 19), (13, 15), (16, 18), (1, 16), (14, 5), (19, 7), (5, 1), (2, 7), (3, 12), (11, 15), (18, 5), (6, 9), (16, 7), (18, 1), (4, 10), (6, 16), (13, 5), (15, 2), (7, 7), (12, 9), (19, 17), (10, 6), (9, 4), (6, 2), (5, 13), (14, 20), (17, 2), (2, 4), (0, 20), (17, 8), (11, 16), (13, 6), (19, 12), (21, 7), (8, 8), (21, 19), (10, 10), (7, 19), (18, 3), (22, 1), (19, 11), (16, 20), (2, 11), (6, 17), (12, 4), (21, 17), (2, 8), (8, 5), (13, 17), (15, 11), (15, 12), (12, 3), (14, 19), (4, 20), (1, 9), (1, 15), (4, 11), (14, 6), (4, 19), (8, 17), (1, 3), (6, 7), (3, 18), (1, 11), (17, 14), (5, 0), (14, 2), (17, 15), (10, 14), (20, 15), (16, 19), (22, 14), (5, 9), (11, 2), (16, 12), (8, 3), (21, 4), (17, 3), (20, 3), (8, 14), (11, 20), (19, 0), (1, 13), (8, 7), (4, 18), (18, 20), (12, 8), (19, 6), (20, 4), (4, 15), (2, 18), (0, 6), (8, 16), (3, 6), (2, 9), (9, 7), (1, 4), (11, 3), (4, 12), (16, 2), (10, 13), (8, 6), (10, 18), (22, 17), (12, 17), (13, 18), (6, 3), (17, 13), (7, 1), (1, 0), (11, 12)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
695196
|
OMEGA_500_SAMPLE
|
omega_500_052
|
geometry_polygon_chords_level_4_0069
|
geometry_polygon_chords
| 4 |
geometry_polygon_chords
|
4
|
[
{
"role": "user",
"content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 7, from vertex 10 to vertex 7, from vertex 5 to vertex 9, from vertex 6 to vertex 11, from vertex 6 to vertex 12, from vertex 6 to vertex 10, from vertex 11 to vertex 4, from vertex 3 to vertex 11, from vertex 2 to vertex 12, from vertex 1 to vertex 8, from vertex 2 to vertex 5, from vertex 5 to vertex 11, and from vertex 3 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
3
|
OMEGA_500_SAMPLE
|
omega_500_053
|
logic_puzzles_grid_rook_level_1_0078
|
logic_puzzles_grid_rook
| 1 |
logic_puzzles_grid_rook
|
1
|
[
{
"role": "user",
"content": "In a 9x8 grid, what is the fewest number of steps it takes to get from the bottom left (0, 0) to the top right (8, 7), 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 cells: (1, 7), (7, 4), (2, 0), (3, 1), (3, 4), (4, 0), (4, 5), (3, 0), (8, 1)? If it's impossible, answer with -1.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
3
|
OMEGA_500_SAMPLE
|
omega_500_054
|
arithmetic_matrix_eigenvalues_level_4_0026
|
arithmetic_matrix_eigenvalues
| 4 |
arithmetic_matrix_eigenvalues
|
4
|
[
{
"role": "user",
"content": "Find the eigenvalues of the matrix and calculate the largest eigenvalue by absolute value:\n[[18, 0, -11, -4, 2, 2, 2],\n [0, 6, 4, -4, -6, 5, 0],\n [-11, 4, 15, -4, 4, -12, -4],\n [-4, -4, -4, 18, 2, 7, 2],\n [2, -6, 4, 2, -16, 8, 8],\n [2, 5, -12, 7, 8, -3, -2],\n [2, 0, -4, 2, 8, -2, -14]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
32
|
OMEGA_500_SAMPLE
|
omega_500_055
|
combinatory_pattern_matching_level_4_0077
|
combinatory_pattern_matching
| 4 |
combinatory_pattern_matching
|
4
|
[
{
"role": "user",
"content": "If you form a word of length 5 by randomly selecting letters from the multiset {p: 4, g: 2, o: 2, n: 2}, what is the expected number of times the pattern 'g+n' appears? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0.19
|
OMEGA_500_SAMPLE
|
omega_500_056
|
combinatory_probability_at_least_n_specific_fixed_level_1_0024
|
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: 1, f: 1, k: 2}, shuffle the letters in the word, what is the probability of at least 2 letter 'k' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
OMEGA_500_SAMPLE
|
omega_500_057
|
combinatory_distribution_level_2_0022
|
combinatory_distribution
| 2 |
combinatory_distribution
|
2
|
[
{
"role": "user",
"content": "You have {'j': 1, 'b': 3, 'z': 2} and need to distribute them into 3 identical boxes with capacities [3, 2, 1]. How many distinct distributions are possible?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
12
|
OMEGA_500_SAMPLE
|
omega_500_058
|
arithmetic_matrix_power_level_6_0063
|
arithmetic_matrix_power
| 6 |
arithmetic_matrix_power
|
6
|
[
{
"role": "user",
"content": "Calculate the matrix power A^6 and find the value at position (4, 10) of the result:\nA =\n[[-1, 0, 2, -2, 2, -1, 6, 9, -4, -6],\n [0, -7, -3, 1, -2, -6, 9, 4, 12, 5],\n [2, -3, -10, -2, -2, -2, -10, 6, 8, 0],\n [-2, 1, -2, -4, 8, 3, -6, 14, -4, 7],\n [2, -2, -2, 8, 1, 6, -1, -8, -2, -6],\n [-1, -6, -2, 3, 6, 6, 4, 8, -10, -2],\n [6, 9, -10, -6, -1, 4, 8, 9, -2, -2],\n [9, 4, 6, 14, -8, 8, 9, -3, -3, 4],\n [-4, 12, 8, -4, -2, -10, -2, -3, -3, 6],\n [-6, 5, 0, 7, -6, -2, -2, 4, 6, -10]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-52811412
|
OMEGA_500_SAMPLE
|
omega_500_059
|
geometry_polygon_color_level_3_0067
|
geometry_polygon_color
| 3 |
geometry_polygon_color
|
3
|
[
{
"role": "user",
"content": "A 11-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is red, vertex 8 is red, vertex 9 is red, vertex 10 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 11-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
3
|
OMEGA_500_SAMPLE
|
omega_500_060
|
number_theory_prime_mod_level_2_0043
|
number_theory_prime_mod
| 2 |
number_theory_prime_mod
|
2
|
[
{
"role": "user",
"content": "let p=17. How many solutions m in [1, 2, ..., 17^2] satisfy m^2+1=0 (mod 17^2)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
2
|
OMEGA_500_SAMPLE
|
omega_500_061
|
geometry_polygon_color_level_5_0028
|
geometry_polygon_color
| 5 |
geometry_polygon_color
|
5
|
[
{
"role": "user",
"content": "A 14-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 uncolored, vertex 4 is uncolored, vertex 5 is red, vertex 6 is red, vertex 7 is blue, vertex 8 is blue, vertex 9 is uncolored, vertex 10 is red, vertex 11 is red, vertex 12 is blue, vertex 13 is red. How many valid colorings of the uncolored vertices exist such that the 14-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0
|
OMEGA_500_SAMPLE
|
omega_500_062
|
algebra_func_extrema_coords_level_2_0079
|
algebra_func_extrema_coords
| 2 |
algebra_func_extrema_coords
|
2
|
[
{
"role": "user",
"content": "Find the x-coordinate of the local extremum with the smallest y-value of the function f(x) = (-3(1|(1cos(2πx + 2) + 2) + 3|) - 1)/(-2(1|(1cos(2πx + 2) + 2) + 3|) + 3) in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-9
|
OMEGA_500_SAMPLE
|
omega_500_063
|
number_theory_prime_mod_level_5_0018
|
number_theory_prime_mod
| 5 |
number_theory_prime_mod
|
5
|
[
{
"role": "user",
"content": "Let p be a prime with p=1 (mod 3). For the least p where, for some integer n, n^6=-1 (mod p^3) has a solution, find the minimal n for this p.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
239
|
OMEGA_500_SAMPLE
|
omega_500_064
|
arithmetic_gcd_level_3_0093
|
arithmetic_gcd
| 3 |
arithmetic_gcd
|
3
|
[
{
"role": "user",
"content": "What is the greatest common factor of 919701 and 70003522538540887?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
34063
|
OMEGA_500_SAMPLE
|
omega_500_065
|
logic_puzzles_blocked_grid_level_1_0084
|
logic_puzzles_blocked_grid
| 1 |
logic_puzzles_blocked_grid
|
1
|
[
{
"role": "user",
"content": "In a 7x8 grid, how many different paths are there from the bottom left (0, 0) to the top right (6, 7), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (6, 3), (3, 1), (1, 6), (4, 5), (1, 1), (1, 2), (5, 5), (0, 4), (5, 6), (2, 6)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
40
|
OMEGA_500_SAMPLE
|
omega_500_066
|
arithmetic_mixed_level_3_0048
|
arithmetic_mixed
| 3 |
arithmetic_mixed
|
3
|
[
{
"role": "user",
"content": "Calculate (-874 - -2997)*(3047 - 3048). 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}"
}
] |
-2123
|
OMEGA_500_SAMPLE
|
omega_500_067
|
number_theory_prime_mod_level_5_0011
|
number_theory_prime_mod
| 5 |
number_theory_prime_mod
|
5
|
[
{
"role": "user",
"content": "Let p be the least prime number for which there exists a positive integer n such that n^2+(2) is divisible by p^4, and that n^2+(2) is also divisible by p^6. Find the smallest positive integer m such that m^2+(2) is divisible by p^4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
221
|
OMEGA_500_SAMPLE
|
omega_500_068
|
logic_puzzles_grid_knight_level_4_0015
|
logic_puzzles_grid_knight
| 4 |
logic_puzzles_grid_knight
|
4
|
[
{
"role": "user",
"content": "Supposing you start in the bottom left (0, 0) cell of a 34x36 grid, what is minimum number of steps you can take to get to (33, 32), if your only valid moves are like a chess knight, i.e, over two squares in one direction and over one square in a perpendicular direction in one move?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
23
|
OMEGA_500_SAMPLE
|
omega_500_069
|
geometry_polygon_chords_level_3_0071
|
geometry_polygon_chords
| 3 |
geometry_polygon_chords
|
3
|
[
{
"role": "user",
"content": "Suppose you have a 14-gon, with vertices numbered 1 through 14 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 6, from vertex 5 to vertex 10, from vertex 13 to vertex 10, from vertex 12 to vertex 3, from vertex 14 to vertex 2, from vertex 7 to vertex 9, from vertex 9 to vertex 14, from vertex 6 to vertex 9, from vertex 2 to vertex 10, and from vertex 8 to vertex 5. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
OMEGA_500_SAMPLE
|
omega_500_070
|
logic_puzzles_grid_rook_level_1_0092
|
logic_puzzles_grid_rook
| 1 |
logic_puzzles_grid_rook
|
1
|
[
{
"role": "user",
"content": "In a 9x6 grid, what is the fewest number of steps it takes to get from the bottom left (0, 0) to the top right (8, 5), 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 cells: (1, 0), (2, 3), (7, 5), (0, 2), (7, 2), (5, 4), (5, 0), (2, 1), (6, 2), (0, 3)? If it's impossible, answer with -1.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
OMEGA_500_SAMPLE
|
omega_500_071
|
arithmetic_matrix_svd_level_4_0002
|
arithmetic_matrix_svd
| 4 |
arithmetic_matrix_svd
|
4
|
[
{
"role": "user",
"content": "Calculate the rounded largest singular value from the SVD of:\n[[0, -12, 12, -6, -4, 11, -12],\n [-1, -5, 11, -2, 6, 4, -5],\n [-10, -10, -12, -8, -3, -6, -4],\n [-6, -4, -5, -1, -11, -12, 3],\n [10, 10, 11, -8, -10, -1, -5],\n [9, -10, -12, -10, -8, 2, 1],\n [-10, -12, -8, 10, 1, -6, -4]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
36
|
OMEGA_500_SAMPLE
|
omega_500_072
|
combinatory_pattern_matching_level_3_0078
|
combinatory_pattern_matching
| 3 |
combinatory_pattern_matching
|
3
|
[
{
"role": "user",
"content": "When randomly selecting 4 letters from the multiset {b: 4, l: 3, q: 2} to form a word, what is the expected number of matches of the pattern 'lb+'? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0.67
|
OMEGA_500_SAMPLE
|
omega_500_073
|
logic_puzzles_zebralogic_level_4_0077
|
logic_puzzles_zebralogic
| 4 |
logic_puzzles_zebralogic
|
4
|
[
{
"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 - Each person has an occupation: `engineer`, `doctor`\n - People have unique favorite sports: `basketball`, `soccer`\n - The people are of nationalities: `brit`, `dane`\n\n## Clues:\n1. Arnold is in the second house.\n2. The person who loves basketball is the person who is a doctor.\n3. The person who is a doctor is directly left of the Dane.\n\n\nChoices:\n0. soccer\n1. basketball\n\nWhen answering, please specify the number corresponding to the correct choice, instead of the choice itself.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0
|
OMEGA_500_SAMPLE
|
omega_500_074
|
algebra_func_derivative_sign_level_2_0023
|
algebra_func_derivative_sign
| 2 |
algebra_func_derivative_sign
|
2
|
[
{
"role": "user",
"content": "For the function f(x) = 3|(-2(2cos(2πx - 3) + 2) - 1) - 3| + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
21
|
OMEGA_500_SAMPLE
|
omega_500_075
|
geometry_polygon_color_level_5_0003
|
geometry_polygon_color
| 5 |
geometry_polygon_color
|
5
|
[
{
"role": "user",
"content": "A 15-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 blue, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is red, vertex 9 is blue, vertex 10 is red, vertex 11 is blue, vertex 12 is red, vertex 13 is red, vertex 14 is red. The 15-gon is then rotated clockwise by 6 positions, so that the vertex that was originally in position 0 is now in position 6. How many red vertices are now in positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
3
|
OMEGA_500_SAMPLE
|
omega_500_076
|
number_theory_digit_sum_level_1_0015
|
number_theory_digit_sum
| 1 |
number_theory_digit_sum
|
1
|
[
{
"role": "user",
"content": "let N be the third greatest 2-digit positive integer with the property that whenever you swap the positions of two of its adjacent digits, the resulting number is a multiple of 6. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
81
|
OMEGA_500_SAMPLE
|
omega_500_077
|
number_theory_digit_sum_level_3_0095
|
number_theory_digit_sum
| 3 |
number_theory_digit_sum
|
3
|
[
{
"role": "user",
"content": "let N be the greatest 4-digit positive integer with the property that both the number and the number formed by reversing its digits are divisible by 6. Let Q and R be the quotient and remainder, respectively, when N is divided by 1000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
8994
|
OMEGA_500_SAMPLE
|
omega_500_078
|
number_theory_digit_sum_level_3_0008
|
number_theory_digit_sum
| 3 |
number_theory_digit_sum
|
3
|
[
{
"role": "user",
"content": "let N be the tenth greatest 4-digit positive integer with the property that the sum of the digits of n is divisible by 3. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
9972
|
OMEGA_500_SAMPLE
|
omega_500_079
|
combinatory_probability_no_fixed_points_level_5_0070
|
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 {r: 2, l: 1, i: 3, w: 1, h: 4}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
2633
|
OMEGA_500_SAMPLE
|
omega_500_080
|
arithmetic_matrix_multiplication_level_1_0011
|
arithmetic_matrix_multiplication
| 1 |
arithmetic_matrix_multiplication
|
1
|
[
{
"role": "user",
"content": "What is the value at position (2, 3) when matrix A is multiplied by matrix B?\nA =\n[[7, 7, -2, -5],\n [-8, -4, 1, -4],\n [-5, -7, 1, -8],\n [-4, 4, -5, 7]]\nB =\n[[7, -7, 8, 3],\n [-6, -8, -8, 2],\n [-4, 3, -6, -8],\n [-8, -1, 1, 2]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-42
|
OMEGA_500_SAMPLE
|
omega_500_081
|
algebra_func_extrema_level_1_0004
|
algebra_func_extrema
| 1 |
algebra_func_extrema
|
1
|
[
{
"role": "user",
"content": "Find the number of local maximum points for f(x) = -1e^(2(-2|x|)) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
11
|
OMEGA_500_SAMPLE
|
omega_500_082
|
arithmetic_matrix_eigenvalues_level_1_0053
|
arithmetic_matrix_eigenvalues
| 1 |
arithmetic_matrix_eigenvalues
|
1
|
[
{
"role": "user",
"content": "Calculate the sum of absolute values of all eigenvalues for the matrix:\n[[-7, 7, 6],\n [0, -6, 1],\n [0, 0, -8]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
21
|
OMEGA_500_SAMPLE
|
omega_500_083
|
logic_puzzles_grid_rook_level_1_0033
|
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 cells: (7, 3), (1, 8), (0, 4), (6, 2), (5, 4), (2, 0), (2, 7), (7, 6), (4, 3), (0, 2), (3, 4), (2, 2), (3, 3)? If it's impossible, answer with -1.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
OMEGA_500_SAMPLE
|
omega_500_084
|
algebra_func_extrema_coords_level_3_0057
|
algebra_func_extrema_coords
| 3 |
algebra_func_extrema_coords
|
3
|
[
{
"role": "user",
"content": "How many local extrema (both minima and maxima) does the function f(x) = 2|(1(2cos(2π(-5x + 2) - 2) - 2) + 3) + 2| + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
200
|
OMEGA_500_SAMPLE
|
omega_500_085
|
arithmetic_matrix_rank_level_7_0090
|
arithmetic_matrix_rank
| 7 |
arithmetic_matrix_rank
|
7
|
[
{
"role": "user",
"content": "Determine the rank of the following matrix:\n[[236, -107, -248, 3, -567, 40, -393, 210, -22, 259, 484, 464],\n [-240, -122, -86, 142, 706, -336, -144, -290, -66, -662, -462, -690],\n [-646, 506, 73, 513, -182, -135, 400, 22, 545, 84, -224, 128],\n [88, -115, -125, -7, -33, -56, -224, 9, -71, -69, 95, 1],\n [-230, -179, -326, 284, 443, -451, -480, -211, -20, -671, -247, -521],\n [-370, 450, 330, 155, -100, 145, 675, 35, 350, 300, -215, 155],\n [-282, 336, 221, 137, -108, 95, 472, 36, 273, 226, -138, 138],\n [-212, 402, -17, 296, -872, 132, 191, 307, 439, 614, 371, 769],\n [-268, 115, 25, 177, 213, -134, 114, -99, 131, -191, -245, -211],\n [494, -543, -83, -438, 581, -31, -415, -184, -565, -427, -56, -508],\n [318, -139, -48, -198, -277, 151, -160, 125, -150, 229, 309, 267],\n [-124, 489, 91, 207, -1129, 314, 362, 419, 473, 923, 517, 1043]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
3
|
OMEGA_500_SAMPLE
|
omega_500_086
|
combinatory_probability_no_fixed_points_level_1_0056
|
combinatory_probability_no_fixed_points
| 1 |
combinatory_probability_no_fixed_points
|
1
|
[
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {h: 1, u: 2, f: 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 written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
17
|
OMEGA_500_SAMPLE
|
omega_500_087
|
arithmetic_matrix_eigenvalues_level_4_0019
|
arithmetic_matrix_eigenvalues
| 4 |
arithmetic_matrix_eigenvalues
|
4
|
[
{
"role": "user",
"content": "Determine the sum of absolute values of all eigenvalues of the matrix:\n[[2, -2, 4, 1, -4, -2, -2],\n [-2, -10, -13, 6, 10, 4, 9],\n [4, -13, -12, 2, 8, 9, 6],\n [1, 6, 2, -12, -2, 7, 7],\n [-4, 10, 8, -2, 2, -12, 6],\n [-2, 4, 9, 7, -12, -18, 12],\n [-2, 9, 6, 7, 6, 12, 10]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
116
|
OMEGA_500_SAMPLE
|
omega_500_088
|
logic_puzzles_blocked_grid_level_1_0094
|
logic_puzzles_blocked_grid
| 1 |
logic_puzzles_blocked_grid
|
1
|
[
{
"role": "user",
"content": "In a 8x7 grid, how many different paths are there from the bottom left (0, 0) to the top right (7, 6), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (7, 1), (0, 1), (0, 4), (1, 4), (1, 3), (7, 4), (4, 1), (2, 0), (3, 4), (4, 0)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
167
|
OMEGA_500_SAMPLE
|
omega_500_089
|
combinatory_probability_no_specific_letter_fixed_level_2_0011
|
combinatory_probability_no_specific_letter_fixed
| 2 |
combinatory_probability_no_specific_letter_fixed
|
2
|
[
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {l: 3, y: 1, t: 2}, shuffle the letters in the word, what is the probability of no letter 't' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
8
|
OMEGA_500_SAMPLE
|
omega_500_090
|
algebra_linear_equation_level_1_0069
|
algebra_linear_equation
| 1 |
algebra_linear_equation
|
1
|
[
{
"role": "user",
"content": "Solve 4*u - 342 = 5*k, 18626*u = 18626*u - 5*k - 350 for u.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-2
|
OMEGA_500_SAMPLE
|
omega_500_091
|
logic_puzzles_grid_knight_level_5_0021
|
logic_puzzles_grid_knight
| 5 |
logic_puzzles_grid_knight
|
5
|
[
{
"role": "user",
"content": "Supposing you start in the bottom left (0, 0) cell of a 42x42 grid, what is minimum number of steps you can take to get to (40, 38), if your only valid moves are like a chess knight, i.e, over two squares in one direction and over one square in a perpendicular direction in one move?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
26
|
OMEGA_500_SAMPLE
|
omega_500_092
|
algebra_linear_equation_level_2_0078
|
algebra_linear_equation
| 2 |
algebra_linear_equation
|
2
|
[
{
"role": "user",
"content": "Let s = 720 + -712. Suppose -4*v = -3*b + 14, -5*b + s = -0*v + v. Solve 4*d = c + b*c - 4*h - 15, c + 20 = 3*h - 4*d, -4*c - h - 13 = 0 for c.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-3
|
OMEGA_500_SAMPLE
|
omega_500_093
|
algebra_func_area_level_4_0006
|
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(-1(-1(5x - 4) + 3)^2 + 5(-1(5x - 4) + 3) + 1) - 1) - 2)/(-2(1(-1(-1(5x - 4) + 3)^2 + 5(-1(5x - 4) + 3) + 1) - 1)) and g(x) = -1|(4(-2(-2|x + 1| - 1)^2 + 1(-2|x + 1| - 1) + 4) + 5)| between the vertical lines x = -1.2 and x = -0.8. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
1.6
|
OMEGA_500_SAMPLE
|
omega_500_094
|
logic_puzzles_grid_chip_level_1_0041
|
logic_puzzles_grid_chip
| 1 |
logic_puzzles_grid_chip
|
1
|
[
{
"role": "user",
"content": "Chips, colored either black or white, are placed in 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. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 2) is black, cell (2, 2) is black, cell (3, 3) is black, cell (0, 0) is black. What is the maximum number of white chips that can be placed on the grid?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
7
|
OMEGA_500_SAMPLE
|
omega_500_095
|
arithmetic_list_prime_factors_level_5_0033
|
arithmetic_list_prime_factors
| 5 |
arithmetic_list_prime_factors
|
5
|
[
{
"role": "user",
"content": "Let s = 5713184494 - 4163212106. Find the second-largest prime factor of s.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
2969
|
OMEGA_500_SAMPLE
|
omega_500_096
|
logic_puzzles_grid_knight_level_5_0008
|
logic_puzzles_grid_knight
| 5 |
logic_puzzles_grid_knight
|
5
|
[
{
"role": "user",
"content": "Supposing you start in the bottom left (0, 0) cell of a 47x47 grid, what is minimum number of steps you can take to get to (45, 44), if your only valid moves are to move over 4 squares in one direction and over 3 squares in a perpendicular direction (at once)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
15
|
OMEGA_500_SAMPLE
|
omega_500_097
|
arithmetic_matrix_inverse_level_5_0054
|
arithmetic_matrix_inverse
| 5 |
arithmetic_matrix_inverse
|
5
|
[
{
"role": "user",
"content": "Find the inverse of 1/450 times the matrix and calculate the sum of absolute values of all entries of the inverse. Round your answer to the nearest integer:\n[[1, 15, -6, 18, -7, -6, 9, 21, -24],\n [4, 8, -4, -23, -20, -11, 1, -16, -17],\n [7, 6, 22, -20, 15, 10, -24, 3, -22],\n [-25, -7, 0, -1, -16, -11, -4, 25, 22],\n [-5, -4, 10, 8, 4, -1, 19, 2, -25],\n [23, 19, -22, -19, 23, -17, 10, -7, -25],\n [-16, 10, -17, 19, 20, 16, 6, 6, -20],\n [22, 12, -9, -16, 24, 1, 24, 9, -5],\n [-1, 2, 21, -11, -22, 3, -17, 1, 0]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
1398
|
OMEGA_500_SAMPLE
|
omega_500_098
|
combinatory_distribution_level_1_0033
|
combinatory_distribution
| 1 |
combinatory_distribution
|
1
|
[
{
"role": "user",
"content": "Arrange the letters {'v': 2, 'b': 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_099
|
algebra_func_zeros_level_1_0000
|
algebra_func_zeros
| 1 |
algebra_func_zeros
|
1
|
[
{
"role": "user",
"content": "How many zeros does the function f(x) = -4(-2sin(1πx - 2) + 2) + 5 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
20
|
OMEGA_500_SAMPLE
|
End of preview. Expand
in Data Studio
Omega-500: Random Sample of Mathematical Problems
This dataset contains a random sample of 500 mathematical problems selected from the comprehensive OMEGA problem families dataset. It provides a diverse, manageable subset for quick evaluation and experimentation across multiple mathematical domains and difficulty levels.
Overview
Omega-500 is designed for:
- Quick Evaluation: Fast assessment of model capabilities across math domains
- Prototyping: Testing new approaches before scaling to larger datasets
- Benchmarking: Standardized subset for fair model comparisons
- Research: Focused analysis on a balanced mathematical problem set
The sample maintains diversity across mathematical domains and difficulty levels while keeping the dataset size manageable for rapid iteration.
Quick Start
from datasets import load_dataset
# Load the Omega-500 sample
dataset = load_dataset("allenai/omega-500")
problems = dataset["train"]
# Access individual problems
first_problem = problems[0]
print("Problem:", first_problem["messages"][0]["content"])
print("Answer:", first_problem["ground_truth"])
print("Family:", first_problem["family"])
print("Difficulty:", first_problem["difficulty_level"])
Dataset Composition
Total Problems: 500
Domain Distribution:
- Algebra: 92 problems (18.4%)
- Arithmetic: 173 problems (34.6%)
- Combinatorics: 84 problems (16.8%)
- Geometry: 45 problems (9.0%)
- Logic: 61 problems (12.2%)
- Number Theory: 45 problems (9.0%)
Data Fields
Each problem contains:
id: Unique identifier for this sampleoriginal_id: Original identifier from source datasetfamily: Problem family (e.g., "algebra_func_area")difficulty_level: Numeric difficulty level from sourcesource_family: Source family directory namesource_level: Source difficulty level namemessages: Problem statement in chat formatground_truth: Correct answerdataset: Dataset identifier ("OMEGA_500_SAMPLE")
Citation
If you use this dataset, please cite the original OMEGA work:
@article{sun2024omega,
title = {OMEGA: Can LLMs Reason Outside the Box in Math? Evaluating Exploratory, Compositional, and Transformative Generalization},
author = {Yiyou Sun and Shawn Hu and Georgia Zhou and Ken Zheng and Hannaneh Hajishirzi and Nouha Dziri and Dawn Song},
journal = {arXiv preprint arXiv:2506.18880},
year = {2024},
}
Related Resources
- Full Problem Families: See omega-problems for the complete dataset
- Explorative Dataset: See omega-explorative for explorative reasoning challenges
- Compositional Dataset: See omega-compositional for compositional reasoning challenges
- Transformative Dataset: See omega-transformative for transformative reasoning challenges
- Paper: See the full details in paper
- Code Repository: See generation code on github
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