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_200
|
algebra_func_extrema_level_5_0088
|
algebra_func_extrema
| 5 |
algebra_func_extrema
|
5
|
[
{
"role": "user",
"content": "How many local maxima does the function f(x) = 1|(1|((2(-4(-3|(1cos(3πx - 2) + 1)| + 3)^2 + 3) - 3)/(3(-4(-3|(1cos(3πx - 2) + 1)| + 3)^2 + 3) - 1))|)| - 2 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
102
|
OMEGA_500_SAMPLE
|
omega_500_201
|
number_theory_triple_count_level_2_0028
|
number_theory_triple_count
| 2 |
number_theory_triple_count
|
2
|
[
{
"role": "user",
"content": "Assume N is the number of integers a which are >= 0 where a <= 4^4 and a^4 is divisible by 4^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
64
|
OMEGA_500_SAMPLE
|
omega_500_202
|
number_theory_digit_sum_level_4_0029
|
number_theory_digit_sum
| 4 |
number_theory_digit_sum
|
4
|
[
{
"role": "user",
"content": "let N be the tenth smallest 5-digit positive integer with the property that the sum of the digits of n is divisible by 9. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
10089
|
OMEGA_500_SAMPLE
|
omega_500_203
|
logic_puzzles_grid_rook_level_3_0080
|
logic_puzzles_grid_rook
| 3 |
logic_puzzles_grid_rook
|
3
|
[
{
"role": "user",
"content": "In a 25x28 grid, what is the fewest number of steps it takes to get from the bottom left (0, 0) to the top right (24, 27), 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, 13), (18, 6), (16, 7), (23, 6), (19, 22), (18, 16), (2, 1), (17, 25), (2, 17), (14, 11), (18, 23), (4, 27), (5, 4), (18, 21), (12, 1), (10, 15), (20, 7), (20, 24), (19, 9), (23, 27), (10, 26), (12, 9), (3, 15), (24, 12), (14, 19), (14, 9), (7, 20), (2, 16), (22, 5), (16, 26), (19, 3), (12, 12), (23, 22), (14, 6), (11, 7), (18, 24), (18, 27), (15, 25), (11, 5), (10, 27), (3, 27), (13, 23), (21, 18), (17, 16), (9, 7), (21, 6), (13, 14), (7, 9), (2, 8), (6, 17), (17, 20), (1, 9), (12, 16), (2, 3), (3, 18), (8, 10), (5, 13), (8, 15), (1, 7), (10, 19), (11, 20), (12, 19), (5, 21), (14, 16), (21, 17), (20, 18), (17, 18), (1, 2), (16, 9), (4, 21), (23, 7), (10, 25), (12, 5), (8, 19), (5, 24), (0, 11), (13, 3), (24, 15), (20, 0), (1, 21), (13, 21), (0, 3), (4, 15), (22, 17), (1, 10), (19, 14), (10, 12)? If it's impossible, answer with -1.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
OMEGA_500_SAMPLE
|
omega_500_204
|
logic_puzzles_grid_rook_level_4_0066
|
logic_puzzles_grid_rook
| 4 |
logic_puzzles_grid_rook
|
4
|
[
{
"role": "user",
"content": "In a 37x35 grid, what is the fewest number of steps it takes to get from the bottom left (0, 0) to the top right (36, 34), 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, 5), (2, 17), (33, 5), (19, 11), (31, 14), (31, 31), (7, 8), (5, 11), (36, 28), (35, 21), (8, 31), (23, 5), (4, 30), (33, 34), (2, 12), (15, 20), (22, 9), (14, 3), (27, 28), (14, 10), (19, 14), (26, 17), (0, 10), (32, 21), (32, 7), (27, 13), (0, 22), (3, 2), (35, 23), (34, 24), (10, 30), (13, 28), (13, 20), (34, 29), (4, 13), (31, 6), (26, 8), (27, 32), (0, 23), (26, 10), (8, 15), (32, 5), (9, 15), (10, 24), (19, 16), (28, 24), (29, 28), (8, 26), (14, 24), (17, 15), (34, 33), (18, 7), (13, 11), (17, 6), (34, 15), (17, 10), (22, 5), (29, 16), (23, 10), (32, 31), (25, 27), (18, 29), (29, 29), (28, 5), (34, 27), (30, 31), (31, 10), (1, 11), (36, 16), (34, 4), (29, 26), (9, 23), (12, 9), (14, 22), (5, 23), (15, 23), (2, 33), (27, 27), (3, 30), (23, 17), (12, 17), (7, 34), (5, 28), (15, 21), (7, 18), (20, 25), (19, 25), (28, 25), (25, 11), (13, 27), (33, 20), (7, 7), (11, 16), (13, 5), (36, 20), (35, 6), (4, 19), (30, 18), (28, 29), (12, 3), (35, 31), (5, 32), (1, 26), (29, 6), (30, 7), (25, 6), (13, 21), (29, 27), (28, 28), (17, 21), (9, 0), (25, 18), (4, 31), (30, 26), (21, 1), (13, 24), (34, 12), (34, 23), (14, 21), (17, 14), (32, 24), (12, 18), (36, 23), (10, 6), (20, 19), (31, 22), (33, 17), (12, 24), (36, 33), (18, 16), (14, 31), (22, 29), (24, 8), (19, 24), (12, 8), (4, 34), (6, 14), (32, 0), (1, 10), (2, 21), (25, 34), (10, 0), (8, 22), (12, 34), (22, 30), (16, 14), (0, 28), (16, 16), (24, 14), (19, 30), (10, 5), (25, 5), (28, 32), (31, 20), (23, 18), (26, 30), (5, 34), (27, 22), (23, 0), (9, 21), (16, 8), (28, 19), (3, 23), (1, 2), (23, 7), (26, 0), (13, 17), (34, 20), (16, 12), (24, 18), (25, 1), (22, 18), (36, 25), (29, 7), (4, 10), (36, 32), (13, 0), (2, 34), (32, 17), (12, 1), (5, 30), (15, 30), (15, 7), (20, 30), (7, 20), (1, 28), (21, 16), (16, 13), (1, 9), (16, 1), (4, 5), (20, 4), (8, 2), (21, 30), (32, 10), (1, 29), (5, 2), (13, 25), (30, 13), (13, 14), (33, 19), (23, 30), (4, 27), (19, 12), (1, 16), (33, 27), (19, 32), (36, 7), (33, 9), (32, 13), (33, 2), (36, 5), (17, 17), (32, 25), (22, 22), (24, 5), (8, 25), (6, 10), (10, 26), (16, 2), (21, 19), (35, 0), (22, 6), (33, 7), (26, 9), (21, 6), (14, 23)? If it's impossible, answer with -1.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
11
|
OMEGA_500_SAMPLE
|
omega_500_205
|
arithmetic_list_prime_factors_level_4_0014
|
arithmetic_list_prime_factors
| 4 |
arithmetic_list_prime_factors
|
4
|
[
{
"role": "user",
"content": "Find the second-largest prime factor of 505191540603401792.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
13255663
|
OMEGA_500_SAMPLE
|
omega_500_206
|
algebra_func_zeros_level_5_0099
|
algebra_func_zeros
| 5 |
algebra_func_zeros
|
5
|
[
{
"role": "user",
"content": "Find the number of x-intercepts of the function f(x) = -5(-1sin(3π((3((-1((-2(1x^2 + 1x - 4) + 2)/(3(1x^2 + 1x - 4) + 2)))/(-2((-2(1x^2 + 1x - 4) + 2)/(3(1x^2 + 1x - 4) + 2)) - 2)) - 2)/(-1((-1((-2(1x^2 + 1x - 4) + 2)/(3(1x^2 + 1x - 4) + 2)))/(-2((-2(1x^2 + 1x - 4) + 2)/(3(1x^2 + 1x - 4) + 2)) - 2)) + 2)) + 1) - 1) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
110
|
OMEGA_500_SAMPLE
|
omega_500_207
|
arithmetic_list_prime_factors_level_2_0069
|
arithmetic_list_prime_factors
| 2 |
arithmetic_list_prime_factors
|
2
|
[
{
"role": "user",
"content": "What is the second-largest prime factor of 96560261?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
277
|
OMEGA_500_SAMPLE
|
omega_500_208
|
number_theory_triple_count_level_1_0060
|
number_theory_triple_count
| 1 |
number_theory_triple_count
|
1
|
[
{
"role": "user",
"content": "Let N be the number of integers a which are non-negative where a <= 2^3 and a^2 is a multiple of 2^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
4
|
OMEGA_500_SAMPLE
|
omega_500_209
|
arithmetic_matrix_inverse_level_3_0093
|
arithmetic_matrix_inverse
| 3 |
arithmetic_matrix_inverse
|
3
|
[
{
"role": "user",
"content": "For the inverse of 1/180 times the matrix below, what is the trace (sum of diagonal elements) rounded to the nearest integer?\n[[15, -2, -12, -5, 4, -1],\n [3, -9, 3, 12, 3, -5],\n [10, -11, -11, 14, 6, -8],\n [-6, -10, -1, 0, 12, 1],\n [15, -8, -8, 9, -7, -14],\n [-1, 2, -13, 1, -12, 14]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-26
|
OMEGA_500_SAMPLE
|
omega_500_210
|
algebra_func_intersection_level_1_0065
|
algebra_func_intersection
| 1 |
algebra_func_intersection
|
1
|
[
{
"role": "user",
"content": "How many solutions does the equation 2cos(3π(-1x + 1) - 1) - 3 = -2|x + 1| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
14
|
OMEGA_500_SAMPLE
|
omega_500_211
|
arithmetic_matrix_power_level_5_0014
|
arithmetic_matrix_power
| 5 |
arithmetic_matrix_power
|
5
|
[
{
"role": "user",
"content": "Calculate the matrix power A^7 and find the sum of absolute values of all entries of the result:\nA =\n[[-8, 5, -3, 9, 8, -7, -12, -8],\n [-4, -1, 1, -11, 4, 1, 0, -4],\n [12, 2, 12, 9, 2, 11, -2, 3],\n [4, -9, -12, -5, 4, -10, 7, 2],\n [2, 11, 9, 1, 10, 1, 11, -11],\n [0, 1, 11, 6, -6, -10, 2, 10],\n [1, -2, 2, 0, 5, -7, -10, 6],\n [2, 12, 10, -8, -3, -6, 12, 4]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
13305833360
|
OMEGA_500_SAMPLE
|
omega_500_212
|
algebra_func_intersection_level_4_0076
|
algebra_func_intersection
| 4 |
algebra_func_intersection
|
4
|
[
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = -5((2(-4(4(-2cos(3πx - 1) - 2)^2 + 1(-2cos(3πx - 1) - 2) - 4) - 3) + 2)/(1(-4(4(-2cos(3πx - 1) - 2)^2 + 1(-2cos(3πx - 1) - 2) - 4) - 3) - 3))^2 + 4((2(-4(4(-2cos(3πx - 1) - 2)^2 + 1(-2cos(3πx - 1) - 2) - 4) - 3) + 2)/(1(-4(4(-2cos(3πx - 1) - 2)^2 + 1(-2cos(3πx - 1) - 2) - 4) - 3) - 3)) + 5 and g(x) = (-2(3sin(1π(-1|(1|x - 1| - 1) - 2|) + 1) + 2) - 3)/(-2(3sin(1π(-1|(1|x - 1| - 1) - 2|) + 1) + 2) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
108
|
OMEGA_500_SAMPLE
|
omega_500_213
|
combinatory_probability_no_specific_letter_fixed_level_4_0083
|
combinatory_probability_no_specific_letter_fixed
| 4 |
combinatory_probability_no_specific_letter_fixed
|
4
|
[
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {h: 1, n: 2, d: 3, u: 4}, shuffle the letters in the word, what is the probability of no letter 'u' 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}"
}
] |
10
|
OMEGA_500_SAMPLE
|
omega_500_214
|
algebra_func_intersection_coords_level_2_0008
|
algebra_func_intersection_coords
| 2 |
algebra_func_intersection_coords
|
2
|
[
{
"role": "user",
"content": "Find the integer closest to the smallest x-value where the functions f(x) = -5(-1cos(1π(1x + 2) + 2) + 2) - 4 and g(x) = -2|((-2x + 3)/(2x + 1)) - 1| - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-1
|
OMEGA_500_SAMPLE
|
omega_500_215
|
arithmetic_matrix_power_level_6_0070
|
arithmetic_matrix_power
| 6 |
arithmetic_matrix_power
|
6
|
[
{
"role": "user",
"content": "For the matrix power A^7 where A =\n[[3, -7, 0, -8, -4, -2, 10, 1, 0, -2],\n [-7, 0, 1, -8, 13, 8, 12, 0, 0, 1],\n [0, 1, -2, -8, -1, 0, 4, 1, -6, -7],\n [-8, -8, -8, -13, -3, 2, 4, 4, 0, 0],\n [-4, 13, -1, -3, -12, -9, -6, -2, 0, 0],\n [-2, 8, 0, 2, -9, -4, -6, -1, -7, -9],\n [10, 12, 4, 4, -6, -6, 14, 2, -4, -8],\n [1, 0, 1, 4, -2, -1, 2, 5, -10, 4],\n [0, 0, -6, 0, 0, -7, -4, -10, -11, -6],\n [-2, 1, -7, 0, 0, -9, -8, 4, -6, 5]], determine the sum of all entries.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-81812077230
|
OMEGA_500_SAMPLE
|
omega_500_216
|
algebra_func_derivative_sign_level_3_0037
|
algebra_func_derivative_sign
| 3 |
algebra_func_derivative_sign
|
3
|
[
{
"role": "user",
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = -1|(2|(4(-1sin(2πx + 1) - 3) - 3)| - 1)| + 3 is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
20
|
OMEGA_500_SAMPLE
|
omega_500_217
|
number_theory_digit_sum_level_4_0068
|
number_theory_digit_sum
| 4 |
number_theory_digit_sum
|
4
|
[
{
"role": "user",
"content": "let N be the tenth greatest 5-digit positive integer with the property that the sum of the digits of n is divisible by 4. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
99963
|
OMEGA_500_SAMPLE
|
omega_500_218
|
number_theory_digit_sum_level_4_0022
|
number_theory_digit_sum
| 4 |
number_theory_digit_sum
|
4
|
[
{
"role": "user",
"content": "let N be the tenth greatest 5-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}"
}
] |
99972
|
OMEGA_500_SAMPLE
|
omega_500_219
|
logic_puzzles_blocked_grid_level_2_0053
|
logic_puzzles_blocked_grid
| 2 |
logic_puzzles_blocked_grid
|
2
|
[
{
"role": "user",
"content": "In a 14x12 grid, how many different paths are there from the bottom left (0, 0) to the top right (13, 11), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (7, 9), (3, 7), (2, 3), (2, 10), (4, 0), (5, 2), (1, 3), (8, 11), (0, 4), (7, 8), (3, 8)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
636418
|
OMEGA_500_SAMPLE
|
omega_500_220
|
geometry_polygon_chords_level_4_0090
|
geometry_polygon_chords
| 4 |
geometry_polygon_chords
|
4
|
[
{
"role": "user",
"content": "Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 3, from vertex 2 to vertex 10, from vertex 5 to vertex 10, from vertex 4 to vertex 11, from vertex 6 to vertex 9, from vertex 11 to vertex 8, from vertex 9 to vertex 11, from vertex 10 to vertex 7, from vertex 4 to vertex 9, from vertex 2 to vertex 4, from vertex 4 to vertex 7, and from vertex 5 to vertex 3. 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}"
}
] |
0
|
OMEGA_500_SAMPLE
|
omega_500_221
|
arithmetic_matrix_rank_level_2_0083
|
arithmetic_matrix_rank
| 2 |
arithmetic_matrix_rank
|
2
|
[
{
"role": "user",
"content": "Calculate the rank of matrix A where A =\n[[18, -17, -10, 14, -18],\n [24, 36, 38, 26, -2],\n [-42, 5, -7, -37, 29],\n [12, 2, 5, 11, -7],\n [-12, 30, 23, -7, 19]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
2
|
OMEGA_500_SAMPLE
|
omega_500_222
|
arithmetic_list_prime_factors_level_2_0014
|
arithmetic_list_prime_factors
| 2 |
arithmetic_list_prime_factors
|
2
|
[
{
"role": "user",
"content": "Find the second-largest prime factor of 28280581499.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
1597
|
OMEGA_500_SAMPLE
|
omega_500_223
|
number_theory_digit_sum_level_1_0002
|
number_theory_digit_sum
| 1 |
number_theory_digit_sum
|
1
|
[
{
"role": "user",
"content": "let N be the maximum 2-digit positive integer with the property that whenever one of its digits changes to 8, the resulting number is a multiple of 7. Let Q and R be the quotient and remainder, respectively, when N is divided by 10. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
13
|
OMEGA_500_SAMPLE
|
omega_500_224
|
logic_puzzles_grid_knight_level_2_0017
|
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 13x14 grid, what is minimum number of steps you can take to get to (11, 10), 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}"
}
] |
3
|
OMEGA_500_SAMPLE
|
omega_500_225
|
arithmetic_matrix_inverse_level_7_0049
|
arithmetic_matrix_inverse
| 7 |
arithmetic_matrix_inverse
|
7
|
[
{
"role": "user",
"content": "Find the inverse of 1/839 times the matrix and calculate the sum of absolute values of all entries of the inverse. Round your answer to the nearest integer:\n[[-19, -13, 6, 9, 4, 21, 4, -14, -12, -2, 14, 7],\n [-13, 29, -4, -4, -24, 2, 22, 3, 32, 10, 10, -12],\n [6, -4, 2, -22, -8, 20, 0, 12, -12, -4, 3, 6],\n [9, -4, -22, 5, 8, 0, -24, -14, 24, -10, -6, -15],\n [4, -24, -8, 8, 34, 6, -7, 14, -3, -30, 11, 10],\n [21, 2, 20, 0, 6, -34, -4, 12, 17, -12, -14, 0],\n [4, 22, 0, -24, -7, -4, 25, -14, 20, 0, -10, -22],\n [-14, 3, 12, -14, 14, 12, -14, 24, 6, -17, 12, 2],\n [-12, 32, -12, 24, -3, 17, 20, 6, 1, 16, 17, -8],\n [-2, 10, -4, -10, -30, -12, 0, -17, 16, 6, -16, 2],\n [14, 10, 3, -6, 11, -14, -10, 12, 17, -16, -14, 0],\n [7, -12, 6, -15, 10, 0, -22, 2, -8, 2, 0, -30]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5989
|
OMEGA_500_SAMPLE
|
omega_500_226
|
algebra_func_derivative_sign_level_3_0075
|
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) = -1|(1sin(1π(-2|((3x - 1)/(3x + 1)) + 1| + 2) - 3) + 1) - 2| negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
OMEGA_500_SAMPLE
|
omega_500_227
|
arithmetic_matrix_eigenvalues_level_7_0060
|
arithmetic_matrix_eigenvalues
| 7 |
arithmetic_matrix_eigenvalues
|
7
|
[
{
"role": "user",
"content": "Find the eigenvalues of the matrix and calculate the largest eigenvalue by absolute value:\n[[7, 24, 21, 35, -19, 14, -4, -14, 19, 21, -23, 11],\n [0, 18, -7, -14, 29, -7, 12, -31, -1, 35, 10, -1],\n [0, 0, -26, 30, 34, -11, 8, 24, 11, 4, -29, 10],\n [0, 0, 0, -15, -35, -17, 24, -3, -32, -33, 16, -8],\n [0, 0, 0, 0, -20, -12, 33, 5, 11, 33, -5, -6],\n [0, 0, 0, 0, 0, -18, 20, -5, 23, -34, 3, 25],\n [0, 0, 0, 0, 0, 0, -1, -25, 25, 8, 32, -11],\n [0, 0, 0, 0, 0, 0, 0, -33, 26, 28, -19, 26],\n [0, 0, 0, 0, 0, 0, 0, 0, -20, -18, -29, 9],\n [0, 0, 0, 0, 0, 0, 0, 0, 0, 19, 20, 8],\n [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -33],\n [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
33
|
OMEGA_500_SAMPLE
|
omega_500_228
|
geometry_polygon_chords_level_3_0010
|
geometry_polygon_chords
| 3 |
geometry_polygon_chords
|
3
|
[
{
"role": "user",
"content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 5, from vertex 2 to vertex 5, from vertex 1 to vertex 7, from vertex 2 to vertex 6, from vertex 4 to vertex 8, from vertex 8 to vertex 6, from vertex 7 to vertex 2, from vertex 8 to vertex 3, from vertex 4 to vertex 1, 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}"
}
] |
6
|
OMEGA_500_SAMPLE
|
omega_500_229
|
number_theory_prime_mod_level_2_0013
|
number_theory_prime_mod
| 2 |
number_theory_prime_mod
|
2
|
[
{
"role": "user",
"content": "Let p be the least prime number for which there exists a positive integer n such that n^5+(-1) is divisible by p^5. Find the least positive integer m such that m^5+(-1) is divisible by p^5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
1
|
OMEGA_500_SAMPLE
|
omega_500_230
|
logic_puzzles_grid_rook_level_4_0098
|
logic_puzzles_grid_rook
| 4 |
logic_puzzles_grid_rook
|
4
|
[
{
"role": "user",
"content": "In a 35x38 grid, what is the fewest number of steps it takes to get from the bottom left (0, 0) to the top right (34, 37), 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: (26, 27), (3, 4), (30, 6), (26, 21), (2, 7), (11, 27), (2, 3), (11, 32), (22, 16), (31, 30), (1, 25), (11, 5), (0, 34), (0, 7), (31, 18), (3, 31), (10, 7), (6, 17), (27, 23), (33, 34), (17, 18), (32, 34), (6, 13), (25, 33), (7, 9), (2, 15), (34, 15), (14, 21), (12, 11), (17, 20), (21, 23), (10, 0), (24, 23), (5, 34), (12, 7), (12, 29), (23, 0), (3, 18), (9, 8), (14, 22), (8, 18), (34, 11), (15, 34), (30, 17), (7, 1), (24, 36), (22, 1), (12, 9), (33, 18), (5, 24), (27, 12), (13, 7), (4, 0), (27, 22), (33, 32), (23, 31), (19, 8), (31, 34), (19, 30), (19, 21), (27, 15), (1, 23), (26, 36), (6, 35), (19, 15), (28, 26), (26, 3), (29, 20), (13, 6), (16, 6), (11, 35), (24, 15), (31, 19), (1, 29), (17, 28), (1, 9), (28, 33), (12, 25), (24, 10), (27, 21), (29, 7), (16, 9), (5, 32), (28, 5), (22, 2), (6, 23), (11, 29), (8, 19), (6, 15), (33, 30), (0, 32), (2, 18), (2, 25), (29, 2), (3, 13), (17, 7), (25, 30), (4, 25), (13, 31), (34, 36), (23, 14), (32, 28), (18, 31), (2, 27), (9, 23), (10, 22), (25, 34), (14, 4), (8, 0), (19, 6), (10, 24), (21, 27), (25, 23), (29, 22), (2, 26), (22, 14), (9, 31), (8, 25), (32, 26), (13, 11), (17, 12), (10, 23), (30, 22), (6, 18), (3, 20), (25, 22), (10, 29), (33, 16), (20, 3), (20, 18), (27, 13), (8, 30), (30, 33), (22, 20), (26, 14), (25, 6), (23, 2), (17, 9), (33, 3), (22, 28), (11, 28), (0, 36), (19, 36), (14, 35), (25, 18), (1, 11), (21, 32), (18, 2), (32, 16), (6, 12), (33, 23), (7, 35), (22, 9), (4, 16), (33, 19), (27, 37), (28, 36), (12, 6), (33, 25), (5, 12), (0, 37), (3, 9), (27, 33), (18, 4), (25, 3), (7, 31), (13, 5), (29, 6), (32, 30), (15, 1), (19, 7), (10, 28), (14, 15), (28, 15), (22, 29), (29, 18), (24, 37), (0, 35), (0, 8), (11, 7), (29, 14), (7, 3), (22, 13), (5, 22), (8, 24), (15, 29), (33, 14), (26, 5), (10, 17), (24, 6), (15, 7), (24, 7), (30, 23), (1, 15), (18, 35), (21, 35), (18, 16), (9, 6), (8, 35), (13, 16), (12, 26), (8, 33), (27, 27), (28, 20), (34, 24), (19, 12), (27, 16), (12, 3), (17, 31), (14, 16), (20, 24), (32, 31), (33, 17), (13, 28), (26, 13), (34, 23), (9, 16), (5, 2), (22, 4), (14, 9), (34, 8), (10, 11), (0, 29), (2, 37), (17, 5), (4, 35), (26, 18), (7, 29), (9, 35), (11, 21), (5, 25), (30, 3), (5, 9), (14, 29), (17, 34), (14, 8), (18, 3), (20, 14), (4, 5), (17, 17), (31, 24), (34, 32), (19, 0), (9, 2), (26, 23), (25, 4), (2, 23), (11, 0), (15, 25)? If it's impossible, answer with -1.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
8
|
OMEGA_500_SAMPLE
|
omega_500_231
|
algebra_func_intersection_level_2_0047
|
algebra_func_intersection
| 2 |
algebra_func_intersection
|
2
|
[
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -1sin(2π((1(1|x + 1| + 1) - 2)/(-1(1|x + 1| + 1) + 2)) + 1) - 1 and g(x) = -3(-3sin(2πx + 3) + 2)^2 - 3(-3sin(2πx + 3) + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
80
|
OMEGA_500_SAMPLE
|
omega_500_232
|
logic_puzzles_blocked_grid_level_1_0093
|
logic_puzzles_blocked_grid
| 1 |
logic_puzzles_blocked_grid
|
1
|
[
{
"role": "user",
"content": "In a 6x7 grid, how many different paths are there from the bottom left (0, 0) to the top right (5, 6), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (0, 2), (5, 2), (3, 5), (2, 1), (1, 0), (0, 4), (4, 3), (4, 2), (1, 3)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
10
|
OMEGA_500_SAMPLE
|
omega_500_233
|
geometry_rotation_level_4_0073
|
geometry_rotation
| 4 |
geometry_rotation
|
4
|
[
{
"role": "user",
"content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 4, from vertex 4 to vertex 2, from vertex 1 to vertex 7, from vertex 2 to vertex 7, and from vertex 10 to vertex 6. 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 10-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}"
}
] |
22
|
OMEGA_500_SAMPLE
|
omega_500_234
|
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_235
|
algebra_func_area_level_4_0035
|
algebra_func_area
| 4 |
algebra_func_area
|
4
|
[
{
"role": "user",
"content": "Find the area of the region bounded by the curves f(x) = -2e^(1(-5(-4(-1(1x^2 - 4x + 2)^2 - 1(1x^2 - 4x + 2))^2 + 3(-1(1x^2 - 4x + 2)^2 - 1(1x^2 - 4x + 2)) + 2))) + 2, g(x) = (-2(1(-2(3cos(1πx + 2) + 1)^2 - 3(3cos(1πx + 2) + 1) - 5) - 5) + 1)/(-1(1(-2(3cos(1πx + 2) + 1)^2 - 3(3cos(1πx + 2) + 1) - 5) - 5) + 2), x = 2.8, and x = 3.5. Round your answer to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0.2
|
OMEGA_500_SAMPLE
|
omega_500_236
|
arithmetic_matrix_determinant_level_6_0039
|
arithmetic_matrix_determinant
| 6 |
arithmetic_matrix_determinant
|
6
|
[
{
"role": "user",
"content": "Find the determinant of the following matrix. Round to the nearest integer, then give your answer modulo 1000:\n[[11, -30, 18, 6, 18, 4, 10, -28, 4, -2],\n [0, 6, 29, 25, 11, 16, -6, -19, 7, -24],\n [0, 0, -27, 30, 14, 16, 0, 19, 19, 8],\n [0, 0, 0, -5, -8, -2, -23, -12, -2, 12],\n [0, 0, 0, 0, 25, -21, 20, 29, -17, 21],\n [0, 0, 0, 0, 0, 6, 17, 27, 8, -3],\n [0, 0, 0, 0, 0, 0, 1, 21, -26, -28],\n [0, 0, 0, 0, 0, 0, 0, -2, 17, 7],\n [0, 0, 0, 0, 0, 0, 0, 0, -5, -14],\n [0, 0, 0, 0, 0, 0, 0, 0, 0, -14]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0
|
OMEGA_500_SAMPLE
|
omega_500_237
|
number_theory_triple_count_level_2_0007
|
number_theory_triple_count
| 2 |
number_theory_triple_count
|
2
|
[
{
"role": "user",
"content": "Assume N is the number of integers a which are >= 0 where a <= 3^4 and a^3 is divisible by 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
27
|
OMEGA_500_SAMPLE
|
omega_500_238
|
combinatory_distribution_level_5_0003
|
combinatory_distribution
| 5 |
combinatory_distribution
|
5
|
[
{
"role": "user",
"content": "Given the letters {'e': 4, 't': 3, 'x': 3, 'n': 2}, place them into 4 labeled boxes with sizes [4, 4, 1, 3]. How many distinct arrangements are possible?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
906
|
OMEGA_500_SAMPLE
|
omega_500_239
|
algebra_func_extrema_level_3_0063
|
algebra_func_extrema
| 3 |
algebra_func_extrema
|
3
|
[
{
"role": "user",
"content": "How many local maxima does the function f(x) = -4(-3sin(1π(5(1|x + 2| + 1) + 2) + 2) - 1) + 1 have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
50
|
OMEGA_500_SAMPLE
|
omega_500_240
|
logic_puzzles_grid_chip_level_2_0094
|
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 (4, 3) is white, cell (2, 4) is white, cell (3, 0) is white, cell (3, 1) is white. What is the maximum number of black chips that can be placed on the grid?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
9
|
OMEGA_500_SAMPLE
|
omega_500_241
|
combinatory_probability_no_specific_letter_fixed_level_4_0031
|
combinatory_probability_no_specific_letter_fixed
| 4 |
combinatory_probability_no_specific_letter_fixed
|
4
|
[
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {y: 3, h: 3, g: 2, t: 2}, shuffle the letters in the word, what is the probability of no letter 'h' 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}"
}
] |
191
|
OMEGA_500_SAMPLE
|
omega_500_242
|
algebra_func_intersection_level_4_0011
|
algebra_func_intersection
| 4 |
algebra_func_intersection
|
4
|
[
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = (-1(-3cos(3π(2|((3(1x^2 - 5x) - 3)/(-1(1x^2 - 5x) - 2)) - 2| + 2) + 2) + 3) - 2)/(-3(-3cos(3π(2|((3(1x^2 - 5x) - 3)/(-1(1x^2 - 5x) - 2)) - 2| + 2) + 2) + 3) - 1) and g(x) = -1(2((-2(2|x| + 2) + 3)/(-2(2|x| + 2) - 3)) - 5) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
72
|
OMEGA_500_SAMPLE
|
omega_500_243
|
arithmetic_matrix_determinant_level_1_0084
|
arithmetic_matrix_determinant
| 1 |
arithmetic_matrix_determinant
|
1
|
[
{
"role": "user",
"content": "Calculate the determinant of the matrix, round to the nearest integer, then find the result modulo 1000:\n[[4, -9, 4],\n [0, -3, 6],\n [0, 0, -1]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
12
|
OMEGA_500_SAMPLE
|
omega_500_244
|
arithmetic_matrix_svd_level_7_0008
|
arithmetic_matrix_svd
| 7 |
arithmetic_matrix_svd
|
7
|
[
{
"role": "user",
"content": "Find the Singular Value Decomposition (SVD) of the matrix and determine the rounded largest singular value:\n[[11, 8, 3, 4, 22, -22, 17, 23, -13, 26, -16, -18],\n [7, -15, 8, 8, -26, 13, -5, 2, -19, -16, 14, 10],\n [-13, 3, 8, -6, 10, -22, -8, -13, -1, -23, -2, 18],\n [15, -2, -9, 13, -19, 12, 13, -13, 5, 24, 11, 6],\n [-4, -12, 6, -2, -10, 6, 20, -25, -13, 26, 13, 13],\n [24, 12, 24, -21, -21, -24, -20, 24, -19, 15, 25, -12],\n [20, 2, 6, 25, 3, 12, 0, 9, 2, 11, 6, 10],\n [0, 6, -23, -5, -25, -17, -22, -17, 6, 11, -14, 4],\n [20, 9, 18, -3, 25, -12, 2, -19, -22, 2, 20, -23],\n [-15, 18, -25, 0, 4, 24, 24, 9, 9, -1, 16, 0],\n [-22, -7, -16, -17, 13, 11, -21, -19, -4, 20, -1, 19],\n [16, -15, -1, -14, 13, -9, -2, 6, 20, 13, 16, -15]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
91
|
OMEGA_500_SAMPLE
|
omega_500_245
|
geometry_rotation_level_5_0070
|
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 6 to vertex 2, from vertex 5 to vertex 3, from vertex 2 to vertex 5, from vertex 1 to vertex 3, from vertex 5 to vertex 1, and from vertex 3 to vertex 6. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 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_246
|
combinatory_probability_no_specific_letter_fixed_level_5_0083
|
combinatory_probability_no_specific_letter_fixed
| 5 |
combinatory_probability_no_specific_letter_fixed
|
5
|
[
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 5 letters from the multiset {y: 2, a: 2, d: 2, e: 2, c: 3}, shuffle the letters in the word, what is the probability of no letter 'e' 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}"
}
] |
97
|
OMEGA_500_SAMPLE
|
omega_500_247
|
arithmetic_matrix_multiplication_level_6_0080
|
arithmetic_matrix_multiplication
| 6 |
arithmetic_matrix_multiplication
|
6
|
[
{
"role": "user",
"content": "What is the sum of absolute values of all entries when matrix A is multiplied by matrix B?\nA =\n[[8, 10, -24, -8, -13, 25, -21, -6, -24, -15],\n [4, 6, 6, 24, 12, 2, -16, -4, -9, -4],\n [17, -7, -9, -2, 10, 6, 20, 10, 2, -12],\n [10, -20, -17, -5, -14, 4, 5, 14, -22, 5],\n [1, -1, -21, -20, -21, 25, 1, -18, -17, -11],\n [10, 15, -19, 6, -18, 21, 25, -11, 11, -7],\n [13, -20, -7, 2, -9, 16, -7, 5, -15, -11],\n [-16, 17, -3, -23, -23, -8, -1, 17, -6, 3],\n [-4, 24, -11, 13, -16, -5, 3, 6, -16, -13],\n [21, -3, 24, -3, 5, -22, 23, -23, -5, -24],\n [-22, -25, -11, 7, -20, -19, -7, 18, -25, 13]]\nB =\n[[-7, 19, 0, 18, -14, -4, -13, 2, -12, -4, 4],\n [18, -6, 18, 1, 12, 16, 15, 8, 13, 15, 18],\n [11, -25, -11, 6, -18, 7, -8, 19, 3, 6, 7],\n [-21, -8, 22, -16, 10, 7, -3, 14, -21, 8, -2],\n [-9, 5, -19, -9, 2, 12, -4, -11, 11, -11, 24],\n [24, -1, -5, 12, -24, 1, 23, 6, -5, -22, 18],\n [-7, -24, 0, 8, -12, -11, -6, 8, 10, -7, -17],\n [-19, 4, -13, 1, 18, 9, -13, -11, 20, -17, 19],\n [-3, 10, 14, -23, 8, -7, 2, -3, 14, 23, 19],\n [-19, 15, -14, -12, 20, -11, -23, -11, 25, -2, 20]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
65512
|
OMEGA_500_SAMPLE
|
omega_500_248
|
arithmetic_matrix_inverse_level_2_0099
|
arithmetic_matrix_inverse
| 2 |
arithmetic_matrix_inverse
|
2
|
[
{
"role": "user",
"content": "Find the value of the entry with the largest absolute value of the inverse of matrix 1/96 * A where A =\n[[1, -8, -11, -2],\n [0, 11, -10, -4],\n [0, 0, -11, -1],\n [0, 0, 0, -5]]\nRound your answer to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-159
|
OMEGA_500_SAMPLE
|
omega_500_249
|
logic_puzzles_grid_knight_level_3_0037
|
logic_puzzles_grid_knight
| 3 |
logic_puzzles_grid_knight
|
3
|
[
{
"role": "user",
"content": "Supposing you start in the bottom left (0, 0) cell of a 27x28 grid, what is minimum number of steps you can take to get to (27, 25), if your only valid moves are to move over 2 squares in one direction and over 3 squares in a perpendicular direction (at once)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
12
|
OMEGA_500_SAMPLE
|
omega_500_250
|
arithmetic_list_prime_factors_level_2_0051
|
arithmetic_list_prime_factors
| 2 |
arithmetic_list_prime_factors
|
2
|
[
{
"role": "user",
"content": "What is the second-largest prime factor of 11672055465?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
443
|
OMEGA_500_SAMPLE
|
omega_500_251
|
number_theory_prime_mod_level_1_0091
|
number_theory_prime_mod
| 1 |
number_theory_prime_mod
|
1
|
[
{
"role": "user",
"content": "Let p be the minimum prime where n^1+-2=0 (mod p^2) 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}"
}
] |
2
|
OMEGA_500_SAMPLE
|
omega_500_252
|
combinatory_distribution_level_5_0037
|
combinatory_distribution
| 5 |
combinatory_distribution
|
5
|
[
{
"role": "user",
"content": "Divide the letters from {'n': 3, 't': 2, 'p': 4, 'a': 3} into 4 undistinguishable boxes with sizes [4, 2, 4, 2]. How many ways?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
316
|
OMEGA_500_SAMPLE
|
omega_500_253
|
arithmetic_matrix_rank_level_4_0062
|
arithmetic_matrix_rank
| 4 |
arithmetic_matrix_rank
|
4
|
[
{
"role": "user",
"content": "Determine the rank of the following matrix:\n[[8, -24, 10, -2, -24, -10, -18],\n [12, -36, 15, -3, -36, -15, -27],\n [-8, 24, -10, 2, 24, 10, 18],\n [-16, 48, -20, 4, 48, 20, 36],\n [20, -60, 25, -5, -60, -25, -45],\n [-20, 60, -25, 5, 60, 25, 45],\n [-40, 120, -50, 10, 120, 50, 90]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
1
|
OMEGA_500_SAMPLE
|
omega_500_254
|
number_theory_digit_sum_level_4_0012
|
number_theory_digit_sum
| 4 |
number_theory_digit_sum
|
4
|
[
{
"role": "user",
"content": "let N be the least 5-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 10000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
20004
|
OMEGA_500_SAMPLE
|
omega_500_255
|
geometry_polygon_color_level_1_0012
|
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 uncolored, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is uncolored, vertex 5 is blue, vertex 6 is uncolored. How many valid colorings of the uncolored vertices exist such that the 7-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}"
}
] |
3
|
OMEGA_500_SAMPLE
|
omega_500_256
|
logic_puzzles_grid_chip_level_3_0023
|
logic_puzzles_grid_chip
| 3 |
logic_puzzles_grid_chip
|
3
|
[
{
"role": "user",
"content": "Chips, colored either black or white, are placed in 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. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 4) is white, cell (4, 5) is white, cell (1, 1) is white, cell (5, 2) is white. What is the maximum number of black chips that can be placed on the grid?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
12
|
OMEGA_500_SAMPLE
|
omega_500_257
|
algebra_func_intersection_level_4_0006
|
algebra_func_intersection
| 4 |
algebra_func_intersection
|
4
|
[
{
"role": "user",
"content": "Determine the number of intersection points between the functions f(x) = 2(-4((-2(2|(1|x + 3| - 2) - 2| - 1) + 2)/(-3(2|(1|x + 3| - 2) - 2| - 1)))^2 + 5((-2(2|(1|x + 3| - 2) - 2| - 1) + 2)/(-3(2|(1|x + 3| - 2) - 2| - 1))) - 4) - 3 and g(x) = -3((-2(1((3x - 3)/(-1x - 1)) - 3) + 1)/(-1(1((3x - 3)/(-1x - 1)) - 3) - 2)) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
8
|
OMEGA_500_SAMPLE
|
omega_500_258
|
logic_puzzles_blocked_grid_level_3_0046
|
logic_puzzles_blocked_grid
| 3 |
logic_puzzles_blocked_grid
|
3
|
[
{
"role": "user",
"content": "In a 28x20 grid, how many different paths are there from the bottom left (0, 0) to the top right (27, 19), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (24, 3), (10, 14), (22, 13), (7, 16), (22, 17), (25, 1), (18, 5), (8, 18), (27, 3), (2, 12), (23, 9), (14, 5), (9, 16), (5, 11), (11, 18), (13, 8), (4, 19), (21, 13), (4, 6), (3, 3), (20, 14), (16, 3), (9, 4), (10, 10), (25, 4), (17, 7), (9, 13), (23, 10), (16, 16), (1, 12), (20, 5), (22, 11), (23, 1), (13, 17), (11, 2), (11, 17), (1, 1), (17, 17), (14, 10), (18, 1), (24, 9), (23, 17), (9, 14), (9, 15), (0, 3), (5, 0), (7, 17), (10, 1), (14, 14), (12, 9), (23, 3), (24, 19), (3, 4), (26, 14), (11, 9), (20, 4), (7, 9), (16, 9), (14, 3), (4, 13), (24, 14), (14, 9), (7, 6), (15, 10), (20, 17), (3, 6), (11, 16), (1, 6), (22, 18), (15, 14), (24, 8), (10, 15), (19, 4), (18, 7), (27, 7), (7, 11), (23, 2), (21, 16), (4, 8), (25, 13), (16, 12), (4, 11), (20, 2), (17, 14), (1, 0), (18, 8), (17, 19), (8, 15), (23, 4), (2, 7), (25, 0), (13, 6), (22, 5), (24, 7), (6, 15), (20, 16), (7, 19), (8, 0), (14, 19), (9, 1), (3, 11), (19, 14), (11, 7), (6, 12), (2, 19), (17, 0), (8, 14), (10, 12), (27, 8), (12, 2), (21, 10), (5, 9), (19, 2), (18, 11), (7, 13), (5, 7), (22, 4), (11, 13), (6, 1), (8, 11), (5, 13), (5, 14), (3, 2)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
388696
|
OMEGA_500_SAMPLE
|
omega_500_259
|
algebra_func_intersection_level_4_0087
|
algebra_func_intersection
| 4 |
algebra_func_intersection
|
4
|
[
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = -2|(-2cos(2π(1(4(1|x| - 1)^2 + 5(1|x| - 1) + 3)^2 - 5(4(1|x| - 1)^2 + 5(1|x| - 1) + 3) - 4)) + 3) + 1| - 2 and g(x) = -1|((-2(2sin(1π(-1x^2 - 5x) + 2) - 1))/(1(2sin(1π(-1x^2 - 5x) + 2) - 1) + 3)) + 2| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
105
|
OMEGA_500_SAMPLE
|
omega_500_260
|
combinatory_pattern_matching_level_1_0066
|
combinatory_pattern_matching
| 1 |
combinatory_pattern_matching
|
1
|
[
{
"role": "user",
"content": "If you form a word of length 2 by randomly selecting letters from the multiset {f: 1, h: 2, i: 1}, what is the expected number of times the pattern 'f.*f' appears? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0.0
|
OMEGA_500_SAMPLE
|
omega_500_261
|
algebra_func_intersection_level_3_0063
|
algebra_func_intersection
| 3 |
algebra_func_intersection
|
3
|
[
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = 1|(4(-1|((-1x + 3)/(-1x - 2)) + 3|) - 5)| - 1 and g(x) = (1(-3(3cos(3πx - 2) + 3) + 2) - 2)/(2(-3(3cos(3πx - 2) + 3) + 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
19
|
OMEGA_500_SAMPLE
|
omega_500_262
|
algebra_func_intersection_level_2_0033
|
algebra_func_intersection
| 2 |
algebra_func_intersection
|
2
|
[
{
"role": "user",
"content": "Find the number of times the graphs of f(x) = -2sin(3π(3(-1|x - 3| - 1)^2 - 3) + 1) and g(x) = 3sin(2π(-2|x - 2| - 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
607
|
OMEGA_500_SAMPLE
|
omega_500_263
|
algebra_func_extrema_level_1_0013
|
algebra_func_extrema
| 1 |
algebra_func_extrema
|
1
|
[
{
"role": "user",
"content": "Determine the count of local maxima for the function f(x) = (1(-1sin(1πx) + 3) + 2)/(-1(-1sin(1πx) + 3) + 2) in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
19
|
OMEGA_500_SAMPLE
|
omega_500_264
|
combinatory_probability_at_least_n_specific_fixed_level_5_0057
|
combinatory_probability_at_least_n_specific_fixed
| 5 |
combinatory_probability_at_least_n_specific_fixed
|
5
|
[
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 5 letters from the multiset {b: 2, d: 2, r: 2, y: 3, c: 2}, shuffle the letters in the word, what is the probability of at least 2 letter 'y' 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}"
}
] |
178
|
OMEGA_500_SAMPLE
|
omega_500_265
|
algebra_linear_equation_level_4_0083
|
algebra_linear_equation
| 4 |
algebra_linear_equation
|
4
|
[
{
"role": "user",
"content": "Suppose 2413505 - 2415157 = -413*g. Solve -4*d - 3*q + 3 = 5*z + o - 3*a, -4*d - 4*q + 2*o + 18 = -2*z, -3*d - 2 + 0 = z + 3*q + g*o + a, -4*o - 4*a = -4*d + 20, -3*d + 2*z - q - a = -2*o - 23 for d.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
OMEGA_500_SAMPLE
|
omega_500_266
|
arithmetic_matrix_determinant_level_5_0048
|
arithmetic_matrix_determinant
| 5 |
arithmetic_matrix_determinant
|
5
|
[
{
"role": "user",
"content": "Find the determinant of the following matrix. Round to the nearest integer, then give your answer modulo 1000:\n[[19, 9, 11, -25, 21, 8, -5, 0, -12],\n [-8, -2, 9, -25, 21, 17, -10, 20, 13],\n [12, -5, -2, 0, 21, 20, 13, -8, 15],\n [-2, 18, 2, -6, 13, -9, -5, 24, -8],\n [24, -24, 11, 10, -9, 17, -20, 9, 14],\n [8, -4, 11, 6, -16, -20, -13, -20, 10],\n [7, -24, -10, 14, -1, -6, 1, 20, -11],\n [14, -16, 12, 1, -9, -4, 13, -12, 23],\n [17, 2, -18, -18, 3, 15, -25, 19, -17]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
734
|
OMEGA_500_SAMPLE
|
omega_500_267
|
algebra_func_intersection_level_3_0053
|
algebra_func_intersection
| 3 |
algebra_func_intersection
|
3
|
[
{
"role": "user",
"content": "How many points of intersection exist between the functions f(x) = 1|((-3(-3cos(3π((-1x - 2)/(-2x + 1)) - 1) - 1) + 3)/(1(-3cos(3π((-1x - 2)/(-2x + 1)) - 1) - 1))) - 3| + 3 and g(x) = (3((-1((2x + 3)/(-2x + 3)) + 3)/(2((2x + 3)/(-2x + 3)) + 1)) - 3)/(-2((-1((2x + 3)/(-2x + 3)) + 3)/(2((2x + 3)/(-2x + 3)) + 1)) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
5
|
OMEGA_500_SAMPLE
|
omega_500_268
|
combinatory_probability_at_least_n_specific_fixed_level_1_0020
|
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 {h: 2, p: 1, v: 1}, shuffle the letters in the word, what is the probability of at least 1 letter 'v' 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}"
}
] |
5
|
OMEGA_500_SAMPLE
|
omega_500_269
|
arithmetic_matrix_eigenvalues_level_4_0015
|
arithmetic_matrix_eigenvalues
| 4 |
arithmetic_matrix_eigenvalues
|
4
|
[
{
"role": "user",
"content": "Calculate the sum of absolute values of all eigenvalues for the matrix:\n[[6, -15, -9, -16, -1, 17, 3],\n [0, -11, 0, 18, -13, 7, 15],\n [0, 0, 14, 3, 2, -13, -13],\n [0, 0, 0, 11, 12, 5, -10],\n [0, 0, 0, 0, 5, 12, -13],\n [0, 0, 0, 0, 0, -4, -13],\n [0, 0, 0, 0, 0, 0, -14]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
65
|
OMEGA_500_SAMPLE
|
omega_500_270
|
number_theory_digit_sum_level_1_0079
|
number_theory_digit_sum
| 1 |
number_theory_digit_sum
|
1
|
[
{
"role": "user",
"content": "let N be the third smallest 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 7. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
24
|
OMEGA_500_SAMPLE
|
omega_500_271
|
arithmetic_matrix_eigenvalues_level_1_0035
|
arithmetic_matrix_eigenvalues
| 1 |
arithmetic_matrix_eigenvalues
|
1
|
[
{
"role": "user",
"content": "Find the eigenvalues of the matrix and calculate the largest eigenvalue by absolute value:\n[[-1, -2, -6],\n [-2, 7, 2],\n [-6, 2, -9]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
12
|
OMEGA_500_SAMPLE
|
omega_500_272
|
arithmetic_matrix_eigenvalues_level_1_0016
|
arithmetic_matrix_eigenvalues
| 1 |
arithmetic_matrix_eigenvalues
|
1
|
[
{
"role": "user",
"content": "Find the eigenvalues of the matrix and calculate the largest eigenvalue by absolute value:\n[[10, -5, 1],\n [0, 9, 0],\n [0, 0, -10]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
10
|
OMEGA_500_SAMPLE
|
omega_500_273
|
algebra_func_area_level_3_0097
|
algebra_func_area
| 3 |
algebra_func_area
|
3
|
[
{
"role": "user",
"content": "Determine the area of the region enclosed by the graphs of f(x) = 1cos(1π(4(-5(-4x^2 - 1x + 5)^2 - 4(-4x^2 - 1x + 5) - 4) - 2) - 2) + 2 and g(x) = 3(-1|(-3x^2 - 1x + 4) + 1| - 2)^2 + 4(-1|(-3x^2 - 1x + 4) + 1| - 2) - 2 between the vertical lines x = 1.1 and x = 1.2000000000000002. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0.2
|
OMEGA_500_SAMPLE
|
omega_500_274
|
number_theory_triple_count_level_5_0020
|
number_theory_triple_count
| 5 |
number_theory_triple_count
|
5
|
[
{
"role": "user",
"content": "Find the number of ordered triples T of positive integers (a, b, c) satisfying three conditions: 1. a, b, c <= 3^3, 2. a, b, c are relatively coprime, and 3. (0)a^3 + (-1)b^3 + (-1)c^3 is a multiple of 3^3. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
600
|
OMEGA_500_SAMPLE
|
omega_500_275
|
algebra_func_extrema_coords_level_5_0018
|
algebra_func_extrema_coords
| 5 |
algebra_func_extrema_coords
|
5
|
[
{
"role": "user",
"content": "Find the x-coordinate of the local maximum with the largest y-value of the function f(x) = -4(2(-2|(-2((-3((2x - 3)/(3x + 2)) + 3)/(-3((2x - 3)/(3x + 2)) - 2))^2 - 5((-3((2x - 3)/(3x + 2)) + 3)/(-3((2x - 3)/(3x + 2)) - 2)) + 4) + 2| - 2) + 5)^2 + 4(2(-2|(-2((-3((2x - 3)/(3x + 2)) + 3)/(-3((2x - 3)/(3x + 2)) - 2))^2 - 5((-3((2x - 3)/(3x + 2)) + 3)/(-3((2x - 3)/(3x + 2)) - 2)) + 4) + 2| - 2) + 5) - 2 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-1
|
OMEGA_500_SAMPLE
|
omega_500_276
|
logic_puzzles_zebralogic_level_1_0047
|
logic_puzzles_zebralogic
| 1 |
logic_puzzles_zebralogic
|
1
|
[
{
"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 keep unique animals: `horse`, `cat`\n\n## Clues:\n1. Eric is not in the first house.\n2. The cat lover is not in the first house.\n\n\nChoices:\n0. Arnold\n1. Eric\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_277
|
algebra_func_derivative_sign_level_1_0057
|
algebra_func_derivative_sign
| 1 |
algebra_func_derivative_sign
|
1
|
[
{
"role": "user",
"content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 5(2sin(1πx))^2 + 3(2sin(1πx)) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
21
|
OMEGA_500_SAMPLE
|
omega_500_278
|
combinatory_probability_no_fixed_points_level_1_0092
|
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 {t: 2, v: 1, x: 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_279
|
arithmetic_matrix_eigenvalues_level_4_0076
|
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[[11, -3, 7, 12, -4, 10, -8],\n [-3, 17, 6, -5, -4, -5, -12],\n [7, 6, 9, -6, 2, -11, 6],\n [12, -5, -6, -9, -4, 2, 10],\n [-4, -4, 2, -4, -6, -2, -6],\n [10, -5, -11, 2, -2, 7, -10],\n [-8, -12, 6, 10, -6, -10, 10]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
30
|
OMEGA_500_SAMPLE
|
omega_500_280
|
arithmetic_list_prime_factors_level_5_0027
|
arithmetic_list_prime_factors
| 5 |
arithmetic_list_prime_factors
|
5
|
[
{
"role": "user",
"content": "Find the second-largest prime factor of 6612512741273908315709.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
96893
|
OMEGA_500_SAMPLE
|
omega_500_281
|
logic_puzzles_zebralogic_level_2_0001
|
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 unique hair colors: `brown`, `black`\n\n## Clues:\n1. Arnold is in the first house.\n2. The person who has brown hair is not in the first house.\n\n\nChoices:\n0. Arnold\n1. Eric\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}"
}
] |
1
|
OMEGA_500_SAMPLE
|
omega_500_282
|
algebra_linear_equation_level_1_0010
|
algebra_linear_equation
| 1 |
algebra_linear_equation
|
1
|
[
{
"role": "user",
"content": "Solve 15*q = -4*q - 4*r + 13, -4*q - 39 = 4*r - 73 + 6 for q.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-1
|
OMEGA_500_SAMPLE
|
omega_500_283
|
combinatory_pattern_matching_level_3_0056
|
combinatory_pattern_matching
| 3 |
combinatory_pattern_matching
|
3
|
[
{
"role": "user",
"content": "When randomly selecting 4 letters from the multiset {j: 1, t: 1, f: 1, v: 3, u: 3} to form a word, what is the expected number of matches of the pattern 'u.*u'? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0.5
|
OMEGA_500_SAMPLE
|
omega_500_284
|
arithmetic_matrix_multiplication_level_1_0034
|
arithmetic_matrix_multiplication
| 1 |
arithmetic_matrix_multiplication
|
1
|
[
{
"role": "user",
"content": "Find the sum of all entries of the product of the following matrices:\nA =\n[[4, -8, 7, 4, 1],\n [3, -7, -6, 1, -7],\n [4, 4, -6, -2, 5]]\nB =\n[[8, 3, 7, 3, 6],\n [1, -2, -4, -8, 8],\n [-6, 4, 5, -5, -4],\n [7, -8, 5, 6, 0],\n [6, 8, -5, 3, -4]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
404
|
OMEGA_500_SAMPLE
|
omega_500_285
|
arithmetic_mixed_level_3_0083
|
arithmetic_mixed
| 3 |
arithmetic_mixed
|
3
|
[
{
"role": "user",
"content": "Evaluate ((-39)/(-12575))/((-5675250)/461587 + (-10455820)/(-844850)) + (-1 - 77/(-80)). 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}"
}
] |
3/3521
|
OMEGA_500_SAMPLE
|
omega_500_286
|
combinatory_pattern_matching_level_1_0085
|
combinatory_pattern_matching
| 1 |
combinatory_pattern_matching
|
1
|
[
{
"role": "user",
"content": "When randomly selecting 2 letters from the multiset {v: 1, t: 1, n: 1, w: 1} to form a word, what is the expected number of matches of the pattern 'w.*w'? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
0.0
|
OMEGA_500_SAMPLE
|
omega_500_287
|
algebra_func_derivative_sign_level_5_0011
|
algebra_func_derivative_sign
| 5 |
algebra_func_derivative_sign
|
5
|
[
{
"role": "user",
"content": "For the function f(x) = 2|(4(2((2(2(1cos(3πx - 3) + 3) - 1) + 1)/(-2(2(1cos(3πx - 3) + 3) - 1) - 3))) - 1)| + 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}"
}
] |
30
|
OMEGA_500_SAMPLE
|
omega_500_288
|
arithmetic_matrix_multiplication_level_6_0093
|
arithmetic_matrix_multiplication
| 6 |
arithmetic_matrix_multiplication
|
6
|
[
{
"role": "user",
"content": "Calculate the matrix product AB and find the value at position (7, 5) of the result:\nA =\n[[12, 1, 12, 14, 19, 24, -3, 5, 10, -9, 11],\n [-23, 10, -21, 20, -11, 23, -20, -16, 6, -3, 22],\n [-9, 20, -6, 18, 15, -11, 18, 20, -4, 24, -20],\n [-22, -3, 18, 19, -7, 19, 5, 18, -23, 2, -23],\n [-12, -8, 1, 25, -16, 23, -11, -7, -17, -22, -11],\n [25, 22, -18, 23, -25, -19, 5, -8, -14, 22, 4],\n [7, 21, -19, 2, 23, -23, 17, -20, 5, 12, -11],\n [-19, 17, 4, 1, -9, -8, -4, 0, 25, 1, 14],\n [-2, -18, -5, 3, 17, -5, -23, 5, 1, 20, 1],\n [-17, 7, 7, 0, 2, 18, 17, 5, -25, -6, -14],\n [-9, -11, 8, 24, -10, -22, -18, 8, 20, -1, -11]]\nB =\n[[2, -17, -24, -9, -16, -17, 5, -24, -2, -7, -9],\n [17, 23, -16, 24, 2, 12, 22, 2, 24, 24, -25],\n [-23, -11, -16, -5, 11, -11, -3, -12, -11, -16, -17],\n [18, 13, -21, -1, 16, 0, -18, -4, -1, 22, -14],\n [-17, -16, 22, -16, -17, 0, 9, 1, 5, -7, 17],\n [12, -20, -11, 3, 21, 19, -21, -12, -24, -3, 18],\n [20, -7, -2, -6, 21, 1, -2, -9, -1, 20, 5],\n [2, 24, 2, -4, -21, 18, 16, -18, 12, 2, 21],\n [-14, -1, 12, 18, -11, 18, 15, 25, 8, 19, -13],\n [-18, -13, 10, -20, 12, 6, -5, 18, 18, 23, -19],\n [0, -11, 23, 16, -3, 12, 15, -11, 22, 14, -10]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-222
|
OMEGA_500_SAMPLE
|
omega_500_289
|
combinatory_probability_no_specific_letter_fixed_level_1_0059
|
combinatory_probability_no_specific_letter_fixed
| 1 |
combinatory_probability_no_specific_letter_fixed
|
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, t: 1, s: 2}, shuffle the letters in the word, what is the probability of no letter 's' 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}"
}
] |
3
|
OMEGA_500_SAMPLE
|
omega_500_290
|
algebra_func_extrema_coords_level_3_0078
|
algebra_func_extrema_coords
| 3 |
algebra_func_extrema_coords
|
3
|
[
{
"role": "user",
"content": "Find the x-coordinate of the local extremum with the smallest y-value of the function f(x) = 3(2|(1(1x + 2)^2 - 3) + 1| + 3)^2 - 1(2|(1(1x + 2)^2 - 3) + 1| + 3) - 4 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-3
|
OMEGA_500_SAMPLE
|
omega_500_291
|
geometry_polygon_color_level_3_0020
|
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 red, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is blue, vertex 8 is blue, vertex 9 is red, vertex 10 is red. The 11-gon is then rotated clockwise by 8 positions, so that the vertex that was originally in position 0 is now in position 8. 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_292
|
arithmetic_matrix_inverse_level_1_0072
|
arithmetic_matrix_inverse
| 1 |
arithmetic_matrix_inverse
|
1
|
[
{
"role": "user",
"content": "For the inverse of 1/60 times the matrix below, what is the trace (sum of diagonal elements) rounded to the nearest integer?\n[[-10, 6, -3],\n [-9, -3, -4],\n [-9, -8, 7]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-4
|
OMEGA_500_SAMPLE
|
omega_500_293
|
combinatory_pattern_matching_level_3_0050
|
combinatory_pattern_matching
| 3 |
combinatory_pattern_matching
|
3
|
[
{
"role": "user",
"content": "When randomly selecting 4 letters from the multiset {k: 3, u: 6} to form a word, what is the expected number of matches of the pattern 'u.*u'? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
2.5
|
OMEGA_500_SAMPLE
|
omega_500_294
|
arithmetic_matrix_eigenvalues_level_6_0067
|
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[[-11, -4, -6, 6, -10, -10, 12, -2, 0, -14],\n [-4, 6, 20, 7, -18, 3, 13, 26, 4, 16],\n [-6, 20, -23, 14, 3, 20, 23, 2, -2, 2],\n [6, 7, 14, -18, -3, 3, 18, 3, 6, -22],\n [-10, -18, 3, -3, 25, -17, -8, -4, -10, 5],\n [-10, 3, 20, 3, -17, 7, -18, 17, 12, -3],\n [12, 13, 23, 18, -8, -18, -30, 14, -7, -11],\n [-2, 26, 2, 3, -4, 17, 14, -22, -12, -4],\n [0, 4, -2, 6, -10, 12, -7, -12, 11, -14],\n [-14, 16, 2, -22, 5, -3, -11, -4, -14, -5]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
71
|
OMEGA_500_SAMPLE
|
omega_500_295
|
geometry_rotation_level_6_0005
|
geometry_rotation
| 6 |
geometry_rotation
|
6
|
[
{
"role": "user",
"content": "Suppose you have a 23-gon, with vertices numbered 1 through 23 in counterclockwise order. Draw the diagonal from vertex 19 to vertex 15, from vertex 10 to vertex 2, from vertex 5 to vertex 11, from vertex 23 to vertex 12, from vertex 9 to vertex 12, from vertex 3 to vertex 20, from vertex 17 to vertex 1, from vertex 7 to vertex 20, from vertex 15 to vertex 11, from vertex 20 to vertex 9, and from vertex 14 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 18 vertices counterclockwise (so that vertex 1 ends up where vertex 19 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 23-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 with 6 sides?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
1
|
OMEGA_500_SAMPLE
|
omega_500_296
|
arithmetic_matrix_inverse_level_6_0007
|
arithmetic_matrix_inverse
| 6 |
arithmetic_matrix_inverse
|
6
|
[
{
"role": "user",
"content": "Find the value of the entry with the largest absolute value of the inverse of matrix 1/600 * A where A =\n[[16, -19, -22, -21, 27, 13, -14, 30, 7, -24],\n [15, -18, 9, 11, -22, 19, -4, 27, -29, -26],\n [-2, 6, 7, -12, -23, 17, 14, -30, -9, 21],\n [-14, -24, -6, 14, -27, 5, -25, 29, 0, -12],\n [30, 13, 23, 8, -4, -21, -5, -12, 8, -28],\n [14, -18, 19, -3, 27, -11, -3, -23, 30, 10],\n [8, -30, -28, -18, -3, 26, 29, 18, -6, 25],\n [2, 7, 22, -25, 27, 13, 14, 1, 14, 30],\n [16, -10, -15, 23, -10, -20, 29, 6, 5, 4],\n [-12, -11, 26, -13, 16, 10, 18, -17, -16, 0]]\nRound your answer to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
-88
|
OMEGA_500_SAMPLE
|
omega_500_297
|
logic_puzzles_zebralogic_level_5_0028
|
logic_puzzles_zebralogic
| 5 |
logic_puzzles_zebralogic
|
5
|
[
{
"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: `Peter`, `Arnold`, `Eric`\n - People have unique favorite book genres: `science fiction`, `mystery`, `romance`\n - Everyone has a favorite smoothie: `watermelon`, `desert`, `cherry`\n - Each person has a unique birthday month: `april`, `jan`, `sept`\n - People have unique heights: `average`, `very short`, `short`\n\n## Clues:\n1. The person who likes Cherry smoothies is not in the second house.\n2. Arnold is the person who loves mystery books.\n3. The person whose birthday is in January is not in the first house.\n4. The person who is very short is the person who loves romance books.\n5. The person who loves mystery books is the person whose birthday is in September.\n6. The person who has an average height is the Desert smoothie lover.\n7. Eric is in the first house.\n8. The Watermelon smoothie lover is the person who is short.\n9. The Watermelon smoothie lover is Eric.\n\n\nChoices:\n0. Arnold\n1. Eric\n2. Peter\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}"
}
] |
2
|
OMEGA_500_SAMPLE
|
omega_500_298
|
arithmetic_matrix_power_level_5_0066
|
arithmetic_matrix_power
| 5 |
arithmetic_matrix_power
|
5
|
[
{
"role": "user",
"content": "For the matrix power A^6 where A =\n[[10, 1, 2, 10, 0, 7, 1, -2],\n [1, -9, -2, -8, 0, 4, 0, -2],\n [2, -2, -12, 2, 6, 8, -4, 4],\n [10, -8, 2, -10, -6, -6, 0, -2],\n [0, 0, 6, -6, -5, -1, 11, -6],\n [7, 4, 8, -6, -1, 9, -4, -2],\n [1, 0, -4, 0, 11, -4, 0, 4],\n [-2, -2, 4, -2, -6, -2, 4, -2]], determine the sum of all entries.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
386675826
|
OMEGA_500_SAMPLE
|
omega_500_299
|
arithmetic_matrix_eigenvalues_level_6_0090
|
arithmetic_matrix_eigenvalues
| 6 |
arithmetic_matrix_eigenvalues
|
6
|
[
{
"role": "user",
"content": "Calculate the largest eigenvalue by absolute value for the matrix:\n[[12, 10, 10, 8, 8, 6, 26, 4, 10, 24],\n [10, 2, -13, -10, -6, 16, -14, -16, -14, -1],\n [10, -13, -4, 20, 5, 6, 14, -6, 0, 15],\n [8, -10, 20, -25, 5, -6, -18, 23, 3, -16],\n [8, -6, 5, 5, -13, 4, -23, -6, -6, 0],\n [6, 16, 6, -6, 4, -13, -16, -2, -1, -4],\n [26, -14, 14, -18, -23, -16, 25, 1, 20, -14],\n [4, -16, -6, 23, -6, -2, 1, 18, 18, 13],\n [10, -14, 0, 3, -6, -1, 20, 18, -19, -6],\n [24, -1, 15, -16, 0, -4, -14, 13, -6, -30]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] |
80
|
OMEGA_500_SAMPLE
|
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