sunyiyou/omega-500 · Datasets at Hugging Face

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_400	arithmetic_gcd_level_3_0041	arithmetic_gcd	3	arithmetic_gcd	3	[ { "role": "user", "content": "What is the greatest common factor of 46346951 and 63820609435667?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	140021	OMEGA_500_SAMPLE
omega_500_401	combinatory_probability_exactly_n_specific_fixed_level_3_0094	combinatory_probability_exactly_n_specific_fixed	3	combinatory_probability_exactly_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 {a: 4, e: 2, c: 2}, shuffle the letters in the word, what is the probability of exact 3 letter 'a' 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}" } ]	37	OMEGA_500_SAMPLE
omega_500_402	arithmetic_matrix_power_level_3_0069	arithmetic_matrix_power	3	arithmetic_matrix_power	3	[ { "role": "user", "content": "Calculate the matrix power A^5 and find the sum of all entries of the result:\nA =\n[[0, 1, -1, 1, 0],\n [1, 2, 0, -5, 1],\n [-1, 0, -4, 0, 1],\n [1, -5, 0, 3, 2],\n [0, 1, 1, 2, 3]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	2895	OMEGA_500_SAMPLE
omega_500_403	arithmetic_matrix_multiplication_level_1_0049	arithmetic_matrix_multiplication	1	arithmetic_matrix_multiplication	1	[ { "role": "user", "content": "Find the value at position (4, 4) of the product of the following matrices:\nA =\n[[8, -6, 3, -3],\n [2, -6, -1, 5],\n [5, 1, -7, 5],\n [0, -6, -1, 7]]\nB =\n[[-7, -3, 3, -4],\n [-7, 7, -4, 6],\n [3, -5, 6, 2],\n [7, -3, 5, 3]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	-17	OMEGA_500_SAMPLE
omega_500_404	number_theory_digit_sum_level_5_0041	number_theory_digit_sum	5	number_theory_digit_sum	5	[ { "role": "user", "content": "let N be the maximum 6-digit positive integer with the property that whenever one of its digits changes to 1, the resulting number is a multiple of 11. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	11112	OMEGA_500_SAMPLE
omega_500_405	arithmetic_matrix_multiplication_level_6_0088	arithmetic_matrix_multiplication	6	arithmetic_matrix_multiplication	6	[ { "role": "user", "content": "Calculate the matrix product AB and find the value at position (7, 8) of the result:\nA =\n[[24, -2, -24, 9, 2, -7, 7, -5, -16, 20, 19],\n [-17, 12, -20, 25, 16, 9, -8, -7, -21, -1, -15],\n [-16, -9, 10, 0, 5, 6, -5, -8, -8, -25, -12],\n [-18, -24, 25, -11, 8, 10, -25, 25, -5, 16, -25],\n [6, -6, 21, 10, 13, 4, -19, -4, 24, 19, -16],\n [-18, 1, -16, -12, 20, -15, -20, -5, 7, -7, 13],\n [-2, -8, 9, 17, 11, -7, -2, 8, -4, -22, -16],\n [-24, -18, 7, 12, 17, 2, -16, 19, -25, 8, -23],\n [-17, -21, 16, -16, -14, -8, -5, -5, -25, -23, -21],\n [-1, 23, -18, -9, -17, 6, -20, 23, 14, -5, -19]]\nB =\n[[-19, 21, 13, 1, -18, -17, 2, -11, -24, -19],\n [21, -8, -4, 24, -16, 24, 7, -19, -23, -12],\n [18, 20, 10, -13, -10, -10, -21, -1, 12, 12],\n [24, 12, 13, 22, 15, 0, -13, 18, 25, 0],\n [-25, -18, -22, -5, -4, 25, 11, 23, -23, -7],\n [-25, 3, -3, 7, 15, -3, -21, -19, -20, 23],\n [-4, 23, -13, -14, -22, -18, 11, 5, 11, 19],\n [0, 17, -18, 14, 7, 19, 3, -21, 2, 14],\n [-9, -3, 19, -17, -24, 6, 25, 23, 5, -19],\n [4, -23, 25, 25, -17, -16, 20, 24, 21, 0],\n [-25, 4, 12, -11, -1, -20, -8, -25, -23, -9]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	459	OMEGA_500_SAMPLE
omega_500_406	combinatory_probability_no_fixed_points_level_1_0043	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 {a: 1, z: 1, t: 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}" } ]	17	OMEGA_500_SAMPLE
omega_500_407	arithmetic_matrix_power_level_6_0042	arithmetic_matrix_power	6	arithmetic_matrix_power	6	[ { "role": "user", "content": "Find the sum of all entries of the 7th power of the matrix:\n[[-1, -2, -14, -7, -3, 4, -3, -7, -7, 14],\n [6, -2, -3, 14, 1, -14, -2, 11, -8, 13],\n [11, -10, -7, 10, 12, -1, 8, -2, 0, 8],\n [2, 0, 5, 3, 3, -13, -11, -14, -11, 3],\n [-10, 4, 0, 13, -7, 5, 2, -14, -3, 11],\n [4, 0, -6, -8, -13, 14, -6, -5, -13, 4],\n [-4, -2, -5, -1, 3, 4, 7, -6, 10, -8],\n [13, -12, 5, -9, 12, 6, -13, 14, -9, -12],\n [1, -7, -13, 2, -6, -3, -12, -4, -14, 8],\n [-2, 0, 11, -10, 6, 2, 10, 8, 9, 10]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	377808268	OMEGA_500_SAMPLE
omega_500_408	geometry_polygon_chords_level_5_0029	geometry_polygon_chords	5	geometry_polygon_chords	5	[ { "role": "user", "content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 1, from vertex 7 to vertex 2, from vertex 5 to vertex 3, from vertex 12 to vertex 2, from vertex 6 to vertex 3, from vertex 2 to vertex 6, from vertex 3 to vertex 7, from vertex 7 to vertex 11, from vertex 3 to vertex 1, from vertex 8 to vertex 4, from vertex 6 to vertex 10, from vertex 11 to vertex 3, from vertex 2 to vertex 4, and from vertex 11 to vertex 1. 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}" } ]	10	OMEGA_500_SAMPLE
omega_500_409	algebra_func_intersection_coords_level_3_0017	algebra_func_intersection_coords	3	algebra_func_intersection_coords	3	[ { "role": "user", "content": "Find the integer value closest to the x-coordinate where f(x) = 3(3(-2((-2x - 3)/(-3x - 1)) - 3) + 4) + 1 and g(x) = 3cos(2π(2\|(-4x^2 - 5x - 2) + 3\| - 3) + 1) + 1 intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	-1	OMEGA_500_SAMPLE
omega_500_410	combinatory_pattern_matching_level_1_0068	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 {j: 1, b: 2, l: 1}, what is the expected number of times the pattern 'lb+' appears? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	0.17	OMEGA_500_SAMPLE
omega_500_411	geometry_polygon_chords_level_5_0089	geometry_polygon_chords	5	geometry_polygon_chords	5	[ { "role": "user", "content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 4, from vertex 3 to vertex 6, from vertex 2 to vertex 4, from vertex 2 to vertex 5, from vertex 1 to vertex 3, from vertex 8 to vertex 6, from vertex 7 to vertex 4, from vertex 7 to vertex 5, from vertex 8 to vertex 3, from vertex 3 to vertex 5, from vertex 8 to vertex 5, from vertex 4 to vertex 1, from vertex 2 to vertex 6, from vertex 7 to vertex 2, and from vertex 2 to vertex 8. 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}" } ]	14	OMEGA_500_SAMPLE
omega_500_412	combinatory_probability_at_least_n_specific_fixed_level_3_0038	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 {e: 4, w: 3, f: 1}, shuffle the letters in the word, what is the probability of at least 3 letter 'e' 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}" } ]	15	OMEGA_500_SAMPLE
omega_500_413	arithmetic_matrix_power_level_3_0079	arithmetic_matrix_power	3	arithmetic_matrix_power	3	[ { "role": "user", "content": "For the matrix power A^5 where A =\n[[1, 2, 0, -1, 0],\n [-4, -5, 1, 1, 4],\n [5, 4, -4, -4, -5],\n [4, 2, 2, -5, 5],\n [2, 3, -2, -5, -5]], determine the sum of all entries.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	13843	OMEGA_500_SAMPLE
omega_500_414	arithmetic_matrix_svd_level_4_0012	arithmetic_matrix_svd	4	arithmetic_matrix_svd	4	[ { "role": "user", "content": "What is the rounded difference between the largest and smallest singular values in the SVD of the following matrix?\n[[12, -10, -4, 4, 4, 7, 3],\n [12, 9, 0, 6, 4, -9, -1],\n [-4, 6, -1, -4, -6, 1, 7],\n [6, 2, 3, 8, -8, -10, -1],\n [12, 7, 8, -9, 8, 10, 3],\n [12, -6, 0, 8, -3, -6, 1],\n [-8, -10, 10, -2, -2, 5, 2]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	28	OMEGA_500_SAMPLE
omega_500_415	number_theory_digit_sum_level_4_0018	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 9. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	99918	OMEGA_500_SAMPLE
omega_500_416	arithmetic_matrix_multiplication_level_5_0036	arithmetic_matrix_multiplication	5	arithmetic_matrix_multiplication	5	[ { "role": "user", "content": "Compute the sum of all entries of the product matrix AB given:\nA =\n[[1, -11, -15, -4, 17, 3, -11, -13, 0],\n [-15, 0, 18, 3, 12, -3, 7, 11, -9],\n [6, -11, 11, 5, 4, -10, 2, 13, 19],\n [2, 8, 10, -14, 19, 12, 0, -19, -3],\n [12, 16, 13, -1, 14, -3, 3, -13, 9],\n [-11, 2, -10, -19, 5, 15, 12, -8, 19],\n [4, -17, -2, -3, 16, 9, -2, 2, -5],\n [-20, -9, 3, 13, -13, -3, -16, 5, 9],\n [-8, -2, 8, -18, 5, 19, -9, -19, -10]]\nB =\n[[7, -3, -8, 16, 1, -9, -19, 16, -14, 5],\n [5, 3, -9, -5, 2, 19, -12, 11, -6, 16],\n [-8, -16, -10, -11, -8, 20, -6, -14, 19, 7],\n [11, 14, -6, 15, 0, -15, -20, -17, -10, 5],\n [16, 2, 8, 0, 6, 18, 19, -8, 2, -5],\n [9, -17, 19, -5, 10, 4, -9, 5, 16, 12],\n [7, -12, -9, -16, 7, 14, -16, 13, 13, -8],\n [18, -7, -10, 17, -17, 12, -14, 20, -13, 16],\n [-11, -6, 17, -6, -16, 8, -2, -11, -9, -17]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	3515	OMEGA_500_SAMPLE
omega_500_417	combinatory_probability_no_fixed_points_level_4_0044	combinatory_probability_no_fixed_points	4	combinatory_probability_no_fixed_points	4	[ { "role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {v: 2, o: 3, q: 2, m: 3}, 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}" } ]	1501	OMEGA_500_SAMPLE
omega_500_418	geometry_polygon_chords_level_5_0078	geometry_polygon_chords	5	geometry_polygon_chords	5	[ { "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 10, from vertex 5 to vertex 8, from vertex 1 to vertex 5, from vertex 9 to vertex 2, from vertex 2 to vertex 8, from vertex 8 to vertex 6, from vertex 5 to vertex 9, from vertex 2 to vertex 10, from vertex 6 to vertex 2, from vertex 4 to vertex 1, from vertex 3 to vertex 8, from vertex 10 to vertex 5, from vertex 7 to vertex 9, and from vertex 8 to vertex 1. 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}" } ]	11	OMEGA_500_SAMPLE
omega_500_419	geometry_rotation_level_5_0055	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 2 to vertex 6, from vertex 2 to vertex 4, from vertex 2 to vertex 5, from vertex 1 to vertex 4, from vertex 1 to vertex 5, 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 with 3 sides?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	18	OMEGA_500_SAMPLE
omega_500_420	algebra_linear_equation_level_2_0016	algebra_linear_equation	2	algebra_linear_equation	2	[ { "role": "user", "content": "Suppose -1050 = 101a - 96a - 55a. Solve 3d - c + 2h = -0c + 17, -34 = -3d - 4c - h, -4c = 4d - 5*h - a for d.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	5	OMEGA_500_SAMPLE
omega_500_421	logic_puzzles_blocked_grid_level_1_0023	logic_puzzles_blocked_grid	1	logic_puzzles_blocked_grid	1	[ { "role": "user", "content": "In a 9x9 grid, how many different paths are there from the bottom left (0, 0) to the top right (8, 8), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (7, 8), (1, 5), (0, 6), (4, 6), (5, 0), (2, 7), (6, 7), (3, 0), (8, 2), (8, 6)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	1285	OMEGA_500_SAMPLE
omega_500_422	arithmetic_mixed_level_2_0046	arithmetic_mixed	2	arithmetic_mixed	2	[ { "role": "user", "content": "What is (-4 - -28) + (-3 - 39) + (141 - (-320 + 1371))? 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}" } ]	-928	OMEGA_500_SAMPLE
omega_500_423	algebra_func_zeros_level_4_0024	algebra_func_zeros	4	algebra_func_zeros	4	[ { "role": "user", "content": "How many solutions does the equation 1\|(4(4(1\|(2\|x - 3\|) - 2\| - 1) + 1) + 1) + 3\| - 1 = 0 have in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	8	OMEGA_500_SAMPLE
omega_500_424	arithmetic_list_prime_factors_level_4_0022	arithmetic_list_prime_factors	4	arithmetic_list_prime_factors	4	[ { "role": "user", "content": "What is the second-largest prime factor of 4086778025543372?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	601813	OMEGA_500_SAMPLE
omega_500_425	arithmetic_matrix_eigenvalues_level_1_0082	arithmetic_matrix_eigenvalues	1	arithmetic_matrix_eigenvalues	1	[ { "role": "user", "content": "What is the sum of absolute values of all eigenvalues for the following matrix?\n[[-7, -6, 0],\n [-6, -8, 8],\n [0, 8, 10]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	32	OMEGA_500_SAMPLE
omega_500_426	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_427	combinatory_pattern_matching_level_1_0076	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 {j: 1, h: 1, s: 1, w: 1}, what is the expected number of times the pattern 'w+s' appears? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	0.08	OMEGA_500_SAMPLE
omega_500_428	algebra_func_extrema_coords_level_2_0073	algebra_func_extrema_coords	2	algebra_func_extrema_coords	2	[ { "role": "user", "content": "How many local minima does the function f(x) = (-2((-1(-1x^2 - 1x) - 3)/(-2(-1x^2 - 1x) + 3)) + 1)/(2((-1(-1x^2 - 1x) - 3)/(-2(-1x^2 - 1x) + 3))) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	0	OMEGA_500_SAMPLE
omega_500_429	geometry_polygon_color_level_3_0030	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 red, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is red, vertex 10 is red. The 11-gon is then rotated clockwise by 4 positions, so that the vertex that was originally in position 0 is now in position 4. 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_430	combinatory_probability_at_least_n_specific_fixed_level_2_0097	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 {v: 1, t: 3, x: 2}, 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}" } ]	7	OMEGA_500_SAMPLE
omega_500_431	combinatory_pattern_matching_level_2_0024	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 {i: 1, r: 1, c: 2, w: 1, l: 1}, what is the expected number of times the pattern 'i+w' appears? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	0.07	OMEGA_500_SAMPLE
omega_500_432	number_theory_triple_count_level_2_0072	number_theory_triple_count	2	number_theory_triple_count	2	[ { "role": "user", "content": "Let N denote the number of ordered pairs of positive integers (a, b) such that a, b <= 4^3 and (4)a^4 + (4)b^4 is divisible by 4^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	24	OMEGA_500_SAMPLE
omega_500_433	arithmetic_gcd_level_2_0078	arithmetic_gcd	2	arithmetic_gcd	2	[ { "role": "user", "content": "Calculate the greatest common factor of 1902 and 1164257312.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	634	OMEGA_500_SAMPLE
omega_500_434	combinatory_probability_no_specific_letter_fixed_level_4_0075	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 {a: 3, h: 3, w: 1, r: 3}, 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_435	geometry_polygon_color_level_2_0078	geometry_polygon_color	2	geometry_polygon_color	2	[ { "role": "user", "content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is uncolored, vertex 7 is red, vertex 8 is red. How many valid colorings of the uncolored vertices exist such that the 9-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_436	number_theory_prime_mod_level_1_0046	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_437	algebra_func_extrema_coords_level_1_0060	algebra_func_extrema_coords	1	algebra_func_extrema_coords	1	[ { "role": "user", "content": "Find the x-coordinate of the local maximum of the function f(x) = -2\|(3x^2 + 3x + 2) - 1\| - 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	0	OMEGA_500_SAMPLE
omega_500_438	number_theory_digit_sum_level_4_0076	number_theory_digit_sum	4	number_theory_digit_sum	4	[ { "role": "user", "content": "let N be the maximum 5-digit positive integer with the property that whenever one of its digits changes to 8, the resulting number is a multiple of 3. 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}" } ]	7784	OMEGA_500_SAMPLE
omega_500_439	arithmetic_list_prime_factors_level_5_0067	arithmetic_list_prime_factors	5	arithmetic_list_prime_factors	5	[ { "role": "user", "content": "Find the second-largest prime factor of 140068604170336290418.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	362707	OMEGA_500_SAMPLE
omega_500_440	arithmetic_matrix_svd_level_7_0013	arithmetic_matrix_svd	7	arithmetic_matrix_svd	7	[ { "role": "user", "content": "Calculate the rounded largest singular value from the SVD of:\n[[-11, 13, 1, 6, 21, 14, -14, 16, 18, -2, 26, 15],\n [9, 0, 11, -20, -22, 16, 18, 24, 4, -4, 15, -17],\n [5, 16, 26, 3, 12, 4, -9, 9, -25, -13, -14, 2],\n [-15, 12, -25, 5, -14, 19, 0, 22, 14, -10, 13, 4],\n [7, -3, 18, 8, 21, -21, -8, 3, -20, -17, 14, 5],\n [7, 1, -12, 3, 10, 2, -5, -23, -7, 5, 9, 23],\n [-23, 0, -10, -15, -5, -10, -17, -26, -5, -1, -9, 23],\n [-13, 20, 2, -15, 9, 8, 25, -15, -20, 2, -16, 6],\n [-21, 20, -12, 5, 24, -16, -4, -1, 7, 19, 7, 2],\n [-23, -10, 8, -14, 14, 12, -12, 2, 2, -16, 25, 5],\n [-25, 20, -13, -7, -5, -16, 0, -24, -10, 17, 1, 2],\n [-6, 20, -11, 10, -7, -21, -8, -22, -24, -26, 11, -12]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	90	OMEGA_500_SAMPLE
omega_500_441	arithmetic_matrix_determinant_level_3_0092	arithmetic_matrix_determinant	3	arithmetic_matrix_determinant	3	[ { "role": "user", "content": "Determine the value of det(A) where A =\n[[10, -1, -9, 3, 4, 6],\n [14, -5, 7, -2, -3, 4],\n [10, 11, 9, -7, -7, 0],\n [-4, -6, -10, 2, -14, 11],\n [4, -7, 3, -10, -14, -1],\n [1, 0, 12, 5, -8, -6]]\nRound to the nearest integer, then give your answer as det(A) mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	636	OMEGA_500_SAMPLE
omega_500_442	logic_puzzles_zebralogic_level_1_0028	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 - Each person has an occupation: `engineer`, `doctor`\n\n## Clues:\n1. Eric is directly left of Arnold.\n2. The person who is a doctor is not in the first house.\n\n\nChoices:\n0. doctor\n1. engineer\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_443	geometry_polygon_color_level_3_0051	geometry_polygon_color	3	geometry_polygon_color	3	[ { "role": "user", "content": "A 10-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 uncolored, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is red, vertex 7 is uncolored, vertex 8 is uncolored, vertex 9 is red. How many valid colorings of the uncolored vertices exist such that the 10-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_444	number_theory_triple_count_level_5_0096	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 <= 4^3, 2. a, b, c are relatively coprime, and 3. (1)a^4 + (1)b^4 + (-1)c^4 is divisible by 4^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	434	OMEGA_500_SAMPLE
omega_500_445	number_theory_prime_mod_level_5_0034	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^4) 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_446	geometry_rotation_level_5_0013	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 2 to vertex 5, from vertex 4 to vertex 2, from vertex 3 to vertex 5, from vertex 1 to vertex 5, from vertex 4 to vertex 6, and from vertex 4 to vertex 1. 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 with 3 sides?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	18	OMEGA_500_SAMPLE
omega_500_447	geometry_polygon_color_level_4_0047	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 blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, 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_448	logic_puzzles_grid_knight_level_3_0091	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 29x27 grid, what is minimum number of steps you can take to get to (26, 23), if your only valid moves are to move over 6 squares in one direction and over 1 squares in a perpendicular direction (at once)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	9	OMEGA_500_SAMPLE
omega_500_449	logic_puzzles_grid_chip_level_5_0077	logic_puzzles_grid_chip	5	logic_puzzles_grid_chip	5	[ { "role": "user", "content": "Chips, colored either black or white, are placed in a 8x8 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 (0, 7) is black, cell (4, 7) is black, cell (6, 3) is black, cell (2, 1) is black, cell (6, 5) is black, cell (7, 2) is black, cell (1, 4) 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}" } ]	13	OMEGA_500_SAMPLE
omega_500_450	arithmetic_list_prime_factors_level_3_0076	arithmetic_list_prime_factors	3	arithmetic_list_prime_factors	3	[ { "role": "user", "content": "Find the second-largest prime factor of 167604909357658.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	6619	OMEGA_500_SAMPLE
omega_500_451	arithmetic_gcd_level_5_0049	arithmetic_gcd	5	arithmetic_gcd	5	[ { "role": "user", "content": "Calculate the highest common factor of 484131285 and 977648343511861119594815.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	53792365	OMEGA_500_SAMPLE
omega_500_452	arithmetic_matrix_inverse_level_3_0026	arithmetic_matrix_inverse	3	arithmetic_matrix_inverse	3	[ { "role": "user", "content": "Find the inverse of 1/180 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, -8, -14, 12, 15, -8],\n [-9, 10, 10, -14, 6, 6],\n [-13, 11, 2, -4, -15, 6],\n [-4, -11, 14, 14, 11, 9],\n [1, 0, 15, -1, 7, -1],\n [-11, 5, -2, -14, -5, 3]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	246	OMEGA_500_SAMPLE
omega_500_453	geometry_polygon_chords_level_1_0066	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 1 to vertex 5, from vertex 4 to vertex 7, from vertex 1 to vertex 7, from vertex 5 to vertex 7, from vertex 8 to vertex 6, and from vertex 7 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}" } ]	3	OMEGA_500_SAMPLE
omega_500_454	geometry_polygon_chords_level_1_0078	geometry_polygon_chords	1	geometry_polygon_chords	1	[ { "role": "user", "content": "Suppose you have a 14-gon, with vertices numbered 1 through 14 in counterclockwise order. Draw the diagonal from vertex 11 to vertex 9, from vertex 6 to vertex 12, from vertex 10 to vertex 1, from vertex 8 to vertex 1, from vertex 14 to vertex 2, from vertex 7 to vertex 13, and from vertex 14 to vertex 7. 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}" } ]	2	OMEGA_500_SAMPLE
omega_500_455	logic_puzzles_grid_chip_level_2_0092	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 (3, 4) is white, cell (1, 0) is white, cell (0, 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_456	combinatory_probability_at_least_n_specific_fixed_level_3_0030	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 {b: 2, t: 3, u: 3}, shuffle the letters in the word, what is the probability of at least 2 letter 'b' 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_457	algebra_func_extrema_level_5_0095	algebra_func_extrema	5	algebra_func_extrema	5	[ { "role": "user", "content": "How many local extrema (minima and maxima combined) does the function f(x) = (1(3cos(1π(-3(-4(2\|(-5x - 3) + 3\| - 1)^2 + 2(2\|(-5x - 3) + 3\| - 1) + 4)^2 + 4(-4(2\|(-5x - 3) + 3\| - 1)^2 + 2(2\|(-5x - 3) + 3\| - 1) + 4) - 4) - 3)) + 3)/(1(3cos(1π(-3(-4(2\|(-5x - 3) + 3\| - 1)^2 + 2(2\|(-5x - 3) + 3\| - 1) + 4)^2 + 4(-4(2\|(-5x - 3) + 3\| - 1)^2 + 2(2\|(-5x - 3) + 3\| - 1) + 4) - 4) - 3)) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	17	OMEGA_500_SAMPLE
omega_500_458	arithmetic_matrix_determinant_level_7_0050	arithmetic_matrix_determinant	7	arithmetic_matrix_determinant	7	[ { "role": "user", "content": "What is the determinant of the matrix:\n[[2, 11, 18, 19, 12, -21, 35, -20, 7, -29, 7, -8],\n [30, 24, -33, -35, -5, 20, 10, -4, -18, 32, -1, -13],\n [-23, -22, 1, -31, -14, -8, -32, 14, -7, 1, -1, -16],\n [-1, -34, 22, -18, 20, 28, 16, -14, 24, -26, -30, -30],\n [-8, 6, 5, 27, -14, -7, 35, -2, -34, -10, -17, -16],\n [-3, -32, -4, -17, -33, 25, 2, -30, 20, 4, 33, 6],\n [-34, -8, -19, 34, 33, 3, -29, -4, -32, -15, -4, -22],\n [34, 10, -16, 22, 12, -29, 4, 23, 0, 8, -5, 6],\n [8, 7, 7, 11, -30, 30, -27, -25, -18, -11, 22, -6],\n [-35, 27, -7, 30, 17, -17, 30, 16, -5, 24, 32, -6],\n [-23, -21, -16, -2, -16, 14, -9, -6, 20, 29, 19, -12],\n [-11, 17, -25, 24, 11, -11, -11, -19, -24, 16, -25, -26]]? Round to the nearest integer, then express your answer modulo 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	408	OMEGA_500_SAMPLE
omega_500_459	logic_puzzles_grid_rook_level_5_0068	logic_puzzles_grid_rook	5	logic_puzzles_grid_rook	5	[ { "role": "user", "content": "In a 40x43 grid, what is the fewest number of steps it takes to get from the bottom left (0, 0) to the top right (39, 42), 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: (32, 27), (35, 19), (27, 21), (1, 27), (26, 23), (0, 18), (15, 26), (29, 2), (7, 22), (32, 7), (25, 7), (22, 13), (20, 13), (28, 13), (7, 38), (14, 22), (3, 0), (13, 6), (35, 32), (39, 3), (26, 20), (15, 25), (31, 9), (12, 11), (14, 34), (34, 0), (21, 23), (12, 36), (13, 24), (12, 39), (38, 25), (37, 10), (13, 17), (21, 42), (37, 7), (18, 7), (12, 1), (14, 39), (30, 2), (3, 42), (29, 31), (17, 5), (16, 33), (36, 32), (15, 23), (12, 15), (17, 8), (4, 37), (3, 38), (2, 9), (8, 16), (10, 38), (33, 6), (36, 4), (16, 35), (11, 21), (10, 18), (33, 11), (10, 1), (25, 29), (23, 32), (20, 31), (33, 18), (25, 9), (6, 5), (13, 35), (4, 39), (6, 36), (0, 40), (19, 34), (3, 4), (30, 40), (25, 19), (11, 1), (27, 1), (6, 41), (21, 1), (0, 21), (7, 3), (39, 27), (34, 26), (29, 24), (28, 23), (27, 17), (38, 13), (31, 34), (37, 2), (1, 23), (19, 33), (36, 1), (4, 15), (24, 38), (37, 40), (26, 10), (29, 17), (33, 22), (37, 42), (2, 21), (38, 5), (7, 32), (31, 32), (17, 0), (26, 41), (10, 30), (19, 22), (38, 31), (17, 24), (2, 18), (26, 18), (29, 5), (3, 9), (16, 41), (17, 41), (9, 15), (39, 26), (8, 21), (15, 10), (13, 33), (0, 36), (9, 7), (5, 35), (35, 8), (26, 26), (11, 2), (12, 40), (16, 27), (6, 12), (11, 6), (37, 20), (2, 29), (0, 27), (9, 32), (20, 18), (30, 14), (21, 8), (34, 27), (28, 15), (11, 26), (36, 22), (23, 8), (19, 23), (36, 41), (33, 38), (17, 6), (7, 23), (23, 14), (6, 22), (36, 40), (13, 7), (22, 26), (14, 11), (20, 36), (7, 13), (15, 4), (33, 27), (14, 26), (25, 36), (11, 39), (1, 41), (15, 19), (34, 9), (17, 25), (27, 2), (17, 36), (15, 3), (1, 7), (38, 28), (0, 38), (31, 20), (32, 26), (34, 29), (22, 22), (18, 2), (30, 41), (6, 31), (7, 15), (19, 35), (8, 39), (2, 10), (10, 33), (12, 34), (12, 41), (32, 11), (32, 5), (34, 32), (20, 15), (15, 1), (0, 25), (34, 3), (15, 2), (37, 12), (33, 36), (4, 36), (1, 42), (17, 34), (18, 4), (6, 0), (36, 16), (12, 10), (20, 33), (15, 11), (21, 19), (3, 35), (34, 21), (37, 36), (25, 17), (39, 17), (15, 31), (35, 41), (19, 2), (6, 3), (27, 31), (33, 32), (16, 8), (5, 40), (19, 27), (8, 34), (24, 1), (39, 1), (10, 39), (27, 3), (19, 30), (39, 24), (6, 10), (2, 6), (26, 8), (35, 25), (16, 7), (10, 19), (30, 0), (1, 20), (33, 24), (11, 0), (1, 36), (26, 1), (24, 30), (9, 6), (23, 33), (18, 28), (21, 34), (28, 40), (9, 33), (22, 16), (23, 16), (16, 0), (5, 4), (1, 22), (20, 9), (5, 28), (29, 12), (39, 29), (36, 5), (1, 18), (3, 20), (24, 37), (33, 3), (3, 24), (26, 30), (4, 32), (39, 35), (18, 24), (30, 24), (29, 1), (39, 13), (20, 24)? If it's impossible, answer with -1.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	7	OMEGA_500_SAMPLE
omega_500_460	logic_puzzles_grid_knight_level_1_0089	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 9x7 grid, what is minimum number of steps you can take to get to (5, 3), if your only valid moves are to move over 3 squares in one direction and over 1 squares in a perpendicular direction (at once)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	3	OMEGA_500_SAMPLE
omega_500_461	combinatory_probability_at_least_n_specific_fixed_level_4_0070	combinatory_probability_at_least_n_specific_fixed	4	combinatory_probability_at_least_n_specific_fixed	4	[ { "role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {a: 2, l: 3, o: 4, i: 1}, shuffle the letters in the word, what is the probability of at least 2 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}" } ]	13	OMEGA_500_SAMPLE
omega_500_462	arithmetic_gcd_level_4_0073	arithmetic_gcd	4	arithmetic_gcd	4	[ { "role": "user", "content": "What is the highest common divisor of 38663909736056969524 and 80928908?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	4760524	OMEGA_500_SAMPLE
omega_500_463	logic_puzzles_grid_rook_level_3_0050	logic_puzzles_grid_rook	3	logic_puzzles_grid_rook	3	[ { "role": "user", "content": "In a 25x26 grid, what is the fewest number of steps it takes to get from the bottom left (0, 0) to the top right (24, 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: (7, 22), (1, 19), (9, 2), (8, 17), (18, 3), (4, 20), (2, 8), (18, 24), (6, 9), (18, 23), (4, 7), (19, 0), (10, 5), (22, 13), (12, 6), (12, 7), (21, 21), (9, 4), (10, 6), (8, 14), (24, 19), (7, 14), (6, 14), (8, 7), (7, 12), (3, 19), (2, 21), (0, 11), (3, 9), (16, 10), (23, 16), (10, 13), (13, 15), (22, 22), (12, 10), (15, 12), (22, 5), (10, 7), (6, 15), (1, 11), (3, 11), (17, 3), (13, 4), (21, 11), (3, 12), (23, 6), (13, 18), (7, 25), (3, 18), (0, 14), (5, 23), (8, 16), (0, 10), (15, 11), (13, 13), (22, 9), (14, 13), (14, 10), (19, 9), (18, 11), (23, 8), (15, 23), (10, 11), (1, 14), (5, 20), (20, 5), (6, 3), (2, 22), (0, 6), (14, 24), (9, 0), (18, 9), (2, 12), (20, 11), (9, 25), (6, 8), (21, 23), (10, 22), (1, 7), (24, 23), (20, 3), (23, 22), (13, 5), (16, 23), (5, 0), (9, 19), (2, 9), (1, 22), (18, 18), (7, 11), (23, 20), (20, 10), (12, 17), (16, 7), (6, 24), (0, 13), (4, 10), (21, 12), (19, 1), (6, 22), (24, 22), (24, 5), (15, 5), (9, 15), (5, 2), (19, 22), (8, 12), (4, 5), (17, 11), (5, 19), (1, 15), (0, 9), (6, 23), (23, 24), (11, 9), (2, 3), (24, 14), (14, 7), (20, 23), (15, 1), (4, 2), (10, 8), (6, 6), (5, 18)? If it's impossible, answer with -1.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	6	OMEGA_500_SAMPLE
omega_500_464	algebra_func_area_level_3_0086	algebra_func_area	3	algebra_func_area	3	[ { "role": "user", "content": "Calculate the area enclosed between the functions f(x) = -3\|(5(1(-3sin(3πx + 1) - 1)^2 + 1(-3sin(3πx + 1) - 1) - 5) + 5) + 1\| + 1 and g(x) = -3\|((2(3\|x - 3\| - 1) - 2)/(-2(3\|x - 3\| - 1) + 1))\| + 2 from x = -1.8 to x = -1.7. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	5.0	OMEGA_500_SAMPLE
omega_500_465	combinatory_pattern_matching_level_2_0061	combinatory_pattern_matching	2	combinatory_pattern_matching	2	[ { "role": "user", "content": "When randomly selecting 3 letters from the multiset {p: 1, j: 2, z: 3} to form a word, what is the expected number of matches of the pattern 'pz+'? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	0.25	OMEGA_500_SAMPLE
omega_500_466	algebra_func_area_level_5_0051	algebra_func_area	5	algebra_func_area	5	[ { "role": "user", "content": "Determine the area of the region enclosed by the graphs of f(x) = (-1(-1((-1((-3(2\|(3\|x - 1\| - 1) + 2\| - 3))/(-1(2\|(3\|x - 1\| - 1) + 2\| - 3) + 2)) + 3)/(2((-3(2\|(3\|x - 1\| - 1) + 2\| - 3))/(-1(2\|(3\|x - 1\| - 1) + 2\| - 3) + 2))))^2 + 4((-1((-3(2\|(3\|x - 1\| - 1) + 2\| - 3))/(-1(2\|(3\|x - 1\| - 1) + 2\| - 3) + 2)) + 3)/(2((-3(2\|(3\|x - 1\| - 1) + 2\| - 3))/(-1(2\|(3\|x - 1\| - 1) + 2\| - 3) + 2)))) + 3) - 3)/(-2(-1((-1((-3(2\|(3\|x - 1\| - 1) + 2\| - 3))/(-1(2\|(3\|x - 1\| - 1) + 2\| - 3) + 2)) + 3)/(2((-3(2\|(3\|x - 1\| - 1) + 2\| - 3))/(-1(2\|(3\|x - 1\| - 1) + 2\| - 3) + 2))))^2 + 4((-1((-3(2\|(3\|x - 1\| - 1) + 2\| - 3))/(-1(2\|(3\|x - 1\| - 1) + 2\| - 3) + 2)) + 3)/(2((-3(2\|(3\|x - 1\| - 1) + 2\| - 3))/(-1(2\|(3\|x - 1\| - 1) + 2\| - 3) + 2)))) + 3) + 3) and g(x) = -1cos(3π(5(4(-1\|((3x - 2)/(2x + 1))\| - 2)^2 + 3(-1\|((3x - 2)/(2x + 1))\| - 2))^2 + 5)) - 1 between the vertical lines x = 0.6 and x = 0.7. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	0.1	OMEGA_500_SAMPLE
omega_500_467	arithmetic_mixed_level_1_0015	arithmetic_mixed	1	arithmetic_mixed	1	[ { "role": "user", "content": "Evaluate 64/180(-42)/(-7)(-5)/20. 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}" } ]	-8/15	OMEGA_500_SAMPLE
omega_500_468	arithmetic_matrix_rank_level_7_0018	arithmetic_matrix_rank	7	arithmetic_matrix_rank	7	[ { "role": "user", "content": "Find the rank of the matrix:\n[[885, 812, 154, 26, 195, -335, -87, 517, -315, 104, -514, 167],\n [-1051, -715, -560, -625, -455, 474, -106, -815, 639, 43, 264, -738],\n [-246, -567, 780, 689, 531, 515, 36, -824, -547, 4, 268, 339],\n [311, 592, 125, 365, 353, -772, 92, 1151, -138, 97, 488, 987],\n [-110, -54, -147, -883, -331, -969, -98, 547, 473, 218, -144, -244],\n [0, -110, -262, -396, -354, -928, -42, 926, 210, -446, 224, 42],\n [-285, -62, -729, -179, -493, 397, 106, -29, 463, -194, 49, -422],\n [-21, 597, -855, 178, -6, 407, -414, 283, -211, -833, 626, 74],\n [-572, -1261, -184, 325, -664, 1096, 515, -995, 324, -430, -177, -770],\n [413, -722, -18, 530, -882, -309, 1122, 964, 505, -309, -488, -9],\n [-735, 6, -558, 962, 242, 938, 102, -84, -36, -400, 1287, 698],\n [268, 128, 559, 1213, 577, -79, 358, 609, -533, -116, 616, 1220]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	6	OMEGA_500_SAMPLE
omega_500_469	combinatory_probability_exactly_n_specific_fixed_level_1_0090	combinatory_probability_exactly_n_specific_fixed	1	combinatory_probability_exactly_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 {q: 1, k: 1, b: 2}, shuffle the letters in the word, what is the probability of exact 1 letter 'b' 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}" } ]	4	OMEGA_500_SAMPLE
omega_500_470	algebra_func_intersection_level_5_0004	algebra_func_intersection	5	algebra_func_intersection	5	[ { "role": "user", "content": "How many solutions does the equation (-1(3\|(1sin(1π(3(1\|((2x + 3)/(-1x + 2)) + 3\|) - 2) + 2) + 3) - 2\| + 1))/(2(3\|(1sin(1π(3(1\|((2x + 3)/(-1x + 2)) + 3\|) - 2) + 2) + 3) - 2\| + 1) - 3) = 4((-2(-3(-3cos(1π(2x + 2) - 1) + 1) - 4) - 1)/(1(-3(-3cos(1π(2x + 2) - 1) + 1) - 4) + 2)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	78	OMEGA_500_SAMPLE
omega_500_471	algebra_func_derivative_sign_level_1_0013	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) = 3sin(1π(2x + 4)) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	21	OMEGA_500_SAMPLE
omega_500_472	algebra_func_extrema_level_2_0059	algebra_func_extrema	2	algebra_func_extrema	2	[ { "role": "user", "content": "Determine the count of local maxima for the function f(x) = 2(2\|(3cos(3πx + 3) + 2) + 3\| + 1)^2 - 1(2\|(3cos(3πx + 3) + 2) + 3\| + 1) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	30	OMEGA_500_SAMPLE
omega_500_473	arithmetic_matrix_eigenvalues_level_6_0076	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[[-18, -7, -6, -2, -14, 6, 17, 4, 6, -10],\n [-7, 5, -8, 6, 11, -20, -5, -14, 12, 3],\n [-6, -8, -12, 6, 10, -14, -3, 12, -16, -5],\n [-2, 6, 6, 16, -6, -15, 10, 12, 8, -4],\n [-14, 11, 10, -6, 26, -16, -22, 14, -3, 17],\n [6, -20, -14, -15, -16, 6, 1, 8, -24, 3],\n [17, -5, -3, 10, -22, 1, 0, 3, -28, 5],\n [4, -14, 12, 12, 14, 8, 3, 0, -20, 0],\n [6, 12, -16, 8, -3, -24, -28, -20, 29, 2],\n [-10, 3, -5, -4, 17, 3, 5, 0, 2, -7]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	77	OMEGA_500_SAMPLE
omega_500_474	arithmetic_matrix_inverse_level_1_0013	arithmetic_matrix_inverse	1	arithmetic_matrix_inverse	1	[ { "role": "user", "content": "Find the inverse of 1/60 times the matrix and calculate the value of the entry with the largest absolute value of the inverse. Round your answer to the nearest integer:\n[[4, 0, -7],\n [2, -4, 8],\n [-9, -1, 2]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	-17	OMEGA_500_SAMPLE
omega_500_475	number_theory_triple_count_level_3_0070	number_theory_triple_count	3	number_theory_triple_count	3	[ { "role": "user", "content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and (-4)a^3 + (-4)b^3 + (-2)c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	81	OMEGA_500_SAMPLE
omega_500_476	logic_puzzles_zebralogic_level_4_0078	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 a unique level of education: `associate`, `high school`\n - Everyone has something unique for lunch: `grilled cheese`, `pizza`\n - They all have a unique favorite flower: `daffodils`, `carnations`\n - People have unique favorite music genres: `rock`, `pop`\n\n## Clues:\n1. The person who loves a bouquet of daffodils is not in the first house.\n2. The person who is a pizza lover is the person who loves a bouquet of daffodils.\n3. The person who loves a bouquet of daffodils is Eric.\n4. The person with a high school diploma is the person who loves a bouquet of daffodils.\n5. The person who loves pop music is the person who loves a bouquet of daffodils.\n\n\nChoices:\n0. pizza\n1. grilled cheese\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_477	arithmetic_mixed_level_5_0061	arithmetic_mixed	5	arithmetic_mixed	5	[ { "role": "user", "content": "Let c be -532 + (-28307728)/6494 - 333204/(-68). What is (-5 - -1)/((-191 + 230482/1212)/(345/c))? 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}" } ]	184	OMEGA_500_SAMPLE
omega_500_478	arithmetic_matrix_svd_level_3_0072	arithmetic_matrix_svd	3	arithmetic_matrix_svd	3	[ { "role": "user", "content": "What is the rounded difference between the largest and smallest singular values in the SVD of the following matrix?\n[[3, -1, 5, 5, -1, 2],\n [-5, 1, -2, 2, -7, 0],\n [9, 7, 0, 10, -7, 7],\n [-10, -4, -5, 4, 7, 0],\n [9, 4, 5, -4, 10, -1],\n [1, 3, -6, 10, 7, -6]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	19	OMEGA_500_SAMPLE
omega_500_479	combinatory_distribution_level_1_0089	combinatory_distribution	1	combinatory_distribution	1	[ { "role": "user", "content": "You have {'p': 2, 'u': 2} and need to distribute them into 2 identical boxes with capacities [3, 1]. How many distinct distributions are possible?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	2	OMEGA_500_SAMPLE
omega_500_480	combinatory_probability_no_specific_letter_fixed_level_3_0085	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 {c: 4, s: 1, a: 3}, shuffle the letters in the word, what is the probability of no letter 'a' 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_481	algebra_func_derivative_sign_level_2_0084	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) = -1sin(2π(-3(-3x - 4)^2 + 2(-3x - 4) + 2) - 3) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	23	OMEGA_500_SAMPLE
omega_500_482	arithmetic_matrix_rank_level_7_0025	arithmetic_matrix_rank	7	arithmetic_matrix_rank	7	[ { "role": "user", "content": "Compute the rank of the given 12x12 matrix:\n[[-236, -140, 196, -268, -470, 766, -308, 324, -214, 570, 334, -420],\n [85, -292, 91, -319, 25, -601, 538, -336, -148, -238, -153, 486],\n [-261, -452, 357, -657, -645, 565, 30, 168, -432, 602, 341, -174],\n [237, 302, -273, 465, 540, -616, 108, -222, 321, -557, -320, 264],\n [-229, -786, 497, -1049, -730, 126, 512, -102, -635, 491, 262, 228],\n [103, -46, -35, -13, 160, -436, 268, -210, 23, -259, -156, 288],\n [-66, -368, 210, -474, -270, -98, 324, -120, -276, 128, 62, 204],\n [5, -436, 203, -527, -175, -433, 554, -288, -284, -54, -49, 438],\n [-9, -212, 105, -261, -105, -167, 246, -120, -144, 2, -7, 186],\n [-156, -174, 168, -276, -345, 429, -102, 162, -195, 369, 213, -198],\n [-164, -46, 112, -124, -305, 581, -278, 258, -115, 401, 237, -342],\n [-206, -620, 406, -838, -620, 196, 352, -36, -514, 450, 244, 120]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	2	OMEGA_500_SAMPLE
omega_500_483	geometry_rotation_level_2_0081	geometry_rotation	2	geometry_rotation	2	[ { "role": "user", "content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 1, from vertex 5 to vertex 3, and from vertex 8 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 8-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 3 sides?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	8	OMEGA_500_SAMPLE
omega_500_484	arithmetic_matrix_power_level_1_0082	arithmetic_matrix_power	1	arithmetic_matrix_power	1	[ { "role": "user", "content": "Find the sum of absolute values of all entries of the 2th power of the matrix:\n[[-1, -1, 0],\n [-1, -3, -1],\n [0, -1, -3]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	45	OMEGA_500_SAMPLE
omega_500_485	arithmetic_list_prime_factors_level_1_0021	arithmetic_list_prime_factors	1	arithmetic_list_prime_factors	1	[ { "role": "user", "content": "Find the second-largest prime factor of 20732.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	71	OMEGA_500_SAMPLE
omega_500_486	logic_puzzles_grid_chip_level_3_0060	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 (1, 3) is black, cell (1, 3) is black, cell (3, 4) is black, cell (0, 0) is black, cell (3, 2) 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}" } ]	10	OMEGA_500_SAMPLE
omega_500_487	arithmetic_matrix_determinant_level_4_0053	arithmetic_matrix_determinant	4	arithmetic_matrix_determinant	4	[ { "role": "user", "content": "Find the determinant of the following matrix. Round to the nearest integer, then give your answer modulo 1000:\n[[-3, -8, 9, 3, 8, -2, -11],\n [-11, 4, 13, -4, 8, -15, 13],\n [18, -9, 9, 13, 13, -4, -13],\n [-10, 11, -3, 6, -9, 11, -10],\n [-12, -7, -2, 6, -12, 0, -2],\n [13, 4, 13, 7, -11, 3, -11],\n [-14, -7, 8, 1, -8, 17, -17]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	712	OMEGA_500_SAMPLE
omega_500_488	algebra_linear_equation_level_5_0052	algebra_linear_equation	5	algebra_linear_equation	5	[ { "role": "user", "content": "Let g be 3 - 752/564(-4)/(-3)((-3056)/3840 + 287/1230). Let z = 4 - 4. Suppose 5h - 21 = 3k, z = -h - 0k - k + 1. Solve 2s + u = -5t - 5f - 3r - 5p - 66, -5f + 5p = 2t - 5s + 5u + 33, -5f + 5r - 2p = -2u + 34, 2f + s + 4r - gu = -4t - 8, 5s - 2u + 3u + 6 = -t - 2f + 4r + p, -2f - 3s + 4u = -4t - hr - 3p - 36 for t.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	-4	OMEGA_500_SAMPLE
omega_500_489	arithmetic_matrix_inverse_level_3_0012	arithmetic_matrix_inverse	3	arithmetic_matrix_inverse	3	[ { "role": "user", "content": "Find the trace (sum of diagonal elements) of the inverse of matrix 1/180 * A where A =\n[[-2, 10, 2, 1, -1, 0],\n [10, -8, -2, 12, -6, 5],\n [2, -2, 12, 0, -10, -10],\n [1, 12, 0, 3, 0, -13],\n [-1, -6, -10, 0, 14, 12],\n [0, 5, -10, -13, 12, -7]]\nRound your answer to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	117	OMEGA_500_SAMPLE
omega_500_490	algebra_func_intersection_level_2_0035	algebra_func_intersection	2	algebra_func_intersection	2	[ { "role": "user", "content": "How many solutions does the equation 1\|(1(2\|x - 2\|)) - 3\| + 2 = 3(-4x^2 + 4x + 5)^2 + 1(-4x^2 + 4x + 5) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	4	OMEGA_500_SAMPLE
omega_500_491	number_theory_prime_mod_level_3_0045	number_theory_prime_mod	3	number_theory_prime_mod	3	[ { "role": "user", "content": "Let p be the smallest prime number for which there exists a positive integer n where n^5+(-2) is a multiple of p^3, and n^5+(-2) is also a multiple of p^5. Find the largest positive integer m such that m^5+(-2) is a multiple of p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	23	OMEGA_500_SAMPLE
omega_500_492	algebra_func_derivative_sign_level_3_0055	algebra_func_derivative_sign	3	algebra_func_derivative_sign	3	[ { "role": "user", "content": "For the function f(x) = 4(4(-1(-3cos(3πx) - 3) - 1) - 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_493	algebra_func_extrema_coords_level_5_0066	algebra_func_extrema_coords	5	algebra_func_extrema_coords	5	[ { "role": "user", "content": "Find the x-coordinate of the leftmost local extremum of the function f(x) = -2\|(-2(2\|(2\|(-5((2x - 2)/(3x - 2))^2 - 1((2x - 2)/(3x - 2)) + 2) - 2\| + 1) - 3\| - 2)^2 + 2(2\|(2\|(-5((2x - 2)/(3x - 2))^2 - 1((2x - 2)/(3x - 2)) + 2) - 2\| + 1) - 3\| - 2)) + 1\| - 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_494	arithmetic_matrix_inverse_level_1_0047	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[[3, -9, 8],\n [0, -8, 6],\n [0, 0, -9]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	6	OMEGA_500_SAMPLE
omega_500_495	algebra_func_extrema_coords_level_4_0059	algebra_func_extrema_coords	4	algebra_func_extrema_coords	4	[ { "role": "user", "content": "How many local minima does the function f(x) = -1\|(5(1((-2(-4x^2 + 1x - 5))/(-3(-4x^2 + 1x - 5) + 4))^2 + 4((-2(-4x^2 + 1x - 5))/(-3(-4x^2 + 1x - 5) + 4)) + 1) + 1) + 3\| have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	0	OMEGA_500_SAMPLE
omega_500_496	arithmetic_matrix_inverse_level_3_0015	arithmetic_matrix_inverse	3	arithmetic_matrix_inverse	3	[ { "role": "user", "content": "Calculate the inverse matrix of 1/180 times the following and determine the value at position (3, 5) of the resulting matrix. Your answer should be rounded to the nearest integer:\n[[-14, 7, 15, 6, 3, 9],\n [0, 15, 14, 13, 6, 10],\n [0, 0, 1, 4, -6, -10],\n [0, 0, 0, 12, -11, -15],\n [0, 0, 0, 0, -4, -11],\n [0, 0, 0, 0, 0, 14]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	-105	OMEGA_500_SAMPLE
omega_500_497	algebra_linear_equation_level_4_0084	algebra_linear_equation	4	algebra_linear_equation	4	[ { "role": "user", "content": "Let m be 4((-2)/((-144)/(-204)) + 2 + (-341)/(-132)). Solve -4g - 10q + u = -9q - 2z - 3l + 13, 5q - z - 5l - 42 = 2g + 3u, q - u - 3l - 29 = 4g, -2g + z + 2u + 2l = m, -3q + z + 4l = -3g - 33 for g.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	-5	OMEGA_500_SAMPLE
omega_500_498	number_theory_prime_mod_level_3_0094	number_theory_prime_mod	3	number_theory_prime_mod	3	[ { "role": "user", "content": "let p=19. Let N be the number of solutions m in [1, 2, ..., 19^1] that satisfy m^3+1=0 (mod 19^1). Let M be the minimum of these solutions. What is N+M?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	11	OMEGA_500_SAMPLE
omega_500_499	combinatory_pattern_matching_level_3_0087	combinatory_pattern_matching	3	combinatory_pattern_matching	3	[ { "role": "user", "content": "Form a word by randomly choosing 4 letters from the multiset {h: 1, e: 1, s: 3, u: 1, m: 3}. What is the expected number of occurrences of the pattern 'e.*u' in each word? Round your answer to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]	0.08	OMEGA_500_SAMPLE

omega_500_400

arithmetic_gcd_level_3_0041

arithmetic_gcd

3

arithmetic_gcd

3

[ { "role": "user", "content": "What is the greatest common factor of 46346951 and 63820609435667?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

140021

OMEGA_500_SAMPLE

omega_500_401

combinatory_probability_exactly_n_specific_fixed_level_3_0094

combinatory_probability_exactly_n_specific_fixed

3

combinatory_probability_exactly_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 {a: 4, e: 2, c: 2}, shuffle the letters in the word, what is the probability of exact 3 letter 'a' 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}" } ]

37

OMEGA_500_SAMPLE

omega_500_402

arithmetic_matrix_power_level_3_0069

arithmetic_matrix_power

3

arithmetic_matrix_power

3

[ { "role": "user", "content": "Calculate the matrix power A^5 and find the sum of all entries of the result:\nA =\n[[0, 1, -1, 1, 0],\n [1, 2, 0, -5, 1],\n [-1, 0, -4, 0, 1],\n [1, -5, 0, 3, 2],\n [0, 1, 1, 2, 3]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

2895

OMEGA_500_SAMPLE

omega_500_403

arithmetic_matrix_multiplication_level_1_0049

arithmetic_matrix_multiplication

1

arithmetic_matrix_multiplication

1

[ { "role": "user", "content": "Find the value at position (4, 4) of the product of the following matrices:\nA =\n[[8, -6, 3, -3],\n [2, -6, -1, 5],\n [5, 1, -7, 5],\n [0, -6, -1, 7]]\nB =\n[[-7, -3, 3, -4],\n [-7, 7, -4, 6],\n [3, -5, 6, 2],\n [7, -3, 5, 3]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

-17

OMEGA_500_SAMPLE

omega_500_404

number_theory_digit_sum_level_5_0041

number_theory_digit_sum

5

number_theory_digit_sum

5

[ { "role": "user", "content": "let N be the maximum 6-digit positive integer with the property that whenever one of its digits changes to 1, the resulting number is a multiple of 11. Let Q and R be the quotient and remainder, respectively, when N is divided by 100000. Find Q + R.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

11112

OMEGA_500_SAMPLE

omega_500_405

arithmetic_matrix_multiplication_level_6_0088

arithmetic_matrix_multiplication

6

arithmetic_matrix_multiplication

6

[ { "role": "user", "content": "Calculate the matrix product AB and find the value at position (7, 8) of the result:\nA =\n[[24, -2, -24, 9, 2, -7, 7, -5, -16, 20, 19],\n [-17, 12, -20, 25, 16, 9, -8, -7, -21, -1, -15],\n [-16, -9, 10, 0, 5, 6, -5, -8, -8, -25, -12],\n [-18, -24, 25, -11, 8, 10, -25, 25, -5, 16, -25],\n [6, -6, 21, 10, 13, 4, -19, -4, 24, 19, -16],\n [-18, 1, -16, -12, 20, -15, -20, -5, 7, -7, 13],\n [-2, -8, 9, 17, 11, -7, -2, 8, -4, -22, -16],\n [-24, -18, 7, 12, 17, 2, -16, 19, -25, 8, -23],\n [-17, -21, 16, -16, -14, -8, -5, -5, -25, -23, -21],\n [-1, 23, -18, -9, -17, 6, -20, 23, 14, -5, -19]]\nB =\n[[-19, 21, 13, 1, -18, -17, 2, -11, -24, -19],\n [21, -8, -4, 24, -16, 24, 7, -19, -23, -12],\n [18, 20, 10, -13, -10, -10, -21, -1, 12, 12],\n [24, 12, 13, 22, 15, 0, -13, 18, 25, 0],\n [-25, -18, -22, -5, -4, 25, 11, 23, -23, -7],\n [-25, 3, -3, 7, 15, -3, -21, -19, -20, 23],\n [-4, 23, -13, -14, -22, -18, 11, 5, 11, 19],\n [0, 17, -18, 14, 7, 19, 3, -21, 2, 14],\n [-9, -3, 19, -17, -24, 6, 25, 23, 5, -19],\n [4, -23, 25, 25, -17, -16, 20, 24, 21, 0],\n [-25, 4, 12, -11, -1, -20, -8, -25, -23, -9]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

459

OMEGA_500_SAMPLE

omega_500_406

combinatory_probability_no_fixed_points_level_1_0043

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 {a: 1, z: 1, t: 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}" } ]

17

OMEGA_500_SAMPLE

omega_500_407

arithmetic_matrix_power_level_6_0042

arithmetic_matrix_power

6

arithmetic_matrix_power

6

[ { "role": "user", "content": "Find the sum of all entries of the 7th power of the matrix:\n[[-1, -2, -14, -7, -3, 4, -3, -7, -7, 14],\n [6, -2, -3, 14, 1, -14, -2, 11, -8, 13],\n [11, -10, -7, 10, 12, -1, 8, -2, 0, 8],\n [2, 0, 5, 3, 3, -13, -11, -14, -11, 3],\n [-10, 4, 0, 13, -7, 5, 2, -14, -3, 11],\n [4, 0, -6, -8, -13, 14, -6, -5, -13, 4],\n [-4, -2, -5, -1, 3, 4, 7, -6, 10, -8],\n [13, -12, 5, -9, 12, 6, -13, 14, -9, -12],\n [1, -7, -13, 2, -6, -3, -12, -4, -14, 8],\n [-2, 0, 11, -10, 6, 2, 10, 8, 9, 10]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

377808268

OMEGA_500_SAMPLE

omega_500_408

geometry_polygon_chords_level_5_0029

geometry_polygon_chords

5

geometry_polygon_chords

5

[ { "role": "user", "content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 1, from vertex 7 to vertex 2, from vertex 5 to vertex 3, from vertex 12 to vertex 2, from vertex 6 to vertex 3, from vertex 2 to vertex 6, from vertex 3 to vertex 7, from vertex 7 to vertex 11, from vertex 3 to vertex 1, from vertex 8 to vertex 4, from vertex 6 to vertex 10, from vertex 11 to vertex 3, from vertex 2 to vertex 4, and from vertex 11 to vertex 1. 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}" } ]

10

OMEGA_500_SAMPLE

omega_500_409

algebra_func_intersection_coords_level_3_0017

algebra_func_intersection_coords

3

algebra_func_intersection_coords

3

[ { "role": "user", "content": "Find the integer value closest to the x-coordinate where f(x) = 3(3(-2((-2x - 3)/(-3x - 1)) - 3) + 4) + 1 and g(x) = 3cos(2π(2|(-4x^2 - 5x - 2) + 3| - 3) + 1) + 1 intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

-1

OMEGA_500_SAMPLE

omega_500_410

combinatory_pattern_matching_level_1_0068

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 {j: 1, b: 2, l: 1}, what is the expected number of times the pattern 'lb+' appears? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

0.17

OMEGA_500_SAMPLE

omega_500_411

geometry_polygon_chords_level_5_0089

geometry_polygon_chords

5

geometry_polygon_chords

5

[ { "role": "user", "content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 4, from vertex 3 to vertex 6, from vertex 2 to vertex 4, from vertex 2 to vertex 5, from vertex 1 to vertex 3, from vertex 8 to vertex 6, from vertex 7 to vertex 4, from vertex 7 to vertex 5, from vertex 8 to vertex 3, from vertex 3 to vertex 5, from vertex 8 to vertex 5, from vertex 4 to vertex 1, from vertex 2 to vertex 6, from vertex 7 to vertex 2, and from vertex 2 to vertex 8. 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}" } ]

14

OMEGA_500_SAMPLE

omega_500_412

combinatory_probability_at_least_n_specific_fixed_level_3_0038

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 {e: 4, w: 3, f: 1}, shuffle the letters in the word, what is the probability of at least 3 letter 'e' 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}" } ]

15

OMEGA_500_SAMPLE

omega_500_413

arithmetic_matrix_power_level_3_0079

arithmetic_matrix_power

3

arithmetic_matrix_power

3

[ { "role": "user", "content": "For the matrix power A^5 where A =\n[[1, 2, 0, -1, 0],\n [-4, -5, 1, 1, 4],\n [5, 4, -4, -4, -5],\n [4, 2, 2, -5, 5],\n [2, 3, -2, -5, -5]], determine the sum of all entries.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

13843

OMEGA_500_SAMPLE

omega_500_414

arithmetic_matrix_svd_level_4_0012

arithmetic_matrix_svd

4

arithmetic_matrix_svd

4

[ { "role": "user", "content": "What is the rounded difference between the largest and smallest singular values in the SVD of the following matrix?\n[[12, -10, -4, 4, 4, 7, 3],\n [12, 9, 0, 6, 4, -9, -1],\n [-4, 6, -1, -4, -6, 1, 7],\n [6, 2, 3, 8, -8, -10, -1],\n [12, 7, 8, -9, 8, 10, 3],\n [12, -6, 0, 8, -3, -6, 1],\n [-8, -10, 10, -2, -2, 5, 2]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

28

OMEGA_500_SAMPLE

omega_500_415

number_theory_digit_sum_level_4_0018

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 9. Find N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

99918

OMEGA_500_SAMPLE

omega_500_416

arithmetic_matrix_multiplication_level_5_0036

arithmetic_matrix_multiplication

5

arithmetic_matrix_multiplication

5

[ { "role": "user", "content": "Compute the sum of all entries of the product matrix AB given:\nA =\n[[1, -11, -15, -4, 17, 3, -11, -13, 0],\n [-15, 0, 18, 3, 12, -3, 7, 11, -9],\n [6, -11, 11, 5, 4, -10, 2, 13, 19],\n [2, 8, 10, -14, 19, 12, 0, -19, -3],\n [12, 16, 13, -1, 14, -3, 3, -13, 9],\n [-11, 2, -10, -19, 5, 15, 12, -8, 19],\n [4, -17, -2, -3, 16, 9, -2, 2, -5],\n [-20, -9, 3, 13, -13, -3, -16, 5, 9],\n [-8, -2, 8, -18, 5, 19, -9, -19, -10]]\nB =\n[[7, -3, -8, 16, 1, -9, -19, 16, -14, 5],\n [5, 3, -9, -5, 2, 19, -12, 11, -6, 16],\n [-8, -16, -10, -11, -8, 20, -6, -14, 19, 7],\n [11, 14, -6, 15, 0, -15, -20, -17, -10, 5],\n [16, 2, 8, 0, 6, 18, 19, -8, 2, -5],\n [9, -17, 19, -5, 10, 4, -9, 5, 16, 12],\n [7, -12, -9, -16, 7, 14, -16, 13, 13, -8],\n [18, -7, -10, 17, -17, 12, -14, 20, -13, 16],\n [-11, -6, 17, -6, -16, 8, -2, -11, -9, -17]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

3515

OMEGA_500_SAMPLE

omega_500_417

combinatory_probability_no_fixed_points_level_4_0044

combinatory_probability_no_fixed_points

4

combinatory_probability_no_fixed_points

4

[ { "role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {v: 2, o: 3, q: 2, m: 3}, 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}" } ]

1501

OMEGA_500_SAMPLE

omega_500_418

geometry_polygon_chords_level_5_0078

geometry_polygon_chords

5

geometry_polygon_chords

5

[ { "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 10, from vertex 5 to vertex 8, from vertex 1 to vertex 5, from vertex 9 to vertex 2, from vertex 2 to vertex 8, from vertex 8 to vertex 6, from vertex 5 to vertex 9, from vertex 2 to vertex 10, from vertex 6 to vertex 2, from vertex 4 to vertex 1, from vertex 3 to vertex 8, from vertex 10 to vertex 5, from vertex 7 to vertex 9, and from vertex 8 to vertex 1. 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}" } ]

11

OMEGA_500_SAMPLE

omega_500_419

geometry_rotation_level_5_0055

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 2 to vertex 6, from vertex 2 to vertex 4, from vertex 2 to vertex 5, from vertex 1 to vertex 4, from vertex 1 to vertex 5, 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 with 3 sides?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

18

OMEGA_500_SAMPLE

omega_500_420

algebra_linear_equation_level_2_0016

algebra_linear_equation

2

algebra_linear_equation

2

[ { "role": "user", "content": "Suppose -1050 = 101*a - 96*a - 55*a. Solve 3*d - c + 2*h = -0*c + 17, -34 = -3*d - 4*c - h, -4*c = 4*d - 5*h - a for d.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

5

OMEGA_500_SAMPLE

omega_500_421

logic_puzzles_blocked_grid_level_1_0023

logic_puzzles_blocked_grid

1

logic_puzzles_blocked_grid

1

[ { "role": "user", "content": "In a 9x9 grid, how many different paths are there from the bottom left (0, 0) to the top right (8, 8), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (7, 8), (1, 5), (0, 6), (4, 6), (5, 0), (2, 7), (6, 7), (3, 0), (8, 2), (8, 6)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

1285

OMEGA_500_SAMPLE

omega_500_422

arithmetic_mixed_level_2_0046

arithmetic_mixed

2

arithmetic_mixed

2

[ { "role": "user", "content": "What is (-4 - -28) + (-3 - 39) + (141 - (-320 + 1371))? 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}" } ]

-928

OMEGA_500_SAMPLE

omega_500_423

algebra_func_zeros_level_4_0024

algebra_func_zeros

4

algebra_func_zeros

4

[ { "role": "user", "content": "How many solutions does the equation 1|(4(4(1|(2|x - 3|) - 2| - 1) + 1) + 1) + 3| - 1 = 0 have in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

8

OMEGA_500_SAMPLE

omega_500_424

arithmetic_list_prime_factors_level_4_0022

arithmetic_list_prime_factors

4

arithmetic_list_prime_factors

4

[ { "role": "user", "content": "What is the second-largest prime factor of 4086778025543372?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

601813

OMEGA_500_SAMPLE

omega_500_425

arithmetic_matrix_eigenvalues_level_1_0082

arithmetic_matrix_eigenvalues

1

arithmetic_matrix_eigenvalues

1

[ { "role": "user", "content": "What is the sum of absolute values of all eigenvalues for the following matrix?\n[[-7, -6, 0],\n [-6, -8, 8],\n [0, 8, 10]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

32

OMEGA_500_SAMPLE

omega_500_426

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_427

combinatory_pattern_matching_level_1_0076

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 {j: 1, h: 1, s: 1, w: 1}, what is the expected number of times the pattern 'w+s' appears? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

0.08

OMEGA_500_SAMPLE

omega_500_428

algebra_func_extrema_coords_level_2_0073

algebra_func_extrema_coords

2

algebra_func_extrema_coords

2

[ { "role": "user", "content": "How many local minima does the function f(x) = (-2((-1(-1x^2 - 1x) - 3)/(-2(-1x^2 - 1x) + 3)) + 1)/(2((-1(-1x^2 - 1x) - 3)/(-2(-1x^2 - 1x) + 3))) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

0

OMEGA_500_SAMPLE

omega_500_429

geometry_polygon_color_level_3_0030

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 red, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is red, vertex 10 is red. The 11-gon is then rotated clockwise by 4 positions, so that the vertex that was originally in position 0 is now in position 4. 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_430

combinatory_probability_at_least_n_specific_fixed_level_2_0097

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 {v: 1, t: 3, x: 2}, 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}" } ]

7

OMEGA_500_SAMPLE

omega_500_431

combinatory_pattern_matching_level_2_0024

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 {i: 1, r: 1, c: 2, w: 1, l: 1}, what is the expected number of times the pattern 'i+w' appears? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

0.07

OMEGA_500_SAMPLE

omega_500_432

number_theory_triple_count_level_2_0072

number_theory_triple_count

2

number_theory_triple_count

2

[ { "role": "user", "content": "Let N denote the number of ordered pairs of positive integers (a, b) such that a, b <= 4^3 and (4)a^4 + (4)b^4 is divisible by 4^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

24

OMEGA_500_SAMPLE

omega_500_433

arithmetic_gcd_level_2_0078

arithmetic_gcd

2

arithmetic_gcd

2

[ { "role": "user", "content": "Calculate the greatest common factor of 1902 and 1164257312.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

634

OMEGA_500_SAMPLE

omega_500_434

combinatory_probability_no_specific_letter_fixed_level_4_0075

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 {a: 3, h: 3, w: 1, r: 3}, 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_435

geometry_polygon_color_level_2_0078

geometry_polygon_color

2

geometry_polygon_color

2

[ { "role": "user", "content": "A 9-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is uncolored, vertex 7 is red, vertex 8 is red. How many valid colorings of the uncolored vertices exist such that the 9-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_436

number_theory_prime_mod_level_1_0046

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_437

algebra_func_extrema_coords_level_1_0060

algebra_func_extrema_coords

1

algebra_func_extrema_coords

1

[ { "role": "user", "content": "Find the x-coordinate of the local maximum of the function f(x) = -2|(3x^2 + 3x + 2) - 1| - 3 in the interval [-10, 10]. Round to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

0

OMEGA_500_SAMPLE

omega_500_438

number_theory_digit_sum_level_4_0076

number_theory_digit_sum

4

number_theory_digit_sum

4

[ { "role": "user", "content": "let N be the maximum 5-digit positive integer with the property that whenever one of its digits changes to 8, the resulting number is a multiple of 3. 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}" } ]

7784

OMEGA_500_SAMPLE

omega_500_439

arithmetic_list_prime_factors_level_5_0067

arithmetic_list_prime_factors

5

arithmetic_list_prime_factors

5

[ { "role": "user", "content": "Find the second-largest prime factor of 140068604170336290418.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

362707

OMEGA_500_SAMPLE

omega_500_440

arithmetic_matrix_svd_level_7_0013

arithmetic_matrix_svd

7

arithmetic_matrix_svd

7

[ { "role": "user", "content": "Calculate the rounded largest singular value from the SVD of:\n[[-11, 13, 1, 6, 21, 14, -14, 16, 18, -2, 26, 15],\n [9, 0, 11, -20, -22, 16, 18, 24, 4, -4, 15, -17],\n [5, 16, 26, 3, 12, 4, -9, 9, -25, -13, -14, 2],\n [-15, 12, -25, 5, -14, 19, 0, 22, 14, -10, 13, 4],\n [7, -3, 18, 8, 21, -21, -8, 3, -20, -17, 14, 5],\n [7, 1, -12, 3, 10, 2, -5, -23, -7, 5, 9, 23],\n [-23, 0, -10, -15, -5, -10, -17, -26, -5, -1, -9, 23],\n [-13, 20, 2, -15, 9, 8, 25, -15, -20, 2, -16, 6],\n [-21, 20, -12, 5, 24, -16, -4, -1, 7, 19, 7, 2],\n [-23, -10, 8, -14, 14, 12, -12, 2, 2, -16, 25, 5],\n [-25, 20, -13, -7, -5, -16, 0, -24, -10, 17, 1, 2],\n [-6, 20, -11, 10, -7, -21, -8, -22, -24, -26, 11, -12]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

90

OMEGA_500_SAMPLE

omega_500_441

arithmetic_matrix_determinant_level_3_0092

arithmetic_matrix_determinant

3

arithmetic_matrix_determinant

3

[ { "role": "user", "content": "Determine the value of det(A) where A =\n[[10, -1, -9, 3, 4, 6],\n [14, -5, 7, -2, -3, 4],\n [10, 11, 9, -7, -7, 0],\n [-4, -6, -10, 2, -14, 11],\n [4, -7, 3, -10, -14, -1],\n [1, 0, 12, 5, -8, -6]]\nRound to the nearest integer, then give your answer as det(A) mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

636

OMEGA_500_SAMPLE

omega_500_442

logic_puzzles_zebralogic_level_1_0028

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 - Each person has an occupation: `engineer`, `doctor`\n\n## Clues:\n1. Eric is directly left of Arnold.\n2. The person who is a doctor is not in the first house.\n\n\nChoices:\n0. doctor\n1. engineer\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_443

geometry_polygon_color_level_3_0051

geometry_polygon_color

3

geometry_polygon_color

3

[ { "role": "user", "content": "A 10-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 uncolored, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is red, vertex 7 is uncolored, vertex 8 is uncolored, vertex 9 is red. How many valid colorings of the uncolored vertices exist such that the 10-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_444

number_theory_triple_count_level_5_0096

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 <= 4^3, 2. a, b, c are relatively coprime, and 3. (1)a^4 + (1)b^4 + (-1)c^4 is divisible by 4^2. Find N mod 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

434

OMEGA_500_SAMPLE

omega_500_445

number_theory_prime_mod_level_5_0034

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^4) 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_446

geometry_rotation_level_5_0013

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 2 to vertex 5, from vertex 4 to vertex 2, from vertex 3 to vertex 5, from vertex 1 to vertex 5, from vertex 4 to vertex 6, and from vertex 4 to vertex 1. 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 with 3 sides?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

18

OMEGA_500_SAMPLE

omega_500_447

geometry_polygon_color_level_4_0047

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 blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, 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_448

logic_puzzles_grid_knight_level_3_0091

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 29x27 grid, what is minimum number of steps you can take to get to (26, 23), if your only valid moves are to move over 6 squares in one direction and over 1 squares in a perpendicular direction (at once)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

9

OMEGA_500_SAMPLE

omega_500_449

logic_puzzles_grid_chip_level_5_0077

logic_puzzles_grid_chip

5

logic_puzzles_grid_chip

5

[ { "role": "user", "content": "Chips, colored either black or white, are placed in a 8x8 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 (0, 7) is black, cell (4, 7) is black, cell (6, 3) is black, cell (2, 1) is black, cell (6, 5) is black, cell (7, 2) is black, cell (1, 4) 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}" } ]

13

OMEGA_500_SAMPLE

omega_500_450

arithmetic_list_prime_factors_level_3_0076

arithmetic_list_prime_factors

3

arithmetic_list_prime_factors

3

[ { "role": "user", "content": "Find the second-largest prime factor of 167604909357658.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

6619

OMEGA_500_SAMPLE

omega_500_451

arithmetic_gcd_level_5_0049

arithmetic_gcd

5

arithmetic_gcd

5

[ { "role": "user", "content": "Calculate the highest common factor of 484131285 and 977648343511861119594815.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

53792365

OMEGA_500_SAMPLE

omega_500_452

arithmetic_matrix_inverse_level_3_0026

arithmetic_matrix_inverse

3

arithmetic_matrix_inverse

3

[ { "role": "user", "content": "Find the inverse of 1/180 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, -8, -14, 12, 15, -8],\n [-9, 10, 10, -14, 6, 6],\n [-13, 11, 2, -4, -15, 6],\n [-4, -11, 14, 14, 11, 9],\n [1, 0, 15, -1, 7, -1],\n [-11, 5, -2, -14, -5, 3]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

246

OMEGA_500_SAMPLE

omega_500_453

geometry_polygon_chords_level_1_0066

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 1 to vertex 5, from vertex 4 to vertex 7, from vertex 1 to vertex 7, from vertex 5 to vertex 7, from vertex 8 to vertex 6, and from vertex 7 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}" } ]

3

OMEGA_500_SAMPLE

omega_500_454

geometry_polygon_chords_level_1_0078

geometry_polygon_chords

1

geometry_polygon_chords

1

[ { "role": "user", "content": "Suppose you have a 14-gon, with vertices numbered 1 through 14 in counterclockwise order. Draw the diagonal from vertex 11 to vertex 9, from vertex 6 to vertex 12, from vertex 10 to vertex 1, from vertex 8 to vertex 1, from vertex 14 to vertex 2, from vertex 7 to vertex 13, and from vertex 14 to vertex 7. 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}" } ]

2

OMEGA_500_SAMPLE

omega_500_455

logic_puzzles_grid_chip_level_2_0092

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 (3, 4) is white, cell (1, 0) is white, cell (0, 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_456

combinatory_probability_at_least_n_specific_fixed_level_3_0030

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 {b: 2, t: 3, u: 3}, shuffle the letters in the word, what is the probability of at least 2 letter 'b' 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_457

algebra_func_extrema_level_5_0095

algebra_func_extrema

5

algebra_func_extrema

5

[ { "role": "user", "content": "How many local extrema (minima and maxima combined) does the function f(x) = (1(3cos(1π(-3(-4(2|(-5x - 3) + 3| - 1)^2 + 2(2|(-5x - 3) + 3| - 1) + 4)^2 + 4(-4(2|(-5x - 3) + 3| - 1)^2 + 2(2|(-5x - 3) + 3| - 1) + 4) - 4) - 3)) + 3)/(1(3cos(1π(-3(-4(2|(-5x - 3) + 3| - 1)^2 + 2(2|(-5x - 3) + 3| - 1) + 4)^2 + 4(-4(2|(-5x - 3) + 3| - 1)^2 + 2(2|(-5x - 3) + 3| - 1) + 4) - 4) - 3)) - 2) have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

17

OMEGA_500_SAMPLE

omega_500_458

arithmetic_matrix_determinant_level_7_0050

arithmetic_matrix_determinant

7

arithmetic_matrix_determinant

7

[ { "role": "user", "content": "What is the determinant of the matrix:\n[[2, 11, 18, 19, 12, -21, 35, -20, 7, -29, 7, -8],\n [30, 24, -33, -35, -5, 20, 10, -4, -18, 32, -1, -13],\n [-23, -22, 1, -31, -14, -8, -32, 14, -7, 1, -1, -16],\n [-1, -34, 22, -18, 20, 28, 16, -14, 24, -26, -30, -30],\n [-8, 6, 5, 27, -14, -7, 35, -2, -34, -10, -17, -16],\n [-3, -32, -4, -17, -33, 25, 2, -30, 20, 4, 33, 6],\n [-34, -8, -19, 34, 33, 3, -29, -4, -32, -15, -4, -22],\n [34, 10, -16, 22, 12, -29, 4, 23, 0, 8, -5, 6],\n [8, 7, 7, 11, -30, 30, -27, -25, -18, -11, 22, -6],\n [-35, 27, -7, 30, 17, -17, 30, 16, -5, 24, 32, -6],\n [-23, -21, -16, -2, -16, 14, -9, -6, 20, 29, 19, -12],\n [-11, 17, -25, 24, 11, -11, -11, -19, -24, 16, -25, -26]]? Round to the nearest integer, then express your answer modulo 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

408

OMEGA_500_SAMPLE

omega_500_459

logic_puzzles_grid_rook_level_5_0068

logic_puzzles_grid_rook

5

logic_puzzles_grid_rook

5

[ { "role": "user", "content": "In a 40x43 grid, what is the fewest number of steps it takes to get from the bottom left (0, 0) to the top right (39, 42), 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: (32, 27), (35, 19), (27, 21), (1, 27), (26, 23), (0, 18), (15, 26), (29, 2), (7, 22), (32, 7), (25, 7), (22, 13), (20, 13), (28, 13), (7, 38), (14, 22), (3, 0), (13, 6), (35, 32), (39, 3), (26, 20), (15, 25), (31, 9), (12, 11), (14, 34), (34, 0), (21, 23), (12, 36), (13, 24), (12, 39), (38, 25), (37, 10), (13, 17), (21, 42), (37, 7), (18, 7), (12, 1), (14, 39), (30, 2), (3, 42), (29, 31), (17, 5), (16, 33), (36, 32), (15, 23), (12, 15), (17, 8), (4, 37), (3, 38), (2, 9), (8, 16), (10, 38), (33, 6), (36, 4), (16, 35), (11, 21), (10, 18), (33, 11), (10, 1), (25, 29), (23, 32), (20, 31), (33, 18), (25, 9), (6, 5), (13, 35), (4, 39), (6, 36), (0, 40), (19, 34), (3, 4), (30, 40), (25, 19), (11, 1), (27, 1), (6, 41), (21, 1), (0, 21), (7, 3), (39, 27), (34, 26), (29, 24), (28, 23), (27, 17), (38, 13), (31, 34), (37, 2), (1, 23), (19, 33), (36, 1), (4, 15), (24, 38), (37, 40), (26, 10), (29, 17), (33, 22), (37, 42), (2, 21), (38, 5), (7, 32), (31, 32), (17, 0), (26, 41), (10, 30), (19, 22), (38, 31), (17, 24), (2, 18), (26, 18), (29, 5), (3, 9), (16, 41), (17, 41), (9, 15), (39, 26), (8, 21), (15, 10), (13, 33), (0, 36), (9, 7), (5, 35), (35, 8), (26, 26), (11, 2), (12, 40), (16, 27), (6, 12), (11, 6), (37, 20), (2, 29), (0, 27), (9, 32), (20, 18), (30, 14), (21, 8), (34, 27), (28, 15), (11, 26), (36, 22), (23, 8), (19, 23), (36, 41), (33, 38), (17, 6), (7, 23), (23, 14), (6, 22), (36, 40), (13, 7), (22, 26), (14, 11), (20, 36), (7, 13), (15, 4), (33, 27), (14, 26), (25, 36), (11, 39), (1, 41), (15, 19), (34, 9), (17, 25), (27, 2), (17, 36), (15, 3), (1, 7), (38, 28), (0, 38), (31, 20), (32, 26), (34, 29), (22, 22), (18, 2), (30, 41), (6, 31), (7, 15), (19, 35), (8, 39), (2, 10), (10, 33), (12, 34), (12, 41), (32, 11), (32, 5), (34, 32), (20, 15), (15, 1), (0, 25), (34, 3), (15, 2), (37, 12), (33, 36), (4, 36), (1, 42), (17, 34), (18, 4), (6, 0), (36, 16), (12, 10), (20, 33), (15, 11), (21, 19), (3, 35), (34, 21), (37, 36), (25, 17), (39, 17), (15, 31), (35, 41), (19, 2), (6, 3), (27, 31), (33, 32), (16, 8), (5, 40), (19, 27), (8, 34), (24, 1), (39, 1), (10, 39), (27, 3), (19, 30), (39, 24), (6, 10), (2, 6), (26, 8), (35, 25), (16, 7), (10, 19), (30, 0), (1, 20), (33, 24), (11, 0), (1, 36), (26, 1), (24, 30), (9, 6), (23, 33), (18, 28), (21, 34), (28, 40), (9, 33), (22, 16), (23, 16), (16, 0), (5, 4), (1, 22), (20, 9), (5, 28), (29, 12), (39, 29), (36, 5), (1, 18), (3, 20), (24, 37), (33, 3), (3, 24), (26, 30), (4, 32), (39, 35), (18, 24), (30, 24), (29, 1), (39, 13), (20, 24)? If it's impossible, answer with -1.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

7

OMEGA_500_SAMPLE

omega_500_460

logic_puzzles_grid_knight_level_1_0089

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 9x7 grid, what is minimum number of steps you can take to get to (5, 3), if your only valid moves are to move over 3 squares in one direction and over 1 squares in a perpendicular direction (at once)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

3

OMEGA_500_SAMPLE

omega_500_461

combinatory_probability_at_least_n_specific_fixed_level_4_0070

combinatory_probability_at_least_n_specific_fixed

4

combinatory_probability_at_least_n_specific_fixed

4

[ { "role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {a: 2, l: 3, o: 4, i: 1}, shuffle the letters in the word, what is the probability of at least 2 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}" } ]

13

OMEGA_500_SAMPLE

omega_500_462

arithmetic_gcd_level_4_0073

arithmetic_gcd

4

arithmetic_gcd

4

[ { "role": "user", "content": "What is the highest common divisor of 38663909736056969524 and 80928908?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

4760524

OMEGA_500_SAMPLE

omega_500_463

logic_puzzles_grid_rook_level_3_0050

logic_puzzles_grid_rook

3

logic_puzzles_grid_rook

3

[ { "role": "user", "content": "In a 25x26 grid, what is the fewest number of steps it takes to get from the bottom left (0, 0) to the top right (24, 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: (7, 22), (1, 19), (9, 2), (8, 17), (18, 3), (4, 20), (2, 8), (18, 24), (6, 9), (18, 23), (4, 7), (19, 0), (10, 5), (22, 13), (12, 6), (12, 7), (21, 21), (9, 4), (10, 6), (8, 14), (24, 19), (7, 14), (6, 14), (8, 7), (7, 12), (3, 19), (2, 21), (0, 11), (3, 9), (16, 10), (23, 16), (10, 13), (13, 15), (22, 22), (12, 10), (15, 12), (22, 5), (10, 7), (6, 15), (1, 11), (3, 11), (17, 3), (13, 4), (21, 11), (3, 12), (23, 6), (13, 18), (7, 25), (3, 18), (0, 14), (5, 23), (8, 16), (0, 10), (15, 11), (13, 13), (22, 9), (14, 13), (14, 10), (19, 9), (18, 11), (23, 8), (15, 23), (10, 11), (1, 14), (5, 20), (20, 5), (6, 3), (2, 22), (0, 6), (14, 24), (9, 0), (18, 9), (2, 12), (20, 11), (9, 25), (6, 8), (21, 23), (10, 22), (1, 7), (24, 23), (20, 3), (23, 22), (13, 5), (16, 23), (5, 0), (9, 19), (2, 9), (1, 22), (18, 18), (7, 11), (23, 20), (20, 10), (12, 17), (16, 7), (6, 24), (0, 13), (4, 10), (21, 12), (19, 1), (6, 22), (24, 22), (24, 5), (15, 5), (9, 15), (5, 2), (19, 22), (8, 12), (4, 5), (17, 11), (5, 19), (1, 15), (0, 9), (6, 23), (23, 24), (11, 9), (2, 3), (24, 14), (14, 7), (20, 23), (15, 1), (4, 2), (10, 8), (6, 6), (5, 18)? If it's impossible, answer with -1.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

6

OMEGA_500_SAMPLE

omega_500_464

algebra_func_area_level_3_0086

algebra_func_area

3

algebra_func_area

3

[ { "role": "user", "content": "Calculate the area enclosed between the functions f(x) = -3|(5(1(-3sin(3πx + 1) - 1)^2 + 1(-3sin(3πx + 1) - 1) - 5) + 5) + 1| + 1 and g(x) = -3|((2(3|x - 3| - 1) - 2)/(-2(3|x - 3| - 1) + 1))| + 2 from x = -1.8 to x = -1.7. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

5.0

OMEGA_500_SAMPLE

omega_500_465

combinatory_pattern_matching_level_2_0061

combinatory_pattern_matching

2

combinatory_pattern_matching

2

[ { "role": "user", "content": "When randomly selecting 3 letters from the multiset {p: 1, j: 2, z: 3} to form a word, what is the expected number of matches of the pattern 'pz+'? Round to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

0.25

OMEGA_500_SAMPLE

omega_500_466

algebra_func_area_level_5_0051

algebra_func_area

5

algebra_func_area

5

[ { "role": "user", "content": "Determine the area of the region enclosed by the graphs of f(x) = (-1(-1((-1((-3(2|(3|x - 1| - 1) + 2| - 3))/(-1(2|(3|x - 1| - 1) + 2| - 3) + 2)) + 3)/(2((-3(2|(3|x - 1| - 1) + 2| - 3))/(-1(2|(3|x - 1| - 1) + 2| - 3) + 2))))^2 + 4((-1((-3(2|(3|x - 1| - 1) + 2| - 3))/(-1(2|(3|x - 1| - 1) + 2| - 3) + 2)) + 3)/(2((-3(2|(3|x - 1| - 1) + 2| - 3))/(-1(2|(3|x - 1| - 1) + 2| - 3) + 2)))) + 3) - 3)/(-2(-1((-1((-3(2|(3|x - 1| - 1) + 2| - 3))/(-1(2|(3|x - 1| - 1) + 2| - 3) + 2)) + 3)/(2((-3(2|(3|x - 1| - 1) + 2| - 3))/(-1(2|(3|x - 1| - 1) + 2| - 3) + 2))))^2 + 4((-1((-3(2|(3|x - 1| - 1) + 2| - 3))/(-1(2|(3|x - 1| - 1) + 2| - 3) + 2)) + 3)/(2((-3(2|(3|x - 1| - 1) + 2| - 3))/(-1(2|(3|x - 1| - 1) + 2| - 3) + 2)))) + 3) + 3) and g(x) = -1cos(3π(5(4(-1|((3x - 2)/(2x + 1))| - 2)^2 + 3(-1|((3x - 2)/(2x + 1))| - 2))^2 + 5)) - 1 between the vertical lines x = 0.6 and x = 0.7. Round to 1 decimal place.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

0.1

OMEGA_500_SAMPLE

omega_500_467

arithmetic_mixed_level_1_0015

arithmetic_mixed

1

arithmetic_mixed

1

[ { "role": "user", "content": "Evaluate 64/180*(-42)/(-7)*(-5)/20. 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}" } ]

-8/15

OMEGA_500_SAMPLE

omega_500_468

arithmetic_matrix_rank_level_7_0018

arithmetic_matrix_rank

7

arithmetic_matrix_rank

7

[ { "role": "user", "content": "Find the rank of the matrix:\n[[885, 812, 154, 26, 195, -335, -87, 517, -315, 104, -514, 167],\n [-1051, -715, -560, -625, -455, 474, -106, -815, 639, 43, 264, -738],\n [-246, -567, 780, 689, 531, 515, 36, -824, -547, 4, 268, 339],\n [311, 592, 125, 365, 353, -772, 92, 1151, -138, 97, 488, 987],\n [-110, -54, -147, -883, -331, -969, -98, 547, 473, 218, -144, -244],\n [0, -110, -262, -396, -354, -928, -42, 926, 210, -446, 224, 42],\n [-285, -62, -729, -179, -493, 397, 106, -29, 463, -194, 49, -422],\n [-21, 597, -855, 178, -6, 407, -414, 283, -211, -833, 626, 74],\n [-572, -1261, -184, 325, -664, 1096, 515, -995, 324, -430, -177, -770],\n [413, -722, -18, 530, -882, -309, 1122, 964, 505, -309, -488, -9],\n [-735, 6, -558, 962, 242, 938, 102, -84, -36, -400, 1287, 698],\n [268, 128, 559, 1213, 577, -79, 358, 609, -533, -116, 616, 1220]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

6

OMEGA_500_SAMPLE

omega_500_469

combinatory_probability_exactly_n_specific_fixed_level_1_0090

combinatory_probability_exactly_n_specific_fixed

1

combinatory_probability_exactly_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 {q: 1, k: 1, b: 2}, shuffle the letters in the word, what is the probability of exact 1 letter 'b' 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}" } ]

4

OMEGA_500_SAMPLE

omega_500_470

algebra_func_intersection_level_5_0004

algebra_func_intersection

5

algebra_func_intersection

5

[ { "role": "user", "content": "How many solutions does the equation (-1(3|(1sin(1π(3(1|((2x + 3)/(-1x + 2)) + 3|) - 2) + 2) + 3) - 2| + 1))/(2(3|(1sin(1π(3(1|((2x + 3)/(-1x + 2)) + 3|) - 2) + 2) + 3) - 2| + 1) - 3) = 4((-2(-3(-3cos(1π(2x + 2) - 1) + 1) - 4) - 1)/(1(-3(-3cos(1π(2x + 2) - 1) + 1) - 4) + 2)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

78

OMEGA_500_SAMPLE

omega_500_471

algebra_func_derivative_sign_level_1_0013

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) = 3sin(1π(2x + 4)) is strictly increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

21

OMEGA_500_SAMPLE

omega_500_472

algebra_func_extrema_level_2_0059

algebra_func_extrema

2

algebra_func_extrema

2

[ { "role": "user", "content": "Determine the count of local maxima for the function f(x) = 2(2|(3cos(3πx + 3) + 2) + 3| + 1)^2 - 1(2|(3cos(3πx + 3) + 2) + 3| + 1) + 5 in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

30

OMEGA_500_SAMPLE

omega_500_473

arithmetic_matrix_eigenvalues_level_6_0076

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[[-18, -7, -6, -2, -14, 6, 17, 4, 6, -10],\n [-7, 5, -8, 6, 11, -20, -5, -14, 12, 3],\n [-6, -8, -12, 6, 10, -14, -3, 12, -16, -5],\n [-2, 6, 6, 16, -6, -15, 10, 12, 8, -4],\n [-14, 11, 10, -6, 26, -16, -22, 14, -3, 17],\n [6, -20, -14, -15, -16, 6, 1, 8, -24, 3],\n [17, -5, -3, 10, -22, 1, 0, 3, -28, 5],\n [4, -14, 12, 12, 14, 8, 3, 0, -20, 0],\n [6, 12, -16, 8, -3, -24, -28, -20, 29, 2],\n [-10, 3, -5, -4, 17, 3, 5, 0, 2, -7]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

77

OMEGA_500_SAMPLE

omega_500_474

arithmetic_matrix_inverse_level_1_0013

arithmetic_matrix_inverse

1

arithmetic_matrix_inverse

1

[ { "role": "user", "content": "Find the inverse of 1/60 times the matrix and calculate the value of the entry with the largest absolute value of the inverse. Round your answer to the nearest integer:\n[[4, 0, -7],\n [2, -4, 8],\n [-9, -1, 2]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

-17

OMEGA_500_SAMPLE

omega_500_475

number_theory_triple_count_level_3_0070

number_theory_triple_count

3

number_theory_triple_count

3

[ { "role": "user", "content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and (-4)a^3 + (-4)b^3 + (-2)c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

81

OMEGA_500_SAMPLE

omega_500_476

logic_puzzles_zebralogic_level_4_0078

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 a unique level of education: `associate`, `high school`\n - Everyone has something unique for lunch: `grilled cheese`, `pizza`\n - They all have a unique favorite flower: `daffodils`, `carnations`\n - People have unique favorite music genres: `rock`, `pop`\n\n## Clues:\n1. The person who loves a bouquet of daffodils is not in the first house.\n2. The person who is a pizza lover is the person who loves a bouquet of daffodils.\n3. The person who loves a bouquet of daffodils is Eric.\n4. The person with a high school diploma is the person who loves a bouquet of daffodils.\n5. The person who loves pop music is the person who loves a bouquet of daffodils.\n\n\nChoices:\n0. pizza\n1. grilled cheese\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_477

arithmetic_mixed_level_5_0061

arithmetic_mixed

5

arithmetic_mixed

5

[ { "role": "user", "content": "Let c be -532 + (-28307728)/6494 - 333204/(-68). What is (-5 - -1)/((-191 + 230482/1212)/(345/c))? 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}" } ]

184

OMEGA_500_SAMPLE

omega_500_478

arithmetic_matrix_svd_level_3_0072

arithmetic_matrix_svd

3

arithmetic_matrix_svd

3

[ { "role": "user", "content": "What is the rounded difference between the largest and smallest singular values in the SVD of the following matrix?\n[[3, -1, 5, 5, -1, 2],\n [-5, 1, -2, 2, -7, 0],\n [9, 7, 0, 10, -7, 7],\n [-10, -4, -5, 4, 7, 0],\n [9, 4, 5, -4, 10, -1],\n [1, 3, -6, 10, 7, -6]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

19

OMEGA_500_SAMPLE

omega_500_479

combinatory_distribution_level_1_0089

combinatory_distribution

1

combinatory_distribution

1

[ { "role": "user", "content": "You have {'p': 2, 'u': 2} and need to distribute them into 2 identical boxes with capacities [3, 1]. How many distinct distributions are possible?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

2

OMEGA_500_SAMPLE

omega_500_480

combinatory_probability_no_specific_letter_fixed_level_3_0085

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 {c: 4, s: 1, a: 3}, shuffle the letters in the word, what is the probability of no letter 'a' 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_481

algebra_func_derivative_sign_level_2_0084

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) = -1sin(2π(-3(-3x - 4)^2 + 2(-3x - 4) + 2) - 3) + 1 negative?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

23

OMEGA_500_SAMPLE

omega_500_482

arithmetic_matrix_rank_level_7_0025

arithmetic_matrix_rank

7

arithmetic_matrix_rank

7

[ { "role": "user", "content": "Compute the rank of the given 12x12 matrix:\n[[-236, -140, 196, -268, -470, 766, -308, 324, -214, 570, 334, -420],\n [85, -292, 91, -319, 25, -601, 538, -336, -148, -238, -153, 486],\n [-261, -452, 357, -657, -645, 565, 30, 168, -432, 602, 341, -174],\n [237, 302, -273, 465, 540, -616, 108, -222, 321, -557, -320, 264],\n [-229, -786, 497, -1049, -730, 126, 512, -102, -635, 491, 262, 228],\n [103, -46, -35, -13, 160, -436, 268, -210, 23, -259, -156, 288],\n [-66, -368, 210, -474, -270, -98, 324, -120, -276, 128, 62, 204],\n [5, -436, 203, -527, -175, -433, 554, -288, -284, -54, -49, 438],\n [-9, -212, 105, -261, -105, -167, 246, -120, -144, 2, -7, 186],\n [-156, -174, 168, -276, -345, 429, -102, 162, -195, 369, 213, -198],\n [-164, -46, 112, -124, -305, 581, -278, 258, -115, 401, 237, -342],\n [-206, -620, 406, -838, -620, 196, 352, -36, -514, 450, 244, 120]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

2

OMEGA_500_SAMPLE

omega_500_483

geometry_rotation_level_2_0081

geometry_rotation

2

geometry_rotation

2

[ { "role": "user", "content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 1, from vertex 5 to vertex 3, and from vertex 8 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 8-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 3 sides?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

8

OMEGA_500_SAMPLE

omega_500_484

arithmetic_matrix_power_level_1_0082

arithmetic_matrix_power

1

arithmetic_matrix_power

1

[ { "role": "user", "content": "Find the sum of absolute values of all entries of the 2th power of the matrix:\n[[-1, -1, 0],\n [-1, -3, -1],\n [0, -1, -3]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

45

OMEGA_500_SAMPLE

omega_500_485

arithmetic_list_prime_factors_level_1_0021

arithmetic_list_prime_factors

1

arithmetic_list_prime_factors

1

[ { "role": "user", "content": "Find the second-largest prime factor of 20732.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

71

OMEGA_500_SAMPLE

omega_500_486

logic_puzzles_grid_chip_level_3_0060

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 (1, 3) is black, cell (1, 3) is black, cell (3, 4) is black, cell (0, 0) is black, cell (3, 2) 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}" } ]

10

OMEGA_500_SAMPLE

omega_500_487

arithmetic_matrix_determinant_level_4_0053

arithmetic_matrix_determinant

4

arithmetic_matrix_determinant

4

[ { "role": "user", "content": "Find the determinant of the following matrix. Round to the nearest integer, then give your answer modulo 1000:\n[[-3, -8, 9, 3, 8, -2, -11],\n [-11, 4, 13, -4, 8, -15, 13],\n [18, -9, 9, 13, 13, -4, -13],\n [-10, 11, -3, 6, -9, 11, -10],\n [-12, -7, -2, 6, -12, 0, -2],\n [13, 4, 13, 7, -11, 3, -11],\n [-14, -7, 8, 1, -8, 17, -17]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

712

OMEGA_500_SAMPLE

omega_500_488

algebra_linear_equation_level_5_0052

algebra_linear_equation

5

algebra_linear_equation

5

[ { "role": "user", "content": "Let g be 3 - 752/564*(-4)/(-3)*((-3056)/3840 + 287/1230). Let z = 4 - 4. Suppose 5*h - 21 = 3*k, z = -h - 0*k - k + 1. Solve 2*s + u = -5*t - 5*f - 3*r - 5*p - 66, -5*f + 5*p = 2*t - 5*s + 5*u + 33, -5*f + 5*r - 2*p = -2*u + 34, 2*f + s + 4*r - g*u = -4*t - 8, 5*s - 2*u + 3*u + 6 = -t - 2*f + 4*r + p, -2*f - 3*s + 4*u = -4*t - h*r - 3*p - 36 for t.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

-4

OMEGA_500_SAMPLE

omega_500_489

arithmetic_matrix_inverse_level_3_0012

arithmetic_matrix_inverse

3

arithmetic_matrix_inverse

3

[ { "role": "user", "content": "Find the trace (sum of diagonal elements) of the inverse of matrix 1/180 * A where A =\n[[-2, 10, 2, 1, -1, 0],\n [10, -8, -2, 12, -6, 5],\n [2, -2, 12, 0, -10, -10],\n [1, 12, 0, 3, 0, -13],\n [-1, -6, -10, 0, 14, 12],\n [0, 5, -10, -13, 12, -7]]\nRound your answer to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

117

OMEGA_500_SAMPLE

omega_500_490

algebra_func_intersection_level_2_0035

algebra_func_intersection

2

algebra_func_intersection

2

[ { "role": "user", "content": "How many solutions does the equation 1|(1(2|x - 2|)) - 3| + 2 = 3(-4x^2 + 4x + 5)^2 + 1(-4x^2 + 4x + 5) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

4

OMEGA_500_SAMPLE

omega_500_491

number_theory_prime_mod_level_3_0045

number_theory_prime_mod

3

number_theory_prime_mod

3

[ { "role": "user", "content": "Let p be the smallest prime number for which there exists a positive integer n where n^5+(-2) is a multiple of p^3, and n^5+(-2) is also a multiple of p^5. Find the largest positive integer m such that m^5+(-2) is a multiple of p^3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

23

OMEGA_500_SAMPLE

omega_500_492

algebra_func_derivative_sign_level_3_0055

algebra_func_derivative_sign

3

algebra_func_derivative_sign

3

[ { "role": "user", "content": "For the function f(x) = 4(4(-1(-3cos(3πx) - 3) - 1) - 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_493

algebra_func_extrema_coords_level_5_0066

algebra_func_extrema_coords

5

algebra_func_extrema_coords

5

[ { "role": "user", "content": "Find the x-coordinate of the leftmost local extremum of the function f(x) = -2|(-2(2|(2|(-5((2x - 2)/(3x - 2))^2 - 1((2x - 2)/(3x - 2)) + 2) - 2| + 1) - 3| - 2)^2 + 2(2|(2|(-5((2x - 2)/(3x - 2))^2 - 1((2x - 2)/(3x - 2)) + 2) - 2| + 1) - 3| - 2)) + 1| - 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_494

arithmetic_matrix_inverse_level_1_0047

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[[3, -9, 8],\n [0, -8, 6],\n [0, 0, -9]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

6

OMEGA_500_SAMPLE

omega_500_495

algebra_func_extrema_coords_level_4_0059

algebra_func_extrema_coords

4

algebra_func_extrema_coords

4

[ { "role": "user", "content": "How many local minima does the function f(x) = -1|(5(1((-2(-4x^2 + 1x - 5))/(-3(-4x^2 + 1x - 5) + 4))^2 + 4((-2(-4x^2 + 1x - 5))/(-3(-4x^2 + 1x - 5) + 4)) + 1) + 1) + 3| have in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

0

OMEGA_500_SAMPLE

omega_500_496

arithmetic_matrix_inverse_level_3_0015

arithmetic_matrix_inverse

3

arithmetic_matrix_inverse

3

[ { "role": "user", "content": "Calculate the inverse matrix of 1/180 times the following and determine the value at position (3, 5) of the resulting matrix. Your answer should be rounded to the nearest integer:\n[[-14, 7, 15, 6, 3, 9],\n [0, 15, 14, 13, 6, 10],\n [0, 0, 1, 4, -6, -10],\n [0, 0, 0, 12, -11, -15],\n [0, 0, 0, 0, -4, -11],\n [0, 0, 0, 0, 0, 14]]\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

-105

OMEGA_500_SAMPLE

omega_500_497

algebra_linear_equation_level_4_0084

algebra_linear_equation

4

algebra_linear_equation

4

[ { "role": "user", "content": "Let m be 4*((-2)/((-144)/(-204)) + 2 + (-341)/(-132)). Solve -4*g - 10*q + u = -9*q - 2*z - 3*l + 13, 5*q - z - 5*l - 42 = 2*g + 3*u, q - u - 3*l - 29 = 4*g, -2*g + z + 2*u + 2*l = m, -3*q + z + 4*l = -3*g - 33 for g.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

-5

OMEGA_500_SAMPLE

omega_500_498

number_theory_prime_mod_level_3_0094

number_theory_prime_mod

3

number_theory_prime_mod

3

[ { "role": "user", "content": "let p=19. Let N be the number of solutions m in [1, 2, ..., 19^1] that satisfy m^3+1=0 (mod 19^1). Let M be the minimum of these solutions. What is N+M?\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

11

OMEGA_500_SAMPLE

omega_500_499

combinatory_pattern_matching_level_3_0087

combinatory_pattern_matching

3

combinatory_pattern_matching

3

[ { "role": "user", "content": "Form a word by randomly choosing 4 letters from the multiset {h: 1, e: 1, s: 3, u: 1, m: 3}. What is the expected number of occurrences of the pattern 'e.*u' in each word? Round your answer to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}" } ]

0.08

OMEGA_500_SAMPLE