{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(-4(-2sin(1\u03c0x - 2) + 2) + 5) - 1| - 3 and g(x) = (-2(-5x^2 - 4x - 2) + 1)/(1(-5x^2 - 4x - 2) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(3|(4x^2 - 1x - 5) - 3| - 2)^2 - 1 = (-1(-3sin(1\u03c0x) - 3) + 3)/(-1(-3sin(1\u03c0x) - 3) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2((1(2e^(2x)))/(-1(2e^(2x)) - 2)) + 1)/(-2((1(2e^(2x)))/(-1(2e^(2x)) - 2)) + 3) and g(x) = -2sin(2\u03c0(5x^2 - 3x + 5)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1481", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3((-3((2x - 3)/(-2x - 2)) - 1)/(-3((2x - 3)/(-2x - 2)) + 1)) - 2 = -2sin(1\u03c0(5x - 5) - 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(5((3x + 2)/(-3x + 2)) + 5) - 3)/(1(5((3x + 2)/(-3x + 2)) + 5) + 3) and g(x) = -1sin(2\u03c0(-2x + 3) - 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(3sin(1\u03c0(-3|x - 2| + 1) - 3) + 3)^2 - 3(3sin(1\u03c0(-3|x - 2| + 1) - 3) + 3) + 1 and g(x) = (-2((-2x + 3)/(-2x - 2)))/(-1((-2x + 3)/(-2x - 2)) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(2|(1cos(3\u03c0x + 1) - 3) - 3| + 1) + 5 and g(x) = (-2(-3sin(1\u03c0x + 2) + 1) - 1)/(-1(-3sin(1\u03c0x + 2) + 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(-2(1x^2 - 5x + 2) - 2)^2 - 1 = 3sin(1\u03c0((2x - 2)/(1x)) + 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(2|(3x^2 + 2x - 3) - 3|)^2 - 1(2|(3x^2 + 2x - 3) - 3|) - 2 = (-1(5x - 2))/(-3(5x - 2) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-2cos(2\u03c0(4x + 1) - 1) + 3) - 1| + 2 and g(x) = -2cos(2\u03c0(-1x + 1) - 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2cos(1\u03c0(-2|((-2x - 3)/(2x)) - 3|) - 1) = (-1((1x - 1)/(3x)) + 3)/(3((1x - 1)/(3x))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(1\u03c0(-4(-1|x + 2| + 2) + 5) - 2) - 2 and g(x) = -2e^(2(-2x - 3)) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(-3|(2sin(2\u03c0x - 1) - 2) + 1| + 3) - 4 = (-3(-3cos(3\u03c0x + 2) + 2) - 2)/(-2(-3cos(3\u03c0x + 2) + 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1sin(2\u03c0(1|(1|x - 1| + 3) + 2| + 3)) and g(x) = (1((-1x + 1)/(1x - 3)) + 2)/(-3((-1x + 1)/(1x - 3)) + 4) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(-2cos(3\u03c0(-3x - 2) - 3) - 2)| and g(x) = (-1((-3x - 2)/(-1x + 2)) + 2)/(3((-3x - 2)/(-1x + 2))) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "124", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(1cos(1\u03c0(1x - 4) + 2) - 2) - 3 and g(x) = 3sin(3\u03c0((3x - 1)/(1x + 2)) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "154", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4((1(2|x| - 1) + 1)/(2(2|x| - 1) - 3))^2 - 5((1(2|x| - 1) + 1)/(2(2|x| - 1) - 3)) + 1 and g(x) = 2|(-5x^2 - 4x - 2) - 3| + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|(5(-3x^2 + 2x + 3) + 2) + 2| + 1 = 1(-3|x| - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(3(-1cos(2\u03c0x + 1) + 3) + 1) - 3| - 2 and g(x) = 1(-2e^(1x) - 3)^2 + 1(-2e^(1x) - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-4(3x^2 + 4x)^2 + 2(3x^2 + 4x) - 4))/(-3(-4(3x^2 + 4x)^2 + 2(3x^2 + 4x) - 4) + 2) and g(x) = -1cos(3\u03c0((-3x)/(3x - 1)) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-1(-1x + 2) - 2) + 3| - 2 and g(x) = -3|(4x^2 - 1x - 1) - 3| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|(1cos(1\u03c0(3|x| + 1) + 2) - 1)| = 2cos(2\u03c0(5x^2 + 3) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "658", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(2sin(3\u03c0(5x^2 - 2x + 5) - 3) - 3) + 1| - 2 and g(x) = 1e^(-1(-4x^2 + 3x - 2)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(2cos(3\u03c0(1|x - 3|) - 3) - 1)^2 + 5 and g(x) = 2(-3x + 5)^2 - 4(-3x + 5) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1cos(3\u03c0(-5(-3x + 3) + 4) - 1) + 2 = (3(3cos(3\u03c0x + 3)) + 2)/(-3(3cos(3\u03c0x + 3)) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-3(-5x - 3)^2 + 4(-5x - 3) + 5)^2 - 4(-3(-5x - 3)^2 + 4(-5x - 3) + 5) - 3 = 1sin(1\u03c0(1x + 1) - 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(2\u03c0(5(3|x - 1| - 3)^2 - 4(3|x - 1| - 3) - 1) + 1) and g(x) = -4(-2|x - 3| + 1)^2 + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "122", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(2((-1x + 1)/(-3x - 1)) + 2)^2 - 5(2((-1x + 1)/(-3x - 1)) + 2) and g(x) = -3|(2|x - 3| + 3) - 1| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(2(-1|x + 1| + 1)^2 + 3(-1|x + 1| + 1) - 3))/(-3(2(-1|x + 1| + 1)^2 + 3(-1|x + 1| + 1) - 3) + 4) and g(x) = -3|(1|x - 2| - 2) + 2| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(1e^(2(3x^2 - 4)) + 1) - 1| + 1 and g(x) = (-1(-3cos(3\u03c0x) - 1) + 2)/(-3(-3cos(3\u03c0x) - 1) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(2(-1x - 1)^2 - 2(-1x - 1) + 4) + 2 = -2sin(3\u03c0(3|x + 3|)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(1((1x - 3)/(-3x - 1))^2 + 2((1x - 3)/(-3x - 1)) + 3)^2 + 1(1((1x - 3)/(-3x - 1))^2 + 2((1x - 3)/(-3x - 1)) + 3) + 4 and g(x) = (2(2sin(1\u03c0x)) + 1)/(-1(2sin(1\u03c0x)) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(-1cos(1\u03c0(-4x^2 + 3x - 3) - 2) - 1)^2 + 3 and g(x) = 2|(-1x - 2) - 1| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "108", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(3\u03c0(3(-1|x - 3| - 1)^2 - 3) + 1) and g(x) = 3sin(2\u03c0(-2|x - 2| - 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "607", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|((3(-2cos(2\u03c0x + 1) + 2) - 1)/(1(-2cos(2\u03c0x + 1) + 2))) - 3| + 1 = -4((-2x - 1)/(-1x + 2))^2 + 1((-2x - 1)/(-1x + 2)) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"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}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1cos(3\u03c0(2(-4x) + 5)) - 2 and g(x) = 4(1e^(-1x) - 3)^2 + 4(1e^(-1x) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1((-2(2cos(3\u03c0x + 1)) + 3)/(1(2cos(3\u03c0x + 1))))^2 - 2((-2(2cos(3\u03c0x + 1)) + 3)/(1(2cos(3\u03c0x + 1)))) - 4 = -1(2|x| + 2)^2 - 3(2|x| + 2) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(1\u03c0(4(-5x^2 + 5x + 1) - 5) - 3) - 3 and g(x) = 2cos(1\u03c0(-2x - 5) + 1) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "981", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(1cos(2\u03c0(3|x| - 3)) + 1)^2 - 3(1cos(2\u03c0(3|x| - 3)) + 1) + 1 and g(x) = 1cos(1\u03c0(-1|x - 2| - 3) + 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(3cos(3\u03c0(3|x + 3| - 2) + 2) + 2) - 2)/(-1(3cos(3\u03c0(3|x + 3| - 2) + 2) + 2) + 3) and g(x) = -2(-5x - 2)^2 - 4(-5x - 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(2\u03c0(2|((3x - 1)/(2x + 1)) + 1| - 1) + 1) and g(x) = -1|(2x^2 - 2) + 1| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(2\u03c0((3(4x - 3) + 3)/(2(4x - 3) + 3)) - 2) + 1 and g(x) = 1(2e^(2x) - 1) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(-3|(-1cos(2\u03c0x - 3)) + 1|) - 4 and g(x) = 3(-1cos(2\u03c0x + 3) - 2)^2 + 2(-1cos(2\u03c0x + 3) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(2cos(1\u03c0(-3x^2 + 5x + 5) + 2) + 1))/(1(2cos(1\u03c0(-3x^2 + 5x + 5) + 2) + 1)) and g(x) = -1|(2|x + 2| - 2) - 3| + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(-1(-2x + 3)^2 - 3(-2x + 3) + 2)^2 + 2(-1(-2x + 3)^2 - 3(-2x + 3) + 2) + 3 and g(x) = 3e^(-1(-1x + 3)) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3cos(1\u03c0(2(-1|x - 3| - 3)^2 - 1(-1|x - 3| - 3))) + 3 and g(x) = 2sin(2\u03c0(4x^2 + 4) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "760", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1sin(2\u03c0((1(1|x + 1| + 1) - 2)/(-1(1|x + 1| + 1) + 2)) + 1) - 1 and g(x) = -3(-3sin(2\u03c0x + 3) + 2)^2 - 3(-3sin(2\u03c0x + 3) + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1sin(1\u03c0(1(3|x + 3| + 1)^2 + 1) - 2) - 1 and g(x) = 5(3sin(3\u03c0x - 3) + 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "167", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2cos(1\u03c0(-5(4x - 4) - 1) - 3) + 3 and g(x) = 1(5x^2 + 3x + 1)^2 + 1(5x^2 + 3x + 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4((2(-1x + 5) + 2)/(2(-1x + 5) - 1))^2 - 5((2(-1x + 5) + 2)/(2(-1x + 5) - 1)) - 3 = -1(-1sin(3\u03c0x + 1) + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(1|(1x - 4) + 1| - 3) + 5 = -2sin(2\u03c0(3|x - 1| - 3) + 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(1((-2x + 3)/(1x)) - 4) - 3)/(1(1((-2x + 3)/(1x)) - 4) - 2) and g(x) = 1(4x^2 + 4x - 5)^2 + 5(4x^2 + 4x - 5) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4((3(-1cos(1\u03c0x - 3) - 2) + 2)/(-1(-1cos(1\u03c0x - 3) - 2) + 3))^2 - 5((3(-1cos(1\u03c0x - 3) - 2) + 2)/(-1(-1cos(1\u03c0x - 3) - 2) + 3)) - 2 = 1((-3x + 2)/(-2x - 1))^2 + 3((-3x + 2)/(-2x - 1)) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4((1(1x + 2) + 2)/(1(1x + 2) - 2)) + 2 = -2sin(1\u03c0(-5x^2 - 5x - 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "384", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(3sin(2\u03c0(-5x^2 - 1x - 5) + 1) - 1) - 1| - 2 and g(x) = 1sin(3\u03c0(2|x + 2| - 1) - 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1422", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(1(-2|x + 2| + 1)^2 + 4(-2|x + 2| + 1) - 5)^2 + 4(1(-2|x + 2| + 1)^2 + 4(-2|x + 2| + 1) - 5) - 2 and g(x) = -5(-3x + 2) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(3\u03c0(-1((2x - 3)/(-2x - 1)) + 5) + 2) + 2 = -4((-3x - 1)/(-2x - 2)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-1(-1|x + 2| + 3)^2 + 4(-1|x + 2| + 3) - 5) + 5 and g(x) = 1|(3sin(3\u03c0x) + 3) + 2| + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(3\u03c0(4(-2x^2 + 5x - 3) - 1) + 3) - 3 and g(x) = 3(1cos(2\u03c0x + 3) - 1) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "437", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(1(-2sin(3\u03c0x + 1) + 2)^2 - 5(-2sin(3\u03c0x + 1) + 2) + 1)^2 - 5(1(-2sin(3\u03c0x + 1) + 2)^2 - 5(-2sin(3\u03c0x + 1) + 2) + 1) + 2 and g(x) = -1((-3x - 1)/(3x + 3))^2 - 1((-3x - 1)/(3x + 3)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1sin(3\u03c0(-1|(2x) - 2|) - 3) - 2 and g(x) = 3(1x + 2)^2 - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1e^(-2(-1((2x - 2)/(2x - 1))^2 + 5((2x - 2)/(2x - 1)) + 1)) + 3 = -3cos(1\u03c0((2x - 2)/(1x + 3)) - 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "87", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3cos(1\u03c0(2|(-2|x - 2| - 1) - 3| + 1) + 2) - 1 = 1(-2sin(3\u03c0x + 3) - 1)^2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-3(-2cos(2\u03c0x + 3) + 2)^2 - 4(-2cos(2\u03c0x + 3) + 2) + 2) and g(x) = (-2(-1sin(1\u03c0x + 3) + 3) + 1)/(3(-1sin(1\u03c0x + 3) + 3) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(2|(2sin(2\u03c0x)) + 3| - 3) - 2 and g(x) = -3|(-1x^2 - 5x + 5) - 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(-2sin(3\u03c0((-2x - 2)/(2x - 1)) - 3) + 1) - 3 and g(x) = 3cos(3\u03c0(1x^2 + 5x - 1) + 1) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "254", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1e^(1(-4((2x - 3)/(2x)) + 3)) + 2 and g(x) = -3sin(3\u03c0(-1|x| + 2) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-3sin(2\u03c0((-1x - 1)/(1x + 2)) + 3) - 1) - 3)/(2(-3sin(2\u03c0((-1x - 1)/(1x + 2)) + 3) - 1)) and g(x) = (3((2x + 3)/(2x + 2)) - 2)/(-3((2x + 3)/(2x + 2)) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(1|(2|x - 3| - 1) - 1| + 2)^2 - 4(1|(2|x - 3| - 1) - 1| + 2) + 1 and g(x) = (-2(2cos(3\u03c0x + 2)) + 3)/(3(2cos(3\u03c0x + 2)) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2((-2(-1cos(2\u03c0x + 3) - 2))/(1(-1cos(2\u03c0x + 3) - 2) + 2)) + 1)/(-1((-2(-1cos(2\u03c0x + 3) - 2))/(1(-1cos(2\u03c0x + 3) - 2) + 2)) + 3) = -1(-2x^2 - 2x + 4) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(-1|(4x^2 - 1x - 5)| + 2) - 5 = 3cos(3\u03c0(5x + 4) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(2\u03c0(-1(3x^2 + 3x - 4)^2 - 2(3x^2 + 3x - 4)) + 2) + 3 and g(x) = -2sin(1\u03c0((-2x + 1)/(3x - 1)) - 2) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3((1(2x + 2) + 3)/(-1(2x + 2)))^2 - 4((1(2x + 2) + 3)/(-1(2x + 2))) - 3 and g(x) = -1cos(3\u03c0(3x - 2) + 2) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-2(-2|x - 3|) - 4) = 2sin(1\u03c0(3x^2 + 1) - 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(-3cos(3\u03c0(1|x - 1| + 3) + 2)) + 1 and g(x) = -2(3x + 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(2(1x^2 - 3x - 1) - 2)^2 - 1(2(1x^2 - 3x - 1) - 2) + 4 and g(x) = -5((-2x - 3)/(2x + 2))^2 + 4((-2x - 3)/(2x + 2)) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-2cos(2\u03c0(1x^2 - 4) + 2) + 1) - 1)/(3(-2cos(2\u03c0(1x^2 - 4) + 2) + 1) + 1) and g(x) = -1|(1|x + 1| + 2)| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "73", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(3|(2|x - 1| - 3) - 3| - 2) + 2 and g(x) = (-1(-2x^2 + 5x + 3) - 2)/(-3(-2x^2 + 5x + 3) + 4) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(-2(-1e^(-1x) + 1)^2 + 2(-1e^(-1x) + 1) + 1) + 3| - 1 = -1cos(2\u03c0(3x^2 - 4x + 3) - 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "526", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(5(-2sin(3\u03c0x - 1))^2 - 3(-2sin(3\u03c0x - 1)) + 2)^2 + 3(5(-2sin(3\u03c0x - 1))^2 - 3(-2sin(3\u03c0x - 1)) + 2) + 2 = 4((-3x + 2)/(-2x)) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(-2|(-3x^2 + 3x - 1) + 1|) - 1)/(-1(-2|(-3x^2 + 3x - 1) + 1|) - 2) and g(x) = 1(-2|x| + 2)^2 + 1(-2|x| + 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(3|(-1x + 3) - 1| - 3)^2 - 4(3|(-1x + 3) - 1| - 3) - 3 = (-1(-2sin(2\u03c0x + 2) - 1) - 1)/(-1(-2sin(2\u03c0x + 2) - 1) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(2(4x + 3) + 1) + 2| + 3 and g(x) = 2sin(3\u03c0(5x - 1) + 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1cos(1\u03c0(-3|((-3x)/(1x - 3)) - 3|) - 3) - 1 and g(x) = -1|(-1sin(1\u03c0x - 2) + 2)| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "76", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(1(-3x^2 + 5x + 1)^2 - 2(-3x^2 + 5x + 1) - 1)^2 + 1(1(-3x^2 + 5x + 1)^2 - 2(-3x^2 + 5x + 1) - 1) + 5 and g(x) = -1(-1e^(2x) + 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2sin(3\u03c0(1|(-4x^2 + 5x + 2)| - 1)) - 2 = -2|((-2x - 3)/(1x - 3)) + 3| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "319", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|((-2(-3|x + 1| + 2) + 2)/(1(-3|x + 1| + 2) + 3)) + 2| and g(x) = (-3(3cos(2\u03c0x + 2) - 2) + 2)/(1(3cos(2\u03c0x + 2) - 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2e^(2((1(-1x - 1) - 1)/(-2(-1x - 1) + 3))) and g(x) = -4(3sin(3\u03c0x - 2) - 1)^2 + 5(3sin(3\u03c0x - 2) - 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "118", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(3cos(3\u03c0(1x) - 3) + 3) - 1)/(1(3cos(3\u03c0(1x) - 3) + 3) - 3) and g(x) = -3cos(1\u03c0(-2x + 3) + 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(2sin(3\u03c0((1x + 2)/(-2x + 1))) + 2)^2 + 5(2sin(3\u03c0((1x + 2)/(-2x + 1))) + 2) + 4 and g(x) = (-3(5x + 1) - 2)/(3(5x + 1) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "67", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4((-3(2x + 1) + 2)/(-3(2x + 1) - 1))^2 - 5((-3(2x + 1) + 2)/(-3(2x + 1) - 1)) + 3 = 3cos(2\u03c0(-1|x + 2| - 2) - 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|(-3sin(2\u03c0((2x + 3)/(2x)) + 1) - 3) - 1| + 1 and g(x) = 2cos(3\u03c0((-2x - 3)/(2x + 2)) + 1) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2((-3(1x^2 + 4x + 3) - 2)/(-2(1x^2 + 4x + 3) + 1)) + 3 and g(x) = -1sin(2\u03c0(3x - 1)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(3(-1cos(3\u03c0x - 1) - 2)^2 - 1(-1cos(3\u03c0x - 1) - 2) - 5)^2 - 1(3(-1cos(3\u03c0x - 1) - 2)^2 - 1(-1cos(3\u03c0x - 1) - 2) - 5) + 1 and g(x) = 1(-2e^(-1x) - 1) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "64", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(1(1x^2 - 2x + 1)^2 - 5(1x^2 - 2x + 1) + 4) - 3| and g(x) = 1|(1|x - 2| + 1) - 1| + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(-2cos(3\u03c0(-2|x - 1| + 1)) + 1) - 5 and g(x) = 1(1cos(2\u03c0x) - 2) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(4(-1sin(2\u03c0x + 2) + 1) - 5) + 2| - 2 and g(x) = -3((2x)/(-2x + 3)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "73", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2((-1(-1|x + 2| - 2) - 1)/(1(-1|x + 2| - 2))) + 5 and g(x) = -3cos(3\u03c0(-3x^2 - 1x - 4) + 1) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1136", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(-1(-1e^(-1x) + 2) + 4) - 3 and g(x) = -5(1cos(2\u03c0x - 3) - 2)^2 + 2(1cos(2\u03c0x - 3) - 2) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(-4(-5x^2 - 4x + 2)^2 + 2)^2 - 3(-4(-5x^2 - 4x + 2)^2 + 2) - 1 = (3(1|x + 3|) + 2)/(3(1|x + 3|) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(3(2cos(1\u03c0x + 1) - 2)^2 - 3(2cos(1\u03c0x + 1) - 2) + 2)^2 - 4(3(2cos(1\u03c0x + 1) - 2)^2 - 3(2cos(1\u03c0x + 1) - 2) + 2) and g(x) = 3|(3e^(1x) + 1) + 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3e^(-1((2(-1|x| + 2) - 3)/(-3(-1|x| + 2) - 2))) + 1 and g(x) = -1sin(2\u03c0(2x + 3) + 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-1(-1sin(3\u03c0x + 3) + 2) + 1)^2 + 4(-1(-1sin(3\u03c0x + 3) + 2) + 1) - 5 = -1(-1x - 5)^2 - 5(-1x - 5) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2cos(2\u03c0(5(1x^2 + 4x + 1)^2 + 1) + 1) - 2 and g(x) = 3cos(2\u03c0(-2|x - 1|) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "624", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1sin(1\u03c0(5(-4x^2 + 3x - 4) + 2) + 2) - 1 and g(x) = -4(-5x^2 - 5x + 4) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(1|(3x^2 + 1x - 2) - 2| - 3) + 3| - 3 and g(x) = -2|(3sin(2\u03c0x - 3)) - 2| + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|(5(1x - 5)^2 + 5(1x - 5) - 4) - 1| + 3 and g(x) = (2(2x - 1) - 2)/(-2(2x - 1) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(-2e^(1((-3x - 3)/(2x - 1))) + 1) - 2)/(2(-2e^(1((-3x - 3)/(2x - 1))) + 1) + 2) and g(x) = 3sin(1\u03c0(5x^2 - 5x - 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "868", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1sin(3\u03c0((-3((1x + 3)/(1x - 2)) + 3)/(-2((1x + 3)/(1x - 2)))) + 2) and g(x) = (1(5x + 4) - 2)/(-1(5x + 4) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "98", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(-3sin(3\u03c0(-5x^2 - 5x - 3) + 2) + 1)| + 2 and g(x) = 1|((-2x - 2)/(2x - 1))| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "930", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-2((-1x + 2)/(-1x - 1))) - 3)/(2(-2((-1x + 2)/(-1x - 1))) + 1) and g(x) = 2cos(1\u03c0(-3|x - 3|) + 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(-2sin(2\u03c0(1x - 2) - 1) - 2) - 3)/(-3(-2sin(2\u03c0(1x - 2) - 1) - 2)) and g(x) = 4(2x + 4) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(5((-2x + 3)/(3x - 1))^2 + 2((-2x + 3)/(3x - 1)) + 5) + 5 and g(x) = (3(1cos(1\u03c0x - 1)) - 1)/(2(1cos(1\u03c0x - 1)) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1cos(2\u03c0(4(5x + 2)^2 - 2(5x + 2) - 2) - 1) + 1 = (-1(-5x^2 - 4x))/(-2(-5x^2 - 4x)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1442", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(1\u03c0(-1(4x^2 - 3x + 2)) + 1) + 2 and g(x) = (-1(-1e^(-1x) - 1) + 1)/(-1(-1e^(-1x) - 1) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "409", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2((2(1|x + 2| - 3) + 1)/(1(1|x + 2| - 3))) + 2)/(-1((2(1|x + 2| - 3) + 1)/(1(1|x + 2| - 3)))) and g(x) = (2(4x^2 + 2x) - 3)/(2(4x^2 + 2x)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(4(3sin(3\u03c0x - 1) - 3)^2 + 1(3sin(3\u03c0x - 1) - 3) - 4)| + 2 and g(x) = 4(-3x^2 + 3x + 2)^2 - 3(-3x^2 + 3x + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(3\u03c0((-3((-3x - 3)/(-1x + 2)) + 3)/(-3((-3x - 3)/(-1x + 2)) + 4)) - 2) + 1 = -1(-3sin(1\u03c0x - 2) + 3)^2 + 4(-3sin(1\u03c0x - 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-2|(-1e^(2x) - 2) + 2| + 3)^2 + 1(-2|(-1e^(2x) - 2) + 2| + 3) - 2 = -3cos(3\u03c0((3x - 1)/(-3x)) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((2(3sin(2\u03c0x - 2) - 1))/(1(3sin(2\u03c0x - 2) - 1) + 1))^2 - 5((2(3sin(2\u03c0x - 2) - 1))/(1(3sin(2\u03c0x - 2) - 1) + 1)) + 1 and g(x) = -5(-1cos(2\u03c0x)) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(3|(1x^2 - 4x - 5) + 2| - 2) + 1| + 2 = 5(2sin(3\u03c0x + 3) - 1)^2 + 4(2sin(3\u03c0x + 3) - 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(2cos(2\u03c0(2x + 2) + 1) + 1) + 3)/(1(2cos(2\u03c0(2x + 2) + 1) + 1) + 1) and g(x) = (-1(-3cos(1\u03c0x + 1) - 1) + 2)/(2(-3cos(1\u03c0x + 1) - 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-4(-1x^2 + 2x + 1)^2 + 2(-1x^2 + 2x + 1) - 1)^2 - 3(-4(-1x^2 + 2x + 1)^2 + 2(-1x^2 + 2x + 1) - 1) + 5 and g(x) = (1(1|x| - 1) - 3)/(1(1|x| - 1) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(4(-3e^(1x) + 2)^2 - 1(-3e^(1x) + 2)) + 1)/(-3(4(-3e^(1x) + 2)^2 - 1(-3e^(1x) + 2)) + 4) and g(x) = -5(-2sin(1\u03c0x + 3)) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|(-4(-2sin(2\u03c0x) - 3) - 3) - 2| + 2 = 1|(2x^2 - 5x + 2) + 2| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2((2(-1x - 4))/(-2(-1x - 4) - 2)) - 2)/(1((2(-1x - 4))/(-2(-1x - 4) - 2))) = -5(-1cos(1\u03c0x - 2) + 1)^2 - 5(-1cos(1\u03c0x - 2) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(3(-2|x - 3| + 3) - 5) + 3| - 1 and g(x) = 3(1|x + 2| - 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2((-1((2x + 1)/(2x + 3)) + 1)/(2((2x + 1)/(2x + 3)))))/(2((-1((2x + 1)/(2x + 3)) + 1)/(2((2x + 1)/(2x + 3))))) and g(x) = -2sin(3\u03c0(2x + 2) + 1) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(1((1x)/(-1x + 3)) + 2) + 2)/(3(1((1x)/(-1x + 3)) + 2) + 1) = -2|(3sin(2\u03c0x + 2)) - 3| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(-2|(-3sin(3\u03c0x + 1) - 1)| + 3) and g(x) = (-1(-2|x - 3| + 1))/(-1(-2|x - 3| + 1) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "116", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1e^(1(3|(-2|x + 2| - 1) + 3| - 2)) + 2 = -1|(-2sin(1\u03c0x + 2) - 3)| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(4(-2cos(1\u03c0x - 2) - 2) + 4))/(1(4(-2cos(1\u03c0x - 2) - 2) + 4) + 3) and g(x) = 2(-1|x + 3| - 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2cos(1\u03c0(-1|(2|x + 2| - 3) - 1| + 1) - 2) + 3 and g(x) = -3(-2cos(3\u03c0x + 1) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5((1(-3e^(-2x) + 3) + 2)/(-2(-3e^(-2x) + 3) - 3))^2 + 2((1(-3e^(-2x) + 3) + 2)/(-2(-3e^(-2x) + 3) - 3)) - 1 and g(x) = -1(-1x^2 + 4x + 3)^2 - 5(-1x^2 + 4x + 3) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(-1e^(-2(1x^2 + 4x)) - 3) + 2| = -2sin(1\u03c0(-1x^2 + 5x) + 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "180", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2((2(-3sin(2\u03c0x) + 2) + 1)/(-2(-3sin(2\u03c0x) + 2) - 3))^2 + 5((2(-3sin(2\u03c0x) + 2) + 1)/(-2(-3sin(2\u03c0x) + 2) - 3)) + 1 and g(x) = 2|(-2cos(3\u03c0x - 2) + 2) - 1| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(2(-3|x - 2| + 1)^2 + 2(-3|x - 2| + 1)) - 3| - 3 and g(x) = (-2(2sin(2\u03c0x - 3)) - 1)/(-3(2sin(2\u03c0x - 3)) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(3sin(2\u03c0(-1|x - 3| - 3) + 1) + 3)^2 - 4(3sin(2\u03c0(-1|x - 3| - 3) + 1) + 3) + 4 and g(x) = -4(5x^2 + 2x + 4) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-3cos(2\u03c0(-3x^2 - 5x + 2) + 1) + 2) - 4 and g(x) = 3e^(1(3x^2 - 5x - 1)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(1(3cos(3\u03c0x) - 2)^2 - 3(3cos(3\u03c0x) - 2) - 5) - 3)/(-3(1(3cos(3\u03c0x) - 2)^2 - 3(3cos(3\u03c0x) - 2) - 5) - 1) = -2e^(2(-2|x + 1|)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3sin(3\u03c0((-1(-1|x + 2| - 2) + 2)/(-2(-1|x + 2| - 2) - 1)) - 2) - 2 and g(x) = 3sin(3\u03c0(-1x^2 - 4x - 5) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3((3(2|x + 2| - 3) + 1)/(1(2|x + 2| - 3) - 2)) + 1)/(-1((3(2|x + 2| - 3) + 1)/(1(2|x + 2| - 3) - 2)) + 3) and g(x) = 3cos(1\u03c0(-5x^2 + 5x - 1) + 3) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "102", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(1|(3x + 3) - 2| - 3) + 1| - 2 = -1sin(1\u03c0(-1x^2 + 3x - 2) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(-3(-1|x + 2| + 1) - 1) + 3| = (2(-2x + 1) + 3)/(3(-2x + 1) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1((-1(2x - 5) + 1)/(-3(2x - 5) - 2)) + 3)/(2((-1(2x - 5) + 1)/(-3(2x - 5) - 2)) - 1) = -2|(-2sin(2\u03c0x + 3) - 2) - 3| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(-3e^(-2((1x)/(-3x + 1))) + 3) - 2)/(-1(-3e^(-2((1x)/(-3x + 1))) + 3) - 2) and g(x) = -1sin(1\u03c0(1x^2 - 5x - 5) + 2) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "106", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(5(-2cos(1\u03c0x) - 2) - 5) + 4 and g(x) = 2|(-3x^2 - 1x) - 3| + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(2\u03c0(5((-1x - 1)/(1x))^2 - 2((-1x - 1)/(1x)) - 1) - 3) - 1 and g(x) = (2(-3x + 2) + 1)/(1(-3x + 2) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "118", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(1\u03c0(2|(-5x^2 + 4x - 2) - 2| + 3) - 3) - 1 = 1e^(-2(-4x - 2)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "771", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(1sin(2\u03c0(-1x + 5) - 2)))/(2(1sin(2\u03c0(-1x + 5) - 2)) + 1) and g(x) = 2sin(1\u03c0(-2|x - 3| + 2) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2((3((2x + 2)/(3x + 1)))/(3((2x + 2)/(3x + 1)) - 2)) - 1 and g(x) = -1cos(3\u03c0(-4x - 4) + 1) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(3|(2|x| - 1)|) - 1| + 2 = (1((2x - 2)/(2x + 3)))/(-3((2x - 2)/(2x + 3)) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(1(-5x - 4)^2 - 3(-5x - 4) - 5) - 1| + 2 and g(x) = -2((-2x + 2)/(2x))^2 - 5((-2x + 2)/(2x)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(2|(-3sin(3\u03c0x + 2) - 1) - 3| - 2) + 1)/(2(2|(-3sin(3\u03c0x + 2) - 1) - 3| - 2) - 1) and g(x) = -1((3x)/(1x)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(-2cos(1\u03c0(-3|x|) + 1) + 1)^2 + 1(-2cos(1\u03c0(-3|x|) + 1) + 1) - 4 = -2|((-2x + 3)/(2x - 1))| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3((2(-3|x + 3| - 2) + 2)/(1(-3|x + 3| - 2) - 3)) + 1)/(2((2(-3|x + 3| - 2) + 2)/(1(-3|x + 3| - 2) - 3)) + 3) and g(x) = (-1(-3sin(1\u03c0x + 1) + 1) - 1)/(2(-3sin(1\u03c0x + 1) + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1sin(2\u03c0(3|(1x^2 + 3x - 1) - 3| + 3) - 1) and g(x) = -3cos(1\u03c0((2x + 1)/(1x + 3)) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "224", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3((3(-3|x - 3| - 3) + 3)/(2(-3|x - 3| - 3) - 3)) - 2 = 3cos(1\u03c0(5x^2 + 4x - 5) + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "292", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-2(-3x - 5)^2 - 5(-3x - 5) - 2) + 1 and g(x) = (2(-2cos(3\u03c0x) + 2) + 3)/(-3(-2cos(3\u03c0x) + 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3sin(3\u03c0(-3|((-1x - 3)/(1x)) + 1| - 3) - 2) - 2 = -1|(1|x + 2| + 2) - 2| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "142", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(5(-1x + 2)^2 - 1)^2 - 5(5(-1x + 2)^2 - 1) - 3 and g(x) = 3cos(2\u03c0(2|x - 2| + 3) + 3) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(1sin(2\u03c0(-2x^2 - 5x) - 2) - 3) + 1| - 1 and g(x) = -2|(-2cos(3\u03c0x) - 2) - 3| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "243", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(2sin(1\u03c0((-2x + 2)/(1x - 3)) + 2) + 1) + 2 and g(x) = -3sin(1\u03c0(-1x^2 - 4x - 2) + 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "132", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(2\u03c0((2((3x + 2)/(2x - 1)) + 2)/(2((3x + 2)/(2x - 1)) - 1)) + 2) = -1cos(3\u03c0(-4x - 4) + 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "127", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-4(-1sin(2\u03c0x + 2) - 2)^2 - 3(-1sin(2\u03c0x + 2) - 2) - 4)^2 + 1(-4(-1sin(2\u03c0x + 2) - 2)^2 - 3(-1sin(2\u03c0x + 2) - 2) - 4) - 4 and g(x) = -2|(3|x| + 1) - 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|(1(3sin(2\u03c0x - 2) - 2) + 2) - 2| and g(x) = 4((3x - 3)/(1x + 3)) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-2(-3|x + 1| + 3) + 3) - 1)/(2(-2(-3|x + 1| + 3) + 3) - 1) = (-2(5x^2 - 2x - 5) - 3)/(-3(5x^2 - 2x - 5) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1cos(3\u03c0((-3((2x + 3)/(-2x + 1)) + 2)/(1((2x + 3)/(-2x + 1)) + 3)) - 2) - 1 and g(x) = 2sin(3\u03c0(-5x^2 + 1x - 5) - 2) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "732", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1e^(-1(-3|(-3|x| + 1) + 3| - 2)) - 3 and g(x) = -2|(3cos(1\u03c0x) - 3)| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(3(-2cos(1\u03c0x - 1) + 2)^2 + 5(-2cos(1\u03c0x - 1) + 2) - 1) + 2 = -3(-2x - 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(-4(-1sin(1\u03c0x + 3) - 1) - 5) + 2)/(-3(-4(-1sin(1\u03c0x + 3) - 1) - 5) + 2) and g(x) = 3|(2x + 4) + 1| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(-3sin(2\u03c0(2|x + 3| - 2) - 2) - 1))/(2(-3sin(2\u03c0(2|x + 3| - 2) - 2) - 1) + 1) and g(x) = -4(-1|x + 3| + 1)^2 + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|(-2(2|x + 2| - 3) - 4) + 3| + 1 and g(x) = (-1(-4x^2 + 3x - 4) - 2)/(1(-4x^2 + 3x - 4) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-1|(-2x - 3) - 2| - 3) - 2)/(-1(-1|(-2x - 3) - 2| - 3)) and g(x) = -4(1sin(1\u03c0x - 2) - 1)^2 - 3(1sin(1\u03c0x - 2) - 1) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-1(2sin(3\u03c0x - 1) + 1)^2 + 5(2sin(3\u03c0x - 1) + 1) + 2) and g(x) = 2|(3|x + 1| - 1) + 1| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(3cos(1\u03c0(-3x + 1) + 2) - 2))/(-3(3cos(1\u03c0(-3x + 1) + 2) - 2) - 2) and g(x) = -2|(-2x^2 - 4x - 2) + 1| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(-2sin(3\u03c0(2x + 5) - 1) + 1) - 2| - 2 and g(x) = -5(-1x^2 - 4x - 1) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-2((3x + 1)/(-2x + 3))^2 - 1((3x + 1)/(-2x + 3)) - 3) + 1| - 1 and g(x) = 1(1cos(1\u03c0x) + 3)^2 - 4(1cos(1\u03c0x) + 3) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(1|(-3sin(3\u03c0x) - 3) + 1| - 3) + 2)/(2(1|(-3sin(3\u03c0x) - 3) + 1| - 3) + 1) = -1(-1x^2 + 2x + 2)^2 - 5(-1x^2 + 2x + 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(2(-2x^2 - 1x + 2) + 5) - 2)/(-2(2(-2x^2 - 1x + 2) + 5) + 3) and g(x) = 2cos(3\u03c0(-4x^2 + 4x - 5) - 3) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(2cos(3\u03c0(-1|x + 2| + 1) - 3) + 2)^2 - 4(2cos(3\u03c0(-1|x + 2| + 1) - 3) + 2) - 1 = -1|(3cos(1\u03c0x) + 2) + 1| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(2\u03c0(-4(2|x - 3|) - 5) - 1) + 1 and g(x) = 4((3x - 3)/(-3x - 1))^2 + 1((3x - 3)/(-3x - 1)) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-5(3cos(2\u03c0x + 3) + 2)^2 - 4(3cos(2\u03c0x + 3) + 2) + 3) + 5 and g(x) = 4(1x + 4)^2 + 3(1x + 4) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(2(3sin(2\u03c0x + 3) - 2) + 2)^2 + 2(2(3sin(2\u03c0x + 3) - 2) + 2) - 5 and g(x) = 3(-1|x| + 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|((2(4x - 5) + 2)/(-2(4x - 5))) - 2| - 3 and g(x) = 3(1sin(3\u03c0x - 3) + 3) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(2\u03c0(1(1x^2 - 3)^2 + 3(1x^2 - 3) - 2) - 3) - 1 and g(x) = 1((2x - 3)/(-3x + 2)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1045", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1((-1(-2x + 5) - 2)/(-3(-2x + 5) - 2)) + 3)/(1((-1(-2x + 5) - 2)/(-3(-2x + 5) - 2)) + 3) and g(x) = -2sin(1\u03c0(1|x - 1| + 3)) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(2\u03c0(1(-1|x - 2| - 1)^2 + 1(-1|x - 2| - 1) + 2) - 3) - 1 and g(x) = -3|(-1|x - 2| + 3) + 1| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "69", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1e^(2(1|((3x + 2)/(-3x + 1)) + 1| - 2)) - 3 = 2cos(1\u03c0(1x^2 + 3x - 4) + 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "202", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(3|(3sin(3\u03c0x - 2) - 1) + 2| - 1))/(-2(3|(3sin(3\u03c0x - 2) - 1) + 2| - 1) + 1) and g(x) = 1sin(1\u03c0(3|x + 3| + 3) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "137", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(2|(-3sin(1\u03c0x + 2) + 3) + 2| - 3)^2 - 4(2|(-3sin(1\u03c0x + 2) + 3) + 2| - 3) + 1 and g(x) = -4(-1x - 4) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(3sin(2\u03c0(-3|x - 2| + 1) - 2) + 3) + 3)/(-1(3sin(2\u03c0(-3|x - 2| + 1) - 2) + 3) + 3) and g(x) = -1(1|x - 2| - 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "74", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(1cos(2\u03c0(-2|x - 2| - 3) - 1) - 3) - 1| = (2(2sin(3\u03c0x - 2) - 1) - 3)/(-2(2sin(3\u03c0x - 2) - 1) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|((3(2sin(2\u03c0x - 1) + 1) - 3)/(-2(2sin(2\u03c0x - 1) + 1) + 1)) - 3| and g(x) = 1|(-1x^2 + 3x) + 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(2sin(1\u03c0((1x + 2)/(-2x + 3)) + 3) + 3) + 3| and g(x) = (-3(2sin(3\u03c0x - 3) - 1) + 3)/(1(2sin(3\u03c0x - 3) - 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(1sin(2\u03c0((-1x + 1)/(-2x + 3)) - 2) + 2) + 3)/(3(1sin(2\u03c0((-1x + 1)/(-2x + 3)) - 2) + 2) - 2) and g(x) = (-3(3x) - 1)/(-2(3x) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3sin(1\u03c0(1|(-4x^2 - 3x) - 2| - 3) + 3) - 3 and g(x) = (3(-2e^(-1x) - 1) + 2)/(2(-2e^(-1x) - 1) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "387", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2e^(-1((-3(-2x^2 + 2x - 1) - 3)/(1(-2x^2 + 2x - 1) + 2))) and g(x) = 4(-1cos(3\u03c0x) + 2)^2 - 5(-1cos(3\u03c0x) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((2(2|x + 1| - 1) - 3)/(-2(2|x + 1| - 1) - 1))^2 - 3((2(2|x + 1| - 1) - 3)/(-2(2|x + 1| - 1) - 1)) + 2 and g(x) = (1(3sin(3\u03c0x) + 1) + 2)/(2(3sin(3\u03c0x) + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(-4(3cos(2\u03c0x + 2) - 2)^2 + 1(3cos(2\u03c0x + 2) - 2) - 3) + 1 and g(x) = 4(5x + 1) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(-1cos(2\u03c0(1x^2 - 3x - 1) + 2) + 3)^2 - 5(-1cos(2\u03c0(1x^2 - 3x - 1) + 2) + 3) - 1 and g(x) = 2|(-3x + 1) - 1| + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1cos(1\u03c0(-1(4x - 1)^2 + 4(4x - 1) + 3) + 2) + 3 = 2|(-2cos(2\u03c0x - 1) + 2) - 3| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "381", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(-1((-1x + 1)/(-1x + 2)) - 1)^2 + 4(-1((-1x + 1)/(-1x + 2)) - 1) + 5 = -3(2x^2 + 5x - 1)^2 - 3(2x^2 + 5x - 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2((-2(1x^2 + 2x + 5) + 3)/(1(1x^2 + 2x + 5))))/(-2((-2(1x^2 + 2x + 5) + 3)/(1(1x^2 + 2x + 5))) + 3) and g(x) = -2cos(1\u03c0(-1x - 2) - 1) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2((1(-3sin(1\u03c0x - 2) - 2) + 3)/(1(-3sin(1\u03c0x - 2) - 2))) - 4 and g(x) = -2cos(2\u03c0(-2x^2 + 2x - 4) - 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "385", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(2\u03c0((1(-1x^2 + 3x) - 3)/(-1(-1x^2 + 3x) + 2)) + 3) + 2 and g(x) = (-1(3|x + 2| + 1) + 1)/(-2(3|x + 2| + 1) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "87", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(1((-2x - 1)/(-1x)) - 2)^2 - 1(1((-2x - 1)/(-1x)) - 2) - 1 and g(x) = 2sin(1\u03c0(5x^2 + 1x + 1) + 3) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "433", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1((-3(-3cos(3\u03c0x + 3)) - 3)/(-2(-3cos(3\u03c0x + 3)) - 1))^2 - 3((-3(-3cos(3\u03c0x + 3)) - 3)/(-2(-3cos(3\u03c0x + 3)) - 1)) - 1 = 5(1e^(1x) + 2)^2 - 2(1e^(1x) + 2) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "70", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(-3sin(2\u03c0(-2|x - 3| - 2) + 2) + 1) + 2)/(-1(-3sin(2\u03c0(-2|x - 3| - 2) + 2) + 1)) and g(x) = 1sin(3\u03c0(3|x - 1| - 1) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "118", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-1(3cos(1\u03c0x - 1) - 1)^2 - 4(3cos(1\u03c0x - 1) - 1) + 1) + 2)/(-1(-1(3cos(1\u03c0x - 1) - 1)^2 - 4(3cos(1\u03c0x - 1) - 1) + 1) - 2) and g(x) = -3(-2e^(1x) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(3sin(1\u03c0(-2|x - 2|) - 3) + 1) + 5 and g(x) = (-1((1x)/(3x - 2)) - 2)/(1((1x)/(3x - 2)) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|((2((1x + 2)/(-2x + 1)) + 3)/(-3((1x + 2)/(-2x + 1)) + 2))| + 2 = -3sin(1\u03c0(-5x^2 - 3x + 5) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2cos(3\u03c0((3(3x^2 - 1x - 3) - 1)/(-1(3x^2 - 1x - 3))) - 2) and g(x) = 2sin(3\u03c0(3|x + 3| - 2) - 1) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "197", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(2(-1|x - 2| + 3) + 2)^2 + 3(2(-1|x - 2| + 3) + 2) - 5 = -3((2x - 3)/(-2x + 1)) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(3cos(3\u03c0(2x^2 - 1x) - 1) - 1) - 1 and g(x) = 5(-1|x| + 2)^2 - 1(-1|x| + 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "129", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3((3(3|x + 1| + 3) - 2)/(3(3|x + 1| + 3))) - 3)/(-1((3(3|x + 1| + 3) - 2)/(3(3|x + 1| + 3))) - 1) and g(x) = 1(-3cos(3\u03c0x - 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3cos(1\u03c0(2(2x^2)) - 2) - 1 and g(x) = 2sin(3\u03c0(1|x + 1| + 1) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "786", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(1|(3sin(2\u03c0x) + 1) - 1| - 2) + 3)/(1(1|(3sin(2\u03c0x) + 1) - 1| - 2) + 1) = 1|(-1x^2 - 1) + 1| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "68", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|(-3cos(3\u03c0(-4x^2 + 5x) - 1) + 1) - 2| - 3 = (-2(-3x + 1) + 2)/(1(-3x + 1) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2e^(1(-3(1|x - 3| + 3)^2 - 3(1|x - 3| + 3) - 5)) + 2 and g(x) = 2cos(3\u03c0(-4x^2 + 1x + 4)) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1432", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4((-1(-2x^2 + 5x + 4) + 1)/(-2(-2x^2 + 5x + 4) - 1))^2 + 5((-1(-2x^2 + 5x + 4) + 1)/(-2(-2x^2 + 5x + 4) - 1)) - 3 and g(x) = -3(2x^2 - 3x + 3) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(2\u03c0(-1|(-2|x + 2| + 3) - 3|) - 3) and g(x) = 1e^(2((2x + 2)/(-1x + 2))) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "67", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(2cos(2\u03c0(2x^2 - 2x - 1)))^2 - 4(2cos(2\u03c0(2x^2 - 2x - 1))) + 2 = 1cos(2\u03c0(1x + 4) + 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "802", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2cos(2\u03c0(-2|(5x)| + 3) + 3) = 2sin(2\u03c0(1|x + 2| + 1)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "200", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-2sin(1\u03c0(2|x + 3| - 3) + 3)) - 4 and g(x) = -3(1sin(2\u03c0x) + 2)^2 + 2(1sin(2\u03c0x) + 2) + 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|((-1(-3sin(3\u03c0x - 3) - 2) + 3)/(-1(-3sin(3\u03c0x - 3) - 2) - 2))| - 2 and g(x) = -2(-2cos(2\u03c0x + 1) - 3)^2 - 3(-2cos(2\u03c0x + 1) - 3) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(2|(-3e^(-1x) - 3) + 2| - 3)^2 - 5(2|(-3e^(-1x) - 3) + 2| - 3) + 4 and g(x) = -3cos(3\u03c0(-2x + 5) + 1) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3((-1(1x^2 - 1x - 3) + 2)/(-3(1x^2 - 1x - 3))) - 1)/(-2((-1(1x^2 - 1x - 3) + 2)/(-3(1x^2 - 1x - 3))) - 3) and g(x) = (-2(3x^2 - 2x + 1) + 3)/(2(3x^2 - 2x + 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(5(-3x^2 + 4x - 1)^2 + 1(-3x^2 + 4x - 1) - 3)^2 + 5(5(-3x^2 + 4x - 1)^2 + 1(-3x^2 + 4x - 1) - 3) - 2 and g(x) = 1((2x + 3)/(3x - 2)) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3sin(3\u03c0(-2((-3x - 2)/(1x + 3))^2 - 4((-3x - 2)/(1x + 3)) - 4) - 3) + 2 = (3(-2|x - 1| + 2))/(2(-2|x - 1| + 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "448", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-2cos(1\u03c0(2|x - 1| + 1)) + 1) - 1)/(1(-2cos(1\u03c0(2|x - 1| + 1)) + 1) - 3) and g(x) = (2((1x - 1)/(-3x + 1)) + 1)/(2((1x - 1)/(-3x + 1)) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(-5(-1x^2 - 3x) + 2)^2 - 2(-5(-1x^2 - 3x) + 2) + 5 = -3(3cos(3\u03c0x) + 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(-3(-1e^(-1x) + 1)) - 4 and g(x) = 1(-1cos(1\u03c0x - 1) + 3)^2 + 3(-1cos(1\u03c0x - 1) + 3) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4((2(1x^2 + 1x - 4) - 1)/(-2(1x^2 + 1x - 4))) + 5 and g(x) = -3|(-3x - 4)| + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(2sin(3\u03c0(-1|x|) - 2) - 3)^2 - 2 and g(x) = -5(-3cos(2\u03c0x) + 2) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2cos(1\u03c0(5((-1x - 3)/(2x + 2))^2 - 5((-1x - 3)/(2x + 2)) + 2) - 3) - 3 and g(x) = -2|(-1cos(1\u03c0x + 2) + 1) - 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(1(2|x + 1| - 2)^2 - 1(2|x + 1| - 2) - 2) + 5 and g(x) = -1sin(1\u03c0(-5x - 5) - 2) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(2cos(2\u03c0(1x) - 3) + 2) - 1)/(-1(2cos(2\u03c0(1x) - 3) + 2)) and g(x) = 4(-1|x| + 1) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(2|(-4x - 2) - 3| + 2) - 3)/(2(2|(-4x - 2) - 3| + 2) + 2) and g(x) = 1(4x^2 + 2x - 5)^2 + 3(4x^2 + 2x - 5) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(-5(1x^2 - 5x + 1))| - 3 and g(x) = (3(2e^(2x) + 3) - 2)/(2(2e^(2x) + 3) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(4(1x^2 - 1x - 1) + 1) - 2| - 2 and g(x) = (3(-2e^(1x) - 2))/(1(-2e^(1x) - 2) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(2\u03c0((-3(3x + 1) - 3)/(-2(3x + 1) + 3))) + 1 and g(x) = 1(-2x^2 - 1x - 5)^2 + 5(-2x^2 - 1x - 5) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(-2sin(1\u03c0(3x^2 + 4x - 5) + 1) + 2) + 1| + 3 and g(x) = 1(-4x - 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3((1(-2|x + 2| + 3) + 2)/(1(-2|x + 2| + 3) + 1)) - 2 and g(x) = -3(-3cos(1\u03c0x - 3) + 2)^2 - 4(-3cos(1\u03c0x - 3) + 2) + 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(3sin(1\u03c0((-2x + 3)/(-3x - 1)) + 3) - 1) - 3| - 3 = -2cos(1\u03c0(-3x^2 - 5x + 2) - 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "584", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1sin(2\u03c0(4(3|x + 1| - 3)^2 + 1(3|x + 1| - 3) + 1) - 3) - 1 and g(x) = (3(-5x^2 - 2x + 1) + 2)/(3(-5x^2 - 2x + 1) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(2cos(2\u03c0(-1|x|) - 3) + 1)^2 + 3 and g(x) = -1((3x + 3)/(-2x - 1))^2 - 1((3x + 3)/(-2x - 1)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(2\u03c0(5((-3x - 3)/(1x - 2))^2 + 3((-3x - 3)/(1x - 2)))) - 1 and g(x) = (-1(1|x + 1| - 1) - 3)/(3(1|x + 1| - 1) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "616", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2sin(1\u03c0((-3(-1|x| - 2) - 3)/(-1(-1|x| - 2) - 3))) + 1 and g(x) = -2cos(1\u03c0((1x - 2)/(-3x - 3)) - 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "133", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1sin(1\u03c0((1(2x^2 - 4x - 1) - 1)/(1(2x^2 - 4x - 1) + 3)) + 3) + 1 and g(x) = -1(-4x^2 + 2x + 4) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1((-3(4x) + 2)/(-1(4x) + 2)) - 5 and g(x) = -3|(-3|x + 2| + 3) - 1| + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(3e^(1(-1x^2 - 4x - 1)) - 3)^2 + 4(3e^(1(-1x^2 - 4x - 1)) - 3) - 2 = 1sin(3\u03c0((-1x - 1)/(3x + 3)) + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-4((3x - 3)/(-1x))^2 + 5((3x - 3)/(-1x))) + 2)/(-3(-4((3x - 3)/(-1x))^2 + 5((3x - 3)/(-1x))) - 2) and g(x) = -3sin(2\u03c0(-4x + 2) + 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "162", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1cos(3\u03c0(-3(-3|x + 1| - 2) + 5) + 3) - 3 and g(x) = 2cos(3\u03c0((2x + 2)/(-2x - 2)) + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "540", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(3(1e^(-1x)) - 2)^2 + 1(3(1e^(-1x)) - 2) - 2 and g(x) = -2|((3x + 1)/(-1x - 1))| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(2sin(1\u03c0(-2x^2 - 2x - 5) + 2)) - 2)/(-1(2sin(1\u03c0(-2x^2 - 2x - 5) + 2)) - 3) and g(x) = 2e^(-2(2|x + 3| + 3)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "402", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(5(-2|x + 2| + 2)^2 - 4(-2|x + 2| + 2))^2 + 1(5(-2|x + 2| + 2)^2 - 4(-2|x + 2| + 2)) + 5 and g(x) = 5(2|x + 2| + 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2cos(3\u03c0(1((-2x - 1)/(-1x - 2)) - 5)) - 1 and g(x) = -1|(-1|x - 2| + 3) - 3| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(1|(-2cos(1\u03c0x - 1) - 2) + 3| - 3)^2 + 4(1|(-2cos(1\u03c0x - 1) - 2) + 3| - 3) + 3 and g(x) = (2(2sin(2\u03c0x + 1) - 2) + 3)/(2(2sin(2\u03c0x + 1) - 2) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "57", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(1\u03c0(-3|(-3|x + 2|)| - 2) - 1) + 2 and g(x) = (1(-2|x - 3| + 1) + 2)/(1(-2|x - 3| + 1) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "152", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2cos(3\u03c0(-5(3x^2 + 5x) + 5) - 3) + 2 and g(x) = 3|(-2cos(3\u03c0x + 2) - 1) + 1| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "684", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(-1e^(-2(1x^2 + 5)) - 1) - 1 and g(x) = -3(1cos(1\u03c0x - 2)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3((3((-2x - 3)/(-2x)) + 2)/(-2((-2x - 3)/(-2x)) - 1)) + 1 = (3(3sin(2\u03c0x - 3)) - 3)/(3(3sin(2\u03c0x - 3))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(2e^(-1(-1|x + 2|)) - 3) - 5 and g(x) = 2(3sin(2\u03c0x) - 3)^2 + 3(3sin(2\u03c0x) - 3) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3sin(3\u03c0(-4(-5x + 4)^2 + 1) - 2) + 3 = 2(-1sin(2\u03c0x + 2) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1421", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-4(-1sin(2\u03c0x) - 3)^2 - 1(-1sin(2\u03c0x) - 3) - 5)^2 - 3(-4(-1sin(2\u03c0x) - 3)^2 - 1(-1sin(2\u03c0x) - 3) - 5) + 3 and g(x) = -5(3|x + 1| + 2)^2 - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-3(-2x - 4) - 2)^2 and g(x) = 2cos(3\u03c0((-2x)/(2x + 3)) + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(1|(-3|x + 2| - 1) + 3| + 3) + 2 and g(x) = (1(-2cos(2\u03c0x + 3) + 2) + 2)/(1(-2cos(2\u03c0x + 3) + 2) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1sin(2\u03c0(-5(1|x - 2| - 3)^2 - 4(1|x - 2| - 3) + 5) + 2) + 2 and g(x) = 1e^(2(3x)) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "695", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3sin(2\u03c0(-1|(-3x - 2) - 3| - 2)) + 1 and g(x) = -2cos(2\u03c0(1|x| - 2) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "122", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(1cos(1\u03c0((-2x + 2)/(1x + 1)) + 1) + 3) - 3| - 2 = -3(3x + 5)^2 + 1(3x + 5) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(-4(-1e^(1x) + 2) - 2) + 1)/(2(-4(-1e^(1x) + 2) - 2) - 3) and g(x) = -1(1sin(2\u03c0x + 3) - 3)^2 - 2(1sin(2\u03c0x + 3) - 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2cos(3\u03c0(2(2x + 5)^2 + 1(2x + 5) - 5) + 2) + 1 and g(x) = 1cos(2\u03c0(1x^2 + 3x + 5) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1cos(2\u03c0((-1(-3x) + 2)/(-2(-3x) - 2)) - 3) and g(x) = 2cos(1\u03c0(-2|x - 3| - 2) - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(4(4x^2 - 5x - 3)^2 - 5)^2 + 1(4(4x^2 - 5x - 3)^2 - 5) - 1 and g(x) = -1|(-1sin(2\u03c0x - 3) - 2) - 2| - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(3sin(3\u03c0(4x^2 - 2x + 5)) + 2)^2 - 1(3sin(3\u03c0(4x^2 - 2x + 5)) + 2) + 3 and g(x) = 3|(-1|x - 1| - 1) - 2| + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1511", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5((3(2cos(3\u03c0x + 1)) + 3)/(-3(2cos(3\u03c0x + 1)) + 2)) + 2 and g(x) = 2cos(2\u03c0(-2x^2 + 3x + 1) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "309", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(1cos(1\u03c0(-2|x - 3|) - 3) + 1)^2 - 3(1cos(1\u03c0(-2|x - 3|) - 3) + 1) + 5 and g(x) = 2cos(3\u03c0((-3x + 3)/(-1x - 2)) + 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "86", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3((-1(5x^2 - 4x - 1) - 1)/(3(5x^2 - 4x - 1) + 1))^2 + 4((-1(5x^2 - 4x - 1) - 1)/(3(5x^2 - 4x - 1) + 1)) + 1 and g(x) = -2((-2x)/(3x + 2))^2 + 1((-2x)/(3x + 2)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(-3(-1e^(-2x) + 1) + 2) - 2| and g(x) = (-1(2sin(2\u03c0x - 1) + 2) + 1)/(2(2sin(2\u03c0x - 1) + 2) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(3\u03c0(-3|(-1x + 4) - 1| + 3) + 2) and g(x) = (-3(-4x + 1) + 1)/(3(-4x + 1) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "181", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2((-2(-3cos(1\u03c0x + 1) - 2) - 2)/(-3(-3cos(1\u03c0x + 1) - 2) - 1))^2 + 5((-2(-3cos(1\u03c0x + 1) - 2) - 2)/(-3(-3cos(1\u03c0x + 1) - 2) - 1)) + 4 and g(x) = -1((2x - 3)/(-2x)) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-1(2e^(-2x) - 3)^2 - 2(2e^(-2x) - 3) + 2) + 5 and g(x) = 2sin(1\u03c0(-5x + 4) - 2) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3e^(-1(5(1x - 5)^2 - 3(1x - 5) - 3)) - 2 and g(x) = -4(2e^(-1x) - 3)^2 + 4(2e^(-1x) - 3) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2((-3(-3sin(1\u03c0x - 2) - 3) + 1)/(2(-3sin(1\u03c0x - 2) - 3) - 3))^2 - 3((-3(-3sin(1\u03c0x - 2) - 3) + 1)/(2(-3sin(1\u03c0x - 2) - 3) - 3)) and g(x) = 2|(-2x + 4) - 2| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(2\u03c0(-2|(1|x + 3|) + 2| - 3) - 3) - 3 and g(x) = 1((1x - 1)/(1x - 2)) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3sin(1\u03c0((-2((1x - 3)/(-1x + 2)) + 3)/(2((1x - 3)/(-1x + 2)) + 2)) + 1) and g(x) = -1|(4x + 5)| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(-1e^(2(1x^2 - 3x - 3)) + 3)^2 + 5(-1e^(2(1x^2 - 3x - 3)) + 3) + 1 and g(x) = -2((-3x + 1)/(1x + 3)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(1\u03c0((1((-1x + 1)/(1x - 3)) + 3)/(3((-1x + 1)/(1x - 3)) + 2)) - 1) - 1 and g(x) = (-3((-2x - 1)/(-3x + 1)))/(-1((-2x - 1)/(-3x + 1)) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "134", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(4(-1x^2 + 3x + 1)^2 - 1(-1x^2 + 3x + 1) - 4) + 1| + 1 and g(x) = -3|(5x^2 - 5x - 4) - 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|((1(-2cos(1\u03c0x - 1)) + 3)/(2(-2cos(1\u03c0x - 1)) + 3)) - 1| and g(x) = 3|(1x^2 - 5x)| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(1|(3cos(2\u03c0x) + 3) - 1| - 3) + 2 and g(x) = 3(-2x - 2)^2 + 4(-2x - 2) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|(-2|(3sin(2\u03c0x) + 3) - 3| + 2) - 3| = 5(-3|x + 3|) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(2\u03c0(-4(-3x - 3) - 3) - 2) + 3 and g(x) = -3(2|x + 1|)^2 + 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3sin(1\u03c0((-2((-3x + 3)/(-3x - 3)) - 2)/(-2((-3x + 3)/(-3x - 3)) - 3)) + 2) + 1 and g(x) = (-3(2|x + 3| - 3) + 1)/(3(2|x + 3| - 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-2cos(2\u03c0(1x^2 + 1x + 3) + 1) - 2)^2 + 3(-2cos(2\u03c0(1x^2 + 1x + 3) + 1) - 2) + 4 and g(x) = (1(2|x - 3| - 3) - 2)/(1(2|x - 3| - 3)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-1cos(1\u03c0(-4x^2 - 4x + 4) + 1) - 2))/(3(-1cos(1\u03c0(-4x^2 - 4x + 4) + 1) - 2) - 2) and g(x) = 2cos(1\u03c0(1x - 4) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-3(-4x^2 + 4x + 3)^2 + 2(-4x^2 + 4x + 3) - 2)^2 + 3(-3(-4x^2 + 4x + 3)^2 + 2(-4x^2 + 4x + 3) - 2) - 4 and g(x) = -3e^(-1(-3|x + 1| - 1)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3(3sin(3\u03c0(-3x^2 + 2x) - 1) + 3) - 3)/(-2(3sin(3\u03c0(-3x^2 + 2x) - 1) + 3) + 3) = -2((-3x - 1)/(-3x - 1))^2 - 3((-3x - 1)/(-3x - 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "428", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(2(-3sin(1\u03c0x - 3))^2 + 4(-3sin(1\u03c0x - 3)) + 4) + 5 and g(x) = 3(3cos(3\u03c0x - 3) - 3) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-5((1x - 1)/(-2x + 1)) + 5) + 2)/(1(-5((1x - 1)/(-2x + 1)) + 5) + 2) and g(x) = 3sin(1\u03c0(-1x) - 3) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(3e^(2(-1x^2 - 5x + 5)) + 1) - 2)/(-3(3e^(2(-1x^2 - 5x + 5)) + 1) - 2) = 1|(2sin(1\u03c0x + 3) - 2) + 3| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(1|(-2sin(3\u03c0x - 1) - 1) - 1| - 3) and g(x) = 1(-3x - 4) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(1\u03c0(-2|((2x - 2)/(1x - 3))| - 1) + 1) + 2 and g(x) = 2sin(3\u03c0(-1|x + 2| + 3) + 3) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(2\u03c0((2((-2x + 3)/(-1x + 3)) - 1)/(2((-2x + 3)/(-1x + 3)) + 1)) - 1) + 2 and g(x) = 2sin(3\u03c0(3x^2 + 2x + 4) - 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "872", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1e^(-1((1(-4x + 5) + 1)/(2(-4x + 5)))) - 2 and g(x) = -2|(-3x^2 - 1x + 2) + 1| + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(1((2x + 1)/(3x + 1))^2 - 4((2x + 1)/(3x + 1)) + 1) - 1 = 1sin(3\u03c0(1x^2 - 5) - 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "596", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(-3sin(2\u03c0(-1|x + 2|) - 3) + 2) + 5 and g(x) = -1sin(1\u03c0((-2x - 1)/(3x - 1)) + 2) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(1((-1x + 2)/(-2x + 1))^2 - 3((-1x + 2)/(-2x + 1)) - 5)^2 + 5(1((-1x + 2)/(-2x + 1))^2 - 3((-1x + 2)/(-2x + 1)) - 5) - 3 and g(x) = (-1(-1|x + 2| + 2) + 1)/(-1(-1|x + 2| + 2) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(1|(-2x) + 2| - 2) - 2)/(2(1|(-2x) + 2| - 2) - 1) = 1sin(2\u03c0(2|x + 3| + 3) - 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "66", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(-1sin(2\u03c0(3x - 5) + 2) + 3) - 3)/(3(-1sin(2\u03c0(3x - 5) + 2) + 3) - 2) = -4(-2cos(3\u03c0x - 1) + 3)^2 + 5(-2cos(3\u03c0x - 1) + 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(2sin(1\u03c0((-2x + 3)/(-1x)) + 1) + 3) + 5 = 1(4x^2 + 2x + 5)^2 - 4(4x^2 + 2x + 5) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(3\u03c0(-4(-3x^2 + 5x - 5) + 4)) - 3 and g(x) = -2e^(2(-2|x + 1| - 3)) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1460", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(3(1x^2 - 5x + 4) + 2) + 3| - 2 and g(x) = (3((-1x - 3)/(3x + 1)) - 2)/(-2((-1x - 3)/(3x + 1))) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2(2cos(2\u03c0((3x + 1)/(-2x + 3)) - 1) + 1)^2 - 1(2cos(2\u03c0((3x + 1)/(-2x + 3)) - 1) + 1) - 2 and g(x) = 5(-1|x + 3| + 1) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-3e^(1(1|x - 1| - 2)) + 1) - 2)/(-1(-3e^(1(1|x - 1| - 2)) + 1) + 2) = -4(1x^2 - 2x - 4)^2 - 2(1x^2 - 2x - 4) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(-2sin(3\u03c0(-2|x - 3|) + 2) - 2)^2 - 2(-2sin(3\u03c0(-2|x - 3|) + 2) - 2) - 2 = (-1(-1x^2 - 3x - 1) + 2)/(-3(-1x^2 - 3x - 1) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(3|(-5x^2 + 2x + 1) + 1| + 2)| + 2 = 4(-1cos(2\u03c0x) - 3)^2 + 4(-1cos(2\u03c0x) - 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(-2cos(3\u03c0(3x + 2)) + 2) and g(x) = 1sin(3\u03c0((-1x + 1)/(-3x + 4)) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-2((1x + 2)/(2x - 1))^2 + 2((1x + 2)/(2x - 1)) - 2) - 2)/(-2(-2((1x + 2)/(2x - 1))^2 + 2((1x + 2)/(2x - 1)) - 2) + 3) and g(x) = -3sin(1\u03c0(-4x - 4) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|(2cos(2\u03c0((-2x - 3)/(-1x)) + 2) + 3) - 1| + 2 = 3|(-2x^2 - 5x + 3) + 2| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(2(-1e^(-1x))^2 + 2(-1e^(-1x)))^2 - 3(2(-1e^(-1x))^2 + 2(-1e^(-1x))) and g(x) = -4(-3sin(2\u03c0x - 1) + 1)^2 - 3(-3sin(2\u03c0x - 1) + 1) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(2\u03c0(3(-4x^2 - 1)^2 + 5(-4x^2 - 1) - 4) + 1) - 2 and g(x) = (-1(1|x|) - 3)/(-3(1|x|) + 4) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "104", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5((3(1x^2 - 1x - 1) - 2)/(3(1x^2 - 1x - 1) + 3)) - 5 = 2cos(1\u03c0(4x^2 + 3x + 2) + 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "759", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(4(-2cos(2\u03c0x + 1) + 1) + 4) + 4 and g(x) = 2e^(2((1x + 2)/(-2x - 3))) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(-2|(-3cos(3\u03c0x + 2) + 2) - 2| - 3) - 5 and g(x) = -1(-3x + 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(3sin(3\u03c0(-4x^2 + 5x + 3) + 2) - 2) - 2)/(-2(3sin(3\u03c0(-4x^2 + 5x + 3) + 2) - 2) + 1) and g(x) = -5((2x + 3)/(2x + 2))^2 + 1((2x + 3)/(2x + 2)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "482", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(4(1x^2 + 5x)^2 - 3(1x^2 + 5x) - 5) + 1)/(1(4(1x^2 + 5x)^2 - 3(1x^2 + 5x) - 5) - 3) and g(x) = (-2(-4x + 4))/(1(-4x + 4)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(3|(-1sin(1\u03c0x + 1) - 1) - 1| - 2)^2 + 3(3|(-1sin(1\u03c0x + 1) - 1) - 1| - 2) + 5 and g(x) = 1|(2x^2 + 4x - 5) - 3| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|((3(1x^2 + 3x + 1) - 1)/(-2(1x^2 + 3x + 1) - 1))| - 2 = 2(-1sin(3\u03c0x + 1)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(-1cos(3\u03c0(4x^2 + 3) + 2) - 1) and g(x) = -3(-2sin(3\u03c0x + 3) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "476", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1((-2(-2x^2 + 5) - 1)/(3(-2x^2 + 5) - 2))^2 + 2((-2(-2x^2 + 5) - 1)/(3(-2x^2 + 5) - 2)) + 3 and g(x) = 1(2x - 4) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-3e^(-2(-1|x + 1| + 1)) + 2) + 1)/(2(-3e^(-2(-1|x + 1| + 1)) + 2) + 3) and g(x) = 3(-2sin(2\u03c0x - 1) - 2)^2 + 1(-2sin(2\u03c0x - 1) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1sin(2\u03c0(-2|(4x^2 + 2x - 1) - 2| + 3) - 3) - 2 and g(x) = -1((1x - 3)/(2x + 1))^2 + 3((1x - 3)/(2x + 1)) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "755", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(2sin(1\u03c0(3|x - 1| - 3) + 2) - 1) = 5((-1x + 2)/(-1x))^2 - 2((-1x + 2)/(-1x)) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(-1(1e^(2x))) + 1| - 3 and g(x) = (-1(-2cos(2\u03c0x - 3)) - 2)/(-3(-2cos(2\u03c0x - 3)) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(-5(3sin(3\u03c0x + 3) + 3) - 2) + 2| + 1 and g(x) = 1|(-2e^(2x)) - 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3(3sin(1\u03c0(2x - 1) - 2) - 1) - 1)/(3(3sin(1\u03c0(2x - 1) - 2) - 1) + 2) = (1(-3sin(1\u03c0x + 1) + 3) - 3)/(-3(-3sin(1\u03c0x + 1) + 3) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|(2((-3x + 2)/(-1x - 3)) - 5) - 1| = -5(-3|x + 2| + 3)^2 - 5(-3|x + 2| + 3) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(1cos(1\u03c0(4x - 4)) - 1) + 3)/(1(1cos(1\u03c0(4x - 4)) - 1) - 2) and g(x) = (2(-3cos(1\u03c0x) + 1) + 2)/(3(-3cos(1\u03c0x) + 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(3\u03c0(-2|(3x^2 + 3x + 1) - 3| - 1)) and g(x) = -3e^(2(4x - 5)) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "773", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-3sin(1\u03c0(-4x^2 - 2x + 2) + 1) - 3) - 3)/(2(-3sin(1\u03c0(-4x^2 - 2x + 2) + 1) - 3) + 1) = (-2(-1x^2 + 4x - 1) - 3)/(-2(-1x^2 + 4x - 1) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "530", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(2sin(1\u03c0(-1x - 1)) - 1)^2 + 4(2sin(1\u03c0(-1x - 1)) - 1) + 4 and g(x) = 2|((1x + 2)/(-1x + 3))| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(-3sin(1\u03c0(-1|x - 2| + 2) + 2) + 3) - 4 and g(x) = 3sin(2\u03c0(1|x| + 1) - 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(-1(1sin(1\u03c0x) - 1) + 2)^2 + 5 and g(x) = -5(-2cos(1\u03c0x - 1))^2 - 5(-2cos(1\u03c0x - 1)) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(-2|(-1x^2 + 4x) + 1| - 1) - 3 and g(x) = -3|(3cos(2\u03c0x + 2) + 3) - 3| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(3|(2|x + 1| + 3) + 3| - 1) - 3)/(3(3|(2|x + 1| + 3) + 3| - 1) - 2) and g(x) = -2sin(3\u03c0(-3x^2 - 1) + 2) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "135", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(3\u03c0(-1(1x - 5) + 4)) - 3 and g(x) = 2cos(3\u03c0(-3|x - 2| + 2)) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "180", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|(3sin(2\u03c0((1x + 1)/(2x + 3)) + 1) + 2)| - 2 and g(x) = -3|(-2x + 2)| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|((3(2sin(2\u03c0x - 2) + 3) - 2)/(-2(2sin(2\u03c0x - 2) + 3))) + 2| + 3 and g(x) = 2cos(2\u03c0(3|x| + 1) + 1) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(3cos(1\u03c0((1x - 1)/(-1x + 2)) - 2) + 1) - 3)/(2(3cos(1\u03c0((1x - 1)/(-1x + 2)) - 2) + 1) - 1) and g(x) = -2sin(2\u03c0(4x^2 + 4) - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "445", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(-1|(5x + 3)| - 2) + 3 and g(x) = -1|(2x^2 + 1x + 1) + 2| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-2cos(2\u03c0(-2x^2 - 4x + 1) + 3) - 3) + 2)/(-3(-2cos(2\u03c0(-2x^2 - 4x + 1) + 3) - 3) - 3) = (3(-2|x + 2|))/(-2(-2|x + 2|) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(2sin(1\u03c0(1x^2 - 2x) - 1) + 3) - 3| and g(x) = 5(2sin(2\u03c0x)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2e^(-1((1(-1x + 3) - 3)/(1(-1x + 3) - 2))) - 1 and g(x) = -2sin(2\u03c0(-2|x + 3| - 2) - 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "77", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3sin(3\u03c0((-2((-1x + 3)/(-2x)) + 1)/(-1((-1x + 3)/(-2x)) - 3)) - 3) - 1 = -2sin(3\u03c0(-1x^2 - 1x - 4) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "328", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2e^(-2((-3(-1|x - 1| + 3) + 1)/(-2(-1|x - 1| + 3) - 2))) + 2 = -2|(-1sin(1\u03c0x + 2))| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|((-1((1x + 3)/(2x + 1)) - 1)/(-1((1x + 3)/(2x + 1)) - 1)) - 2| - 1 = 2sin(2\u03c0(1|x - 1| - 2) + 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(1\u03c0(-3((-3x - 1)/(1x + 1))^2 + 4((-3x - 1)/(1x + 1)) - 5) + 1) + 2 and g(x) = -5((-2x + 3)/(2x)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(3|(-1x - 4)| - 2) - 1)/(-1(3|(-1x - 4)| - 2)) and g(x) = 1|((1x + 3)/(1x)) - 3| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3((3(-2x + 1) + 3)/(-2(-2x + 1))) + 4 and g(x) = -1(-2cos(1\u03c0x + 1))^2 - 4(-2cos(1\u03c0x + 1)) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(2e^(1(3x)) + 2) - 2)/(-3(2e^(1(3x)) + 2)) and g(x) = 3|(-1x^2 - 5x + 4) + 2| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2cos(1\u03c0(4((2x - 2)/(1x))^2 + 3((2x - 2)/(1x)) - 2)) and g(x) = (2(-5x^2 + 1x + 2) + 3)/(-3(-5x^2 + 1x + 2) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "151", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3((-1(1|x| - 2) + 2)/(3(1|x| - 2) - 2)) + 4 = -1(-5x^2 + 5)^2 + 5(-5x^2 + 5) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3((1(-1|x - 3| + 1) - 2)/(3(-1|x - 3| + 1) + 3))^2 + 4((1(-1|x - 3| + 1) - 2)/(3(-1|x - 3| + 1) + 3)) - 4 and g(x) = 2cos(2\u03c0(1|x + 2|) + 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2sin(1\u03c0((1((-1x - 2)/(-3x - 2)) - 1)/(1((-1x - 2)/(-3x - 2)))) + 1) = (1(-3x^2 - 1x - 2) - 2)/(3(-3x^2 - 1x - 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "72", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(3cos(1\u03c0((2x + 3)/(1x + 1)) + 2) + 1)^2 - 1 and g(x) = 3(3sin(3\u03c0x - 1) + 3)^2 + 3(3sin(3\u03c0x - 1) + 3) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(-5(5x^2 - 3x - 3) - 1) - 1| + 1 and g(x) = (-2((-2x)/(1x + 1)) - 3)/(2((-2x)/(1x + 1))) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(-1cos(3\u03c0(-1|x + 3| - 2) - 2) - 3) - 1 and g(x) = -2(-2sin(3\u03c0x + 1) + 2)^2 + 4(-2sin(3\u03c0x + 1) + 2) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2cos(3\u03c0(-1|(-5x - 4) + 1| - 3) + 2) and g(x) = -3|((3x + 1)/(2x + 2)) - 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(-3cos(1\u03c0(-2x - 1) - 2) + 1)^2 + 2(-3cos(1\u03c0(-2x - 1) - 2) + 1) = 4(-2e^(-2x) + 1)^2 + 3(-2e^(-2x) + 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-3((2x + 3)/(2x + 2))^2 - 1((2x + 3)/(2x + 2)) - 5) + 1)/(2(-3((2x + 3)/(2x + 2))^2 - 1((2x + 3)/(2x + 2)) - 5) + 2) and g(x) = -3sin(3\u03c0(-2x^2 - 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1146", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2cos(1\u03c0(4(-5x^2 - 3x)^2 + 4(-5x^2 - 3x) + 4) + 1) - 2 = (-2(-2x^2 + 2x - 5) - 1)/(2(-2x^2 + 2x - 5) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "686", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(-5(-1x^2 + 3x + 4)^2 - 1(-1x^2 + 3x + 4) + 2)| + 2 and g(x) = (-3(3|x + 2| - 1) - 3)/(2(3|x + 2| - 1) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(1|(3x + 4)| - 3)^2 - 3(1|(3x + 4)| - 3) - 5 and g(x) = 3e^(-1(3x^2 + 5x - 1)) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(-2|(-3cos(2\u03c0x - 2) - 3) - 2|) + 2)/(1(-2|(-3cos(2\u03c0x - 2) - 3) - 2|)) and g(x) = (1(2sin(1\u03c0x - 2)))/(-3(2sin(1\u03c0x - 2)) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(3sin(3\u03c0((1x - 3)/(-2x - 3)) - 2) + 2)^2 - 3(3sin(3\u03c0((1x - 3)/(-2x - 3)) - 2) + 2) and g(x) = 2(-1sin(1\u03c0x + 2) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "85", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-4(-2cos(1\u03c0x + 3)) + 1) - 2)/(-1(-4(-2cos(1\u03c0x + 3)) + 1) + 2) and g(x) = -1(2x - 5) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(-1|((3x - 3)/(-2x + 3)) - 3| + 1) and g(x) = (1(-2|x + 1| + 1) - 1)/(1(-2|x + 1| + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(1(-4x + 3) - 1) + 1)/(2(1(-4x + 3) - 1) - 1) = 1sin(2\u03c0(4x^2 + 5) + 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1373", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(-1sin(3\u03c0(2|x + 3|) - 2) + 2)^2 + 5(-1sin(3\u03c0(2|x + 3|) - 2) + 2) - 4 and g(x) = 2(3x - 4)^2 + 3(3x - 4) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1((-1(1x + 4))/(1(1x + 4) + 1)) + 3)/(-3((-1(1x + 4))/(1(1x + 4) + 1))) and g(x) = -3sin(2\u03c0((-1x - 2)/(2x)) + 1) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(1(1x + 2) - 2) + 1)/(-3(1(1x + 2) - 2) + 4) and g(x) = -1cos(3\u03c0(2x^2 + 3x - 4) - 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "979", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-5(2sin(2\u03c0x + 1)) - 2)^2 + 1(-5(2sin(2\u03c0x + 1)) - 2) + 2 and g(x) = 1(3x + 4)^2 + 3(3x + 4) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1((-3(-1|x + 2| + 2) + 1)/(-1(-1|x + 2| + 2) + 2))^2 - 1((-3(-1|x + 2| + 2) + 1)/(-1(-1|x + 2| + 2) + 2)) - 1 = -3|(1cos(3\u03c0x - 2) - 3) + 1| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(-2(3x^2 - 5x - 3)^2 + 5(3x^2 - 5x - 3) - 1) + 3)/(2(-2(3x^2 - 5x - 3)^2 + 5(3x^2 - 5x - 3) - 1) - 1) = 1(1|x + 1| - 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2((-2(-1sin(1\u03c0x - 3)) - 2)/(1(-1sin(1\u03c0x - 3)))) - 2 = 4(3|x + 3| - 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2sin(3\u03c0(3((-1x - 3)/(1x - 3))) - 1) + 1 = 2(3e^(1x) - 3)^2 - 2(3e^(1x) - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(1|(1x^2 + 2x) - 2| - 3)| + 3 and g(x) = (-1(-1|x + 2| + 2) - 1)/(3(-1|x + 2| + 2)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2((-1(1|x - 3| - 3) + 1)/(-3(1|x - 3| - 3) - 3))^2 + 3((-1(1|x - 3| - 3) + 1)/(-3(1|x - 3| - 3) - 3)) = -2(-1sin(1\u03c0x + 2) + 2)^2 - 1(-1sin(1\u03c0x + 2) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-2|(-2|x - 1| + 1) - 1| - 3))/(3(-2|(-2|x - 1| + 1) - 1| - 3) - 2) and g(x) = -4(-1sin(2\u03c0x) + 2)^2 + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2sin(1\u03c0((2(-1|x + 2| + 3) + 1)/(-2(-1|x + 2| + 3) - 2)) - 1) - 3 and g(x) = 2cos(1\u03c0(4x^2 - 3x - 3) - 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "116", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(2\u03c0(3|(4x^2 - 1x) - 3|) - 1) - 2 and g(x) = 3cos(2\u03c0(1|x| - 1) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "716", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(-5(-1sin(2\u03c0x - 2) + 2) + 4) - 2)/(2(-5(-1sin(2\u03c0x - 2) + 2) + 4) + 2) and g(x) = 5((-1x + 2)/(1x + 1))^2 + 5((-1x + 2)/(1x + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(-3(-2|x + 1| + 2)^2 + 1(-2|x + 1| + 2) + 1) + 3)/(-1(-3(-2|x + 1| + 2)^2 + 1(-2|x + 1| + 2) + 1) + 2) and g(x) = (3(3cos(2\u03c0x - 2) - 1) - 2)/(3(3cos(2\u03c0x - 2) - 1) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-1|(2e^(-1x) - 2) + 2| + 2)^2 + 2(-1|(2e^(-1x) - 2) + 2| + 2) + 4 and g(x) = 2cos(2\u03c0((-1x)/(-3x - 2)) + 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(1\u03c0(-5(-3x + 1)^2 + 2(-3x + 1) - 3) - 3) - 1 and g(x) = 3(1|x - 2| - 3) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "244", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-1(-2|x + 3| - 2) - 4)^2 - 5(-1(-2|x + 3| - 2) - 4) + 2 and g(x) = -2cos(1\u03c0(-2x + 3) - 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3e^(2(-2|(-4x - 2) - 3| + 2)) + 2 and g(x) = (2(-3x^2 - 1x + 2) + 3)/(-3(-3x^2 - 1x + 2) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2((1(-2|x - 3| + 1) - 1)/(-2(-2|x - 3| + 1))) - 3)/(3((1(-2|x - 3| + 1) - 1)/(-2(-2|x - 3| + 1))) + 1) and g(x) = 1(3cos(3\u03c0x - 1))^2 - 1(3cos(3\u03c0x - 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "104", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(1sin(3\u03c0(-5x + 5) - 3) + 2) + 3)/(1(1sin(3\u03c0(-5x + 5) - 3) + 2)) = -3sin(3\u03c0((-3x - 1)/(-1x - 3)) - 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "135", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(2\u03c0(2|(-4x^2 + 4x + 5) - 2| - 1) + 2) and g(x) = -3(3cos(3\u03c0x - 1) - 2) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "395", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-2e^(-2(2x^2 - 3x + 3)))^2 - 3(-2e^(-2(2x^2 - 3x + 3))) - 2 and g(x) = -2|(3sin(3\u03c0x - 2) - 2) + 2| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(-2cos(3\u03c0((2x + 2)/(2x - 1)) + 3)) - 2)/(-3(-2cos(3\u03c0((2x + 2)/(2x - 1)) + 3)) - 2) = -1sin(3\u03c0(-1|x + 3| + 1) - 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2cos(1\u03c0(2|(2x + 3) - 2| - 3)) + 1 and g(x) = 1e^(1((-3x - 2)/(3x + 2))) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(-3(2|x + 1| - 1)^2 + 3(2|x + 1| - 1) + 4) + 1)/(-2(-3(2|x + 1| - 1)^2 + 3(2|x + 1| - 1) + 4)) = -1e^(-1(1x^2 + 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(-1(1x^2 + 2) + 4)| - 2 and g(x) = -1|((-3x - 2)/(-1x - 2))| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1((-1((2x + 1)/(1x - 3)) - 3)/(1((2x + 1)/(1x - 3)) - 2)) - 2)/(3((-1((2x + 1)/(1x - 3)) - 3)/(1((2x + 1)/(1x - 3)) - 2)) - 2) and g(x) = -3cos(2\u03c0((2x + 1)/(3x - 2)) + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-3sin(3\u03c0(2|x - 1| - 3) - 2) + 2) - 2)/(-1(-3sin(3\u03c0(2|x - 1| - 3) - 2) + 2) + 2) and g(x) = -3|(-3cos(3\u03c0x - 3) - 3) - 2| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "118", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(2\u03c0(2(5x^2 - 4x + 1)^2 + 5(5x^2 - 4x + 1) + 3) - 3) and g(x) = (-2((-2x - 1)/(-2x - 2)) - 2)/(2((-2x - 1)/(-2x - 2))) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "187", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|((-2(2|x + 2| + 1) + 2)/(-1(2|x + 2| + 1) + 2)) - 1| = (1(3cos(1\u03c0x - 3) - 2) - 3)/(-1(3cos(1\u03c0x - 3) - 2)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(2\u03c0(3(-2|x - 2| + 1)^2 - 3(-2|x - 2| + 1) - 4) + 1) + 2 and g(x) = 2sin(3\u03c0((-1x + 2)/(-3x + 1)) - 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(-1(1sin(2\u03c0x - 2))^2 + 3(1sin(2\u03c0x - 2)) - 2) - 2 and g(x) = (1((2x + 1)/(-2x + 3)) - 2)/(2((2x + 1)/(-2x + 3)) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1((3(-2x^2 + 1x - 5) - 1)/(2(-2x^2 + 1x - 5) + 2)) + 4 = 1|(4x^2 - 3x - 5) + 1| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3sin(3\u03c0(-1|(-1x + 4) - 3| + 3) - 2) - 1 and g(x) = -1e^(1(-4x^2 - 4x)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5((1(1|x + 3| - 2) + 2)/(1(1|x + 3| - 2) + 1))^2 - 5((1(1|x + 3| - 2) + 2)/(1(1|x + 3| - 2) + 1)) - 1 and g(x) = (-1(-5x^2 - 1x - 5) - 2)/(-3(-5x^2 - 1x - 5) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(4(-2x - 2) + 5) + 3)/(3(4(-2x - 2) + 5) + 1) and g(x) = 2cos(1\u03c0(1x + 1) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2cos(2\u03c0(-3(-4x - 5)^2 - 1(-4x - 5) - 3) + 2) - 1 and g(x) = (-1(5x - 5) + 2)/(2(5x - 5)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1510", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(1cos(1\u03c0(-3x^2 - 3x - 4) + 3) - 1)^2 + 4(1cos(1\u03c0(-3x^2 - 3x - 4) + 3) - 1) - 5 and g(x) = -2|(1x^2 + 2x + 5)| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(2|((-3x - 1)/(-3x + 2)) + 3| - 3))/(2(2|((-3x - 1)/(-3x + 2)) + 3| - 3) + 3) and g(x) = -3cos(1\u03c0(-5x + 3)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "98", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(-3(-3cos(2\u03c0x + 3) - 1) - 3) + 5 and g(x) = (-1(1cos(3\u03c0x - 2) + 3) + 1)/(1(1cos(3\u03c0x - 2) + 3) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(1(2|x + 2|)^2 + 1(2|x + 2|) - 1) - 1| + 3 = (-1(2e^(2x) - 3) - 3)/(-3(2e^(2x) - 3) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(1\u03c0(4(1x + 3)^2 - 2(1x + 3) + 4) + 1) + 3 and g(x) = 2(-1x^2 + 3x + 3)^2 - 4(-1x^2 + 3x + 3) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(5(-3|x + 3| + 3))^2 - 5(5(-3|x + 3| + 3)) + 4 and g(x) = (1(1cos(3\u03c0x + 2) - 2))/(1(1cos(3\u03c0x + 2) - 2) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(2(-3cos(3\u03c0x + 3) - 1) - 4) - 2| + 3 and g(x) = 1(1x^2 + 5x + 4) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2((-2(-1sin(3\u03c0x + 3) - 3) + 3)/(-2(-1sin(3\u03c0x + 3) - 3) - 1)) + 3 and g(x) = 5(3cos(1\u03c0x + 2) - 1) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(-1cos(1\u03c0(1x - 1) + 3) + 2) + 3 = -2e^(1(-4x^2 - 2x - 1)) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(2|(-2cos(1\u03c0x - 3) + 2) - 3| - 2) + 1)/(2(2|(-2cos(1\u03c0x - 3) + 2) - 3| - 2) - 3) and g(x) = -3cos(3\u03c0(-1|x - 2| + 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(-1e^(1(-2|x - 3| - 1)))| - 3 = -2cos(2\u03c0((1x + 1)/(2x)) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(1\u03c0(-5(-1|x + 1| - 1)^2 - 5(-1|x + 1| - 1) - 3) + 3) + 2 = (1(1|x| + 1) - 3)/(-2(1|x| + 1) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "865", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(2|(1|x + 1| + 1) - 3|) - 2)/(2(2|(1|x + 1| + 1) - 3|) + 3) and g(x) = 2cos(3\u03c0(4x^2 - 2x - 1) - 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1576", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2(-2sin(3\u03c0(-5x - 4) + 2) - 2)^2 - 3(-2sin(3\u03c0(-5x - 4) + 2) - 2) - 4 and g(x) = -3cos(1\u03c0((-2x - 2)/(-1x)) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "282", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2sin(1\u03c0(-3(3x^2 + 5x + 3)^2 - 2) + 3) + 1 and g(x) = 3e^(1((3x + 2)/(-2x + 3))) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "241", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(-2sin(3\u03c0(5x + 2)) + 1)| + 2 and g(x) = -3|(2|x + 2| - 3) + 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(-3cos(2\u03c0((3x - 3)/(-2x)) - 2) - 1)^2 - 5 = -5(-2|x + 2|) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(3sin(3\u03c0(4x - 1) - 1) - 1) and g(x) = 1|(-3x + 4) + 1| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|((-1(-1x + 1))/(-1(-1x + 1) + 2)) + 3| - 1 = 5(-2sin(3\u03c0x - 1) - 2)^2 + 5(-2sin(3\u03c0x - 1) - 2) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(2sin(3\u03c0(-1|x - 2| + 3) - 1))^2 - 4(2sin(3\u03c0(-1|x - 2| + 3) - 1)) + 1 and g(x) = 3sin(2\u03c0(1|x| - 1)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(2\u03c0((3(2x^2 - 2) + 2)/(2(2x^2 - 2) - 1)) - 2) + 3 = -3cos(2\u03c0(3|x - 3| - 2) + 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "155", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(2cos(3\u03c0((-1x + 2)/(3x - 2)) - 3) - 3) - 2 = 2((2x - 3)/(-1x - 3))^2 + 5((2x - 3)/(-1x - 3)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5((2(-2cos(3\u03c0x + 1) + 1) - 3)/(-2(-2cos(3\u03c0x + 1) + 1)))^2 - 1 and g(x) = (-3(-3cos(3\u03c0x + 2) + 1) + 1)/(3(-3cos(3\u03c0x + 2) + 1) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "93", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1e^(-1(1(5x^2 - 2x))) + 2 and g(x) = 2sin(2\u03c0(-2x + 2) + 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(-5(-1|x + 2| - 1)^2 + 4(-1|x + 2| - 1)) - 3| + 3 and g(x) = -2(2x^2 + 5x - 5)^2 + 4(2x^2 + 5x - 5) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1((3((-2x)/(2x + 3)) + 2)/(-3((-2x)/(2x + 3)) + 4))^2 + 4((3((-2x)/(2x + 3)) + 2)/(-3((-2x)/(2x + 3)) + 4)) - 5 and g(x) = -2(-3cos(1\u03c0x - 3) + 3)^2 + 3(-3cos(1\u03c0x - 3) + 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((2(3cos(2\u03c0x)) - 1)/(-3(3cos(2\u03c0x)) + 2))^2 + 3((2(3cos(2\u03c0x)) - 1)/(-3(3cos(2\u03c0x)) + 2)) + 3 and g(x) = (1(5x^2 - 4x + 1) - 1)/(1(5x^2 - 4x + 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "64", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1cos(3\u03c0(-1|(2x + 3) - 2| - 3) + 1) + 1 = 3|(-1x + 2) - 3| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(2|(-1|x - 3| + 1) + 1| - 2) - 1)/(3(2|(-1|x - 3| + 1) + 1| - 2) - 2) and g(x) = (3(3|x + 1| + 3) - 3)/(3(3|x + 1| + 3) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-3sin(3\u03c0(1x + 1) + 2) + 2) - 3)/(3(-3sin(3\u03c0(1x + 1) + 2) + 2) - 2) and g(x) = (2(-5x + 4) - 3)/(-3(-5x + 4) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2cos(2\u03c0(3(-3|x + 2| - 3) + 1)) - 1 and g(x) = -4(-2x^2 + 3x + 4) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(3(3|x + 1| - 3)^2 + 5(3|x + 1| - 3) + 1)^2 - 1(3(3|x + 1| - 3)^2 + 5(3|x + 1| - 3) + 1) + 2 and g(x) = -2(-3|x + 2| - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3sin(2\u03c0(-2|(3x) + 1|) + 1) + 3 and g(x) = 3cos(3\u03c0(4x + 3) - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "240", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(-5(-1cos(3\u03c0x + 3) + 1) + 4) - 1)/(2(-5(-1cos(3\u03c0x + 3) + 1) + 4) + 2) and g(x) = (3(-1sin(2\u03c0x + 3) + 1) + 3)/(-3(-1sin(2\u03c0x + 3) + 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(3|(-1x^2 - 3x + 4) + 2| - 1)^2 - 4(3|(-1x^2 - 3x + 4) + 2| - 1) + 4 and g(x) = -5(2|x - 3| - 2)^2 + 4(2|x - 3| - 2) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(2|(1x^2 - 4x + 3) + 2| + 1) - 4 and g(x) = (-1(-1cos(1\u03c0x - 3)) + 2)/(-3(-1cos(1\u03c0x - 3)) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3cos(1\u03c0(4(2|x| + 2) - 4) + 3) + 3 and g(x) = 1|(-2sin(3\u03c0x - 3)) + 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "160", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(2|(4x + 2) + 2| - 1)^2 + 3(2|(4x + 2) + 2| - 1) + 2 and g(x) = (2(-3sin(2\u03c0x - 1) - 2) + 3)/(-3(-3sin(2\u03c0x - 1) - 2) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(5(-1x - 4) + 3) - 1| + 1 = 1|((1x - 2)/(-3x - 3)) - 2| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(-3(-2x + 4) + 1)^2 + 4(-3(-2x + 4) + 1) + 1 = -3((2x + 3)/(-1x))^2 - 5((2x + 3)/(-1x)) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(4(1cos(3\u03c0x - 3) - 3) - 3) + 1)/(3(4(1cos(3\u03c0x - 3) - 3) - 3) - 1) = -2|(2|x - 2| + 2) - 3| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(1cos(2\u03c0(1|x + 1|) + 2)) + 2 and g(x) = 2|(-3cos(3\u03c0x + 3) + 1) - 1| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(-2(2cos(3\u03c0x - 1) + 2)^2 + 5(2cos(3\u03c0x - 1) + 2) - 1) + 3)/(2(-2(2cos(3\u03c0x - 1) + 2)^2 + 5(2cos(3\u03c0x - 1) + 2) - 1) + 2) = 2|((-1x + 3)/(-1x - 2)) + 3| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "97", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(-2cos(2\u03c0(-1x^2 - 4x) + 1)) - 3)/(2(-2cos(2\u03c0(-1x^2 - 4x) + 1)) - 3) = (3(2sin(2\u03c0x)) - 2)/(3(2sin(2\u03c0x)) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "404", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(3cos(1\u03c0(2|x - 3| - 3)) + 2)^2 + 1(3cos(1\u03c0(2|x - 3| - 3)) + 2) + 2 = 3((2x + 2)/(1x - 2)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(3cos(1\u03c0(-4x^2 + 4) + 2) + 1)^2 + 1(3cos(1\u03c0(-4x^2 + 4) + 2) + 1) + 2 = 1(-1|x + 2| + 3)^2 + 2(-1|x + 2| + 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "561", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3sin(1\u03c0(1(4x^2 - 2x + 2)^2 + 3(4x^2 - 2x + 2) - 1) + 1) + 3 and g(x) = (2(-4x^2 - 1x + 1) - 2)/(-2(-4x^2 - 1x + 1) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-1sin(2\u03c0(1x^2 - 1x + 5) - 1) + 1) + 2 = 1(1|x - 1| - 3) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(1e^(-2(1|x - 3| + 3)) + 1) + 2)/(3(1e^(-2(1|x - 3| + 3)) + 1)) and g(x) = (-3(-2sin(2\u03c0x + 3) + 2) + 2)/(1(-2sin(2\u03c0x + 3) + 2)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(-1((-1x + 3)/(2x + 2)) - 5) + 2)/(-3(-1((-1x + 3)/(2x + 2)) - 5) - 1) = -2sin(1\u03c0(3x - 2) + 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "57", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5((3(-3cos(3\u03c0x) + 3))/(2(-3cos(3\u03c0x) + 3) + 3))^2 + 1((3(-3cos(3\u03c0x) + 3))/(2(-3cos(3\u03c0x) + 3) + 3)) - 5 and g(x) = 3|(-2cos(2\u03c0x - 1) + 2) - 2| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3((-2(3sin(1\u03c0x - 3) - 2) - 2)/(-1(3sin(1\u03c0x - 3) - 2) + 2)))/(2((-2(3sin(1\u03c0x - 3) - 2) - 2)/(-1(3sin(1\u03c0x - 3) - 2) + 2)) + 3) = -3|(2x^2 - 1x) - 3| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(2(4x - 1)^2 + 1(4x - 1) - 4) - 4 = (2(1cos(2\u03c0x - 1) - 1) + 2)/(-1(1cos(2\u03c0x - 1) - 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(1(1x^2 - 5x + 1)^2 + 4(1x^2 - 5x + 1) + 4)^2 + 5(1(1x^2 - 5x + 1)^2 + 4(1x^2 - 5x + 1) + 4) + 1 and g(x) = -3sin(3\u03c0(1x^2 + 3x + 2) + 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(-2(-3|x| + 3)) - 2)/(1(-2(-3|x| + 3)) + 1) and g(x) = 1(-3x^2 + 5x - 5) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(-2((-2x - 2)/(2x + 3)) - 5) + 2| - 2 and g(x) = -3|(-2cos(2\u03c0x + 3) + 2) - 3| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "68", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(3(1cos(3\u03c0x - 1) - 1)^2 + 2(1cos(3\u03c0x - 1) - 1))^2 - 4(3(1cos(3\u03c0x - 1) - 1)^2 + 2(1cos(3\u03c0x - 1) - 1)) - 4 and g(x) = 1cos(1\u03c0(1x^2 - 1x - 3) - 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-5(-1|x + 1| + 3) + 2) + 3)/(2(-5(-1|x + 1| + 3) + 2) + 1) and g(x) = 2|((2x - 2)/(-3x - 2)) - 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1sin(3\u03c0(1((-3x - 3)/(1x + 2))^2 + 1((-3x - 3)/(1x + 2)) - 2) + 2) - 2 = -2(-2|x - 1|) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1((3(3cos(2\u03c0x - 2) + 1) - 3)/(2(3cos(2\u03c0x - 2) + 1) - 1)) - 3)/(-2((3(3cos(2\u03c0x - 2) + 1) - 3)/(2(3cos(2\u03c0x - 2) + 1) - 1)) - 2) and g(x) = -3sin(2\u03c0(2x - 5) - 3) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "98", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(-1(5x^2 - 4x - 5)^2 + 1(5x^2 - 4x - 5) + 1) + 3)/(1(-1(5x^2 - 4x - 5)^2 + 1(5x^2 - 4x - 5) + 1) + 2) = 1|(-2sin(2\u03c0x + 2) + 2)| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(1\u03c0(1(-3|x| + 1)^2 + 3(-3|x| + 1) - 2) + 1) - 1 and g(x) = (-2(-2x^2 + 3x - 2))/(-3(-2x^2 + 3x - 2) + 4) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "907", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|((-2(-3|x + 1| + 1) - 3)/(3(-3|x + 1| + 1))) - 3| and g(x) = (-1(3|x + 1| - 2) + 1)/(3(3|x + 1| - 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(4(-3x)^2 - 4(-3x) + 3) + 2| + 2 and g(x) = (2(2sin(3\u03c0x + 3)))/(-2(2sin(3\u03c0x + 3)) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(4(-3x^2 - 1)^2 - 3(-3x^2 - 1) - 5))/(3(4(-3x^2 - 1)^2 - 3(-3x^2 - 1) - 5) + 1) and g(x) = -3sin(3\u03c0(-1x^2 + 5x - 5) - 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "638", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(3sin(1\u03c0(2x - 3)) - 1) - 1 and g(x) = -5(1x + 4)^2 - 5(1x + 4) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(-2sin(2\u03c0((-2x - 2)/(-3x)) - 1) - 2) + 3)/(2(-2sin(2\u03c0((-2x - 2)/(-3x)) - 1) - 2) - 1) and g(x) = -3sin(1\u03c0(1x - 1) - 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(3|(1x^2 + 3)| - 3) + 1)/(-1(3|(1x^2 + 3)| - 3) + 2) = 3cos(3\u03c0(-3x + 5) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "162", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(-2(-3sin(2\u03c0x + 2) + 2) - 1)^2 - 1(-2(-3sin(2\u03c0x + 2) + 2) - 1) - 4 and g(x) = 1(1x) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(1(-4x^2 + 3x + 5)^2 + 4(-4x^2 + 3x + 5) - 5) - 2| - 3 and g(x) = -5((-1x)/(-1x + 2)) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1e^(-1((-1(-5x - 3) + 1)/(3(-5x - 3) - 1))) - 2 and g(x) = -2(-1|x + 3| + 2)^2 - 4(-1|x + 3| + 2) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(-1((3x)/(-3x - 1)) + 3) - 3)/(3(-1((3x)/(-3x - 1)) + 3) - 2) and g(x) = 3cos(2\u03c0(-3x) - 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "121", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(2cos(3\u03c0((1x + 2)/(2x - 3)) + 1) + 1)^2 + 4(2cos(3\u03c0((1x + 2)/(2x - 3)) + 1) + 1) + 2 = 3(-2x + 3)^2 + 2(-2x + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "77", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2((3(-2|x + 3| - 2) + 3)/(3(-2|x + 3| - 2) - 1))^2 + 3((3(-2|x + 3| - 2) + 3)/(3(-2|x + 3| - 2) - 1)) - 2 = -2sin(2\u03c0(4x^2 - 4x - 1) - 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "489", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(5(-3sin(2\u03c0x + 1) + 1)^2 + 1(-3sin(2\u03c0x + 1) + 1))| + 3 and g(x) = -1|(3x + 4) + 1| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-2((-1x + 3)/(-3x + 1))^2 + 5((-1x + 3)/(-3x + 1)) + 4)^2 - 5(-2((-1x + 3)/(-3x + 1))^2 + 5((-1x + 3)/(-3x + 1)) + 4) + 1 and g(x) = 2|(-2sin(3\u03c0x + 1) + 3) + 3| + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-3sin(3\u03c0(-1|x + 3| - 1) + 2) - 2)^2 + 5 and g(x) = -3cos(3\u03c0(-1x^2 - 1x + 2) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "144", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5((3(-2|x + 1| + 2) + 1)/(-1(-2|x + 1| + 2) - 3)) - 2 and g(x) = 2(3cos(2\u03c0x - 1) - 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3e^(1(1|((1x - 3)/(2x - 1)) + 3| - 2)) - 1 = 3|(-3sin(1\u03c0x - 2)) - 3| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(2sin(1\u03c0((-1x)/(-1x - 3)) - 3) - 3)^2 + 4(2sin(1\u03c0((-1x)/(-1x - 3)) - 3) - 3) + 1 and g(x) = 1((-2x - 2)/(-1x - 2)) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1((1(-1x^2 + 2x - 1))/(1(-1x^2 + 2x - 1) + 1)) + 1)/(-1((1(-1x^2 + 2x - 1))/(1(-1x^2 + 2x - 1) + 1)) + 2) and g(x) = 3cos(1\u03c0(5x^2 - 2x + 1) - 3) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-1sin(2\u03c0(-3x - 4) - 1) - 3)^2 + 4(-1sin(2\u03c0(-3x - 4) - 1) - 3) + 1 and g(x) = (2((-2x - 2)/(-2x + 3)) + 1)/(-1((-2x - 2)/(-2x + 3)) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4((3(-2x^2 + 3x - 1) - 1)/(3(-2x^2 + 3x - 1) + 2)) + 5 and g(x) = -3sin(3\u03c0(-2x - 5) + 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3((-3(3e^(2x) + 1) + 3)/(2(3e^(2x) + 1) + 2))^2 + 1((-3(3e^(2x) + 1) + 3)/(2(3e^(2x) + 1) + 2)) + 2 and g(x) = -1sin(3\u03c0(-4x - 5) - 1) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1((3(1|x - 2|) - 1)/(-3(1|x - 2|))) - 4 and g(x) = -3sin(3\u03c0((-2x)/(-1x + 3)) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(3(-2x^2 + 5x + 1) + 1)^2 - 4(3(-2x^2 + 5x + 1) + 1) - 4 and g(x) = -1cos(1\u03c0(-2|x + 3|) + 1) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(-1sin(2\u03c0(-3x^2 - 5x - 1) - 2) - 3) + 2)/(2(-1sin(2\u03c0(-3x^2 - 5x - 1) - 2) - 3) + 2) and g(x) = -3sin(3\u03c0((-2x - 1)/(1x)) + 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "89", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(1cos(1\u03c0(-5x^2 + 3x - 3) + 2) - 1) + 1)/(-2(1cos(1\u03c0(-5x^2 + 3x - 3) + 2) - 1) + 1) and g(x) = -4((3x + 3)/(-2x + 3)) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "176", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1sin(2\u03c0(-1|(-2x + 1) - 3| - 1) - 2) - 2 = 1cos(2\u03c0(-5x - 4) + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "102", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(-2|(3|x - 2| - 1)| - 1) - 2)/(1(-2|(3|x - 2| - 1)| - 1) - 3) and g(x) = 3|(3|x - 1| - 3) - 1| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(1|(-5x^2 - 5x + 5) + 1| - 1)^2 - 4(1|(-5x^2 - 5x + 5) + 1| - 1) and g(x) = 3|(-1cos(1\u03c0x + 1)) - 2| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3e^(2(-3|(5x + 3) + 1| + 1)) - 2 = (-1(2|x + 1| - 2) - 1)/(2(2|x + 1| - 2) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3((-3(-1x^2 + 4x) + 3)/(-3(-1x^2 + 4x) - 3)) - 1)/(3((-3(-1x^2 + 4x) + 3)/(-3(-1x^2 + 4x) - 3)) - 2) and g(x) = (3(3x^2 + 5x + 1) - 2)/(2(3x^2 + 5x + 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(1sin(3\u03c0(3x^2 + 4x)) - 3) + 3)/(-3(1sin(3\u03c0(3x^2 + 4x)) - 3) - 1) and g(x) = 2sin(1\u03c0(-2x - 2) - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "85", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-3(2|x - 2| - 3)^2 + 5(2|x - 2| - 3) + 3) - 1| + 1 and g(x) = (-3(-1sin(2\u03c0x - 3) + 2) + 2)/(-2(-1sin(2\u03c0x - 3) + 2) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2sin(2\u03c0((-1(-1|x + 2| - 2) - 1)/(2(-1|x + 2| - 2) + 2)) + 3) - 3 and g(x) = -5(-3x^2 - 1x + 4)^2 - 2(-3x^2 - 1x + 4) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(2\u03c0(-2|(-1x - 4)| - 1) + 1) - 1 and g(x) = (1(-4x^2 - 3x - 4) + 1)/(-2(-4x^2 - 3x - 4) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(-3cos(2\u03c0(3|x| - 1) + 2))^2 - 3(-3cos(2\u03c0(3|x| - 1) + 2)) - 3 = -3e^(1(2|x - 2| + 2)) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(1((-2x + 2)/(-3x - 1))^2) and g(x) = -5(-2sin(3\u03c0x - 1)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2e^(2((3((1x + 2)/(2x + 3)) - 3)/(2((1x + 2)/(2x + 3)) + 1))) and g(x) = -3sin(2\u03c0(3|x + 1| - 2) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "116", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(-2|((-2x)/(1x + 1)) + 3| - 2) - 3 and g(x) = (-1(-1|x - 2| + 2) - 3)/(3(-1|x - 2| + 2)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(3\u03c0(2((-1x + 2)/(-1x + 2))^2 - 3((-1x + 2)/(-1x + 2)) + 5) - 2) + 2 and g(x) = 1sin(2\u03c0(-5x^2 + 5x - 3) - 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1494", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(3sin(3\u03c0(1|x + 3| - 1) - 2)) - 3)/(3(3sin(3\u03c0(1|x + 3| - 1) - 2)) + 3) and g(x) = 2((-2x - 2)/(-1x + 3))^2 + 1((-2x - 2)/(-1x + 3)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3cos(1\u03c0(1|(2x - 4) - 2|) - 3) - 1 and g(x) = (-3((-1x)/(-1x + 2)) + 3)/(-1((-1x)/(-1x + 2)) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(-3cos(1\u03c0(1|x - 1| - 3) + 3) - 1) - 1| - 1 and g(x) = 5(1|x + 2|) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(2(3cos(3\u03c0x + 2) - 1) - 2))/(-2(2(3cos(3\u03c0x + 2) - 1) - 2) - 2) and g(x) = -5(2cos(3\u03c0x + 3) - 3)^2 - 1(2cos(3\u03c0x + 3) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(1(1x^2 + 5x - 1)^2 + 5(1x^2 + 5x - 1) + 2) + 1)/(1(1(1x^2 + 5x - 1)^2 + 5(1x^2 + 5x - 1) + 2)) and g(x) = 2|(1cos(3\u03c0x - 1) + 2)| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(2sin(1\u03c0(4x - 4) - 1) + 1) + 1| - 1 = (3(-5x^2 - 5x + 5))/(-2(-5x^2 - 5x + 5) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(2e^(2(2|x + 3| - 2))) - 3| + 3 and g(x) = (-2(3x - 2) - 1)/(-2(3x - 2) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-1(-1x - 1)^2 - 5(-1x - 1) - 2) + 5 and g(x) = (-2(3cos(3\u03c0x - 1)) - 3)/(1(3cos(3\u03c0x - 1)) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(-2|(1|x + 1| - 3) - 3| + 2) + 3 and g(x) = -1cos(3\u03c0(1x + 4) + 1) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(3\u03c0(-3|(-5x^2 - 3x + 4) - 2| + 2) - 3) - 2 and g(x) = (-2(1sin(2\u03c0x + 1) - 3) + 1)/(1(1sin(2\u03c0x + 1) - 3) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "864", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(3sin(3\u03c0(-2|x - 3|)))^2 - 1(3sin(3\u03c0(-2|x - 3|))) - 4 and g(x) = -2(-5x + 5) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(1(3sin(2\u03c0x - 3) - 3) + 1) + 3)/(3(1(3sin(2\u03c0x - 3) - 3) + 1)) and g(x) = -2((3x + 2)/(-2x + 3))^2 - 2((3x + 2)/(-2x + 3)) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(2\u03c0(3|(3x - 5) + 3| + 1) + 2) - 3 and g(x) = -2|(-2x - 3) + 1| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(1|(5x) + 1| + 2) - 1)/(-1(1|(5x) + 1| + 2) + 3) and g(x) = 3sin(1\u03c0(4x + 1) - 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "74", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(2e^(-2(1x^2 - 4x - 4)) - 3))/(-3(2e^(-2(1x^2 - 4x - 4)) - 3) + 4) and g(x) = -2cos(2\u03c0(-3|x - 1| + 1) + 2) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "118", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(-1|((-3x + 3)/(1x - 2)) + 3| + 3)^2 + 4(-1|((-3x + 3)/(1x - 2)) + 3| + 3) - 4 = (-2((-3x + 1)/(-3x - 3)))/(3((-3x + 1)/(-3x - 3)) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1((1(-1x - 1) + 2)/(2(-1x - 1) + 1))^2 - 3((1(-1x - 1) + 2)/(2(-1x - 1) + 1)) - 4 and g(x) = (1(3|x| - 2) - 2)/(1(3|x| - 2) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(-4(2sin(3\u03c0x + 1)) + 5) - 5 = -3(-3sin(2\u03c0x + 1) - 3)^2 + 5(-3sin(2\u03c0x + 1) - 3) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-5(-3x^2 - 5x + 2)^2 + 4(-3x^2 - 5x + 2) + 3) - 1)/(1(-5(-3x^2 - 5x + 2)^2 + 4(-3x^2 - 5x + 2) + 3)) and g(x) = 1(1e^(2x))^2 + 5(1e^(2x)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(-2(-1x^2 + 2) - 2) + 1)/(2(-2(-1x^2 + 2) - 2) + 3) and g(x) = -2(3cos(1\u03c0x + 2)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1sin(3\u03c0((-1(1x^2 - 2x - 2) - 2)/(-2(1x^2 - 2x - 2) + 1))) - 2 and g(x) = -2cos(2\u03c0((3x + 3)/(3x + 3)) + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2((2(2cos(3\u03c0x + 1)) + 2)/(3(2cos(3\u03c0x + 1)) - 2)) - 2)/(-1((2(2cos(3\u03c0x + 1)) + 2)/(3(2cos(3\u03c0x + 1)) - 2))) and g(x) = 1|((-3x + 3)/(-1x + 3)) - 1| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(2((-2x - 3)/(-2x + 1)) - 2) - 3)/(-2(2((-2x - 3)/(-2x + 1)) - 2) + 3) and g(x) = 2sin(2\u03c0((2x + 3)/(-1x)) - 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "82", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-3(5x^2 + 2x - 2) - 3) + 2| - 1 and g(x) = 1((1x - 2)/(1x - 3)) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-3sin(2\u03c0(-5x + 4) + 1) + 1)^2 + 1(-3sin(2\u03c0(-5x + 4) + 1) + 1) - 5 and g(x) = (-1(-3x + 1) - 1)/(3(-3x + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "199", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2sin(2\u03c0((1((-3x + 3)/(-2x + 3)) + 3)/(2((-3x + 3)/(-2x + 3)))) - 1) + 1 = (-3(5x^2 + 4x + 5) - 3)/(3(5x^2 + 4x + 5) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(3cos(3\u03c0((-2x - 2)/(-1x - 2))) + 2) - 2| = -3sin(1\u03c0((1x - 1)/(-1x + 2)) - 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1((-2(-4x^2 - 5x - 4) + 3)/(-1(-4x^2 - 5x - 4) + 3)) - 4 and g(x) = 3cos(1\u03c0(1x + 5) + 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-3(1cos(3\u03c0x - 1)) + 2) + 3 and g(x) = (-1(-3cos(1\u03c0x + 2) - 2) + 2)/(-3(-3cos(1\u03c0x + 2) - 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(1\u03c0(-3|(1x^2 + 4x - 3)| + 3) - 3) + 3 and g(x) = (2(1|x| + 1) + 2)/(1(1|x| + 1) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "614", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(2\u03c0(3(1|x - 3| - 3) - 5) + 2) + 3 and g(x) = 2|(1|x - 2| - 3) - 1| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(-3|(-2cos(3\u03c0x + 2))| - 1) + 2)/(1(-3|(-2cos(3\u03c0x + 2))| - 1) + 1) and g(x) = 1(-2sin(3\u03c0x + 3))^2 - 4(-2sin(3\u03c0x + 3)) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(2|(-2e^(-1x) - 1) + 2| - 2) - 2)/(2(2|(-2e^(-1x) - 1) + 2| - 2) + 2) = 2sin(1\u03c0((-2x + 1)/(-1x - 1)) + 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3cos(1\u03c0(-1|(3x - 2) - 2| + 3) + 2) + 3 = -2|(1|x - 3| + 1) - 2| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(2\u03c0(-2|(1x + 3) + 2| - 3) + 1) + 1 and g(x) = 2e^(2((2x - 3)/(-3x + 4))) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1sin(2\u03c0(-2|(-3x^2 - 3x - 3) - 3| + 1) + 1) - 3 and g(x) = -4((-1x)/(-2x + 3)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1140", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(2|(-1x + 2) + 1| - 3)^2 - 5(2|(-1x + 2) + 1| - 3) - 2 and g(x) = 5((1x)/(-3x - 3)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(3sin(3\u03c0(-4x - 3) + 2) + 1) - 4 and g(x) = 2(3e^(2x) - 2)^2 + 3(3e^(2x) - 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "119", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|((-1(-1x^2 - 4x + 2) + 1)/(-3(-1x^2 - 4x + 2) + 1)) + 2| + 3 and g(x) = 2(-2e^(1x) + 3)^2 - 3(-2e^(1x) + 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(1|((3x - 3)/(-1x + 2)) + 3| - 3)^2 + 3(1|((3x - 3)/(-1x + 2)) + 3| - 3) + 1 and g(x) = -5(3cos(2\u03c0x + 2) + 1)^2 - 1(3cos(2\u03c0x + 2) + 1) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1cos(3\u03c0(5(-1x - 2)^2 - 3(-1x - 2) - 5) + 3) = (1(-1x))/(-3(-1x) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1560", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(2(1|x + 2| - 2)^2 + 2(1|x + 2| - 2) - 5) - 3)/(2(2(1|x + 2| - 2)^2 + 2(1|x + 2| - 2) - 5) - 1) and g(x) = (-1(3|x| + 2) + 2)/(3(3|x| + 2) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|(1((-2x + 3)/(3x - 2))^2 - 4((-2x + 3)/(3x - 2)) - 1) + 2| - 1 = 3(-1|x - 2|)^2 + 4(-1|x - 2|) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(2cos(2\u03c0(3|x - 1| + 2)) + 1)^2 + 3(2cos(2\u03c0(3|x - 1| + 2)) + 1) - 2 and g(x) = (1(1sin(2\u03c0x) - 2) + 2)/(-1(1sin(2\u03c0x) - 2) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "200", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(-3|((1x + 1)/(3x - 2)) - 2| - 1) + 2| + 3 and g(x) = -3(-1sin(1\u03c0x - 3) + 1)^2 - 2(-1sin(1\u03c0x - 3) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3((-2(-3sin(1\u03c0x - 1) - 2) - 3)/(2(-3sin(1\u03c0x - 1) - 2) + 1))^2 - 5((-2(-3sin(1\u03c0x - 1) - 2) - 3)/(2(-3sin(1\u03c0x - 1) - 2) + 1)) + 1 = (-3(1sin(1\u03c0x + 3) + 2) + 1)/(-2(1sin(1\u03c0x + 3) + 2) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(1cos(1\u03c0(-2x - 5) + 1)) and g(x) = (-2(2|x - 1|) + 2)/(3(2|x - 1|) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1((-1(-1cos(3\u03c0x - 2) + 3) - 1)/(-1(-1cos(3\u03c0x - 2) + 3) + 2)) + 1 and g(x) = 1(1sin(3\u03c0x - 2) - 2)^2 + 1(1sin(3\u03c0x - 2) - 2) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(2cos(3\u03c0((-3x + 1)/(2x - 1)) + 2) - 2) - 2)/(1(2cos(3\u03c0((-3x + 1)/(2x - 1)) + 2) - 2)) and g(x) = -2cos(3\u03c0((-2x - 3)/(2x + 2)) + 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2(2cos(2\u03c0(-3x^2 - 1x - 3)) - 1) - 1 and g(x) = 2(-1e^(-1x) - 1) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(2|(-1cos(2\u03c0x + 1) + 3) + 1| + 1) - 2| + 2 and g(x) = -1|(3|x - 3| - 1) - 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1cos(2\u03c0(1(1x^2 - 4x - 1)^2 - 2) - 2) - 3 and g(x) = 1cos(2\u03c0(-4x + 5)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(3\u03c0(-3|(-2x) - 2| + 1) + 1) - 1 = (-2(-1cos(3\u03c0x - 3) - 2) - 1)/(3(-1cos(3\u03c0x - 3) - 2) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "360", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(1\u03c0(1|((-2x - 3)/(1x - 2)) + 2| + 3)) and g(x) = (-2((-2x - 2)/(-2x - 2)))/(-1((-2x - 2)/(-2x - 2)) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "94", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(1(2x) - 3) + 2 = 1sin(1\u03c0((2x + 2)/(-3x - 2)) - 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-1sin(2\u03c0(3|x + 2| - 3) + 2) + 3)^2 - 2(-1sin(2\u03c0(3|x + 2| - 3) + 2) + 3) + 2 and g(x) = -2(-3sin(2\u03c0x + 3) - 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(2sin(2\u03c0(-3|x - 2| - 1) + 3) + 1)^2 + 1(2sin(2\u03c0(-3|x - 2| - 1) + 3) + 1) + 4 = 1sin(3\u03c0(2x^2 + 4x - 2)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(3\u03c0(3(2|x - 2| - 2)^2 - 5(2|x - 2| - 2)) + 2) + 2 and g(x) = (3(2x + 2) - 3)/(-3(2x + 2) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "304", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4((-2(1|x - 3| - 2) - 3)/(-1(1|x - 3| - 2) + 3)) - 2 and g(x) = 4(-2|x - 3| + 2) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(1sin(1\u03c0((3x - 3)/(3x - 2)) - 3) + 2) - 1 and g(x) = (-2(-2|x| + 3) + 1)/(-3(-2|x| + 3) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2(-3sin(1\u03c0((2x - 3)/(-2x + 3)) + 3) + 1)^2 + 3(-3sin(1\u03c0((2x - 3)/(-2x + 3)) + 3) + 1) - 2 and g(x) = 2(1cos(1\u03c0x + 1) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(-2cos(3\u03c0(1x^2 - 3x - 5) + 2) - 1) - 1| and g(x) = -3(3x^2 - 4x - 4) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(-4(-1x) + 2)^2 + 4(-4(-1x) + 2) - 3 = 2sin(2\u03c0(-4x + 2) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(1|(2sin(3\u03c0x + 3) + 3) - 3| + 3) + 5 = 4(2cos(3\u03c0x) + 3)^2 - 4(2cos(3\u03c0x) + 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2sin(3\u03c0(-1(-2x^2 - 1x + 5)) + 2) - 3 = (1(1e^(-2x) + 3) - 2)/(-1(1e^(-2x) + 3) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "147", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1e^(2(2(4x)^2 + 5(4x))) - 1 = -2sin(3\u03c0(2|x - 3| + 3) + 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-1(-2cos(3\u03c0x + 2) - 1)^2 - 4(-2cos(3\u03c0x + 2) - 1) - 5)^2 - 5(-1(-2cos(3\u03c0x + 2) - 1)^2 - 4(-2cos(3\u03c0x + 2) - 1) - 5) - 1 and g(x) = -1|(-1cos(3\u03c0x - 2) - 1) - 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3sin(3\u03c0((-2((2x)/(-3x + 4)) - 1)/(1((2x)/(-3x + 4)))) - 3) + 3 = 1(3|x - 2| - 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-2((-1x - 3)/(-2x - 3))^2 - 1) + 4 = -2sin(2\u03c0(1|x + 1| - 3) - 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2((-1(-3cos(3\u03c0x - 1) + 3))/(2(-3cos(3\u03c0x - 1) + 3) - 3)) + 2)/(-3((-1(-3cos(3\u03c0x - 1) + 3))/(2(-3cos(3\u03c0x - 1) + 3) - 3))) = -1sin(1\u03c0((-3x)/(1x + 3))) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(1cos(1\u03c0(5x^2 + 2) + 1) - 3) - 1 = 2sin(3\u03c0((3x + 1)/(3x - 2)) - 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "125", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(4(1x) - 1) - 3)/(2(4(1x) - 1) + 1) = -2cos(3\u03c0(4x - 4)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "235", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-2sin(3\u03c0(-1x - 3))) - 3)/(3(-2sin(3\u03c0(-1x - 3)))) and g(x) = (3(-5x - 1) - 2)/(1(-5x - 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1cos(2\u03c0((3((2x + 2)/(3x + 3)) + 1)/(2((2x + 2)/(3x + 3)) + 3)) + 3) and g(x) = -4(-2cos(2\u03c0x - 1) - 1) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(2cos(1\u03c0(1|x - 1| - 1) - 2) + 1)^2 - 4(2cos(1\u03c0(1|x - 1| - 1) - 2) + 1) + 3 and g(x) = 3e^(1((-1x + 2)/(2x - 1))) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(-2cos(2\u03c0(-1|x + 2| + 2)) + 3)^2 - 2(-2cos(2\u03c0(-1|x + 2| + 2)) + 3) - 4 and g(x) = (2(-3|x - 3| - 3))/(2(-3|x - 3| - 3) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3e^(1(-2|(-2|x + 1| + 2) + 1| - 3)) + 3 = (1(-1cos(1\u03c0x + 2) + 1) + 3)/(-1(-1cos(1\u03c0x + 2) + 1) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(-3sin(3\u03c0((-1x - 3)/(2x - 1)) - 3)) - 2)/(1(-3sin(3\u03c0((-1x - 3)/(2x - 1)) - 3))) and g(x) = 5(2|x - 1|) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(5(2|x - 3| - 1) - 5) + 5 and g(x) = 4(3cos(2\u03c0x + 1) - 1) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3sin(3\u03c0(5(2x^2 - 3x + 5)^2 - 5(2x^2 - 3x + 5) - 1) - 1) and g(x) = 2sin(2\u03c0(-1x^2 + 1x - 4) - 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "602", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(1|(-1|x| + 3) + 2| - 1) - 1)/(3(1|(-1|x| + 3) + 2| - 1) + 1) and g(x) = 1sin(1\u03c0(-3|x + 3| + 1) - 2) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2((2(2x^2 + 5x - 2) + 3)/(-1(2x^2 + 5x - 2) + 3))^2 + 3((2(2x^2 + 5x - 2) + 3)/(-1(2x^2 + 5x - 2) + 3)) - 1 = (-3(-3cos(1\u03c0x + 3)) + 2)/(-1(-3cos(1\u03c0x + 3)) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(3|((-2x + 1)/(-1x + 3)) + 1| - 1))/(3(3|((-2x + 1)/(-1x + 3)) + 1| - 1) + 3) and g(x) = 5(1x^2 + 3x - 3)^2 + 4(1x^2 + 3x - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(3\u03c0(1|(-1x^2 + 2x + 5) + 3| + 1) + 3) and g(x) = -2|(3cos(2\u03c0x - 2) - 1) + 1| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "280", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3sin(2\u03c0(3|(-5x^2 + 2x - 3) + 1| - 1) + 2) - 2 and g(x) = (1(-4x^2 + 1x - 5) + 3)/(1(-4x^2 + 1x - 5) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "982", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(3\u03c0(5((-3x + 2)/(3x - 2))^2 + 4((-3x + 2)/(3x - 2)) - 1) + 2) + 1 and g(x) = 4(-2|x| + 1)^2 - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-3(2cos(3\u03c0x + 3) + 2) + 2)^2 + 3(-3(2cos(3\u03c0x + 3) + 2) + 2) - 2 and g(x) = -2(-3sin(1\u03c0x - 2) + 1)^2 + 1(-3sin(1\u03c0x - 2) + 1) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(3cos(1\u03c0((3x - 2)/(2x + 3)) - 3) - 2))/(-2(3cos(1\u03c0((3x - 2)/(2x + 3)) - 3) - 2) - 1) = (-2(1e^(-1x) + 3) + 2)/(2(1e^(-1x) + 3)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(-1|(1|x - 2| + 2) - 1|) - 3)/(-2(-1|(1|x - 2| + 2) - 1|) - 2) and g(x) = (1(3sin(3\u03c0x + 3) + 1) + 2)/(1(3sin(3\u03c0x + 3) + 1) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1((2(-3sin(1\u03c0x - 3) - 3))/(2(-3sin(1\u03c0x - 3) - 3) - 1)) + 2)/(3((2(-3sin(1\u03c0x - 3) - 3))/(2(-3sin(1\u03c0x - 3) - 3) - 1))) and g(x) = (1((2x + 1)/(-1x - 3)) - 1)/(1((2x + 1)/(-1x - 3)) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(2cos(3\u03c0(-5x^2 - 1x - 2) - 1) + 3) + 1| - 3 and g(x) = (1(-3x^2 - 4x + 2) + 2)/(-2(-3x^2 - 4x + 2) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(3(-1cos(3\u03c0x + 2) + 1) - 3)^2 - 2(3(-1cos(3\u03c0x + 2) + 1) - 3) + 2 = (2(-3cos(2\u03c0x - 3)) + 1)/(-2(-3cos(2\u03c0x - 3))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "117", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(3\u03c0(2(1|x - 3| - 1)^2 + 5(1|x - 3| - 1) + 3) - 1) + 3 and g(x) = -2cos(2\u03c0(-2|x - 2| - 1) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "528", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5((2(1cos(2\u03c0x + 3) - 1) + 3)/(-2(1cos(2\u03c0x + 3) - 1) - 1))^2 - 1((2(1cos(2\u03c0x + 3) - 1) + 3)/(-2(1cos(2\u03c0x + 3) - 1) - 1)) - 4 and g(x) = (1((-2x - 1)/(-1x)) + 1)/(3((-2x - 1)/(-1x)) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(2(-1cos(2\u03c0x + 2))^2 + 5(-1cos(2\u03c0x + 2)) + 2) = 1((2x + 3)/(-1x))^2 - 5((2x + 3)/(-1x)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3sin(1\u03c0(-5((1x + 3)/(-2x - 2)) - 4) + 3) + 3 and g(x) = -1|((1x - 1)/(1x + 1)) + 1| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(-3(2|x + 1| - 1)^2 + 4(2|x + 1| - 1) + 1) and g(x) = 2cos(3\u03c0((2x - 1)/(-1x)) - 2) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-5(3sin(1\u03c0x + 1) + 3)^2 + 5(3sin(1\u03c0x + 1) + 3) + 4) + 3)/(2(-5(3sin(1\u03c0x + 1) + 3)^2 + 5(3sin(1\u03c0x + 1) + 3) + 4) + 2) = -2sin(1\u03c0(3x^2 + 5x + 2) - 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3e^(1(1(-1|x - 2| + 3) - 3)) and g(x) = 2sin(1\u03c0(-4x^2 + 1x) + 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(3(-1sin(2\u03c0x + 3) + 1) - 1) + 5 = -1(-5x - 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(-3sin(3\u03c0(2x + 2) + 1) - 2)^2 + 5(-3sin(3\u03c0(2x + 2) + 1) - 2) and g(x) = 1(-1sin(2\u03c0x) - 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3cos(2\u03c0(1(1x - 2)^2 - 1(1x - 2) + 3)) - 3 and g(x) = 4(-2cos(2\u03c0x + 2) - 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "140", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4((1(-2cos(1\u03c0x + 2) + 3) - 3)/(3(-2cos(1\u03c0x + 2) + 3) - 2)) + 4 = -3(-3sin(3\u03c0x + 1) - 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(2cos(3\u03c0(-3x^2 + 2x + 1) - 2) - 1) + 2 = (-3(1e^(1x) - 3) + 1)/(-2(1e^(1x) - 3) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1380", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(-4(1|x - 1| - 2)^2 - 5(1|x - 1| - 2) + 4) and g(x) = 1cos(3\u03c0(4x^2 - 3x - 3) - 1) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(1|(-2|x + 3|) + 2| - 1)^2 + 1(1|(-2|x + 3|) + 2| - 1) + 1 = (1(1|x + 1| + 2) - 2)/(-2(1|x + 1| + 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(3|(-3cos(3\u03c0x - 3)) - 1|) + 1)/(3(3|(-3cos(3\u03c0x - 3)) - 1|) + 1) = -1|(4x - 3) + 1| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(1(-3sin(3\u03c0x) + 1)^2 - 4(-3sin(3\u03c0x) + 1) - 2) + 1| - 1 and g(x) = 1e^(-1(5x^2 - 2x - 4)) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "142", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(1\u03c0(-4(-2x + 4)^2 - 3) - 2) - 2 and g(x) = -1|(5x - 4) - 3| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4((-3(1x^2 + 4x - 5) + 2)/(-3(1x^2 + 4x - 5) + 2))^2 + 5((-3(1x^2 + 4x - 5) + 2)/(-3(1x^2 + 4x - 5) + 2)) + 4 = -3(3cos(2\u03c0x + 3) - 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(-3(3sin(3\u03c0x + 1)) + 4) - 2)/(-1(-3(3sin(3\u03c0x + 1)) + 4) - 3) and g(x) = (-1(1|x| + 3) + 3)/(-2(1|x| + 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2sin(1\u03c0((-1(-3|x - 1| + 1) - 3)/(1(-3|x - 1| + 1) - 2)) - 1) + 2 = 3cos(1\u03c0((-3x + 1)/(1x)) - 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-3sin(2\u03c0(1|x + 3| - 3) + 3) - 1) + 2 = (1(2x^2 + 1x - 4) + 3)/(-1(2x^2 + 1x - 4) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1cos(1\u03c0(-1((1x - 3)/(-3x))^2 + 2((1x - 3)/(-3x)) + 5) - 1) + 2 and g(x) = 1|(-3x + 4) - 1| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(5(-2x - 5) - 4) - 2| and g(x) = -4(2cos(3\u03c0x + 1))^2 - 5(2cos(3\u03c0x + 1)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|((2(3sin(1\u03c0x - 1) + 1) - 3)/(-2(3sin(1\u03c0x - 1) + 1) - 1))| + 1 and g(x) = 3(-1|x + 2| + 3)^2 - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(1(2x - 4) - 5) - 2 and g(x) = 5(-1sin(2\u03c0x - 3) + 1)^2 - 1(-1sin(2\u03c0x - 3) + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(-2sin(2\u03c0(-2x^2 + 5x - 4) + 2) + 2) - 1)/(2(-2sin(2\u03c0(-2x^2 + 5x - 4) + 2) + 2) + 1) and g(x) = -5((2x + 3)/(-3x)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(-1((-3x)/(2x + 2))^2 + 4((-3x)/(2x + 2)) - 1) + 3)/(3(-1((-3x)/(2x + 2))^2 + 4((-3x)/(2x + 2)) - 1)) and g(x) = -3(1x^2 - 1x)^2 - 2(1x^2 - 1x) + 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1cos(2\u03c0(-3(-2|x + 1| + 3)^2 + 3(-2|x + 1| + 3) + 1) - 1) = -1(3sin(2\u03c0x + 1) + 2)^2 - 4(3sin(2\u03c0x + 1) + 2) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(-1cos(3\u03c0(5x^2 + 2x - 2) + 3)) + 5 and g(x) = (-3(-3sin(1\u03c0x + 2) - 1) + 2)/(-1(-3sin(1\u03c0x + 2) - 1) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "768", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(1(-3cos(3\u03c0x - 1) - 1)^2 + 5(-3cos(3\u03c0x - 1) - 1) + 1) + 3)/(2(1(-3cos(3\u03c0x - 1) - 1)^2 + 5(-3cos(3\u03c0x - 1) - 1) + 1) + 1) and g(x) = 1(4x - 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(2cos(2\u03c0(3x) - 2) + 3)^2 - 5(2cos(2\u03c0(3x) - 2) + 3) + 1 = 2sin(1\u03c0((-3x)/(-2x - 1)) + 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(2sin(3\u03c0(5x^2 + 1x + 5) + 1) - 1)^2 + 5(2sin(3\u03c0(5x^2 + 1x + 5) + 1) - 1) + 1 and g(x) = -2e^(-2(3x)) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "674", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2e^(1(-5(2|x - 3| + 1) - 2)) and g(x) = 2|(-2x^2 + 1x + 5) - 1| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(3|(-1sin(1\u03c0x - 1) + 3) - 1| - 1) + 1)/(1(3|(-1sin(1\u03c0x - 1) + 3) - 1| - 1) - 3) and g(x) = 1|(4x^2 + 3x - 2) - 1| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1cos(3\u03c0(2(4x^2 + 2x + 2) + 1) - 3) - 2 and g(x) = 3((1x)/(1x - 2)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "200", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3((2(5x^2 - 1x + 4) + 3)/(-3(5x^2 - 1x + 4) + 4))^2 + 4((2(5x^2 - 1x + 4) + 3)/(-3(5x^2 - 1x + 4) + 4)) - 1 and g(x) = -1(-2|x - 2| + 3)^2 + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(2|(-2|x + 2| + 1) - 2| + 2) + 1 and g(x) = -3|(-3sin(1\u03c0x + 2))| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-3|(-3x^2 - 4x - 2) + 1| - 1) + 2)/(1(-3|(-3x^2 - 4x - 2) + 1| - 1) + 2) and g(x) = -3|(-1x - 4) - 1| + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2sin(3\u03c0((-2((2x)/(2x + 3)) + 2)/(-1((2x)/(2x + 3)) + 3)) + 1) - 1 and g(x) = 4(-3sin(3\u03c0x - 2) + 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(-2(-2x^2 + 5x - 5) + 1))/(2(-2(-2x^2 + 5x - 5) + 1) + 1) and g(x) = -1|(4x^2 + 5x - 3) - 3| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-1sin(3\u03c0(-1x + 1) - 3) - 2))/(-1(-1sin(3\u03c0(-1x + 1) - 3) - 2) + 2) and g(x) = 1(-2x^2 + 5x - 1)^2 - 2(-2x^2 + 5x - 1) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(-2sin(3\u03c0(2x - 4) + 1) + 2)^2 + 1(-2sin(3\u03c0(2x - 4) + 1) + 2) - 3 and g(x) = -3((2x - 1)/(2x + 2))^2 - 2((2x - 1)/(2x + 2)) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3((3(2sin(2\u03c0x) + 1))/(3(2sin(2\u03c0x) + 1) + 3))^2 + 1((3(2sin(2\u03c0x) + 1))/(3(2sin(2\u03c0x) + 1) + 3)) + 3 = -2e^(-1(-3|x - 3| + 1)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(3(-1cos(2\u03c0x - 3))^2 + 4(-1cos(2\u03c0x - 3)) + 1) - 3 and g(x) = -1sin(1\u03c0(3|x - 2| + 3) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(-2sin(2\u03c0(1x + 1)) - 2)^2 + 3(-2sin(2\u03c0(1x + 1)) - 2) - 1 = (3(-1|x + 3| - 1) - 3)/(1(-1|x + 3| - 1) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-2e^(-2(-1x^2 + 4x - 3)) + 3)^2 + 3(-2e^(-2(-1x^2 + 4x - 3)) + 3) + 5 and g(x) = 1cos(3\u03c0(-5x + 3) - 1) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2((-1((-1x + 1)/(-3x + 4)) - 3)/(1((-1x + 1)/(-3x + 4)) - 2)) + 2 = -3|(-4x^2 - 5x + 4)| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-5(4x^2 + 1x - 4) + 5) + 3)/(2(-5(4x^2 + 1x - 4) + 5) - 1) and g(x) = 4(-2cos(2\u03c0x + 1) + 3)^2 - 4(-2cos(2\u03c0x + 1) + 3) - 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(3|(-4x + 1) - 1| + 3) - 1)/(1(3|(-4x + 1) - 1| + 3) + 2) and g(x) = (2(-3cos(1\u03c0x - 3) + 2) - 1)/(-3(-3cos(1\u03c0x - 3) + 2) + 4) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(-1|(-2x - 1) - 1| + 2) + 4 = -1cos(2\u03c0(5x + 4)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(-1|((-1x + 2)/(3x + 2)) + 3| + 2) + 2| and g(x) = (-1(-3sin(3\u03c0x + 1) + 3) - 3)/(3(-3sin(3\u03c0x + 1) + 3) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "57", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2((-3(1|x + 1| - 2) + 3)/(-2(1|x + 1| - 2)))^2 - 1((-3(1|x + 1| - 2) + 3)/(-2(1|x + 1| - 2))) - 1 and g(x) = (2(5x^2 - 4x + 4) + 2)/(-2(5x^2 - 4x + 4) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(-3sin(3\u03c0(-1|x + 3| + 2) + 3) - 3)^2 + 5 and g(x) = -2e^(-1(-1x - 5)) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(2(1x^2 - 3x - 5) - 1) + 1)/(3(2(1x^2 - 3x - 5) - 1) + 1) and g(x) = -2|(4x^2 - 4x - 3) - 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(2\u03c0((1(-1x^2 + 1x))/(2(-1x^2 + 1x) + 3))) - 1 and g(x) = 4(-2x^2 - 5x - 1)^2 + 4(-2x^2 - 5x - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-3(-1cos(2\u03c0x + 1) - 2)^2 + 5))/(3(-3(-1cos(2\u03c0x + 1) - 2)^2 + 5) - 2) and g(x) = (-2(-3x - 3))/(1(-3x - 3) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(-5((2x + 2)/(-2x + 3)) - 4) - 2| + 3 and g(x) = -1cos(1\u03c0(3x^2 - 2x + 3) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "92", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(1|((2x + 1)/(2x + 2)) - 2| - 2) - 1 and g(x) = -3(-2sin(3\u03c0x - 2) + 1)^2 - 3(-2sin(3\u03c0x - 2) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3cos(3\u03c0(-1|((-3x - 1)/(1x + 1)) + 3| - 1) + 3) + 3 = (1(-2e^(-2x) + 2) - 3)/(-3(-2e^(-2x) + 2) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(1\u03c0(-2|(-3|x - 1|) - 2| - 1) + 2) - 2 and g(x) = 1((3x - 2)/(-1x - 2))^2 + 2((3x - 2)/(-1x - 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-3|(1x^2 + 1x - 2) + 3| + 2))/(-2(-3|(1x^2 + 1x - 2) + 3| + 2) + 3) and g(x) = -5(2sin(1\u03c0x) - 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1((-1((2x)/(2x - 3)) - 1)/(-1((2x)/(2x - 3)) + 2)) + 1)/(2((-1((2x)/(2x - 3)) - 1)/(-1((2x)/(2x - 3)) + 2)) + 3) and g(x) = -2cos(2\u03c0((-2x + 2)/(1x + 1)) - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|((-3((-1x + 2)/(-1x - 1)))/(2((-1x + 2)/(-1x - 1)) - 1)) + 2| - 3 and g(x) = 3(2sin(3\u03c0x - 3) + 1) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(3(-1x^2 + 5x)^2 - 4(-1x^2 + 5x) + 5) - 1)/(1(3(-1x^2 + 5x)^2 - 4(-1x^2 + 5x) + 5) - 3) and g(x) = 1(2sin(2\u03c0x - 1)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(3|((-1x + 3)/(1x)) + 1| - 2)^2 - 2(3|((-1x + 3)/(1x)) + 1| - 2) - 1 and g(x) = -2(-1x^2 + 4x + 1)^2 - 2(-1x^2 + 4x + 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(4((-3x)/(-2x + 3))^2 - 3))/(-1(4((-3x)/(-2x + 3))^2 - 3)) and g(x) = 5(-1x^2 - 2x + 2)^2 - 3(-1x^2 - 2x + 2) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3(2cos(2\u03c0(-2x + 3) + 2)))/(2(2cos(2\u03c0(-2x + 3) + 2)) - 1) = -1(2x^2 - 2x)^2 - 4(2x^2 - 2x) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(-3|(-1|x + 3| - 1) + 2| - 1) + 3)/(3(-3|(-1|x + 3| - 1) + 2| - 1) + 2) = (-2(3sin(2\u03c0x + 1)) - 2)/(-1(3sin(2\u03c0x + 1))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-5((2x + 1)/(-3x + 4))^2 - 5((2x + 1)/(-3x + 4)) - 1)^2 + 4(-5((2x + 1)/(-3x + 4))^2 - 5((2x + 1)/(-3x + 4)) - 1) + 3 and g(x) = -5(-1cos(2\u03c0x - 3) - 1)^2 + 5(-1cos(2\u03c0x - 3) - 1) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3cos(1\u03c0((1(2x + 1) + 1)/(2(2x + 1) - 1)) + 3) + 3 and g(x) = (-1((3x + 2)/(2x + 1)))/(-1((3x + 2)/(2x + 1)) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1((-3(-1|x - 2| + 2) - 1)/(1(-1|x - 2| + 2) + 3)) - 5 = -1cos(1\u03c0(-2|x + 1| - 2) - 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(2\u03c0(1|(-2x - 5) - 2|) + 3) - 2 and g(x) = -3sin(1\u03c0(4x - 3) - 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(-2sin(2\u03c0((-3x + 2)/(-3x + 4)) + 1) + 2) - 3)/(-1(-2sin(2\u03c0((-3x + 2)/(-3x + 4)) + 1) + 2) - 2) = 2sin(2\u03c0(3|x - 3| - 1) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "124", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|((-1(-2|x - 3|) - 2)/(-3(-2|x - 3|))) + 1| - 3 = -2sin(3\u03c0(-5x - 4)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-4(-5x)^2 + 4(-5x) + 5) - 1| - 1 and g(x) = -3e^(1(1|x - 2| - 3)) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(2e^(2(-3x - 1)) - 1)^2 + 1(2e^(2(-3x - 1)) - 1) + 2 = 2sin(2\u03c0(2x - 4) + 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2((-3(-3|x + 2| + 2) - 3)/(-3(-3|x + 2| + 2) + 2)))/(-2((-3(-3|x + 2| + 2) - 3)/(-3(-3|x + 2| + 2) + 2)) + 1) and g(x) = -3sin(1\u03c0(2x - 3) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(2(-3x - 4) - 1)^2 - 1(2(-3x - 4) - 1) and g(x) = 3sin(2\u03c0(1x^2 - 1x + 4) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|((1(5x^2 + 2x + 3))/(2(5x^2 + 2x + 3))) + 3| + 1 = 5(-3sin(3\u03c0x + 2) - 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(3\u03c0(-1(5x + 2) - 1) - 2) and g(x) = 1sin(2\u03c0(3x) + 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "300", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(-3(2|x - 1| - 1) + 4)^2 + 5 = -1|(4x) - 1| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5((-1(2|x - 1| - 3) + 2)/(1(2|x - 1| - 3) + 1))^2 - 3((-1(2|x - 1| - 3) + 2)/(1(2|x - 1| - 3) + 1)) - 1 and g(x) = 3cos(2\u03c0(-4x^2 + 4x) - 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1176", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(2e^(-2(3x + 3))) - 2 and g(x) = 1sin(2\u03c0(5x^2 - 4x + 2) + 1) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "724", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(3((1x - 1)/(-1x - 2)) + 5))/(2(3((1x - 1)/(-1x - 2)) + 5) + 3) and g(x) = -3sin(3\u03c0((-3x - 2)/(-1x - 3)) + 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "149", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3sin(2\u03c0(3|(-3|x - 1|) + 2| + 1) + 2) + 3 and g(x) = 1|((3x + 3)/(3x + 2)) + 1| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "172", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(2\u03c0(1|(-2|x - 2| - 1) + 1|) + 3) - 1 and g(x) = -3|(-5x^2 + 2x + 2) + 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(-1|((2x + 1)/(-1x)) + 1| + 1) + 2| and g(x) = 3|(3cos(2\u03c0x - 3) - 1) - 3| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(3sin(1\u03c0(-1|x - 2| - 3) - 1) + 2) + 2| and g(x) = 2(-2|x - 3| - 2) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(1|(1|x + 1|) - 3| + 2) - 5 = 4(-3cos(3\u03c0x - 3) + 1)^2 + 1(-3cos(3\u03c0x - 3) + 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "98", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(1(-3|x + 3| + 2) + 5) + 1)/(2(1(-3|x + 3| + 2) + 5) + 1) and g(x) = 3sin(3\u03c0(-3|x + 2| - 1) - 3) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "178", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3((1(-1x^2 - 1))/(-1(-1x^2 - 1) - 1))^2 + 2((1(-1x^2 - 1))/(-1(-1x^2 - 1) - 1)) = 4(-3cos(2\u03c0x + 1) + 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-3sin(1\u03c0((-3x + 1)/(2x)) - 1) - 1)^2 - 4(-3sin(1\u03c0((-3x + 1)/(2x)) - 1) - 1) + 1 and g(x) = (1(-1x + 1))/(-1(-1x + 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(-3|(2|x| + 3) - 2| + 1) + 1)/(-3(-3|(2|x| + 3) - 2| + 1) + 1) and g(x) = -1cos(3\u03c0(1x^2 - 1x - 3) + 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "280", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(-1|(3cos(1\u03c0x + 1) + 3)| + 3))/(-1(-1|(3cos(1\u03c0x + 1) + 3)| + 3) + 2) and g(x) = 3|(3sin(3\u03c0x) + 2) + 2| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(-3(-2sin(2\u03c0x + 2)) + 3) + 2)/(2(-3(-2sin(2\u03c0x + 2)) + 3) + 3) = (1(1cos(2\u03c0x + 2) + 1))/(-3(1cos(2\u03c0x + 2) + 1) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(1(2x - 4) - 1) + 3| + 1 and g(x) = -1cos(1\u03c0(2|x| + 1) + 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(1cos(1\u03c0((-1x - 3)/(-3x)) - 1))^2 + 2(1cos(1\u03c0((-1x - 3)/(-3x)) - 1)) + 3 and g(x) = 2cos(1\u03c0(-2|x + 3|) - 3) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(1cos(2\u03c0(1|x| + 3) - 1) - 2) - 4 and g(x) = 1|(-2|x + 3| + 2) + 3| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(-2cos(1\u03c0(3|x + 1| + 3) + 3) + 1) - 5 and g(x) = -1|(-5x - 4) - 2| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2((-2(1cos(2\u03c0x - 3) - 1) - 1)/(-3(1cos(2\u03c0x - 3) - 1))) + 1)/(2((-2(1cos(2\u03c0x - 3) - 1) - 1)/(-3(1cos(2\u03c0x - 3) - 1))) + 1) and g(x) = 1cos(1\u03c0((3x - 3)/(2x + 2))) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2((-1(-5x + 1) + 1)/(3(-5x + 1) + 3)) - 2)/(1((-1(-5x + 1) + 1)/(3(-5x + 1) + 3)) + 1) and g(x) = 3(-2cos(3\u03c0x + 3) - 1) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(1|((-2x)/(2x + 1)) + 3| - 2)^2 + 1(1|((-2x)/(2x + 1)) + 3| - 2) - 1 and g(x) = 5(-2x - 5)^2 - 5(-2x - 5) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3((1(1e^(1x) - 2) + 1)/(3(1e^(1x) - 2) - 2)) - 1 and g(x) = -3(-3cos(2\u03c0x - 2) + 1)^2 + 3(-3cos(2\u03c0x - 2) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "77", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(4(-1|x - 1|)^2 - 2(-1|x - 1|) - 5)^2 - 3(4(-1|x - 1|)^2 - 2(-1|x - 1|) - 5) + 5 and g(x) = 1((3x - 2)/(1x - 3))^2 - 5((3x - 2)/(1x - 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|((-1((1x - 1)/(-2x + 3)))/(-3((1x - 1)/(-2x + 3)) + 2))| - 1 and g(x) = -1|((-1x)/(1x - 3)) + 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(2cos(2\u03c0(4x^2 + 3x) - 1) + 1)^2 + 3(2cos(2\u03c0(4x^2 + 3x) - 1) + 1) + 4 and g(x) = 3cos(2\u03c0(-2|x + 1|) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(1|((3x + 2)/(-2x)) + 2| + 2) - 1)/(2(1|((3x + 2)/(-2x)) + 2| + 2) + 3) and g(x) = 2cos(3\u03c0(-3x + 1) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "180", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(4(-1|x| + 1) - 1) - 1)/(-2(4(-1|x| + 1) - 1) - 2) and g(x) = -1cos(3\u03c0((-1x + 2)/(3x - 2)) + 3) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(2|((3x)/(-3x - 2)) + 1| - 3) + 2| - 1 = (1(-1sin(1\u03c0x - 2)) - 2)/(-1(-1sin(1\u03c0x - 2)) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2sin(1\u03c0(5(-1x^2 - 2x - 5) - 2) + 1) and g(x) = -1(-1|x - 2| + 3)^2 - 2(-1|x - 2| + 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "139", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(-1cos(3\u03c0(-4x^2 - 2x + 4) + 2) - 1) + 1| and g(x) = (-2(4x^2 - 5x + 2) + 2)/(2(4x^2 - 5x + 2) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1318", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2e^(-1(-2|(5x^2 - 4x - 4)| - 1)) - 3 and g(x) = 4(-1cos(2\u03c0x + 1) + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(-2(1cos(1\u03c0x + 2) + 2)^2 + 4(1cos(1\u03c0x + 2) + 2) + 4)^2 + 1(-2(1cos(1\u03c0x + 2) + 2)^2 + 4(1cos(1\u03c0x + 2) + 2) + 4) - 2 and g(x) = 1(3|x - 1| - 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(2(3cos(3\u03c0x + 3) + 2)^2 - 4(3cos(3\u03c0x + 3) + 2))^2 - 2 and g(x) = 1sin(1\u03c0(5x + 2) - 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "240", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(4(1x - 4)^2 + 1(1x - 4) - 4))/(1(4(1x - 4)^2 + 1(1x - 4) - 4) + 2) and g(x) = -3sin(1\u03c0(3|x + 3| + 3) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(-1|(1|x - 2| - 3) + 2|)| + 1 and g(x) = -2|(2sin(3\u03c0x + 3) + 2) - 2| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(1e^(-1(3x^2 - 1x - 5)) + 3) + 2 = 1(3cos(1\u03c0x + 2) + 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(2|(2|x + 3| + 1) - 2|)| + 3 and g(x) = (-1(-3sin(3\u03c0x - 1) - 2) - 3)/(3(-3sin(3\u03c0x - 1) - 2) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(-1(-1cos(2\u03c0x) + 2)^2 + 2(-1cos(2\u03c0x) + 2)) + 1| + 2 = 2((2x - 2)/(1x + 2)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(-3|(-5x + 2) - 3| + 3)| = -3cos(2\u03c0(-2x^2 + 1x + 2) + 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4((-3(-3x) + 1)/(1(-3x) - 3))^2 - 5((-3(-3x) + 1)/(1(-3x) - 3)) - 3 and g(x) = -2|((-2x - 1)/(1x + 3)) + 2| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1sin(2\u03c0(1(-4x - 4)^2 + 2(-4x - 4) + 1) + 3) - 3 and g(x) = 1(-2|x - 1| + 1) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "136", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(3sin(3\u03c0(4x^2 + 2)) - 1)^2 - 3(3sin(3\u03c0(4x^2 + 2)) - 1) + 3 = -3cos(2\u03c0(-2x^2 - 5) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "502", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(2\u03c0(-5((3x + 1)/(-1x)) + 3) - 1) + 1 and g(x) = 2e^(-2(1x^2 - 3x - 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(1(3x^2 + 3x + 5) - 2) - 4 = 3|(-3cos(3\u03c0x - 3) + 3) - 2| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|((-3(3x^2 - 5x + 2) + 1)/(2(3x^2 - 5x + 2))) - 2| and g(x) = (-1(-1e^(-2x) - 3) + 1)/(2(-1e^(-2x) - 3) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1cos(2\u03c0(-3(-3|x + 1| + 2) + 2) + 2) and g(x) = -2sin(3\u03c0(1x + 5) - 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "164", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(1|((-3x - 3)/(-2x - 1)) - 2| - 2) + 1 and g(x) = 1(1sin(1\u03c0x + 1) + 2)^2 - 1(1sin(1\u03c0x + 1) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(1(3|x + 2| - 2) + 2) - 2)/(3(1(3|x + 2| - 2) + 2) + 1) = 3sin(2\u03c0(2|x - 1| - 3) - 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(1(1sin(2\u03c0x - 1) + 3)^2 + 3(1sin(2\u03c0x - 1) + 3) - 1) - 3)/(1(1(1sin(2\u03c0x - 1) + 3)^2 + 3(1sin(2\u03c0x - 1) + 3) - 1) - 3) and g(x) = -3|(-1sin(2\u03c0x - 2) + 1) - 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(1\u03c0(-1((-1x + 1)/(2x)) + 1) - 3) - 2 and g(x) = -3sin(2\u03c0(-4x^2 + 2x + 3) + 3) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(-1(3|x + 2|) - 4) and g(x) = (-3(-2sin(1\u03c0x + 3) - 1))/(-1(-2sin(1\u03c0x + 3) - 1) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(1\u03c0(-2|(-5x^2 + 4x) + 3| - 3)) + 1 and g(x) = -2cos(1\u03c0(-2x^2 + 1x + 5) - 2) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1440", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-3sin(2\u03c0(-1|x + 1| - 3) + 3) + 3) - 1)/(-2(-3sin(2\u03c0(-1|x + 1| - 3) + 3) + 3)) and g(x) = (-1(2sin(1\u03c0x + 3) + 2) - 3)/(-2(2sin(1\u03c0x + 3) + 2) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1((2(1x - 1) - 2)/(-3(1x - 1) - 3)) - 3)/(3((2(1x - 1) - 2)/(-3(1x - 1) - 3)) + 2) and g(x) = 1sin(1\u03c0(-2|x + 3| - 3) + 1) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(2(3|x - 1| - 3))^2 + 2(2(3|x - 1| - 3)) and g(x) = 1e^(-2(1x + 2)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(1cos(3\u03c0((2x - 2)/(2x + 2)) - 1) - 2)^2 + 5(1cos(3\u03c0((2x - 2)/(2x + 2)) - 1) - 2) + 1 and g(x) = 3((1x + 2)/(3x + 2)) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "67", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2sin(2\u03c0(-3|(-1x^2 + 3x + 2) + 3| + 3) + 3) + 2 = -1(-1x^2 + 2x + 1)^2 - 4(-1x^2 + 2x + 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(1(-1cos(2\u03c0x - 2) - 1) - 1) - 3| and g(x) = -3|(-2e^(1x) + 2) + 3| + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(-2|(-4x + 1)| + 1) - 1)/(3(-2|(-4x + 1)| + 1) + 1) and g(x) = -2(1|x + 3|)^2 + 4(1|x + 3|) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|((-2(-5x + 3) + 3)/(1(-5x + 3))) - 2| - 1 and g(x) = 3|((-2x + 3)/(-3x + 4))| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1cos(1\u03c0(-2(4x^2 - 5x)^2 - 5(4x^2 - 5x) + 1) + 2) + 2 and g(x) = -3sin(3\u03c0((1x + 1)/(-1x - 3))) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "233", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(-3sin(2\u03c0(4x + 5))) - 1)/(2(-3sin(2\u03c0(4x + 5))) + 3) and g(x) = -1cos(1\u03c0(5x - 2) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "140", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(1(-1|x + 3| - 1)^2 + 1(-1|x + 3| - 1) - 1)^2 - 2(1(-1|x + 3| - 1)^2 + 1(-1|x + 3| - 1) - 1) - 4 and g(x) = (-2(1cos(1\u03c0x)))/(2(1cos(1\u03c0x)) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|((-3(-2e^(2x) - 2) - 3)/(-1(-2e^(2x) - 2) + 2)) + 1| and g(x) = 5(-2cos(3\u03c0x + 1) - 3)^2 - 1(-2cos(3\u03c0x + 1) - 3) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(3(3cos(3\u03c0x - 2) + 1) - 3)| - 2 and g(x) = 2(-3|x - 3| + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(-3sin(1\u03c0(-4x + 3)) - 2) + 4 and g(x) = -2e^(-2((-1x - 1)/(-3x + 2))) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "74", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3cos(3\u03c0(-3|(-2|x - 1| - 1) - 2|)) = (-2(-1sin(3\u03c0x - 3)))/(2(-1sin(3\u03c0x - 3)) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "300", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(-2sin(1\u03c0((-2x + 1)/(-3x)) + 1) + 1)^2 - 2(-2sin(1\u03c0((-2x + 1)/(-3x)) + 1) + 1) + 2 and g(x) = (3(2e^(2x) - 2) - 3)/(3(2e^(2x) - 2) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(-1cos(1\u03c0(-1|x| + 1) + 1) - 1) + 1 and g(x) = -2sin(3\u03c0(-5x + 2) - 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "276", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(1(-1x - 4)^2 + 4(-1x - 4) + 3)^2 + 5(1(-1x - 4)^2 + 4(-1x - 4) + 3) - 2 = -2(2|x + 2| + 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(1\u03c0(-1((3x + 1)/(-2x + 1))^2 + 1((3x + 1)/(-2x + 1)) - 4) - 2) and g(x) = (-1(3|x + 2| + 1) - 3)/(-2(3|x + 2| + 1) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "83", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(-2|(-1|x + 3| + 2)| + 1) - 1| - 3 and g(x) = 1|(-1x - 5)| + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(1(1x) + 1) - 3)/(-1(1(1x) + 1) - 1) = 3cos(3\u03c0(-1|x + 2| - 2) + 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(3\u03c0(-1(3x + 2)^2 - 4)) - 3 and g(x) = 2cos(3\u03c0(-3x - 3) + 1) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "890", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(-2((-3x + 2)/(-2x - 3))^2 - 1((-3x + 2)/(-2x - 3))) - 3| and g(x) = 3(-2sin(2\u03c0x)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(3sin(3\u03c0(1x^2 + 2) - 2)) - 3)/(2(3sin(3\u03c0(1x^2 + 2) - 2)) + 2) and g(x) = -2((3x - 3)/(-1x - 3)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "215", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(-2|(-3sin(2\u03c0x + 1)) + 3| + 3) + 2)/(-2(-2|(-3sin(2\u03c0x + 1)) + 3| + 3) + 1) and g(x) = -2sin(1\u03c0(-1x^2 + 5x + 5) + 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "203", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(-5(-1x^2 + 4x + 3) + 5) + 3| and g(x) = 4(-3|x + 1|) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4((1((2x)/(3x - 1)) - 1)/(2((2x)/(3x - 1)) - 1))^2 - 3((1((2x)/(3x - 1)) - 1)/(2((2x)/(3x - 1)) - 1)) - 5 and g(x) = -1|(3cos(2\u03c0x - 3) + 2) - 2| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-2cos(2\u03c0(-3|x| + 3) - 2) + 2) + 3)/(-3(-2cos(2\u03c0(-3|x| + 3) - 2) + 2) + 1) and g(x) = 2|(1e^(-2x) + 1) - 1| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3sin(2\u03c0(-1(3x^2 - 5x - 1) - 2) - 1) + 2 and g(x) = 2((-2x - 3)/(-3x - 2)) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(2sin(2\u03c0(3|x - 2| + 3) - 2) - 1) - 4 and g(x) = 2|(2|x + 2| + 3)| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-3cos(3\u03c0(-1|x - 2| - 1))) - 3 and g(x) = 1|(2e^(-1x) - 3)| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(-4((1x + 3)/(-1x + 3)) + 1) + 2| - 3 and g(x) = -1|((-3x)/(-3x - 2)) - 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3e^(-1(2|(1x^2 - 1x + 3) + 3| + 2)) and g(x) = 2|(-2|x|) + 3| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(-5(-4x^2 + 4x + 3) + 2) + 3)/(3(-5(-4x^2 + 4x + 3) + 2) + 2) and g(x) = 3cos(3\u03c0(1|x - 2| - 2) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3e^(-1((1(-1|x + 1| + 3) + 1)/(-1(-1|x + 1| + 3)))) + 1 and g(x) = 4(2cos(1\u03c0x + 2) + 1) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(3cos(3\u03c0(1|x + 1| - 1) - 1) + 1) - 1 and g(x) = (-1(-5x^2 + 4x - 3) + 1)/(3(-5x^2 + 4x - 3) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(1sin(3\u03c0(-1x^2 - 3x - 1) + 2) - 1)| + 2 and g(x) = 2sin(1\u03c0(-1|x + 3| + 1)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "323", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(1(3cos(2\u03c0x - 3) + 1) + 2)^2 - 1(1(3cos(2\u03c0x - 3) + 1) + 2) + 3 and g(x) = (2((-1x)/(-1x - 3)) + 1)/(-2((-1x)/(-1x - 3)) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3cos(3\u03c0(-2(2|x + 2| - 2)^2 - 5(2|x + 2| - 2))) - 1 and g(x) = 3((-2x + 3)/(3x)) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(-3sin(1\u03c0(4x^2 + 1x + 5) + 1) + 2) - 2| + 1 and g(x) = (-2(4x^2 - 1x - 3) + 1)/(1(4x^2 - 1x - 3) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1cos(3\u03c0(3|(3x^2 + 4x - 5) - 1| + 1) - 2) and g(x) = 1cos(2\u03c0(5x + 1)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "673", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|((-1(-2x - 5))/(3(-2x - 5) - 2)) - 3| + 1 and g(x) = (3(-3cos(1\u03c0x + 2) - 2) + 2)/(-1(-3cos(1\u03c0x + 2) - 2) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(-1sin(2\u03c0(2|x + 2| - 1) + 3)) + 3| + 3 = -2(2x - 1)^2 - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(3|(-2x - 5)| - 3)^2 - 2(3|(-2x - 5)| - 3) + 4 and g(x) = -2|(-1sin(3\u03c0x + 1) + 1)| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(1\u03c0(3(-5x^2 + 1) + 5) - 1) - 1 and g(x) = 4(-2|x - 1| - 1) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1cos(1\u03c0(1((-1x - 2)/(-3x - 3))^2 - 2((-1x - 2)/(-3x - 3)) - 4) + 3) - 3 = -2(3|x - 3| - 3)^2 + 1(3|x - 3| - 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1e^(1(-3|(2|x + 2| + 3) - 3| - 2)) - 3 and g(x) = 2sin(1\u03c0(-3x^2 + 3x + 1) - 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "602", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1((2(-1|x - 1| - 1) + 2)/(1(-1|x - 1| - 1))) + 2 = 3|(-5x^2 + 3x + 4)| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2e^(-1(1|(-1|x| - 2) + 2| - 2)) + 1 and g(x) = 2sin(1\u03c0((-2x + 1)/(-1x - 2)) - 2) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "78", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5((3((2x - 3)/(3x + 1)) + 2)/(-1((2x - 3)/(3x + 1)) + 3)) + 5 and g(x) = -3(-3cos(1\u03c0x - 1) - 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(-1(-1sin(1\u03c0x - 1) - 2)^2 - 2(-1sin(1\u03c0x - 1) - 2) + 3) - 3| + 1 = -3e^(2(-1|x + 2| + 3)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-3(-3x)^2 + 5(-3x) + 5) + 3)/(3(-3(-3x)^2 + 5(-3x) + 5) + 2) = -3|((3x - 3)/(2x + 2)) - 3| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3cos(3\u03c0((-1(-2|x - 3| + 2) - 2)/(3(-2|x - 3| + 2) - 2)) - 3) - 1 and g(x) = (-1((2x - 3)/(1x + 2)) - 2)/(2((2x - 3)/(1x + 2)) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(1\u03c0(2(-1x + 4) - 1)) and g(x) = (2((-1x - 2)/(1x)) + 3)/(-2((-1x - 2)/(1x)) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|(-1cos(1\u03c0(-3|x - 2|) - 1) - 3) + 3| - 3 = -2|((2x + 3)/(-1x + 2)) + 3| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|((1((-3x - 1)/(2x + 1)))/(1((-3x - 1)/(2x + 1))))| = -1(3cos(2\u03c0x + 2) - 3) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(3\u03c0((-1((-2x + 1)/(-1x - 3)) + 2)/(-2((-2x + 1)/(-1x - 3)))) - 2) - 3 and g(x) = -3|(3x^2 + 1x - 4) + 1| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(2sin(1\u03c0(4x) + 2) - 2) - 3)/(2(2sin(1\u03c0(4x) + 2) - 2) + 1) = -5(-4x - 4) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2cos(1\u03c0(-5(-4x + 1)^2 - 5) + 1) + 2 = 5(-3cos(3\u03c0x - 3) + 3)^2 + 3(-3cos(3\u03c0x - 3) + 3) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "88", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(-2|(-1|x - 2| + 3) - 2| + 3) + 1| + 3 = (3(5x + 5) - 2)/(-2(5x + 5) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(3\u03c0(1|((2x + 2)/(1x - 2)) + 2|) - 3) - 2 and g(x) = -3cos(1\u03c0(3|x - 3| + 2)) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(3(-3sin(3\u03c0x + 3) - 2) + 5) + 1 = 2|(4x^2 - 3x - 4) - 1| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(-1|((3x + 2)/(1x)) + 3| + 1) + 2| + 2 and g(x) = -3(4x^2 - 5x - 5) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(1(-3cos(1\u03c0x + 1) + 2)) + 2| + 2 and g(x) = -3|(2sin(1\u03c0x - 3))| + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(2|(3sin(3\u03c0x - 3) + 2) - 3| - 3))/(1(2|(3sin(3\u03c0x - 3) + 2) - 3| - 3)) and g(x) = 5(-2sin(3\u03c0x - 1)) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1e^(1(-2|(2x^2 - 2x - 2) + 1| - 2)) - 1 and g(x) = 1cos(2\u03c0(1x^2 - 2x - 3)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "189", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(-2|(5x - 3) - 2| - 3)^2 - 1(-2|(5x - 3) - 2| - 3) + 2 and g(x) = (-1(-1cos(2\u03c0x + 3)) + 1)/(2(-1cos(2\u03c0x + 3)) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(-2(-1sin(2\u03c0x - 3) - 1) + 3) + 1| + 2 = 2|(-3e^(-1x) - 1) - 2| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(2\u03c0(-1(-2x^2 + 3x)^2 - 1(-2x^2 + 3x) - 3) - 3) - 2 and g(x) = 3(-1x + 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "126", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(2cos(2\u03c0(-2x + 3) - 2) - 3) - 3)/(2(2cos(2\u03c0(-2x + 3) - 2) - 3) + 2) and g(x) = 3sin(3\u03c0(1x^2 + 1x + 1) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "514", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3((3(3cos(3\u03c0x + 3) + 3))/(-2(3cos(3\u03c0x + 3) + 3) - 2)) + 2)/(3((3(3cos(3\u03c0x + 3) + 3))/(-2(3cos(3\u03c0x + 3) + 3) - 2))) and g(x) = -1(1cos(3\u03c0x + 2) - 2) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2(4(1cos(1\u03c0x - 3) + 1)^2 - 2(1cos(1\u03c0x - 3) + 1) + 3)^2 - 3(4(1cos(1\u03c0x - 3) + 1)^2 - 2(1cos(1\u03c0x - 3) + 1) + 3) - 4 and g(x) = 2sin(3\u03c0(-2x^2 + 3x - 3) + 1) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(-4(3cos(2\u03c0x + 1) - 3)^2 - 2(3cos(2\u03c0x + 1) - 3) - 4)^2 - 2(-4(3cos(2\u03c0x + 1) - 3)^2 - 2(3cos(2\u03c0x + 1) - 3) - 4) - 2 and g(x) = -3((-1x)/(-1x - 2))^2 - 4((-1x)/(-1x - 2)) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-1sin(3\u03c0(-4x - 4) - 2) + 2)^2 - 5(-1sin(3\u03c0(-4x - 4) - 2) + 2) + 4 = (-1(-2|x - 1| - 1) + 2)/(3(-2|x - 1| - 1) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "240", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|((-1(-2x^2 - 5x + 2) + 1)/(-2(-2x^2 - 5x + 2) + 3)) + 1| - 3 and g(x) = -4((-1x - 3)/(-1x - 3))^2 + 3((-1x - 3)/(-1x - 3)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-3(-1x^2 + 4x + 1))^2 - 4(-3(-1x^2 + 4x + 1)) - 1 and g(x) = 3(-3x^2 + 5x + 4)^2 + 2(-3x^2 + 5x + 4) - 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(5(-1cos(2\u03c0x + 2))^2 + 2(-1cos(2\u03c0x + 2)) - 2) - 2 and g(x) = -1|(-3cos(2\u03c0x - 2) - 1)| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3sin(1\u03c0(-1(1x - 3)^2 - 5(1x - 3) - 2) - 1) + 3 = -2sin(1\u03c0(-3|x + 3| - 3) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "98", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(3|((-3x - 2)/(2x)) + 1| + 3) + 1)/(-1(3|((-3x - 2)/(2x)) + 1| + 3) - 2) and g(x) = -2(-2x^2 + 2x + 4)^2 + 1(-2x^2 + 2x + 4) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(-1|(-2x + 2) + 2| + 1) - 3 and g(x) = -3sin(2\u03c0(-5x^2 - 1x - 4) - 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "75", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|((2((1x - 2)/(-2x - 3)) + 1)/(-1((1x - 2)/(-2x - 3)))) - 3| + 1 and g(x) = -3sin(2\u03c0(-1x^2 + 1x) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(1(1|x| - 3)^2 - 2) - 1)/(2(1(1|x| - 3)^2 - 2) + 3) and g(x) = -2cos(1\u03c0((-2x - 3)/(-2x - 1)) + 3) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(-1|(2|x - 3| + 1) - 1| + 3)^2 - 5(-1|(2|x - 3| + 1) - 1| + 3) - 4 and g(x) = 2sin(3\u03c0((-3x + 2)/(-1x - 2)) - 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3sin(3\u03c0(-4(5x + 1)^2 + 1(5x + 1) + 4) + 3) - 3 = (-2((2x)/(-3x - 3)) + 1)/(3((2x)/(-3x - 3))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1292", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-2(4x^2 - 5x - 1)^2 + 2(4x^2 - 5x - 1) + 2)^2 - 3(-2(4x^2 - 5x - 1)^2 + 2(4x^2 - 5x - 1) + 2) + 4 and g(x) = (-3(-5x) - 2)/(-1(-5x)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(4(-2cos(1\u03c0x - 2) + 3) - 3)^2 + 3(4(-2cos(1\u03c0x - 2) + 3) - 3) - 1 = -3|(-1e^(1x))| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(2cos(3\u03c0(-2|x + 1| + 3) - 2) + 3)^2 + 1(2cos(3\u03c0(-2|x + 1| + 3) - 2) + 3) + 3 = 3(5x^2 + 3x - 5)^2 - 5(5x^2 + 3x - 5) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2sin(2\u03c0(5((3x - 3)/(-1x - 2)) - 4) - 1) = 2cos(1\u03c0(4x - 3) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "275", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3e^(2(-3|(-2x + 3) - 2| + 2)) - 3 and g(x) = -4(-2cos(1\u03c0x) - 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|((-2(-2e^(-2x) + 2) + 1)/(2(-2e^(-2x) + 2) + 2))| - 2 and g(x) = -1|(-2cos(3\u03c0x - 1) + 3) - 3| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(-2cos(3\u03c0(2|x - 1| + 1) + 2) - 2)^2 + 1(-2cos(3\u03c0(2|x - 1| + 1) + 2) - 2) + 3 = -5(-1e^(2x) + 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "64", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(1(-2x^2 - 2x + 3)^2 - 4(-2x^2 - 2x + 3) - 2) + 2)/(-1(1(-2x^2 - 2x + 3)^2 - 4(-2x^2 - 2x + 3) - 2) + 3) and g(x) = -5(1x^2 - 4x + 1)^2 - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(3cos(3\u03c0(2x + 5) + 3) - 2) + 2 = 1(2sin(2\u03c0x) - 3) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(3\u03c0(-3|(-3x^2 + 4x - 2) - 1| + 3)) and g(x) = 2(3x^2 + 1x - 1)^2 - 1(3x^2 + 1x - 1) - 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(4(-1cos(2\u03c0x))^2 + 5(-1cos(2\u03c0x)) - 5) + 1| - 3 and g(x) = 1(-1x + 4) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(3(2sin(2\u03c0x - 2) + 1) - 3) - 3)/(-2(3(2sin(2\u03c0x - 2) + 1) - 3) - 2) = -1(3x - 2)^2 - 2(3x - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(-1cos(1\u03c0((2x + 3)/(3x + 1)) - 3) - 1) + 2)/(-3(-1cos(1\u03c0((2x + 3)/(3x + 1)) - 3) - 1) - 1) and g(x) = -5(-1sin(2\u03c0x + 2) + 3) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(1|(-3x + 4) + 3| + 2) + 1| - 1 and g(x) = -5(1sin(1\u03c0x + 2) + 1)^2 - 1(1sin(1\u03c0x + 2) + 1) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(-5((2x - 3)/(2x + 3))^2 + 3((2x - 3)/(2x + 3)) - 5)^2 - 2(-5((2x - 3)/(2x + 3))^2 + 3((2x - 3)/(2x + 3)) - 5) + 3 and g(x) = (3(-3cos(3\u03c0x - 2) + 3) + 2)/(3(-3cos(3\u03c0x - 2) + 3) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3cos(2\u03c0((1(-2x^2 + 3x - 1) - 2)/(-3(-2x^2 + 3x - 1) - 2)) + 2) - 1 and g(x) = 5(-3cos(3\u03c0x - 3) + 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(-2|(3x^2 + 1x + 4) - 1| - 1) - 1| + 2 and g(x) = -2(3sin(3\u03c0x) - 2)^2 - 2(3sin(3\u03c0x) - 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2((-2(2sin(3\u03c0x + 1) + 1) + 1)/(-3(2sin(3\u03c0x + 1) + 1) - 3))^2 - 3((-2(2sin(3\u03c0x + 1) + 1) + 1)/(-3(2sin(3\u03c0x + 1) + 1) - 3)) - 5 and g(x) = 5(2|x + 3| - 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(1\u03c0(2(-3x^2 + 4x)^2)) - 2 and g(x) = 5((2x - 3)/(2x + 3))^2 - 2((2x - 3)/(2x + 3)) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "481", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(-2sin(2\u03c0((2x)/(-1x - 3)) - 1)) - 4 = (1(-1sin(3\u03c0x + 2) - 1) - 2)/(2(-1sin(3\u03c0x + 2) - 1) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3((3(-4x^2 + 1x - 3))/(-1(-4x^2 + 1x - 3) - 3)) + 3)/(2((3(-4x^2 + 1x - 3))/(-1(-4x^2 + 1x - 3) - 3)) + 2) and g(x) = -3sin(1\u03c0(-2|x + 1| - 2) + 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1((3(-1sin(3\u03c0x + 3)) - 2)/(2(-1sin(3\u03c0x + 3)) - 1)) - 1 and g(x) = -4(4x^2 + 2x + 5) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5((-2(2sin(3\u03c0x - 1) - 3) - 3)/(1(2sin(3\u03c0x - 1) - 3) + 2))^2 - 4((-2(2sin(3\u03c0x - 1) - 3) - 3)/(1(2sin(3\u03c0x - 1) - 3) + 2)) + 3 = (3(3cos(3\u03c0x) - 2) - 3)/(-3(3cos(3\u03c0x) - 2) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-2cos(2\u03c0((-3x - 3)/(-2x + 3)) + 3) - 3) + 2| - 3 and g(x) = 1|((1x + 2)/(3x)) + 3| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3cos(1\u03c0((1(3x^2 + 3x - 4) + 1)/(-1(3x^2 + 3x - 4))) + 3) - 1 and g(x) = -5(1e^(2x) - 3) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(-5(-1x^2 - 4x + 2) - 4)^2 + 5(-5(-1x^2 - 4x + 2) - 4) - 2 and g(x) = (-2(1sin(1\u03c0x - 1) - 3) - 3)/(3(1sin(1\u03c0x - 1) - 3) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(-2cos(1\u03c0(2x - 2) - 1) + 1) - 2)/(-1(-2cos(1\u03c0(2x - 2) - 1) + 1) + 3) and g(x) = 1sin(3\u03c0(5x^2 - 3)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "177", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(-1|(-3cos(2\u03c0x - 1) - 1)| + 2) - 1)/(1(-1|(-3cos(2\u03c0x - 1) - 1)| + 2) + 1) and g(x) = 3(-2cos(2\u03c0x - 2) + 2)^2 + 3(-2cos(2\u03c0x - 2) + 2) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(3(-2sin(3\u03c0x - 3)) - 2) - 3 = 2|(-2cos(1\u03c0x + 2) + 1) - 2| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(3\u03c0(2|(-1x + 4)| + 2)) - 2 = -2cos(1\u03c0(4x + 5) - 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-2sin(2\u03c0(-2x^2 - 4x + 1) + 3)) - 3)/(-3(-2sin(2\u03c0(-2x^2 - 4x + 1) + 3)) - 3) and g(x) = 1(3sin(2\u03c0x + 3) + 1)^2 - 1(3sin(2\u03c0x + 3) + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "254", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(3sin(2\u03c0(-1|x + 3|) + 2) - 1) - 2| - 2 and g(x) = (-2(-3x + 4) - 1)/(-1(-3x + 4) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3cos(3\u03c0((-2(-3|x - 2| + 2) - 1)/(3(-3|x - 2| + 2) + 3)) + 1) = 3sin(2\u03c0((-2x - 3)/(-1x)) - 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "112", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(1(-1cos(3\u03c0x + 2) - 2) - 1)^2 + 5(1(-1cos(3\u03c0x + 2) - 2) - 1) + 3 = -3((3x + 1)/(2x - 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(4(-3cos(3\u03c0x + 2) + 2) + 4) - 2)/(1(4(-3cos(3\u03c0x + 2) + 2) + 4) - 2) = -2cos(3\u03c0(-3|x + 3| + 2) + 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "180", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(1|(-3|x - 3| + 3) - 3|) - 5 and g(x) = 4(3|x + 2| - 3)^2 - 2(3|x + 2| - 3) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(3cos(3\u03c0(-2|x - 2| + 2) + 1) + 1) - 5 and g(x) = -2(-3sin(1\u03c0x - 3) + 3)^2 + 2(-3sin(1\u03c0x - 3) + 3) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|((-3(1|x| - 3) - 3)/(-3(1|x| - 3))) - 1| + 2 and g(x) = 1|(-1|x + 3| - 3) - 2| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1sin(1\u03c0((-2(-3x^2 - 3x - 4) + 2)/(-1(-3x^2 - 3x - 4) + 3)) - 1) - 3 = 2sin(2\u03c0(3|x - 1| - 3) + 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "114", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-3|(2sin(1\u03c0x - 3) + 3) + 1| + 1)^2 - 4(-3|(2sin(1\u03c0x - 3) + 3) + 1| + 1) - 4 = -2|(3|x + 2| + 1) - 2| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|((-1(3e^(-1x) + 1) + 1)/(2(3e^(-1x) + 1)))| = 3sin(1\u03c0(2x - 5) + 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(-1(-1x - 3) + 5) + 2| = 2(-2sin(1\u03c0x - 3) + 2)^2 + 5(-2sin(1\u03c0x - 3) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(1\u03c0(4(-3x^2 + 2x + 3)^2 - 2(-3x^2 + 2x + 3) + 5) + 1) + 3 and g(x) = 3|((-2x + 3)/(3x - 2))| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "629", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1sin(2\u03c0(-5(2x^2 - 2x - 3)^2 + 5(2x^2 - 2x - 3) + 1) + 1) - 1 and g(x) = 5(2cos(1\u03c0x + 1) - 3)^2 + 3(2cos(1\u03c0x + 1) - 3) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "131", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(5(2x + 2)^2 - 2(2x + 2) - 4)^2 - 3 and g(x) = -2|(1|x - 2| + 3) + 3| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(-2(-3sin(2\u03c0x - 2) + 2)) - 2 = -3((-1x + 3)/(1x + 2)) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(3(-1cos(1\u03c0x + 1))^2 + 2(-1cos(1\u03c0x + 1)) - 4) - 4 and g(x) = 1cos(3\u03c0(-1|x - 2| + 2) + 1) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1sin(1\u03c0(5(-1|x| + 3)^2 + 3(-1|x| + 3)) - 2) and g(x) = 3sin(1\u03c0(1x + 5) + 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "188", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1sin(3\u03c0(-1(-4x + 3) - 3) - 3) and g(x) = (3((-3x - 2)/(2x - 3)))/(2((-3x - 2)/(2x - 3)) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(1|(1x^2 - 4) + 2| - 3)^2 - 2(1|(1x^2 - 4) + 2| - 3) - 2 = -2cos(1\u03c0(-2x + 1) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(3(-3sin(2\u03c0x + 2) - 2)^2 + 3(-3sin(2\u03c0x + 2) - 2) - 4) - 2)/(2(3(-3sin(2\u03c0x + 2) - 2)^2 + 3(-3sin(2\u03c0x + 2) - 2) - 4) + 2) and g(x) = 1cos(1\u03c0((1x - 3)/(1x + 1)) + 1) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(-1(-4x + 1)^2 - 4(-4x + 1) + 3)^2 + 2(-1(-4x + 1)^2 - 4(-4x + 1) + 3) and g(x) = -3|((2x - 3)/(-3x + 1)) - 1| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(-3cos(3\u03c0(-4x - 2) + 3) + 3) - 1| - 2 = (2(-2sin(3\u03c0x + 3) + 2) + 3)/(-2(-2sin(3\u03c0x + 3) + 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1cos(1\u03c0(1((-1x + 1)/(2x - 1)) - 3) + 3) + 2 and g(x) = -2cos(1\u03c0(-2x^2 - 5) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "276", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-2((-1x + 2)/(2x + 2))^2 + 3((-1x + 2)/(2x + 2)) + 2) - 1)/(-1(-2((-1x + 2)/(2x + 2))^2 + 3((-1x + 2)/(2x + 2)) + 2)) and g(x) = -1|((-2x - 1)/(-3x - 2)) - 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(-5(3x^2 - 3x - 5)^2 - 2(3x^2 - 3x - 5) + 5) - 3)/(1(-5(3x^2 - 3x - 5)^2 - 2(3x^2 - 3x - 5) + 5) + 2) and g(x) = 2cos(1\u03c0((-2x)/(-2x + 3)) + 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1e^(1((1(1x^2 + 1x - 1) - 3)/(3(1x^2 + 1x - 1) + 1))) + 1 and g(x) = -1(1|x - 1| + 1)^2 + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|((-1(-4x - 4) + 3)/(1(-4x - 4) - 3))| - 3 and g(x) = 2cos(3\u03c0(-2x + 5) + 3) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(1(3x + 2) - 4)^2 + 4(1(3x + 2) - 4) + 2 = -2((-1x - 3)/(2x + 3))^2 - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|((3(-1x - 5) - 3)/(3(-1x - 5) - 2)) + 3| + 2 and g(x) = -1(-2|x| + 2)^2 - 3(-2|x| + 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(2\u03c0((2((1x - 3)/(1x - 2)) - 2)/(-2((1x - 3)/(1x - 2)) - 1)) + 3) + 3 and g(x) = -3(-2cos(3\u03c0x + 2) - 1) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-1((1x - 3)/(1x + 1)) + 4)^2 + 4(-1((1x - 3)/(1x + 1)) + 4) + 3 and g(x) = -1|((-3x - 2)/(-2x + 1)) + 3| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((-2(3sin(1\u03c0x - 3) - 3) - 1)/(-1(3sin(1\u03c0x - 3) - 3) + 2)) + 4 and g(x) = (-1(-1sin(2\u03c0x - 1)) - 2)/(-2(-1sin(2\u03c0x - 1)) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(1(-5x - 5)^2 - 2(-5x - 5) + 3) - 1)/(1(1(-5x - 5)^2 - 2(-5x - 5) + 3) + 3) and g(x) = 1(-1sin(1\u03c0x + 2) + 1)^2 + 3(-1sin(1\u03c0x + 2) + 1) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(1\u03c0(-1|(3x^2 - 5x + 5) - 3| - 2) + 2) - 1 and g(x) = -1e^(1(-4x^2 + 5x - 2)) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "273", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(1(1|x + 2| - 2)^2 + 5(1|x + 2| - 2) + 3) + 5 and g(x) = -1sin(3\u03c0(2|x - 2| + 3) + 2) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-1(-2x + 2)^2 - 5(-2x + 2) - 4) + 4 and g(x) = -1(2cos(1\u03c0x) + 2)^2 - 4(2cos(1\u03c0x) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(4(1sin(2\u03c0x - 3) + 2)^2 - 4(1sin(2\u03c0x - 3) + 2) - 2) - 4 and g(x) = -2|(-5x^2 + 4x - 5) - 1| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-3sin(2\u03c0(4x + 4) - 3) + 3)^2 + 1(-3sin(2\u03c0(4x + 4) - 3) + 3) - 1 = (2(-3|x + 2| - 3) - 3)/(-2(-3|x + 2| - 3) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "160", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3cos(1\u03c0(-5(1|x - 3| - 1) - 1) - 2) + 2 and g(x) = (-2(-1x^2 - 1x + 2) - 1)/(-2(-1x^2 - 1x + 2) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(2sin(1\u03c0(3x^2 + 5x + 4) - 2) - 2)^2 - 1(2sin(1\u03c0(3x^2 + 5x + 4) - 2) - 2) - 2 = 5((1x + 1)/(-2x + 1))^2 + 4((1x + 1)/(-2x + 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "584", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3((-1(-3e^(1x) + 3))/(-3(-3e^(1x) + 3) - 3)) - 5 and g(x) = -2sin(1\u03c0(3|x + 3| - 2) + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(-1cos(2\u03c0(-4x - 1) - 2)) and g(x) = 3(-3cos(3\u03c0x))^2 + 2(-3cos(3\u03c0x)) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(-2|(3sin(3\u03c0x + 1) + 2) + 3| - 3) + 1| = 3(-2e^(-1x) - 2)^2 - 4(-2e^(-1x) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(3sin(3\u03c0(-3x + 3) + 1)) + 1 and g(x) = (2(3|x - 1| + 3) - 1)/(-2(3|x - 1| + 3) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "180", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(2|(-3|x - 2| + 2)| - 1)^2 - 1(2|(-3|x - 2| + 2)| - 1) + 4 and g(x) = 1((-2x + 3)/(1x))^2 - 2((-2x + 3)/(1x)) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(3(4x^2 - 2x)^2 - 1(4x^2 - 2x) - 2) - 3 and g(x) = -3(1sin(2\u03c0x + 3) - 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(-1(-1sin(2\u03c0x - 3) + 2))^2 - 3(-1(-1sin(2\u03c0x - 3) + 2)) + 3 and g(x) = -1sin(3\u03c0((-1x + 2)/(-2x + 3)) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-2sin(1\u03c0((2x + 1)/(2x - 1)) + 1) + 3) + 1 = (3((-1x - 1)/(2x - 1)) + 1)/(1((-1x - 1)/(2x - 1))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(-3sin(2\u03c0((-1x)/(-2x + 3))) + 3) - 3)/(-2(-3sin(2\u03c0((-1x)/(-2x + 3))) + 3)) and g(x) = 1cos(1\u03c0((1x + 3)/(2x - 3)) + 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(-5(2cos(1\u03c0x - 1) - 3)^2 + 5(2cos(1\u03c0x - 1) - 3) - 1) - 3)/(2(-5(2cos(1\u03c0x - 1) - 3)^2 + 5(2cos(1\u03c0x - 1) - 3) - 1) + 1) and g(x) = 2sin(1\u03c0(-4x^2 - 2x - 4) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "800", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-4(-3cos(2\u03c0x - 3))^2 + 1(-3cos(2\u03c0x - 3)) + 5) + 3 and g(x) = 1|(-1|x - 3| + 2) - 2| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "68", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2cos(2\u03c0(3((2x + 1)/(3x - 1)) + 4) + 1) - 1 and g(x) = -3|(-3cos(2\u03c0x + 2)) - 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(2|(1x^2 - 1x - 4)| - 1))/(-2(2|(1x^2 - 1x - 4)| - 1) - 1) and g(x) = (-3(-4x^2 - 4x - 3))/(-3(-4x^2 - 4x - 3) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-5((3x + 3)/(-3x + 2)) - 4) + 2| - 2 and g(x) = -2sin(2\u03c0(-1|x + 3| - 3) + 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(1\u03c0(1(4x^2 + 1x - 4)^2 - 2(4x^2 + 1x - 4) + 1)) + 2 and g(x) = (3((-2x + 1)/(1x + 2)) - 2)/(2((-2x + 1)/(1x + 2))) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1323", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(3cos(3\u03c0(-1|x + 1| + 3) - 3) + 1) + 3)/(3(3cos(3\u03c0(-1|x + 1| + 3) - 3) + 1) + 2) and g(x) = (3(3x^2 + 4x) - 3)/(-3(3x^2 + 4x) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3cos(2\u03c0((2((-1x - 1)/(2x + 2)) - 2)/(3((-1x - 1)/(2x + 2)) - 1)) + 3) - 2 and g(x) = 1cos(3\u03c0(-1|x + 2| - 3) + 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(-1(2|x| + 1)^2 + 1(2|x| + 1) + 1) + 1)/(-1(-1(2|x| + 1)^2 + 1(2|x| + 1) + 1) - 2) and g(x) = -2cos(2\u03c0(-1x^2 + 3x - 1) + 2) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-2|(-2|x - 1| - 2) - 1| - 3) - 2)/(2(-2|(-2|x - 1| - 2) - 1| - 3) + 2) and g(x) = 3e^(-1(1x^2 + 1x + 4)) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4((-1(-3sin(1\u03c0x) + 1) - 3)/(-1(-3sin(1\u03c0x) + 1) - 1)) + 1 and g(x) = (-3(-2|x| - 3) + 3)/(1(-2|x| - 3) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(-2|(5x + 1)|) - 1| + 2 and g(x) = 2(2cos(2\u03c0x - 1) + 2)^2 + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3sin(3\u03c0(3|(1x + 3)| - 2) - 3) - 2 = (3((1x - 1)/(1x + 2)))/(-2((1x - 1)/(1x + 2)) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "118", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1cos(2\u03c0(-5(1|x - 1|)^2 + 1(1|x - 1|) + 5) + 1) + 2 and g(x) = 2e^(-1(4x + 1)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "649", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(-1(-1|x - 3|) - 4)^2 - 2(-1(-1|x - 3|) - 4) + 4 = 3(-1cos(1\u03c0x + 2))^2 + 3(-1cos(1\u03c0x + 2)) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1((2(3sin(1\u03c0x + 2) + 2) - 1)/(-2(3sin(1\u03c0x + 2) + 2) + 3))^2 - 4((2(3sin(1\u03c0x + 2) + 2) - 1)/(-2(3sin(1\u03c0x + 2) + 2) + 3)) + 5 = 2|(3|x - 3| + 2) + 1| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(2(-2sin(2\u03c0x + 1) - 1)^2 - 4(-2sin(2\u03c0x + 1) - 1) - 5))/(-2(2(-2sin(2\u03c0x + 1) - 1)^2 - 4(-2sin(2\u03c0x + 1) - 1) - 5) + 1) and g(x) = 3sin(3\u03c0(-3x) + 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(1cos(3\u03c0((-1x - 3)/(-2x)) + 2)) + 3| - 1 and g(x) = 5(2|x - 1| - 3) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(-4((1x - 3)/(2x - 1))^2 + 3((1x - 3)/(2x - 1)) - 2) + 2)/(1(-4((1x - 3)/(2x - 1))^2 + 3((1x - 3)/(2x - 1)) - 2) - 2) = -1cos(2\u03c0((2x + 2)/(1x))) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(-2sin(3\u03c0(5x^2 + 3x - 4) - 1) + 2) - 1)/(3(-2sin(3\u03c0(5x^2 + 3x - 4) - 1) + 2) - 2) = 1(4x^2 - 2x - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5((3(2|x + 2| - 2) + 3)/(-2(2|x + 2| - 2) - 3))^2 + 1((3(2|x + 2| - 2) + 3)/(-2(2|x + 2| - 2) - 3)) and g(x) = -3cos(1\u03c0(1x^2 + 4x + 2)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(-4(1|x - 1| - 2)^2 + 4(1|x - 1| - 2) - 3)| + 3 and g(x) = (1(-1x^2 - 5x + 2) + 3)/(2(-1x^2 - 5x + 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3e^(-1((-2(1x + 5))/(-2(1x + 5) - 3))) - 1 and g(x) = -2sin(2\u03c0(2x - 2) - 1) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "70", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5((-1((2x + 1)/(-3x + 1)) + 2)/(1((2x + 1)/(-3x + 1)) - 3))^2 - 5((-1((2x + 1)/(-3x + 1)) + 2)/(1((2x + 1)/(-3x + 1)) - 3)) - 5 = 2sin(1\u03c0(-2x^2 - 5x + 5) + 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "406", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1cos(3\u03c0(1|(-4x^2 - 2x + 5) + 2| - 3) + 2) + 1 and g(x) = -1e^(-1(2x^2 + 2x - 3)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "896", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(-1|((-2x + 1)/(2x + 2))| + 1)^2 - 4(-1|((-2x + 1)/(2x + 2))| + 1) and g(x) = -1(-1sin(3\u03c0x + 3)) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5((-2(2cos(3\u03c0x - 3) + 1) + 2)/(2(2cos(3\u03c0x - 3) + 1) + 1))^2 + 3((-2(2cos(3\u03c0x - 3) + 1) + 2)/(2(2cos(3\u03c0x - 3) + 1) + 1)) - 2 and g(x) = -3e^(2(1x^2 + 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(2e^(2(-2|x| + 1)) + 2) - 3| + 2 = 2cos(3\u03c0(-2x + 5) - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "112", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(3cos(2\u03c0(4x^2 + 2x - 1) - 3) + 2) - 1 and g(x) = (2(-4x - 4))/(3(-4x - 4) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1307", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(-2(3|x - 1|) - 1) + 1 and g(x) = 1(4x^2 - 1x - 4)^2 - 3(4x^2 - 1x - 4) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3((2((2x - 1)/(-2x - 1)) + 3)/(-3((2x - 1)/(-2x - 1)) + 4)) - 1 and g(x) = 1cos(3\u03c0(4x - 3) - 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1cos(2\u03c0(3((1x - 2)/(-2x + 3)) + 1) + 2) + 3 and g(x) = 2sin(1\u03c0(-1x + 4) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(-3sin(3\u03c0((3x - 2)/(3x + 1)) - 2) + 2) + 2| - 1 and g(x) = (2(2|x - 1| - 3))/(-3(2|x - 1| - 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-3sin(2\u03c0((-1x - 1)/(-1x + 3)) - 2)) + 1)/(-2(-3sin(2\u03c0((-1x - 1)/(-1x + 3)) - 2)) - 2) and g(x) = 2|(5x + 5) - 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-1sin(3\u03c0((2x + 3)/(2x - 3)) + 3) - 1))/(-2(-1sin(3\u03c0((2x + 3)/(2x - 3)) + 3) - 1) - 1) and g(x) = 3(-2|x + 1| - 2) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2e^(1(-3(5x^2 + 4x - 1) - 2)) and g(x) = 3sin(3\u03c0(-3x + 2) - 2) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "172", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3cos(3\u03c0((-2(5x + 2) - 3)/(-2(5x + 2))) - 3) and g(x) = -1cos(1\u03c0(1x + 3) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(2sin(2\u03c0((-1x + 1)/(-2x - 3)) - 2) - 2)^2 - 5(2sin(2\u03c0((-1x + 1)/(-2x - 3)) - 2) - 2) = (2(1cos(3\u03c0x + 1) + 2) - 2)/(3(1cos(3\u03c0x + 1) + 2) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|(-1e^(-1(-2|x + 3| - 1)) + 3) + 3| + 3 and g(x) = (1(1|x - 2| - 3) - 2)/(2(1|x - 2| - 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(1((-3x)/(-2x - 2))^2 + 5((-3x)/(-2x - 2)) + 1) + 1)/(-3(1((-3x)/(-2x - 2))^2 + 5((-3x)/(-2x - 2)) + 1) - 1) and g(x) = -3sin(3\u03c0(3|x| - 1) - 3) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "182", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3sin(2\u03c0(-2|(1x^2 - 2x + 3) + 2| - 1) - 2) - 2 and g(x) = (-3(-1x^2 + 4x) + 2)/(1(-1x^2 + 4x) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "774", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4((1((1x - 3)/(-3x - 3)))/(1((1x - 3)/(-3x - 3)) + 1))^2 + 4((1((1x - 3)/(-3x - 3)))/(1((1x - 3)/(-3x - 3)) + 1)) + 3 and g(x) = 1sin(1\u03c0(-4x^2 + 1x + 2)) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "188", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(2(4x - 4) + 4)^2 - 4 = 1cos(1\u03c0((3x + 3)/(-3x + 1)) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(3sin(3\u03c0(-3|x| + 2) + 2) + 1))/(2(3sin(3\u03c0(-3|x| + 2) + 2) + 1) + 2) and g(x) = -3sin(3\u03c0(-3|x - 1|) + 3) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "167", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-2(2x^2 + 1x - 3)^2 - 3(2x^2 + 1x - 3))^2 - 4(-2(2x^2 + 1x - 3)^2 - 3(2x^2 + 1x - 3)) + 5 and g(x) = -3|(2sin(1\u03c0x - 1) + 3) - 1| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(1(-4x + 4)^2 - 4(-4x + 4) - 1) - 3| + 1 = 1(-3sin(3\u03c0x + 1) - 3) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-2cos(3\u03c0(-4x) + 3) + 1)^2 - 1(-2cos(3\u03c0(-4x) + 3) + 1) and g(x) = -2(1x + 1)^2 + 5(1x + 1) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-2|(-3|x - 1| + 1) - 2| - 2) - 3)/(3(-2|(-3|x - 1| + 1) - 2| - 2) + 2) and g(x) = 2cos(2\u03c0((1x - 1)/(-1x + 2)) + 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(-1(4x^2 + 1x - 3) + 1)^2 - 4(-1(4x^2 + 1x - 3) + 1) - 4 = -2(-3x - 5) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(-2cos(3\u03c0(-3x^2 + 2x + 3) - 2) + 2)^2 - 5(-2cos(3\u03c0(-3x^2 + 2x + 3) - 2) + 2) - 3 and g(x) = 3(-2|x + 1| - 1) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(5((-1x)/(-1x + 2)) - 4))/(-3(5((-1x)/(-1x + 2)) - 4) + 2) and g(x) = 3(1|x - 2| - 3)^2 - 3(1|x - 2| - 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(2|(-1|x + 2|) + 3|)^2 - 2(2|(-1|x + 2|) + 3|) - 2 and g(x) = 1cos(3\u03c0(1|x + 2|)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(2|(-1e^(1x) + 2) + 2| - 1) - 1| + 3 and g(x) = 2(-2sin(1\u03c0x + 3))^2 + 2(-2sin(1\u03c0x + 3)) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(1((-2x + 1)/(-1x - 1))^2 - 5) + 2| + 2 and g(x) = 1sin(3\u03c0(1|x - 2| + 3) + 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1sin(3\u03c0(1|((-1x + 1)/(-2x - 3)) - 1| + 1) - 2) - 3 and g(x) = -2e^(-1(-3x - 5)) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3((1(-2e^(-2x) - 3) + 3)/(2(-2e^(-2x) - 3) - 1)) and g(x) = -1|(2|x + 2| - 3) + 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-2e^(-2(1x^2 - 1x - 2)) - 3) + 1)/(3(-2e^(-2(1x^2 - 1x - 2)) - 3) - 1) and g(x) = 1cos(2\u03c0(2x^2 - 4x + 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "808", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5((-2(-1|x - 2| + 2) - 2)/(-1(-1|x - 2| + 2) - 3)) - 4 = -2sin(3\u03c0(5x^2 + 3x + 3) - 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2e^(1(-1|(-2|x + 2| + 1) + 1| - 3)) + 1 and g(x) = 2cos(2\u03c0(2|x| + 1) + 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(1|(-1x - 1) + 3| - 1)^2 + 2(1|(-1x - 1) + 3| - 1) + 5 and g(x) = 3cos(1\u03c0(1x^2 - 5x + 3) - 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-3sin(1\u03c0(-3x^2 - 3x - 3) + 1) + 1)^2 + 1(-3sin(1\u03c0(-3x^2 - 3x - 3) + 1) + 1) + 4 and g(x) = 3|(3sin(3\u03c0x + 3) - 2) + 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "86", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2(1sin(3\u03c0(-5x^2 - 1x + 3) + 2) + 1) - 1 and g(x) = 4(-2|x - 2| - 1)^2 + 5(-2|x - 2| - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(-2(3cos(3\u03c0x - 3) + 3)^2 + 4(3cos(3\u03c0x - 3) + 3) + 5) + 3| + 3 and g(x) = -1e^(-1(-2|x - 3| - 1)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-1(-4x^2 - 3x + 4)^2 - 1(-4x^2 - 3x + 4) + 3) + 2)/(3(-1(-4x^2 - 3x + 4)^2 - 1(-4x^2 - 3x + 4) + 3) + 2) and g(x) = (-2(2e^(1x) - 2) + 2)/(-2(2e^(1x) - 2) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-1|(2sin(2\u03c0x + 3) - 1) - 2| + 2) + 2)/(-1(-1|(2sin(2\u03c0x + 3) - 1) - 2| + 2) + 3) and g(x) = (-1(-2|x| + 1) - 1)/(-1(-2|x| + 1) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(3cos(3\u03c0(1|x| + 2) + 1)) + 2| + 3 and g(x) = (-3(5x) - 2)/(-2(5x) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "119", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(2\u03c0(4(-3x - 3) - 4) + 1) - 3 and g(x) = 3cos(2\u03c0(-1|x - 2| + 2) - 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "240", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(-5(2sin(1\u03c0x) + 1)^2 + 3(2sin(1\u03c0x) + 1) + 3)| + 3 and g(x) = 1(4x^2 + 4x - 4) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(2\u03c0(2|(-1x^2 - 1x + 4) + 2| - 3) - 1) - 3 and g(x) = -3e^(1(1|x + 3| - 1)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5((-2(2cos(3\u03c0x - 3) - 3) - 3)/(2(2cos(3\u03c0x - 3) - 3) + 1))^2 + 4((-2(2cos(3\u03c0x - 3) - 3) - 3)/(2(2cos(3\u03c0x - 3) - 3) + 1)) - 4 and g(x) = -1|(-2|x + 1| - 2) - 1| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3((-2(-1sin(2\u03c0x + 1) + 2) - 1)/(3(-1sin(2\u03c0x + 1) + 2) + 1)) - 1)/(-1((-2(-1sin(2\u03c0x + 1) + 2) - 1)/(3(-1sin(2\u03c0x + 1) + 2) + 1)) - 1) and g(x) = -5(-2cos(1\u03c0x - 3) - 2) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(5(-3|x + 2| + 3)^2 + 1(-3|x + 2| + 3) - 5)^2 - 3(5(-3|x + 2| + 3)^2 + 1(-3|x + 2| + 3) - 5) - 4 and g(x) = 1sin(1\u03c0((-3x - 1)/(2x + 1)) - 3) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|((1(-5x - 1) - 2)/(2(-5x - 1) - 3))| = -2cos(3\u03c0(-2|x + 1| - 1) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "118", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(-1|(1sin(3\u03c0x + 1)) + 1| + 1) - 1 and g(x) = 3(-2sin(2\u03c0x)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|((-1(-4x + 2) + 2)/(-1(-4x + 2) - 2)) - 2| = -5(2e^(1x) - 2)^2 + 1(2e^(1x) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(-2(2cos(1\u03c0x + 1) + 1) + 1)| = -2e^(-2(1|x + 3| - 1)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2cos(2\u03c0((-3(-3x^2 + 4x + 2) + 3)/(-3(-3x^2 + 4x + 2) + 2)) + 1) - 2 and g(x) = -3(-2cos(3\u03c0x + 3) - 1) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1sin(3\u03c0(3(-5x^2 - 2x - 1) + 4) - 3) - 1 and g(x) = -3((3x - 2)/(-3x)) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1177", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-3|((3x + 3)/(-3x + 4)) - 1| + 2)^2 + 2(-3|((3x + 3)/(-3x + 4)) - 1| + 2) - 5 = -5(1e^(-1x) - 1)^2 + 5(1e^(-1x) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(3sin(2\u03c0(1x^2 - 5x + 2) - 1) + 3) + 2)/(1(3sin(2\u03c0(1x^2 - 5x + 2) - 1) + 3) + 3) and g(x) = -2(2|x - 1| - 1)^2 + 1(2|x - 1| - 1) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3((2(-2sin(3\u03c0x - 1) + 1) - 3)/(3(-2sin(3\u03c0x - 1) + 1)))^2 - 5((2(-2sin(3\u03c0x - 1) + 1) - 3)/(3(-2sin(3\u03c0x - 1) + 1))) + 2 and g(x) = 1(2|x - 1| - 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-3|(-2sin(1\u03c0x - 3) + 1)| + 3)^2 + 2(-3|(-2sin(1\u03c0x - 3) + 1)| + 3) - 3 = 3|(2x^2 + 4x - 2) - 2| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(4(-1sin(3\u03c0x) + 2) - 5) + 2| + 1 and g(x) = (-2(-4x - 2) - 1)/(1(-4x - 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(-2(1cos(2\u03c0x + 2) + 3)^2 + 1(1cos(2\u03c0x + 2) + 3))^2 + 1(-2(1cos(2\u03c0x + 2) + 3)^2 + 1(1cos(2\u03c0x + 2) + 3)) + 4 and g(x) = 3(4x^2 + 4x + 2)^2 - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(-1((2x - 2)/(-2x + 3)) - 3) + 1)/(3(-1((2x - 2)/(-2x + 3)) - 3) - 2) and g(x) = 1|(-1sin(3\u03c0x - 3)) + 2| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|((-2(-1|x - 2| + 1) - 2)/(2(-1|x - 2| + 1) - 1)) + 3| + 1 and g(x) = 2((-2x - 1)/(1x))^2 + 1((-2x - 1)/(1x)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(-3sin(2\u03c0(-2x^2 - 1x - 1))) + 1 = 1e^(-2((-1x + 2)/(-2x + 3))) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "798", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-3(-2sin(2\u03c0x - 3) - 1)^2 - 3(-2sin(2\u03c0x - 3) - 1) + 2)^2 - 5(-3(-2sin(2\u03c0x - 3) - 1)^2 - 3(-2sin(2\u03c0x - 3) - 1) + 2) - 1 and g(x) = (1(-2|x - 3| - 2) + 2)/(-1(-2|x - 3| - 2) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(2((-3x + 2)/(-2x - 1))^2 + 5((-3x + 2)/(-2x - 1)) - 1) + 3| + 1 and g(x) = -2sin(2\u03c0(-1x^2 + 5x + 4)) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(3sin(2\u03c0(1|x + 2| - 2) + 1))^2 + 4(3sin(2\u03c0(1|x + 2| - 2) + 1)) - 2 and g(x) = -3cos(3\u03c0(-4x^2 - 2x + 3) - 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "705", "dataset": "MATH"}