{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(-2sin(1\u03c0x - 2) + 2) + 2)/(2(-2sin(1\u03c0x - 2) + 2) + 1) and g(x) = (1x)/(-3x - 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 2sin(3\u03c0(3x - 2) + 2) + 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": "19", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-1sin(2\u03c0x - 2) - 2)^2 - 4(-1sin(2\u03c0x - 2) - 2) + 1 = 1x + 4 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) = -2sin(2\u03c0((2x - 2)/(3x + 2))) + 1 and g(x) = (1x - 1)/(1x + 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": "How many points of intersection exist between the functions f(x) = (3(-3|x + 3| - 1) + 3)/(-1(-3|x + 3| - 1) - 2) and g(x) = -2x^2 + 4x - 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(1\u03c0(4x^2 + 3x - 2) - 3) - 3 = -2x^2 - 4x - 5 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 -2sin(3\u03c0(2|x - 3| - 2) - 2) + 1 = 4x^2 - 4x + 1 have for x 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 solutions does the equation -2sin(2\u03c0(3x - 1) - 3) - 2 = -1|x + 2| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1sin(3\u03c0(3x^2 - 5x) - 2) = -5x - 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": "Find the number of times the graphs of f(x) = -3cos(2\u03c0(2|x + 2| + 2) + 3) + 1 and g(x) = 4x - 3 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 -2|(2sin(2\u03c0x + 2) + 2) - 1| + 1 = 1x^2 + 3x - 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 points of intersection exist between the functions f(x) = -2(-3cos(1\u03c0x + 2) - 1) - 2 and g(x) = (-3x - 3)/(-2x) 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) = 4(1x^2 - 2x - 4)^2 + 5(1x^2 - 2x - 4) - 2 and g(x) = -1x^2 + 5x + 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 2|(2cos(2\u03c0x - 3) + 1) + 1| - 3 = 2x^2 - 4x + 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": "How many points of intersection exist between the functions f(x) = (-3(2sin(2\u03c0x - 1) - 2) - 1)/(3(2sin(2\u03c0x - 1) - 2)) and g(x) = (3x - 3)/(2x - 3) 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": "Determine the number of intersection points between the functions f(x) = -2|((-1x + 3)/(-2x + 3)) - 1| and g(x) = 1x^2 - 5x 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(-2sin(3\u03c0x - 3))^2 + 5(-2sin(3\u03c0x - 3)) - 4 and g(x) = -2|x - 2| - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "76", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(1cos(2\u03c0x + 2))^2 + 2(1cos(2\u03c0x + 2)) - 1 = (2x - 3)/(-2x) 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": "Find the number of times the graphs of f(x) = -3(-3cos(3\u03c0x - 1) + 2) - 4 and g(x) = (-1x + 2)/(-1x - 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) = 3cos(1\u03c0(-3x^2 + 3x - 1) - 3) + 2 and g(x) = -3x^2 - 5x 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) = 2cos(3\u03c0(4x - 5) + 1) - 1 and g(x) = 5x - 5 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": "How many solutions does the equation (-2(-1cos(1\u03c0x + 1) + 1) - 2)/(2(-1cos(1\u03c0x + 1) + 1)) = -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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(2cos(2\u03c0x) + 1)^2 + 1(2cos(2\u03c0x) + 1) - 1 and g(x) = 3x^2 - 4x 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": "Find the number of times the graphs of f(x) = -2|(1cos(3\u03c0x - 1) - 3)| + 1 and g(x) = -3|x + 2| - 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": "How many solutions does the equation 1(-1cos(1\u03c0x - 3))^2 - 3(-1cos(1\u03c0x - 3)) - 3 = 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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1cos(3\u03c0((-2x)/(-3x + 4))) + 1 = 2|x - 2| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(2cos(3\u03c0x + 3) - 2)^2 - 4(2cos(3\u03c0x + 3) - 2) - 3 and g(x) = (-2x - 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": "65", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(3x^2 - 3x - 5)^2 - 4(3x^2 - 3x - 5) and g(x) = 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": "Find the number of times the graphs of f(x) = 1cos(2\u03c0(3x + 5) + 2) + 3 and g(x) = -1x^2 + 4x + 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": "Find the number of times the graphs of f(x) = 3sin(1\u03c0(-4x - 1) + 1) + 3 and g(x) = 5x^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) = (-2(3sin(2\u03c0x + 2) + 3) + 3)/(2(3sin(2\u03c0x + 2) + 3)) and g(x) = 4x^2 - 5x - 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 points of intersection exist between the functions f(x) = 4(-3cos(3\u03c0x - 1) - 2)^2 + 1(-3cos(3\u03c0x - 1) - 2) - 3 and g(x) = 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": "90", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1sin(1\u03c0(2x - 5) - 3) + 3 and g(x) = (-3x - 1)/(-1x + 2) 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) = 5(-1sin(3\u03c0x + 3) + 2)^2 - 3(-1sin(3\u03c0x + 3) + 2) - 5 and g(x) = 3|x - 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": "How many points of intersection exist between the functions f(x) = -2cos(3\u03c0((2x + 2)/(1x)) + 2) + 1 and g(x) = -3x + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "75", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(2cos(2\u03c0x) - 2) + 1 and g(x) = 2x^2 - 4x - 5 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 -3cos(1\u03c0((-2x - 1)/(-1x - 2)) - 1) = -2x^2 - 3x 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|(2sin(2\u03c0x + 1) - 3) + 3| and g(x) = -1x 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": "Determine the number of intersection points between the functions f(x) = -1(-2cos(2\u03c0x + 1) + 2) + 4 and g(x) = 3x^2 + 3x + 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(3sin(2\u03c0x + 1) - 1))/(2(3sin(2\u03c0x + 1) - 1) - 3) and g(x) = (-2x)/(-1x - 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": "Determine the number of intersection points between the functions f(x) = 3|(1sin(1\u03c0x + 2) - 2) + 3| + 3 and g(x) = -1x + 2 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 solutions does the equation 4(-2sin(1\u03c0x - 2) + 3) + 5 = 2x^2 - 2x 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 2sin(3\u03c0(-4x^2 + 2x - 2) + 2) - 2 = -2|x - 1| - 1 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": "Find the number of times the graphs of f(x) = 3|(3cos(3\u03c0x - 3) + 3) - 3| + 3 and g(x) = 5x + 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": "Determine the number of intersection points between the functions f(x) = 2|((-1x + 2)/(-2x + 3)) + 3| and g(x) = 4x^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": "Determine the number of intersection points between the functions f(x) = -2|((-1x - 3)/(-2x - 1)) + 3| + 3 and g(x) = -1|x + 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) = 5(-2cos(2\u03c0x - 3) - 1) + 2 and g(x) = (1x)/(1x) 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 -1sin(3\u03c0(3|x + 3| + 1) - 1) + 1 = 1x - 1 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": "Find the number of times the graphs of f(x) = (-2(-3sin(2\u03c0x + 3) + 2) + 2)/(3(-3sin(2\u03c0x + 3) + 2) - 2) and g(x) = (-2x + 2)/(-1x + 3) 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) = -1|(-3x^2 + 4x + 5) - 2| and g(x) = 2x - 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 points of intersection exist between the functions f(x) = (2(-1x^2 - 5x + 1) - 1)/(2(-1x^2 - 5x + 1) + 2) and g(x) = 1x^2 + 5x - 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) = -3cos(3\u03c0(-2|x + 2|) + 3) - 3 and g(x) = 1x - 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": "How many points of intersection exist between the functions f(x) = (2(2cos(3\u03c0x + 1) + 3) + 2)/(-2(2cos(3\u03c0x + 1) + 3) + 3) and g(x) = -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": "39", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(-2sin(2\u03c0x + 2))^2 - 5(-2sin(2\u03c0x + 2)) - 2 and g(x) = 1x^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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(1\u03c0((-3x + 3)/(3x - 2))) - 1 and g(x) = -2|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": "21", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(-3sin(1\u03c0x + 1) - 1) - 5 = (-3x + 3)/(2x) 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": "Determine the number of intersection points between the functions f(x) = -2sin(3\u03c0(2x^2 + 2x) - 1) + 3 and g(x) = 2x - 1 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) = 1(-3sin(1\u03c0x + 1)) + 2 and g(x) = 1x + 5 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) = -1cos(1\u03c0(-2|x + 1| - 1) + 1) + 1 and g(x) = (2x - 1)/(2x + 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": "Determine the number of intersection points between the functions f(x) = 3(-1sin(2\u03c0x - 2) + 2) - 2 and g(x) = 1x 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(3cos(3\u03c0x - 2) - 2) - 2 and g(x) = -1x + 3 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": "How many points of intersection exist between the functions f(x) = 3|(-3sin(2\u03c0x + 1) - 2) + 2| - 1 and g(x) = (3x + 3)/(3x - 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": "How many solutions does the equation 3sin(2\u03c0(-4x + 3) - 3) - 1 = 1|x + 3| - 1 have for x 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) = (1(2sin(1\u03c0x)) + 1)/(-2(2sin(1\u03c0x)) + 3) and g(x) = 4x + 5 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": "How many solutions does the equation 1cos(2\u03c0(-1x^2 + 5x + 5) - 2) - 3 = -2x^2 - 3x + 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 2cos(3\u03c0(-1x + 1) - 1) - 3 = -2|x + 1| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(3\u03c0(-3x - 5)) - 2 and g(x) = (3x + 1)/(-1x - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "157", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1cos(1\u03c0(-2|x - 1| + 1) - 3) + 1 and g(x) = -1x^2 + 3x 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) = 3|(-2sin(2\u03c0x + 2)) - 1| + 3 and g(x) = (3x - 3)/(1x + 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 points of intersection exist between the functions f(x) = -3(-2cos(3\u03c0x + 1) - 3)^2 - 2(-2cos(3\u03c0x + 1) - 3) - 2 and g(x) = -2x^2 - 5x + 2 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": "How many solutions does the equation (2(2cos(3\u03c0x + 2) - 1) + 3)/(-1(2cos(3\u03c0x + 2) - 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": "55", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(3\u03c0(5x - 4) + 2) and g(x) = -3|x - 3| + 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 solutions does the equation -3cos(1\u03c0(1|x + 1|) + 3) - 2 = -1x 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 points of intersection exist between the functions f(x) = 2sin(3\u03c0(3|x + 2| - 3) + 2) and g(x) = 4x^2 + 1x 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": "Find the number of times the graphs of f(x) = -4(-3cos(2\u03c0x) - 3)^2 - 1(-3cos(2\u03c0x) - 3) + 5 and g(x) = -5x^2 + 5x + 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": "Determine the number of intersection points between the functions f(x) = (1(3cos(3\u03c0x + 3) - 1) + 1)/(-1(3cos(3\u03c0x + 3) - 1) + 2) and g(x) = -5x - 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": "Find the number of times the graphs of f(x) = 1cos(2\u03c0(3x - 5) + 3) + 1 and g(x) = 3x - 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": "Find the number of times the graphs of f(x) = (1(2sin(3\u03c0x + 1) - 1))/(-1(2sin(3\u03c0x + 1) - 1) - 1) and g(x) = 1x + 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": "How many solutions does the equation 2|(1cos(3\u03c0x - 3) - 3) + 1| - 2 = 3|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": "Find the number of times the graphs of f(x) = -2|(2cos(3\u03c0x - 3)) - 1| - 2 and g(x) = 1x^2 + 3x - 1 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": "How many points of intersection exist between the functions f(x) = 5(3x^2 - 2x - 1)^2 - 4(3x^2 - 2x - 1) - 4 and g(x) = 3x + 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) = -2cos(3\u03c0((1x + 1)/(-3x + 2)) + 2) - 2 and g(x) = 2x^2 - 3x - 3 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 solutions does the equation -1(1x^2 + 5x + 1)^2 + 3(1x^2 + 5x + 1) - 1 = (-2x + 1)/(1x + 3) 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) = -1|((1x - 3)/(1x - 2)) + 2| - 3 and g(x) = -2x^2 - 3x - 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) = 4(-2sin(3\u03c0x) + 1)^2 + 5(-2sin(3\u03c0x) + 1) - 5 and g(x) = -4x^2 - 2x + 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": "How many points of intersection exist between the functions f(x) = 1(4x^2 + 4x - 4)^2 - 3(4x^2 + 4x - 4) - 1 and g(x) = 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 points of intersection exist between the functions f(x) = -3cos(2\u03c0(5x^2 - 5x - 3) + 3) and g(x) = (2x)/(-1x + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "934", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(2sin(3\u03c0x + 1) - 1) + 3)/(-1(2sin(3\u03c0x + 1) - 1) - 2) and g(x) = 2x^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": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(3sin(1\u03c0x + 1))^2 - 2(3sin(1\u03c0x + 1)) - 3 and g(x) = 2x^2 - 5x - 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 solutions does the equation -2cos(2\u03c0(2|x + 2| - 3)) + 3 = 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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(-1cos(3\u03c0x + 1)) + 2)/(-2(-1cos(3\u03c0x + 1))) = -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": "58", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(1sin(1\u03c0x - 3) + 2)^2 - 1(1sin(1\u03c0x - 3) + 2) - 2 and g(x) = (2x - 1)/(-1x - 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 solutions does the equation (1(3sin(2\u03c0x + 3)) + 1)/(-1(3sin(2\u03c0x + 3)) + 2) = (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": "41", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(3sin(2\u03c0x - 2) - 1) + 1 = (1x + 3)/(3x + 1) 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": "How many points of intersection exist between the functions f(x) = -2sin(2\u03c0(-2|x + 3|)) - 1 and g(x) = 1x^2 - 5x 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) = (-1(3sin(2\u03c0x - 3) + 1) - 2)/(3(3sin(2\u03c0x - 3) + 1) - 1) and g(x) = -5x - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(5x^2 - 2x - 4)^2 + 5(5x^2 - 2x - 4) + 1 and g(x) = (-3x - 2)/(3x - 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 3cos(2\u03c0(5x + 3) - 2) + 1 = 3x^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": "10", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3sin(3\u03c0(2x^2 - 2x - 4) + 2) + 3 = 5x + 2 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 -3cos(1\u03c0(4x - 4) + 2) = 2x^2 + 5x - 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 points of intersection exist between the functions f(x) = (-3(5x^2 + 4x - 4) + 2)/(2(5x^2 + 4x - 4) - 1) and g(x) = -2x^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) = 2|(1sin(3\u03c0x + 1) + 3) - 2| - 2 and g(x) = -2x^2 + 4x - 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": "Determine the number of intersection points between the functions f(x) = 2sin(2\u03c0(2x^2 + 4x + 2) + 3) and g(x) = 1x^2 + 1x - 5 where -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 solutions does the equation 3(2sin(1\u03c0x + 1)) + 5 = (-2x - 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": "11", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(2cos(3\u03c0x + 2) + 3) - 4 = -3|x - 2| 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) = 4(2sin(2\u03c0x - 3) + 1)^2 - 5(2sin(2\u03c0x - 3) + 1) - 2 and g(x) = 4x + 5 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 3(1x^2 - 4x + 1)^2 - 4(1x^2 - 4x + 1) - 1 = (-2x - 3)/(2x + 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 -4(2cos(1\u03c0x - 1)) + 2 = -1|x| + 3 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 solutions does the equation -3cos(1\u03c0(-3x - 1) + 1) + 1 = (-2x + 1)/(-1x + 2) 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 -2sin(3\u03c0(2x^2 + 1x + 1)) + 2 = (-2x + 1)/(2x + 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 points of intersection exist between the functions f(x) = (1(-3sin(3\u03c0x - 3) + 2) - 2)/(2(-3sin(3\u03c0x - 3) + 2) - 1) and g(x) = -2x - 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 points of intersection exist between the functions f(x) = -2cos(3\u03c0(-1|x - 1|)) - 3 and g(x) = (1x + 3)/(-1x + 2) 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) = 3(2sin(2\u03c0x - 3) + 1) and g(x) = 5x^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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-3cos(2\u03c0x + 3) + 3)^2 + 5(-3cos(2\u03c0x + 3) + 3) - 2 and g(x) = 1|x + 3| intersect 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 points of intersection exist between the functions f(x) = -1sin(1\u03c0((-3x - 2)/(2x - 3))) + 1 and g(x) = 3|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": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1cos(3\u03c0(-4x - 2) + 2) and g(x) = -3|x - 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": "Find the number of times the graphs of f(x) = 3cos(1\u03c0(-4x^2 - 2x + 1)) - 2 and g(x) = 2|x + 3| intersect in the interval [-10, 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) = 4(5x^2 + 4x - 3)^2 + 3(5x^2 + 4x - 3) - 2 and g(x) = 2|x - 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(3cos(2\u03c0x + 2) + 1) + 1)/(3(3cos(2\u03c0x + 2) + 1) - 2) and g(x) = 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": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(-3sin(3\u03c0x - 2)) - 3| - 2 and g(x) = -1x^2 + 1x + 5 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) = -5(-1cos(1\u03c0x - 2) + 1) + 5 and g(x) = (-1x - 2)/(-1x) intersect in the interval [-10, 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 5(1cos(3\u03c0x + 1) - 3)^2 + 1 = 2|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) = (3(-1sin(3\u03c0x) - 3) + 3)/(-1(-1sin(3\u03c0x) - 3) + 3) and g(x) = 2|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) = 5(2cos(3\u03c0x) - 3) + 5 and g(x) = (-1x)/(2x + 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": "Determine the number of intersection points between the functions f(x) = 4(-2sin(3\u03c0x + 2) - 1) + 2 and g(x) = 3|x + 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": "Find the number of times the graphs of f(x) = 3sin(2\u03c0(3x^2 - 1x)) - 1 and g(x) = -4x - 2 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 points of intersection exist between the functions f(x) = -3sin(1\u03c0(-4x^2 + 1x - 4) + 3) and g(x) = (-3x - 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": "797", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(3x^2 + 4x - 2)^2 + 4(3x^2 + 4x - 2) - 4 and g(x) = (3x + 1)/(3x - 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": "How many points of intersection exist between the functions f(x) = -3cos(1\u03c0((3x + 1)/(-3x + 2)) + 1) + 3 and g(x) = -1x^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": "18", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(1cos(3\u03c0x + 2) - 3)^2 - 3(1cos(3\u03c0x + 2) - 3) + 1 and g(x) = 4x - 4 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": "How many solutions does the equation -3cos(1\u03c0(-1x^2 + 4x + 1) + 3) = -1x^2 + 4x + 4 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) = (-2(-2cos(2\u03c0x + 2)) + 1)/(2(-2cos(2\u03c0x + 2))) and g(x) = 2x^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": "15", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(1\u03c0(-4x^2 - 3x - 2) + 3) + 3 and g(x) = 2x + 2 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": "How many points of intersection exist between the functions f(x) = (-1(2x^2 + 5x - 2) + 3)/(-2(2x^2 + 5x - 2) - 1) and g(x) = 1x^2 - 5x + 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 solutions does the equation -3cos(2\u03c0(-1x - 3) + 1) = -3x - 4 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) = -1cos(2\u03c0(1x^2 - 3x - 5) - 2) + 1 and g(x) = (-2x - 1)/(1x + 2) 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) = 5(-2sin(2\u03c0x + 2))^2 - 2(-2sin(2\u03c0x + 2)) + 1 and g(x) = 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": "68", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(2\u03c0(-5x - 4) + 2) - 1 and g(x) = -1|x + 3| + 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 1(-2cos(1\u03c0x + 3) + 3)^2 - 4(-2cos(1\u03c0x + 3) + 3) - 5 = (-2x)/(3x) 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 5(2sin(3\u03c0x + 3) - 1)^2 + 4(2sin(3\u03c0x + 3) - 1) - 3 = 2|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": "68", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(2cos(2\u03c0x + 1) + 1) - 4 and g(x) = -1x + 1 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) = -2cos(3\u03c0((-3x)/(-3x - 3))) and g(x) = (3x + 2)/(2x + 1) where -10 \u2264 x \u2264 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) = (-1(2sin(1\u03c0x) - 1) + 1)/(-2(2sin(1\u03c0x) - 1) + 1) and g(x) = 4x^2 + 5x + 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 3(-3cos(3\u03c0x - 2) - 3)^2 - 2(-3cos(3\u03c0x - 2) - 3) + 2 = 1x^2 - 5x + 3 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 solutions does the equation 2cos(3\u03c0(1x - 5) - 2) + 1 = 5x^2 + 5x + 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 3(-2sin(2\u03c0x + 2))^2 - 4(-2sin(2\u03c0x + 2)) - 5 = -1x^2 + 5x + 3 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": "Find the number of times the graphs of f(x) = -3sin(2\u03c0(1|x - 1| + 2) - 1) - 2 and g(x) = (-1x - 1)/(-1x + 2) 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": "Find the number of times the graphs of f(x) = -2sin(2\u03c0(-1x - 3) + 3) - 3 and g(x) = -1x + 1 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 points of intersection exist between the functions f(x) = (-2(3x^2 + 1x - 4))/(2(3x^2 + 1x - 4) - 1) and g(x) = -1x^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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(1cos(2\u03c0x + 3) + 2)^2 + 5 = 4x + 3 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 2sin(1\u03c0(-1|x + 3|) + 3) - 2 = (-1x + 1)/(1x + 3) 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": "How many solutions does the equation -2|(-3sin(2\u03c0x + 1) + 2) + 3| + 2 = 3x - 5 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 points of intersection exist between the functions f(x) = 5(-5x^2 + 3x + 4)^2 + 5(-5x^2 + 3x + 4) - 3 and g(x) = -2|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": "Find the number of times the graphs of f(x) = -2cos(2\u03c0(3x - 4)) - 2 and g(x) = -3|x + 3| - 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": "Find the number of times the graphs of f(x) = 3sin(1\u03c0(3|x - 1| - 1) - 1) - 2 and g(x) = (2x + 3)/(-1x + 3) 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) = -1sin(2\u03c0(-2x^2 + 2x - 2)) and g(x) = 1x + 3 in the range -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) = 2(1sin(1\u03c0x + 1) - 3)^2 + 3(1sin(1\u03c0x + 1) - 3) + 4 and g(x) = 2|x| 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": "Find the number of times the graphs of f(x) = -3cos(1\u03c0(3|x + 1| - 2) - 3) - 1 and g(x) = 1|x + 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": "Find the number of times the graphs of f(x) = -2|(3cos(2\u03c0x + 3))| + 2 and g(x) = 1x^2 - 2x - 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": "Determine the number of intersection points between the functions f(x) = (1(1sin(3\u03c0x + 3) + 1) + 1)/(3(1sin(3\u03c0x + 3) + 1)) and g(x) = (-2x)/(-3x + 1) 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": "Determine the number of intersection points between the functions f(x) = -3cos(1\u03c0(1x^2 + 5x - 1) - 3) + 2 and g(x) = -3|x - 3| + 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 points of intersection exist between the functions f(x) = -2sin(1\u03c0(-3|x| - 1) - 1) + 2 and g(x) = 2x^2 - 5x 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) = 2cos(2\u03c0((2x - 2)/(-1x + 2))) and g(x) = -3|x - 1| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1sin(1\u03c0(5x^2 - 1x - 3) - 2) + 2 and g(x) = 5x^2 + 3x - 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(-2sin(1\u03c0x - 3) - 3)^2 - 5(-2sin(1\u03c0x - 3) - 3) - 5 and g(x) = -2x + 5 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) = 1sin(1\u03c0((2x - 2)/(-3x)) + 3) - 1 and g(x) = 3x^2 + 5x - 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": "Find the number of times the graphs of f(x) = 2cos(2\u03c0(2x^2 - 2x - 4) + 2) + 1 and g(x) = -5x - 4 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": "Determine the number of intersection points between the functions f(x) = -2sin(3\u03c0(2|x - 2|) - 2) and g(x) = 1x^2 - 1x - 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 solutions does the equation 3sin(2\u03c0(-3x^2 - 2x - 1)) + 2 = (3x + 2)/(-2x - 3) 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 points of intersection exist between the functions f(x) = -3cos(3\u03c0(-1|x + 3| - 1) - 2) and g(x) = (2x + 3)/(-3x - 3) in the range -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|(5x^2 - 5) - 2| - 1 and g(x) = -2x + 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": "Determine the number of intersection points between the functions f(x) = -2sin(1\u03c0(5x^2 - 4x - 4) - 1) and g(x) = -2|x - 2| - 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 points of intersection exist between the functions f(x) = -4(2sin(3\u03c0x - 2) + 2) + 4 and g(x) = (-2x + 3)/(-1x) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-3sin(3\u03c0x + 3)) + 2| + 3 and g(x) = -2x + 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": "How many solutions does the equation 1cos(2\u03c0(-2x^2 - 1x - 5) + 2) + 1 = -3x^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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2cos(2\u03c0((3x - 3)/(-2x + 1)) + 1) + 3 = -4x + 3 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": "Find the number of times the graphs of f(x) = 1sin(2\u03c0(-1|x - 3| - 2) - 3) and g(x) = (-1x - 1)/(-3x - 1) 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) = (-2(-1cos(2\u03c0x - 1)) + 1)/(3(-1cos(2\u03c0x - 1)) - 1) and g(x) = (2x - 2)/(-2x - 1) where -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) = -2cos(2\u03c0((-3x - 2)/(-1x - 3))) + 2 and g(x) = -3|x + 2| + 2 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": "Find the number of times the graphs of f(x) = 2cos(2\u03c0(-1x - 2) - 2) and g(x) = 3x^2 + 4x - 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(1cos(2\u03c0x) + 1)^2 - 2(1cos(2\u03c0x) + 1) - 3 and g(x) = -2|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": "18", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2sin(3\u03c0(-1x - 3) - 3) + 3 = 3|x + 3| + 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": "How many points of intersection exist between the functions f(x) = -1sin(2\u03c0(2|x - 3| - 2)) - 2 and g(x) = -5x^2 + 3x - 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) = 3sin(2\u03c0((1x - 1)/(-3x + 1)) - 2) - 3 and g(x) = -5x - 2 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": "How many solutions does the equation -1|(-3cos(3\u03c0x - 1) - 3) + 3| - 3 = -3x^2 + 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": "Find the number of times the graphs of f(x) = -3(-3x^2 - 1x + 2)^2 + 3(-3x^2 - 1x + 2) + 1 and g(x) = -2x^2 - 2x - 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) = 1cos(2\u03c0(-1x - 4) - 2) + 2 and g(x) = (-2x - 2)/(-1x + 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": "Find the number of times the graphs of f(x) = -1sin(1\u03c0((-1x + 1)/(3x + 3)) - 2) - 1 and g(x) = 1|x + 3| - 3 intersect in the interval [-10, 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) = 2cos(1\u03c0(3x - 2) - 2) - 1 and g(x) = 1|x - 1| - 3 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) = -1(-3sin(2\u03c0x - 3) - 1)^2 + 1(-3sin(2\u03c0x - 3) - 1) + 3 and g(x) = 1x 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": "How many points of intersection exist between the functions f(x) = 3sin(1\u03c0(1x + 3) + 1) and g(x) = (-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": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(2cos(1\u03c0x - 1) + 3) + 3 and g(x) = (-1x + 2)/(-1x - 1) 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) = (-1(1sin(1\u03c0x - 3) - 2) - 1)/(-2(1sin(1\u03c0x - 3) - 2) - 3) and g(x) = (1x + 3)/(-2x) 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 solutions does the equation 2cos(1\u03c0(-2|x + 3| + 2) - 1) - 1 = -2|x + 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) = -3cos(2\u03c0(5x + 4) - 1) - 2 and g(x) = -3|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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(2sin(3\u03c0x + 1) - 2))/(-2(2sin(3\u03c0x + 1) - 2) - 3) and g(x) = (-1x + 2)/(-1x + 3) 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": "How many points of intersection exist between the functions f(x) = 1(-2cos(1\u03c0x) + 1)^2 + 3(-2cos(1\u03c0x) + 1) and g(x) = 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": "20", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1sin(3\u03c0(-1x^2 + 2) - 1) + 2 = (-1x - 1)/(-2x + 3) have for x in the interval [-10, 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) = (-1(3sin(1\u03c0x + 2) - 2))/(-3(3sin(1\u03c0x + 2) - 2) - 2) and g(x) = -1x^2 - 4x - 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) = (1(-2sin(3\u03c0x - 2)) - 3)/(-1(-2sin(3\u03c0x - 2)) - 2) and g(x) = (1x - 2)/(3x + 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": "Determine the number of intersection points between the functions f(x) = -1cos(3\u03c0((-1x - 1)/(-1x - 3)) + 2) and g(x) = 2|x + 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": "Determine the number of intersection points between the functions f(x) = -3cos(1\u03c0(2|x - 3| - 3) - 1) + 3 and g(x) = -1x^2 - 1x + 4 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) = 1|(-4x^2 + 4x + 5) - 2| + 2 and g(x) = 2|x + 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": "Determine the number of intersection points between the functions f(x) = -3sin(3\u03c0(-5x^2 - 3x + 4) - 3) - 2 and g(x) = -4x^2 + 2x - 5 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) = -2cos(3\u03c0(5x^2 - 5x - 3) + 2) - 2 and g(x) = 3x + 3 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": "Determine the number of intersection points between the functions f(x) = (-3(-3sin(1\u03c0x) - 2) + 1)/(1(-3sin(1\u03c0x) - 2)) and g(x) = -2x^2 - 1x - 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": "How many solutions does the equation -1|(-5x^2 + 1) + 2| - 1 = -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": "How many points of intersection exist between the functions f(x) = -5(-3sin(3\u03c0x + 1) + 2) and g(x) = 1x + 4 in the range -10 \u2264 x \u2264 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) = 1|(-3cos(1\u03c0x - 2) - 3) + 3| - 2 and g(x) = (2x + 3)/(3x + 3) in the range -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) = -3cos(1\u03c0((1x)/(-2x + 3)) - 1) + 1 and g(x) = 2x 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 points of intersection exist between the functions f(x) = (-3(2cos(3\u03c0x) - 1) - 2)/(-2(2cos(3\u03c0x) - 1) + 3) and g(x) = (-1x + 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": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(-2sin(3\u03c0x) + 1) - 2| - 1 and g(x) = 3x 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) = 3|((3x + 3)/(1x - 2)) + 3| - 1 and g(x) = 3|x| + 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 -3sin(1\u03c0(-1x^2 + 5x - 1) + 3) + 1 = 1x - 5 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": "How many points of intersection exist between the functions f(x) = 2|(1sin(2\u03c0x + 3) + 3)| - 3 and g(x) = -1x - 3 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 points of intersection exist between the functions f(x) = 1cos(3\u03c0((-3x - 2)/(1x + 1)) + 1) + 2 and g(x) = 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(1x^2 - 4x - 4) + 1)/(-2(1x^2 - 4x - 4) + 1) and g(x) = 2|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": "How many solutions does the equation 2sin(2\u03c0(1|x - 3| + 1) - 2) = -1x^2 - 4x 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(3\u03c0(1|x - 1| - 1)) = -1|x - 1| + 2 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 points of intersection exist between the functions f(x) = 1cos(2\u03c0(-1x - 1) + 2) and g(x) = 3|x| - 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 solutions does the equation -2cos(3\u03c0(1|x + 1| - 1) + 2) - 1 = -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": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(1cos(1\u03c0x + 2)) - 2)/(1(1cos(1\u03c0x + 2)) + 2) and g(x) = (-1x + 2)/(-1x + 3) 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 solutions does the equation -1|(2|x + 2| - 3) + 1| + 1 = (3x + 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": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2sin(1\u03c0(3x^2 - 4x + 3) - 2) - 1 and g(x) = -5x^2 - 2x + 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": "Determine the number of intersection points between the functions f(x) = 2cos(1\u03c0(-5x^2 - 3x + 3) - 2) - 2 and g(x) = -4x^2 - 5x + 1 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": "Find the number of times the graphs of f(x) = -3cos(2\u03c0(-3x^2 + 3x + 2)) - 2 and g(x) = -2|x + 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "158", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(1cos(1\u03c0x - 3) + 1) and g(x) = -1|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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(2cos(1\u03c0x + 1) + 2) + 1)/(-3(2cos(1\u03c0x + 1) + 2) + 2) and g(x) = 1x + 5 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": "Find the number of times the graphs of f(x) = -2sin(2\u03c0(2|x + 1| - 3) - 3) - 1 and g(x) = 2|x - 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) = 2(1cos(2\u03c0x - 3) - 1) + 4 and g(x) = 1|x| - 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": "Find the number of times the graphs of f(x) = 5(1cos(2\u03c0x - 1) + 3) + 3 and g(x) = 5x + 5 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) = -3(3sin(1\u03c0x - 3) - 1) and g(x) = -1|x| - 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 -3sin(2\u03c0(-1x + 1)) + 2 = 1x - 3 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(2sin(3\u03c0x - 1) - 2) + 3)/(1(2sin(3\u03c0x - 1) - 2) + 2) and g(x) = -2|x| - 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": "How many solutions does the equation 1(-1cos(1\u03c0x + 1) + 2)^2 + 1(-1cos(1\u03c0x + 1) + 2) = 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(1\u03c0(3x^2 - 4x - 3) + 2) and g(x) = (-1x - 1)/(-1x + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "462", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-2|x - 1| + 2)^2 + 5(-2|x - 1| + 2) + 1 and g(x) = (-2x + 2)/(3x + 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) = 1|(-3|x + 2| + 2) + 1| - 1 and g(x) = (3x - 1)/(2x - 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": "Find the number of times the graphs of f(x) = -2cos(3\u03c0(-1x^2 - 4x - 5)) and g(x) = 2|x - 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": "Find the number of times the graphs of f(x) = -1(-2sin(3\u03c0x + 3) - 3)^2 - 2(-2sin(3\u03c0x + 3) - 3) + 2 and g(x) = (-2x)/(-2x - 3) 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": "How many solutions does the equation 2(2sin(2\u03c0x + 2) + 1) - 5 = 2x - 2 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": "Find the number of times the graphs of f(x) = -2(-1sin(2\u03c0x - 1) - 1)^2 + 5(-1sin(2\u03c0x - 1) - 1) - 1 and g(x) = 4x - 5 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 points of intersection exist between the functions f(x) = -2sin(1\u03c0((1x + 2)/(-1x)) - 3) + 3 and g(x) = (1x + 1)/(-3x) 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) = -1sin(3\u03c0(5x - 1)) + 3 and g(x) = (-1x - 2)/(1x - 2) 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": "Find the number of times the graphs of f(x) = (3(-3sin(2\u03c0x - 3) + 2) - 2)/(-2(-3sin(2\u03c0x - 3) + 2) + 1) and g(x) = 3|x - 1| + 2 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) = -3sin(1\u03c0(-1x^2 - 4x + 1)) and g(x) = 3x + 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": "How many points of intersection exist between the functions f(x) = -2sin(3\u03c0((1x - 2)/(3x - 2)) - 3) + 2 and g(x) = (2x)/(-1x + 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": "Determine the number of intersection points between the functions f(x) = 3sin(1\u03c0(-2|x - 2| - 3) - 2) and g(x) = -3|x - 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(2\u03c0x) + 2) + 3)/(-1(3cos(2\u03c0x) + 2) - 2) and g(x) = -1x^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 -2sin(3\u03c0(2|x - 1|) - 2) - 1 = 3|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": "12", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(3\u03c0((1x - 2)/(3x + 3)) - 3) + 1 and g(x) = 3|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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-2sin(2\u03c0x + 3) - 2) + 3)/(-2(-2sin(2\u03c0x + 3) - 2) - 2) = 4x^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": "22", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2(3sin(2\u03c0x - 2) - 1) - 2 and g(x) = 1x + 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": "Find the number of times the graphs of f(x) = 1cos(3\u03c0(3x + 5)) - 3 and g(x) = -2x^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) = -1sin(3\u03c0(-2|x - 1| + 1) + 1) + 2 and g(x) = -1x - 2 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) = 2cos(2\u03c0(3|x + 3| - 2)) - 2 and g(x) = 4x 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": "Find the number of times the graphs of f(x) = -3cos(1\u03c0((1x - 1)/(-3x - 1)) - 2) + 1 and g(x) = 5x + 1 intersect in the interval [-10, 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 -5(-2cos(3\u03c0x - 1) + 1)^2 + 2(-2cos(3\u03c0x - 1) + 1) = -3|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": "70", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(-2cos(2\u03c0x)) + 5 = 1x 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": "Find the number of times the graphs of f(x) = 2(-2|x - 1| + 2)^2 + 5(-2|x - 1| + 2) and g(x) = 1|x + 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) = -1|(1sin(3\u03c0x))| + 2 and g(x) = (-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": "How many solutions does the equation -2(-1cos(1\u03c0x - 1) + 2)^2 + 5 = -1x + 3 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 solutions does the equation -3cos(3\u03c0(-1|x - 1| - 1)) = (-3x)/(-1x + 3) have for x in the interval [-10, 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 2cos(1\u03c0(2x^2 - 2x - 2) - 1) - 1 = 3x^2 - 3x - 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(-3sin(2\u03c0x)) - 3 and g(x) = (-3x - 1)/(-2x + 3) 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) = -3sin(3\u03c0(-1|x + 3| - 3) - 2) and g(x) = -1x^2 - 5x + 4 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) = 2|(-2sin(3\u03c0x + 3) - 2) + 3| + 3 and g(x) = 2x - 3 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": "How many points of intersection exist between the functions f(x) = -2(3|x| - 1)^2 + 3(3|x| - 1) + 5 and g(x) = (3x - 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": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(2cos(2\u03c0x - 2) + 1) - 1)/(-1(2cos(2\u03c0x - 2) + 1) + 2) and g(x) = -2x + 4 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 points of intersection exist between the functions f(x) = -4(-2cos(3\u03c0x) + 3) - 4 and g(x) = -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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(2\u03c0(-1x^2 - 3x + 1) - 3) + 1 and g(x) = (-3x + 1)/(2x - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "400", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(1\u03c0((3x + 1)/(-2x + 3))) + 1 and g(x) = 1|x + 2| - 2 where -10 \u2264 x \u2264 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) = -2cos(2\u03c0(2|x + 3| - 3)) and g(x) = 3x^2 + 4x + 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 (2(-3cos(3\u03c0x + 1) - 2) + 2)/(-3(-3cos(3\u03c0x + 1) - 2)) = 1x^2 + 4x have for x in the interval [-10, 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) = 2|(2x + 1)| and g(x) = 1x^2 - 1x 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) = -1sin(3\u03c0(4x^2 + 5x - 3)) and g(x) = -2x^2 - 5x - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(3sin(1\u03c0x + 1) - 3)| + 1 and g(x) = 2|x + 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) = (1(3sin(1\u03c0x + 3) - 1) + 3)/(-3(3sin(1\u03c0x + 3) - 1) + 1) and g(x) = (-1x - 2)/(-3x) 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": "How many points of intersection exist between the functions f(x) = 2(1sin(2\u03c0x) - 2) - 3 and g(x) = -2x - 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": "How many solutions does the equation 3cos(2\u03c0((-3x - 1)/(2x + 2)) + 1) + 1 = 1|x| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(-2sin(1\u03c0x + 2) + 1) and g(x) = (2x)/(1x) 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(2sin(1\u03c0x + 3) - 1) + 1 and g(x) = 1x - 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) = -2sin(3\u03c0(-4x - 5)) and g(x) = 5x^2 - 3x - 5 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(1\u03c0(-5x + 4) + 3) - 2 = 2x^2 - 4x - 1 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": "How many solutions does the equation -2sin(1\u03c0(4x - 4)) - 2 = 2|x + 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 -5(-2sin(2\u03c0x - 3))^2 - 4(-2sin(2\u03c0x - 3)) - 1 = (-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": "75", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3sin(2\u03c0(3|x + 3| - 2) - 2) - 3 and g(x) = (-1x - 3)/(1x + 3) 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": "How many points of intersection exist between the functions f(x) = -4(3cos(1\u03c0x - 1) + 3) and g(x) = -1|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": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(3cos(3\u03c0x + 3) + 1) + 3 and g(x) = 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": "12", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-3sin(2\u03c0x + 3) - 1)^2 + 3(-3sin(2\u03c0x + 3) - 1) - 4 and g(x) = -3x + 4 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 solutions does the equation -2(4x^2 + 5x + 1)^2 + 3(4x^2 + 5x + 1) - 1 = (-2x)/(1x) 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(2x^2 - 3x - 5)^2 + 5(2x^2 - 3x - 5) + 2 and g(x) = -2x^2 + 3x + 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(3cos(1\u03c0x + 3) + 2) + 3)/(1(3cos(1\u03c0x + 3) + 2) + 1) and g(x) = 2|x| + 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": "How many solutions does the equation (3(2sin(2\u03c0x - 1) + 2) - 1)/(1(2sin(2\u03c0x - 1) + 2) - 2) = 5x^2 + 4x + 4 have for x in the interval [-10, 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) = -2(2sin(1\u03c0x - 2) + 2)^2 - 2(2sin(1\u03c0x - 2) + 2) and g(x) = -5x^2 - 1x - 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 points of intersection exist between the functions f(x) = 1sin(2\u03c0(4x^2 - 1x - 5) + 3) and g(x) = -5x^2 - 4x + 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": "How many solutions does the equation -3cos(1\u03c0(5x + 4) + 2) - 2 = 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(1sin(3\u03c0x + 3) - 3)^2 + 2(1sin(3\u03c0x + 3) - 3) - 5 and g(x) = (-2x - 3)/(1x - 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": "Find the number of times the graphs of f(x) = -1|(-2sin(2\u03c0x + 1) - 2) + 1| + 2 and g(x) = -1|x + 1| + 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 solutions does the equation 5(1|x - 2| - 3)^2 + 2(1|x - 2| - 3) - 5 = (-2x - 2)/(2x + 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(3cos(3\u03c0x - 2) + 1)^2 + 2(3cos(3\u03c0x - 2) + 1) - 2 and g(x) = 1x - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3sin(1\u03c0(-1|x - 1| - 3) + 3) + 1 and g(x) = (-3x + 1)/(-1x - 2) 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 -1cos(3\u03c0(2|x| - 3) - 2) - 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": "5", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(3\u03c0(5x^2 + 2) - 3) + 1 and g(x) = 3x 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": "Find the number of times the graphs of f(x) = -1(-1x^2 - 5x + 3)^2 + 4(-1x^2 - 5x + 3) + 4 and g(x) = -2|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": "How many points of intersection exist between the functions f(x) = 1cos(3\u03c0(1x + 1) - 3) and g(x) = (1x)/(-2x) 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) = 3(3cos(1\u03c0x + 3) + 2) - 2 and g(x) = (2x)/(2x - 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": "Find the number of times the graphs of f(x) = -5(2|x + 1| - 1)^2 + 2(2|x + 1| - 1) + 2 and g(x) = 1|x + 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) = -2sin(1\u03c0(-2x + 2) - 2) - 1 and g(x) = -3x^2 - 2x 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\u03c0x - 3) + 3)| - 1 and g(x) = -3|x + 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": "How many solutions does the equation 2cos(2\u03c0(-2x + 3)) - 2 = -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": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|(-3cos(1\u03c0x + 3) + 1) - 1| + 2 and g(x) = (-1x - 1)/(-3x + 2) 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": "How many solutions does the equation 2sin(3\u03c0(-1x^2 + 1x + 2) - 3) - 2 = (1x - 3)/(-3x + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "525", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2sin(2\u03c0(3|x - 2| - 3) + 1) - 3 and g(x) = -2|x + 3| + 1 intersect 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) = 4(3sin(3\u03c0x + 2) - 2)^2 + 5(3sin(3\u03c0x + 2) - 2) + 4 and g(x) = 3|x + 3| - 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 solutions does the equation -3sin(3\u03c0(1x - 3)) + 3 = 5x^2 + 4x - 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 solutions does the equation -3(2cos(3\u03c0x) - 2)^2 + 1(2cos(3\u03c0x) - 2) - 1 = -5x + 4 have for x in the interval [-10, 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) = (-3(-2sin(3\u03c0x - 2) - 2) - 3)/(-1(-2sin(3\u03c0x - 2) - 2) + 3) and g(x) = (1x + 3)/(2x) where -10 \u2264 x \u2264 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) = -4(-1sin(1\u03c0x) + 2)^2 + 4(-1sin(1\u03c0x) + 2) + 4 and g(x) = -4x + 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 points of intersection exist between the functions f(x) = (3(-2sin(2\u03c0x - 1) + 2) + 2)/(1(-2sin(2\u03c0x - 1) + 2) - 2) and g(x) = 4x - 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) = 2sin(1\u03c0(-5x^2 - 4x + 4)) - 2 and g(x) = 5x + 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": "Find the number of times the graphs of f(x) = -2cos(1\u03c0(4x^2 + 5x) + 1) - 1 and g(x) = -3x^2 - 5x + 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 points of intersection exist between the functions f(x) = 5(-1x^2 - 3x + 5)^2 + 5(-1x^2 - 3x + 5) - 5 and g(x) = (2x + 2)/(3x - 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 points of intersection exist between the functions f(x) = 2sin(3\u03c0(-3|x - 1| - 1) + 3) + 2 and g(x) = 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": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(2\u03c0(1x^2 + 3x - 2)) - 1 and g(x) = -1|x| + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "107", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(1\u03c0(-2|x - 1| - 3) - 2) - 3 and g(x) = -2x 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(1cos(2\u03c0x) - 1) + 3)/(-1(1cos(2\u03c0x) - 1) - 3) and g(x) = (-3x - 3)/(1x) intersect in the interval [-10, 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 -3sin(2\u03c0(2x - 1)) + 1 = (-2x - 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": "77", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(-2x^2 + 3x + 5) - 1| - 1 and g(x) = 1|x| + 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) = 4(2sin(3\u03c0x + 3) + 1) + 3 and g(x) = (-2x - 3)/(-3x) 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": "How many points of intersection exist between the functions f(x) = 3((3x + 3)/(3x - 2))^2 + 1((3x + 3)/(3x - 2)) - 1 and g(x) = 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": "Find the number of times the graphs of f(x) = -1sin(3\u03c0(1x^2 + 4x - 4)) - 3 and g(x) = (-2x + 1)/(1x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "131", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(-2cos(1\u03c0x - 1)) + 3)/(2(-2cos(1\u03c0x - 1)) + 3) = 2|x| - 3 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 solutions does the equation 3sin(2\u03c0(-1|x + 2| + 3)) = (-1x)/(-2x + 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 points of intersection exist between the functions f(x) = 1sin(3\u03c0(-4x + 2)) and g(x) = (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": "231", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(2\u03c0(1|x + 1| - 2) - 1) - 2 and g(x) = -1|x - 2| + 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) = (-1(3cos(3\u03c0x - 2) - 1) + 1)/(-2(3cos(3\u03c0x - 2) - 1) - 1) and g(x) = -4x + 5 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": "Find the number of times the graphs of f(x) = 2(-1sin(3\u03c0x + 1) + 3) + 5 and g(x) = 2x^2 + 2x + 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) = 2|(-2cos(2\u03c0x - 1) - 1) + 1| - 1 and g(x) = -2|x - 3| + 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": "Find the number of times the graphs of f(x) = -2sin(1\u03c0(2|x - 1| + 1) - 2) - 3 and g(x) = (3x - 3)/(-3x - 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": "Determine the number of intersection points between the functions f(x) = 2sin(2\u03c0(-4x + 3)) - 3 and g(x) = -5x^2 + 3x + 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 points of intersection exist between the functions f(x) = -5(4x^2 - 4x)^2 + 4 and g(x) = -3x^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 -3sin(1\u03c0((-2x + 3)/(1x + 2))) - 1 = 3x^2 + 4x - 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) = -3cos(1\u03c0(-1x + 3) + 3) + 2 and g(x) = (-3x - 1)/(-1x - 3) 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 points of intersection exist between the functions f(x) = -3(-4x^2 + 1)^2 - 2(-4x^2 + 1) + 4 and g(x) = 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(-3cos(3\u03c0x + 2) - 3) + 1| + 3 and g(x) = (2x)/(2x) in the range -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(-3sin(2\u03c0x - 2) + 3)^2 + 5 = 2x^2 + 5x + 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) = 2(-3cos(2\u03c0x - 3) + 3)^2 - 2(-3cos(2\u03c0x - 3) + 3) - 3 and g(x) = 4x + 3 in the range -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 solutions does the equation (-1(-3cos(1\u03c0x - 2) + 2) + 3)/(-3(-3cos(1\u03c0x - 2) + 2) - 3) = -2x^2 + 1x 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 2|(2sin(1\u03c0x + 3) - 3) + 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": "9", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(2sin(1\u03c0x - 3) + 2)^2 + 5(2sin(1\u03c0x - 3) + 2) - 3 and g(x) = -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": "12", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3sin(2\u03c0(-1x^2 - 2x + 3) - 3) + 1 and g(x) = (-1x - 2)/(1x - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "235", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-3cos(3\u03c0x - 3) - 1) + 3 and g(x) = 1x + 1 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": "How many solutions does the equation -3cos(3\u03c0(2|x| - 2) + 1) + 3 = 1x^2 - 5x - 1 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": "Determine the number of intersection points between the functions f(x) = -1sin(1\u03c0(-3|x| + 3) - 2) - 1 and g(x) = -1x - 3 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 solutions does the equation 1cos(3\u03c0(5x^2 - 1x + 5) - 2) - 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": "102", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-2x^2 - 5x + 2))/(-2(-2x^2 - 5x + 2) + 3) and g(x) = -2|x + 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 5(-1cos(3\u03c0x)) = (-2x - 2)/(-2x - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(-1sin(2\u03c0x + 2) + 2) - 3| + 3 and g(x) = 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": "Find the number of times the graphs of f(x) = 2(-3cos(3\u03c0x - 1) - 3)^2 + 1(-3cos(3\u03c0x - 1) - 3) - 4 and g(x) = 3x^2 + 2x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(3cos(2\u03c0x - 2) - 2)^2 - 2(3cos(2\u03c0x - 2) - 2) - 3 and g(x) = -3x^2 + 3x - 5 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": "How many solutions does the equation -2sin(3\u03c0(-2x^2 + 1x - 3) + 3) - 1 = -2x^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": "22", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(2\u03c0((-2x + 1)/(-2x + 3)) + 1) and g(x) = (1x + 1)/(2x) 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(2cos(1\u03c0x) - 1) + 1)/(1(2cos(1\u03c0x) - 1) + 1) = 5x^2 + 3x + 1 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": "Determine the number of intersection points between the functions f(x) = 1(1cos(2\u03c0x + 1) + 2)^2 + 2(1cos(2\u03c0x + 1) + 2) + 1 and g(x) = 1|x| + 1 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) = -2cos(2\u03c0(-1|x + 1| + 2) - 2) - 3 and g(x) = -1|x - 1| 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": "Find the number of times the graphs of f(x) = 1cos(2\u03c0(-5x) + 2) - 3 and g(x) = 1x - 4 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": "How many solutions does the equation -2sin(3\u03c0((1x - 1)/(1x)) + 2) = 4x^2 + 5x + 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": "Find the number of times the graphs of f(x) = -1sin(2\u03c0(-5x - 5) + 1) and g(x) = (-2x - 2)/(2x + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(2cos(3\u03c0x + 3)) - 1| and g(x) = 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": "24", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|(-2sin(1\u03c0x + 3)) - 2| + 2 = (-3x + 3)/(-2x - 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 points of intersection exist between the functions f(x) = 3(3x^2 - 5x - 5)^2 + 5(3x^2 - 5x - 5) and g(x) = 4x^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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-2sin(3\u03c0x + 3) + 2) - 4 and g(x) = -3x - 3 intersect in the interval [-10, 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 -3cos(1\u03c0(-3x^2 - 4x - 3) - 3) - 3 = (1x - 3)/(-2x + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "513", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(2\u03c0((1x + 1)/(1x)) - 2) + 1 and g(x) = (2x + 3)/(-3x) 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 -2sin(1\u03c0(2x^2 + 1x - 1) + 1) = -2|x + 3| - 1 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) = 1(-3sin(1\u03c0x - 1) + 3)^2 - 2(-3sin(1\u03c0x - 1) + 3) - 5 and g(x) = 1|x + 1| - 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) = 2sin(2\u03c0(-3x^2 - 4x) + 1) + 2 and g(x) = 2|x + 1| + 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": "Find the number of times the graphs of f(x) = -1sin(1\u03c0(-1x^2 - 3) + 1) - 2 and g(x) = -1|x - 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": "Find the number of times the graphs of f(x) = -1sin(3\u03c0((-1x + 1)/(3x - 1)) - 3) + 3 and g(x) = -5x^2 + 3x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(-1cos(3\u03c0x + 3) + 1) - 2 and g(x) = -1|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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(3\u03c0(-1x^2 - 2x + 5) - 1) and g(x) = -1|x| - 1 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 points of intersection exist between the functions f(x) = -1(3|x - 2| - 2)^2 + 2(3|x - 2| - 2) + 3 and g(x) = (-3x + 1)/(1x + 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": "Determine the number of intersection points between the functions f(x) = (-1(2|x| - 2) - 1)/(3(2|x| - 2)) and g(x) = -4x^2 + 2x + 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((-1x + 1)/(-1x + 3))^2 + 5 and g(x) = 3x^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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-3sin(3\u03c0x + 3) + 2) + 2)/(1(-3sin(3\u03c0x + 3) + 2) + 1) = (3x + 2)/(-2x + 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": "Find the number of times the graphs of f(x) = -3|(-3sin(3\u03c0x + 2) - 3) - 1| - 2 and g(x) = -5x - 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": "Determine the number of intersection points between the functions f(x) = (-2(-2cos(2\u03c0x + 3)))/(-3(-2cos(2\u03c0x + 3)) - 2) and g(x) = 3x + 4 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 -5(-2sin(3\u03c0x - 3) + 2)^2 + 5(-2sin(3\u03c0x - 3) + 2) = 1x^2 - 4x 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(1\u03c0(5x^2 + 3x + 3)) - 2 and g(x) = -1x^2 - 3x + 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": "Find the number of times the graphs of f(x) = -3cos(2\u03c0(3|x - 1| + 3) + 2) - 1 and g(x) = (-2x + 1)/(3x + 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": "Determine the number of intersection points between the functions f(x) = -1|(1sin(3\u03c0x) + 3) - 3| - 1 and g(x) = -4x^2 - 2x 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) = -2cos(1\u03c0((1x - 2)/(-3x + 2)) + 3) and g(x) = (3x + 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": "25", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(1sin(2\u03c0x - 3) + 1)^2 + 2(1sin(2\u03c0x - 3) + 1) - 3 and g(x) = -1x + 4 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": "How many points of intersection exist between the functions f(x) = -5(2sin(3\u03c0x - 1) - 3)^2 - 4(2sin(3\u03c0x - 1) - 3) + 2 and g(x) = -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": "30", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2cos(2\u03c0(2x^2 - 1x - 1) - 3) - 2 and g(x) = -1x^2 - 1x - 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": "Determine the number of intersection points between the functions f(x) = -1(-2sin(1\u03c0x + 1) + 1)^2 - 1(-2sin(1\u03c0x + 1) + 1) + 2 and g(x) = (-3x + 1)/(2x - 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": "Find the number of times the graphs of f(x) = 3(1sin(3\u03c0x + 2) + 3) - 2 and g(x) = 4x^2 + 5x - 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) = -1sin(1\u03c0((-2x - 3)/(1x - 3)) + 2) + 1 and g(x) = -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": "How many solutions does the equation (-2(2cos(2\u03c0x - 2)) + 3)/(-1(2cos(2\u03c0x - 2)) + 2) = 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": "16", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1cos(3\u03c0((-2x - 3)/(2x - 1)) - 1) - 3 = (2x)/(1x - 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) = -1sin(3\u03c0((-1x - 2)/(-3x - 1)) - 1) and g(x) = -1x^2 - 2x 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) = 2cos(3\u03c0(4x + 2) + 3) + 3 and g(x) = 4x - 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 solutions does the equation -4(2cos(2\u03c0x + 1) - 2) - 3 = 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": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(-3sin(1\u03c0x - 2) - 1) - 4 and g(x) = -3|x - 3| + 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": "Determine the number of intersection points between the functions f(x) = -4(-2cos(2\u03c0x + 1) - 2) - 5 and g(x) = -3x^2 + 3x + 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": "Find the number of times the graphs of f(x) = -1|(2sin(3\u03c0x - 2) - 3) + 3| + 2 and g(x) = 3|x| - 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": "Find the number of times the graphs of f(x) = 3sin(3\u03c0(4x)) + 1 and g(x) = -1|x + 1| - 1 intersect in the interval [-10, 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(2x^2 + 5x - 5) - 3)/(1(2x^2 + 5x - 5)) and g(x) = 3x^2 - 5x - 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) = 5(1cos(1\u03c0x - 3)) + 1 and g(x) = -3x + 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": "Determine the number of intersection points between the functions f(x) = -1cos(2\u03c0(3x^2 + 1x + 2) + 1) - 2 and g(x) = 3x + 1 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) = 2cos(3\u03c0(-1|x - 3| - 3) + 3) + 1 and g(x) = -1|x - 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": "Find the number of times the graphs of f(x) = -3(-2sin(2\u03c0x + 2) + 3)^2 + 2(-2sin(2\u03c0x + 2) + 3) - 2 and g(x) = -5x - 3 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": "Determine the number of intersection points between the functions f(x) = 3cos(2\u03c0(-3|x - 3| - 3) + 3) - 1 and g(x) = -2|x| - 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": "Find the number of times the graphs of f(x) = -4(-2sin(3\u03c0x + 1) - 1) - 1 and g(x) = 3x^2 + 1x + 4 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 points of intersection exist between the functions f(x) = 3cos(1\u03c0(2x^2 - 3x + 3) - 2) + 1 and g(x) = (3x)/(-2x - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "386", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(2\u03c0(-1|x + 3| - 3) + 2) + 3 and g(x) = (-1x + 3)/(-3x - 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": "Find the number of times the graphs of f(x) = (1(3sin(2\u03c0x - 3) - 2) - 3)/(1(3sin(2\u03c0x - 3) - 2) + 3) and g(x) = (-1x + 3)/(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) = 3sin(2\u03c0(1x^2 - 3x + 4) + 2) - 3 and g(x) = 2x 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": "Find the number of times the graphs of f(x) = (1(-2sin(3\u03c0x - 2) - 3) - 2)/(2(-2sin(3\u03c0x - 2) - 3)) and g(x) = 1|x| - 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) = -1sin(2\u03c0(5x + 3) + 3) - 2 and g(x) = -5x^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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(2\u03c0(1x^2 - 5x) - 2) + 2 = (-3x + 1)/(2x - 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": "Find the number of times the graphs of f(x) = 1sin(2\u03c0(4x + 4) - 1) - 3 and g(x) = 3|x - 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) = -3(3sin(1\u03c0x + 1) + 1) - 3 and g(x) = (-1x + 3)/(-2x - 3) 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) = -1(-2sin(2\u03c0x - 2) - 2) + 1 and g(x) = 2|x - 2| - 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": "Find the number of times the graphs of f(x) = 2sin(3\u03c0((3x)/(-3x - 3)) - 1) - 1 and g(x) = 1x - 1 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": "How many points of intersection exist between the functions f(x) = -2(3cos(3\u03c0x - 1) + 2) + 1 and g(x) = 1x^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": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(-3cos(3\u03c0x + 3)) and g(x) = -1x + 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) = -2|(3cos(1\u03c0x + 1)) + 1| + 2 and g(x) = (2x + 1)/(2x + 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) = 1cos(2\u03c0(3x^2 + 1x - 1) - 2) and g(x) = -3x - 5 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 solutions does the equation (-3(3sin(3\u03c0x) - 2) - 1)/(-1(3sin(3\u03c0x) - 2) + 2) = (-1x - 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": "59", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2sin(2\u03c0(5x - 4)) + 1 = 1x^2 + 5x + 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": "Determine the number of intersection points between the functions f(x) = -3sin(2\u03c0((1x + 2)/(3x + 3)) - 2) and g(x) = 1|x + 2| + 1 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) = -5(2sin(1\u03c0x - 3) + 1) + 3 and g(x) = (3x - 3)/(-2x + 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": "Determine the number of intersection points between the functions f(x) = 2(3sin(3\u03c0x + 1))^2 - 2(3sin(3\u03c0x + 1)) + 3 and g(x) = -1x + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(1sin(3\u03c0x - 2) - 3)^2 + 1(1sin(3\u03c0x - 2) - 3) + 1 = 2|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": "30", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(2\u03c0(-3|x + 3| - 3) - 3) + 2 and g(x) = 1x - 3 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 solutions does the equation -1|(-2cos(3\u03c0x) + 1)| - 2 = -5x^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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(2sin(1\u03c0x + 2) - 2))/(-1(2sin(1\u03c0x + 2) - 2) - 3) and g(x) = 1x - 4 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) = 1|(3sin(3\u03c0x + 2) + 3) - 1| + 2 and g(x) = 4x^2 - 2x + 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 points of intersection exist between the functions f(x) = -3(-3sin(1\u03c0x - 1) + 2)^2 - 1(-3sin(1\u03c0x - 1) + 2) - 1 and g(x) = (3x)/(-1x + 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) = -4(-2cos(3\u03c0x + 2) - 1)^2 + 1(-2cos(3\u03c0x + 2) - 1) - 1 and g(x) = 4x^2 - 2x - 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) = -1(1cos(2\u03c0x + 1) - 3)^2 + 1(1cos(2\u03c0x + 1) - 3) - 2 and g(x) = 5x - 4 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) = -2cos(3\u03c0(-3|x + 2| + 2) + 1) - 1 and g(x) = -3x 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) = -3sin(3\u03c0(-2|x - 2| - 2) - 3) - 1 and g(x) = -4x^2 + 1x + 4 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": "Find the number of times the graphs of f(x) = 3sin(2\u03c0((-3x - 3)/(-3x + 2)) + 3) - 1 and g(x) = -2x^2 + 5x - 3 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|(-1x^2 + 4x + 3) - 1| + 2 and g(x) = -1x - 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(2x + 3) + 2) = (-1x + 2)/(1x + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2cos(3\u03c0(1x^2 - 4x - 2) - 1) + 1 = 1x^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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1sin(3\u03c0(1x^2)) + 2 and g(x) = -1x 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 -1sin(3\u03c0(1x^2 + 3x + 2) + 3) + 3 = (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": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1cos(1\u03c0(2|x + 1| - 1) - 2) - 2 and g(x) = (3x - 1)/(-1x - 3) 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) = -2cos(1\u03c0(-3x + 5) + 2) + 1 and g(x) = (-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": "59", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2sin(2\u03c0(1|x - 2|)) - 1 = (1x + 1)/(-1x - 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 points of intersection exist between the functions f(x) = 2(2sin(1\u03c0x - 2) + 2)^2 - 4(2sin(1\u03c0x - 2) + 2) + 2 and g(x) = (-2x - 2)/(-2x) in the range -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 points of intersection exist between the functions f(x) = -3|(3sin(3\u03c0x + 3) + 3) - 1| + 3 and g(x) = -5x^2 + 5x + 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": "Determine the number of intersection points between the functions f(x) = (3(-1cos(3\u03c0x - 3) + 2) + 3)/(-1(-1cos(3\u03c0x - 3) + 2)) and g(x) = -1x^2 + 5x - 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) = -5(2x^2 + 4x - 1)^2 - 2(2x^2 + 4x - 1) + 1 and g(x) = -1|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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-3sin(2\u03c0x + 3) + 2) - 3)/(1(-3sin(2\u03c0x + 3) + 2)) and g(x) = 3x^2 - 2x + 4 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": "How many points of intersection exist between the functions f(x) = -1sin(1\u03c0(4x^2 - 5x - 1) - 3) - 1 and g(x) = -1x + 1 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) = 4(-2sin(1\u03c0x - 1) - 2) - 4 and g(x) = -3|x + 3| - 1 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) = -4(2sin(1\u03c0x - 3) - 1) - 1 and g(x) = 2|x - 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": "How many points of intersection exist between the functions f(x) = 3(-1sin(1\u03c0x + 3) + 2)^2 - 3(-1sin(1\u03c0x + 3) + 2) - 3 and g(x) = (3x - 2)/(-1x - 3) 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": "How many solutions does the equation -3sin(1\u03c0((-1x - 2)/(1x)) - 2) - 2 = 1x^2 + 5x - 1 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 solutions does the equation (3(-2sin(3\u03c0x) - 1) + 3)/(-1(-2sin(3\u03c0x) - 1) + 2) = 1x^2 - 5x - 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(3sin(3\u03c0x + 2)) + 5 = 1x^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": "16", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(1\u03c0(2x - 4) - 3) and g(x) = 2x^2 - 3x + 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 solutions does the equation (1(-5x^2 - 4x - 1) + 1)/(2(-5x^2 - 4x - 1) + 1) = 3x^2 - 3x + 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(1\u03c0(-4x^2 - 1x + 1) + 1) + 3 and g(x) = 3|x - 3| where -10 \u2264 x \u2264 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) = 5(3cos(1\u03c0x + 2) + 1)^2 - 3(3cos(1\u03c0x + 2) + 1) + 1 and g(x) = 1x^2 - 2x - 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) = -2cos(3\u03c0((-3x)/(3x + 2)) + 1) - 1 and g(x) = (3x + 3)/(-3x - 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(-2cos(2\u03c0x + 2))^2 + 5(-2cos(2\u03c0x + 2)) - 2 = 1|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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3sin(2\u03c0(3x + 3) + 3) and g(x) = 1x + 3 in the range -10 \u2264 x \u2264 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) = -3cos(1\u03c0(-2x^2 + 3x + 1) - 1) + 3 and g(x) = -4x - 4 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 -3sin(1\u03c0(-1x - 4) + 3) + 1 = 3|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": "Determine the number of intersection points between the functions f(x) = -3sin(1\u03c0(-4x^2 - 1) + 2) - 2 and g(x) = (-1x - 2)/(-1x - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "257", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(3\u03c0(3|x - 2| + 1)) - 1 and g(x) = 1x + 4 in the range -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 points of intersection exist between the functions f(x) = -2|(2|x + 3| - 2) + 1| + 1 and g(x) = (-2x - 1)/(-2x + 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": "How many solutions does the equation -2(-3cos(1\u03c0x + 3) - 2) + 5 = 4x^2 + 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(-1sin(3\u03c0x + 1) + 1))/(-2(-1sin(3\u03c0x + 1) + 1) - 3) and g(x) = (-1x - 2)/(1x - 2) 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": "Determine the number of intersection points between the functions f(x) = 3cos(1\u03c0(1x^2 + 2x + 3) - 3) - 1 and g(x) = 4x^2 - 3x + 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) = 3cos(2\u03c0(1x^2 - 2x - 3) + 3) - 3 and g(x) = -4x - 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": "How many solutions does the equation -2cos(3\u03c0(2x + 4) - 3) - 3 = (2x - 3)/(-2x + 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": "Determine the number of intersection points between the functions f(x) = -3sin(3\u03c0((3x + 1)/(-2x + 1)) + 2) - 1 and g(x) = 3x - 5 where -10 \u2264 x \u2264 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) = -2(2cos(2\u03c0x - 2) - 1) - 3 and g(x) = -2x + 5 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) = -2cos(3\u03c0(-3|x + 1| + 3) - 1) - 2 and g(x) = -5x^2 - 5x - 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": "Find the number of times the graphs of f(x) = 3(1sin(3\u03c0x + 2) + 2) + 1 and g(x) = 4x - 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": "Determine the number of intersection points between the functions f(x) = -3|(3x^2 - 3x - 5) + 3| + 3 and g(x) = 5x - 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 -2(-2sin(2\u03c0x - 1) - 1)^2 - 2(-2sin(2\u03c0x - 1) - 1) + 2 = 5x + 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": "Determine the number of intersection points between the functions f(x) = -3cos(1\u03c0(3|x - 2| + 3) - 3) - 1 and g(x) = 1x - 4 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(-3sin(2\u03c0x - 2) - 3)^2 + 1(-3sin(2\u03c0x - 2) - 3) + 3 and g(x) = -4x + 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": "How many points of intersection exist between the functions f(x) = -3cos(2\u03c0(2|x + 1| + 3) + 1) - 2 and g(x) = -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": "18", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-2sin(2\u03c0x + 3))^2 + 4(-2sin(2\u03c0x + 3)) + 3 = (-1x - 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": "39", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-2sin(3\u03c0x - 1) - 1)^2 - 4(-2sin(3\u03c0x - 1) - 1) - 2 and g(x) = 3x^2 + 2x - 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": "Find the number of times the graphs of f(x) = 3(-2cos(2\u03c0x) - 2)^2 + 1(-2cos(2\u03c0x) - 2) + 5 and g(x) = -1x + 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) = 2sin(1\u03c0(-4x + 2) + 3) + 2 and g(x) = 1x^2 - 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 points of intersection exist between the functions f(x) = 4(3cos(3\u03c0x - 2))^2 - 2(3cos(3\u03c0x - 2)) - 5 and g(x) = -5x^2 - 5x - 5 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(3sin(3\u03c0x - 2)) - 3 and g(x) = -3x + 1 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|(1cos(3\u03c0x - 2) + 3) - 1| + 1 and g(x) = -3x^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) = -3cos(2\u03c0(-3|x - 2| - 1) - 3) + 3 and g(x) = -1|x| + 1 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) = -4(3sin(2\u03c0x - 1) + 3)^2 + 3(3sin(2\u03c0x - 1) + 3) + 3 and g(x) = -4x + 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": "Find the number of times the graphs of f(x) = -1(-5x^2 - 5x + 5)^2 - 5(-5x^2 - 5x + 5) + 4 and g(x) = -1x - 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 1(3sin(1\u03c0x) + 2)^2 - 3(3sin(1\u03c0x) + 2) - 1 = -1|x + 2| 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": "How many points of intersection exist between the functions f(x) = (2(3|x + 2| - 1) - 1)/(-1(3|x + 2| - 1) - 1) and g(x) = -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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2cos(3\u03c0(5x)) + 2 and g(x) = 1x + 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) = -4(-3cos(3\u03c0x - 2) + 1)^2 - 1(-3cos(3\u03c0x - 2) + 1) - 2 and g(x) = -2x - 4 intersect in the interval [-10, 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|(-3sin(2\u03c0x + 2)) + 1| - 1 and g(x) = -1x^2 - 3x + 5 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) = -2sin(3\u03c0(5x^2 + 4x + 2) + 2) - 3 and g(x) = -5x^2 + 1x - 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) = (3(-2cos(3\u03c0x)))/(-3(-2cos(3\u03c0x)) + 4) and g(x) = -1x + 4 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": "How many points of intersection exist between the functions f(x) = 1sin(3\u03c0(-4x - 2) + 1) - 2 and g(x) = -2x - 3 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) = (1(-1x^2 + 1))/(-2(-1x^2 + 1) + 1) and g(x) = -1|x + 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) = (2(-2cos(1\u03c0x)) + 2)/(3(-2cos(1\u03c0x)) - 2) and g(x) = -3x + 1 intersect 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) = 3cos(3\u03c0(-2x + 3)) + 2 and g(x) = (3x - 3)/(-2x + 3) 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 points of intersection exist between the functions f(x) = -1sin(2\u03c0(-1|x - 1| - 2) + 2) - 2 and g(x) = -1x - 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) = -2cos(3\u03c0(-3|x - 1| + 3) + 3) + 2 and g(x) = (-1x + 2)/(-3x - 3) 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": "Determine the number of intersection points between the functions f(x) = 2cos(3\u03c0((2x - 1)/(-2x + 3)) - 2) - 3 and g(x) = -2x - 2 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": "Find the number of times the graphs of f(x) = -1sin(2\u03c0(2x^2 + 4x + 3) + 1) - 3 and g(x) = -1x^2 + 4x - 5 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) = 5(-3sin(2\u03c0x - 1) + 3) - 3 and g(x) = -1x + 5 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": "Find the number of times the graphs of f(x) = -3cos(3\u03c0((2x + 3)/(3x - 2)) - 2) - 1 and g(x) = 1x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "71", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-3sin(3\u03c0x - 1) - 3)^2 + 5(-3sin(3\u03c0x - 1) - 3) + 3 and g(x) = 4x^2 + 5x - 5 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) = 3sin(1\u03c0(-3|x - 3| - 3)) + 1 and g(x) = 1|x + 1| - 3 intersect 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(-2cos(3\u03c0x))^2 + 2(-2cos(3\u03c0x)) - 5 and g(x) = (1x)/(-2x - 3) 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) = -1(-1x^2 - 1x + 5)^2 - 2(-1x^2 - 1x + 5) + 5 and g(x) = (2x + 2)/(2x + 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 -1cos(3\u03c0(1x^2 + 2x + 3) - 3) + 1 = 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(5x^2 - 2x - 1)^2 - 3 = (3x - 2)/(2x + 3) 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 -2(1cos(3\u03c0x + 1) - 2) - 1 = -1x + 3 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": "How many solutions does the equation 1|(3sin(2\u03c0x - 3) - 3) + 3| + 2 = 3x^2 - 2x + 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 4(3cos(2\u03c0x + 2) - 2) + 3 = -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": "28", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(1\u03c0(-2x^2 - 4x + 3) - 1) and g(x) = (-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": "402", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(1sin(2\u03c0x - 2) - 1)^2 + 3 = -1x + 4 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 points of intersection exist between the functions f(x) = -3cos(2\u03c0(-5x - 2) - 1) - 3 and g(x) = 2x + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(-1sin(3\u03c0x + 1) - 1) + 3)/(-1(-1sin(3\u03c0x + 1) - 1) - 1) = 2|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": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(4x - 4) + 3| + 3 and g(x) = -3x^2 + 5x - 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) = 2cos(2\u03c0((-1x - 2)/(2x - 3)) - 1) + 2 and g(x) = -3x + 5 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 points of intersection exist between the functions f(x) = -1(-3sin(2\u03c0x + 3) - 1) + 2 and g(x) = (2x + 2)/(1x) in the range -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 1cos(2\u03c0(5x^2 + 3x + 4)) = (-2x - 3)/(-3x) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1076", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(2sin(1\u03c0x) + 2) + 2| + 3 and g(x) = (-3x - 3)/(3x + 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": "Determine the number of intersection points between the functions f(x) = -4(-3|x - 1| + 2)^2 - 1(-3|x - 1| + 2) + 2 and g(x) = -1x - 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(-2sin(3\u03c0x - 1) + 3)^2 + 3(-2sin(3\u03c0x - 1) + 3) + 1 and g(x) = 3|x - 3| - 1 where -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) = 1cos(2\u03c0(-5x^2 + 1x + 3)) + 1 and g(x) = 2|x - 3| where -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 solutions does the equation -1cos(2\u03c0(-4x^2 - 2) + 3) + 1 = (2x - 3)/(-1x - 1) 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": "Find the number of times the graphs of f(x) = (2(3x^2 - 2x - 5))/(1(3x^2 - 2x - 5) + 3) and g(x) = 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": "Determine the number of intersection points between the functions f(x) = 1|(-1sin(2\u03c0x + 1) - 1) + 3| - 2 and g(x) = -1|x - 2| + 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": "How many points of intersection exist between the functions f(x) = (-2(-3sin(1\u03c0x + 1) + 3) - 3)/(-3(-3sin(1\u03c0x + 1) + 3)) and g(x) = 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": "12", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2sin(1\u03c0(-4x - 1) + 1) = -3x^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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(3cos(3\u03c0x - 2) - 1)^2 + 4(3cos(3\u03c0x - 2) - 1) + 1 and g(x) = -4x - 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 points of intersection exist between the functions f(x) = -3|(-4x^2 - 4x + 3) - 3| + 3 and g(x) = (-3x - 2)/(2x - 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 points of intersection exist between the functions f(x) = 2cos(2\u03c0(-4x^2 + 4x - 2) + 2) - 2 and g(x) = -3x + 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 solutions does the equation -2cos(3\u03c0((3x + 1)/(3x - 1)) + 3) + 1 = -5x + 1 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) = -4(-3|x + 2| + 2)^2 - 1(-3|x + 2| + 2) - 5 and g(x) = -3x^2 + 3x - 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 -2(-3cos(2\u03c0x + 2) - 3)^2 - 5(-3cos(2\u03c0x + 2) - 3) = 1|x| - 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) = -3|(-2cos(2\u03c0x + 3) + 2) - 3| + 3 and g(x) = (1x - 1)/(2x - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-3sin(3\u03c0x - 3) + 1)^2 + 2(-3sin(3\u03c0x - 3) + 1) + 4 and g(x) = 5x - 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": "Find the number of times the graphs of f(x) = -3cos(1\u03c0(4x - 3) + 3) + 1 and g(x) = (1x + 2)/(2x - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "75", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3sin(2\u03c0(1|x| + 3) + 1) + 2 and g(x) = (3x)/(1x) 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) = 2|(-2|x - 3| + 2)| - 2 and g(x) = (-1x - 2)/(2x - 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 3(-1cos(1\u03c0x + 2) + 1)^2 - 2 = (-2x - 3)/(-2x - 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) = -3|(-1sin(2\u03c0x - 1) - 3) - 1| + 1 and g(x) = -1|x - 3| - 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": "Find the number of times the graphs of f(x) = 3|(-3|x|) + 2| - 2 and g(x) = -1x + 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 2cos(3\u03c0(1x^2 - 3x - 4) + 3) + 1 = 1x - 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": "Determine the number of intersection points between the functions f(x) = 3sin(2\u03c0(-4x - 1) + 2) - 3 and g(x) = (2x - 2)/(1x - 2) 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) = -4(2cos(3\u03c0x + 2) - 1) + 3 and g(x) = 1x + 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(-2sin(3\u03c0x) - 2) + 5 = 5x + 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": "Find the number of times the graphs of f(x) = 2|(-2sin(3\u03c0x - 2) + 3) + 3| - 3 and g(x) = 1|x + 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3sin(2\u03c0(-2x - 1)) + 2 = (-2x - 2)/(-1x - 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": "Determine the number of intersection points between the functions f(x) = -2cos(3\u03c0(1|x - 3| + 2) - 3) - 1 and g(x) = 1x + 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) = 2sin(1\u03c0(-4x^2 + 3x - 5)) + 2 and g(x) = 2|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": "5", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(3\u03c0((-1x - 3)/(-2x + 3)) - 3) - 3 and g(x) = -1x^2 - 5x 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) = 3sin(2\u03c0(-2x^2 - 4x - 1) + 3) and g(x) = (-1x)/(-2x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "795", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(2\u03c0((-2x)/(-1x - 3))) - 1 = 2x + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "106", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(-1cos(1\u03c0x + 3) - 1) + 3)/(1(-1cos(1\u03c0x + 3) - 1)) and g(x) = -1x^2 - 2x - 2 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": "Determine the number of intersection points between the functions f(x) = -1cos(3\u03c0(-5x - 5) + 2) + 3 and g(x) = 2|x + 1| - 2 where -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 solutions does the equation (3(2cos(3\u03c0x - 2) - 2) + 2)/(3(2cos(3\u03c0x - 2) - 2) + 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": "39", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2cos(1\u03c0((3x - 1)/(-1x - 3))) - 2 and g(x) = (-3x)/(2x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "96", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3sin(1\u03c0(5x - 4) + 2) + 3 and g(x) = (-3x + 3)/(-1x) in the range -10 \u2264 x \u2264 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) = -2sin(1\u03c0(-2|x + 3|) + 1) + 2 and g(x) = -1x - 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(-1x^2 + 4x + 5) - 2)/(-1(-1x^2 + 4x + 5) + 2) and g(x) = -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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2cos(3\u03c0(1x^2 + 2x - 5) + 1) and g(x) = -1|x| + 2 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": "Find the number of times the graphs of f(x) = -3(-2cos(1\u03c0x + 2) + 1)^2 + 4(-2cos(1\u03c0x + 2) + 1) - 3 and g(x) = 5x + 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(-2sin(3\u03c0x) - 1)^2 + 3(-2sin(3\u03c0x) - 1) and g(x) = -3x - 5 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": "How many solutions does the equation -5(-3|x - 2| + 2)^2 - 5(-3|x - 2| + 2) = (-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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(1\u03c0(1x^2 - 3x - 2) - 2) + 2 and g(x) = -1x + 2 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": "Determine the number of intersection points between the functions f(x) = -3(2cos(3\u03c0x + 1) + 1) + 1 and g(x) = -2x - 4 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": "How many points of intersection exist between the functions f(x) = 2(2sin(2\u03c0x + 1)) and g(x) = 2x + 4 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) = -2|(-5x^2 - 5x + 3) + 1| and g(x) = 5x - 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) = 2cos(2\u03c0(5x) - 3) + 3 and g(x) = -2x^2 + 3x + 5 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": "Find the number of times the graphs of f(x) = (-1(-1cos(3\u03c0x + 3) - 1) + 2)/(1(-1cos(3\u03c0x + 3) - 1)) and g(x) = (2x + 1)/(-1x + 3) intersect in the interval [-10, 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) = -2cos(3\u03c0(-2|x + 3| - 3) + 3) - 2 and g(x) = -1x^2 + 4x - 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": "Determine the number of intersection points between the functions f(x) = -3cos(1\u03c0(-3x^2 - 3x + 1) + 1) - 2 and g(x) = (1x - 2)/(2x) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "549", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(2cos(2\u03c0x))^2 + 3(2cos(2\u03c0x)) - 1 = 2x - 5 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": "Determine the number of intersection points between the functions f(x) = 3(-2sin(2\u03c0x - 3) + 2) + 1 and g(x) = -1|x| + 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(5x^2 + 1x + 1) + 3) + 1 and g(x) = 5x - 4 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) = -3cos(3\u03c0(-4x^2 + 4x + 4) + 1) + 2 and g(x) = 3x - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "83", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-3cos(3\u03c0x - 2) - 3)^2 - 3(-3cos(3\u03c0x - 2) - 3) and g(x) = 1|x| - 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) = 1sin(2\u03c0(-5x^2 + 5x - 3) - 1) + 1 and g(x) = (-1x + 2)/(1x + 2) 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) = 3sin(2\u03c0((-1x + 3)/(2x + 3)) + 2) + 1 and g(x) = (-2x + 2)/(2x) in the range -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) = -3sin(3\u03c0((-1x + 3)/(1x)) + 2) - 2 and g(x) = 5x + 2 intersect in the interval [-10, 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) = -3(3sin(2\u03c0x - 1) + 1) + 4 and g(x) = (3x + 1)/(3x - 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(2cos(2\u03c0x + 1) + 2)^2 - 1(2cos(2\u03c0x + 1) + 2) + 2 = 1x - 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": "Find the number of times the graphs of f(x) = -4(-1cos(3\u03c0x + 3) - 1) + 3 and g(x) = -3x^2 + 5x + 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": "Find the number of times the graphs of f(x) = (-2(-3x^2 + 3x + 3) + 2)/(3(-3x^2 + 3x + 3) + 1) and g(x) = 2x^2 - 5x 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(-2cos(2\u03c0x + 1) - 1) + 1)/(-1(-2cos(2\u03c0x + 1) - 1) - 3) and g(x) = 3|x| - 2 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) = 2sin(1\u03c0(4x - 4) - 1) + 1 and g(x) = (-1x + 1)/(2x - 3) 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": "Find the number of times the graphs of f(x) = -1(-3cos(3\u03c0x + 1) + 1)^2 - 5(-3cos(3\u03c0x + 1) + 1) - 4 and g(x) = (2x - 3)/(-2x + 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) = 3sin(2\u03c0(-4x - 1)) - 1 and g(x) = -1x^2 - 5x - 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 solutions does the equation (-2(3cos(3\u03c0x - 1)) - 3)/(1(3cos(3\u03c0x - 1)) + 2) = -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": "54", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(1cos(2\u03c0x + 1) + 1) + 2)/(-3(1cos(2\u03c0x + 1) + 1) - 1) and g(x) = 4x^2 - 1x - 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) = 1|(-1sin(2\u03c0x - 3) - 3) - 1| - 3 and g(x) = (-2x + 3)/(-1x + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(-2sin(2\u03c0x + 3) - 2) - 4 and g(x) = -5x - 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": "How many points of intersection exist between the functions f(x) = (-3(-2cos(2\u03c0x + 2) + 2))/(-3(-2cos(2\u03c0x + 2) + 2) - 3) and g(x) = (2x + 1)/(1x - 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": "How many points of intersection exist between the functions f(x) = -2|(1cos(2\u03c0x) - 3) + 1| + 1 and g(x) = 1x 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) = (2(3sin(1\u03c0x - 3)) - 3)/(-3(3sin(1\u03c0x - 3)) - 1) and g(x) = 1x + 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": "Determine the number of intersection points between the functions f(x) = 1cos(2\u03c0(-5x^2 + 1x) - 1) - 1 and g(x) = (1x + 2)/(-2x - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1380", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(1\u03c0(-5x - 1)) - 1 and g(x) = -1|x - 1| - 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) = -1cos(2\u03c0(-4x + 2) - 2) + 1 and g(x) = 2x^2 + 1x - 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": "Determine the number of intersection points between the functions f(x) = -2sin(1\u03c0(4x^2 + 1x + 5)) + 3 and g(x) = -2x + 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": "How many solutions does the equation 3cos(1\u03c0(3x^2 + 5) + 1) - 1 = (-3x + 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": "497", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(1cos(3\u03c0x + 3) - 1)^2 - 3(1cos(3\u03c0x + 3) - 1) - 2 and g(x) = 5x^2 - 4x + 1 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) = 2cos(3\u03c0(-3x + 5) - 1) + 2 and g(x) = (2x - 3)/(2x + 2) 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": "Find the number of times the graphs of f(x) = (3(1cos(2\u03c0x - 3) + 2) - 1)/(-2(1cos(2\u03c0x - 3) + 2) + 3) and g(x) = 1x^2 - 3x + 3 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) = 2cos(2\u03c0(5x^2 + 3x + 2) + 3) and g(x) = 4x + 3 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": "Determine the number of intersection points between the functions f(x) = 1|(3sin(3\u03c0x - 3) + 1) + 2| - 1 and g(x) = 1|x + 2| + 2 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(1|x| - 2))/(-1(1|x| - 2) + 2) and g(x) = -4x^2 + 4x + 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 1sin(1\u03c0(1x^2 - 3x + 1) - 3) - 2 = 1x + 5 have for x in the interval [-10, 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) = -2sin(1\u03c0((2x - 2)/(-2x - 1)) + 1) - 1 and g(x) = (-2x + 3)/(3x + 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 points of intersection exist between the functions f(x) = 3(-2cos(2\u03c0x - 3) + 1)^2 + 5(-2cos(2\u03c0x - 3) + 1) + 5 and g(x) = 5x + 5 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) = -2|(-3cos(2\u03c0x - 2) + 2) - 2| + 3 and g(x) = -2x^2 + 5x + 4 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) = 1sin(2\u03c0(-4x + 4) + 3) + 1 and g(x) = 1x - 4 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": "Determine the number of intersection points between the functions f(x) = 2cos(3\u03c0(2|x| + 2)) + 2 and g(x) = 2|x + 2| - 1 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": "Find the number of times the graphs of f(x) = 1sin(3\u03c0(-3|x - 1| + 3) + 1) + 3 and g(x) = -1|x| + 3 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 solutions does the equation -1|(1cos(2\u03c0x) - 3) - 1| + 3 = (-2x)/(1x) 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) = (3(2sin(2\u03c0x - 1) + 1))/(2(2sin(2\u03c0x - 1) + 1) + 1) and g(x) = (2x + 1)/(2x - 3) 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 points of intersection exist between the functions f(x) = 3sin(2\u03c0(-4x^2 - 3x - 1) + 3) and g(x) = 3x^2 - 4x 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": "Determine the number of intersection points between the functions f(x) = (3(-3cos(3\u03c0x - 3) + 2) + 2)/(3(-3cos(3\u03c0x - 3) + 2) + 1) and g(x) = 4x^2 + 1x + 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 points of intersection exist between the functions f(x) = 1sin(2\u03c0((-3x + 3)/(-2x - 1)) - 2) - 2 and g(x) = 4x^2 - 1x - 5 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 points of intersection exist between the functions f(x) = 1cos(2\u03c0(2|x + 1| + 1)) - 2 and g(x) = -1x + 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": "Find the number of times the graphs of f(x) = 3(-2cos(1\u03c0x - 3) - 3)^2 + 1(-2cos(1\u03c0x - 3) - 3) - 5 and g(x) = 3x^2 - 4x - 4 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) = 3(3|x + 3| - 3)^2 + 3 and g(x) = 4x^2 + 1x - 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) = 1sin(1\u03c0(-4x - 5) - 2) + 1 and g(x) = -1x - 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 points of intersection exist between the functions f(x) = 1sin(1\u03c0(-4x + 1) - 2) - 3 and g(x) = -2|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": "Find the number of times the graphs of f(x) = -1(3sin(3\u03c0x - 2) - 1) - 1 and g(x) = -2|x - 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": "Determine the number of intersection points between the functions f(x) = 3(-1cos(1\u03c0x + 1) + 2) - 3 and g(x) = (-2x - 3)/(-1x) 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) = 4(1sin(3\u03c0x - 2))^2 - 1(1sin(3\u03c0x - 2)) - 2 and g(x) = 1x^2 - 2x + 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 (-1(3cos(2\u03c0x - 2)) + 2)/(3(3cos(2\u03c0x - 2)) - 2) = (-2x + 3)/(-3x - 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": "How many points of intersection exist between the functions f(x) = -2sin(2\u03c0((-2x - 1)/(2x - 1)) + 1) + 3 and g(x) = 4x + 3 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) = 2cos(1\u03c0((-2x - 1)/(1x + 2)) + 3) and g(x) = 5x^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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(2\u03c0(-3|x + 2| + 3) - 2) + 2 and g(x) = (-1x + 1)/(-1x) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "115", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(2\u03c0(1|x - 1| - 3) + 3) and g(x) = -1x - 5 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(3\u03c0(-3|x + 3| + 3)) - 3 and g(x) = (3x + 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) = 2cos(1\u03c0((-1x)/(-2x + 1))) and g(x) = -1|x| 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": "How many solutions does the equation 2(1sin(2\u03c0x + 1) + 1)^2 - 3(1sin(2\u03c0x + 1) + 1) - 1 = 1|x| - 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": "How many solutions does the equation (-2(-2|x - 2| + 3) + 2)/(3(-2|x - 2| + 3) + 3) = -5x^2 + 4x + 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 1(3cos(2\u03c0x + 2))^2 - 1(3cos(2\u03c0x + 2)) = 5x^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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1sin(3\u03c0(-2x^2 + 1x + 4) + 1) - 3 and g(x) = 3|x - 3| - 3 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": "How many solutions does the equation -3cos(2\u03c0(-5x - 1) - 3) + 2 = (-2x - 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": "193", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1cos(3\u03c0(-5x^2 - 3x + 1) - 2) + 3 and g(x) = 5x^2 + 4x + 1 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(1cos(3\u03c0x) - 1) + 3)/(2(1cos(3\u03c0x) - 1) + 2) and g(x) = (-2x - 2)/(3x + 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|(3sin(1\u03c0x - 3) - 3) + 2| + 2 and g(x) = (-2x)/(3x + 1) 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": "How many points of intersection exist between the functions f(x) = (1(1|x - 1| - 1))/(2(1|x - 1| - 1) - 1) and g(x) = -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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(-1|x + 3| + 1)^2 + 5(-1|x + 3| + 1) + 4 and g(x) = 2|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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1sin(2\u03c0(2x + 5) - 3) + 3 and g(x) = -3x^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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(3cos(1\u03c0x + 3) - 2)| - 2 = (-3x - 3)/(-2x + 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": "Determine the number of intersection points between the functions f(x) = -1(-1cos(2\u03c0x + 1) - 2)^2 and g(x) = (3x - 1)/(-1x - 3) 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 1(3sin(3\u03c0x) + 2) + 1 = (1x + 3)/(3x + 1) have for x 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) = -2cos(2\u03c0(-5x + 3) - 2) - 3 and g(x) = -3|x + 1| - 1 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": "Determine the number of intersection points between the functions f(x) = (-3(3cos(3\u03c0x - 2) + 1) - 3)/(-3(3cos(3\u03c0x - 2) + 1) - 1) and g(x) = -2|x + 3| - 2 where -10 \u2264 x \u2264 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) = 5(-1cos(2\u03c0x + 1) + 3) and g(x) = 2|x - 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 -3cos(1\u03c0(-2|x + 2| + 1)) - 3 = (2x + 3)/(-2x - 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 solutions does the equation 2sin(2\u03c0(-3x - 4) - 1) + 3 = 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": "16", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|(2cos(2\u03c0x + 2) - 1)| + 2 = -5x^2 + 4x + 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(-2cos(2\u03c0x) - 3)^2 + 5(-2cos(2\u03c0x) - 3) - 3 = 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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2sin(2\u03c0(-1|x| - 1) - 3) + 1 = (2x - 1)/(-3x) 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 points of intersection exist between the functions f(x) = -4(2cos(2\u03c0x + 3) - 3)^2 - 1(2cos(2\u03c0x + 3) - 3) + 5 and g(x) = (-2x + 2)/(-1x - 3) 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 5(-2cos(3\u03c0x + 3) + 1)^2 + 2(-2cos(3\u03c0x + 3) + 1) - 1 = (-1x - 3)/(-3x) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "115", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(-1sin(2\u03c0x + 2) + 1) - 3 and g(x) = 2x - 4 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) = (1(-2sin(2\u03c0x + 3) + 2))/(-1(-2sin(2\u03c0x + 3) + 2) + 2) and g(x) = 1x + 4 intersect in the interval [-10, 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 2sin(3\u03c0(3|x - 3| + 3) - 2) - 1 = -1x + 3 have for x in the interval [-10, 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) = -1sin(2\u03c0(1x^2 - 5x - 2) - 1) and g(x) = (2x - 3)/(3x + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "265", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(2sin(2\u03c0x + 3)) - 3 = 5x - 2 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 2cos(2\u03c0(3|x| - 2) - 2) - 1 = (-2x + 3)/(1x) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "96", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1cos(3\u03c0(-1|x + 2| - 3) + 3) + 2 and g(x) = 2|x + 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 solutions does the equation (-2(2cos(2\u03c0x + 1) - 3) + 2)/(-1(2cos(2\u03c0x + 1) - 3) + 2) = (1x - 2)/(1x - 3) 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(-2x^2 - 4x)^2 + 3(-2x^2 - 4x) - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(3\u03c0(-1x^2 - 1x - 3)) and g(x) = 2x^2 + 2x + 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": "Find the number of times the graphs of f(x) = 3cos(3\u03c0((-3x + 2)/(2x + 3)) + 2) + 3 and g(x) = -3x^2 - 1x + 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": "How many points of intersection exist between the functions f(x) = 3cos(2\u03c0(2|x - 2| - 2) + 1) + 2 and g(x) = 1|x + 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": "Find the number of times the graphs of f(x) = 1sin(3\u03c0(2|x| + 2) - 2) + 2 and g(x) = 2|x + 2| - 1 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 solutions does the equation -3sin(3\u03c0((3x - 1)/(1x)) + 3) + 3 = 3x - 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 points of intersection exist between the functions f(x) = -1sin(3\u03c0(1x^2 + 4) + 3) - 3 and g(x) = -2|x - 3| + 1 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 2e^(-2(2|x - 3| - 1)) - 2 = -1|x| + 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 3|(-1sin(2\u03c0x + 1) - 2) + 2| + 1 = 2x - 4 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 -1cos(1\u03c0(2x^2 + 3x + 5) - 3) + 2 = -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": "22", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3sin(1\u03c0(2|x + 3|) + 2) + 2 = (-1x - 3)/(3x + 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": "Find the number of times the graphs of f(x) = -4(1sin(2\u03c0x)) + 3 and g(x) = (-1x - 1)/(-2x) 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) = -1((3x)/(-3x + 2))^2 + 4((3x)/(-3x + 2)) + 3 and g(x) = -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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2cos(1\u03c0((3x)/(-1x - 2)) - 1) = -1|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": "33", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(1\u03c0((-1x - 3)/(2x + 2)) - 3) - 3 and g(x) = -1x - 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": "Determine the number of intersection points between the functions f(x) = 5(3sin(1\u03c0x - 2) - 3)^2 - 1(3sin(1\u03c0x - 2) - 3) - 1 and g(x) = 2|x - 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 2cos(2\u03c0(4x + 5) + 3) - 2 = (-1x + 3)/(3x + 3) 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": "Determine the number of intersection points between the functions f(x) = 2cos(2\u03c0(-1|x - 2| + 2)) - 1 and g(x) = -1x + 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) = 2|(1cos(2\u03c0x - 2) + 2)| - 1 and g(x) = 3|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) = 3|(-3|x - 2|) + 3| - 2 and g(x) = -1x + 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(-2|x - 3| + 3)) = 1x^2 + 4x 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) = 2cos(3\u03c0(5x^2 + 5x - 1)) + 1 and g(x) = 3|x - 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "208", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(3sin(3\u03c0x + 3) + 1) + 2)/(1(3sin(3\u03c0x + 3) + 1) - 3) and g(x) = (-1x)/(3x + 1) 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) = 3cos(3\u03c0(2|x - 1| - 1) - 1) - 3 and g(x) = 1x - 3 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) = 4(-1|x| + 2)^2 - 3(-1|x| + 2) - 5 and g(x) = 2|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(-3cos(1\u03c0x - 1)) - 3)/(-2(-3cos(1\u03c0x - 1)) + 3) = -5x^2 - 5x + 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": "How many solutions does the equation 2sin(3\u03c0((3x - 1)/(-1x)) - 3) + 2 = -5x - 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 (3(-4x^2 + 5x + 5) + 3)/(-1(-4x^2 + 5x + 5) + 2) = 1x^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": "Find the number of times the graphs of f(x) = -2cos(1\u03c0((2x - 1)/(2x + 1)) + 2) - 2 and g(x) = -2x - 2 intersect 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 3cos(1\u03c0(-3|x| + 2) - 2) + 3 = 5x^2 - 5x - 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 (1(-1x^2 + 1) + 1)/(3(-1x^2 + 1) - 2) = -4x^2 + 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 points of intersection exist between the functions f(x) = (-3(3cos(2\u03c0x - 2) + 1) - 2)/(-2(3cos(2\u03c0x - 2) + 1) + 3) and g(x) = -1x - 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": "How many points of intersection exist between the functions f(x) = 3sin(1\u03c0(4x^2 - 2x - 2) - 3) - 3 and g(x) = (3x + 3)/(1x) 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 -2cos(3\u03c0(2x^2 + 4x + 3) + 1) = 5x - 2 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) = 4(-2cos(2\u03c0x) + 1) - 5 and g(x) = 3|x + 3| - 2 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": "Find the number of times the graphs of f(x) = -2sin(2\u03c0(-1x^2 + 3x - 1) - 3) and g(x) = (-1x - 3)/(3x + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "401", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2sin(1\u03c0(4x - 5) + 2) + 1 and g(x) = -2x^2 + 5x 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 (1(2cos(3\u03c0x - 2)) - 1)/(-1(2cos(3\u03c0x - 2))) = (-3x)/(-2x) 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) = -2cos(2\u03c0(5x - 4) + 3) + 3 and g(x) = 5x - 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) = 2cos(1\u03c0(-2|x - 2| + 3) + 2) - 3 and g(x) = (2x - 2)/(-2x) 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": "How many points of intersection exist between the functions f(x) = 3cos(3\u03c0(2x^2 + 3x - 5) + 1) and g(x) = 5x^2 + 2x - 4 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": "Find the number of times the graphs of f(x) = (-2(-2sin(1\u03c0x + 1) + 1) - 3)/(-3(-2sin(1\u03c0x + 1) + 1) + 1) and g(x) = -1|x - 2| + 3 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 solutions does the equation -2(1cos(2\u03c0x - 3) + 3) + 5 = -4x^2 + 3x + 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) = 2(1sin(3\u03c0x - 2) - 1) + 1 and g(x) = (3x - 2)/(2x + 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) = -3sin(1\u03c0(-1x - 2) - 2) + 3 and g(x) = 2|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": "How many points of intersection exist between the functions f(x) = -2cos(1\u03c0(4x + 3) - 2) and g(x) = -3x - 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 points of intersection exist between the functions f(x) = -1|(3x^2 + 2x - 4) + 3| - 1 and g(x) = (2x)/(-1x) 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) = -2cos(2\u03c0(3x^2 + 2x - 5) - 2) + 1 and g(x) = (-3x + 3)/(-1x - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "589", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(1sin(3\u03c0x - 1) - 2) - 2 and g(x) = 1x + 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": "How many points of intersection exist between the functions f(x) = -1sin(1\u03c0(-5x^2 + 4x + 4)) + 2 and g(x) = -2|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": "32", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(3cos(2\u03c0x - 3) + 1) + 1)/(-1(3cos(2\u03c0x - 3) + 1) - 2) and g(x) = 1x + 2 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) = 4(1sin(2\u03c0x)) + 2 and g(x) = (2x - 3)/(1x + 2) intersect 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) = -4(-3sin(3\u03c0x - 2) + 3) + 2 and g(x) = (-1x - 1)/(-3x) 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 solutions does the equation -1sin(2\u03c0(1x - 5) - 1) - 1 = 3|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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(-2cos(1\u03c0x - 3) - 3)^2 - 1 = -1x + 4 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": "Find the number of times the graphs of f(x) = 3(-1cos(1\u03c0x + 1) + 1)^2 + 2(-1cos(1\u03c0x + 1) + 1) - 3 and g(x) = 2x^2 - 2x 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) = -1sin(1\u03c0(-2|x + 1|) + 2) and g(x) = (-2x - 1)/(-2x + 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": "Determine the number of intersection points between the functions f(x) = (2(-3cos(2\u03c0x) + 2) + 1)/(-2(-3cos(2\u03c0x) + 2) + 3) and g(x) = 1x + 4 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 (-3(3x^2 - 1x - 3) - 3)/(2(3x^2 - 1x - 3) - 3) = -3|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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(-3cos(3\u03c0x) - 2)| - 2 and g(x) = 3x + 1 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": "How many points of intersection exist between the functions f(x) = -2|(-3|x + 1| + 1) - 1| - 1 and g(x) = -1x^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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(2cos(1\u03c0x + 3) - 2)^2 + 3 and g(x) = 1|x - 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(3sin(3\u03c0x + 1) + 3) + 1 and g(x) = (3x + 1)/(3x + 2) in the range -10 \u2264 x \u2264 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 -2sin(2\u03c0(-2x^2 + 5x - 4) + 2) + 2 = -1x^2 + 5x + 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|(3cos(2\u03c0x - 2) - 1) - 2| - 3 = -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": "22", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(3cos(1\u03c0x - 2) - 1) + 2| + 1 and g(x) = 1|x - 3| + 1 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": "Determine the number of intersection points between the functions f(x) = 2|(-4x^2 + 1x + 5) + 1| + 1 and g(x) = 4x^2 - 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 points of intersection exist between the functions f(x) = 2cos(3\u03c0((-1x + 3)/(-2x + 3))) and g(x) = 1x^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": "48", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(1|x - 2| - 1) - 3)/(-3(1|x - 2| - 1) + 2) and g(x) = 1x^2 + 2x - 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": "Find the number of times the graphs of f(x) = 2(-1cos(3\u03c0x + 2) - 1)^2 + 2(-1cos(3\u03c0x + 2) - 1) and g(x) = 1|x - 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": "How many solutions does the equation -3(2cos(2\u03c0x - 2) + 1)^2 - 3(2cos(2\u03c0x - 2) + 1) + 1 = 4x + 5 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) = (-2(3sin(3\u03c0x + 3) - 2) - 3)/(1(3sin(3\u03c0x + 3) - 2) + 3) and g(x) = (-3x - 2)/(2x) 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": "How many points of intersection exist between the functions f(x) = 2|(-4x^2 + 3x) + 2| and g(x) = 3x^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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(-2sin(2\u03c0x + 3) - 1) - 2 and g(x) = 1x^2 + 2x - 5 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 -3sin(2\u03c0(4x^2 + 5x - 5) - 2) + 1 = (-1x)/(-1x + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1314", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(3cos(3\u03c0x - 3) + 1) + 2| + 2 and g(x) = -4x^2 - 3x 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) = 1e^(1(-3x^2 - 2x + 3)) + 2 and g(x) = 2|x - 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) = 1cos(3\u03c0(-2|x - 2| + 3) - 2) + 2 and g(x) = 2|x + 3| - 2 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) = -3sin(2\u03c0(3|x - 3| - 3) + 1) and g(x) = 1x^2 + 2x + 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": "Determine the number of intersection points between the functions f(x) = -1(-1sin(2\u03c0x - 1) - 3)^2 - 2(-1sin(2\u03c0x - 1) - 3) + 5 and g(x) = (-2x)/(-3x) 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) = (2(5x^2 - 5))/(3(5x^2 - 5) + 3) and g(x) = 1|x - 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) = -3cos(3\u03c0(3x + 3) - 2) + 3 and g(x) = -1|x - 2| + 2 intersect 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) = -1cos(2\u03c0(-5x - 2) - 2) - 3 and g(x) = -2|x - 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) = -3sin(1\u03c0((3x + 1)/(-1x))) - 2 and g(x) = -2|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": "12", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(-2cos(2\u03c0x) - 2) + 3 = 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(3sin(1\u03c0x - 1))^2 and g(x) = -4x - 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) = -3cos(1\u03c0(-3x^2 - 1x + 4) + 1) - 1 and g(x) = (-3x - 2)/(3x + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "600", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1cos(3\u03c0(-4x - 4) + 3) + 1 and g(x) = 3|x - 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": "Find the number of times the graphs of f(x) = -5((-1x + 1)/(-2x + 3))^2 + 5((-1x + 1)/(-2x + 3)) - 2 and g(x) = -5x^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) = 3(-1sin(2\u03c0x + 2) - 2)^2 - 4(-1sin(2\u03c0x + 2) - 2) + 3 and g(x) = 4x^2 - 4x - 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 solutions does the equation -5(-3sin(2\u03c0x - 1)) - 3 = -4x^2 - 5x + 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|(-2cos(3\u03c0x) + 1) + 2| - 2 and g(x) = -1x^2 - 3x + 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": "Determine the number of intersection points between the functions f(x) = 2|(2cos(2\u03c0x - 1)) + 1| + 1 and g(x) = 3x^2 + 5x - 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|(-2sin(3\u03c0x + 2) + 2) + 1| + 2 and g(x) = 1x + 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": "Determine the number of intersection points between the functions f(x) = (-3(3cos(1\u03c0x - 3) - 1) - 2)/(-3(3cos(1\u03c0x - 3) - 1)) and g(x) = (3x)/(1x - 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": "How many points of intersection exist between the functions f(x) = 2cos(3\u03c0(-2|x| + 1) + 2) and g(x) = (2x - 2)/(2x - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "117", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(3\u03c0((-2x + 3)/(3x + 2))) and g(x) = (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": "73", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(3|x| - 2)^2 - 5(3|x| - 2) - 5 = 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": "How many points of intersection exist between the functions f(x) = -3cos(3\u03c0(2|x - 2| - 1) - 2) and g(x) = 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": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(3cos(3\u03c0x + 1) - 3) + 2| - 3 and g(x) = -4x^2 - 4x + 1 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 1(1sin(3\u03c0x + 3) - 2)^2 - 3(1sin(3\u03c0x + 3) - 2) - 5 = 3|x - 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": "Determine the number of intersection points between the functions f(x) = 2|(2sin(2\u03c0x - 1)) + 1| and g(x) = 1x^2 + 1x 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) = -1sin(1\u03c0(-1x^2 + 4x + 4) - 1) - 2 and g(x) = -1x^2 + 5x - 4 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) = -1sin(3\u03c0(2x - 3) + 1) and g(x) = -1x^2 - 4x - 3 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": "Find the number of times the graphs of f(x) = 1(3sin(2\u03c0x) - 1)^2 - 4(3sin(2\u03c0x) - 1) + 3 and g(x) = 2|x + 1| + 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) = 2cos(2\u03c0((1x - 2)/(-1x - 1)) + 3) - 2 and g(x) = 3x^2 + 3x 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": "Find the number of times the graphs of f(x) = 4(1sin(2\u03c0x - 1)) - 2 and g(x) = (-1x + 2)/(1x) 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": "How many points of intersection exist between the functions f(x) = -2cos(3\u03c0((3x + 2)/(-1x + 2)) - 3) - 3 and g(x) = -3x - 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": "Find the number of times the graphs of f(x) = (-2(-2sin(3\u03c0x - 2) + 1) - 2)/(-1(-2sin(3\u03c0x - 2) + 1) - 2) and g(x) = (3x)/(2x + 3) 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 (1(1sin(3\u03c0x + 2) + 2) + 3)/(1(1sin(3\u03c0x + 2) + 2) - 2) = 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": "41", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2cos(2\u03c0(-3|x + 2| + 2) + 3) + 3 and g(x) = 2x + 5 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": "Determine the number of intersection points between the functions f(x) = 3sin(2\u03c0((-2x - 2)/(-1x))) - 3 and g(x) = -4x - 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": "How many points of intersection exist between the functions f(x) = -1sin(3\u03c0(1x^2 - 3x + 3)) + 2 and g(x) = 3|x| - 2 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 -2cos(2\u03c0(1x^2 - 1x + 2) + 1) + 2 = (3x + 2)/(1x) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "393", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1cos(1\u03c0(-3x^2 - 2x + 2) + 1) + 2 = (-2x - 1)/(-1x - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "389", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-3sin(2\u03c0x + 2) - 3)| - 2 and g(x) = -3|x - 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) = 2sin(1\u03c0(3|x - 1| - 3) - 3) - 1 and g(x) = -1|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": "12", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-2cos(1\u03c0x - 1) + 3) + 2)/(-1(-2cos(1\u03c0x - 1) + 3) + 2) and g(x) = -1x^2 + 4x + 1 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": "Determine the number of intersection points between the functions f(x) = 2|(2cos(1\u03c0x + 2) - 1) + 3| - 3 and g(x) = (1x - 2)/(1x + 1) 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": "How many solutions does the equation 3(1cos(1\u03c0x + 2) + 1)^2 + 2(1cos(1\u03c0x + 2) + 1) + 5 = 1|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": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(2cos(2\u03c0x + 3) - 2)^2 - 5(2cos(2\u03c0x + 3) - 2) and g(x) = 1x 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) = 3cos(2\u03c0(4x^2 - 5) + 2) + 2 and g(x) = -1|x + 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "345", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(-2cos(2\u03c0x + 1) + 3)^2 - 3(-2cos(2\u03c0x + 1) + 3) - 4 and g(x) = 4x^2 - 3x - 1 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) = -2(1sin(2\u03c0x + 2) + 1)^2 + 1(1sin(2\u03c0x + 2) + 1) + 3 and g(x) = 2|x + 2| - 2 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) = 5(-3sin(2\u03c0x + 3) + 2)^2 - 3(-3sin(2\u03c0x + 3) + 2) - 1 and g(x) = 5x + 5 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": "How many points of intersection exist between the functions f(x) = -5(2|x - 1| - 3)^2 - 5(2|x - 1| - 3) - 1 and g(x) = -2|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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(3sin(1\u03c0x + 1) + 1) + 1 and g(x) = 1|x - 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": "Determine the number of intersection points between the functions f(x) = -5(3sin(2\u03c0x - 3))^2 - 1(3sin(2\u03c0x - 3)) - 3 and g(x) = -4x^2 - 1x - 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 solutions does the equation -3(-1cos(2\u03c0x - 3) + 3) + 1 = -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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(-3sin(3\u03c0x - 1) - 1)^2 - 5(-3sin(3\u03c0x - 1) - 1) + 4 = -1x^2 - 3x + 5 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": "Find the number of times the graphs of f(x) = 3(-3cos(2\u03c0x) + 1) + 5 and g(x) = 1|x + 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 solutions does the equation 1(1sin(3\u03c0x) - 2)^2 + 1(1sin(3\u03c0x) - 2) + 3 = 4x^2 - 5x + 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 points of intersection exist between the functions f(x) = -4(-1cos(1\u03c0x - 3) - 2)^2 + 1(-1cos(1\u03c0x - 3) - 2) - 4 and g(x) = -4x^2 + 2x + 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": "Determine the number of intersection points between the functions f(x) = (2(-3sin(3\u03c0x + 3) + 1) - 2)/(3(-3sin(3\u03c0x + 3) + 1)) and g(x) = -2x^2 - 5 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 points of intersection exist between the functions f(x) = -3sin(2\u03c0(3x^2 - 1x - 5) + 3) and g(x) = (2x - 3)/(-1x + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "972", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2cos(1\u03c0(2x^2 - 2x + 2) + 1) + 2 and g(x) = -4x - 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 points of intersection exist between the functions f(x) = -3sin(2\u03c0(1|x + 2| - 3) + 3) + 3 and g(x) = -2|x| + 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) = -2(-3cos(1\u03c0x + 1) + 2)^2 - 2(-3cos(1\u03c0x + 1) + 2) and g(x) = -3|x - 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) = (-2(2sin(1\u03c0x + 2) - 1) + 3)/(-1(2sin(1\u03c0x + 2) - 1) - 1) and g(x) = -4x^2 + 2x + 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": "Determine the number of intersection points between the functions f(x) = 2|(-1x^2 - 4x - 2) + 3| - 2 and g(x) = (3x + 1)/(3x + 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) = 3|(3sin(3\u03c0x + 1) + 1) - 3| and g(x) = (-2x)/(-3x - 1) 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 points of intersection exist between the functions f(x) = (-2(-1cos(2\u03c0x)) - 3)/(2(-1cos(2\u03c0x)) - 1) and g(x) = -4x^2 + 2x - 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) = (-1(-1cos(2\u03c0x + 2)) - 1)/(3(-1cos(2\u03c0x + 2)) + 1) and g(x) = (-2x - 2)/(3x - 2) where -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 solutions does the equation (1(-4x^2 - 4x + 1) - 1)/(-2(-4x^2 - 4x + 1) - 2) = -3|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": "Find the number of times the graphs of f(x) = -3(2sin(2\u03c0x) - 1)^2 - 4(2sin(2\u03c0x) - 1) + 2 and g(x) = (1x - 1)/(-2x + 3) 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) = -5(1cos(2\u03c0x - 3)) and g(x) = -4x - 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": "Find the number of times the graphs of f(x) = 1cos(2\u03c0(5x^2 + 2x + 3) + 2) and g(x) = -3|x - 1| + 3 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": "How many solutions does the equation 2cos(2\u03c0(5x) + 2) - 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": "107", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|(-2cos(2\u03c0x + 3) + 1) + 3| + 2 = 3x^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 points of intersection exist between the functions f(x) = -3|(3sin(2\u03c0x - 3) - 1) - 3| + 1 and g(x) = -2|x + 3| + 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": "Determine the number of intersection points between the functions f(x) = (1(-3cos(3\u03c0x - 1)) - 2)/(1(-3cos(3\u03c0x - 1)) - 3) and g(x) = 2x^2 - 3x + 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 solutions does the equation 2cos(1\u03c0(4x + 4) - 1) - 3 = 2|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": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2sin(2\u03c0(-4x^2 - 3x) - 2) - 1 and g(x) = (3x + 2)/(2x + 3) 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": "How many solutions does the equation 3sin(2\u03c0((-3x + 1)/(-1x + 2)) + 2) + 3 = (-3x + 3)/(3x + 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": "Find the number of times the graphs of f(x) = -2|(3sin(3\u03c0x + 1) - 2)| + 2 and g(x) = 3x - 5 intersect in the interval [-10, 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) = (-1(-3cos(1\u03c0x - 1) - 2) - 1)/(2(-3cos(1\u03c0x - 1) - 2) + 1) and g(x) = -2x^2 - 1x + 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 points of intersection exist between the functions f(x) = 2|(2cos(1\u03c0x + 2) + 2)| - 2 and g(x) = -4x^2 - 4x + 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 points of intersection exist between the functions f(x) = 2cos(1\u03c0(1x^2 + 2x + 4) - 2) - 3 and g(x) = -2|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": "24", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-3cos(3\u03c0x - 3) + 3)^2 - 1(-3cos(3\u03c0x - 3) + 3) + 1 = -1x + 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": "Determine the number of intersection points between the functions f(x) = -3sin(2\u03c0(-3|x + 1| - 3) + 2) and g(x) = 3|x + 1| - 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": "Determine the number of intersection points between the functions f(x) = 1(-1cos(3\u03c0x + 1) - 1)^2 + 5 and g(x) = 1x + 5 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) = -4(2sin(1\u03c0x - 3) + 1) - 2 and g(x) = -1x + 4 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": "Find the number of times the graphs of f(x) = -3cos(3\u03c0((-3x + 2)/(-3x - 3)) - 1) + 1 and g(x) = (-1x)/(-1x - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "71", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(3sin(3\u03c0x + 2) - 3) - 3| + 3 and g(x) = -3|x - 2| - 1 intersect 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 points of intersection exist between the functions f(x) = 3cos(2\u03c0(-1x - 1) - 3) - 3 and g(x) = (3x - 3)/(-1x - 1) in the range -10 \u2264 x \u2264 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) = -1cos(3\u03c0(-4x^2 - 4x + 4) + 3) + 2 and g(x) = (-3x + 3)/(-2x) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1216", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(2cos(3\u03c0x + 2) + 1) + 3)/(2(2cos(3\u03c0x + 2) + 1) + 3) and g(x) = 1x^2 - 4x + 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": "How many points of intersection exist between the functions f(x) = 2(-2sin(1\u03c0x + 2) + 3) and g(x) = 2|x - 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) = 3sin(2\u03c0((2x + 3)/(-2x - 1)) + 3) - 2 and g(x) = 1|x + 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) = 3cos(2\u03c0(-3x^2 - 2) + 1) and g(x) = (-2x + 3)/(-3x + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1145", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3sin(2\u03c0(-1x^2 + 3x) + 3) - 1 and g(x) = -4x^2 - 2x 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) = 4(-2sin(3\u03c0x + 3) - 1)^2 - 4(-2sin(3\u03c0x + 3) - 1) - 3 and g(x) = 3x^2 - 4x + 4 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 points of intersection exist between the functions f(x) = -3|(2cos(1\u03c0x + 3) + 2) - 1| and g(x) = -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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2sin(1\u03c0(-1x^2 - 3x - 2) + 2) + 1 and g(x) = -4x + 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 1cos(2\u03c0(-2|x - 2| - 2) + 2) + 1 = (-1x)/(1x - 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": "Find the number of times the graphs of f(x) = -2|(-3cos(1\u03c0x) + 3) + 2| and g(x) = -1|x - 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": "Determine the number of intersection points between the functions f(x) = 1cos(3\u03c0(-2x - 1)) + 2 and g(x) = 2|x - 2| - 2 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": "How many points of intersection exist between the functions f(x) = 2sin(3\u03c0(-2x - 4) + 2) - 3 and g(x) = 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(-2cos(1\u03c0x + 2) + 2) - 2 and g(x) = (-1x + 3)/(-2x - 3) 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": "How many solutions does the equation 3cos(3\u03c0(-5x^2 + 5)) = -3x^2 - 5x + 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": "Find the number of times the graphs of f(x) = (3(2cos(2\u03c0x - 1) - 3) - 3)/(-1(2cos(2\u03c0x - 1) - 3)) and g(x) = -1x^2 + 1x + 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) = 1sin(3\u03c0(-1|x| - 1) + 1) - 3 and g(x) = -2|x| + 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) = 5(3cos(2\u03c0x + 3)) - 5 and g(x) = -3|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": "28", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1sin(2\u03c0(-4x + 2) + 3) = (2x + 2)/(3x) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "141", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(-2sin(3\u03c0x - 2) - 1)^2 + 5(-2sin(3\u03c0x - 2) - 1) + 3 and g(x) = 5x^2 - 2x - 5 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) = 4(-3sin(1\u03c0x + 1) - 1) and g(x) = 2|x - 3| + 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 solutions does the equation 4(2sin(2\u03c0x) + 1)^2 + 4(2sin(2\u03c0x) + 1) - 4 = 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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(-1|x| + 1)^2 - 1(-1|x| + 1) and g(x) = 2|x| - 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) = 3|(-1sin(2\u03c0x + 2) + 2) - 3| - 2 and g(x) = 1|x| - 2 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) = (1(3sin(2\u03c0x - 3) - 1) + 2)/(1(3sin(2\u03c0x - 3) - 1)) and g(x) = 1|x| + 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": "Find the number of times the graphs of f(x) = -4(2cos(1\u03c0x - 2) - 2)^2 + 5(2cos(1\u03c0x - 2) - 2) + 2 and g(x) = -3|x - 3| + 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 -2sin(2\u03c0(-5x^2 - 5x - 5) + 1) = (2x + 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": "1473", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(2cos(1\u03c0x + 3) - 1) + 2)/(2(2cos(1\u03c0x + 3) - 1)) and g(x) = -5x^2 + 5x + 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": "How many solutions does the equation 2sin(2\u03c0(-3x + 1) - 1) = -1x + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(-2sin(2\u03c0x - 2) + 1) - 5 and g(x) = 2|x - 1| - 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": "Determine the number of intersection points between the functions f(x) = -3cos(2\u03c0(-2x^2 + 1x + 2) + 1) - 3 and g(x) = -3|x + 3| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "101", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3(3cos(1\u03c0x + 2) + 2) - 3)/(3(3cos(1\u03c0x + 2) + 2) - 1) = 2x^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": "13", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3cos(2\u03c0(-3|x - 1| + 1) + 2) - 3 = 3|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": "12", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(3sin(2\u03c0x - 3) + 1) + 1| + 2 and g(x) = (1x - 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": "60", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1cos(3\u03c0(3x^2 + 1x - 4) + 1) + 1 = -3x^2 - 3x + 5 have for x 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 solutions does the equation 1(2sin(2\u03c0x + 3) + 2)^2 + 3(2sin(2\u03c0x + 3) + 2) - 5 = 3|x - 1| - 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": "How many points of intersection exist between the functions f(x) = 2|((3x)/(3x - 2)) - 2| - 1 and g(x) = 2|x + 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) = (-2(1cos(1\u03c0x + 3) + 2))/(-2(1cos(1\u03c0x + 3) + 2) + 3) and g(x) = 5x^2 - 4x + 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": "Find the number of times the graphs of f(x) = 2sin(3\u03c0(5x - 2)) + 3 and g(x) = 2x intersect 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) = -3cos(2\u03c0((1x + 1)/(2x)) + 2) - 2 and g(x) = (1x - 1)/(2x) 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": "Determine the number of intersection points between the functions f(x) = -2(1sin(1\u03c0x - 3))^2 - 5(1sin(1\u03c0x - 3)) - 1 and g(x) = 1x^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) = -1cos(1\u03c0((3x - 1)/(-3x - 2)) - 2) + 1 and g(x) = (-3x - 2)/(2x) 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": "How many points of intersection exist between the functions f(x) = -3sin(2\u03c0(4x^2 - 1x - 5)) + 1 and g(x) = -4x^2 - 2x + 1 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) = 1cos(1\u03c0(2|x - 1| - 1) - 2) + 1 and g(x) = (1x - 1)/(3x + 1) in the range -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|(-3sin(2\u03c0x - 2) + 2) + 1| - 3 and g(x) = -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 solutions does the equation -5(-1cos(1\u03c0x + 3) + 3)^2 - 2(-1cos(1\u03c0x + 3) + 3) + 4 = 5x 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) = 1sin(1\u03c0(-5x - 4) + 1) + 1 and g(x) = 1x - 5 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": "Find the number of times the graphs of f(x) = 3|(3cos(1\u03c0x + 2) + 2) - 2| and g(x) = (2x + 1)/(1x) 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) = 2sin(2\u03c0(4x^2 + 1x + 5)) - 3 and g(x) = -5x^2 + 1x - 1 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": "How many solutions does the equation (-2(-5x^2 + 1x + 2) - 3)/(2(-5x^2 + 1x + 2) + 2) = 3|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": "Determine the number of intersection points between the functions f(x) = 1|(1x^2 - 5x + 1) + 3| - 1 and g(x) = (1x)/(3x) 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) = -2|(1cos(3\u03c0x + 3) - 2) - 1| + 3 and g(x) = -2|x - 2| + 2 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": "Find the number of times the graphs of f(x) = 2|(-2|x + 1| + 3)| + 2 and g(x) = 3|x| + 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) = (-1(-3|x + 1| + 3) - 1)/(3(-3|x + 1| + 3)) and g(x) = 2x^2 - 2x - 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) = -1cos(3\u03c0(2x - 2)) - 3 and g(x) = -2x + 1 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 points of intersection exist between the functions f(x) = -2cos(2\u03c0(-4x^2 + 3x - 4) - 2) + 3 and g(x) = 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": "514", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2sin(3\u03c0(-3x + 2) - 3) - 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": "179", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(3\u03c0(-2x^2 - 3x - 5) - 1) + 3 and g(x) = -4x + 5 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 2cos(1\u03c0(4x - 2) - 2) - 2 = (-3x - 2)/(2x) 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": "Determine the number of intersection points between the functions f(x) = -4(-2sin(2\u03c0x + 3) + 2)^2 + 5(-2sin(2\u03c0x + 3) + 2) and g(x) = -4x + 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) = -2(-5x^2 + 1x + 3)^2 - 1(-5x^2 + 1x + 3) - 3 and g(x) = -5x^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) = (-2(-1sin(1\u03c0x - 3) + 2) - 3)/(3(-1sin(1\u03c0x - 3) + 2) - 2) and g(x) = 1x + 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) = -1cos(1\u03c0(1x^2 + 1x + 3) - 3) and g(x) = (1x)/(3x + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "201", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2sin(2\u03c0(-3|x - 3| - 1) - 3) - 1 = (-1x + 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": "99", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3sin(1\u03c0((3x - 1)/(1x - 2)) + 1) - 1 and g(x) = 1x + 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) = 2cos(2\u03c0(4x^2 - 5) - 1) - 3 and g(x) = (-2x - 3)/(1x) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1164", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(3|x + 1| - 2) + 3)/(2(3|x + 1| - 2) - 1) and g(x) = -1|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 points of intersection exist between the functions f(x) = -1|(-1sin(3\u03c0x - 2) - 3)| + 3 and g(x) = 2x^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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2sin(1\u03c0((-3x + 3)/(-2x + 3)) + 1) + 2 = 2|x| + 1 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) = 2cos(2\u03c0(5x^2 + 1x + 1) - 1) and g(x) = -3|x| + 2 intersect 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 points of intersection exist between the functions f(x) = -3sin(3\u03c0(-3|x - 2|) + 3) + 1 and g(x) = (-2x - 3)/(-1x - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "173", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-3cos(3\u03c0x)) - 2| - 1 and g(x) = -5x + 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": "How many points of intersection exist between the functions f(x) = 5(-2sin(3\u03c0x - 3) + 1) - 1 and g(x) = (1x + 3)/(3x + 1) 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 solutions does the equation 3((-2x - 2)/(3x - 2))^2 - 5((-2x - 2)/(3x - 2)) - 1 = 3|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": "Determine the number of intersection points between the functions f(x) = 2(1sin(3\u03c0x - 2) - 3) - 5 and g(x) = -3|x + 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) = -2cos(1\u03c0((1x + 1)/(-1x - 2)) + 2) - 1 and g(x) = -1|x + 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) = -1cos(3\u03c0(3x^2 + 3x - 4) + 2) + 2 and g(x) = -2|x - 2| + 3 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": "How many solutions does the equation 3cos(1\u03c0(2|x + 2| - 1) - 1) - 3 = 1x 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) = -5(-1sin(2\u03c0x - 2)) - 5 and g(x) = -1x^2 + 5x - 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": "Find the number of times the graphs of f(x) = 3sin(3\u03c0((-3x)/(-3x - 3)) - 3) + 3 and g(x) = 1|x - 2| - 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) = -3cos(2\u03c0(5x + 4) - 1) + 1 and g(x) = -1x^2 - 5x + 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 solutions does the equation (1(-1|x - 2| + 1) + 2)/(3(-1|x - 2| + 1) - 2) = -4x^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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1sin(2\u03c0(2x^2 + 5x - 3) - 1) + 3 = (1x + 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": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3sin(3\u03c0(1x^2 - 3x + 3)) - 1 and g(x) = 2x + 5 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 -2cos(1\u03c0(-5x + 5) - 2) - 3 = 4x + 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) = 1cos(2\u03c0((-1x - 2)/(2x))) - 1 and g(x) = 5x - 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": "Determine the number of intersection points between the functions f(x) = -2|(2sin(3\u03c0x - 3) + 2) + 1| - 1 and g(x) = 2x^2 - 4x - 4 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) = 2sin(1\u03c0(-3|x| - 1) - 2) + 1 and g(x) = (-3x)/(-3x + 4) 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": "How many points of intersection exist between the functions f(x) = (3(-3sin(2\u03c0x + 2) - 1) + 3)/(-2(-3sin(2\u03c0x + 2) - 1) - 3) and g(x) = -3|x + 3| - 2 in the range -10 \u2264 x \u2264 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(2cos(3\u03c0x + 1) - 2) + 3)/(2(2cos(3\u03c0x + 1) - 2) - 1) and g(x) = -1x - 5 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) = 1(1cos(3\u03c0x - 2) - 2)^2 + 1 and g(x) = 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1cos(3\u03c0(-5x - 5) - 3) + 1 and g(x) = (3x - 3)/(3x) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "269", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1cos(2\u03c0(-3x^2 + 5x - 1) - 2) - 2 = 4x^2 - 3x - 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(3\u03c0(3|x + 3| + 1) - 1) - 1 and g(x) = (-1x + 1)/(-1x - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "107", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(2sin(2\u03c0x)) + 1)/(-3(2sin(2\u03c0x)) + 4) and g(x) = -5x 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) = 3sin(2\u03c0(2x^2 - 5x + 5) + 2) - 2 and g(x) = -5x^2 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|(2sin(3\u03c0x + 1)) + 1| - 3 and g(x) = 1x - 5 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 solutions does the equation -1cos(3\u03c0(-3|x + 3| + 2)) + 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": "18", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(3\u03c0(-5x + 1) + 3) = -5x + 5 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) = -3sin(3\u03c0(-1x - 5) + 1) + 2 and g(x) = -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": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3sin(3\u03c0(1x^2 + 4x - 3) - 2) + 2 and g(x) = 1x^2 + 1x - 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": "Determine the number of intersection points between the functions f(x) = 3sin(2\u03c0(2x^2 - 3x + 4) + 2) - 3 and g(x) = 1x + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "317", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(2\u03c0(3|x - 1| + 1) - 3) and g(x) = 1x^2 + 5x + 2 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 (-3(2sin(3\u03c0x - 2) - 1) + 2)/(2(2sin(3\u03c0x - 2) - 1) + 1) = 1x + 1 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 points of intersection exist between the functions f(x) = 2cos(3\u03c0(-1x^2 - 3x + 4) + 1) - 3 and g(x) = 1x - 4 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 solutions does the equation 1sin(3\u03c0(-2x + 1)) = 1x + 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) = 2cos(3\u03c0(-1|x + 2| - 1)) - 1 and g(x) = 1|x + 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) = -3sin(1\u03c0((-2x + 3)/(3x + 2))) + 1 and g(x) = -5x - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(1\u03c0(2x^2 + 2x - 3) - 1) - 1 and g(x) = (-2x + 2)/(-2x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "165", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(4x^2 + 4x - 2)^2 + 5(4x^2 + 4x - 2) + 4 and g(x) = -1x 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(2\u03c0x - 3) + 3) - 4 and g(x) = 2x + 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) = -2cos(3\u03c0(-3|x| - 3) - 3) + 3 and g(x) = (2x + 1)/(2x) 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": "Find the number of times the graphs of f(x) = 5(-3sin(3\u03c0x + 2) + 3)^2 - 2(-3sin(3\u03c0x + 2) + 3) - 2 and g(x) = (3x - 1)/(-3x - 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": "How many points of intersection exist between the functions f(x) = 2cos(2\u03c0((2x + 3)/(1x)) - 1) - 1 and g(x) = (-2x - 2)/(-1x) 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": "Determine the number of intersection points between the functions f(x) = 3cos(2\u03c0(-1x + 2) - 3) - 3 and g(x) = (2x - 2)/(-2x - 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": "How many points of intersection exist between the functions f(x) = 2cos(2\u03c0(-1x + 5) + 1) + 2 and g(x) = 2|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": "How many solutions does the equation -2(-3cos(1\u03c0x - 1) - 2)^2 - 3(-3cos(1\u03c0x - 1) - 2) - 4 = -3|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": "22", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-3sin(3\u03c0x + 3) + 2) - 1 and g(x) = -2|x| + 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": "Determine the number of intersection points between the functions f(x) = (2(-2x^2 - 5x - 1) + 3)/(-3(-2x^2 - 5x - 1) - 3) and g(x) = 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 points of intersection exist between the functions f(x) = -2sin(2\u03c0(-4x^2 - 1x - 2) - 3) - 1 and g(x) = -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": "27", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(-1cos(3\u03c0x + 1) - 2) - 3| + 3 and g(x) = 4x - 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 points of intersection exist between the functions f(x) = 2|(1|x + 1| - 1) - 3| - 2 and g(x) = (1x)/(-2x - 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": "Determine the number of intersection points between the functions f(x) = -2cos(2\u03c0(-1x + 4)) - 1 and g(x) = 1x - 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": "How many points of intersection exist between the functions f(x) = (1(-3sin(2\u03c0x + 3) + 2) - 3)/(-3(-3sin(2\u03c0x + 3) + 2) + 2) and g(x) = -1|x + 2| + 1 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 2sin(3\u03c0(1x + 1) - 3) + 1 = (-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": "7", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(-2cos(3\u03c0x + 1))^2 = -4x - 1 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": "How many points of intersection exist between the functions f(x) = 3(2cos(3\u03c0x - 2) + 1)^2 - 5(2cos(3\u03c0x - 2) + 1) + 5 and g(x) = -5x^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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2sin(3\u03c0(-1x^2 + 1) - 1) - 2 = -2x^2 - 5x - 3 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) = -3sin(1\u03c0(-5x - 4) - 3) - 1 and g(x) = -1x^2 - 1x - 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": "Find the number of times the graphs of f(x) = 2cos(2\u03c0(-1x^2 - 3x - 3) + 1) + 2 and g(x) = (-2x + 2)/(2x - 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 -3cos(2\u03c0(-2|x + 3|)) + 2 = -1x + 5 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) = -2|(-2cos(1\u03c0x) + 3) - 2| + 3 and g(x) = 1x + 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": "Find the number of times the graphs of f(x) = (-1(-4x^2 + 4x + 1) + 1)/(-2(-4x^2 + 4x + 1) - 1) and g(x) = -1x^2 - 4x - 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(3\u03c0(5x^2 - 3x - 1) + 2) + 1 and g(x) = 3x - 1 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": "Find the number of times the graphs of f(x) = 1cos(2\u03c0(-3x^2 + 1x - 4) + 2) - 1 and g(x) = 2x + 4 intersect 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 (-1(-2sin(2\u03c0x - 3) - 2))/(3(-2sin(2\u03c0x - 3) - 2) + 3) = -2x^2 + 2x - 3 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) = (2(-3cos(1\u03c0x - 1) + 1) - 2)/(2(-3cos(1\u03c0x - 1) + 1)) and g(x) = 4x - 4 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": "How many points of intersection exist between the functions f(x) = 1cos(1\u03c0((-2x - 1)/(3x + 2)) + 3) and g(x) = (1x)/(-2x + 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": "Find the number of times the graphs of f(x) = (1(3cos(1\u03c0x + 1) - 2) - 3)/(-2(3cos(1\u03c0x + 1) - 2) - 2) and g(x) = 3|x + 3| - 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(3\u03c0(-4x^2 + 2x + 2) - 1) - 2 = (-2x - 3)/(3x - 2) 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": "How many solutions does the equation -1(-1sin(2\u03c0x + 3) - 3) + 1 = 1x - 4 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) = (3(2sin(2\u03c0x - 2) + 1) + 1)/(-2(2sin(2\u03c0x - 2) + 1) - 2) and g(x) = -2x^2 - 3x + 4 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) = 3(-2cos(2\u03c0x) - 2) + 2 and g(x) = -4x + 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": "Find the number of times the graphs of f(x) = -1|(-3cos(2\u03c0x + 2) - 3) + 3| + 2 and g(x) = 2x - 5 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 -3sin(3\u03c0(4x^2 - 3x) - 2) + 1 = -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": "31", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3sin(2\u03c0((-1x)/(-1x + 2)) - 2) + 1 = 2|x + 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": "How many solutions does the equation -3|(1sin(1\u03c0x + 2) + 1) + 1| + 1 = 1x - 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": "Find the number of times the graphs of f(x) = 1sin(3\u03c0(-4x^2 + 1x - 5) + 3) + 2 and g(x) = 4x - 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": "Find the number of times the graphs of f(x) = -2sin(2\u03c0(3x + 3) + 1) + 2 and g(x) = -3|x - 3| + 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": "Determine the number of intersection points between the functions f(x) = 3sin(1\u03c0((1x - 2)/(2x - 1)) + 3) - 1 and g(x) = -3x - 1 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) = 2cos(3\u03c0(5x - 5) + 2) - 2 and g(x) = (2x - 3)/(3x - 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 solutions does the equation 1(-1cos(3\u03c0x - 1) + 1)^2 - 1(-1cos(3\u03c0x - 1) + 1) + 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": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3cos(3\u03c0((3x)/(-1x - 3)) + 2) - 2 and g(x) = 5x^2 + 4x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}