{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(-1cos(3\u03c0x + 1) + 2)^2 + 4(-1cos(3\u03c0x + 1) + 2) + 5 and g(x) = (-3x - 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": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5((-2(-3e^(-1x)) + 1)/(1(-3e^(-1x)) - 3))^2 - 1((-2(-3e^(-1x)) + 1)/(1(-3e^(-1x)) - 3)) + 1 and g(x) = 1|(4x - 2) - 1| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(-3sin(3\u03c0x - 1) - 1)^2 - 2(-3sin(3\u03c0x - 1) - 1) - 2 = -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": "12", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|((-1(-1sin(1\u03c0(2x^2 + 2x - 3) + 1) - 3) + 1)/(-1(-1sin(1\u03c0(2x^2 + 2x - 3) + 1) - 3) - 1))| + 2 and g(x) = 3((-3(2|x - 1| + 1) - 1)/(2(2|x - 1| + 1) + 1)) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "399", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(3(1(-2|x - 1| + 1) + 4)^2 + 1(1(-2|x - 1| + 1) + 4) + 4) - 1)/(2(3(1(-2|x - 1| + 1) + 4)^2 + 1(1(-2|x - 1| + 1) + 4) + 4) - 1) and g(x) = -3(-5((3x + 2)/(-1x)) + 4)^2 + 4(-5((3x + 2)/(-1x)) + 4) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(-5(-2cos(3\u03c0x + 3) - 3)^2 + 5(-2cos(3\u03c0x + 3) - 3) - 4) + 1 and g(x) = (2(-1|x - 3| + 3) + 3)/(-1(-1|x - 3| + 3) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(-4x + 5) + 2| + 1 and g(x) = (1x)/(2x + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2((-2(2|(-2cos(2\u03c0x + 2) - 3) + 2| - 1) - 2)/(2(2|(-2cos(2\u03c0x + 2) - 3) + 2| - 1) + 1)) + 3 and g(x) = 1cos(3\u03c0((-3(-3x^2 + 4x + 3) - 2)/(-3(-3x^2 + 4x + 3) + 4)) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(2sin(2\u03c0x - 1) + 1) + 5 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": "36", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(3sin(3\u03c0x - 1) + 3) - 3)/(3(3sin(3\u03c0x - 1) + 3) + 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(3x^2 + 3x - 5))/(-2(3x^2 + 3x - 5) - 1) and g(x) = 4x^2 + 4x - 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) = 4(2|(-3x + 1) - 1| - 2)^2 - 1(2|(-3x + 1) - 1| - 2) - 1 and g(x) = -2cos(3\u03c0((2x - 2)/(-2x + 1)) - 1) - 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(3(-2(2|x - 2|) - 4)^2 + 2(-2(2|x - 2|) - 4) - 3) + 3)/(3(3(-2(2|x - 2|) - 4)^2 + 2(-2(2|x - 2|) - 4) - 3) + 2) = 1(1((-1x + 1)/(-3x + 1)) + 5)^2 - 2(1((-1x + 1)/(-3x + 1)) + 5) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(2e^(1x) - 1) + 2)/(-2(2e^(1x) - 1) - 1) 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(2|((1(1x^2 - 2x + 3))/(-1(1x^2 - 2x + 3) - 3)) + 3| - 3) + 3 and g(x) = -1cos(3\u03c0((2(3x^2 - 1x - 5) + 3)/(-3(3x^2 - 1x - 5) + 4)) - 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(2e^(-2(-4(-1|x + 1| + 3) + 5)) - 3) + 2)/(-2(2e^(-2(-4(-1|x + 1| + 3) + 5)) - 3) - 1) and g(x) = -1|((-2(-4x^2 - 4x + 5) + 3)/(-3(-4x^2 - 4x + 5) + 4)) + 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(-3cos(1\u03c0x - 2) + 3) + 1 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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(5((-1x + 1)/(2x + 1))^2 - 5((-1x + 1)/(2x + 1)) - 1) - 3| - 1 and g(x) = -3e^(-2(1x^2 - 4x - 3)) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1cos(3\u03c0(-2((-3x + 3)/(2x - 1)) + 4) - 1) - 3 and g(x) = 1cos(1\u03c0((3x + 1)/(-2x - 1)) - 2) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(4(2x - 4)^2)| + 3 and g(x) = 1(1e^(-1x) - 2) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(1\u03c0(3|x - 3| - 1) + 3) - 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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3((-3x - 3)/(2x - 1))^2 - 4((-3x - 3)/(2x - 1)) + 1 and g(x) = 1x^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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(-3((-3x + 1)/(3x - 1)) + 3)^2 + 2(-3((-3x + 1)/(3x - 1)) + 3) - 2 and g(x) = (2(2|x + 2| - 3))/(2(2|x + 2| - 3) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(-3sin(1\u03c0(-1x^2 + 1x + 5) + 3) + 1) + 2 and g(x) = (1(1|x + 3| - 1) - 2)/(3(1|x + 3| - 1) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "202", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(1sin(3\u03c0(3|x| + 3) + 2) + 3) + 5 and g(x) = -2|(-2cos(1\u03c0x - 2) - 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": "Determine the number of intersection points between the functions f(x) = -3sin(2\u03c0(-2((-3x - 1)/(2x + 3)) - 2) + 3) - 3 and g(x) = (2(-5x^2 + 4x + 2) - 3)/(1(-5x^2 + 4x + 2) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(-3sin(1\u03c0(5(2|x - 2| + 1) - 5)) + 3)^2 - 2(-3sin(1\u03c0(5(2|x - 2| + 1) - 5)) + 3) - 3 = 1sin(3\u03c0(1(2|x - 1| - 2) - 4) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "200", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2sin(2\u03c0((-2x + 2)/(-2x + 3)) - 3) - 1 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(-5x^2 - 4x + 4) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(2|(-1((-1x + 3)/(-3x))^2 + 2((-1x + 3)/(-3x)) + 3) - 1| - 2) - 5 and g(x) = -2cos(3\u03c0(-2((-2x - 3)/(1x - 3)) + 5)) - 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) = (2(3(2|x - 2| - 1)^2 - 3(2|x - 2| - 1) - 4) - 2)/(-3(3(2|x - 2| - 1)^2 - 3(2|x - 2| - 1) - 4)) and g(x) = 1|(-3x^2 + 1x + 4)| + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3cos(2\u03c0(2x + 1) + 1) - 3 and g(x) = -5x^2 + 3x - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((2x + 2)/(2x - 1))^2 + 5((2x + 2)/(2x - 1)) - 1 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1sin(1\u03c0(-1x^2 + 4x + 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": "9", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-2sin(3\u03c0(3x^2 - 4x + 4) - 1) - 1) + 1)/(1(-2sin(3\u03c0(3x^2 - 4x + 4) - 1) - 1) + 2) and g(x) = -1sin(3\u03c0(2|x + 3| - 1) - 1) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "854", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(2\u03c0((-3x + 3)/(-2x)) - 2) - 2 and g(x) = 3|x + 1| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1((-1(-2e^(1x) + 3) - 2)/(2(-2e^(1x) + 3) + 3))^2 - 4((-1(-2e^(1x) + 3) - 2)/(2(-2e^(1x) + 3) + 3)) + 3 and g(x) = (1(-3x - 5) + 1)/(3(-3x - 5) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(-1sin(1\u03c0((-1x)/(-2x - 1)) + 3))^2 - 2 and g(x) = 2|((2x - 1)/(2x - 1)) - 1| - 3 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) = 1|(1((-3(-3x - 2) - 1)/(-2(-3x - 2) + 3))^2 - 1((-3(-3x - 2) - 1)/(-2(-3x - 2) + 3)) + 2) - 1| - 2 and g(x) = -1(2|(2x^2 - 3x + 1) + 2| + 1) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-3(1cos(3\u03c0(1|x| - 2) + 2) + 2)^2 - 2) = 5((1(2e^(1x)) - 1)/(1(2e^(1x)) + 3))^2 + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|((-2(1cos(1\u03c0x - 1) - 1) - 1)/(2(1cos(1\u03c0x - 1) - 1) - 3)) - 1| + 3 and g(x) = 4((1x - 1)/(-3x + 2))^2 - 5((1x - 1)/(-3x + 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-1e^(-1x) + 1)^2 - 3(-1e^(-1x) + 1) + 5 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(1|(1e^(-1x) + 2) + 1| - 2) - 2)/(2(1|(1e^(-1x) + 2) + 1| - 2) + 3) and g(x) = 1(3x^2 + 5x - 4) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(3|((-1(-5x^2 + 4x - 4) + 1)/(1(-5x^2 + 4x - 4) - 2))| - 1) and g(x) = 1|(-1(-1x - 4)^2 + 2(-1x - 4) + 4) - 3| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(1(1|x| - 2) - 3) - 3)/(3(1(1|x| - 2) - 3)) and g(x) = (-2(-3|x| - 1) + 1)/(2(-3|x| - 1) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(3|((1x - 1)/(-1x - 1)) + 3| - 1) and g(x) = 2sin(2\u03c0(5x + 4) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(-2x + 1) - 3| + 3 and g(x) = 5x^2 + 3x intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1sin(1\u03c0(1|(3(5x - 1)^2 - 3(5x - 1)) - 2| - 1) + 2) - 3 and g(x) = 2((1(-1|x - 1|) - 1)/(-1(-1|x - 1|) + 3)) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "77", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|((3(-2(1x^2 - 2x + 3) + 5) + 1)/(2(-2(1x^2 - 2x + 3) + 5) + 3)) - 3| - 3 and g(x) = -1|(-2e^(2((3x)/(-2x + 1))) + 1) - 3| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4((-3(-2|x| - 2) - 1)/(-2(-2|x| - 2) + 3)) - 5 and g(x) = -1(4x^2 - 3x - 1) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(2x^2 + 5x + 2) + 1 = (2x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(-3cos(1\u03c0(1|x - 2| + 3) + 3) - 1) - 2)/(-2(-3cos(1\u03c0(1|x - 2| + 3) + 3) - 1) + 1) and g(x) = -3|(2sin(1\u03c0x - 2) + 1) + 2| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|(-3sin(2\u03c0x - 2) - 3) - 3| + 3 = -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 -1|((-3x - 1)/(-1x - 1)) + 1| + 3 = -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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2((-2(-3|x - 1| + 1) + 2)/(-2(-3|x - 1| + 1) - 2)) + 4 = 3cos(2\u03c0((2x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4((2(-1|(-1x^2 - 4x - 2) - 3| - 3) + 2)/(1(-1|(-1x^2 - 4x - 2) - 3| - 3) - 3)) - 4 and g(x) = -1(-3|(-1cos(2\u03c0x + 2) + 2) + 1| + 3) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(-3cos(2\u03c0(-1|x + 3| - 3) - 3) - 2) + 4 = -2cos(3\u03c0(1|x| - 2) - 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(3|(-1sin(2\u03c0((-3x - 1)/(-1x + 2)) + 3) + 1)|) - 4 and g(x) = -3((-2(-3|x - 3| + 3) - 3)/(2(-3|x - 3| + 3) + 1)) - 4 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": "How many solutions does the equation 3|(5(2cos(2\u03c0x + 2) - 2)) - 1| + 3 = 1|((2x)/(-1x - 2))| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|((-1(-5x^2 - 1x + 4))/(-3(-5x^2 - 1x + 4) + 1)) + 1| + 2 and g(x) = (-1(5x^2) + 2)/(-2(5x^2) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-2(4x^2 - 4x - 1) + 3) - 2)/(-2(-2(4x^2 - 4x - 1) + 3) + 3) and g(x) = -3((-3x + 1)/(3x - 2))^2 + 5((-3x + 1)/(3x - 2)) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(1e^(-1x) - 3) - 5 = -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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3cos(3\u03c0(-1|x + 1| - 1) - 3) - 3 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-1(-1sin(3\u03c0(4x - 1) - 3) + 1)^2 - 4(-1sin(3\u03c0(4x - 1) - 3) + 1) + 3) - 2)/(-1(-1(-1sin(3\u03c0(4x - 1) - 3) + 1)^2 - 4(-1sin(3\u03c0(4x - 1) - 3) + 1) + 3) + 2) and g(x) = 4(5(3|x| - 2)^2 - 3(3|x| - 2) - 4) + 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 points of intersection exist between the functions f(x) = (-1(2e^(1(-2x - 1)) + 1) - 1)/(3(2e^(1(-2x - 1)) + 1) - 2) and g(x) = -3cos(1\u03c0((-2x + 2)/(3x - 2)) - 3) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-1x^2 - 1x - 4) + 3)/(-1(-1x^2 - 1x - 4)) and g(x) = -4x^2 - 1x + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(1sin(1\u03c0x - 3))^2 + 5(1sin(1\u03c0x - 3)) + 2 and g(x) = -1x^2 + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-1sin(3\u03c0(-4(-5x^2 + 2x + 5) + 3) + 3) + 2) + 3)/(2(-1sin(3\u03c0(-4(-5x^2 + 2x + 5) + 3) + 3) + 2)) and g(x) = 2|(4(-2e^(2x) + 3)^2 + 3(-2e^(2x) + 3) - 4) - 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2sin(2\u03c0(2|(-1(-2x + 5)^2 + 2) - 3| + 2)) + 3 = 3(-5((-3x - 3)/(2x))^2 - 1((-3x - 3)/(2x)) + 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3sin(3\u03c0((-2(2(4x + 4)^2 + 3(4x + 4) + 5))/(-2(2(4x + 4)^2 + 3(4x + 4) + 5) + 3)) - 3) - 2 and g(x) = 3e^(-2((-1(2x - 1) + 2)/(-1(2x - 1) - 1))) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((1(-2|x - 1| + 3) + 1)/(2(-2|x - 1| + 3) + 1))^2 - 1 and g(x) = -2|(2x - 3) + 1| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(-3(-4x^2 + 2x + 3) + 4) - 3)/(1(-3(-4x^2 + 2x + 3) + 4)) and g(x) = 1|((3x + 3)/(-2x + 3)) - 1| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1sin(2\u03c0(3(5(1|x + 2| - 1)^2 - 4(1|x + 2| - 1) + 5) + 2) + 2) and g(x) = -1|(2(-1x)^2 - 1(-1x)) + 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(3e^(-1x) - 2)^2 + 2(3e^(-1x) - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(4(3sin(3\u03c0((2x + 1)/(-1x + 3)) + 3) + 2) + 2)| + 1 = 5(-5(3cos(2\u03c0x - 3) + 1)^2 + 4(3cos(2\u03c0x - 3) + 1) + 2) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "107", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(3(-2e^(1x) - 1)^2 - 5(-2e^(1x) - 1) - 2) - 3| and g(x) = -5((-2x)/(-3x - 1))^2 - 1((-2x)/(-3x - 1)) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(1(-3|((-2x + 3)/(-1x + 3))| + 1)^2 + 1(-3|((-2x + 3)/(-1x + 3))| + 1) - 3) - 3 = (2(-3e^(1(2|x + 1| + 3)) + 1) + 1)/(1(-3e^(1(2|x + 1| + 3)) + 1) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(-2(4x^2 - 1x - 4) - 5) + 1)/(3(-2(4x^2 - 1x - 4) - 5) + 3) and g(x) = 1(4x^2 + 5x - 5) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(3|((2(-3sin(3\u03c0x + 2) + 3) - 2)/(1(-3sin(3\u03c0x + 2) + 3) + 1)) - 1| - 2) + 2 and g(x) = (3(3(-3|x + 3| - 1)^2 - 3(-3|x + 3| - 1) + 5) - 2)/(-3(3(-3|x + 3| - 1)^2 - 3(-3|x + 3| - 1) + 5) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(-1cos(2\u03c0((-1(4x^2 + 1x - 5) - 3)/(3(4x^2 + 1x - 5) - 2)) + 1) - 2) + 2| + 1 and g(x) = -1(-3((3x + 1)/(3x - 2)) + 4)^2 - 3(-3((3x + 1)/(3x - 2)) + 4) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-3(-2x + 4)^2 - 3(-2x + 4) + 4) - 3)/(-3(-3(-2x + 4)^2 - 3(-2x + 4) + 4) - 3) and g(x) = 1sin(1\u03c0(4x^2 - 2x + 5)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(3|x| + 3))/(-1(3|x| + 3) + 2) 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(1sin(1\u03c0x + 3))^2 - 1(1sin(1\u03c0x + 3)) - 5 and g(x) = -2|x - 2| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(2\u03c0(-1|((-1(-3x) - 1)/(-1(-3x) + 2)) + 3|) - 2) + 2 and g(x) = 3sin(1\u03c0(5((-3x + 2)/(-1x + 3))^2 + 4((-3x + 2)/(-1x + 3)) + 3) - 2) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "427", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-1|x - 1| - 1)^2 + 5(-1|x - 1| - 1) + 3 = -1x^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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3((-2(-2|(1x^2 - 5x - 1) + 2| - 1) - 3)/(1(-2|(1x^2 - 5x - 1) + 2| - 1))) + 2 and g(x) = -1cos(3\u03c0(3(1x^2 + 4x - 5) + 3) - 2) + 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": "How many solutions does the equation -3((3(3|(1cos(2\u03c0x - 1) + 1) - 3| - 3) + 3)/(2(3|(1cos(2\u03c0x - 1) + 1) - 3| - 3) + 1)) + 1 = (-3(2|(-2x + 4) - 3| - 3) - 3)/(-1(2|(-2x + 4) - 3| - 3) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(2cos(3\u03c0x - 2) + 3)^2 - 3(2cos(3\u03c0x - 2) + 3) - 4 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": "23", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-5(3|x - 3| + 1)^2 - 2(3|x - 3| + 1) + 4) - 2)/(-2(-5(3|x - 3| + 1)^2 - 2(3|x - 3| + 1) + 4) - 3) = 3(2x^2 - 1x - 3) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3cos(2\u03c0(2(5x - 2)^2 - 1(5x - 2) + 2) + 3) + 2 and g(x) = 1|((-2x - 3)/(-2x + 1)) - 3| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "532", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4((3(-3(-1|x - 2| - 1)^2 - 2(-1|x - 2| - 1) + 1) - 1)/(3(-3(-1|x - 2| - 1)^2 - 2(-1|x - 2| - 1) + 1) - 2)) + 4 and g(x) = 1(2((3x - 1)/(-1x + 2))^2 - 5((3x - 1)/(-1x + 2)) - 2) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(2x - 1) + 3)/(-3(2x - 1) + 1) = 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|((2x + 3)/(2x + 2)) - 1| and g(x) = -1|x + 3| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(1sin(3\u03c0x)) - 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": "Find the number of times the graphs of f(x) = (1(1|(3x + 3) + 1| - 1) - 1)/(2(1|(3x + 3) + 1| - 1) + 1) and g(x) = -3|(-1sin(1\u03c0x + 3) + 1) + 2| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(3\u03c0(5x^2 - 4x + 4) - 2) + 3 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": "120", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|((-2x + 2)/(-2x - 3)) + 2| + 2 and g(x) = 4x^2 - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(-3cos(3\u03c0((-1(-2x - 2) + 1)/(-3(-2x - 2) - 2)) + 3) + 2)^2 - 3(-3cos(3\u03c0((-1(-2x - 2) + 1)/(-3(-2x - 2) - 2)) + 3) + 2) - 5 = -2sin(2\u03c0(-1|(1x + 5) + 3| + 3)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3(-3|(-5(1x^2 + 4x - 5) + 3) + 1| + 2))/(2(-3|(-5(1x^2 + 4x - 5) + 3) + 1| + 2) + 3) = 3((2(2x^2 - 5x) - 1)/(2(2x^2 - 5x) + 3)) + 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": "Determine the number of intersection points between the functions f(x) = 2|(3((3(-5x^2 + 2x - 3) - 1)/(-3(-5x^2 + 2x - 3) - 2)) - 1) + 3| + 2 and g(x) = 2((2(1sin(3\u03c0x + 2) + 1) - 1)/(-2(1sin(3\u03c0x + 2) + 1) + 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3sin(3\u03c0(1(1|x + 2| + 1) + 4) - 1) - 1 and g(x) = -3sin(3\u03c0(4x)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(-1|((-1(-2x + 1) + 3)/(3(-2x + 1) + 2)) + 1| + 1)^2 + 1(-1|((-1(-2x + 1) + 3)/(3(-2x + 1) + 2)) + 1| + 1) and g(x) = -2sin(3\u03c0(-3(3x^2 + 2x + 2) + 5) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1455", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(5x^2 + 2x - 4) + 2)/(2(5x^2 + 2x - 4) - 3) 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(-1(-2|((3x + 3)/(-2x + 3)) - 3|) + 3)^2 - 2(-1(-2|((3x + 3)/(-2x + 3)) - 3|) + 3) + 4 and g(x) = -2e^(2(5(1x^2 + 1x - 5) - 4)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(-1((2(2|x - 2| - 1))/(-3(2|x - 2| - 1) + 1)) - 1)^2 - 2(-1((2(2|x - 2| - 1))/(-3(2|x - 2| - 1) + 1)) - 1) - 3 and g(x) = -3|(5(3x^2 - 3x + 2)) - 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": "Determine the number of intersection points between the functions f(x) = -1(3((2x + 3)/(2x - 1)) - 4)^2 + 3(3((2x + 3)/(2x - 1)) - 4) - 1 and g(x) = 1cos(3\u03c0(1x + 2) + 2) - 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 points of intersection exist between the functions f(x) = 2|((-2(-5(2x^2 + 3x + 3) - 2) - 1)/(2(-5(2x^2 + 3x + 3) - 2) + 3))| + 1 and g(x) = -2(3|((-1x + 2)/(-3x - 1)) - 1| - 1)^2 - 2(3|((-1x + 2)/(-3x - 1)) - 1| - 1) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(-2x + 4) - 2)/(3(-2x + 4) - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(5((1x + 2)/(2x + 3)))^2 - 5(5((1x + 2)/(2x + 3))) - 2 and g(x) = 4(1sin(3\u03c0x) + 3)^2 + 3(1sin(3\u03c0x) + 3) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(1|(3x^2 - 2x + 2) + 2| - 2))/(-2(1|(3x^2 - 2x + 2) + 2| - 2) - 2) and g(x) = -3cos(1\u03c0(-4x^2 - 1) + 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(2(2|(1x^2 + 4)|) + 3) - 1)/(2(2(2|(1x^2 + 4)|) + 3) - 3) and g(x) = 1cos(2\u03c0(-3(1|x + 1| + 3) + 3) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-2|(3((-1x - 2)/(-3x - 3))^2 - 4((-1x - 2)/(-3x - 3)) - 2) - 2| - 2) + 5 = -4(2sin(1\u03c0(-3|x + 3|) - 3) - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(-3cos(2\u03c0x - 2) + 3) - 2| - 1 and g(x) = (-2x + 3)/(3x - 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": "Determine the number of intersection points between the functions f(x) = -4(-2|(-3x^2 - 4x + 3) + 2| + 2) - 1 and g(x) = -1((-2x)/(1x - 3)) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(3sin(1\u03c0x - 2) + 1) - 2 and g(x) = -2|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 solutions does the equation -1sin(3\u03c0((1x - 1)/(1x + 3)) - 2) = (1x + 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": "127", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-2e^(2(2x + 4)) - 2) - 1)/(1(-2e^(2(2x + 4)) - 2) + 1) = (-3(2x^2 + 5x - 3) - 3)/(-2(2x^2 + 5x - 3) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(5(2cos(1\u03c0(-3|x - 1| - 1) + 1) + 2)^2 + 5(2cos(1\u03c0(-3|x - 1| - 1) + 1) + 2) - 1) - 2 and g(x) = (-2(-2|(-4x + 5)| + 1) - 2)/(-2(-2|(-4x + 5)| + 1)) 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": "How many solutions does the equation (1(2cos(2\u03c0x - 1) + 2) + 2)/(-1(2cos(2\u03c0x - 1) + 2) - 3) = -1x^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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(-1x + 3) + 2)/(-3(-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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(-5(-3x + 4)^2 - 2(-3x + 4) - 3))/(3(-5(-3x + 4)^2 - 2(-3x + 4) - 3) + 3) = -2(3sin(3\u03c0x + 3) + 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(4(1|(2sin(2\u03c0x - 2) + 1) + 2| - 1) + 3))/(-3(4(1|(2sin(2\u03c0x - 2) + 1) + 2| - 1) + 3) + 4) and g(x) = 3e^(-1(1(-2|x + 2| + 1))) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|((-3(4x - 2))/(3(4x - 2) + 3)) + 3| + 2 and g(x) = 1e^(1(-2|x + 1| + 2)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|((-2x - 3)/(1x - 2)) - 3| + 3 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(3|(-4(-1x^2 - 5x - 4) + 5) - 3|) + 3)/(1(3|(-4(-1x^2 - 5x - 4) + 5) - 3|) - 2) and g(x) = -1|(3e^(1(1x^2 - 2x - 3)) + 3)| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(1|(1(-3x - 4)^2 + 3(-3x - 4) - 1) + 3| - 2) - 1)/(-2(1|(1(-3x - 4)^2 + 3(-3x - 4) - 1) + 3| - 2)) and g(x) = (1(2cos(1\u03c0(1x + 3) - 3) + 1) + 1)/(-2(2cos(1\u03c0(1x + 3) - 3) + 1) + 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": "Determine the number of intersection points between the functions f(x) = 5(2sin(1\u03c0x + 2) - 2) + 3 and g(x) = (-1x - 3)/(3x) 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) = 1((2(-1|(-4x^2 - 2x - 1) + 2| - 2) + 3)/(2(-1|(-4x^2 - 2x - 1) + 2| - 2) + 3)) - 1 and g(x) = (-2(-5(3|x - 2| - 1) - 2) + 1)/(-3(-5(3|x - 2| - 1) - 2) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-1x - 4))/(-1(-1x - 4) - 1) 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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(-1((-1(-2|x - 1| - 1) - 3)/(-3(-2|x - 1| - 1) - 3))^2 - 3((-1(-2|x - 1| - 1) - 3)/(-3(-2|x - 1| - 1) - 3)) - 3) + 4 = -1|(3((-2x + 3)/(-2x)) - 1) - 2| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(3e^(-1(1(1x + 5)^2 - 1(1x + 5))) + 2) - 2| - 1 and g(x) = (2(5(-1x^2 - 2x - 1) + 5) - 1)/(1(5(-1x^2 - 2x - 1) + 5)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(2(-3(2|x - 2|)^2 + 4(2|x - 2|) - 1) + 5) - 1)/(-2(2(-3(2|x - 2|)^2 + 4(2|x - 2|) - 1) + 5) + 3) and g(x) = (-3(2e^(1(-2|x + 3| + 3)) + 1) + 1)/(-2(2e^(1(-2|x + 3| + 3)) + 1) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-2(-2((3x - 3)/(1x + 1)) + 2)^2 + 2(-2((3x - 3)/(1x + 1)) + 2) + 1) - 3| + 2 and g(x) = -1(1e^(-1(-1|x - 1| + 2)) - 2)^2 - 3(1e^(-1(-1|x - 1| + 2)) - 2) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(3|(-3((-3x + 1)/(3x + 2)) + 4) - 2| - 1)^2 - 3(3|(-3((-3x + 1)/(3x + 2)) + 4) - 2| - 1) - 2 and g(x) = (-3(-1(-2sin(2\u03c0x + 1) + 3) + 2) + 3)/(2(-1(-2sin(2\u03c0x + 1) + 3) + 2)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(1(-2|((-1x - 1)/(-1x + 2)) - 2| + 3) - 5)^2 + 5(1(-2|((-1x - 1)/(-1x + 2)) - 2| + 3) - 5) - 4 = 1|((1(2x - 3) + 2)/(1(2x - 3) - 2)) - 2| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(-4(-2e^(-1x) + 3)^2 + 4)| = 1|(-1x^2 - 5x - 4) + 1| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3((3(4x + 5) + 3)/(3(4x + 5) - 2))^2 + 2((3(4x + 5) + 3)/(3(4x + 5) - 2)) and g(x) = 1cos(1\u03c0(5x^2 + 2x - 2) + 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(2(-3cos(3\u03c0(-1|x + 2| + 1) + 1) - 2) + 3) - 2)/(-3(2(-3cos(3\u03c0(-1|x + 2| + 1) + 1) - 2) + 3) - 3) and g(x) = 1cos(1\u03c0(-5(4x^2 + 2x - 5) - 1) - 1) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|(-1sin(2\u03c0(1x + 2) - 2)) + 3| + 1 = 1(2e^(2x)) - 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 -4(5(-2cos(1\u03c0x + 1) + 1) + 2)^2 - 2(5(-2cos(1\u03c0x + 1) + 1) + 2) + 2 = -3(-3|x - 1| + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(2|(-2sin(1\u03c0((2x - 1)/(3x))) - 2) - 3| - 3) + 4 and g(x) = 3|(-1cos(1\u03c0(4x - 1)) + 3)| + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|((-1(3e^(2x) + 3))/(-2(3e^(2x) + 3) + 3)) - 3| + 2 = 1|(3sin(2\u03c0x) - 1) + 1| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-2cos(3\u03c0(-4(5x^2 - 4x)) - 3) + 2) + 2)/(-1(-2cos(3\u03c0(-4(5x^2 - 4x)) - 3) + 2) + 2) and g(x) = 1(2sin(2\u03c0(-1|x - 2| + 1) - 3) + 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "352", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(4((3x + 2)/(-1x - 1))^2 - 1((3x + 2)/(-1x - 1)) - 1) and g(x) = -3(1sin(1\u03c0x - 1) + 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-1((2(3|x + 3| - 2) + 3)/(1(3|x + 3| - 2) + 3))^2 + 4((2(3|x + 3| - 2) + 3)/(1(3|x + 3| - 2) + 3)) - 2) - 3 = -2|(3(-2e^(2x) + 1) + 4) - 3| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|((-2(4(-5x + 3)^2 - 1(-5x + 3) - 4) + 3)/(3(4(-5x + 3)^2 - 1(-5x + 3) - 4) + 2)) + 1| and g(x) = 1|((-1(4x^2 - 2) + 1)/(2(4x^2 - 2) - 1)) + 2| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-1(1|((-3x - 3)/(2x - 3)) - 2| - 1) + 3)^2 - 4(-1(1|((-3x - 3)/(2x - 3)) - 2| - 1) + 3) - 4 and g(x) = -3cos(2\u03c0(-2((-2x - 3)/(-2x - 1)) - 1) - 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(-2cos(1\u03c0x - 3) + 2)^2 - 3(-2cos(1\u03c0x - 3) + 2) + 5 and g(x) = 1x^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 points of intersection exist between the functions f(x) = -2sin(1\u03c0(1((1(2x^2 - 4) + 2)/(-2(2x^2 - 4) + 3)) - 4)) + 2 and g(x) = (-3(1(1x^2 + 2x - 4) + 4) + 1)/(-2(1(1x^2 + 2x - 4) + 4) + 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) = -5(3sin(1\u03c0x) - 1)^2 - 2(3sin(1\u03c0x) - 1) + 5 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": "26", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|((-2(-4x^2 - 5x - 4) - 2)/(-1(-4x^2 - 5x - 4)))| - 3 and g(x) = -1(-3sin(2\u03c0x + 1) + 2)^2 - 3(-3sin(2\u03c0x + 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": "How many points of intersection exist between the functions f(x) = -1e^(2((-1(5x^2 - 5x - 4) - 2)/(-3(5x^2 - 5x - 4) - 2))) + 2 and g(x) = (2(1sin(3\u03c0x) + 2) - 3)/(2(1sin(3\u03c0x) + 2) + 3) in the range -10 \u2264 x \u2264 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) = (2(2(-2e^(1(-2|x| + 3)) + 1)) - 1)/(2(2(-2e^(1(-2|x| + 3)) + 1)) + 3) and g(x) = -4((3(3x^2 - 1x - 4) + 2)/(-1(3x^2 - 1x - 4) - 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": "Find the number of times the graphs of f(x) = 3(-2|(-4((1x - 1)/(2x)) - 4) + 3| + 2)^2 + 1(-2|(-4((1x - 1)/(2x)) - 4) + 3| + 2) - 4 and g(x) = (1(-1sin(3\u03c0(2|x + 1| + 3) + 2) - 2))/(-2(-1sin(3\u03c0(2|x + 1| + 3) + 2) - 2) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(-2(-1|((-1x + 1)/(-2x + 1)) + 1| - 1) + 3)^2 - 1 and g(x) = 2(5((3x + 3)/(-2x + 3))^2 - 1((3x + 3)/(-2x + 3)) - 1) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(1(1sin(3\u03c0(2x + 4) - 3))^2 - 3(1sin(3\u03c0(2x + 4) - 3)) - 1) + 2| - 2 and g(x) = (-1(1sin(2\u03c0(-1x + 4) + 2) - 2) - 2)/(3(1sin(2\u03c0(-1x + 4) + 2) - 2) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4((-1(-1|(2e^(-1x)) + 1| + 1) + 3)/(2(-1|(2e^(-1x)) + 1| + 1) + 1)) + 4 and g(x) = 1cos(3\u03c0(5(-3|x - 2| + 2) + 2)) where -10 \u2264 x \u2264 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) = 3sin(1\u03c0(3(-2x - 2)^2 + 5(-2x - 2) + 2) + 3) + 3 and g(x) = 3(2sin(3\u03c0x - 3) + 3) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "453", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-1|(5x)| + 1) + 3)/(2(-1|(5x)| + 1) + 2) and g(x) = -4(3|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": "How many points of intersection exist between the functions f(x) = 1cos(3\u03c0((-2(3(-5x + 2)^2 - 3) - 2)/(-2(3(-5x + 2)^2 - 3))) + 3) - 1 and g(x) = 3cos(3\u03c0(-5((3x + 3)/(3x)) + 2) + 2) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(1x + 3) - 1)/(2(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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5((-2(1|(-2x + 5) - 2| - 3) + 3)/(-2(1|(-2x + 5) - 2| - 3) - 1))^2 + 5((-2(1|(-2x + 5) - 2| - 3) + 3)/(-2(1|(-2x + 5) - 2| - 3) - 1)) + 3 = (3(3(4x - 2)^2 - 2(4x - 2) - 1) + 3)/(-1(3(4x - 2)^2 - 2(4x - 2) - 1) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|(-3cos(3\u03c0x + 2) - 2) + 1| + 1 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": "Find the number of times the graphs of f(x) = 2|(1(-3x^2 - 1x + 4) - 4) + 1| - 3 and g(x) = 3sin(2\u03c0(2|x + 2| - 1) + 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": "Determine the number of intersection points between the functions f(x) = -3(-4(3|((-2x - 3)/(3x + 3)) - 1| - 2)^2 - 1(3|((-2x - 3)/(3x + 3)) - 1| - 2) + 1) - 4 and g(x) = 2sin(1\u03c0(-1(1|x - 3| + 3)^2 - 5(1|x - 3| + 3) - 2)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(3cos(3\u03c0(1((1x - 1)/(-3x + 1))^2 + 1((1x - 1)/(-3x + 1)) - 2) + 3) + 2) - 2| + 3 and g(x) = -3(-2cos(3\u03c0(2|x| - 1) + 2) + 1) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "124", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(3|(4(-1x + 3)^2 + 2(-1x + 3) - 4) - 2| - 3) + 1)/(-1(3|(4(-1x + 3)^2 + 2(-1x + 3) - 4) - 2| - 3)) = 1sin(1\u03c0(-2|(-4x^2 + 2x + 4)| - 1) - 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1((-2(5x) + 3)/(2(5x) + 2))^2 - 1((-2(5x) + 3)/(2(5x) + 2)) - 5 and g(x) = (-2(-3x^2 + 4x - 4))/(-1(-3x^2 + 4x - 4) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3cos(3\u03c0(-2x + 1) + 2) 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": "19", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(-2x^2 - 5x - 1) + 2 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(-3|x + 2| + 1) + 2)/(1(-3|x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(2|((-1x)/(2x - 3)) + 1| - 3) + 1 = 1(5x + 3)^2 + 5(5x + 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-4x^2 - 2x + 4) - 1)/(3(-4x^2 - 2x + 4) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(-5(-3sin(3\u03c0x))^2 - 3(-3sin(3\u03c0x)) + 3) + 3| + 3 = (1(-2x^2 + 4x - 1) - 2)/(1(-2x^2 + 4x - 1) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(-4(2sin(2\u03c0(5x - 4) - 3))^2 + 3(2sin(2\u03c0(5x - 4) - 3)) + 3) - 3| - 2 and g(x) = (2(-1sin(3\u03c0(-2|x - 3| - 3) - 2) + 2) + 3)/(3(-1sin(3\u03c0(-2|x - 3| - 3) - 2) + 2) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "400", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|((-2x)/(-3x - 1)) + 2| - 2 = 1x^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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-1|(1x - 1) + 1| + 2) - 1)/(2(-1|(1x - 1) + 1| + 2)) and g(x) = (2(-1cos(3\u03c0x + 1) + 2) + 3)/(-2(-1cos(3\u03c0x + 1) + 2) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|((2(-1(-4x^2 - 5x) + 3) - 2)/(-1(-1(-4x^2 - 5x) + 3) + 2)) + 2| - 2 and g(x) = -1(-2cos(1\u03c0(-3|x - 2| - 3) + 1) - 3)^2 + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(1(-2|x| + 1)^2 - 5(-2|x| + 1) - 2) + 3)/(2(1(-2|x| + 1)^2 - 5(-2|x| + 1) - 2) - 1) and g(x) = 2(-4x - 4)^2 - 5(-4x - 4) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5((3(-2|(2x - 2) + 2| + 3))/(-1(-2|(2x - 2) + 2| + 3) - 1))^2 + 3((3(-2|(2x - 2) + 2| + 3))/(-1(-2|(2x - 2) + 2| + 3) - 1)) + 4 and g(x) = -5(-3|(-3x^2 - 3x - 5)| + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(-2cos(1\u03c0x + 1) - 2) + 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|((1(3(-2x - 4)^2 - 1) - 2)/(-2(3(-2x - 4)^2 - 1) - 2)) - 1| - 2 and g(x) = -1|(-2((2x + 3)/(-2x)) + 5)| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(-1(3|(-1x - 1) - 2| + 2)^2 + 4(3|(-1x - 1) - 2| + 2) + 4) - 1)/(2(-1(3|(-1x - 1) - 2| + 2)^2 + 4(3|(-1x - 1) - 2| + 2) + 4) + 3) = -3((1(-1|x|) - 2)/(-2(-1|x|) + 1))^2 + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(-2sin(1\u03c0x + 2) - 3) + 3| + 2 and g(x) = 1x^2 - 5x + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(-1|(5(-3cos(2\u03c0x + 3) - 3) + 5) + 2| + 2) - 3)/(-1(-1|(5(-3cos(2\u03c0x + 3) - 3) + 5) + 2| + 2) + 2) and g(x) = -2|(2cos(3\u03c0(-2x^2 - 2x + 5) + 2)) + 1| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "78", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(-1(-1e^(-2((-3x)/(-1x))) + 3) - 3) + 1| - 1 and g(x) = 1e^(-2(-1((-3x - 2)/(-3x - 1))^2 + 5((-3x - 2)/(-3x - 1)) - 4)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(-1|(1e^(-1x) - 2) - 2| + 1) and g(x) = -1sin(3\u03c0((2x + 3)/(-3x + 4)) - 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(3x + 4))/(2(3x + 4) + 3) = 1|x - 1| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1sin(3\u03c0(-1x - 4) - 1) 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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(1cos(3\u03c0(-2|((-2x - 1)/(2x + 2)) - 2| - 3) - 1) + 2)^2 + 5 and g(x) = (-3(-3e^(-2(3x)) + 1) + 2)/(3(-3e^(-2(3x)) + 1) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3sin(3\u03c0(2((-2x - 2)/(1x - 3))^2 - 4((-2x - 2)/(1x - 3)) - 5) + 3) - 2 and g(x) = (-1(1e^(2x) + 1) + 1)/(2(1e^(2x) + 1) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "474", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(-1|x + 3| + 1)^2 - 3(-1|x + 3| + 1) + 5 and g(x) = -5x intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(2(-4(3sin(2\u03c0x - 3) + 2)^2 - 2(3sin(2\u03c0x - 3) + 2) + 5) - 1) + 2)/(2(2(-4(3sin(2\u03c0x - 3) + 2)^2 - 2(3sin(2\u03c0x - 3) + 2) + 5) - 1) - 1) and g(x) = 5(-1(-1|x - 1| + 1)^2 + 3(-1|x - 1| + 1) + 3) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(1|(-1(-1x^2 + 4x + 2) + 2) + 1| - 1))/(-3(1|(-1(-1x^2 + 4x + 2) + 2) + 1| - 1) - 3) and g(x) = 3(-1cos(1\u03c0((-3x + 1)/(2x - 1)) - 3) + 3)^2 - 2(-1cos(1\u03c0((-3x + 1)/(2x - 1)) - 3) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5((-1(2(2sin(3\u03c0x - 1) - 1) - 3) - 3)/(1(2(2sin(3\u03c0x - 1) - 1) - 3) - 3))^2 + 3((-1(2(2sin(3\u03c0x - 1) - 1) - 3) - 3)/(1(2(2sin(3\u03c0x - 1) - 1) - 3) - 3)) and g(x) = 2(-5(1|x - 1| - 1)^2 - 5(1|x - 1| - 1) - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-1sin(1\u03c0x) + 1)^2 - 3 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": "16", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(-1sin(1\u03c0x + 3) + 2)^2 - 5(-1sin(1\u03c0x + 3) + 2) + 1 = 2|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": "18", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(-2|((-1x + 1)/(-2x - 2)) + 2| + 2) - 3 and g(x) = 2cos(1\u03c0(-2x^2 + 3x)) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-2(5(1|x + 3| - 3)^2 - 5) + 5) - 2)/(2(-2(5(1|x + 3| - 3)^2 - 5) + 5)) and g(x) = 1(-2e^(-1(-1x - 1)) - 3)^2 + 4(-2e^(-1(-1x - 1)) - 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": "Find the number of times the graphs of f(x) = -1((1(3sin(2\u03c0(2|x - 2| - 2) - 3) + 1) - 3)/(2(3sin(2\u03c0(2|x - 2| - 2) - 3) + 1) - 1)) - 2 and g(x) = 1(-1|(-1e^(1x) + 3) + 1| + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-2cos(1\u03c0(2|x - 3| - 2) + 3) + 2)^2 - 5(-2cos(1\u03c0(2|x - 3| - 2) + 3) + 2) - 1 and g(x) = -1e^(1(2|x + 1|)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(1(-1(-1|x| - 1)^2 - 3(-1|x| - 1) - 4) + 3) + 3)/(3(1(-1(-1|x| - 1)^2 - 3(-1|x| - 1) - 4) + 3) - 2) and g(x) = 5(3|(3e^(2x) - 3) - 3| - 1)^2 - 1(3|(3e^(2x) - 3) - 3| - 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(3((-3x + 2)/(-1x - 2))^2 + 1((-3x + 2)/(-1x - 2)) - 5) + 1 and g(x) = -2(1|x - 1| + 2) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-1(1|(-1cos(3\u03c0x + 1) - 3) + 3| + 2)^2 + 4(1|(-1cos(3\u03c0x + 1) - 3) + 3| + 2) - 3) + 2)/(-2(-1(1|(-1cos(3\u03c0x + 1) - 3) + 3| + 2)^2 + 4(1|(-1cos(3\u03c0x + 1) - 3) + 3| + 2) - 3) + 1) = -2(-4((3x - 2)/(-1x - 1))^2 + 2((3x - 2)/(-1x - 1)) - 4) + 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 2sin(2\u03c0(3x + 2) + 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": "12", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1cos(2\u03c0(2|(-1(1x^2 + 5x - 2) + 4) + 3| - 2) + 1) - 3 = (-3(5(3e^(1x) + 1)^2 - 3(3e^(1x) + 1) - 1) - 1)/(1(5(3e^(1x) + 1)^2 - 3(3e^(1x) + 1) - 1) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "707", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(2|x - 2|) - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5((-2(-3|x - 2| + 2) - 3)/(-1(-3|x - 2| + 2) - 2)) + 4 and g(x) = -1(-3x^2 - 2x - 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4((-3x + 3)/(-3x + 1)) and g(x) = 3|x - 2| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2e^(2((-1x)/(-1x))) 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(1\u03c0(-1(2|((-2x - 3)/(-1x - 3)) + 3| - 3)^2 - 4) - 2) + 2 = (1(4(2|x + 3| + 1) + 4) - 2)/(1(4(2|x + 3| + 1) + 4) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "251", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(1((-2x - 2)/(-3x))^2 - 4((-2x - 2)/(-3x)) - 5) - 4 = 4(-1cos(3\u03c0x - 3) + 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|(-2(-1((-2x + 1)/(-2x - 1)))^2 - 2(-1((-2x + 1)/(-2x - 1))) + 2) - 3| - 1 and g(x) = -3(2(-1|x - 2| - 3)^2 - 4(-1|x - 2| - 3) + 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3e^(-1(-5(-4((3x + 2)/(1x + 1)) - 2)^2 + 3)) - 2 and g(x) = -2((2(1|x + 2| - 2) + 3)/(3(1|x + 2| - 2) - 2))^2 - 4((2(1|x + 2| - 2) + 3)/(3(1|x + 2| - 2) - 2)) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(-1cos(1\u03c0(-1|x + 1| - 1) + 2) + 2) + 1)/(-1(-1cos(1\u03c0(-1|x + 1| - 1) + 2) + 2) + 2) and g(x) = -3e^(-1((3x - 2)/(-1x - 1))) - 2 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": "Determine the number of intersection points between the functions f(x) = (-1(-1sin(1\u03c0(1|(5x^2 - 1x + 5) - 1| + 3) - 1) - 1) - 1)/(-3(-1sin(1\u03c0(1|(5x^2 - 1x + 5) - 1| + 3) - 1) - 1) + 2) and g(x) = -2cos(1\u03c0(-1|((-2x - 3)/(-1x - 3)) + 1| - 3) - 2) 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": "Determine the number of intersection points between the functions f(x) = 1|(1x + 1) - 1| - 2 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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-3cos(1\u03c0(4x^2 + 4x + 4) - 3) + 2) + 4 and g(x) = 3((3x - 2)/(-2x))^2 + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(-3(-4((-1x + 1)/(2x + 2))^2 - 1((-1x + 1)/(2x + 2))) - 4) - 3| - 2 and g(x) = (-1(-1sin(3\u03c0(-3x^2 - 1x - 1) + 1)))/(1(-1sin(3\u03c0(-3x^2 - 1x - 1) + 1))) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(3x + 5) + 3| - 2 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(3cos(3\u03c0x - 3) - 1) + 1)/(-3(3cos(3\u03c0x - 3) - 1) + 2) and g(x) = 2|x - 2| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(-1x^2 - 5x - 2) - 1| - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|((-1x + 1)/(-2x + 3)) - 2| + 2 = -3x^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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(3|(2(4x)^2 + 5(4x) - 3) + 3| - 1) + 1)/(3(3|(2(4x)^2 + 5(4x) - 3) + 3| - 1)) = (1(-3|(1cos(1\u03c0x + 3) + 1) - 1| - 3) - 1)/(3(-3|(1cos(1\u03c0x + 3) + 1) - 1| - 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(-1cos(1\u03c0x + 1) + 1)^2 - 5(-1cos(1\u03c0x + 1) + 1) + 3 = 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(-3|(-2x) + 3| + 1) - 1)/(-1(-3|(-2x) + 3| + 1) + 2) and g(x) = 1sin(1\u03c0(1|x + 1| + 3) - 2) - 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) = -1((3(-1e^(-1x) + 2) + 3)/(-1(-1e^(-1x) + 2) - 2)) - 2 and g(x) = 3cos(3\u03c0(2x + 5) - 3) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "111", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(-1x - 4) - 3)/(2(-1x - 4) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2e^(2(-1|x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-3|(-2e^(-2x) - 1) + 1| + 1) - 2)/(-2(-3|(-2e^(-2x) - 1) + 1| + 1) + 3) and g(x) = -2|(5x) - 3| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(1|(-3cos(3\u03c0(1x + 4)) + 3) + 2| - 1) - 3)/(3(1|(-3cos(3\u03c0(1x + 4)) + 3) + 2| - 1) - 1) = 1|((-2(2x + 3) - 3)/(1(2x + 3) + 1)) + 2| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(-2|(-1(4x - 3)^2 - 4(4x - 3) + 4) - 2| + 1) - 1)/(1(-2|(-1(4x - 3)^2 - 4(4x - 3) + 4) - 2| + 1) - 2) and g(x) = (1(-2(-1e^(2x) - 2)^2 - 4(-1e^(2x) - 2) + 3) + 2)/(-3(-2(-1e^(2x) - 2)^2 - 4(-1e^(2x) - 2) + 3) - 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) = 1((2(-3e^(-2x) + 2) + 3)/(-1(-3e^(-2x) + 2)))^2 + 4((2(-3e^(-2x) + 2) + 3)/(-1(-3e^(-2x) + 2))) and g(x) = 3cos(3\u03c0(2|x + 1| + 2) + 2) + 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": "Find the number of times the graphs of f(x) = -2((-3(-1|(-4x^2 - 1) + 2| + 1) + 1)/(-1(-1|(-4x^2 - 1) + 2| + 1) - 3)) - 4 and g(x) = -2|(2((-3x + 1)/(2x + 1)) - 2) + 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(-4(3sin(2\u03c0x - 3) - 2) - 5) - 3| - 2 and g(x) = -4(1|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) = 1cos(3\u03c0(-3|x - 2| - 2) - 2) and g(x) = (-2x)/(1x - 3) 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 solutions does the equation -5(1((-2(1|x - 1| + 3) - 1)/(1(1|x - 1| + 3) - 3))^2 - 4((-2(1|x - 1| + 3) - 1)/(1(1|x - 1| + 3) - 3))) - 3 = -3|(3(1x^2 + 4x + 4) + 4) + 1| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(3cos(2\u03c0(-4(-4x^2 - 3x - 1) + 5)) + 1) - 1)/(-3(3cos(2\u03c0(-4(-4x^2 - 3x - 1) + 5)) + 1) - 2) and g(x) = (-1(4(-4x - 5)^2 + 5(-4x - 5) + 3) - 3)/(-3(4(-4x - 5)^2 + 5(-4x - 5) + 3)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-3cos(1\u03c0x)) + 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": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(-3cos(3\u03c0x - 1) + 1) + 1 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": "61", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1cos(1\u03c0(-2|((-1x - 2)/(2x + 2)) + 2| + 1) + 2) = 3|((3x + 1)/(-1x + 3)) - 2| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(-2cos(3\u03c0((-3x - 1)/(2x + 3)) - 2)) - 5 and g(x) = (2(2cos(3\u03c0x - 3)) + 3)/(-2(2cos(3\u03c0x - 3)) - 2) 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) = -2((-1(3|(-1cos(3\u03c0x - 3) + 3) - 2| - 1))/(2(3|(-1cos(3\u03c0x - 3) + 3) - 2| - 1))) - 2 and g(x) = (3(-1(-1sin(1\u03c0x + 3) + 1)^2 + 3(-1sin(1\u03c0x + 3) + 1) - 2) - 3)/(3(-1(-1sin(1\u03c0x + 3) + 1)^2 + 3(-1sin(1\u03c0x + 3) + 1) - 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3((3(3|x| - 1) + 2)/(1(3|x| - 1) - 2))^2 + 4((3(3|x| - 1) + 2)/(1(3|x| - 1) - 2)) + 3 and g(x) = 1|(-3sin(2\u03c0x - 3)) + 3| + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1((3(-5(-2|x - 3| + 3) - 2) - 1)/(3(-5(-2|x - 3| + 3) - 2) + 2))^2 + 3((3(-5(-2|x - 3| + 3) - 2) - 1)/(3(-5(-2|x - 3| + 3) - 2) + 2)) - 2 and g(x) = 3cos(3\u03c0(3|(-5x - 1) + 3|) + 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|((-1(-1x + 1) + 3)/(-2(-1x + 1) + 1)) + 2| - 1 and g(x) = 4(1sin(1\u03c0x - 1) + 1)^2 + 3(1sin(1\u03c0x - 1) + 1) - 5 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(4(2(2|x + 2| + 3)^2 + 4(2|x + 2| + 3)) - 4) + 3)/(-2(4(2(2|x + 2| + 3)^2 + 4(2|x + 2| + 3)) - 4) + 3) = 3(1|(-1x + 5) + 2|)^2 - 1(1|(-1x + 5) + 2|) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(2cos(3\u03c0(-3|x|) + 1) + 3) - 3)/(-2(2cos(3\u03c0(-3|x|) + 1) + 3) + 3) = 1(-2|x - 2| - 2) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "159", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(1cos(1\u03c0x - 1) + 1) + 3)/(-3(1cos(1\u03c0x - 1) + 1) + 2) = -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": "10", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(1e^(-2(2x^2 - 5x - 2)) - 1) = 3(-3|x + 2| - 1)^2 + 4(-3|x + 2| - 1) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2((2(2x + 2) + 1)/(2(2x + 2) - 3))^2 - 3((2(2x + 2) + 1)/(2(2x + 2) - 3)) - 4 = 1sin(3\u03c0(4x - 4) + 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|((-1(-1(-2x + 1)^2 + 5(-2x + 1) + 5) - 3)/(-1(-1(-2x + 1)^2 + 5(-2x + 1) + 5) - 1))| - 1 and g(x) = -2cos(1\u03c0(-5((-3x - 3)/(2x - 1))^2 + 4((-3x - 3)/(2x - 1)) + 3) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "180", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(4(-2|x + 2| + 3) - 2)^2 + 1(4(-2|x + 2| + 3) - 2) + 4 = (-3(1cos(3\u03c0x + 1) + 2) + 1)/(2(1cos(3\u03c0x + 1) + 2) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|((3(-4x) - 1)/(3(-4x) - 2)) + 2| - 1 = -5(1e^(1x) - 3)^2 + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3(2(-4x^2 - 2x + 4) - 2) + 1)/(-2(2(-4x^2 - 2x + 4) - 2) + 3) = 2cos(1\u03c0(-5x^2 - 5x - 5) - 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "985", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(-3(2((1x + 3)/(-2x)) - 4)^2 - 3(2((1x + 3)/(-2x)) - 4) + 2) + 1| + 2 and g(x) = -5((1(-3sin(3\u03c0x - 2) - 3) - 3)/(-2(-3sin(3\u03c0x - 2) - 3) + 3))^2 + 2((1(-3sin(3\u03c0x - 2) - 3) - 3)/(-2(-3sin(3\u03c0x - 2) - 3) + 3)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1sin(1\u03c0(-3x^2 - 3x + 3) + 1) + 1 and g(x) = -4x^2 + 4x + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(1\u03c0(-1x^2 - 2x - 4) + 3) - 3 and g(x) = -4x^2 + 4x - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(2e^(1(2x^2 - 1x - 5))) + 2| - 3 and g(x) = -2|(4x^2 - 1x - 1) - 1| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(3|(1x + 4) + 1| - 3) + 3)/(1(3|(1x + 4) + 1| - 3) + 2) and g(x) = (3(1x + 2))/(1(1x + 2) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(1(3|((2x)/(3x - 2)) - 3| - 2)^2 + 4(3|((2x)/(3x - 2)) - 3| - 2) + 2) + 4 and g(x) = (-2(-3(3|x - 1| - 1) - 3) + 2)/(-1(-3(3|x - 1| - 1) - 3) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1((2(-1x^2 - 4x - 1) - 2)/(1(-1x^2 - 4x - 1) + 2)) - 3 and g(x) = (2(3|x - 3| - 3) - 2)/(1(3|x - 3| - 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3sin(2\u03c0(3|x - 2| - 2) + 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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(3\u03c0(-2|(-3x + 5) - 2| - 2) + 2) + 1 and g(x) = 2cos(3\u03c0(3x^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": "1162", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(1|(3(2x + 4)^2 - 5(2x + 4) + 3) - 1|) - 2)/(-2(1|(3(2x + 4)^2 - 5(2x + 4) + 3) - 1|) + 3) and g(x) = (3(-2|(-3x) - 3| - 3))/(-1(-2|(-3x) - 3| - 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(2e^(1(-1(5x + 1)^2 + 2(5x + 1) + 2)) - 2) + 1)/(3(2e^(1(-1(5x + 1)^2 + 2(5x + 1) + 2)) - 2) - 2) and g(x) = (-2(-3(-3e^(-1x) - 2) - 1) + 3)/(-3(-3(-3e^(-1x) - 2) - 1) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(1sin(2\u03c0(3x - 2) - 3) - 1) + 3| - 2 and g(x) = (-1(1x^2 - 3x) + 3)/(-1(1x^2 - 3x) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-5(3cos(2\u03c0(4x + 3)) - 1)^2 + 3(3cos(2\u03c0(4x + 3)) - 1) + 3) + 1)/(-1(-5(3cos(2\u03c0(4x + 3)) - 1)^2 + 3(3cos(2\u03c0(4x + 3)) - 1) + 3) - 1) and g(x) = 3cos(3\u03c0((2(4x^2 - 4x - 5) - 1)/(3(4x^2 - 4x - 5) - 2)) + 2) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(1|(5(-4x + 1)^2 + 4(-4x + 1) - 4)| - 3) + 3)/(-2(1|(5(-4x + 1)^2 + 4(-4x + 1) - 4)| - 3) - 2) and g(x) = -2cos(1\u03c0((3(3x^2 - 1x - 1))/(-1(3x^2 - 1x - 1))) - 3) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(1cos(2\u03c0x + 2) - 1) - 3)/(-1(1cos(2\u03c0x + 2) - 1) + 3) 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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(2\u03c0((1(-2|(-1x + 3) - 2| + 2))/(2(-2|(-1x + 3) - 2| + 2) + 1)) + 1) and g(x) = -1|(1cos(2\u03c0((2x - 1)/(-2x - 1)) - 2) - 2)| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-1|x - 1| + 2) + 1)/(3(-1|x - 1| + 2) - 1) = -3x + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(1sin(1\u03c0(-2|x| - 1) + 2) - 1) - 4 and g(x) = (2(3|x - 2| - 2) - 2)/(-1(3|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) = 2|((-1x + 1)/(-3x + 1)) - 2| + 1 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(2((2x + 3)/(1x - 2))^2 + 1((2x + 3)/(1x - 2)) - 5) - 1 and g(x) = (-2(-3|x - 3| + 1) + 3)/(1(-3|x - 3| + 1) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(-2|((3x)/(-1x)) - 1| + 1)^2 + 5(-2|((3x)/(-1x)) - 1| + 1) - 1 and g(x) = (2(-2sin(1\u03c0x) + 2) - 1)/(3(-2sin(1\u03c0x) + 2) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(-3cos(1\u03c0(4(-3|x - 2|) - 2) - 2) - 1)^2 + 2(-3cos(1\u03c0(4(-3|x - 2|) - 2) - 2) - 1) + 5 = -2((2(-3x - 5) + 2)/(2(-3x - 5) + 1))^2 - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "476", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-2(3|(2x^2 - 3x - 1) + 1| - 3) + 4) + 2)/(-3(-2(3|(2x^2 - 3x - 1) + 1| - 3) + 4) + 4) and g(x) = (-1(-1(3e^(1x) - 3)^2 - 5(3e^(1x) - 3) + 1) - 1)/(2(-1(3e^(1x) - 3)^2 - 5(3e^(1x) - 3) + 1) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(1\u03c0(-3x^2 - 2x + 2)) and g(x) = (2x - 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": "601", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2cos(3\u03c0(1|(1x + 4) + 2|) + 2) and g(x) = -2(1|x - 1| - 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(2|x + 1|) - 4 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(1cos(1\u03c0x - 3) + 1) + 2 = -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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1((1x + 3)/(-2x - 1))^2 + 2((1x + 3)/(-2x - 1)) - 2 = -1x^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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-1(4x + 2)^2 - 3(4x + 2) + 5) + 2)/(2(-1(4x + 2)^2 - 3(4x + 2) + 5) + 1) and g(x) = 1(3cos(2\u03c0x - 3) + 3)^2 + 3(3cos(2\u03c0x - 3) + 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(-3((2x - 1)/(-1x + 3)))| = 5(-1cos(3\u03c0x)) + 1 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) = 4(1x^2 + 5x + 3) and g(x) = -5x^2 - 2x in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1((3x - 2)/(2x - 3)) - 5 = 2x have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2e^(-2(2|(-2(-2x^2 + 5x + 5) + 1)| + 2)) - 1 and g(x) = -1(-1|(1sin(2\u03c0x - 2) + 2)|) - 4 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) = (2(4(-2|(-2x^2 + 2x + 5) - 1| + 3) + 5) + 2)/(2(4(-2|(-2x^2 + 2x + 5) - 1| + 3) + 5) + 2) and g(x) = (1(3|(-2x - 5)| - 1) - 1)/(-2(3|(-2x - 5)| - 1) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(-5(2e^(1x) - 2) + 2)^2 - 3(-5(2e^(1x) - 2) + 2) + 2 and g(x) = 3|((3x - 1)/(-3x - 1)) - 2| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|((-1(3(5x^2 + 1x - 1) + 1) - 2)/(2(3(5x^2 + 1x - 1) + 1) - 3)) - 3| - 3 = (3(-5(2|x - 2| - 1) - 4) - 3)/(2(-5(2|x - 2| - 1) - 4) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(-2(3((1x)/(3x)) + 2)^2 + 4(3((1x)/(3x)) + 2) - 4)| - 2 and g(x) = 1|(3(2x)^2 - 3(2x) - 4) + 2| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3((-2x + 2)/(-1x + 3)) - 4 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(3(-3|(1x^2 + 3x - 4) + 3| + 1) + 3) + 2)/(-1(3(-3|(1x^2 + 3x - 4) + 3| + 1) + 3) - 3) and g(x) = -5(-3cos(3\u03c0(-1x^2 - 2x - 5) - 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "606", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(1|((2(4x - 2) - 3)/(3(4x - 2) - 1)) - 2| - 1)^2 + 2(1|((2(4x - 2) - 3)/(3(4x - 2) - 1)) - 2| - 1) - 2 and g(x) = 3|(-3sin(2\u03c0(1x^2 + 5x - 5) - 1) - 1) + 2| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(-3(-1|x + 3|) - 5) + 3)/(-3(-3(-1|x + 3|) - 5) - 3) and g(x) = (-1(-3|x - 2| + 2) - 2)/(-2(-3|x - 2| + 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-2|x + 2|) + 2)/(3(-2|x + 2|) + 1) 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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|(2e^(1(3x + 2)) + 2)| - 2 and g(x) = -5(2cos(1\u03c0x + 3) + 3)^2 + 5(2cos(1\u03c0x + 3) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(2((1x)/(1x - 2))^2 - 5((1x)/(1x - 2)) + 2) + 3| - 3 and g(x) = -2cos(2\u03c0(-1|x + 3| - 2) + 1) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(2|(5((-1x)/(-1x - 1))^2 + 1((-1x)/(-1x - 1)) + 2) + 3| - 3) - 4 and g(x) = 2(3|(-4x - 1) - 3| + 2)^2 + 5(3|(-4x - 1) - 3| + 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(3cos(2\u03c0(2|((-3x - 1)/(-1x + 2)) + 1| + 3) - 1) + 1) + 2 and g(x) = -2(-1sin(1\u03c0(1|x + 2| + 3) - 2) - 2)^2 - 4(-1sin(1\u03c0(1|x + 2| + 3) - 2) - 2) - 5 intersect 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(-1|(2cos(1\u03c0(1x + 1) + 1) - 1)|)^2 - 1(-1|(2cos(1\u03c0(1x + 1) + 1) - 1)|) + 2 and g(x) = 3|(-2cos(2\u03c0((1x - 1)/(-3x - 2)) + 2) - 1) + 1| - 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": "Find the number of times the graphs of f(x) = 3cos(1\u03c0(-4((2x - 2)/(1x + 1))^2 - 2) + 1) - 2 and g(x) = -1(1sin(1\u03c0x - 2) - 1)^2 + 3(1sin(1\u03c0x - 2) - 1) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "130", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(2|(-5((1x - 1)/(3x + 3)) - 4) + 2| - 2)^2 - 4(2|(-5((1x - 1)/(3x + 3)) - 4) + 2| - 2) + 3 = (-2(4(-2cos(1\u03c0x + 2) + 1) - 1) - 3)/(3(4(-2cos(1\u03c0x + 2) + 1) - 1) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(-3x^2 - 4x + 1) + 3 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(-2sin(2\u03c0(3x^2 + 3x - 2)) + 1) + 5 and g(x) = -1sin(2\u03c0(-1x - 2) + 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1148", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2((2(-3e^(2(2x^2 - 5x + 1)) + 1))/(2(-3e^(2(2x^2 - 5x + 1)) + 1) + 3)) + 1 and g(x) = (-1(2(3cos(1\u03c0x - 2) - 1) + 2) + 2)/(-2(2(3cos(1\u03c0x - 2) - 1) + 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-3sin(2\u03c0(-5x - 5) + 1) + 2) + 2)/(-2(-3sin(2\u03c0(-5x - 5) + 1) + 2) + 1) and g(x) = 1sin(2\u03c0(3|x - 1|)) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "138", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(-4(4x^2 + 1x - 4) - 2) - 1| + 2 and g(x) = -3sin(3\u03c0(-1|x| - 2) - 3) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-2sin(2\u03c0x))^2 - 3(-2sin(2\u03c0x)) + 4 = (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": "33", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(1sin(3\u03c0(-2(-4x - 2)^2 + 4(-4x - 2) - 4) + 3) + 1) - 3| - 1 = 2|(-3((2x + 1)/(-1x)) + 3) - 3| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "146", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(-3|((-2(4x^2 - 2x + 1) + 2)/(-1(4x^2 - 2x + 1) + 3)) + 3| - 1) - 2 = -2(5(1|x - 1| - 2) - 1)^2 + 2(5(1|x - 1| - 2) - 1) - 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": "Determine the number of intersection points between the functions f(x) = 2(1|(4x + 4) + 2| - 1)^2 - 4(1|(4x + 4) + 2| - 1) - 2 and g(x) = -1(1e^(2x) - 1) - 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 solutions does the equation (-3(-1|(-1(5x - 4)^2 + 4(5x - 4) - 2) + 3| + 3) + 3)/(-1(-1|(-1(5x - 4)^2 + 4(5x - 4) - 2) + 3| + 3) + 2) = -1|((3(1cos(3\u03c0x + 2) - 2) + 1)/(1(1cos(3\u03c0x + 2) - 2) - 2)) + 2| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(4(3cos(1\u03c0x + 1)) - 1) - 3| - 3 and g(x) = 4(-1cos(2\u03c0x + 3) + 1)^2 + 1(-1cos(2\u03c0x + 3) + 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(-4(-1x^2 + 5) - 2) + 2)/(1(-4(-1x^2 + 5) - 2)) and g(x) = -3|((1x + 2)/(-3x + 2)) - 1| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-1sin(1\u03c0x - 2) + 1) - 2)/(-1(-1sin(1\u03c0x - 2) + 1) + 2) and g(x) = -1|x - 3| + 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) = -1sin(3\u03c0(-1|x + 2| + 3) + 2) + 3 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1((1(-5x^2 + 5x + 4) - 3)/(2(-5x^2 + 5x + 4) - 1)) - 3 and g(x) = (3(-2cos(3\u03c0x + 3) + 1) - 2)/(-1(-2cos(3\u03c0x + 3) + 1) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(-1sin(3\u03c0((-3(3|x| - 2) - 2)/(-1(3|x| - 2) - 3)) - 3))^2 - 5(-1sin(3\u03c0((-3(3|x| - 2) - 2)/(-1(3|x| - 2) - 3)) - 3)) + 1 = -2e^(1(-3|(2x - 1) + 2| - 3)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1((2x - 2)/(-3x + 2)) - 3 = (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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2((-2(2|(-2x^2 - 4x + 3) - 1|) + 2)/(2(2|(-2x^2 - 4x + 3) - 1|) - 1)) - 1 = -4(-1cos(2\u03c0((2x + 3)/(3x + 1)) + 3) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(3sin(1\u03c0(-1|x + 3| + 3) - 3) + 3) + 5 = (1(3x - 3) - 1)/(-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": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(1((-3x)/(2x - 1))^2 + 2((-3x)/(2x - 1)) - 3) + 1| - 1 and g(x) = 3(3x)^2 + 5(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": "Determine the number of intersection points between the functions f(x) = -2cos(3\u03c0(2(-1|(-3x + 3) - 3| - 3)^2 - 2(-1|(-3x + 3) - 3| - 3) + 4) + 3) - 3 and g(x) = -1(4(1|x - 3| + 1)^2 - 4(1|x - 3| + 1) - 1) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(-2|(-1((3x + 3)/(1x + 2)) + 4)| + 2)^2 + 5(-2|(-1((3x + 3)/(1x + 2)) + 4)| + 2) and g(x) = 2e^(-1(-2(-3x^2 - 2x) + 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2e^(-1(3(3x - 1)^2 + 1(3x - 1) + 2)) + 2 = 5(1|x - 2| + 1) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1((2(-2(-1|x - 3|)^2 - 5(-1|x - 3|) + 2) + 2)/(-1(-2(-1|x - 3|)^2 - 5(-1|x - 3|) + 2))) - 5 and g(x) = -4((-1(3|x - 1| - 1))/(1(3|x - 1| - 1))) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1((-3(-2|(-4x - 5) + 2| + 2) + 2)/(-2(-2|(-4x - 5) + 2| + 2) - 3))^2 - 4((-3(-2|(-4x - 5) + 2| + 2) + 2)/(-2(-2|(-4x - 5) + 2| + 2) - 3)) = -2|(-1(-2sin(2\u03c0x - 1) + 1) - 2) - 1| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-2|(4x^2 + 1x + 1)| - 3) + 1)/(3(-2|(4x^2 + 1x + 1)| - 3) + 2) and g(x) = 3cos(1\u03c0((1x - 3)/(2x + 3)) + 3) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(-4(2|x + 2| - 1)) + 1)/(3(-4(2|x + 2| - 1)) - 2) and g(x) = 3((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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(1(-1e^(-1x) + 1) - 2)^2 - 4(1(-1e^(-1x) + 1) - 2) + 5 = -5(-3|x - 3| - 1) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(5((2(2|x - 3| - 2))/(3(2|x - 3| - 2) + 1)) - 1)^2 + 5(5((2(2|x - 3| - 2))/(3(2|x - 3| - 2) + 1)) - 1) - 4 and g(x) = 3|(-3((2x - 1)/(-2x - 1))^2 + 2((2x - 1)/(-2x - 1))) + 2| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(-1|(-5((-3x)/(-2x - 3))^2 - 3((-3x)/(-2x - 3)) + 4) - 1|) + 4 and g(x) = 5((-2(4x + 2) - 2)/(-1(4x + 2) - 3))^2 + 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|((-2(-3x - 1) - 2)/(-2(-3x - 1) + 3)) - 2| and g(x) = -3|(3e^(2x) - 1) - 1| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|((2(3sin(2\u03c0(3x^2 + 5x + 1) + 3)) + 3)/(-1(3sin(2\u03c0(3x^2 + 5x + 1) + 3)) - 2)) + 2| + 3 and g(x) = 4(-3|(1x + 2) + 2| + 3)^2 - 5(-3|(1x + 2) + 2| + 3) + 4 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 solutions does the equation -1(3((-1(-2|x - 3| - 1) + 3)/(-3(-2|x - 3| - 1) + 2))^2 + 5((-1(-2|x - 3| - 1) + 3)/(-3(-2|x - 3| - 1) + 2)) + 3) + 2 = -4(2|(1x + 5) - 1| - 3)^2 - 4(2|(1x + 5) - 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) = -2(-3cos(2\u03c0x) + 2) - 4 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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(2|(4x - 1) + 2| - 1) + 2)/(1(2|(4x - 1) + 2| - 1) - 2) and g(x) = (1(2e^(1x) + 1) - 3)/(1(2e^(1x) + 1) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(3|(2x^2 + 4x + 3) - 2| + 3) - 3)/(1(3|(2x^2 + 4x + 3) - 2| + 3) - 3) and g(x) = 1(1e^(1x) - 3) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(3sin(1\u03c0(3((-2x - 3)/(-3x + 2))^2 - 4((-2x - 3)/(-3x + 2)) - 3) - 2) + 2) - 3| - 1 and g(x) = 1((-1(4x - 4) + 2)/(3(4x - 4) - 2))^2 + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "87", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1((-3(-3|x + 1| - 1) - 3)/(-3(-3|x + 1| - 1) - 3)) + 1 and g(x) = -2(1|x| - 1)^2 - 3(1|x| - 1) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5((1(-2x^2 + 1x))/(1(-2x^2 + 1x) - 3)) - 3 and g(x) = (1(-1|x - 3| + 2))/(3(-1|x - 3| + 2) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5((2(-1e^(1x) - 1) + 3)/(2(-1e^(1x) - 1) - 1)) + 4 = 1|((-1x + 3)/(3x - 2)) + 1| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-5(3|(-3x - 1)| + 1)^2 - 2(3|(-3x - 1)| + 1) + 2) - 3)/(2(-5(3|(-3x - 1)| + 1)^2 - 2(3|(-3x - 1)| + 1) + 2) - 3) and g(x) = (-3(2(1x^2 + 4x - 5) + 1) - 1)/(-2(2(1x^2 + 4x - 5) + 1) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(2|x - 1| - 1) - 3 and g(x) = 1x^2 + 3x + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(1x^2 - 1) - 2)/(-1(1x^2 - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(-2|x|)^2 + 5(-2|x|) - 5 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((1x - 1)/(1x + 2)) - 1 and g(x) = -1x intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(-5x^2 + 1x + 5) - 2| = -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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(-3(5((-3x - 1)/(-2x)) + 5)^2 - 5(5((-3x - 1)/(-2x)) + 5) + 3) + 3| - 3 and g(x) = 2(4((-3x)/(2x - 3))^2 + 2((-3x)/(2x - 3)) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(1cos(1\u03c0x + 2) + 1) - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-2|x - 1| - 1) + 2)/(-2(-2|x - 1| - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(-3x^2 + 4x + 5) + 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(3(5x^2 + 5x - 1)))/(3(3(5x^2 + 5x - 1)) + 1) and g(x) = 3cos(2\u03c0(4x^2 - 5x - 3) + 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1377", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(-3(-3|x - 3| + 3)^2 + 5(-3|x - 3| + 3) - 1) - 2 and g(x) = 3(2sin(2\u03c0x - 3) + 1) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(1(3x^2 + 4x) + 4) - 1| + 2 and g(x) = -2|(5x + 5) + 2| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1sin(1\u03c0((-1(-1x - 5) + 2)/(-2(-1x - 5) + 3)) + 3) - 2 and g(x) = -2|(1x^2 - 2x - 4) - 3| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(5(2|x + 3| - 3)^2 + 1(2|x + 3| - 3) - 2) + 2 and g(x) = (-1(2e^(-1x) + 2) - 1)/(1(2e^(-1x) + 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1((-1(1(-2|x - 3| - 1) - 1) + 3)/(-3(1(-2|x - 3| - 1) - 1) + 4))^2 + 4((-1(1(-2|x - 3| - 1) - 1) + 3)/(-3(1(-2|x - 3| - 1) - 1) + 4)) - 4 and g(x) = 3cos(1\u03c0(3(-1|x - 2| - 3)^2 - 4(-1|x - 2| - 3) - 3) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1035", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1cos(2\u03c0(-1|(-4x + 2) + 1| - 3) - 3) = 4((-2x - 2)/(-1x + 2))^2 + 4((-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": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1e^(1(-1x^2 - 4x - 2)) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1((-1x - 3)/(1x - 2)) - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3sin(2\u03c0(-1|x| + 3) + 2) + 1 = (1x)/(-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": "How many solutions does the equation -2|((2x - 3)/(3x - 2)) + 1| + 2 = (-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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2cos(3\u03c0(3(4((-1x + 3)/(3x)) + 2)^2 + 3(4((-1x + 3)/(3x)) + 2) - 1) + 3) - 3 and g(x) = -2sin(1\u03c0((-1(2x - 3) + 2)/(1(2x - 3) + 3)) - 2) - 2 where -10 \u2264 x \u2264 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) = 5(-1e^(-2(4x^2 + 1x - 4)) + 2) + 1 and g(x) = (-2(2cos(2\u03c0x - 1) + 2) - 1)/(-2(2cos(2\u03c0x - 1) + 2) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(3\u03c0(2|x - 3|) + 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": "14", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-1|x + 3| - 2) + 2)/(2(-1|x + 3| - 2) + 3) 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(-3|((1x + 1)/(-1x + 2)) + 3| - 1)^2 + 3(-3|((1x + 1)/(-1x + 2)) + 3| - 1) - 2 and g(x) = 5((3x - 2)/(3x + 2))^2 + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4((-3x + 1)/(-1x + 2)) + 3 and g(x) = 1x^2 - 4x + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1sin(3\u03c0(3(-1|x + 1| + 1) - 4) - 1) - 1 = -1e^(-1(2x - 1)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(-3cos(3\u03c0(-1x - 2) + 2) - 1)| - 1 and g(x) = 4(3sin(1\u03c0x + 1) - 1)^2 - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-2cos(1\u03c0x + 1) - 3) + 3| + 2 and g(x) = 4x - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-1|(1(-1x^2 + 3x - 4) + 3) + 2| - 2) + 3)/(2(-1|(1(-1x^2 + 3x - 4) + 3) + 2| - 2) - 1) and g(x) = -4(-1(-3cos(1\u03c0x + 1) - 2)^2 + 3) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(3(2sin(1\u03c0x + 3) + 1)^2 - 1(2sin(1\u03c0x + 3) + 1)) + 3 and g(x) = -3(2cos(3\u03c0x - 1) - 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1e^(2(2((2(5x^2 - 1x - 4) + 1)/(1(5x^2 - 1x - 4) - 2)) - 4)) - 2 and g(x) = -3cos(1\u03c0(-2((-2x + 3)/(-3x - 1)) - 2) + 2) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(-2x + 4)^2 - 1(-2x + 4) and g(x) = 2x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(4(1(-1e^(2x) + 3) - 5)^2 - 4(1(-1e^(2x) + 3) - 5) + 4))/(-2(4(1(-1e^(2x) + 3) - 5)^2 - 4(1(-1e^(2x) + 3) - 5) + 4) + 3) and g(x) = 3sin(3\u03c0(-3|(-3x + 2)|) + 1) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "540", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(-5x^2 - 2x + 4) - 3| - 2 and g(x) = -5x^2 + 5x + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1cos(3\u03c0(-4((3x + 2)/(1x + 2)) + 2) + 3) - 1 and g(x) = 3sin(2\u03c0(3x - 5) + 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "145", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(1cos(1\u03c0(-1|(-5x + 3) + 2| + 3) - 2) + 1) - 1)/(3(1cos(1\u03c0(-1|(-5x + 3) + 2| + 3) - 2) + 1) + 1) and g(x) = (2(1cos(2\u03c0(-1x - 4) + 3) + 1) - 2)/(2(1cos(2\u03c0(-1x - 4) + 3) + 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(2sin(2\u03c0(-4x^2 - 1x - 1) + 2) + 1) - 2)/(1(2sin(2\u03c0(-4x^2 - 1x - 1) + 2) + 1) + 3) and g(x) = 3(-2|x| + 1) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(1\u03c0(1((2(3x^2) - 1)/(3(3x^2) - 2)) - 1) + 1) + 3 and g(x) = -3cos(1\u03c0(-1|(-1x^2 - 3x + 5)| - 2) - 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "217", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(4(1|x + 2| + 1) - 2)^2 + 3(4(1|x + 2| + 1) - 2) - 5 and g(x) = 3|(4x - 3)| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3((-2(1(3|x - 2| - 1)^2 - 4(3|x - 2| - 1)))/(-1(1(3|x - 2| - 1)^2 - 4(3|x - 2| - 1)) + 2)) - 3 and g(x) = (1(1(2sin(2\u03c0x + 1) - 2)^2 + 2(2sin(2\u03c0x + 1) - 2)) - 1)/(2(1(2sin(2\u03c0x + 1) - 2)^2 + 2(2sin(2\u03c0x + 1) - 2)) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(4(-1|x|)^2 + 2(-1|x|) - 1) - 3)/(1(4(-1|x|)^2 + 2(-1|x|) - 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 solutions does the equation 1(-3e^(-2(2|x + 3| + 2)) + 1)^2 - 4(-3e^(-2(2|x + 3| + 2)) + 1) + 4 = (1(4x^2 - 5x + 3) + 3)/(3(4x^2 - 5x + 3) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(-5(-2|(-3sin(2\u03c0x) + 3) - 3| + 1)^2 + 1(-2|(-3sin(2\u03c0x) + 3) - 3| + 1) - 1) - 2)/(-1(-5(-2|(-3sin(2\u03c0x) + 3) - 3| + 1)^2 + 1(-2|(-3sin(2\u03c0x) + 3) - 3| + 1) - 1)) and g(x) = 2|(1sin(2\u03c0(1x^2 + 5x - 4) + 1) - 1)| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "406", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3cos(1\u03c0(5((3(-5x^2 + 1x + 3) + 3)/(-3(-5x^2 + 1x + 3) + 2)) - 2) + 1) - 3 and g(x) = (-1(4(-4x^2 + 2x - 3) + 3) - 2)/(1(4(-4x^2 + 2x - 3) + 3) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "64", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(2cos(3\u03c0(3|(-2x + 2) + 2| - 2) - 1) + 2) - 2)/(-3(2cos(3\u03c0(3|(-2x + 2) + 2| - 2) - 1) + 2) + 4) = -3|(-4(-1sin(3\u03c0x) + 3)^2 - 2(-1sin(3\u03c0x) + 3) - 2) + 3| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1cos(2\u03c0((-2(4(-2|x - 3| + 2)^2 - 5(-2|x - 3| + 2) + 1) - 3)/(-3(4(-2|x - 3| + 2)^2 - 5(-2|x - 3| + 2) + 1) + 1)) - 2) + 2 and g(x) = 3|((-2(-1x + 2))/(1(-1x + 2) + 3)) - 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": "Find the number of times the graphs of f(x) = (-3(-1|(2x - 3)| - 1) - 2)/(-1(-1|(2x - 3)| - 1) - 2) and g(x) = -3(2cos(3\u03c0x + 3) - 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(1\u03c0(2(2|(4x + 5) - 2| + 2)^2 - 5(2|(4x + 5) - 2| + 2) + 1) + 1) + 3 and g(x) = (-3(2|(-2x - 2) + 3| + 3) - 1)/(-2(2|(-2x - 2) + 3| + 3) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1254", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1cos(1\u03c0((-3x)/(3x)) + 1) - 3 and g(x) = 2x^2 + 2x - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(-2|(-2(-3x + 1)^2 - 3(-3x + 1) + 5) - 2| + 1) + 2)/(-2(-2|(-2(-3x + 1)^2 - 3(-3x + 1) + 5) - 2| + 1)) and g(x) = -1|(1cos(2\u03c0((-2x - 1)/(-1x - 3)) + 1) + 3) + 2| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(-1|((-3x - 2)/(-1x + 2)) - 3| + 2) + 4 and g(x) = -4((2x + 2)/(1x + 1)) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-2sin(3\u03c0(5x + 4)) + 3) - 2| - 3 and g(x) = -2|(5x - 5) - 1| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(-4((2(-2x^2 + 2x + 5))/(-3(-2x^2 + 2x + 5) + 4))) + 1| and g(x) = -5(-4(-2|x + 2| + 1)^2 + 1(-2|x + 2| + 1) + 5) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(3\u03c0((-2x - 1)/(-1x - 2)) + 1) - 3 and g(x) = -1x^2 - 1x + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(3(-3sin(1\u03c0x - 3) - 1)^2 - 5(-3sin(1\u03c0x - 3) - 1) - 3) = 5((-3x + 2)/(1x + 3)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3cos(2\u03c0(-3x + 1) - 3) + 1 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": "30", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(1|(-4x) + 1| - 1)^2 + 3(1|(-4x) + 1| - 1) + 5 = -1|(2cos(1\u03c0x + 3) - 2) + 1| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((2(-2sin(3\u03c0x - 2) - 1))/(1(-2sin(3\u03c0x - 2) - 1) + 2))^2 - 4((2(-2sin(3\u03c0x - 2) - 1))/(1(-2sin(3\u03c0x - 2) - 1) + 2)) + 1 and g(x) = -4((-2x + 1)/(2x + 1))^2 + 2((-2x + 1)/(2x + 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "70", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-4(2((-1x)/(-1x + 2)) + 5)^2)| + 1 and g(x) = -1sin(3\u03c0(-5(1|x + 1|)^2 + 3(1|x + 1|) + 2) - 3) + 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) = -2|(1((3(1x^2 + 5x + 3))/(-2(1x^2 + 5x + 3) - 2)) - 1) - 1| + 3 and g(x) = (-3(-4(-1e^(2x) + 1) + 4))/(-1(-4(-1e^(2x) + 1) + 4) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|((3(-5(-2x^2 + 2x + 1) + 4))/(1(-5(-2x^2 + 2x + 1) + 4) - 3)) + 3| + 1 and g(x) = -2|(2(-3e^(1x) + 2) - 3) + 2| - 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 -2e^(-1((-1(-2|(-1x^2 + 1x + 2)|) - 1)/(2(-2|(-1x^2 + 1x + 2)|) - 1))) + 2 = 1|(-1((1x + 1)/(3x)) + 2) - 1| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-2x + 1)^2 + 1(-2x + 1) - 5 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(-2e^(-2x) - 2) + 2| - 3 and g(x) = -3|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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2((-3(3|(3sin(2\u03c0x) + 3) - 2| + 2) + 3)/(-2(3|(3sin(2\u03c0x) + 3) - 2| + 2)))^2 - 4 and g(x) = -5(2(1e^(1x))^2 - 5(1e^(1x)) + 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(1x - 4)^2 + 3(1x - 4) + 1 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(-3|x + 3| + 1)^2 - 5(-3|x + 3| + 1) - 4 and g(x) = 3x^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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(-1((-2x + 3)/(-1x + 2))^2 - 5((-2x + 3)/(-1x + 2)) + 4) + 5 and g(x) = 2(-3|x + 3| - 1)^2 + 2(-3|x + 3| - 1) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|(3((1(3x))/(-1(3x) + 2))^2 - 1((1(3x))/(-1(3x) + 2)) + 3) + 2| + 1 = -3|(2((1x)/(2x + 2)) - 3)| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3((1(4(1|x + 3| + 3)^2) - 2)/(-1(4(1|x + 3| + 3)^2) - 3)) + 5 and g(x) = -3cos(1\u03c0(-1|(5x + 3) + 3|) - 2) 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) = -4(-2sin(2\u03c0x) - 3) - 3 and g(x) = 4x^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) = -2sin(1\u03c0(1x^2 - 3x - 1) - 3) + 3 and g(x) = 3|x + 2| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(1x^2 + 5x + 3) + 4 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(2|(-5x - 2) + 3|) + 2)/(3(2|(-5x - 2) + 3|) - 1) and g(x) = -2|(2x + 4) + 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) = (-2(2e^(2x) + 1) + 3)/(2(2e^(2x) + 1) + 2) 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-3|(1cos(2\u03c0x) + 2) - 3| - 2) + 3)/(3(-3|(1cos(2\u03c0x) + 2) - 3| - 2) + 2) = 1(1x^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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(-4(2|(3x^2 + 3x - 3) - 2| - 1) + 4) + 3)/(-2(-4(2|(3x^2 + 3x - 3) - 2| - 1) + 4) - 1) = -1(-2cos(2\u03c0(-2|x - 2| - 1) + 1) - 1)^2 - 1(-2cos(2\u03c0(-2|x - 2| - 1) + 1) - 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "158", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(-1|(2(4x^2 - 5x - 3) + 1) - 3| + 3) - 3)/(-2(-1|(2(4x^2 - 5x - 3) + 1) - 3| + 3) - 1) and g(x) = -1|((1(-1x^2 + 4x - 3) - 1)/(-3(-1x^2 + 4x - 3)))| - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-1((1x + 1)/(1x)) - 1) - 2| - 3 and g(x) = (-3(-1x^2 - 4x + 2) - 2)/(3(-1x^2 - 4x + 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 2(1|(-2((-2x + 3)/(-1x + 2))) - 3| - 1)^2 - 4(1|(-2((-2x + 3)/(-1x + 2))) - 3| - 1) - 5 = 5(-1e^(1((3x)/(-3x + 1))) + 2)^2 + 1(-1e^(1((3x)/(-3x + 1))) + 2) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(1|(1(-3x - 1)^2 + 1(-3x - 1) - 5) - 1| - 1) - 3)/(3(1|(1(-3x - 1)^2 + 1(-3x - 1) - 5) - 1| - 1) - 2) and g(x) = 3sin(1\u03c0((-2(4x^2 + 2x + 1) - 3)/(-2(4x^2 + 2x + 1) - 3)) - 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(1|(-3x^2 + 5x - 1) + 2| + 2) - 3)/(-3(1|(-3x^2 + 5x - 1) + 2| + 2) - 3) = 3cos(3\u03c0(4x^2 + 3x + 1) - 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1478", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(3x^2 - 3x - 1) - 2 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(-5(3((-2x + 1)/(3x))^2 - 1) - 5) + 3| and g(x) = 2|((1(3cos(1\u03c0x - 2) + 2) + 2)/(2(3cos(1\u03c0x - 2) + 2))) - 1| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(-2|(1((3x - 2)/(2x))^2 + 4((3x - 2)/(2x))) + 1| + 2) + 3 and g(x) = -4(-3cos(1\u03c0(-4x^2 + 2x + 3) - 2) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(-1(-3cos(1\u03c0(-2|x| + 2) + 3) - 1)^2 + 5(-3cos(1\u03c0(-2|x| + 2) + 3) - 1) + 3) - 2)/(-2(-1(-3cos(1\u03c0(-2|x| + 2) + 3) - 1)^2 + 5(-3cos(1\u03c0(-2|x| + 2) + 3) - 1) + 3) - 1) = 2|(-3(-2cos(1\u03c0x + 1) + 2)^2 + 3(-2cos(1\u03c0x + 1) + 2) + 2) + 2| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2cos(1\u03c0((3(3(1x^2 + 1x) - 3) - 2)/(3(3(1x^2 + 1x) - 3) - 2)) + 3) - 2 = -2|(-1((-2x - 3)/(1x))) + 3| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(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": "45", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(2sin(3\u03c0((-3x - 2)/(1x + 2))) + 2) + 2 = 3|(4x^2 + 1x + 3) - 2| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|((3x - 1)/(2x - 1)) + 3| + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(2cos(2\u03c0(-3|x - 3| - 1) - 1) - 2)^2 + 2(2cos(2\u03c0(-3|x - 3| - 1) - 1) - 2) - 3 and g(x) = (2(1x^2 + 3x + 5) - 2)/(1(1x^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": "120", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1sin(1\u03c0(1|x + 2| + 1) - 2) + 3 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4((-2(-1x^2 + 4) - 1)/(3(-1x^2 + 4) - 2)) - 2 and g(x) = 1|(2sin(3\u03c0x - 1) + 2) - 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|((2(-2(-3x + 3)^2 + 2(-3x + 3) + 4) + 3)/(-3(-2(-3x + 3)^2 + 2(-3x + 3) + 4) + 2)) + 2| - 1 and g(x) = (-3(3(4x + 4)^2 + 5(4x + 4) + 3) + 2)/(1(3(4x + 4)^2 + 5(4x + 4) + 3)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(3cos(2\u03c0x + 2) + 3) - 2| + 2 and g(x) = -5x + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3cos(2\u03c0((-2x)/(-1x + 3)) - 2) + 1 and g(x) = 5x^2 + 4x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(-4x^2 - 3x + 4)| - 3 = (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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(1(1|((2x - 1)/(1x)) + 2| - 1)^2 + 2(1|((2x - 1)/(1x)) + 2| - 1) - 2) - 2 and g(x) = -1(1((1x + 1)/(-2x - 1)) + 5)^2 - 4(1((1x + 1)/(-2x - 1)) + 5) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(3e^(1x) - 2) - 1 and g(x) = 5x^2 - 1x - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(2|(4(1e^(-2x) - 1)^2 + 1(1e^(-2x) - 1) - 3) + 2| + 3) + 2)/(-3(2|(4(1e^(-2x) - 1)^2 + 1(1e^(-2x) - 1) - 3) + 2| + 3) - 2) and g(x) = (-2(-1|(1x^2 - 2x - 4) + 3| + 2))/(-1(-1|(1x^2 - 2x - 4) + 3| + 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(-3|x| - 2) + 1)/(-2(-3|x| - 2) + 1) = -1|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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(3(-3x^2 + 1x + 4) + 4) + 2| + 3 and g(x) = -1cos(2\u03c0(-3x^2 + 3x - 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) = 2|((-1x - 3)/(-1x)) + 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(1x^2) - 2| + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1sin(3\u03c0(1|(1x^2 + 5x - 1)| - 1) - 3) + 3 and g(x) = 2cos(2\u03c0(-2x + 5) + 3) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "214", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(-2sin(3\u03c0x + 1) - 1) - 1| + 3 and g(x) = 3|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": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(-2cos(1\u03c0(3|x - 3| - 1)) - 3)^2 - 4(-2cos(1\u03c0(3|x - 3| - 1)) - 3) and g(x) = 2(-2e^(-1x) - 1) - 4 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(2\u03c0(1(1x - 2)^2 + 5(1x - 2) + 1) + 1) - 3 = 1|((2x + 1)/(2x)) - 3| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "187", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(-4(-4x^2 - 2x + 5) + 3) - 2)/(-1(-4(-4x^2 - 2x + 5) + 3) + 3) and g(x) = -4(-5x^2 + 1x - 4) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(2e^(2(-2x^2 - 4x - 5)) + 2) + 3)/(-1(2e^(2(-2x^2 - 4x - 5)) + 2) - 3) and g(x) = -2cos(2\u03c0((3x + 3)/(2x + 3)) - 1) 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) = (3(-1|(5(1x - 1)^2 - 5(1x - 1) + 2) + 2| - 3) + 2)/(-2(-1|(5(1x - 1)^2 - 5(1x - 1) + 2) + 2| - 3) + 1) and g(x) = -1|(2(2x - 4)^2 + 5(2x - 4)) - 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1((-2(5x + 3) + 3)/(1(5x + 3)))^2 + 5((-2(5x + 3) + 3)/(1(5x + 3))) + 5 and g(x) = 5(-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) = -2(2sin(1\u03c0x) - 1)^2 + 2(2sin(1\u03c0x) - 1) - 5 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) = 2cos(2\u03c0(-3x^2 - 5x - 2) + 3) - 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": "419", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(2\u03c0((2x)/(2x))) - 2 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(3\u03c0((-1x - 1)/(2x - 3))) and g(x) = -5x^2 - 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": "Determine the number of intersection points between the functions f(x) = 1(1|(-3e^(-1x) - 1) - 1| + 1) + 1 and g(x) = (-3(-3cos(1\u03c0x + 3) + 3))/(3(-3cos(1\u03c0x + 3) + 3) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(3cos(2\u03c0(5((2x + 2)/(3x + 3))^2 + 4((2x + 2)/(3x + 3))) - 3)) + 5 and g(x) = 1(-3|(1x^2 + 4x - 4) - 3| + 1) - 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) = -3cos(3\u03c0(3(-4((1x - 1)/(2x - 1))^2 + 1((1x - 1)/(2x - 1)) - 2) + 3) + 3) + 1 and g(x) = 2(-4((1x + 3)/(1x - 2))^2 - 5((1x + 3)/(1x - 2)) - 1) + 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) = -2cos(3\u03c0((-3x + 3)/(-2x - 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": "112", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(1(-1|x|)^2 - 5(-1|x|) - 2) - 2)/(3(1(-1|x|)^2 - 5(-1|x|) - 2) + 3) and g(x) = -2|(-1x - 3) - 1| + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(-5x^2 - 2x + 5) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(1|(3(-3x - 3)^2 - 5(-3x - 3) + 1)|) + 2)/(-2(1|(3(-3x - 3)^2 - 5(-3x - 3) + 1)|) + 3) and g(x) = -4(-4((-3x + 1)/(-1x - 3))^2 - 4((-3x + 1)/(-1x - 3)) + 1) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(-3cos(2\u03c0(-5x^2 - 3x + 3) - 1)) - 2 and g(x) = 3|(-1x - 5) + 2| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "228", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(-1|((-2(-3x))/(-2(-3x) + 3)) + 3| + 3)^2 - 2(-1|((-2(-3x))/(-2(-3x) + 3)) + 3| + 3) - 1 and g(x) = 1sin(3\u03c0(-1(1|x - 3| + 1) + 2) + 3) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(3sin(2\u03c0x) + 1)^2 - 4(3sin(2\u03c0x) + 1) + 1 = (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": "74", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(3cos(1\u03c0(-2((-1x + 2)/(2x + 2)) - 4) + 2) + 2) - 1| and g(x) = -2(-2cos(2\u03c0(3|x + 3| - 3) + 2) + 1)^2 + 5 intersect 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) = (2(1|(2x^2 + 3x - 5)| + 2))/(-1(1|(2x^2 + 3x - 5)| + 2) + 3) and g(x) = 3|(5x^2 - 2x + 1) - 3| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2e^(2(-2|x + 1| + 1)) - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(1((-2x + 3)/(-2x - 1)) - 1) + 2| and g(x) = -2(1|x + 3| + 1) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(3x^2 - 2x + 4) + 3)/(-1(3x^2 - 2x + 4) - 2) and g(x) = -1x^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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-2cos(1\u03c0(2|(4x^2 - 4x + 3) + 2| - 3) + 2) - 3) - 1)/(-2(-2cos(1\u03c0(2|(4x^2 - 4x + 3) + 2| - 3) + 2) - 3) + 1) and g(x) = -3(3|(4x^2 - 1x - 5) + 3| - 2) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-2sin(1\u03c0((3(-1x^2 + 5x + 5) - 1)/(1(-1x^2 + 5x + 5) - 2)) - 2) + 3) and g(x) = (-1(3|(-4x) + 1|) - 1)/(3(3|(-4x) + 1|) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(1(-3cos(3\u03c0x - 2))^2 - 1(-3cos(3\u03c0x - 2)) + 4))/(-2(1(-3cos(3\u03c0x - 2))^2 - 1(-3cos(3\u03c0x - 2)) + 4) - 1) and g(x) = -4(-3x^2 - 4x + 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2((-3(-4x - 2) + 1)/(2(-4x - 2)))^2 - 3((-3(-4x - 2) + 1)/(2(-4x - 2))) + 1 and g(x) = -2(1|x + 2| + 1)^2 + 5(1|x + 2| + 1) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|((-2(3x^2 + 2x - 3) - 2)/(-1(3x^2 + 2x - 3))) + 1| + 3 and g(x) = 2(-3x - 3)^2 + 4(-3x - 3) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(-2sin(1\u03c0(1x^2 + 1x + 3) + 1) - 2) - 4 = 1(-5x^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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(-2sin(1\u03c0x + 2) + 3) - 3| - 2 = -1x^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": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2((-2(-1e^(1x) + 2) - 3)/(-1(-1e^(1x) + 2) - 1))^2 + 4((-2(-1e^(1x) + 2) - 3)/(-1(-1e^(1x) + 2) - 1)) + 4 and g(x) = 2(-1cos(1\u03c0x - 1) + 3) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(-2x - 3) - 3| + 3 and g(x) = 3|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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(1x) + 1| + 2 and g(x) = (1x - 1)/(3x - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(3x + 3)^2 - 5(3x + 3) - 4 = (-1x - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-1|(-2cos(3\u03c0(1x + 5)) + 3) - 1| + 2)^2 - 4(-1|(-2cos(3\u03c0(1x + 5)) + 3) - 1| + 2) - 5 and g(x) = (-3(-2|(-5x - 3) - 3| - 1) - 1)/(-3(-2|(-5x - 3) - 3| - 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-2sin(2\u03c0(1|x| - 3) - 2)) - 1)/(-3(-2sin(2\u03c0(1|x| - 3) - 2)) + 1) and g(x) = 3e^(2((2x - 2)/(3x + 1))) + 1 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": "How many points of intersection exist between the functions f(x) = 5(-2(-3sin(3\u03c0x - 1) - 2)^2 - 4(-3sin(3\u03c0x - 1) - 2) + 4) + 2 and g(x) = -1|(3x + 2) + 1| + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "82", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-2cos(2\u03c0x + 2)) - 3)/(-2(-2cos(2\u03c0x + 2)) - 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": "39", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(-3((-1(-4x^2 - 4x + 4) - 2)/(-2(-4x^2 - 4x + 4) + 3)) + 3)| + 3 and g(x) = -1e^(-1((1(1|x + 3| + 1) - 1)/(-2(1|x + 3| + 1) + 1))) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(-3e^(-1(2|x - 1| - 3)) + 2)^2 + 4(-3e^(-1(2|x - 1| - 3)) + 2) + 1 = 4(-2e^(1x) + 3) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1cos(3\u03c0(2|((-2x + 1)/(1x + 1))| - 2)) - 2 and g(x) = -1cos(1\u03c0(5x^2 + 3x + 4) - 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "671", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-5x^2 + 3x + 5))/(-3(-5x^2 + 3x + 5) - 2) 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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(-3sin(3\u03c0(-1|x - 1| - 3)) - 1)^2 - 5(-3sin(3\u03c0(-1|x - 1| - 3)) - 1) + 2 and g(x) = 2|((-2x + 2)/(-1x - 2))| + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3e^(-1(-5(-2|x| - 3) - 3)) and g(x) = -1(2cos(3\u03c0x - 1)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-3|((3x + 1)/(-3x)) - 3| - 1)^2 + 5(-3|((3x + 1)/(-3x)) - 3| - 1) and g(x) = -3(3|x| - 2)^2 + 3(3|x| - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(2|(-3e^(2x) - 3) - 2| + 1) + 3)/(-3(2|(-3e^(2x) - 3) - 2| + 1) + 2) and g(x) = -2((1x + 2)/(1x - 3))^2 - 4((1x + 2)/(1x - 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-5(3|x - 2| - 2)^2 - 4(3|x - 2| - 2) - 1) - 3)/(2(-5(3|x - 2| - 2)^2 - 4(3|x - 2| - 2) - 1)) and g(x) = -2(-2cos(1\u03c0x + 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(-1x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(-2sin(2\u03c0(-2|x + 3| + 1) + 2) - 3) - 3)/(-1(-2sin(2\u03c0(-2|x + 3| + 1) + 2) - 3)) and g(x) = 2(-2x - 2)^2 + 4(-2x - 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) = -2|(4(-1cos(2\u03c0(2x^2 - 2x + 3) + 1) - 3) - 4)| and g(x) = -2|(5((2x)/(-2x + 3))^2 + 5) - 1| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "127", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(-1((3(1|x - 2| - 1) - 2)/(3(1|x - 2| - 1) - 1))^2 + 4((3(1|x - 2| - 1) - 2)/(3(1|x - 2| - 1) - 1)) - 4) and g(x) = -2cos(2\u03c0(3|((1x - 1)/(-2x - 3)) + 2| + 1) + 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": "How many solutions does the equation -1(-3|x - 3| + 2)^2 - 1 = (2x - 2)/(-1x - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(1sin(1\u03c0(1(-5x + 1)^2 + 5(-5x + 1)) - 3) - 2) - 2| = 1(4((-1x - 2)/(2x))^2 - 2((-1x - 2)/(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": "14", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2sin(1\u03c0((-1(-5(-2x^2 - 2x - 5) + 2) + 3)/(-2(-5(-2x^2 - 2x - 5) + 2) - 1))) - 3 = -1|(-1((2x + 1)/(-1x - 1)) - 2) + 1| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1e^(1((-1(2(1|x| - 2) + 2))/(1(2(1|x| - 2) + 2) + 1))) and g(x) = 3|((1(-2x^2 + 4x - 4))/(-3(-2x^2 + 4x - 4) + 4)) - 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(-4((1(2|x + 2| - 3) - 2)/(-2(2|x + 2| - 3) - 3)) + 5)^2 - 2(-4((1(2|x + 2| - 3) - 2)/(-2(2|x + 2| - 3) - 3)) + 5) + 3 and g(x) = 2((-3(1|x + 3|) + 1)/(-1(1|x + 3|) + 3))^2 - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(-2|(-3cos(2\u03c0((1x + 3)/(3x + 3)) - 3) + 2) - 3| + 3)^2 - 4(-2|(-3cos(2\u03c0((1x + 3)/(3x + 3)) - 3) + 2) - 3| + 3) + 1 and g(x) = -3e^(-2(1|(2x^2 - 2) + 2| + 1)) + 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": "How many points of intersection exist between the functions f(x) = 2sin(3\u03c0(1|(1x^2 - 4x - 2)|) - 1) - 2 and g(x) = -3|(5x^2 - 5x + 1) + 2| + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-3sin(3\u03c0x - 3)) - 3 = 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(-3((3(-4x^2 + 3) + 3)/(1(-4x^2 + 3))) + 3) - 3| and g(x) = (3(2|(4x^2 + 1x - 5) + 2| - 3))/(1(2|(4x^2 + 1x - 5) + 2| - 3) + 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) = (-1(2(-1|x| + 2) + 4) + 1)/(1(2(-1|x| + 2) + 4)) and g(x) = 2|(-2sin(2\u03c0x + 3) + 3) + 1| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(2e^(1(3(1|x| - 2) - 5)) - 2) - 1)/(-2(2e^(1(3(1|x| - 2) - 5)) - 2) + 3) and g(x) = (1(-2(-1x)^2 - 4(-1x) - 2) + 3)/(-3(-2(-1x)^2 - 4(-1x) - 2) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-3x^2 + 4x) 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-1sin(1\u03c0(4(4x^2 - 2x - 3) - 5) - 2) + 3) - 2| - 3 and g(x) = -2cos(3\u03c0(2(-4x^2) + 4) + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "711", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|(-2((2x + 1)/(-1x + 2))^2 - 2((2x + 1)/(-1x + 2)) + 3) + 3| - 1 and g(x) = -1(2|x + 3| + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1((3x - 3)/(2x + 2))^2 + 2((3x - 3)/(2x + 2)) - 1 and g(x) = (1x + 3)/(-2x - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(2\u03c0((-3(-1x^2 + 3x - 4) - 2)/(3(-1x^2 + 3x - 4) - 2)) - 3) - 1 and g(x) = 3(-3x^2 + 4x + 3) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-2cos(3\u03c0x + 2) - 2) + 3)/(2(-2cos(3\u03c0x + 2) - 2) + 2) and g(x) = -5x^2 + 4x - 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 points of intersection exist between the functions f(x) = -1(3sin(3\u03c0x + 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": "36", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-1x - 4)^2 - 5(-1x - 4) + 4 = 5x - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-5(-2sin(2\u03c0(1x^2 + 5x) + 3) + 2) + 3) - 2)/(2(-5(-2sin(2\u03c0(1x^2 + 5x) + 3) + 2) + 3) + 1) and g(x) = (1(3|(-3e^(2x) - 1) - 1| - 2) + 2)/(1(3|(-3e^(2x) - 1) - 1| - 2) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-3|x + 3|)^2 - 3(-3|x + 3|) + 3 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(-3x - 2) - 3| + 3 and g(x) = 1x^2 + 5x - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(2sin(3\u03c0((-1x + 2)/(-2x - 1)) + 3)) - 2| - 2 and g(x) = 4(2cos(2\u03c0x + 3) - 3)^2 + 1(2cos(2\u03c0x + 3) - 3) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|((-2x - 3)/(-1x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(1(2cos(1\u03c0(3|x + 2| - 3) - 2) - 2)^2 + 3(2cos(1\u03c0(3|x + 2| - 3) - 2) - 2) + 2) + 2)/(2(1(2cos(1\u03c0(3|x + 2| - 3) - 2) - 2)^2 + 3(2cos(1\u03c0(3|x + 2| - 3) - 2) - 2) + 2) + 3) and g(x) = 3sin(1\u03c0(-5(-2|x - 3| + 2)^2 - 4(-2|x - 3| + 2) - 3) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1450", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(-4((1x + 1)/(2x + 1))^2 - 1((1x + 1)/(2x + 1)) - 2) + 2| and g(x) = (2(-3x + 5))/(1(-3x + 5) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3(-3x - 2) + 1)/(-1(-3x - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(3(-3|x - 3| + 3) + 4) - 2)/(1(3(-3|x - 3| + 3) + 4) - 2) and g(x) = -3((-1x - 3)/(-2x + 3))^2 + 1((-1x - 3)/(-2x + 3)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1sin(2\u03c0(5(-2|((2x + 1)/(-3x + 2)) + 1| + 1)^2 - 4)) + 1 and g(x) = -2|(1((-3x + 3)/(-1x + 2))^2 - 4((-3x + 3)/(-1x + 2)) + 5) - 3| + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(-3(-2sin(3\u03c0x + 2) - 1)^2 - 2(-2sin(3\u03c0x + 2) - 1) + 1) + 1| - 1 and g(x) = 1sin(2\u03c0(-1x + 2) + 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "185", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3e^(1(4((-2x)/(-2x + 3))^2 + 4((-2x)/(-2x + 3)) - 3)) + 2 and g(x) = 4((3x + 1)/(1x + 1)) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4((-3x + 3)/(-2x + 3))^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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(1cos(1\u03c0((-2x + 3)/(1x + 1))) + 3)^2 - 3(1cos(1\u03c0((-2x + 3)/(1x + 1))) + 3) - 1 = -2(1x - 5)^2 + 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 4((-2x + 1)/(-1x + 3))^2 - 4((-2x + 1)/(-1x + 3)) - 2 = -1x^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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3((2(1|(-3e^(1x) + 1) + 3|))/(1(1|(-3e^(1x) + 1) + 3|) + 2))^2 - 1((2(1|(-3e^(1x) + 1) + 3|))/(1(1|(-3e^(1x) + 1) + 3|) + 2)) = -1(2e^(1(5x^2 + 2x - 3)) - 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(2\u03c0(-1((-2x - 2)/(-1x + 2))^2 - 1((-2x - 2)/(-1x + 2)) + 5) + 3) + 2 and g(x) = -1|(-4x^2 - 3x + 2) + 2| + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(-1|((-1(1e^(-2x) + 2) + 2)/(-2(1e^(-2x) + 2) - 2))|)^2 + 4(-1|((-1(1e^(-2x) + 2) + 2)/(-2(1e^(-2x) + 2) - 2))|) and g(x) = 3cos(2\u03c0((-3(4x^2 + 2x + 3) - 3)/(2(4x^2 + 2x + 3) - 1)) + 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(1|(5((2x + 1)/(-1x - 2)) + 1) - 3| + 1)^2 + 1(1|(5((2x + 1)/(-1x - 2)) + 1) - 3| + 1) - 3 = 1((2(2sin(3\u03c0x - 1) - 3) + 3)/(-1(2sin(3\u03c0x - 1) - 3) - 3))^2 + 5((2(2sin(3\u03c0x - 1) - 3) + 3)/(-1(2sin(3\u03c0x - 1) - 3) - 3)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(4x)| + 1 and g(x) = -3x intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(-3sin(3\u03c0x - 3) + 1) + 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": "61", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-2cos(3\u03c0((2(2|x + 1|) - 3)/(2(2|x + 1|))) + 1) + 2)^2 - 5(-2cos(3\u03c0((2(2|x + 1|) - 3)/(2(2|x + 1|))) + 1) + 2) - 1 and g(x) = (3(-3cos(1\u03c0(-3|x - 2|) + 1) + 3) - 3)/(3(-3cos(1\u03c0(-3|x - 2|) + 1) + 3) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(3|(-1(-3cos(2\u03c0x + 2) + 3)^2 + 4(-3cos(2\u03c0x + 2) + 3) + 3) - 1| + 1))/(-2(3|(-1(-3cos(2\u03c0x + 2) + 3)^2 + 4(-3cos(2\u03c0x + 2) + 3) + 3) - 1| + 1) - 2) and g(x) = -1|(3e^(-1((-3x + 1)/(3x + 1))) - 3) - 3| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(3\u03c0((-2x + 3)/(-1x - 3)) - 1) - 3 and g(x) = (-2x + 2)/(3x + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "83", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|((-2(1(5x^2 + 3x - 4) - 4) - 2)/(-1(1(5x^2 + 3x - 4) - 4) + 3)) + 1| - 3 and g(x) = -1|(-3sin(1\u03c0((2x - 1)/(1x - 3)) - 3)) - 2| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(-3cos(1\u03c0x + 2) - 3) + 3)/(-2(-3cos(1\u03c0x + 2) - 3)) 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-3|x - 1| - 1) + 4 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(3sin(1\u03c0(2(3|x - 3| + 2)^2 + 5(3|x - 3| + 2) - 1) + 1) + 3) + 5 and g(x) = -1cos(1\u03c0(-3|(-2x^2 + 5x - 1) + 3| - 2) - 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1167", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(1(-1cos(3\u03c0(3|x - 2| - 1) - 1) - 3) + 4) + 3)/(1(1(-1cos(3\u03c0(3|x - 2| - 1) - 1) - 3) + 4) + 3) and g(x) = 4(2((-2x - 2)/(2x)) - 1)^2 + 2(2((-2x - 2)/(2x)) - 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(1cos(2\u03c0x - 3) + 1) - 3 and g(x) = -2x^2 - 1x intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1sin(2\u03c0((3x + 1)/(-3x)) - 2) - 3 = -5x^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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(-2((-1(2x + 1) + 2)/(-2(2x + 1) - 1))^2 - 4((-1(2x + 1) + 2)/(-2(2x + 1) - 1)) - 3) + 3| and g(x) = 3|((2(-2sin(1\u03c0x + 1) - 2) - 1)/(1(-2sin(1\u03c0x + 1) - 2) - 3)) - 1| - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(2((-1(5x^2 - 5x + 2) - 1)/(-2(5x^2 - 5x + 2) + 1))) - 1| - 1 and g(x) = 3|(1cos(3\u03c0(1x + 1) - 3) + 1) + 3| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(5(2e^(2(-1x - 4)) - 3)^2 + 5(2e^(2(-1x - 4)) - 3) - 2) - 1)/(-3(5(2e^(2(-1x - 4)) - 3)^2 + 5(2e^(2(-1x - 4)) - 3) - 2)) = (1(-4(1cos(2\u03c0x - 1) + 2) + 3) + 3)/(2(-4(1cos(2\u03c0x - 1) + 2) + 3) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(2|((-2x - 2)/(2x - 1)) + 1| - 3) and g(x) = (-1(2sin(3\u03c0x) + 1) - 3)/(-2(2sin(3\u03c0x) + 1) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1((1x + 1)/(-1x - 3))^2 - 3((1x + 1)/(-1x - 3)) + 5 = 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((2(-1|x + 2| + 1) + 1)/(1(-1|x + 2| + 1) - 3))^2 - 1((2(-1|x + 2| + 1) + 1)/(1(-1|x + 2| + 1) - 3)) - 3 and g(x) = -1(4x + 5)^2 + 1(4x + 5) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(1|(3e^(2x) - 1)| - 2) - 3)/(3(1|(3e^(2x) - 1)| - 2) + 2) = 1(-1x^2 + 5x + 3) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4((-3(3|x - 2| - 3) + 1)/(-2(3|x - 2| - 3) + 3))^2 + 2((-3(3|x - 2| - 3) + 1)/(-2(3|x - 2| - 3) + 3)) - 1 and g(x) = -5(1e^(1x))^2 + 3(1e^(1x)) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1sin(1\u03c0(-2|x + 1| + 2) - 3) and g(x) = 2x^2 - 2x - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3sin(2\u03c0((-1(1|(5x - 1) - 2| - 3) + 1)/(3(1|(5x - 1) - 2| - 3) - 2)) - 3) - 2 and g(x) = (-3(-2|(1x - 5) + 3| + 2) + 3)/(1(-2|(1x - 5) + 3| + 2) - 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": "How many points of intersection exist between the functions f(x) = (-1(-5(-2|x + 2| - 2) + 2) - 1)/(2(-5(-2|x + 2| - 2) + 2)) and g(x) = (-2(-2cos(3\u03c0x + 3) + 1) + 2)/(-3(-2cos(3\u03c0x + 3) + 1)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(-2cos(2\u03c0(2x) + 1) - 3) + 2| + 2 and g(x) = 5(3x)^2 - 2(3x) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(5(2e^(1((-3x - 2)/(1x + 1))) + 2)) - 3| + 2 and g(x) = 2e^(2(-5(3|x - 3| - 2) - 4)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3cos(1\u03c0((3(-1x + 5) + 3)/(3(-1x + 5) - 2)) - 2) - 3 = -3sin(2\u03c0((2x - 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": "58", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4((2(-5x + 2) + 1)/(-1(-5x + 2) - 1))^2 + 5((2(-5x + 2) + 1)/(-1(-5x + 2) - 1)) - 2 = -3((-2x + 1)/(-1x + 3)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|((3(-3(-1sin(1\u03c0x)) - 2) - 2)/(1(-3(-1sin(1\u03c0x)) - 2) + 3)) + 2| + 2 and g(x) = 2(1(-3sin(3\u03c0x - 2) + 1) - 3)^2 - 5(1(-3sin(3\u03c0x - 2) + 1) - 3) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(2(-3(-1|x + 3|) - 1)^2 - 1(-3(-1|x + 3|) - 1) - 4) - 1)/(3(2(-3(-1|x + 3|) - 1)^2 - 1(-3(-1|x + 3|) - 1) - 4) - 2) and g(x) = -4((1(1x + 3) + 2)/(2(1x + 3)))^2 - 4((1(1x + 3) + 2)/(2(1x + 3))) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(-1(-2|((-2x + 2)/(-2x)) - 2| + 2)^2 + 2(-2|((-2x + 2)/(-2x)) - 2| + 2) - 2) + 5 and g(x) = -1cos(2\u03c0(2((-3x - 1)/(-3x - 3)) + 5)) - 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) = -1|(-5(2((3x)/(-1x + 2))^2) + 1) + 3| - 1 and g(x) = -4((1(-2x + 4) - 1)/(3(-2x + 4) + 2))^2 + 1((1(-2x + 4) - 1)/(3(-2x + 4) + 2)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2((2x - 3)/(3x + 1))^2 - 1((2x - 3)/(3x + 1)) - 5 = -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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(5(-4x - 1)^2 + 2(-4x - 1)) - 2)/(2(5(-4x - 1)^2 + 2(-4x - 1)) - 1) and g(x) = 4(-1sin(3\u03c0x)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(3\u03c0(-3(-5x^2 + 2x + 5) - 1) + 2) - 2 and g(x) = -1e^(-2(-5x)) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "768", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4((-1(1|(-2sin(1\u03c0x)) + 2| - 2) + 1)/(1(1|(-2sin(1\u03c0x)) + 2| - 2) + 1)) and g(x) = (-2(-1sin(1\u03c0(-2x - 2) - 1) + 2))/(1(-1sin(1\u03c0(-2x - 2) - 1) + 2)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(5x + 5)^2 - 3(5x + 5) 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(-1sin(2\u03c0x - 1) + 3))/(3(-1sin(2\u03c0x - 1) + 3) + 2) = -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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(-2cos(1\u03c0x + 1) - 2)^2 + 5(-2cos(1\u03c0x + 1) - 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": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(2|x - 3| + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-4((-3(5x^2 - 2x + 4) - 3)/(3(5x^2 - 2x + 4))) - 4) + 2| - 3 and g(x) = (1(-1(3sin(2\u03c0x - 3) + 3) + 3) - 3)/(-1(-1(3sin(2\u03c0x - 3) + 3) + 3) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(2|(-1(-4x^2 + 1) - 4) - 3| + 3) + 2)/(1(2|(-1(-4x^2 + 1) - 4) - 3| + 3) - 3) and g(x) = 1|((-3(3e^(1x) - 3) + 3)/(-3(3e^(1x) - 3) - 2)) + 2| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(1cos(2\u03c0(1|x| + 1) + 1) - 3)^2 - 3(1cos(2\u03c0(1|x| + 1) + 1) - 3) + 4 and g(x) = 3|(-1x^2 + 1x) - 2| + 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": "Determine the number of intersection points between the functions f(x) = (-2(2e^(-2x)) + 2)/(-3(2e^(-2x))) and g(x) = (-2x - 1)/(3x) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(2sin(2\u03c0x + 3) + 2) - 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(2sin(3\u03c0((-1(-1x^2 - 2x - 5) - 3)/(-2(-1x^2 - 2x - 5) - 3))) + 3) - 1 and g(x) = (1(-2(-1x - 1)^2 + 4(-1x - 1) + 5) + 1)/(1(-2(-1x - 1)^2 + 4(-1x - 1) + 5) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-2((1x - 1)/(1x + 3)) - 5)^2 + 4(-2((1x - 1)/(1x + 3)) - 5) + 2 and g(x) = (1(1e^(-1x) + 2) - 3)/(1(1e^(-1x) + 2) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2cos(3\u03c0((2x + 1)/(-3x))) - 1 and g(x) = -2x^2 + 2x + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1((-2(-1(-2sin(1\u03c0x - 2) + 2) + 4))/(-1(-1(-2sin(1\u03c0x - 2) + 2) + 4)))^2 - 1((-2(-1(-2sin(1\u03c0x - 2) + 2) + 4))/(-1(-1(-2sin(1\u03c0x - 2) + 2) + 4))) and g(x) = -2((-2(2cos(3\u03c0x - 3) - 2) - 1)/(-1(2cos(3\u03c0x - 3) - 2) - 2))^2 - 3((-2(2cos(3\u03c0x - 3) - 2) - 1)/(-1(2cos(3\u03c0x - 3) - 2) - 2)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(-1cos(2\u03c0x - 1) - 3) - 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": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3cos(3\u03c0((1(-3x - 3) + 3)/(-2(-3x - 3) - 2)) - 2) + 1 and g(x) = (2(-2|x - 1| - 1) + 3)/(3(-2|x - 1| - 1) + 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 solutions does the equation -2((2(-1|(1x^2 + 5x + 4) - 3| - 2) + 1)/(1(-1|(1x^2 + 5x + 4) - 3| - 2) + 3)) = (1(5(-3|x - 2| + 2)^2 - 4(-3|x - 2| + 2) - 5) - 3)/(1(5(-3|x - 2| + 2)^2 - 4(-3|x - 2| + 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": "How many points of intersection exist between the functions f(x) = 2(1(1|x + 3| - 1)^2 + 1(1|x + 3| - 1) - 3) - 5 and g(x) = -1cos(1\u03c0(3|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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(5(2x - 2)^2 + 1(2x - 2)))/(2(5(2x - 2)^2 + 1(2x - 2)) - 3) and g(x) = -1sin(3\u03c0(-2x^2 - 2x - 1) + 3) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4((1(-1sin(3\u03c0(1|x + 1| + 3) - 1) - 2) - 3)/(-2(-1sin(3\u03c0(1|x + 1| + 3) - 1) - 2)))^2 + 4((1(-1sin(3\u03c0(1|x + 1| + 3) - 1) - 2) - 3)/(-2(-1sin(3\u03c0(1|x + 1| + 3) - 1) - 2))) - 1 and g(x) = -2((2(-2|x + 3| + 2) + 1)/(2(-2|x + 3| + 2) - 1))^2 + 4((2(-2|x + 3| + 2) + 1)/(2(-2|x + 3| + 2) - 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5((-1(-1(-1|x| + 3) - 4) - 3)/(3(-1(-1|x| + 3) - 4) + 3))^2 - 1 and g(x) = 1cos(1\u03c0((-1(5x + 5) + 1)/(-2(5x + 5) + 3)) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4((2(2e^(2x) - 2) - 2)/(-3(2e^(2x) - 2) + 2))^2 + 4 and g(x) = (2(3cos(2\u03c0x + 2) + 2) - 2)/(-3(3cos(2\u03c0x + 2) + 2) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(3cos(2\u03c0x + 3) + 3) + 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": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(1cos(3\u03c0(-1x) + 3) + 1) + 1| - 2 and g(x) = (-2(1x) + 2)/(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) = -5(-1e^(2x) + 1)^2 - 3 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(1cos(1\u03c0(3x - 1)) - 1) + 3)/(3(1cos(1\u03c0(3x - 1)) - 1) + 2) and g(x) = 2(2cos(3\u03c0x + 2) - 1)^2 - 4(2cos(3\u03c0x + 2) - 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(5(2|x + 3| + 2) - 3)^2 + 3(5(2|x + 3| + 2) - 3) + 3 and g(x) = 3e^(-1(-3x^2 - 3x + 4)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4((2(-2cos(2\u03c0x) - 1))/(3(-2cos(2\u03c0x) - 1) - 2)) + 1 and g(x) = -3|(1cos(1\u03c0x) + 1) + 3| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(-5(5(2cos(3\u03c0x) - 1)^2 - 3(2cos(3\u03c0x) - 1) + 3) + 5) - 3| - 1 and g(x) = -2(-3e^(-1((3x - 2)/(-3x + 2))) + 3)^2 + 3(-3e^(-1((3x - 2)/(-3x + 2))) + 3) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-3(-2|x - 3| + 3)^2 + 4(-2|x - 3| + 3) + 1) - 1 and g(x) = (3(-1sin(3\u03c0x) - 3) + 2)/(3(-1sin(3\u03c0x) - 3) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|((2(-4(-2x^2 - 3x + 3) - 3) - 2)/(3(-4(-2x^2 - 3x + 3) - 3) + 1)) + 2| + 2 = 1cos(1\u03c0(1((-1x - 3)/(3x + 2)) + 2) + 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(1|((3(2x^2 - 2x - 5))/(2(2x^2 - 2x - 5) - 1)) - 1| - 3) + 4 and g(x) = (2(-1(1|x| - 3) + 3) - 3)/(2(-1(1|x| - 3) + 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": "Find the number of times the graphs of f(x) = 5(1(1|((-2x - 1)/(1x + 3))| - 2)^2 + 1(1|((-2x - 1)/(1x + 3))| - 2) - 1) + 5 and g(x) = (-1(-2|(5x + 5) - 1| + 2) + 1)/(-3(-2|(5x + 5) - 1| + 2) + 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 1(-3|x - 2| - 1) + 1 = -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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|((1(3x + 4) - 2)/(3(3x + 4) - 2)) + 2| - 2 and g(x) = -1(2cos(3\u03c0x + 3) + 3)^2 + 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": "How many solutions does the equation 3sin(1\u03c0(-1(4(1|x + 1| - 1)^2 + 3(1|x + 1| - 1) - 2) + 3) - 3) = 2(5((-2x + 3)/(1x + 3))^2 - 5((-2x + 3)/(1x + 3)) - 2) + 5 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) = -1|(-4(-1sin(3\u03c0x)) - 2) + 2| + 1 and g(x) = (1(1|x - 3| + 2) - 3)/(2(1|x - 3| + 2) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(-2e^(2(-3x^2 + 4x - 3)) + 3) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(2(-3cos(3\u03c0((-2x + 3)/(2x - 3))) + 3)^2 + 1(-3cos(3\u03c0((-2x + 3)/(2x - 3))) + 3) - 4) + 1 and g(x) = 3(-1|((-3x + 1)/(-2x)) + 2| + 3) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|((3(-3(2x)^2 - 5(2x) + 5) + 2)/(-1(-3(2x)^2 - 5(2x) + 5) + 3)) + 2| - 1 and g(x) = (-2(-4(1x^2 - 3) - 3))/(-2(-4(1x^2 - 3) - 3) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-1(-4x^2 - 5x) - 4) + 3| - 1 and g(x) = -2|(5x^2 + 5x - 5)| + 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(3(3(3|x - 2| + 2)^2 + 1(3|x - 2| + 2) + 1) + 2) - 2)/(3(3(3(3|x - 2| + 2)^2 + 1(3|x - 2| + 2) + 1) + 2) + 3) and g(x) = (1(2|(1x - 4) - 1| + 3) - 3)/(2(2|(1x - 4) - 1| + 3) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(1(-3(3|x - 1| - 3)^2 + 4(3|x - 1| - 3) + 2) + 3) + 1)/(-3(1(-3(3|x - 1| - 3)^2 + 4(3|x - 1| - 3) + 2) + 3) + 4) and g(x) = 1cos(2\u03c0(-2(3|x| - 1)^2 + 2(3|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": "88", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2e^(-1(-1(-2|((3x - 1)/(-1x + 2)) + 2| + 1) - 2)) - 3 = -3|(5(-4x^2 + 4x - 1)) + 1| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(1((3x - 3)/(3x - 2)) + 1) - 2| + 3 = -2cos(1\u03c0(-1x^2 + 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": "202", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-1sin(2\u03c0x) - 3) + 2| - 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": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2e^(2(-4((-1(-1|x - 3| + 3) - 3)/(1(-1|x - 3| + 3)))^2 + 1((-1(-1|x - 3| + 3) - 3)/(1(-1|x - 3| + 3))) + 4)) + 1 and g(x) = -3cos(2\u03c0((-2(-2x + 2) + 2)/(2(-2x + 2) - 1)) - 1) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|((2(2(-1x^2 + 4x + 2) - 3) - 1)/(2(2(-1x^2 + 4x + 2) - 3))) - 2| + 3 = (-1(1(-5x + 4)^2 + 1(-5x + 4) - 1) + 1)/(-1(1(-5x + 4)^2 + 1(-5x + 4) - 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 solutions does the equation -3|(-1(1x^2 - 2x + 1) + 3) + 3| + 2 = -2sin(1\u03c0((2x + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(-2(-1((-1x - 3)/(3x - 1)) + 1)^2 + 1) + 2| + 1 = 2(-1|((-3x + 3)/(-1x - 1)) + 1| + 1)^2 - 3(-1|((-3x + 3)/(-1x - 1)) + 1| + 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1cos(2\u03c0(3|(-4x^2 + 1x - 2) - 3| + 1) + 3) + 2 = -3(2x - 4)^2 + 3(2x - 4) + 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": "How many points of intersection exist between the functions f(x) = -2cos(2\u03c0(1|((2x - 3)/(2x)) + 3| + 3) - 1) - 1 and g(x) = -3sin(3\u03c0(-3|x| - 2) - 3) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "126", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(1e^(2x) - 2) - 3| - 2 = -2x have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((-3(-2|x - 3| + 3) - 3)/(-1(-2|x - 3| + 3) + 3)) and g(x) = 4(-2x + 3)^2 + 4(-2x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(2|x - 3| - 1) - 3)/(1(2|x - 3| - 1) - 3) and g(x) = (-3x + 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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3((-1(3cos(3\u03c0(3|x + 1| - 2) + 2) - 1))/(-1(3cos(3\u03c0(3|x + 1| - 2) + 2) - 1) - 3)) - 4 = -1((-2(-2|x + 2| + 3) + 3)/(2(-2|x + 2| + 3) + 1))^2 + 5((-2(-2|x + 2| + 3) + 3)/(2(-2|x + 2| + 3) + 1)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|((1(-3e^(-1x) + 3) + 2)/(3(-3e^(-1x) + 3) + 1)) - 2| + 2 and g(x) = 3sin(1\u03c0(4x^2 - 5x) - 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "770", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(3sin(2\u03c0(2(-5x + 4)^2 + 3(-5x + 4) + 1) - 1) + 3) - 3)/(2(3sin(2\u03c0(2(-5x + 4)^2 + 3(-5x + 4) + 1) - 1) + 3) - 1) and g(x) = 2sin(3\u03c0(-2(3|x + 2| - 3)^2 - 2(3|x + 2| - 3))) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "104", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1e^(1((-2(1|(-5x^2 + 2x - 4) + 2| - 3) + 1)/(2(1|(-5x^2 + 2x - 4) + 2| - 3) + 1))) + 1 and g(x) = -5(3((-2x - 2)/(1x + 3)) - 4)^2 - 5(3((-2x - 2)/(1x + 3)) - 4) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(1sin(3\u03c0x - 1) - 3) + 2)/(-1(1sin(3\u03c0x - 1) - 3) - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(2|(1x + 5)| - 1) + 3)/(1(2|(1x + 5)| - 1) - 2) and g(x) = -3sin(1\u03c0(-3|x - 2| - 2) + 1) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1((-2(-3|(1x + 2) - 3| + 1))/(-2(-3|(1x + 2) - 3| + 1) + 1))^2 - 3((-2(-3|(1x + 2) - 3| + 1))/(-2(-3|(1x + 2) - 3| + 1) + 1)) - 4 and g(x) = -1((-2(-2sin(3\u03c0x - 1) + 2) - 3)/(-1(-2sin(3\u03c0x - 1) + 2) + 2))^2 - 5((-2(-2sin(3\u03c0x - 1) + 2) - 3)/(-1(-2sin(3\u03c0x - 1) + 2) + 2)) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5((-1x - 3)/(-2x + 1)) - 5 and g(x) = (-1x - 1)/(1x + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2((1(-3|(-4x^2 + 2) - 2| + 2) - 2)/(3(-3|(-4x^2 + 2) - 2| + 2) - 2)) - 1 and g(x) = (-2(3(-2x^2 + 4x - 4) - 1))/(1(3(-2x^2 + 4x - 4) - 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2sin(3\u03c0(-3|((-2x + 3)/(-1x - 1)) - 3|) + 3) - 1 = 1((-3x)/(3x - 1))^2 + 1((-3x)/(3x - 1)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "201", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2cos(3\u03c0((1(-3|x - 1| - 1) - 1)/(2(-3|x - 1| - 1) + 1)) - 3) - 3 = -1sin(3\u03c0(4x^2 + 5x + 1) + 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(-3|(2((-2x)/(1x - 3))^2 + 3((-2x)/(1x - 3)) + 4) - 2| + 3) + 3 and g(x) = 1(2cos(2\u03c0(-5x - 3) - 3))^2 - 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-4x + 5) + 1)/(-3(-4x + 5)) and g(x) = 3|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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3((-3(3|x + 1| - 3))/(-2(3|x + 1| - 3) - 2)) - 1 and g(x) = -3(-1x^2 + 4x + 4) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(3(2x^2 + 2x + 4)) + 2)/(3(3(2x^2 + 2x + 4))) and g(x) = 5(1|x| - 2)^2 - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(1(5x - 1)^2 + 5(5x - 1)) + 2)/(2(1(5x - 1)^2 + 5(5x - 1))) = 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(-5(-2cos(3\u03c0(2x^2 + 2x + 5) - 2) + 1) - 1) - 2| - 3 and g(x) = 2cos(3\u03c0((2(-5x^2 + 5x) - 3)/(3(-5x^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": "728", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-1|(5x^2 + 1x - 2) - 2|) - 2)/(-1(-1|(5x^2 + 1x - 2) - 2|) - 2) and g(x) = -4((-2x - 2)/(-1x - 2)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(3|x + 2|) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(1(3x - 2)^2 - 1(3x - 2) - 5) + 1| - 1 = -2(3|x + 1| - 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(1e^(-1x) - 3) - 1)/(2(1e^(-1x) - 3)) 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(-1((3x - 1)/(-3x - 2)) + 1)^2 + 4(-1((3x - 1)/(-3x - 2)) + 1) + 1 and g(x) = 3|(-4x^2 + 1x - 3) + 2| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3e^(-1(-2x^2 + 3x + 4)) - 3 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(2|x - 1| + 2) - 4 and g(x) = 1x^2 + 2x - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(4(-1|((-3x - 3)/(1x + 3)) - 2| + 1) + 3)^2 - 4(4(-1|((-3x - 3)/(1x + 3)) - 2| + 1) + 3) + 2 and g(x) = -1(-2sin(2\u03c0(-1x^2 + 1x - 2) - 2) - 3) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4((1(-1|(-4x + 5)| - 1) - 3)/(3(-1|(-4x + 5)| - 1) - 2))^2 and g(x) = -2(3((-3x + 2)/(2x - 1)) + 2)^2 - 3(3((-3x + 2)/(2x - 1)) + 2) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-1(1|(3e^(1x) - 3) + 2| + 3)))/(3(-1(1|(3e^(1x) - 3) + 2| + 3)) - 1) and g(x) = (3(3|(-2x^2 - 4x + 5)| - 1) - 3)/(1(3|(-2x^2 - 4x + 5)| - 1)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(-3(-2sin(3\u03c0(3x^2 + 3x)) + 3) + 2) + 1)/(-3(-3(-2sin(3\u03c0(3x^2 + 3x)) + 3) + 2) - 2) = 1e^(-2((2(2x^2 - 4x - 1))/(-2(2x^2 - 4x - 1) - 1))) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2((-3x + 1)/(1x))^2 + 1((-3x + 1)/(1x)) - 2 and g(x) = -5x^2 - 2x + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(3((2x)/(-3x - 3)) + 3) - 3| - 2 and g(x) = -3(-3x^2 - 1x + 4) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(4x^2 - 2x - 4) - 1)/(-1(4x^2 - 2x - 4) - 1) 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(2sin(1\u03c0x + 2)) - 2)/(1(2sin(1\u03c0x + 2)) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1sin(1\u03c0(1x^2 + 4x - 3) - 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": "36", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-3(2|(1x - 2) - 1| - 2)^2 - 1(2|(1x - 2) - 1| - 2) + 3) + 2)/(-1(-3(2|(1x - 2) - 1| - 2)^2 - 1(2|(1x - 2) - 1| - 2) + 3) + 3) and g(x) = -2|(5((-3x - 3)/(-3x + 1)) + 3) - 1| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(1(3|((-2x - 2)/(3x + 3)) - 1| - 1)^2 + 5(3|((-2x - 2)/(3x + 3)) - 1| - 1) + 1) - 3 and g(x) = -4((-3(2|x| - 1))/(-1(2|x| - 1) + 3)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(3x^2 - 5) - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(-2cos(2\u03c0(-2|x|) + 3) + 2) + 4 and g(x) = (-2(-3cos(1\u03c0x)) + 1)/(1(-3cos(1\u03c0x)) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(-2x^2 - 5x + 5) - 2 and g(x) = (-2x + 1)/(1x - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-2(3(2|x + 1| + 2) + 2)^2 - 4(3(2|x + 1| + 2) + 2) - 5) - 3)/(1(-2(3(2|x + 1| + 2) + 2)^2 - 4(3(2|x + 1| + 2) + 2) - 5) + 2) and g(x) = 3|(1(-2cos(3\u03c0x + 2) - 3) + 1) - 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(3x + 2)^2 - 3(3x + 2) + 5 and g(x) = -3x^2 - 1x + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(1x^2 + 3x - 5) - 2| - 3 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(3(2sin(1\u03c0(-3|x + 1| - 1) - 1))^2 + 1(2sin(1\u03c0(-3|x + 1| - 1) - 1)) - 3) + 2 and g(x) = 1((2(2|x - 3| + 3) + 2)/(1(2|x - 3| + 3))) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-2(-4x - 1)^2 + 1(-4x - 1) + 3) + 1)/(2(-2(-4x - 1)^2 + 1(-4x - 1) + 3) + 1) and g(x) = 2|(-3x^2 - 4x + 1) - 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(2(-1cos(3\u03c0x - 3) + 2)^2 + 4(-1cos(3\u03c0x - 3) + 2) - 3) - 3| + 3 and g(x) = 2e^(2(1x)) - 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": "How many solutions does the equation -5(-2(1|(-1cos(2\u03c0x) - 3) + 3| - 1) + 3)^2 + 3(-2(1|(-1cos(2\u03c0x) - 3) + 3| - 1) + 3) - 4 = 2(-3((2x + 2)/(-1x - 3))^2 - 2((2x + 2)/(-1x - 3)) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(-1|(-2sin(1\u03c0x) - 3) + 2| + 3)^2 - 3(-1|(-2sin(1\u03c0x) - 3) + 2| + 3) - 5 and g(x) = -1(2sin(1\u03c0x + 1)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(2cos(2\u03c0x + 3) + 2) - 1| + 2 and g(x) = (-3x)/(-2x + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "70", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(2cos(3\u03c0(2x^2 + 1x) + 3) - 1) + 3)/(-3(2cos(3\u03c0(2x^2 + 1x) + 3) - 1)) = -5(3cos(2\u03c0x - 3) - 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "177", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|((2(-1x^2 - 3x + 2) - 3)/(-1(-1x^2 - 3x + 2) + 3)) - 1| - 3 and g(x) = (-3(1x - 5) + 3)/(-3(1x - 5) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3((2(3x - 5) + 2)/(1(3x - 5) + 3))^2 - 4((2(3x - 5) + 2)/(1(3x - 5) + 3)) + 3 and g(x) = -1|((-2x - 2)/(1x - 2)) + 1| + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(5(-2|x| + 1) + 1) + 2)/(-2(5(-2|x| + 1) + 1) - 2) and g(x) = 2cos(3\u03c0(5x^2 + 3) + 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(3cos(3\u03c0x + 3) + 3) + 4 = 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": "24", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(-1sin(2\u03c0(1|x - 3| + 2) - 3) - 1) - 4 and g(x) = (1(1x^2 - 4x - 2) - 2)/(3(1x^2 - 4x - 2) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|((-2(1sin(2\u03c0x) - 1))/(-1(1sin(2\u03c0x) - 1) - 2)) + 1| and g(x) = (-2(-1x^2 + 5x - 4) + 1)/(-1(-1x^2 + 5x - 4) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(-3sin(1\u03c0x + 3)) - 4 = -4x^2 + 2x - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1e^(1(-2|x + 2| + 3)) + 2 = 1x^2 - 2x have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|((1x + 2)/(-3x - 1)) - 1| + 3 and g(x) = 2x^2 + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(1sin(2\u03c0(-1(1x^2 - 4x - 1) + 4) - 1) - 2) - 1| + 3 and g(x) = -3(-5((2x + 2)/(-3x + 4)) - 1)^2 + 4(-5((2x + 2)/(-3x + 4)) - 1) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "152", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(2sin(1\u03c0((-3x + 3)/(-2x - 3)) + 3) + 2)^2 + 3(2sin(1\u03c0((-3x + 3)/(-2x - 3)) + 3) + 2) + 4 = 3|(-1e^(-1x) + 3) + 3| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-2|(2(1x^2 - 2x - 5) + 1) + 2| - 2) - 2)/(2(-2|(2(1x^2 - 2x - 5) + 1) + 2| - 2) + 2) and g(x) = -1|(1cos(2\u03c0((1x + 1)/(-3x - 2)) + 2) - 1) - 2| + 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) = 1e^(-1(1|(-5x - 1)| + 3)) - 1 and g(x) = (-3(1|x + 3| - 1) + 3)/(1(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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-2sin(1\u03c0x - 2)) + 2 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 solutions does the equation -1(-4((-2(1|x + 1|) + 3)/(1(1|x + 1|))) + 4)^2 - 3(-4((-2(1|x + 1|) + 3)/(1(1|x + 1|))) + 4) - 3 = 4(2((-1x - 3)/(-2x - 3))^2 - 5((-1x - 3)/(-2x - 3)) - 2) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(2|x + 1| - 2))/(3(2|x + 1| - 2) + 1) = (2x)/(2x - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-4((-2(1cos(2\u03c0x + 1)) + 1)/(2(1cos(2\u03c0x + 1)) - 1))^2 + 4((-2(1cos(2\u03c0x + 1)) + 1)/(2(1cos(2\u03c0x + 1)) - 1)) + 3) and g(x) = (1(5(2sin(1\u03c0x + 1))) + 1)/(2(5(2sin(1\u03c0x + 1))) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(-2sin(3\u03c0x - 3) - 3)^2 - 3(-2sin(3\u03c0x - 3) - 3) + 5 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": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2cos(1\u03c0((3(2|(-1x^2 - 5x - 1) + 3| - 3) - 1)/(-1(2|(-1x^2 - 5x - 1) + 3| - 3) + 2)) + 3) - 1 and g(x) = 1(2e^(-1(4x^2 + 1x)) + 1) - 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": "How many solutions does the equation 2cos(2\u03c0(1|x - 1| - 3) + 3) + 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": "9", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(2(1((-1x)/(-1x + 2))^2 + 2((-1x)/(-1x + 2)) + 1) - 2) + 1| = -2((-3(-5x^2 + 3x - 2) + 2)/(2(-5x^2 + 3x - 2))) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3((3(-3|x - 1| + 3) + 3)/(3(-3|x - 1| + 3) + 1)) - 2 and g(x) = -3(1cos(3\u03c0x + 2)) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(4(1cos(1\u03c0((1x - 1)/(-3x + 2)) - 1) + 2)^2 + 3(1cos(1\u03c0((1x - 1)/(-3x + 2)) - 1) + 2) + 2) - 5 and g(x) = -1(-2|((2x)/(3x + 2)) - 1| - 2)^2 - 2(-2|((2x)/(3x + 2)) - 1| - 2) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4((1(-3sin(1\u03c0x - 2) + 1) - 1)/(-2(-3sin(1\u03c0x - 2) + 1)))^2 + 1((1(-3sin(1\u03c0x - 2) + 1) - 1)/(-2(-3sin(1\u03c0x - 2) + 1))) + 5 and g(x) = (-1(5x^2 - 5x - 5) + 2)/(-2(5x^2 - 5x - 5) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3(2(-1sin(1\u03c0x + 3) - 1) + 2) - 1)/(2(2(-1sin(1\u03c0x + 3) - 1) + 2) - 1) = (-1(-1e^(-1x) - 3))/(3(-1e^(-1x) - 3) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|(-3((1x - 1)/(-2x + 1))^2 - 2((1x - 1)/(-2x + 1)) - 4) + 2| + 1 = -1(3|x| - 1)^2 + 5(3|x| - 1) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3((-3(-1|(-4x^2 + 2x - 2)| - 2) + 2)/(2(-1|(-4x^2 + 2x - 2)| - 2) + 1)) + 1 and g(x) = -3cos(3\u03c0(2|(2x + 1) - 2| - 3)) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "234", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-3|((1(-1x + 2) + 1)/(3(-1x + 2) + 1)) + 1| - 3)^2 + 1(-3|((1(-1x + 2) + 1)/(3(-1x + 2) + 1)) + 1| - 3) + 3 and g(x) = -4(3|(-3e^(-2x) + 3) + 2| + 3) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(1\u03c0(3x + 3) - 3) + 1 = 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": "How many points of intersection exist between the functions f(x) = -2|(-1sin(2\u03c0x) - 2) - 2| - 1 and g(x) = 4x + 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-3((3x - 3)/(-3x + 1))^2 - 4((3x - 3)/(-3x + 1)) - 4) - 5 and g(x) = -1|((-2x - 2)/(2x + 1)) + 3| + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(2e^(2x) - 3))/(1(2e^(2x) - 3)) 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(-4((-2x + 3)/(2x))^2 - 5((-2x + 3)/(2x)) + 4) + 1| + 1 and g(x) = 3((-1x + 1)/(3x + 1))^2 + 2((-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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(1(-3x^2 + 2x + 4) - 1) + 1)/(-2(1(-3x^2 + 2x + 4) - 1) + 3) and g(x) = 4(2sin(1\u03c0x + 2) - 3) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(-1e^(-1(4(1x - 1)^2 + 3(1x - 1) + 1)) + 3) + 2| and g(x) = 1((-3(3x^2 + 1x - 3) + 1)/(1(3x^2 + 1x - 3) + 3)) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|((1x - 1)/(3x + 3)) - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-2sin(2\u03c0(3x^2 + 1x)) + 1) + 2 and g(x) = 2sin(3\u03c0((1x + 3)/(1x + 3)) - 3) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1139", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2sin(2\u03c0(-1|x| + 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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(-1e^(2x) + 2)^2 - 1(-1e^(2x) + 2) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1((-1(-4x - 5) - 2)/(1(-4x - 5) + 3))^2 + 3((-1(-4x - 5) - 2)/(1(-4x - 5) + 3)) - 5 and g(x) = 2|(-5x - 5) + 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(3|(4x - 3)| - 3) + 2)/(3(3|(4x - 3)| - 3) + 1) and g(x) = (-1(5x - 2) + 3)/(-2(5x - 2) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(-3sin(2\u03c0x + 3) + 1) - 3| - 1 = 4x - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(2cos(2\u03c0((-1(2x - 1))/(1(2x - 1) - 2)) - 1) + 1) + 3| = -5((1(-1|x + 3| + 1))/(2(-1|x + 3| + 1) + 2)) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(-1(-3|(2x^2 + 1x + 2) + 2| - 1) - 5) - 3)/(-3(-1(-3|(2x^2 + 1x + 2) + 2| - 1) - 5) - 1) and g(x) = -3(2((1x + 3)/(-3x - 1))^2 - 1((1x + 3)/(-3x - 1)) - 5) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(-1|((1(-3x^2 + 3x - 4) + 3)/(2(-3x^2 + 3x - 4) - 1)) + 3|) + 5 and g(x) = (1(-5(2|x + 2| - 2) + 1) + 3)/(1(-5(2|x + 2| - 2) + 1) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(3e^(1(-3|((-1x + 2)/(-1x)) - 1| - 2)) - 2) and g(x) = (-1(2(-1x^2 + 3x + 3) - 4))/(2(2(-1x^2 + 3x + 3) - 4) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-2sin(2\u03c0(-1|x + 1| - 3) - 3) - 1) - 3 and g(x) = 3sin(1\u03c0(-3x - 2) - 1) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3cos(2\u03c0(1(-4(-3|x + 3| + 2) - 4)^2 + 5(-4(-3|x + 3| + 2) - 4) - 2) - 1) and g(x) = -2sin(2\u03c0((-1(1|x - 2| - 2) + 2)/(-3(1|x - 2| - 2) - 2)) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1423", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(4(2sin(3\u03c0(-5x - 1) + 3) + 2)^2 - 3) + 2| - 2 = (2(2(-3sin(3\u03c0x - 2)) - 2) - 3)/(-1(2(-3sin(3\u03c0x - 2)) - 2) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "163", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4((1(2(3|x| + 2)^2 - 5(3|x| + 2) + 1) + 3)/(1(2(3|x| + 2)^2 - 5(3|x| + 2) + 1))) + 4 and g(x) = -1e^(2(5(3|x| - 1)^2 - 3(3|x| - 1) - 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(-2(3|(-1x + 5)| + 1)^2 + 3(3|(-1x + 5)| + 1) + 5) + 2)/(-1(-2(3|(-1x + 5)| + 1)^2 + 3(3|(-1x + 5)| + 1) + 5) + 2) and g(x) = -2cos(2\u03c0((3(1|x - 3| + 2) - 1)/(-1(1|x - 3| + 2) - 3)) - 2) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1cos(1\u03c0(3|x - 3| + 3) - 1) + 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": "7", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(-3(1|(-5x^2 - 1x - 4) + 1|) + 1) - 2)/(-2(-3(1|(-5x^2 - 1x - 4) + 1|) + 1) - 2) = 5(2sin(3\u03c0((-3x - 1)/(2x)) + 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(3|x - 1| - 1))/(3(3|x - 1| - 1) + 3) 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(1(1(1|x + 2| - 2) - 2)^2 + 2(1(1|x + 2| - 2) - 2) + 1))/(-1(1(1(1|x + 2| - 2) - 2)^2 + 2(1(1|x + 2| - 2) - 2) + 1) + 2) and g(x) = (-3(3e^(-2(2|x| - 2))) + 2)/(1(3e^(-2(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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(-3((3x - 3)/(-2x - 2)) - 1)^2 - 5(-3((3x - 3)/(-2x - 2)) - 1) + 1 and g(x) = 4(-3x + 5)^2 - 2(-3x + 5) + 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) = 2sin(3\u03c0(-4(-1|(5x - 1) + 2| - 1)^2 + 4(-1|(5x - 1) + 2| - 1) + 3) + 1) - 2 and g(x) = -5(1|((-1x + 2)/(2x + 2)) - 3| - 1)^2 + 1(1|((-1x + 2)/(2x + 2)) - 3| - 1) + 2 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": "Find the number of times the graphs of f(x) = -2((-2(-2|x - 2| - 1) + 1)/(2(-2|x - 2| - 1) - 1)) - 2 and g(x) = (2(-1|x + 1| + 1) - 1)/(1(-1|x + 1| + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(3cos(3\u03c0(-2x^2 + 2) - 2) - 3) + 1 and g(x) = -1|(3sin(2\u03c0x + 1) + 1) - 1| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "970", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(3|(-3(-2x^2 - 5x - 2) - 5) + 3| + 1) + 3)/(-3(3|(-3(-2x^2 - 5x - 2) - 5) + 3| + 1) + 1) = 3sin(2\u03c0(4(-5x^2 - 1x - 2) - 4) - 2) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1433", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(-1cos(3\u03c0x - 1) - 1) + 1)/(-1(-1cos(3\u03c0x - 1) - 1) + 2) 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1sin(1\u03c0(-3(1|x - 1|) + 3) + 1) - 2 and g(x) = 2(2|x + 3|)^2 - 2(2|x + 3|) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4((-1(-1(1|x - 3| + 1)^2 + 2(1|x - 3| + 1) + 1))/(3(-1(1|x - 3| + 1)^2 + 2(1|x - 3| + 1) + 1) + 2)) - 5 and g(x) = -1|(-1(1x - 2)^2 + 2(1x - 2) - 4) - 3| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(-2|(-2e^(1x) + 2) - 1|) + 5 and g(x) = -3sin(1\u03c0(-5x^2 - 1x - 1) - 3) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "479", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-3|(-2sin(1\u03c0(-4x + 2)) + 2) - 1| + 2)^2 - 3(-3|(-2sin(1\u03c0(-4x + 2)) + 2) - 1| + 2) + 1 and g(x) = -3(2((1x + 1)/(1x + 2)))^2 + 1(2((1x + 1)/(1x + 2))) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "198", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(2sin(3\u03c0x + 2) + 1) - 2| - 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": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1cos(2\u03c0(-1x + 3) + 3) + 2 and g(x) = -1x^2 - 2x + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(3(5x + 5)^2 + 3(5x + 5)) - 3)/(-3(3(5x + 5)^2 + 3(5x + 5)) + 2) and g(x) = -3e^(-1((2x - 2)/(3x + 1))) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(1(3x^2 + 2) + 3)| - 3 and g(x) = 3|(5x - 2)| + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3sin(1\u03c0(-5(-1|((3x + 1)/(2x - 1)) - 2| + 1)^2 - 4(-1|((3x + 1)/(2x - 1)) - 2| + 1) + 2)) and g(x) = 1|(1cos(2\u03c0((1x + 3)/(1x + 2))) - 1) - 1| + 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": "Determine the number of intersection points between the functions f(x) = -3((-1(1|(4x + 3) - 2| + 2) + 3)/(-2(1|(4x + 3) - 2| + 2) - 2))^2 - 3((-1(1|(4x + 3) - 2| + 2) + 3)/(-2(1|(4x + 3) - 2| + 2) - 2)) and g(x) = 3((1(-4x^2) + 3)/(1(-4x^2))) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(4(1(-2|x + 2| + 3)^2 + 4(-2|x + 2| + 3) - 3) - 4) - 2)/(-3(4(1(-2|x + 2| + 3)^2 + 4(-2|x + 2| + 3) - 3) - 4)) = 3sin(1\u03c0(-5((2x + 3)/(-2x - 3))^2 + 3((2x + 3)/(-2x - 3)) - 3) - 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-1cos(2\u03c0x - 2))^2 - 5(-1cos(2\u03c0x - 2)) - 3 and g(x) = 4x^2 - 2x - 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(3|(2cos(3\u03c0x + 2) + 1) - 3|) - 3)/(3(3|(2cos(3\u03c0x + 2) + 1) - 3|) + 3) and g(x) = -1cos(3\u03c0(-3x^2 + 2) - 3) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1sin(1\u03c0((-1(1x^2 + 1x + 3) - 2)/(1(1x^2 + 1x + 3))) - 1) - 2 and g(x) = 5(4x + 5)^2 + 4(4x + 5) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(-1x^2 + 2x - 2)| - 2 and g(x) = 3x^2 + 4x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(3\u03c0(4x + 1)) - 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": "71", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(-3cos(3\u03c0x)))/(3(-3cos(3\u03c0x)) + 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": "29", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(-3cos(1\u03c0x + 2) - 3)^2 + 2(-3cos(1\u03c0x + 2) - 3) 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": "19", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(1|(2(2sin(3\u03c0x - 3) + 2)^2 - 1(2sin(3\u03c0x - 3) + 2) - 1)| + 1) + 2 and g(x) = -5(-1sin(2\u03c0(-3|x - 2| + 2) - 2) + 1) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((-2(-1sin(3\u03c0(-1x^2 + 4x - 3) - 2)) - 3)/(3(-1sin(3\u03c0(-1x^2 + 4x - 3) - 2)) - 1)) + 2 and g(x) = -1e^(1((-1(3|x - 3| - 3) - 1)/(1(3|x - 3| - 3) + 2))) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "407", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(-2sin(3\u03c0(-3((1x - 2)/(1x)) - 3) - 2) + 2) - 3| + 2 and g(x) = (-1(4(1cos(2\u03c0x + 1) + 3)^2 - 2(1cos(2\u03c0x + 1) + 3) + 5))/(3(4(1cos(2\u03c0x + 1) + 3)^2 - 2(1cos(2\u03c0x + 1) + 3) + 5) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "146", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3sin(2\u03c0(-2|x - 3| + 2) + 1) + 1 and g(x) = 1x^2 + 4x - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(1((-1x)/(-1x + 2))^2 + 3((-1x)/(-1x + 2)) - 2) - 1| + 3 and g(x) = (2(1x^2 + 3x + 4) + 1)/(-2(1x^2 + 3x + 4) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4((3(2cos(1\u03c0x - 1)) + 2)/(2(2cos(1\u03c0x - 1)) - 1))^2 + 3((3(2cos(1\u03c0x - 1)) + 2)/(2(2cos(1\u03c0x - 1)) - 1)) - 4 and g(x) = (-3(3x^2 - 3x - 3) - 1)/(-1(3x^2 - 3x - 3) + 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": "Determine the number of intersection points between the functions f(x) = -2(-3|x - 1| - 1)^2 - 4(-3|x - 1| - 1) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(4x^2 - 3x - 1) + 5 and g(x) = (-3x)/(-1x + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-1(-2|x + 2| - 3) - 3) + 3)/(3(-1(-2|x + 2| - 3) - 3) + 1) and g(x) = 1cos(1\u03c0((3x - 2)/(-2x + 1)) + 1) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(4(3|x - 3| - 2) - 2) + 1)/(-1(4(3|x - 3| - 2) - 2) - 3) and g(x) = 3cos(3\u03c0(-2x)) 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) = -1e^(1(1|x - 2| - 3)) 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4((-1(1e^(-1x) + 1) + 2)/(2(1e^(-1x) + 1) - 3)) + 1 and g(x) = -1sin(2\u03c0((-1x - 2)/(-1x - 1)) + 2) 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": "How many solutions does the equation 3cos(1\u03c0(-1(2|((1x + 1)/(3x - 2))| - 3)^2 - 3) + 2) + 2 = -4((-2(-1sin(3\u03c0x - 3) + 1))/(-3(-1sin(3\u03c0x - 3) + 1) + 2))^2 - 1((-2(-1sin(3\u03c0x - 3) + 1))/(-3(-1sin(3\u03c0x - 3) + 1) + 2)) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "102", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4((3(-5x - 2) + 1)/(1(-5x - 2) + 1))^2 + 2((3(-5x - 2) + 1)/(1(-5x - 2) + 1)) + 1 and g(x) = -2|(-3x + 5) + 1| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(-3x^2 - 1) - 1)/(-3(-3x^2 - 1) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(-2e^(-1x) + 3)^2 + 5(-2e^(-1x) + 3) - 1 = -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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(1((-1(3x^2 + 1x) - 1)/(2(3x^2 + 1x) + 1)) - 2) - 1| - 1 and g(x) = -3((-2(2|x - 2| - 1) + 1)/(2(2|x - 2| - 1) + 3)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(2\u03c0(1(-4(1|x + 2|) + 3)^2 - 4(-4(1|x + 2|) + 3) + 3) + 3) and g(x) = (-1(-2|(-2sin(1\u03c0x + 1) - 2) + 1|) + 2)/(2(-2|(-2sin(1\u03c0x + 1) - 2) + 1|) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "905", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(1\u03c0(2x^2 + 1x - 5) + 1) and g(x) = (1x - 3)/(2x + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "397", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2sin(2\u03c0((-3(1(2x^2 - 2) - 1) - 2)/(1(1(2x^2 - 2) - 1) - 3)) + 3) + 2 = 1((-1(-2cos(1\u03c0x + 1)) - 2)/(1(-2cos(1\u03c0x + 1)))) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3((2(-2|x - 2| + 3) + 3)/(1(-2|x - 2| + 3) + 3)) - 3 and g(x) = 1(-4x^2 - 2x) + 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": "Find the number of times the graphs of f(x) = -2(-1x + 1)^2 - 4 and g(x) = -2|x + 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(3|(5x^2 + 2x - 1) - 3| - 2))/(-1(3|(5x^2 + 2x - 1) - 3| - 2) + 2) and g(x) = -2|((2x + 3)/(3x - 1)) - 3| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1cos(1\u03c0(-2x^2 - 5x) + 2) - 1 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": "371", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-4(-3(-1|x + 1| - 1) - 2)^2 + 2(-3(-1|x + 1| - 1) - 2) + 5) + 3)/(-2(-4(-3(-1|x + 1| - 1) - 2)^2 + 2(-3(-1|x + 1| - 1) - 2) + 5) - 1) and g(x) = 2|(-2((1x + 2)/(-1x))^2 + 1) - 1| - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-3|(2x^2 - 3x - 3) - 2| + 2) - 1 = 1(-4x - 3)^2 + 1(-4x - 3) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(-5((2(1|x| - 3) - 1)/(-2(1|x| - 3) + 3))^2 - 4) and g(x) = (-3(-3|(-5x + 5) + 2| + 3) - 3)/(-1(-3|(-5x + 5) + 2| + 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(-3(-2|(-3sin(3\u03c0x - 1)) + 3| + 2) - 1)^2 - 5 and g(x) = (1(-1|(-1e^(1x) + 3) - 3| - 2) + 1)/(2(-1|(-1e^(1x) + 3) - 3| - 2) + 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": "Find the number of times the graphs of f(x) = -3sin(3\u03c0((2(3x + 5) - 2)/(3(3x + 5) - 2))) - 1 and g(x) = (-1(2sin(2\u03c0x + 2) - 2) - 2)/(-2(2sin(2\u03c0x + 2) - 2) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(-1(-2x^2 + 3x - 2) - 5) - 3)/(-1(-1(-2x^2 + 3x - 2) - 5) - 2) and g(x) = -2cos(2\u03c0(1x + 4) + 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1((2(3x^2 - 4x) + 2)/(-3(3x^2 - 4x) - 1)) + 1 = 2e^(-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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(-1cos(3\u03c0(3|x + 3| + 3) + 3) + 1)^2 + 4(-1cos(3\u03c0(3|x + 3| + 3) + 3) + 1) + 5 and g(x) = 1|((-3x + 3)/(-2x + 3)) + 2| - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(2(-2sin(3\u03c0x - 3) - 2) + 2)^2 + 5(2(-2sin(3\u03c0x - 3) - 2) + 2) + 3 = 3sin(1\u03c0(-1x + 3)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(4x - 3) + 2| + 1 and g(x) = 4x^2 - 5x where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|(1(2sin(1\u03c0((-1x + 2)/(-2x + 3)) - 1) - 2)^2 - 5(2sin(1\u03c0((-1x + 2)/(-2x + 3)) - 1) - 2) - 5) + 2| - 2 and g(x) = -2cos(3\u03c0(2|((-1x - 3)/(2x - 1)) + 1| + 3) + 2) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-1((2(-1|x - 2| + 1) - 2)/(-3(-1|x - 2| + 1) - 3)) + 3)^2 - 1(-1((2(-1|x - 2| + 1) - 2)/(-3(-1|x - 2| + 1) - 3)) + 3) + 4 and g(x) = -3|(1(-3e^(1x) + 1) + 1) - 3| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1((2(-4(-3|x - 2|) - 5) - 2)/(-3(-4(-3|x - 2|) - 5)))^2 - 1((2(-4(-3|x - 2|) - 5) - 2)/(-3(-4(-3|x - 2|) - 5))) + 3 and g(x) = -1sin(2\u03c0(2(-3|x + 1| + 2)^2 + 3(-3|x + 1| + 2) + 1)) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "116", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(-3sin(3\u03c0(4x + 2)) + 2) - 3)/(2(-3sin(3\u03c0(4x + 2)) + 2) - 3) and g(x) = -3((-3x)/(2x - 1))^2 + 5((-3x)/(2x - 1)) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "83", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(-4x^2 + 5x + 2) - 2 = -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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2((-2(3e^(1x)))/(-2(3e^(1x)) - 3))^2 + 4((-2(3e^(1x)))/(-2(3e^(1x)) - 3)) + 2 and g(x) = 2sin(3\u03c0(5x - 4) - 3) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "190", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(-5((-2x - 2)/(-1x - 2)) + 3)^2 + 4(-5((-2x - 2)/(-1x - 2)) + 3) - 2 and g(x) = -3(-2e^(2x) - 2) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(-3sin(1\u03c0(-3|((3x - 1)/(3x + 3)) - 1| - 1) - 1))^2 + 5(-3sin(1\u03c0(-3|((3x - 1)/(3x + 3)) - 1| - 1) - 1)) - 2 = (2(2cos(1\u03c0(-4x^2 + 5x - 2) - 3) + 1) + 3)/(3(2cos(1\u03c0(-4x^2 + 5x - 2) - 3) + 1) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "341", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2(-2sin(2\u03c0(3|x + 2| + 2) + 2) + 3)^2 + 4(-2sin(2\u03c0(3|x + 2| + 2) + 2) + 3) + 3 and g(x) = 3|(-4x^2 + 2x + 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) = -3e^(-2(-1(5(2|x - 2|)^2 + 1(2|x - 2|) - 1) + 4)) + 1 and g(x) = -2|(-1((3x - 3)/(-2x))^2 - 4((3x - 3)/(-2x)) - 4)| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(-3(-1(2e^(2x) + 2) + 4)^2 + 3(-1(2e^(2x) + 2) + 4) + 1) + 1)/(3(-3(-1(2e^(2x) + 2) + 4)^2 + 3(-1(2e^(2x) + 2) + 4) + 1) + 3) = 2e^(2(2(-2|x - 2| - 3) - 5)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-2(-2|((-2x - 3)/(-3x - 3)) + 1| + 2)^2 - 4(-2|((-2x - 3)/(-3x - 3)) + 1| + 2) + 5) - 3 and g(x) = (-2(2|(-1x^2 - 4x - 2) + 1| + 2) + 2)/(3(2|(-1x^2 - 4x - 2) + 1| + 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(2|((2(-2x + 1) - 2)/(2(-2x + 1))) + 2| + 2)^2 - 4 and g(x) = (2(-1|(1x + 2) + 1| + 3) + 2)/(-2(-1|(1x + 2) + 1| + 3) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(-1|(-2((3x - 3)/(1x - 2)) - 3) - 3| + 2)^2 + 3(-1|(-2((3x - 3)/(1x - 2)) - 3) - 3| + 2) - 4 and g(x) = -1|(1((-1x - 3)/(1x))) - 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": "Determine the number of intersection points between the functions f(x) = -4(-1(-2|x - 1| + 2) + 1)^2 + 5(-1(-2|x - 1| + 2) + 1) + 4 and g(x) = 1(-1e^(1x) - 1)^2 + 3(-1e^(1x) - 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-1x^2 + 3x - 1) - 1)/(3(-1x^2 + 3x - 1) + 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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|(-3x - 1) + 2| - 2 = (3x)/(-1x) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(2(-4(1|x - 3|)^2 - 1(1|x - 3|) - 1)) - 1)/(1(2(-4(1|x - 3|)^2 - 1(1|x - 3|) - 1)) + 1) and g(x) = -3cos(2\u03c0(4((3x - 3)/(1x)) + 4) + 1) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "162", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(-3x - 1)^2 + 3(-3x - 1) - 2 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4((1x + 2)/(-1x + 2))^2 + 2((1x + 2)/(-1x + 2)) + 4 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1cos(2\u03c0(-1x^2 + 3x) + 1) - 1 and g(x) = 4x^2 - 2x - 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(-1(2e^(-1(-1|x - 1| - 2)) + 1)^2 - 5(2e^(-1(-1|x - 1| - 2)) + 1) - 3) - 2 and g(x) = -1((-2(-1x) - 3)/(-2(-1x) - 1))^2 - 2((-2(-1x) - 3)/(-2(-1x) - 1)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(4x + 2)^2 - 4(4x + 2) - 1 and g(x) = 4x^2 + 1x + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(-1x^2 - 5x) - 2)/(2(-1x^2 - 5x) + 1) 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3sin(2\u03c0(-1x^2 + 2x - 1) - 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": "34", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(-4((-1(2cos(1\u03c0x + 2) - 1) + 3)/(3(2cos(1\u03c0x + 2) - 1) - 1))^2 - 4((-1(2cos(1\u03c0x + 2) - 1) + 3)/(3(2cos(1\u03c0x + 2) - 1) - 1)) + 1) - 2| - 1 and g(x) = (-2(2cos(3\u03c0(4x + 4) - 2) - 3))/(1(2cos(3\u03c0(4x + 4) - 2) - 3) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "71", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-1cos(2\u03c0x) - 1)^2 - 1(-1cos(2\u03c0x) - 1) + 3 and g(x) = -3|x + 2| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5((-1(1|x - 3| - 1) + 3)/(1(1|x - 3| - 1)))^2 + 3((-1(1|x - 3| - 1) + 3)/(1(1|x - 3| - 1))) - 3 = (-1(-1x + 3) - 3)/(2(-1x + 3) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1sin(2\u03c0((-2(2x^2 - 1x - 1) + 3)/(-2(2x^2 - 1x - 1) + 3)) + 2) + 1 and g(x) = 5(1|x + 2| - 3)^2 - 5(1|x + 2| - 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3cos(2\u03c0(4x^2 + 5x + 3) - 2) + 3 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) = 2cos(2\u03c0((1(4(-3x + 4)^2 - 2(-3x + 4)) - 3)/(1(4(-3x + 4)^2 - 2(-3x + 4)) - 2)) - 1) and g(x) = 4(1((1x + 2)/(-2x + 3)) - 1)^2 - 5(1((1x + 2)/(-2x + 3)) - 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(2\u03c0(2((-3x)/(2x + 1)) + 5) - 1) - 3 and g(x) = -1((-2x + 2)/(2x + 2)) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4((-2(5(3|x + 2| + 2)^2 - 3(3|x + 2| + 2)) + 1)/(-2(5(3|x + 2| + 2)^2 - 3(3|x + 2| + 2)) - 1)) - 2 and g(x) = (-3(2|(1sin(3\u03c0x + 2) + 2) - 1| + 2))/(2(2|(1sin(3\u03c0x + 2) + 2) - 1| + 2) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(-1cos(3\u03c0x + 3) - 1) - 4 = -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": "24", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(-3|x + 3| - 1) - 4 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4((-3(-1x + 2) + 2)/(3(-1x + 2) - 1))^2 - 2((-3(-1x + 2) + 2)/(3(-1x + 2) - 1)) = (-1(3e^(-2x) - 3) - 3)/(2(3e^(-2x) - 3) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(2(-2(-2|x - 2| - 1)^2 - 2(-2|x - 2| - 1) + 4) + 2))/(1(2(-2(-2|x - 2| - 1)^2 - 2(-2|x - 2| - 1) + 4) + 2) + 3) = (-1(4(3|x - 2|)^2 - 3(3|x - 2|) - 1) + 1)/(-1(4(3|x - 2|)^2 - 3(3|x - 2|) - 1) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(1sin(1\u03c0x + 3) + 3) + 4 = -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": "Determine the number of intersection points between the functions f(x) = (3(4(-3|(2x^2 + 5x - 3) - 3| + 3) - 2))/(3(4(-3|(2x^2 + 5x - 3) - 3| + 3) - 2) - 2) and g(x) = (3(-2(1x + 1)^2 - 5(1x + 1) + 4) + 1)/(-1(-2(1x + 1)^2 - 5(1x + 1) + 4) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|((-2(1x^2 - 3x - 5))/(-3(1x^2 - 3x - 5) + 1)) + 2| = 2|(-4x + 3) + 3| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(-3|(2cos(3\u03c0x + 2) - 2) - 1| + 2) - 2 = (-1(1|x + 3| + 3) + 1)/(1(1|x + 3| + 3) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(3|(-4x^2 - 4x + 2) + 2| - 2) - 5 and g(x) = (-1(1|x - 1| + 1))/(-2(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": "Determine the number of intersection points between the functions f(x) = (-3(1sin(3\u03c0x + 2) - 2) - 1)/(2(1sin(3\u03c0x + 2) - 2) + 2) and g(x) = -1x^2 - 3x + 3 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": "Find the number of times the graphs of f(x) = -1|(-1(3cos(2\u03c0x + 2) - 2)^2 - 3) - 1| + 1 and g(x) = -2((1x)/(1x + 2))^2 - 1((1x)/(1x + 2)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(3|(2sin(3\u03c0(5x^2 + 4x - 1))) + 1| - 2) + 3)/(1(3|(2sin(3\u03c0(5x^2 + 4x - 1))) + 1| - 2)) and g(x) = 5(-1(-2|x - 2| - 3)^2 + 1(-2|x - 2| - 3) - 3) + 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": "Find the number of times the graphs of f(x) = -3|(5(-4x - 3)^2 - 1(-4x - 3) - 5) + 2| - 1 and g(x) = -3e^(-2(-1x^2 + 5x + 2)) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3e^(2((-2x - 1)/(2x - 1))) - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(3x - 2)^2 - 2(3x - 2) + 4 = (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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(1\u03c0(-4x^2 - 3x + 5) - 2) - 1 and g(x) = 3x^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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(-2(-5x^2 + 3) - 2) + 1)/(-2(-2(-5x^2 + 3) - 2) + 3) and g(x) = -1|(1x - 5) + 3| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(3|((3x + 2)/(1x + 3))|)^2 - 2 = 4(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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(1|(2e^(2x) + 1) + 1| - 3) + 1)/(2(1|(2e^(2x) + 1) + 1| - 3) - 1) and g(x) = 3|(4x^2 - 2x - 5) + 1| + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((-1(2cos(1\u03c0(-5x - 2) + 2)) - 1)/(-1(2cos(1\u03c0(-5x - 2) + 2)) - 1))^2 - 2((-1(2cos(1\u03c0(-5x - 2) + 2)) - 1)/(-1(2cos(1\u03c0(-5x - 2) + 2)) - 1)) + 5 and g(x) = 1|(-2(2cos(3\u03c0x + 3) - 3)^2 + 4(2cos(3\u03c0x + 3) - 3) + 5) + 3| intersect in the interval [-10, 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) = -3(3|x + 3| + 1)^2 - 1(3|x + 3| + 1) - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-3(-1|x + 3|) - 3)^2 + 3(-3(-1|x + 3|) - 3) + 1 = -1cos(3\u03c0(1|x - 2| - 3) + 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3(2|x|))/(1(2|x|) - 2) = -2x^2 + 2x - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4((2(-1(2|x + 3| + 1) - 2))/(3(-1(2|x + 3| + 1) - 2) + 1))^2 + 5((2(-1(2|x + 3| + 1) - 2))/(3(-1(2|x + 3| + 1) - 2) + 1)) - 1 and g(x) = -3((-2(2|x + 1| - 2) + 2)/(2(2|x + 1| - 2) - 1))^2 + 5((-2(2|x + 1| - 2) + 2)/(2(2|x + 1| - 2) - 1)) + 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) = (1(1(5x^2 + 2x - 5) + 2) - 2)/(3(1(5x^2 + 2x - 5) + 2) - 2) and g(x) = (-1(1|x - 2| - 3) + 3)/(3(1|x - 2| - 3) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(4((-2(-5x - 4) - 1)/(-3(-5x - 4) - 2))^2 - 4((-2(-5x - 4) - 1)/(-3(-5x - 4) - 2)) + 1) - 1| + 2 and g(x) = -3(2|((2x)/(3x)) + 1| - 2)^2 + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(1\u03c0((1x - 2)/(2x - 1)) - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-1|(-2((1x - 2)/(-1x))^2 + 2((1x - 2)/(-1x)) - 1) + 2| - 2) - 5 and g(x) = 4(-1((-3x + 3)/(-3x - 2))^2 - 3((-3x + 3)/(-3x - 2)) + 2) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(-3sin(3\u03c0(-2(1|x| - 2) + 4) - 1) - 2)^2 + 2(-3sin(3\u03c0(-2(1|x| - 2) + 4) - 1) - 2) + 2 = 2|(-2(-3x^2 - 2x - 3) - 1) + 1| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(-2|((-2(-5x^2 - 5x) + 2)/(3(-5x^2 - 5x) - 2)) - 1| + 2) + 4 and g(x) = -5(1(-1|x - 2| + 3)^2 + 4(-1|x - 2| + 3) - 2) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-3|x - 2|) + 3)/(-3(-3|x - 2|) - 3) and g(x) = -3x - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(-1|(1x^2 + 1x) + 3|) + 4 and g(x) = 2cos(2\u03c0(-2|x + 2|) - 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2((1(1x - 1) + 1)/(3(1x - 1) - 1))^2 - 1((1(1x - 1) + 1)/(3(1x - 1) - 1)) + 4 and g(x) = (-2(2sin(3\u03c0x + 1) + 1) - 2)/(3(2sin(3\u03c0x + 1) + 1) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(1\u03c0(4x^2 + 1x - 5) - 1) + 1 and g(x) = 2x^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": "Determine the number of intersection points between the functions f(x) = -1|(1(-3x - 3)^2 + 5(-3x - 3) - 3) - 1| + 3 and g(x) = -3sin(2\u03c0(1|x + 1| + 3) + 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(3sin(3\u03c0x - 2) - 3) - 2)/(-3(3sin(3\u03c0x - 2) - 3) + 2) and g(x) = (-1x + 2)/(1x + 2) 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": "How many points of intersection exist between the functions f(x) = (1(1|x - 1| - 3) + 1)/(1(1|x - 1| - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(-1((-2(4x + 3) - 3)/(-1(4x + 3) + 3))^2 - 2((-2(4x + 3) - 3)/(-1(4x + 3) + 3)) + 3) + 2| - 2 and g(x) = 5(2|(-1x + 3) - 3| - 1)^2 + 1(2|(-1x + 3) - 3| - 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(2\u03c0(-2|x + 1| + 1) + 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(3cos(1\u03c0x) + 1) - 1 and g(x) = 4x - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(-2sin(3\u03c0(-1(-3x^2 + 5x - 4) + 3)) + 2) + 1| + 1 and g(x) = 1sin(1\u03c0(-1((2x + 1)/(1x + 1)) + 5) + 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1326", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1sin(2\u03c0(-1((2(4x^2 + 4x + 3) + 3)/(-1(4x^2 + 4x + 3) + 3)) - 2) - 3) - 3 and g(x) = (-1(3|(-4x^2 + 5x + 1)|) - 3)/(2(3|(-4x^2 + 5x + 1)|) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(1(-1x^2 + 5x - 3) + 5) - 3)/(1(1(-1x^2 + 5x - 3) + 5) - 2) and g(x) = 4(-3cos(1\u03c0x + 3) + 3)^2 + 4(-3cos(1\u03c0x + 3) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(-2|(1x) + 1| + 1)^2 - 5(-2|(1x) + 1| + 1) + 3 and g(x) = -1(-5x^2 + 2x + 2) + 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1((1x + 1)/(-2x + 1))^2 - 2((1x + 1)/(-2x + 1)) + 4 and g(x) = (3x - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2e^(-2(-3((3(-2x^2 + 3x - 4))/(-1(-2x^2 + 3x - 4) + 2)) - 1)) - 3 and g(x) = -3|(3sin(3\u03c0(2x)) + 1) + 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 2(3|((-1(-2x^2 + 1x + 1) + 3)/(3(-2x^2 + 1x + 1) + 1))| - 2) - 4 = 2|(-2cos(3\u03c0(4x^2 + 4x) - 1) + 1) + 1| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(1\u03c0((2x + 3)/(-3x - 2)) + 3) + 2 and g(x) = (-3x + 3)/(-3x) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(2(1x)^2 - 5(1x) - 2) + 1)/(1(2(1x)^2 - 5(1x) - 2) + 3) and g(x) = 2e^(-1(2|x + 1| - 2)) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(1((1(-5x^2 + 3x + 1) - 2)/(-2(-5x^2 + 3x + 1) + 1)) + 1) + 1| - 2 and g(x) = 1sin(3\u03c0(4(2|x + 3| - 1)^2 + 5(2|x + 3| - 1) + 1) + 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1322", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(3sin(3\u03c0(4(-1x^2 + 3x + 1) - 5) - 3) - 3) - 3| and g(x) = 2((2(5x^2 + 2x - 5) - 3)/(-1(5x^2 + 2x - 5) - 3)) + 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 points of intersection exist between the functions f(x) = (2(2|(-2x^2 - 1x + 4) - 2| - 1) + 1)/(-1(2|(-2x^2 - 1x + 4) - 2| - 1) + 3) and g(x) = (1(-5x - 3) + 3)/(2(-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 solutions does the equation -4((2(-1|(1x^2 + 1x - 2) + 3| + 1) + 1)/(3(-1|(1x^2 + 1x - 2) + 3| + 1) + 3)) + 1 = 1(2e^(-2(5x + 2)) - 2)^2 + 5(2e^(-2(5x + 2)) - 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(4(-2sin(3\u03c0(1x^2 + 4x - 3) - 2) + 2) - 4) + 2| - 1 and g(x) = 4(-1((2x - 3)/(-1x + 2))^2 + 5((2x - 3)/(-1x + 2)) - 2) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(2(-1|x - 3| + 3)^2 + 1(-1|x - 3| + 3) - 5) and g(x) = -2sin(3\u03c0(1x + 5)) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1((-3(-4(-3|x - 2| - 1)^2 + 2(-3|x - 2| - 1) + 4) + 3)/(2(-4(-3|x - 2| - 1)^2 + 2(-3|x - 2| - 1) + 4) - 3)) - 4 and g(x) = -1|(3sin(1\u03c0(-2x + 2)) + 2) + 2| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(4x + 3)^2 + 1(4x + 3) - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(4(2sin(2\u03c0x + 3) + 1) + 2)^2 - 5(4(2sin(2\u03c0x + 3) + 1) + 2) and g(x) = 4((-1x + 2)/(-1x - 3))^2 - 3((-1x + 2)/(-1x - 3)) + 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) = -2|(1e^(1((-3x + 1)/(-1x - 1))) - 1) - 2| - 3 and g(x) = -2|(-1x - 4) + 2| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(-1|(-2x + 1) + 3| + 2)^2 - 1(-1|(-2x + 1) + 3| + 2) + 4 = 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": "5", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-5(-3cos(3\u03c0(3|x| + 2) + 1) + 2)^2 + 5(-3cos(3\u03c0(3|x| + 2) + 1) + 2) + 5) + 4 and g(x) = 1|(-3(-3e^(-1x) + 2)^2 - 1(-3e^(-1x) + 2) - 3) - 3| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "194", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(4x - 3)^2 + 1(4x - 3) + 1 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3((-2(2|(-2x^2 - 4x) - 2| - 1) + 2)/(2(2|(-2x^2 - 4x) - 2| - 1) + 3)) - 5 and g(x) = 1(-3e^(-1(2x + 5)))^2 - 1(-3e^(-1(2x + 5))) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(3\u03c0(-2|(-1x^2 + 3x - 5)| + 3) - 3) + 3 and g(x) = 2|(-2e^(-2x) + 2) + 3| + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(-1sin(2\u03c0(2((-2x - 3)/(-3x + 4))^2 - 5((-2x - 3)/(-3x + 4)) - 2) - 1) + 1) - 1| - 2 and g(x) = (2(3(2cos(2\u03c0x - 2)) - 4) + 2)/(2(3(2cos(2\u03c0x - 2)) - 4) - 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) = 3(1(-1cos(2\u03c0x + 1) - 2)^2 - 5) - 3 and g(x) = 2((2x + 1)/(1x + 1))^2 + 2((2x + 1)/(1x + 1)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-1|(4x^2 + 2x + 5) + 1| - 2) - 2)/(-2(-1|(4x^2 + 2x + 5) + 1| - 2) - 1) = 3cos(2\u03c0((1x - 3)/(-1x)) - 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "68", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(1\u03c0(1x + 1) - 1) + 3 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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3((-1x - 3)/(-1x - 2))^2 - 5((-1x - 3)/(-1x - 2)) - 3 = 3x^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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|(-1sin(3\u03c0x + 3) - 1) - 1| + 1 = -3x have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|((-3(1x - 5) + 3)/(3(1x - 5) + 3)) - 2| - 3 and g(x) = 5(2cos(3\u03c0x - 1) + 2)^2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4((1(-1|x - 1| - 2) + 1)/(1(-1|x - 1| - 2) + 2)) - 1 = -2cos(1\u03c0((-1x - 1)/(-2x - 3))) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(-1(3cos(2\u03c0((-1x + 1)/(-3x - 3)) - 3) - 3)^2 - 5(3cos(2\u03c0((-1x + 1)/(-3x - 3)) - 3) - 3) - 5) - 1 = 2|(1(1x^2 - 5x - 2) - 2) + 1| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(3((-1(2sin(2\u03c0x) + 2) - 3)/(-1(2sin(2\u03c0x) + 2) - 1)) + 2)^2 + 5(3((-1(2sin(2\u03c0x) + 2) - 3)/(-1(2sin(2\u03c0x) + 2) - 1)) + 2) + 1 = -3|((-1(-1e^(2x) + 1))/(3(-1e^(2x) + 1) - 1)) + 2| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-4((1x + 1)/(-3x - 1)) + 1)^2 - 2(-4((1x + 1)/(-3x - 1)) + 1) - 5 = -1|((3x - 3)/(3x - 2)) + 2| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|((3(-5(-3x)^2 + 5(-3x)) - 1)/(1(-5(-3x)^2 + 5(-3x)) - 2))| + 2 and g(x) = -2sin(3\u03c0(1((-2x + 3)/(-1x))^2 + 4((-2x + 3)/(-1x)) + 4)) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(3(3|(4x^2 + 2x + 2)| + 2) + 3) + 3)/(2(3(3|(4x^2 + 2x + 2)| + 2) + 3)) = (-2(-2|(1x^2 - 1x - 3) - 1| + 3) + 1)/(1(-2|(1x^2 - 1x - 3) - 1| + 3) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(4((1x + 2)/(1x))^2 - 4((1x + 2)/(1x))) - 1 and g(x) = -3|((2x - 2)/(1x - 3)) - 3| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(-2(-1e^(-1x))^2 + 5(-1e^(-1x))))/(1(-2(-1e^(-1x))^2 + 5(-1e^(-1x))) + 1) = -5((1x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-2x - 2) - 3)/(-2(-2x - 2) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|((-2(-1(-1x + 2)^2 - 4(-1x + 2) - 3) - 1)/(3(-1(-1x + 2)^2 - 4(-1x + 2) - 3) + 3)) - 3| and g(x) = 1|(-2(-1x^2 - 4x + 3) - 4) - 3| + 1 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(-2sin(3\u03c0x + 3)) - 2 and g(x) = -2|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": "How many solutions does the equation 1|((2(-3(-4x + 2)^2 - 3(-4x + 2) + 4) + 3)/(3(-3(-4x + 2)^2 - 3(-4x + 2) + 4) - 1))| = -2|(1(5x + 3)^2 + 3(5x + 3) - 1) + 1| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3sin(3\u03c0(-2(-1|(5x^2 + 3x + 5) - 3| + 2) + 2) + 1) - 1 and g(x) = -1|(2e^(2(2x + 5)) + 1) - 2| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "597", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|((-2(4(3x^2 - 2x - 2) - 4))/(-3(4(3x^2 - 2x - 2) - 4) + 1)) - 2| + 1 = (-3(1(1|x + 1| + 2) + 4) + 1)/(1(1(1|x + 1| + 2) + 4)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(2sin(2\u03c0(1x^2 - 2x - 5) - 2) - 1) - 3)/(1(2sin(2\u03c0(1x^2 - 2x - 5) - 2) - 1) + 2) and g(x) = -3sin(1\u03c0(1|x + 3| - 1)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "138", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(1\u03c0(-2|x + 3| + 3) - 1) + 2 and g(x) = -1x^2 - 5x - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(4x - 5) - 1)/(2(4x - 5) - 1) and g(x) = (3x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1sin(3\u03c0(4(-3x^2 + 4x + 5) - 3) - 3) = 3cos(3\u03c0(1x^2 - 1x + 2) + 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "547", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2(3|x| - 3) + 4 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3((1x + 1)/(1x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1((-1(-4x - 4) + 3)/(-2(-4x - 4) - 3))^2 - 2((-1(-4x - 4) + 3)/(-2(-4x - 4) - 3)) - 1 and g(x) = -2|(3x + 3) + 3| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3((-2x - 1)/(-1x - 2))^2 - 2((-2x - 1)/(-1x - 2)) + 3 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(3|(-1((-3x + 3)/(2x - 3)) + 4) + 3| + 2)^2 - 4(3|(-1((-3x + 3)/(2x - 3)) + 4) + 3| + 2) - 2 = -4(-2|(-3x^2 - 5x - 5) + 1| - 3) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-1(2|(4x^2 - 4x + 2) + 2| - 3) + 2))/(1(-1(2|(4x^2 - 4x + 2) + 2| - 3) + 2) + 3) and g(x) = 2((-2(3|x + 3| - 2) + 2)/(-2(3|x + 3| - 2) - 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(1((-2x + 3)/(3x - 2))^2 + 4((-2x + 3)/(3x - 2)) - 4) + 2 = (-2(-1|x| - 1))/(-3(-1|x| - 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(2(-3cos(2\u03c0x + 3))^2 - 1(-3cos(2\u03c0x + 3)) + 2) + 1)/(2(2(-3cos(2\u03c0x + 3))^2 - 1(-3cos(2\u03c0x + 3)) + 2) - 3) and g(x) = 1|(2e^(2x) + 2) - 2| - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2cos(3\u03c0(1x + 5) - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1sin(2\u03c0(-3|(-3((2x - 1)/(-2x)) - 5) + 2| - 3) + 1) - 2 = -2|(-1(-1x^2 + 4x + 4) + 3)| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5((-2(-4(3|x + 2|)^2 + 5(3|x + 2|) + 1) + 2)/(3(-4(3|x + 2|)^2 + 5(3|x + 2|) + 1) - 1)) - 3 = 1(-1cos(2\u03c0(2|x - 3| - 2) + 1) - 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(-2((-1(-5x - 5) - 2)/(1(-5x - 5) + 2))^2 - 5((-1(-5x - 5) - 2)/(1(-5x - 5) + 2))) + 2| + 1 and g(x) = (1(4(3sin(1\u03c0x + 1) - 3) + 2))/(2(4(3sin(1\u03c0x + 1) - 3) + 2) + 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 1|(-2cos(2\u03c0x + 3) + 2) + 3| + 1 = -5x - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(3\u03c0(-4(-1|(5x^2 - 5x - 1)| + 2) - 4)) + 3 and g(x) = 4((-3(-3x - 1) - 1)/(3(-3x - 1) + 3))^2 - 5 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) = (-2(-4(-3cos(2\u03c0x) + 1) - 3))/(-1(-4(-3cos(2\u03c0x) + 1) - 3) - 2) and g(x) = -2e^(-1(2x^2 - 4x - 3)) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(1((1x)/(3x + 3)) - 2)^2 + 2(1((1x)/(3x + 3)) - 2) - 1 and g(x) = 2e^(1(-2x^2 + 4x - 5)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1((-1x)/(-1x - 2))^2 - 5 = (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": "3", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|(-2cos(1\u03c0(4(-5x^2 + 1x + 3) + 2) - 1) + 3) - 3| - 3 = (-2(-2(3x - 4)^2 - 5(3x - 4) + 5) + 3)/(-3(-2(3x - 4)^2 - 5(3x - 4) + 5) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(3(3|x - 3|) + 4) - 3)/(-2(3(3|x - 3|) + 4)) and g(x) = 1sin(1\u03c0(-1|x - 3| - 2) + 1) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|((3(2x + 2) - 2)/(-3(2x + 2) + 4))| - 3 and g(x) = -4(-1|x| - 1)^2 + 2(-1|x| - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1((-3(1|(-4x^2 + 2x) + 3| + 3) + 3)/(1(1|(-4x^2 + 2x) + 3| + 3))) + 2 = (-2(2|(-3x^2 - 1x - 2)| - 1) - 1)/(-1(2|(-3x^2 - 1x - 2)| - 1) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3e^(-1(1|((1x)/(2x + 2)) - 1| - 1)) + 3 and g(x) = (2(3|x + 2|) - 3)/(1(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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(3(3x^2 - 2x) - 1) + 1| - 3 and g(x) = -2cos(1\u03c0((-2x)/(2x + 2)) + 2) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(4x^2 - 3x - 2) + 1)/(2(4x^2 - 3x - 2) - 1) 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": "How many solutions does the equation 2sin(2\u03c0(5x^2 - 5) - 2) - 2 = (1x - 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": "32", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3((-3(-3|(2x + 3) + 1|) - 3)/(1(-3|(2x + 3) + 1|) - 2))^2 + 2((-3(-3|(2x + 3) + 1|) - 3)/(1(-3|(2x + 3) + 1|) - 2)) + 5 and g(x) = 3cos(3\u03c0(2((-1x + 3)/(-1x)) - 4)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(3cos(2\u03c0x - 1)) - 1| + 3 = 2x + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5((-1(4(3|x + 1| + 2)^2 + 2(3|x + 1| + 2) - 2) + 1)/(1(4(3|x + 1| + 2)^2 + 2(3|x + 1| + 2) - 2) + 1)) - 1 and g(x) = (2(4(-1|x + 1| + 1)^2 + 5(-1|x + 1| + 1) - 4))/(2(4(-1|x + 1| + 1)^2 + 5(-1|x + 1| + 1) - 4) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-3(-1|(5x^2 - 4x + 3) + 2| + 3) + 5) - 2)/(2(-3(-1|(5x^2 - 4x + 3) + 2| + 3) + 5) + 3) and g(x) = (3(-1cos(3\u03c0(-5x - 3) - 1)) - 1)/(2(-1cos(3\u03c0(-5x - 3) - 1)) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "207", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(-1e^(-2x) + 3)^2 + 5 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(2e^(-2x) + 3) - 1| - 3 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3e^(1(2((1(2|x + 1| + 1) - 3)/(2(2|x + 1| + 1) + 3))^2 + 1((1(2|x + 1| + 1) - 3)/(2(2|x + 1| + 1) + 3)))) - 1 and g(x) = -2e^(1(2|((-1x)/(-2x - 2)) - 1| - 3)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1sin(3\u03c0((2(-3(2|x - 3| + 1) - 3) + 3)/(-1(-3(2|x - 3| + 1) - 3) + 2)) + 1) and g(x) = -3(-3((1x - 3)/(-2x + 3))^2 - 2((1x - 3)/(-2x + 3))) - 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(1((-1x - 1)/(2x - 1)) - 5)^2 - 4(1((-1x - 1)/(2x - 1)) - 5) + 3 = (3(-2|x - 1| + 3) - 2)/(3(-2|x - 1| + 3) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-2sin(3\u03c0(3x^2 + 1x) - 2)) + 2 and g(x) = -3|(4x^2 + 1x + 1) - 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": "How many points of intersection exist between the functions f(x) = (-3(-1cos(1\u03c0(-1(2|x + 3| + 2)^2 - 5(2|x + 3| + 2) - 5) - 1) - 1) - 2)/(2(-1cos(1\u03c0(-1(2|x + 3| + 2)^2 - 5(2|x + 3| + 2) - 5) - 1) - 1) + 1) and g(x) = -1sin(1\u03c0(2(-1|x - 1| - 1)^2 - 1(-1|x - 1| - 1) - 1) - 3) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "165", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(3\u03c0(-4x^2 - 1x - 1) + 3) + 2 and g(x) = -3x^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": "31", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(-3|(3e^(2x) - 2) - 1| + 2) - 2 and g(x) = 2sin(3\u03c0(2|x + 2| + 1) + 1) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|((-1(-5(2x^2 - 3x + 5) + 5) + 3)/(-2(-5(2x^2 - 3x + 5) + 5))) - 3| + 1 and g(x) = (2(2|(-3x - 5) + 3| + 1) - 1)/(1(2|(-3x - 5) + 3| + 1) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5((2x + 1)/(-2x + 3)) + 5 = (-1x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2e^(2(1|(-1x^2 - 2x + 3) + 3|)) - 2 and g(x) = -5(-3|x - 1| - 1) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|((1(-4x^2 - 5x + 1) + 2)/(-2(-4x^2 - 5x + 1) + 3)) + 2| + 3 and g(x) = (-2(-2|x + 2| + 3) - 2)/(-2(-2|x + 2| + 3) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-3e^(-1(4x^2 + 4x))) + 3| + 3 and g(x) = -4(-3|x - 1| - 1)^2 - 1(-3|x - 1| - 1) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(1(-2e^(-2x) + 3) + 3)^2 + 1(1(-2e^(-2x) + 3) + 3) - 1 = 1|(4x^2 + 4x + 4) - 2| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-1sin(3\u03c0(1(-3x^2 + 3x - 3) + 3) - 2)))/(2(-1sin(3\u03c0(1(-3x^2 + 3x - 3) + 3) - 2)) + 3) and g(x) = 1|(1(-5x - 1)^2 - 3(-5x - 1) + 3) - 3| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(1\u03c0((-3x + 3)/(-2x - 3)) - 2) 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 points of intersection exist between the functions f(x) = -3(3|(-1((3x - 1)/(3x - 2)) + 4) + 3| - 3)^2 - 3 and g(x) = -5(2e^(1((2x)/(-1x + 3)))) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(-3sin(3\u03c0x - 3) + 1) - 1)/(2(-3sin(3\u03c0x - 3) + 1) + 3) and g(x) = -3|x + 2| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(-2cos(3\u03c0((1x - 3)/(-2x - 3)) - 3) + 1) = 1sin(3\u03c0((-1x - 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": "96", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(-1((1x - 1)/(1x + 1)) + 1) - 1| and g(x) = (3(3|x + 1| + 2) - 2)/(-2(3|x + 1| + 2) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(-3cos(1\u03c0(2|(5x^2 + 3x + 4) + 2| - 2) + 2)) - 2 = 1cos(1\u03c0(2(1|x + 1| + 2) - 4) - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1499", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(3|(3x^2 - 2x + 2) - 3| - 2))/(2(3|(3x^2 - 2x + 2) - 3| - 2) - 1) = -2|(-1sin(2\u03c0x) - 1) + 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) = -1sin(1\u03c0(-1|x + 2| - 1) - 1) - 2 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(2((-3x)/(3x + 2)) - 1)^2 - 2(2((-3x)/(3x + 2)) - 1) + 2 = (1(3cos(1\u03c0x) - 1) - 3)/(3(3cos(1\u03c0x) - 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(3sin(2\u03c0(3(-2|x| + 2) + 3) - 1) - 2) - 1)/(-2(3sin(2\u03c0(3(-2|x| + 2) + 3) - 1) - 2) - 3) and g(x) = (3(3e^(1(-3|x + 1| + 2))) - 2)/(-1(3e^(1(-3|x + 1| + 2))) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "234", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(2|(2(-5x^2 - 4x + 1) + 4) + 2| - 2) + 1)/(-2(2|(2(-5x^2 - 4x + 1) + 4) + 2| - 2) - 1) and g(x) = 2|(-2sin(2\u03c0((3x + 3)/(1x + 1))) - 1) + 3| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(1((3(-2|x - 2| - 3) - 3)/(-2(-2|x - 2| - 3) + 3)) + 5)^2 + 4(1((3(-2|x - 2| - 3) - 3)/(-2(-2|x - 2| - 3) + 3)) + 5) - 5 and g(x) = 1(-2e^(2(-3|x - 3| + 1)) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(-2|(-3sin(3\u03c0(-2x - 1) + 2) + 2) - 3| + 2)^2 - 1(-2|(-3sin(3\u03c0(-2x - 1) + 2) + 2) - 3| + 2) = (-1(1|(-1x + 2) + 1| - 2))/(1(1|(-1x + 2) + 1| - 2) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "396", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2((-3(4(1|x - 3| + 2)^2 - 5(1|x - 3| + 2) + 4) + 3)/(-3(4(1|x - 3| + 2)^2 - 5(1|x - 3| + 2) + 4) + 2)) - 4 and g(x) = -5(-4(2e^(-1x) - 2) - 2)^2 + 5(-4(2e^(-1x) - 2) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(1\u03c0(5x + 4) + 3) - 2 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": "14", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5((1(1|x - 1| - 3) - 1)/(2(1|x - 1| - 3) + 2))^2 + 2((1(1|x - 1| - 3) - 1)/(2(1|x - 1| - 3) + 2)) + 3 = -2cos(2\u03c0((-3x - 3)/(2x - 1)) + 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "64", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(1(2|((2x - 2)/(-1x + 2))| - 2) + 1)^2 - 1(1(2|((2x - 2)/(-1x + 2))| - 2) + 1) - 1 and g(x) = -5(4((-2x + 1)/(-1x - 3)) - 1)^2 - 3(4((-2x + 1)/(-1x - 3)) - 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|((1(1sin(3\u03c0x + 1) + 2) + 3)/(-1(1sin(3\u03c0x + 1) + 2) + 2)) + 2| + 1 = -3|(-5x + 5) - 3| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "114", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(3\u03c0(-2(-3|((1x + 2)/(1x + 1)) - 3| + 3)^2 + 2(-3|((1x + 2)/(1x + 1)) - 3| + 3) + 1) + 1) - 3 and g(x) = 3cos(1\u03c0((-3(1x + 2) - 3)/(-3(1x + 2) - 2)) - 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "253", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(3(2e^(-1x) - 3) - 2) - 2)/(1(3(2e^(-1x) - 3) - 2)) = -3((3x + 2)/(-1x - 2))^2 + 5((3x + 2)/(-1x - 2)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(1sin(2\u03c0(-2|(3x) + 2| + 1) - 3)))/(-3(1sin(2\u03c0(-2|(3x) + 2| + 1) - 3)) - 1) and g(x) = (1(4(2cos(1\u03c0x - 3) + 2) + 5) + 1)/(-1(4(2cos(1\u03c0x - 3) + 2) + 5) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-3sin(3\u03c0x - 2) - 3) - 2)/(-1(-3sin(3\u03c0x - 2) - 3)) 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": "56", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(-3|((2x - 3)/(-3x + 2)) - 3| + 2)^2 + 3(-3|((2x - 3)/(-3x + 2)) - 3| + 2) - 4 and g(x) = 1cos(2\u03c0(2|x| - 1) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1sin(3\u03c0(2x^2 - 3x - 5) - 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": "287", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1((-3(3(3|x - 1|)^2 - 3(3|x - 1|)) + 3)/(-1(3(3|x - 1|)^2 - 3(3|x - 1|)) + 2)) + 5 = 1|(2cos(3\u03c0(-5x - 1) + 2) + 2) - 1| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(4((2(3|x + 2| + 3) - 3)/(2(3|x + 2| + 3)))^2 + 1) - 4 = -2|(2(2x + 4)^2 - 5(2x + 4) + 3) - 2| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(3\u03c0(-3|x| - 2)) - 2 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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-5(-3|(1sin(1\u03c0x - 1) - 1) - 2| - 2)^2 - 5(-3|(1sin(1\u03c0x - 1) - 1) - 2| - 2) + 4) - 2)/(-1(-5(-3|(1sin(1\u03c0x - 1) - 1) - 2| - 2)^2 - 5(-3|(1sin(1\u03c0x - 1) - 1) - 2| - 2) + 4) + 3) and g(x) = (1(2sin(3\u03c0(1|x - 1| + 2) + 2) + 1) - 1)/(1(2sin(3\u03c0(1|x - 1| + 2) + 2) + 1) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|((2(4x^2 + 2x - 2) - 1)/(2(4x^2 + 2x - 2) - 1)) - 1| + 2 and g(x) = 1(-3x + 5)^2 - 3(-3x + 5) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1((2(-2(1|x + 2| - 1)^2 - 2(1|x + 2| - 1) + 1) + 2)/(1(-2(1|x + 2| - 1)^2 - 2(1|x + 2| - 1) + 1) + 2)) + 2 = 1e^(-2((2(-2|x| + 2) - 2)/(3(-2|x| + 2) + 3))) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(5x^2 + 4x + 1) - 4 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3((3(3|x| + 3) + 1)/(-2(3|x| + 3) - 1))^2 - 3((3(3|x| + 3) + 1)/(-2(3|x| + 3) - 1)) - 5 and g(x) = 3(1cos(2\u03c0x + 3)) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(-2e^(2((1(1x^2 + 4x + 1) + 1)/(3(1x^2 + 4x + 1) + 1))) + 2) - 1| + 2 and g(x) = 2|((3(-1e^(-1x) - 3) - 3)/(-2(-1e^(-1x) - 3) + 3)) - 3| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(2e^(1x) - 2)^2 - 5(2e^(1x) - 2) - 5 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-1|((3(4x + 1) - 2)/(-2(4x + 1) - 2)) + 2| - 3)^2 + 2(-1|((3(4x + 1) - 2)/(-2(4x + 1) - 2)) + 2| - 3) + 2 and g(x) = -2(-2|((2x)/(-2x + 1)) + 3| + 3)^2 + 2(-2|((2x)/(-2x + 1)) + 3| + 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|((1(3x - 3))/(-2(3x - 3) - 3))| + 2 and g(x) = (-3(2sin(1\u03c0x - 3) - 3) + 3)/(-2(2sin(1\u03c0x - 3) - 3) - 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": "How many points of intersection exist between the functions f(x) = -2e^(-1((3(5x - 1))/(-2(5x - 1) + 3))) + 3 and g(x) = (2(3|x + 3| - 3) + 2)/(1(3|x + 3| - 3) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|(-5((-3(-3cos(2\u03c0x - 1) - 1) + 1)/(-1(-3cos(2\u03c0x - 1) - 1) + 2))^2 + 3((-3(-3cos(2\u03c0x - 1) - 1) + 1)/(-1(-3cos(2\u03c0x - 1) - 1) + 2)) + 3) - 1| + 2 and g(x) = (1(3(-2|x - 2| - 1) - 4) - 3)/(1(3(-2|x - 2| - 1) - 4) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "158", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(4x^2 + 2x - 4) - 3 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": "2", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|((3(5(-2x^2 + 3x - 1) - 1) + 1)/(1(5(-2x^2 + 3x - 1) - 1))) - 2| + 1 = (2(2|(-1x + 5) - 3| + 2) - 3)/(1(2|(-1x + 5) - 3| + 2) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}