{"messages": [{"content": "How many solutions does the equation -5(-3sin(3\u03c0(5(-2|x - 2| - 3) + 3)) - 3)^2 - 4(-3sin(3\u03c0(5(-2|x - 2| - 3) + 3)) - 3) - 2 = -2(2e^(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": "492", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(1((-3(-2sin(1\u03c0x + 3) - 1) - 3)/(-2(-2sin(1\u03c0x + 3) - 1) - 1))^2 - 5((-3(-2sin(1\u03c0x + 3) - 1) - 3)/(-2(-2sin(1\u03c0x + 3) - 1) - 1)) + 2) - 4 and g(x) = (-2(3|(3|x - 2| + 1) + 1| + 1) + 2)/(-3(3|(3|x - 2| + 1) + 1| + 1) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(-1|(3((-2x + 2)/(-1x - 1)) + 4) - 1| + 3)^2 - 1(-1|(3((-2x + 2)/(-1x - 1)) + 4) - 1| + 3) + 1 and g(x) = (1(-1|((-3x - 2)/(-2x - 3)) + 3| + 1) - 2)/(1(-1|((-3x - 2)/(-2x - 3)) + 3| + 1) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(-2(-4((-3x)/(2x - 1)) - 2)^2 - 2(-4((-3x)/(2x - 1)) - 2) + 3)^2 - 3(-2(-4((-3x)/(2x - 1)) - 2)^2 - 2(-4((-3x)/(2x - 1)) - 2) + 3) + 1 and g(x) = 5(3cos(3\u03c0(-4x + 2) - 3) - 3)^2 + 3(3cos(3\u03c0(-4x + 2) - 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": "12", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(4(-5(2x^2 - 4) + 2) + 4)^2 - 4(4(-5(2x^2 - 4) + 2) + 4) + 3 and g(x) = (3(-1e^(-1(-1|x + 3| - 2))) + 1)/(2(-1e^(-1(-1|x + 3| - 2))) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(3|((1(-1x^2 + 1x + 1) + 2)/(2(-1x^2 + 1x + 1) + 1)) - 1| - 2)^2 - 2(3|((1(-1x^2 + 1x + 1) + 2)/(2(-1x^2 + 1x + 1) + 1)) - 1| - 2) - 1 and g(x) = (-2(3|(1|x - 2| + 2)| + 2) + 3)/(-1(3|(1|x - 2| + 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": "Find the number of times the graphs of f(x) = -1(-3sin(3\u03c0(2((1x - 2)/(-2x - 2)) + 5) - 2))^2 - 3 and g(x) = -2(1sin(3\u03c0(1|x + 3| - 1))) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "104", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(1|(3((1x + 3)/(-1x + 3))^2 + 1((1x + 3)/(-1x + 3)) + 2) - 3| + 1) - 1 and g(x) = -2(-2e^(-1(-2x + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1((2(-1|(3x^2 + 5x + 2) + 2| + 1) - 3)/(1(-1|(3x^2 + 5x + 2) + 2| + 1))) - 1)/(2((2(-1|(3x^2 + 5x + 2) + 2| + 1) - 3)/(1(-1|(3x^2 + 5x + 2) + 2| + 1))) - 3) = 2(3|(-1cos(3\u03c0x) + 3) - 1| - 3) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(3(-3(5x^2 + 4x - 2) - 1) - 5))/(-3(3(-3(5x^2 + 4x - 2) - 1) - 5) - 2) and g(x) = 2cos(2\u03c0(3(2x - 1) - 1) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "242", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-1sin(3\u03c0((-1(4x^2 + 5x - 2) - 2)/(1(4x^2 + 5x - 2))) - 3) + 3) + 2)/(1(-1sin(3\u03c0((-1(4x^2 + 5x - 2) - 2)/(1(4x^2 + 5x - 2))) - 3) + 3) - 2) and g(x) = 1(2(2cos(2\u03c0x + 2) - 1)^2 - 1(2cos(2\u03c0x + 2) - 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": "66", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5((1((2(2cos(2\u03c0x - 3) + 3))/(-1(2cos(2\u03c0x - 3) + 3) - 3)) + 3)/(2((2(2cos(2\u03c0x - 3) + 3))/(-1(2cos(2\u03c0x - 3) + 3) - 3)) + 3))^2 + 4((1((2(2cos(2\u03c0x - 3) + 3))/(-1(2cos(2\u03c0x - 3) + 3) - 3)) + 3)/(2((2(2cos(2\u03c0x - 3) + 3))/(-1(2cos(2\u03c0x - 3) + 3) - 3)) + 3)) + 2 and g(x) = (2(-1((-2x + 2)/(-2x + 1))^2 + 5((-2x + 2)/(-2x + 1)) + 5) + 3)/(-2(-1((-2x + 2)/(-2x + 1))^2 + 5((-2x + 2)/(-2x + 1)) + 5) + 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": "Find the number of times the graphs of f(x) = -4(-3cos(2\u03c0(1|(-3|x - 2| - 1) - 2| - 1) - 1) - 2) + 2 and g(x) = -2e^(-2(-3((-3x + 3)/(-1x - 2)) + 4)) + 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(5(-2cos(3\u03c0(2x + 3) - 3) + 2)^2 + 3(-2cos(3\u03c0(2x + 3) - 3) + 2) - 1)^2 + 5(5(-2cos(3\u03c0(2x + 3) - 3) + 2)^2 + 3(-2cos(3\u03c0(2x + 3) - 3) + 2) - 1) - 5 and g(x) = -4(2(2cos(1\u03c0x) + 2) - 4) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(-2(-2|(1|x - 1| - 2) - 2|)^2 - 4(-2|(1|x - 1| - 2) - 2|)) - 1| - 1 = (2(3cos(3\u03c0((3x)/(3x + 1)) + 1) + 2) + 1)/(-3(3cos(3\u03c0((3x)/(3x + 1)) + 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": "16", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1sin(2\u03c0((-2(2|((-1x)/(2x))| - 1) + 3)/(1(2|((-1x)/(2x))| - 1) - 3)) - 3) - 1 and g(x) = -5(1(-5x - 2)^2 + 4(-5x - 2) - 5)^2 - 2(1(-5x - 2)^2 + 4(-5x - 2) - 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": "Determine the number of intersection points between the functions f(x) = 1|(-2(-2cos(1\u03c0(-5x^2 + 2x + 1)) + 1) + 3) - 3| + 2 and g(x) = 2|((-2(-5x^2 + 1x - 5) + 3)/(-1(-5x^2 + 1x - 5) + 2)) - 2| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(1(2((-1x - 2)/(3x + 1))^2 - 4)^2 - 4(2((-1x - 2)/(3x + 1))^2 - 4) - 2) + 1)/(-3(1(2((-1x - 2)/(3x + 1))^2 - 4)^2 - 4(2((-1x - 2)/(3x + 1))^2 - 4) - 2)) and g(x) = (2(-2|(-2|x + 1| + 3) + 3| + 1) - 3)/(1(-2|(-2|x + 1| + 3) + 3| + 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1cos(3\u03c0(-3(2|((-1x - 1)/(2x + 3)) - 1| + 3)^2 - 4) - 2) - 1 and g(x) = 5(-1((-2x + 1)/(2x - 3))^2 + 3((-2x + 1)/(2x - 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": "How many solutions does the equation 1|(-2|((2((-1x - 3)/(-3x - 2)) - 3)/(-3((-1x - 3)/(-3x - 2)) + 4)) - 2| + 1) + 2| + 2 = 5(1cos(2\u03c0(4x^2 - 2) - 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": "1348", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1((-2(-4(-1e^(1x) + 3) + 4) + 1)/(-1(-4(-1e^(1x) + 3) + 4))) - 2 and g(x) = -2(-1sin(2\u03c0(2|x - 1| - 3) + 2))^2 + 4(-1sin(2\u03c0(2|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": "79", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(2(1(1x - 4) + 3)^2 + 3(1(1x - 4) + 3) - 1) + 1)/(2(2(1(1x - 4) + 3)^2 + 3(1(1x - 4) + 3) - 1) + 1) and g(x) = 1sin(3\u03c0(3|(1x^2 + 3x) + 2| - 2)) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1363", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1((-1((-2(3sin(2\u03c0x + 1)) + 1)/(1(3sin(2\u03c0x + 1)) - 2)) + 1)/(-1((-2(3sin(2\u03c0x + 1)) + 1)/(1(3sin(2\u03c0x + 1)) - 2)) - 2)) + 1)/(-1((-1((-2(3sin(2\u03c0x + 1)) + 1)/(1(3sin(2\u03c0x + 1)) - 2)) + 1)/(-1((-2(3sin(2\u03c0x + 1)) + 1)/(1(3sin(2\u03c0x + 1)) - 2)) - 2)) + 2) and g(x) = (1(-1(2x^2 - 5x) + 1) - 1)/(3(-1(2x^2 - 5x) + 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": "Determine the number of intersection points between the functions f(x) = 1(2|(-3(-5x^2 + 4x + 2) + 3) - 2| + 1) - 3 and g(x) = -2(-1sin(2\u03c0(-2x^2 - 5x + 2) - 2))^2 + 1(-1sin(2\u03c0(-2x^2 - 5x + 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 solutions does the equation 2|(5(-1(3x - 1)^2 - 4(3x - 1) + 4) + 3) - 3| - 1 = -2cos(1\u03c0(5(-4x^2 + 2)^2 - 1(-4x^2 + 2) + 2) + 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(-2((3(-1x^2 + 5x - 5) - 3)/(2(-1x^2 + 5x - 5) + 1))^2 + 3((3(-1x^2 + 5x - 5) - 3)/(2(-1x^2 + 5x - 5) + 1)) - 5))/(3(-2((3(-1x^2 + 5x - 5) - 3)/(2(-1x^2 + 5x - 5) + 1))^2 + 3((3(-1x^2 + 5x - 5) - 3)/(2(-1x^2 + 5x - 5) + 1)) - 5) - 2) = 4(-3cos(1\u03c0(2|x - 3| - 1) - 1) + 1)^2 - 4(-3cos(1\u03c0(2|x - 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": "80", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3cos(2\u03c0(-2(-5(2x^2 - 2)^2 - 1(2x^2 - 2) + 1) + 2) - 1) - 1 = (-2(-3|(2cos(1\u03c0x - 3) + 2)| - 1))/(2(-3|(2cos(1\u03c0x - 3) + 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": "1183", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(-2|(-3sin(1\u03c0(-2x - 2) + 1) - 3) + 2| + 2))/(-1(-2|(-3sin(1\u03c0(-2x - 2) + 1) - 3) + 2| + 2) + 2) and g(x) = 3sin(3\u03c0((1(-2|x - 3| + 2) - 1)/(1(-2|x - 3| + 2) + 1)) + 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "136", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2((1(-1|(1x - 1) + 3| + 1) + 2)/(-2(-1|(1x - 1) + 3| + 1) + 3)))/(1((1(-1|(1x - 1) + 3| + 1) + 2)/(-2(-1|(1x - 1) + 3| + 1) + 3))) and g(x) = -3((2(3cos(3\u03c0x - 1) + 2) - 2)/(-1(3cos(3\u03c0x - 1) + 2)))^2 + 3((2(3cos(3\u03c0x - 1) + 2) - 2)/(-1(3cos(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": "120", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(2(-3(2x^2 - 3x)^2 - 3(2x^2 - 3x) + 3) - 3) - 2| - 3 and g(x) = 2|(3sin(3\u03c0(-5x^2 + 1x - 3) - 2) - 3) + 2| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4((2(2(-3|x + 1|)^2 + 4(-3|x + 1|) - 2) - 2)/(-1(2(-3|x + 1|)^2 + 4(-3|x + 1|) - 2) + 2))^2 + 5((2(2(-3|x + 1|)^2 + 4(-3|x + 1|) - 2) - 2)/(-1(2(-3|x + 1|)^2 + 4(-3|x + 1|) - 2) + 2)) + 5 = 2(2|(2sin(1\u03c0x + 2) - 3) + 2| - 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-5(-4((1x + 1)/(-2x))^2 - 3((1x + 1)/(-2x)) - 1) + 1)^2 + 2 = -3(-2(-1x^2 + 4)^2 + 2(-1x^2 + 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": "Determine the number of intersection points between the functions f(x) = 5(2((-2(1|x| + 1) - 1)/(-1(1|x| + 1) + 3))^2 + 4((-2(1|x| + 1) - 1)/(-1(1|x| + 1) + 3)) - 3)^2 - 1(2((-2(1|x| + 1) - 1)/(-1(1|x| + 1) + 3))^2 + 4((-2(1|x| + 1) - 1)/(-1(1|x| + 1) + 3)) - 3) + 4 and g(x) = 2e^(-1(-3((-3x - 2)/(-1x - 3)))) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(2sin(2\u03c0(1|(1|x + 1| + 1) + 2| - 1) - 1) + 1) + 3)/(1(2sin(2\u03c0(1|(1|x + 1| + 1) + 2| - 1) - 1) + 1) + 2) = -2(-2sin(3\u03c0(-1x + 5) + 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) = 1(3sin(2\u03c0(-1(-5x^2 + 1x + 3)^2 + 5(-5x^2 + 1x + 3) + 3)) + 2)^2 - 4(3sin(2\u03c0(-1(-5x^2 + 1x + 3)^2 + 5(-5x^2 + 1x + 3) + 3)) + 2) + 5 and g(x) = -1cos(2\u03c0((-2(-4x + 3) + 3)/(-2(-4x + 3) + 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": "325", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3sin(3\u03c0(3|(1(-2x^2 - 5x + 2)^2 - 3(-2x^2 - 5x + 2) - 3) + 3| + 1) + 2) + 2 and g(x) = (1(1(-2x + 1) - 4) - 1)/(-3(1(-2x + 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": "679", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(4((1((-1x + 1)/(1x)) - 2)/(-3((-1x + 1)/(1x)) - 1))^2 - 2((1((-1x + 1)/(1x)) - 2)/(-3((-1x + 1)/(1x)) - 1)) - 5) - 3| - 1 and g(x) = -2|(-3|(3cos(3\u03c0x - 2) - 2) + 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": "21", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(-2|(4(-2|x - 2| + 1)) - 2| + 1) + 2| = 4(-5(-2|x + 2| + 2)^2 + 4(-2|x + 2| + 2) + 1)^2 - 5(-5(-2|x + 2| + 2)^2 + 4(-2|x + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|((3((-1(1cos(2\u03c0x - 3) + 3) - 2)/(-2(1cos(2\u03c0x - 3) + 3) + 3)) - 3)/(-1((-1(1cos(2\u03c0x - 3) + 3) - 2)/(-2(1cos(2\u03c0x - 3) + 3) + 3)) - 3)) + 1| - 3 and g(x) = -2cos(2\u03c0((-3((-2x + 2)/(-3x + 1)) - 1)/(1((-2x + 2)/(-3x + 1)) + 1)) + 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3cos(2\u03c0(-2(3(-3|x - 2| - 1)^2 + 1(-3|x - 2| - 1) + 4) + 3) + 3) + 1 and g(x) = (1(1((1x - 2)/(1x - 3))^2 - 5((1x - 2)/(1x - 3)) + 3) + 1)/(3(1((1x - 2)/(1x - 3))^2 - 5((1x - 2)/(1x - 3)) + 3) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1419", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4((-2(3(-3|x + 1| + 2) + 1) + 3)/(3(3(-3|x + 1| + 2) + 1) + 1))^2 - 5((-2(3(-3|x + 1| + 2) + 1) + 3)/(3(3(-3|x + 1| + 2) + 1) + 1)) - 3 and g(x) = 2sin(1\u03c0(-5(-5x + 2)) - 1) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(1\u03c0(2|(5(-1x^2 + 3x - 3)) + 3| - 2) + 2) + 3 and g(x) = 2|((-2(1x^2 + 4x - 1))/(-3(1x^2 + 4x - 1) + 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": "134", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(3|(1sin(2\u03c0(-3|x| + 1) - 2) - 3) + 3| - 1) + 2 and g(x) = 1(3|(1x) + 2| - 2)^2 - 5(3|(1x) + 2| - 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3sin(3\u03c0((1(5(-3|x + 1| + 2) - 5) + 3)/(1(5(-3|x + 1| + 2) - 5) + 1)) - 3) + 2 and g(x) = -1e^(1(1(-4x^2 + 1x + 2)^2 - 4(-4x^2 + 1x + 2) + 3)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(3(3(-1x - 5) - 2) + 5) + 1 and g(x) = -4(2(3sin(3\u03c0x) - 3)^2 + 2(3sin(3\u03c0x) - 3))^2 + 2(2(3sin(3\u03c0x) - 3)^2 + 2(3sin(3\u03c0x) - 3)) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1((-2((-3(1sin(3\u03c0x - 1) + 2) - 3)/(-2(1sin(3\u03c0x - 1) + 2))) + 3)/(2((-3(1sin(3\u03c0x - 1) + 2) - 3)/(-2(1sin(3\u03c0x - 1) + 2))) + 2))^2 - 4((-2((-3(1sin(3\u03c0x - 1) + 2) - 3)/(-2(1sin(3\u03c0x - 1) + 2))) + 3)/(2((-3(1sin(3\u03c0x - 1) + 2) - 3)/(-2(1sin(3\u03c0x - 1) + 2))) + 2)) - 1 and g(x) = 3sin(2\u03c0((-2((-2x)/(3x + 3)) + 2)/(-1((-2x)/(3x + 3)) - 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": "47", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5((3(-1|(5x + 3) - 1| + 3) + 3)/(2(-1|(5x + 3) - 1| + 3) - 1)) - 5 and g(x) = (-1(-3|((3x + 2)/(-1x + 2)) + 3| + 2) - 2)/(1(-3|((3x + 2)/(-1x + 2)) + 3| + 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) = -4(-2|(-1(2x^2 + 4x + 3)^2 + 3) + 2| + 1)^2 + 3(-2|(-1(2x^2 + 4x + 3)^2 + 3) + 2| + 1) - 3 and g(x) = -1|((-2(-3|x + 1| - 2) + 2)/(-1(-3|x + 1| - 2) - 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|(-2sin(1\u03c0(5x^2 + 4x + 4) + 2) - 1)| - 1) - 1 and g(x) = -3(3(-2e^(-1x) - 1)^2 - 4(-2e^(-1x) - 1) - 4)^2 + 3(3(-2e^(-1x) - 1)^2 - 4(-2e^(-1x) - 1) - 4) + 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": "Determine the number of intersection points between the functions f(x) = -5(2|(1(3sin(1\u03c0x - 2) + 1)^2 + 5(3sin(1\u03c0x - 2) + 1) - 4) - 2| - 2)^2 + 2(2|(1(3sin(1\u03c0x - 2) + 1)^2 + 5(3sin(1\u03c0x - 2) + 1) - 4) - 2| - 2) and g(x) = 3sin(1\u03c0(3(-2|x - 3| - 1)^2 - 4(-2|x - 3| - 1) + 4) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(-3(3cos(3\u03c0(-3x + 1) - 1) + 2) - 1) + 3)/(3(-3(3cos(3\u03c0(-3x + 1) - 1) + 2) - 1) + 2) = 3(4((-3x - 3)/(-3x + 1))^2 + 1((-3x - 3)/(-3x + 1)) + 3) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2((-2(-2|(-2|x|) + 3| - 3) + 1)/(3(-2|(-2|x|) + 3| - 3) + 1)))/(3((-2(-2|(-2|x|) + 3| - 3) + 1)/(3(-2|(-2|x|) + 3| - 3) + 1)) - 2) and g(x) = 5(3sin(1\u03c0(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": "62", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(5(-1(-3x - 4)^2 + 1(-3x - 4)) + 1)^2 + 4(5(-1(-3x - 4)^2 + 1(-3x - 4)) + 1) - 2 and g(x) = (-1(-1e^(2(-4x + 3)) + 1) - 1)/(3(-1e^(2(-4x + 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 points of intersection exist between the functions f(x) = 1|((-3(-3cos(3\u03c0((-1x - 2)/(-2x + 1)) - 1) - 1) + 3)/(1(-3cos(3\u03c0((-1x - 2)/(-2x + 1)) - 1) - 1))) - 3| + 3 and g(x) = (3((-1((2x + 3)/(-2x + 3)) + 3)/(2((2x + 3)/(-2x + 3)) + 1)) - 3)/(-2((-1((2x + 3)/(-2x + 3)) + 3)/(2((2x + 3)/(-2x + 3)) + 1)) - 1) in the range -10 \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) = 5(-2cos(1\u03c0(4(3|x + 2| + 3)) - 3) - 3) + 5 and g(x) = (1(1(3x^2 - 5x + 2) + 2) + 3)/(-1(1(3x^2 - 5x + 2) + 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "240", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1((2(-1sin(3\u03c0(-3x + 1) + 1) + 1) - 1)/(3(-1sin(3\u03c0(-3x + 1) + 1) + 1)))^2 - 1((2(-1sin(3\u03c0(-3x + 1) + 1) + 1) - 1)/(3(-1sin(3\u03c0(-3x + 1) + 1) + 1))) + 3 = -3cos(2\u03c0(2((-3x - 2)/(3x - 1)) + 5) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "179", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(-3|(-2sin(1\u03c0(5x^2 - 1x - 4)) - 3) + 3|) - 5 = (-1(-2cos(1\u03c0(-4x^2 - 1x + 4) + 2)) - 2)/(-2(-2cos(1\u03c0(-4x^2 - 1x + 4) + 2)) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "106", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(-2sin(2\u03c0(5((1x)/(-1x - 2))^2 + 1((1x)/(-1x - 2)) + 1) - 3)) - 2 and g(x) = -1(-3(1x^2 - 2x - 4) - 3)^2 - 4(-3(1x^2 - 2x - 4) - 3) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(-1((2(-1x^2 + 5x - 5) + 1)/(1(-1x^2 + 5x - 5) - 3)) + 2) + 1| - 2 = -3|(1(-3|x + 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": "How many points of intersection exist between the functions f(x) = 3(2sin(2\u03c0(-4(-4x^2 + 2x - 2)^2 - 5(-4x^2 + 2x - 2) + 4) + 2))^2 + 4(2sin(2\u03c0(-4(-4x^2 + 2x - 2)^2 - 5(-4x^2 + 2x - 2) + 4) + 2)) + 4 and g(x) = (-2(-3|((-2x + 1)/(2x - 3)) + 2| + 2))/(3(-3|((-2x + 1)/(2x - 3)) + 2| + 2) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(1((-2(2e^(1x) - 3) + 3)/(-2(2e^(1x) - 3) + 1))^2 + 3((-2(2e^(1x) - 3) + 3)/(-2(2e^(1x) - 3) + 1))) - 3 and g(x) = 5(3|(3x) + 1|)^2 + 1(3|(3x) + 1|) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2((1(-1sin(1\u03c0(-2|x + 1| + 3) + 1)) - 2)/(2(-1sin(1\u03c0(-2|x + 1| + 3) + 1)) - 3)) - 1)/(3((1(-1sin(1\u03c0(-2|x + 1| + 3) + 1)) - 2)/(2(-1sin(1\u03c0(-2|x + 1| + 3) + 1)) - 3))) = (-2(-3cos(3\u03c0(1|x - 2| - 1) + 3) + 1) + 3)/(-3(-3cos(3\u03c0(1|x - 2| - 1) + 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": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(-5(5((3x + 3)/(-2x + 3)) + 4) + 5) + 1 and g(x) = (-2(-2cos(2\u03c0(-1|x - 1| + 1) + 1) + 2) - 2)/(-3(-2cos(2\u03c0(-1|x - 1| + 1) + 1) + 2) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(4(-1|((-1x + 3)/(-1x - 2)) + 3|) - 5)| - 1 and g(x) = (1(-3(3cos(3\u03c0x - 2) + 3) + 2) - 2)/(2(-3(3cos(3\u03c0x - 2) + 3) + 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1sin(1\u03c0(1(-4(-1|x - 2| + 1)^2 - 4(-1|x - 2| + 1) + 5)^2 + 1(-4(-1|x - 2| + 1)^2 - 4(-1|x - 2| + 1) + 5)) - 1) - 1 = -3cos(1\u03c0(4((1x + 3)/(3x))^2 - 3((1x + 3)/(3x)) + 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": "107", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(-2((-2(5x^2 + 1x - 2) - 1)/(-2(5x^2 + 1x - 2) - 2)) + 3) + 3| - 2 and g(x) = -1|((-3(1x - 3) - 1)/(2(1x - 3) - 1)) - 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(-3(-1(1x^2 - 5x + 3) - 2)^2 + 1)^2 - 3(-3(-1(1x^2 - 5x + 3) - 2)^2 + 1) - 4 and g(x) = (-1(-1((-1x + 1)/(-2x))^2 + 4((-1x + 1)/(-2x))) - 2)/(2(-1((-1x + 1)/(-2x))^2 + 4((-1x + 1)/(-2x))) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1sin(3\u03c0(5((-3(-1x - 5))/(-2(-1x - 5) - 2)) - 1) + 2) + 2 and g(x) = -4(1(-3cos(1\u03c0x + 3) - 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": "26", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2((1(1(5x^2) + 4) - 3)/(3(1(5x^2) + 4) - 2))^2 and g(x) = 3(-2|(2|x + 2|) - 2|) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(1cos(2\u03c0(3|(-2x^2 + 2) + 1| - 2) - 2) - 3) - 2 = (3(-2cos(2\u03c0((3x)/(1x - 2))) + 2) + 1)/(-1(-2cos(2\u03c0((3x)/(1x - 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": "502", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-3sin(2\u03c0((2(2x + 1) + 3)/(1(2x + 1) - 3)) - 3) + 2) + 1)/(3(-3sin(2\u03c0((2(2x + 1) + 3)/(1(2x + 1) - 3)) - 3) + 2) - 2) and g(x) = 4(-1((-3x + 1)/(3x + 2)) - 5) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3((1(-1|(5x - 3) + 2| + 1) + 3)/(-1(-1|(5x - 3) + 2| + 1) - 1)) + 2)/(-2((1(-1|(5x - 3) + 2| + 1) + 3)/(-1(-1|(5x - 3) + 2| + 1) - 1)) - 2) and g(x) = -3(2sin(2\u03c0(-2|x - 2|) - 2) - 3) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1e^(1(-3(4(1x^2 + 5x + 5)^2 - 3(1x^2 + 5x + 5) - 4) - 1)) + 2 = -1(5(3e^(1x) + 2))^2 - 3(5(3e^(1x) + 2)) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-2(-2cos(2\u03c0(-1|x - 3| - 1) - 3))^2 + 4(-2cos(2\u03c0(-1|x - 3| - 1) - 3)) + 1) + 5 and g(x) = 2(1|(3x^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": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(1\u03c0(-2|(-2|(5x + 3) - 2| + 1) + 3| - 3)) + 2 and g(x) = -1(-3cos(1\u03c0(-2x^2 + 5x - 3) + 2) - 3) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "265", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3e^(2(2|((1((-1x - 3)/(3x - 2)) + 3)/(3((-1x - 3)/(3x - 2)) + 2))| - 1)) + 1 = -2cos(2\u03c0(-4(1|x + 2| - 1)^2 - 4(1|x + 2| - 1) - 1) - 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2sin(1\u03c0(-2|(5((2x)/(1x - 3))^2 - 1((2x)/(1x - 3))) - 1| - 2) + 1) - 3 and g(x) = (-2(-2|(-1|x + 1| - 3) + 3| + 3) + 2)/(3(-2|(-1|x + 1| - 3) + 3| + 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2sin(1\u03c0(2(1|(-1x - 2) + 1| + 1)^2 - 5(1|(-1x - 2) + 1| + 1) - 4) + 1) - 2 = 1(2sin(2\u03c0(-4x - 2) + 1) + 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "285", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(3(-2sin(1\u03c0((1x - 2)/(1x - 3))) + 2)^2 - 5(-2sin(1\u03c0((1x - 2)/(1x - 3))) + 2) + 1) + 1 and g(x) = -5(-3|(2x^2 + 2x - 2) - 1| + 3)^2 + 1(-3|(2x^2 + 2x - 2) - 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": "How many solutions does the equation (3(3(3cos(2\u03c0(-1x + 1) + 2) + 1)^2 + 4(3cos(2\u03c0(-1x + 1) + 2) + 1) - 2) + 2)/(-3(3(3cos(2\u03c0(-1x + 1) + 2) + 1)^2 + 4(3cos(2\u03c0(-1x + 1) + 2) + 1) - 2) + 1) = 4((-3(1|x - 2| - 1) - 3)/(1(1|x - 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": "24", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2((3((-2(2cos(3\u03c0x - 3) - 1) - 3)/(-3(2cos(3\u03c0x - 3) - 1) + 4)) - 2)/(3((-2(2cos(3\u03c0x - 3) - 1) - 3)/(-3(2cos(3\u03c0x - 3) - 1) + 4)) + 1)) - 1)/(1((3((-2(2cos(3\u03c0x - 3) - 1) - 3)/(-3(2cos(3\u03c0x - 3) - 1) + 4)) - 2)/(3((-2(2cos(3\u03c0x - 3) - 1) - 3)/(-3(2cos(3\u03c0x - 3) - 1) + 4)) + 1)) + 2) = -2((-1((1x + 1)/(-2x + 3)) - 2)/(3((1x + 1)/(-2x + 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": "39", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1sin(1\u03c0(-4(-1|(-1x^2 - 5x - 4) - 2| - 3) - 4) + 1) - 2 and g(x) = -3(3e^(1(-4x - 3)) - 1)^2 + 3(3e^(1(-4x - 3)) - 1) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "600", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1sin(3\u03c0(2|(3|((2x - 2)/(-1x - 2)) - 1| - 3) + 3| - 2)) - 1 and g(x) = -4(-1cos(3\u03c0(-3x^2 + 1) - 2)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1396", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(-3(2|(-1|x - 3|) + 3| - 3) + 3) - 1 and g(x) = 2|(3cos(1\u03c0(-1x^2 + 1x + 2) + 1) + 2)| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-3|((-3((3x - 2)/(1x + 2)) + 2)/(3((3x - 2)/(1x + 2)) + 1)) - 1| - 1) + 3| + 1 and g(x) = (1(3(-1x^2 + 5x - 1) - 1) - 2)/(2(3(-1x^2 + 5x - 1) - 1) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|((-2(-2(-1|x + 3| + 3) - 1) + 3)/(-2(-2(-1|x + 3| + 3) - 1))) + 1| + 3 and g(x) = 1(2|(-4x + 2) - 3| - 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1((-1(3((1x - 2)/(-1x - 2)) + 2))/(-2(3((1x - 2)/(-1x - 2)) + 2) - 3)) + 2)/(1((-1(3((1x - 2)/(-1x - 2)) + 2))/(-2(3((1x - 2)/(-1x - 2)) + 2) - 3))) = 1cos(2\u03c0(-1(-3|x + 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": "25", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(-3(3|((-3x - 1)/(-3x - 2)) + 1| - 2)^2 + 3(3|((-3x - 1)/(-3x - 2)) + 1| - 2) - 1) + 5 and g(x) = -3e^(-1(2(1x^2 - 4x - 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(-1|(-3(-1x - 2)^2 - 4(-1x - 2) + 4) - 3| + 2) + 1 and g(x) = (2(2sin(3\u03c0(-5x^2 - 3x + 5)) + 3) + 2)/(1(2sin(3\u03c0(-5x^2 - 3x + 5)) + 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1sin(2\u03c0(3(-3(-1x^2 + 5) - 3)^2 - 5(-3(-1x^2 + 5) - 3) - 4) - 3) + 1 and g(x) = -1|(3sin(1\u03c0((1x - 1)/(3x - 1)) + 3) + 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": "223", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3sin(3\u03c0(5(-1(4x^2 - 5x + 2)^2 + 4(4x^2 - 5x + 2) + 4)^2 + 5(-1(4x^2 - 5x + 2)^2 + 4(4x^2 - 5x + 2) + 4) + 3) - 2) - 1 and g(x) = -1|(-2((2x)/(1x + 2)) - 4) - 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3((2((1(-2sin(3\u03c0x + 1) + 3) - 3)/(2(-2sin(3\u03c0x + 1) + 3) - 1)) - 3)/(2((1(-2sin(3\u03c0x + 1) + 3) - 3)/(2(-2sin(3\u03c0x + 1) + 3) - 1)) - 1)) = (-3(-3e^(2((-3x - 1)/(1x))) - 2) - 1)/(-1(-3e^(2((-3x - 1)/(1x))) - 2) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(4(-2e^(2(5x^2 + 3x - 4)) + 1) - 5) + 1)/(-2(4(-2e^(2(5x^2 + 3x - 4)) + 1) - 5) + 3) and g(x) = -2(-3(-3|x - 2| + 1)^2 - 5(-3|x - 2| + 1) - 1)^2 - 2(-3(-3|x - 2| + 1)^2 - 5(-3|x - 2| + 1) - 1) + 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) = (-1(-2e^(1(-4((2x - 3)/(-1x)))) - 1))/(3(-2e^(1(-4((2x - 3)/(-1x)))) - 1) - 2) and g(x) = -2sin(2\u03c0((-3(2|x + 3| - 1) - 1)/(2(2|x + 3| - 1) - 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": "92", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3(4(5(2|x - 2| + 1) + 5)^2 + 4(5(2|x - 2| + 1) + 5) - 3) + 1)/(-2(4(5(2|x - 2| + 1) + 5)^2 + 4(5(2|x - 2| + 1) + 5) - 3) + 3) = 5(3sin(3\u03c0(3x^2 - 1x - 2) - 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": "1118", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3((-2(3|(-3x^2 - 2x - 3) + 3| - 3) - 1)/(1(3|(-3x^2 - 2x - 3) + 3| - 3))) + 5 and g(x) = (-1(-1|(-3x + 2) - 1| - 1) - 1)/(-3(-1|(-3x + 2) - 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) = (-2((-3(-3cos(1\u03c0(-3x - 2) + 3) - 1) - 3)/(3(-3cos(1\u03c0(-3x - 2) + 3) - 1) + 2)))/(2((-3(-3cos(1\u03c0(-3x - 2) + 3) - 1) - 3)/(3(-3cos(1\u03c0(-3x - 2) + 3) - 1) + 2)) + 3) and g(x) = 1cos(2\u03c0((2((-1x - 1)/(3x - 1)) - 2)/(1((-1x - 1)/(3x - 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": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-1(-3|(3x - 1) - 3| - 1)^2 + 1))/(2(-1(-3|(3x - 1) - 3| - 1)^2 + 1) + 1) and g(x) = 3cos(3\u03c0(-2|((2x + 2)/(-2x)) - 1| + 1) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(-2e^(-1(-1(-2x^2 - 4x + 1) - 2))) + 1| + 1 and g(x) = (3((2(-1sin(1\u03c0x - 2) + 1) + 2)/(-3(-1sin(1\u03c0x - 2) + 1))) - 2)/(-2((2(-1sin(1\u03c0x - 2) + 1) + 2)/(-3(-1sin(1\u03c0x - 2) + 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": "Determine the number of intersection points between the functions f(x) = (2((-1(-4(-2e^(-1x) - 2)) - 2)/(-3(-4(-2e^(-1x) - 2)))) - 1)/(-3((-1(-4(-2e^(-1x) - 2)) - 2)/(-3(-4(-2e^(-1x) - 2))))) and g(x) = -3cos(1\u03c0(1|((2x + 3)/(-1x + 2)) + 1| - 3)) - 1 where -10 \u2264 x \u2264 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) = -2cos(2\u03c0(-2|(4(-4x - 1) + 3) - 1| + 2)) and g(x) = (-3(4(1x + 3) + 2) - 1)/(-2(4(1x + 3) + 2) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1008", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(1sin(2\u03c0(1((3x + 1)/(2x + 2))^2 + 1((3x + 1)/(2x + 2)) + 1)) - 1)^2 + 4(1sin(2\u03c0(1((3x + 1)/(2x + 2))^2 + 1((3x + 1)/(2x + 2)) + 1)) - 1) - 4 = 1|(-1((-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": "87", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(2|(-1e^(-2((-2x)/(-3x + 4))) + 2) - 2| - 2)^2 + 4(2|(-1e^(-2((-2x)/(-3x + 4))) + 2) - 2| - 2) + 2 and g(x) = 2cos(3\u03c0(5(-5x^2 - 4x - 4)^2 + 1(-5x^2 - 4x - 4) + 1) - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(1|(2(-1x^2 - 3x - 2) - 3) + 3|) - 3| - 2 and g(x) = 2sin(3\u03c0(3|(-2x - 1)|) - 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(1\u03c0(-2(-1|(-1x - 4) + 3| + 1)^2 + 2(-1|(-1x - 4) + 3| + 1) + 2) - 3) - 3 and g(x) = 4((1(-4x - 2) + 3)/(-3(-4x - 2)))^2 + 3((1(-4x - 2) + 3)/(-3(-4x - 2))) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "364", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(1(-1(1x^2 + 5x)^2 + 5) + 1) - 5 and g(x) = -3((2(-3|x| + 3))/(2(-3|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": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(-3(4(2|x| - 1) - 2) + 3) + 3 and g(x) = -4(3sin(2\u03c0((-1x + 3)/(2x + 1)) - 2)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-3|(-1|(3|x + 2| - 2) + 3| - 1) + 3| - 3)^2 - 4(-3|(-1|(3|x + 2| - 2) + 3| - 1) + 3| - 3) + 4 = 1(3((-2x - 1)/(-3x + 1))^2 - 2((-2x - 1)/(-3x + 1)) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(2|(-2(-3cos(2\u03c0x - 2) + 3) - 2) - 2| - 3) - 3| - 2 and g(x) = 1|(-3(-1|x - 3| + 2)^2 - 5(-1|x - 3| + 2) + 3)| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2e^(2(-3(3|(-2x^2 - 3x - 1) - 3| - 2)^2 + 4(3|(-2x^2 - 3x - 1) - 3| - 2) + 3)) - 1 and g(x) = -3(-1(-4x^2 + 3x - 1)^2 + 1(-4x^2 + 3x - 1) + 5)^2 + 1(-1(-4x^2 + 3x - 1)^2 + 1(-4x^2 + 3x - 1) + 5) + 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(2|(2|((1x - 2)/(-3x + 2)) + 1| + 2) - 2| - 2) - 1)/(1(2|(2|((1x - 2)/(-3x + 2)) + 1| + 2) - 2| - 2) + 2) and g(x) = -3cos(3\u03c0((1(1x^2 + 1x - 5) - 2)/(-3(1x^2 + 1x - 5) - 1)) - 3) + 2 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": "How many points of intersection exist between the functions f(x) = -3|(-4((-3(-1|x - 2| + 3) - 1)/(1(-1|x - 2| + 3) + 3))^2 + 5((-3(-1|x - 2| + 3) - 1)/(1(-1|x - 2| + 3) + 3)) - 4) - 2| - 2 and g(x) = (2(-2sin(2\u03c0(-4x^2 - 1x + 3) + 2) + 1) - 2)/(2(-2sin(2\u03c0(-4x^2 - 1x + 3) + 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": "34", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|(-1(-2(-4x^2 + 5)^2 + 3(-4x^2 + 5) - 4)^2 - 1(-2(-4x^2 + 5)^2 + 3(-4x^2 + 5) - 4) + 2) + 1| - 2 = -3|((1(-2e^(-1x)) + 3)/(-2(-2e^(-1x)) - 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 solutions does the equation 3(2(-1((-1x + 2)/(-2x))^2 - 4((-1x + 2)/(-2x)) + 1)^2 - 2(-1((-1x + 2)/(-2x))^2 - 4((-1x + 2)/(-2x)) + 1) - 4) + 3 = -1cos(2\u03c0((-1(-5x^2 - 3x + 5) - 2)/(2(-5x^2 - 3x + 5) - 1))) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(-2cos(2\u03c0(-2((-1x + 2)/(-2x - 1)) - 4) - 3) - 2) - 2| - 1 and g(x) = 1(2sin(3\u03c0(1|x + 3|) + 2) - 2)^2 + 3(2sin(3\u03c0(1|x + 3|) + 2) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(2\u03c0(5(5((-1x + 1)/(-3x + 4))^2 - 2((-1x + 1)/(-3x + 4)) - 5))) - 1 and g(x) = -3(-2|(1x^2 - 2x + 4) - 2| + 1)^2 - 2(-2|(1x^2 - 2x + 4) - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4((1(2|(-2x^2 + 2x - 4) - 2| + 1) - 2)/(-2(2|(-2x^2 + 2x - 4) - 2| + 1) + 3))^2 + 5((1(2|(-2x^2 + 2x - 4) - 2| + 1) - 2)/(-2(2|(-2x^2 + 2x - 4) - 2| + 1) + 3)) + 5 and g(x) = 1sin(1\u03c0(1((-2x - 1)/(-2x - 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": "24", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|((-3(-1cos(2\u03c0(-1|x + 1|) - 1) - 3) + 2)/(-3(-1cos(2\u03c0(-1|x + 1|) - 1) - 3) + 2)) - 3| + 3 and g(x) = (-2((-1(2cos(3\u03c0x + 1) + 1) - 2)/(2(2cos(3\u03c0x + 1) + 1) - 1)) + 1)/(-2((-1(2cos(3\u03c0x + 1) + 1) - 2)/(2(2cos(3\u03c0x + 1) + 1) - 1)) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(2sin(2\u03c0(1|(-4x^2 - 4x + 3) + 1| - 1) + 3) - 2) - 2| - 2 and g(x) = 2sin(3\u03c0(1(3x^2 - 1x + 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": "757", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2((-3(2|(-2|x - 1| - 1) + 2| - 1) + 1)/(-3(2|(-2|x - 1| - 1) + 2| - 1) + 4))^2 + 1((-3(2|(-2|x - 1| - 1) + 2| - 1) + 1)/(-3(2|(-2|x - 1| - 1) + 2| - 1) + 4)) + 2 and g(x) = -3((-1(1sin(2\u03c0x) + 1) - 2)/(-2(1sin(2\u03c0x) + 1) - 3))^2 + 4((-1(1sin(2\u03c0x) + 1) - 2)/(-2(1sin(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": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(-2(1(-2cos(3\u03c0x + 2) + 3)^2 - 3(-2cos(3\u03c0x + 2) + 3) + 2) + 1) - 2)/(-2(-2(1(-2cos(3\u03c0x + 2) + 3)^2 - 3(-2cos(3\u03c0x + 2) + 3) + 2) + 1)) and g(x) = -2cos(2\u03c0((-1((-2x + 1)/(-1x + 2)) - 3)/(-1((-2x + 1)/(-1x + 2)) - 2)) + 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1sin(2\u03c0(-1(2(-3x - 3)^2 - 3(-3x - 3) + 3) + 2) + 3) + 2 and g(x) = -3cos(2\u03c0(-3((-3x + 2)/(-3x + 1))^2 + 3((-3x + 2)/(-3x + 1)) - 3) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "643", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(-1(-1((-3x - 3)/(-1x + 2)) - 2) - 3) - 3)/(-3(-1(-1((-3x - 3)/(-1x + 2)) - 2) - 3) - 2) and g(x) = -3cos(3\u03c0(-2(5x - 5) - 1) + 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "267", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(-4(2sin(3\u03c0(-1|x|) + 3)) - 4)^2 - 4(-4(2sin(3\u03c0(-1|x|) + 3)) - 4) + 1 and g(x) = (2((-1(-1x) + 1)/(-1(-1x) + 2)))/(-3((-1(-1x) + 1)/(-1(-1x) + 2)) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "119", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(-2cos(1\u03c0(-2((3x + 2)/(2x + 1))^2 + 5((3x + 2)/(2x + 1)) - 1) - 3)) + 3)/(-1(-2cos(1\u03c0(-2((3x + 2)/(2x + 1))^2 + 5((3x + 2)/(2x + 1)) - 1) - 3)) - 2) and g(x) = -5(2|(-1cos(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": "21", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2((-1(-2cos(3\u03c0(3x + 4))) + 3)/(2(-2cos(3\u03c0(3x + 4))) + 2)))/(2((-1(-2cos(3\u03c0(3x + 4))) + 3)/(2(-2cos(3\u03c0(3x + 4))) + 2)) - 1) and g(x) = -3sin(2\u03c0(4(5x - 5) + 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": "566", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(-4(2(1x^2 - 2x - 2)^2 + 3(1x^2 - 2x - 2) + 1) - 2)| - 3 = (3(-5(5x + 1)) - 2)/(1(-5(5x + 1))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(3|((-3(-5x + 1) - 3)/(-3(-5x + 1) - 2)) + 1| + 3) + 1)/(3(3|((-3(-5x + 1) - 3)/(-3(-5x + 1) - 2)) + 1| + 3) + 2) and g(x) = 4(1|(4x^2 + 1x - 2)| - 2)^2 + 5(1|(4x^2 + 1x - 2)| - 2) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(1|(4((-3x - 1)/(-1x)) + 4) + 3| - 2) - 2| - 1 = (-1(1sin(1\u03c0(3x^2 - 5x + 3) - 2) + 3) - 3)/(-2(1sin(1\u03c0(3x^2 - 5x + 3) - 2) + 3) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|((-2(-2sin(3\u03c0(-1x - 4) - 1) - 1) + 3)/(-1(-2sin(3\u03c0(-1x - 4) - 1) - 1) - 3)) + 1| + 2 and g(x) = -4(3e^(-1(2x + 4)) + 1)^2 + 1(3e^(-1(2x + 4)) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|(-2(2sin(2\u03c0(4x + 5) + 3) - 1) - 2) - 3| and g(x) = -3(-4(1x - 4) + 2)^2 + 4(-4(1x - 4) + 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": "How many solutions does the equation -1((3(2(-2|x| - 1) + 2) + 2)/(-3(2(-2|x| - 1) + 2) + 4))^2 - 5((3(2(-2|x| - 1) + 2) + 2)/(-3(2(-2|x| - 1) + 2) + 4)) - 2 = -2sin(1\u03c0(1((-1x + 3)/(2x + 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": "43", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2((1(-3(-2sin(2\u03c0x - 2))^2 + 4(-2sin(2\u03c0x - 2)) - 1) + 1)/(2(-3(-2sin(2\u03c0x - 2))^2 + 4(-2sin(2\u03c0x - 2)) - 1) - 3)) - 3)/(2((1(-3(-2sin(2\u03c0x - 2))^2 + 4(-2sin(2\u03c0x - 2)) - 1) + 1)/(2(-3(-2sin(2\u03c0x - 2))^2 + 4(-2sin(2\u03c0x - 2)) - 1) - 3)) - 1) and g(x) = -3sin(2\u03c0((-1(-5x + 4) - 2)/(-3(-5x + 4) + 1)) - 2) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(1e^(1(1(-3|x + 1| + 3)^2 + 2(-3|x + 1| + 3))) - 1) - 5 and g(x) = (2(2(3x^2 + 5x + 4)^2 - 4(3x^2 + 5x + 4) - 1) + 1)/(2(2(3x^2 + 5x + 4)^2 - 4(3x^2 + 5x + 4) - 1) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4((-3((2(-5x^2 + 4x + 3) - 1)/(3(-5x^2 + 4x + 3) + 1)))/(3((2(-5x^2 + 4x + 3) - 1)/(3(-5x^2 + 4x + 3) + 1))))^2 - 1 and g(x) = (1(-2sin(3\u03c0(2x + 5) - 1) + 1) + 2)/(-2(-2sin(3\u03c0(2x + 5) - 1) + 1) - 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 (-1(1|(4((-2x - 2)/(-1x + 3))^2 - 4((-2x - 2)/(-1x + 3)) - 5) + 3|) + 2)/(3(1|(4((-2x - 2)/(-1x + 3))^2 - 4((-2x - 2)/(-1x + 3)) - 5) + 3|) - 2) = 3cos(3\u03c0(3(-1|x - 3| + 3) + 5) - 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "144", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|((2(-3(-2x + 3) + 3))/(-2(-3(-2x + 3) + 3) - 2)) - 2| + 3 and g(x) = -4(4(2cos(1\u03c0x + 3))^2 - 2(2cos(1\u03c0x + 3)) - 4) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3e^(1(1(3(-4x^2 + 2x - 5) + 2) - 5)) and g(x) = 1(5(3sin(3\u03c0x + 2))^2 + 4(3sin(3\u03c0x + 2)) - 4) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(-3|((1((1x - 1)/(-1x + 2)) + 3)/(-2((1x - 1)/(-1x + 2)) + 3))|))/(3(-3|((1((1x - 1)/(-1x + 2)) + 3)/(-2((1x - 1)/(-1x + 2)) + 3))|) - 2) = -1(-5(-3sin(1\u03c0x) - 2)^2 + 4(-3sin(1\u03c0x) - 2) + 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(-2(4((-3x - 2)/(-2x - 3)) - 5)^2 + 1(4((-3x - 2)/(-2x - 3)) - 5) + 3)^2 - 5(-2(4((-3x - 2)/(-2x - 3)) - 5)^2 + 1(4((-3x - 2)/(-2x - 3)) - 5) + 3) - 5 and g(x) = -1sin(2\u03c0(-2|(-3x + 2) + 1| - 3) + 1) + 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((2(-1(-5x + 4)^2 - 1) - 1)/(3(-1(-5x + 4)^2 - 1) + 2)) + 3)/(1((2(-1(-5x + 4)^2 - 1) - 1)/(3(-1(-5x + 4)^2 - 1) + 2)) - 3) and g(x) = -5(2|(3cos(2\u03c0x + 3) - 2) + 3|) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-1((-2(-3e^(2x) + 2) - 3)/(2(-3e^(2x) + 2) - 1))^2 - 5((-2(-3e^(2x) + 2) - 3)/(2(-3e^(2x) + 2) - 1)) + 5)^2 + 4(-1((-2(-3e^(2x) + 2) - 3)/(2(-3e^(2x) + 2) - 1))^2 - 5((-2(-3e^(2x) + 2) - 3)/(2(-3e^(2x) + 2) - 1)) + 5) - 2 = -2(3(1|x| - 3) - 4)^2 - 2(3(1|x| - 3) - 4) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(1(4((-1x - 2)/(-3x - 1))) + 5) - 1 = (3(-3sin(1\u03c0(-1|x + 2| + 1) + 1) - 2) - 1)/(1(-3sin(1\u03c0(-1|x + 2| + 1) + 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": "11", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2sin(3\u03c0((-1(2(5x + 5) + 1))/(-2(2(5x + 5) + 1) + 1)) - 1) + 3 = 1|(3(-1sin(1\u03c0x + 1) + 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": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(1e^(-2(-3((-2x + 1)/(3x + 1)) - 5)) + 1) - 3| - 2 and g(x) = 3cos(2\u03c0(2|(-2|x + 1| + 1) - 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": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2cos(1\u03c0(5(5((-2x + 3)/(1x + 2)) - 5)^2 + 5(5((-2x + 3)/(1x + 2)) - 5) - 5) + 1) and g(x) = -2e^(-2((2((-1x)/(3x - 2)))/(-1((-1x)/(3x - 2)) - 2))) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1420", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2((-1(-4(-2x^2 + 5x + 1)^2 - 3(-2x^2 + 5x + 1) + 2) + 2)/(-3(-4(-2x^2 + 5x + 1)^2 - 3(-2x^2 + 5x + 1) + 2) - 1)))/(2((-1(-4(-2x^2 + 5x + 1)^2 - 3(-2x^2 + 5x + 1) + 2) + 2)/(-3(-4(-2x^2 + 5x + 1)^2 - 3(-2x^2 + 5x + 1) + 2) - 1)) - 1) = 1(-1((2x - 2)/(-3x - 1))^2 + 3((2x - 2)/(-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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(-1(-4(-5x + 1)) + 5) + 1)/(-2(-1(-4(-5x + 1)) + 5) - 1) and g(x) = (-1(3|(3sin(3\u03c0x - 2) - 1) + 2| - 1))/(-2(3|(3sin(3\u03c0x - 2) - 1) + 2| - 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "98", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(-1((2(2|x + 2| - 2) - 3)/(1(2|x + 2| - 2)))^2 - 5((2(2|x + 2| - 2) - 3)/(1(2|x + 2| - 2))) + 4)^2 + 4(-1((2(2|x + 2| - 2) - 3)/(1(2|x + 2| - 2)))^2 - 5((2(2|x + 2| - 2) - 3)/(1(2|x + 2| - 2))) + 4) - 3 = 1(-2cos(2\u03c0(-1|x - 2| - 3) - 1) + 3) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(2|(2sin(3\u03c0(3|x| + 1) + 3) - 1) + 3| + 1)^2 + 5(2|(2sin(3\u03c0(3|x| + 1) + 3) - 1) + 3| + 1) and g(x) = 1|((-1(3sin(3\u03c0x - 1) + 2) + 1)/(1(3sin(3\u03c0x - 1) + 2) - 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": "26", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(1(1|(-1x - 1) + 1| - 3)^2 + 4(1|(-1x - 1) + 1| - 3) - 4)^2 + 2(1(1|(-1x - 1) + 1| - 3)^2 + 4(1|(-1x - 1) + 1| - 3) - 4) - 4 = 3sin(1\u03c0(5(-2|x + 1| + 1)^2 - 5(-2|x + 1| + 1) - 3) - 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2((1(-1((-3x + 1)/(3x - 2)) - 2) - 3)/(-3(-1((-3x + 1)/(3x - 2)) - 2) + 1)) + 2)/(-2((1(-1((-3x + 1)/(3x - 2)) - 2) - 3)/(-3(-1((-3x + 1)/(3x - 2)) - 2) + 1))) = (-1(4(-1cos(3\u03c0x) + 2) - 5) + 2)/(-3(4(-1cos(3\u03c0x) + 2) - 5) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2((-2(1(1sin(1\u03c0x + 1) + 1)^2 - 4(1sin(1\u03c0x + 1) + 1) + 5) - 2)/(-1(1(1sin(1\u03c0x + 1) + 1)^2 - 4(1sin(1\u03c0x + 1) + 1) + 5) - 3))^2 - 1((-2(1(1sin(1\u03c0x + 1) + 1)^2 - 4(1sin(1\u03c0x + 1) + 1) + 5) - 2)/(-1(1(1sin(1\u03c0x + 1) + 1)^2 - 4(1sin(1\u03c0x + 1) + 1) + 5) - 3)) + 1 = 2(-1cos(2\u03c0(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": "20", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(5(-3cos(3\u03c0(-1x^2 + 3x + 5)) - 2) - 1)^2 - 5(5(-3cos(3\u03c0(-1x^2 + 3x + 5)) - 2) - 1) + 1 and g(x) = -5(1sin(1\u03c0(-1x - 1) - 2) - 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": "819", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(3|(-3cos(3\u03c0((-3x)/(-2x - 2)) - 1) - 2) + 2| + 3) + 3 = 5(1(1x^2 + 4x - 1)^2 + 5(1x^2 + 4x - 1) - 4)^2 + 5(1(1x^2 + 4x - 1)^2 + 5(1x^2 + 4x - 1) - 4) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3((-1((2(-3x + 4))/(2(-3x + 4) - 3)) - 3)/(2((2(-3x + 4))/(2(-3x + 4) - 3)) + 1)) + 2)/(-1((-1((2(-3x + 4))/(2(-3x + 4) - 3)) - 3)/(2((2(-3x + 4))/(2(-3x + 4) - 3)) + 1)) + 3) and g(x) = -4(-2cos(2\u03c0(-2|x| + 2) - 2) - 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4((-2((-1(1cos(2\u03c0x) + 3) + 3)/(1(1cos(2\u03c0x) + 3))))/(-2((-1(1cos(2\u03c0x) + 3) + 3)/(1(1cos(2\u03c0x) + 3))) + 3)) and g(x) = -1cos(2\u03c0(-4((-2x - 1)/(1x + 3))^2 + 5((-2x - 1)/(1x + 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": "186", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(3(-1e^(2((-2x - 1)/(1x - 3))) + 1) - 4)^2 + 2(3(-1e^(2((-2x - 1)/(1x - 3))) + 1) - 4) + 4 = -3(-2sin(3\u03c0(5x - 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": "209", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(-2sin(3\u03c0(-1|(3x - 1) - 3| + 1) - 1) + 1) - 1| + 2 and g(x) = 2cos(1\u03c0(-3(1x - 3) + 4) - 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "240", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-1|((-1((-2x + 2)/(2x - 1)) + 3)/(-1((-2x + 2)/(2x - 1)) - 3))| + 3) and g(x) = (2(-2(3sin(2\u03c0x - 3) - 3)^2 - 3(3sin(2\u03c0x - 3) - 3)))/(2(-2(3sin(2\u03c0x - 3) - 3)^2 - 3(3sin(2\u03c0x - 3) - 3)) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-1|((-3(-2e^(1x) - 3) - 3)/(2(-2e^(1x) - 3) - 1)) - 1| - 1) - 3)/(-1(-1|((-3(-2e^(1x) - 3) - 3)/(2(-2e^(1x) - 3) - 1)) - 1| - 1) + 2) and g(x) = 4(4(3cos(1\u03c0x + 3) - 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 3(3sin(1\u03c0(-3|(-3x - 4) - 1| + 1) + 3) - 2)^2 - 4 = -1|(-1(2|x + 2| - 2)^2 - 3(2|x + 2| - 2) + 1) + 2| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-1cos(2\u03c0(-4(5x^2 - 3x)^2 - 3(5x^2 - 3x) - 5)) + 2) + 3)/(-1(-1cos(2\u03c0(-4(5x^2 - 3x)^2 - 3(5x^2 - 3x) - 5)) + 2) + 2) and g(x) = -5((2(-1x^2 + 2x - 5) + 3)/(-2(-1x^2 + 2x - 5))) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "74", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(-5(-2((2x + 3)/(-2x + 3)) - 3)^2 - 3(-2((2x + 3)/(-2x + 3)) - 3) + 5) - 3| + 3 = 2|(-1(-3sin(3\u03c0x))^2 + 5(-3sin(3\u03c0x)) + 4) + 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(5(-3(-1x^2 - 4x + 5) + 5)^2 + 4(-3(-1x^2 - 4x + 5) + 5) - 2) + 3 and g(x) = -3((1((-3x)/(-1x)))/(2((-3x)/(-1x)) - 1))^2 - 1((1((-3x)/(-1x)))/(2((-3x)/(-1x)) - 1)) - 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-2|(-1|((-3x + 3)/(-1x - 2)) - 3| + 3)| + 3) + 2)/(2(-2|(-1|((-3x + 3)/(-1x - 2)) - 3| + 3)| + 3) + 1) and g(x) = (2(-3sin(3\u03c0((-2x - 1)/(-2x)) + 2) + 1) - 2)/(-2(-3sin(3\u03c0((-2x - 1)/(-2x)) + 2) + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(1(-1(2|x| + 3)^2 + 2(2|x| + 3) - 4) + 4) + 2)/(3(1(-1(2|x| + 3)^2 + 2(2|x| + 3) - 4) + 4) + 3) and g(x) = (-2(1cos(3\u03c0(2|x + 3| + 2)) + 2) + 3)/(-1(1cos(3\u03c0(2|x + 3| + 2)) + 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 5(4((-1(3|x + 2| - 2) - 3)/(-3(3|x + 2| - 2) - 2))^2 - 1((-1(3|x + 2| - 2) - 3)/(-3(3|x + 2| - 2) - 2)) + 3) + 3 = 4(-2|(1x^2 + 4x - 4) + 3| - 3)^2 + 5(-2|(1x^2 + 4x - 4) + 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": "How many solutions does the equation (-2(2cos(3\u03c0(1|((-1x - 2)/(-3x + 4))| + 3) + 1) + 1))/(-2(2cos(3\u03c0(1|((-1x - 2)/(-3x + 4))| + 3) + 1) + 1) + 3) = -1|(1(2|x + 1| + 2)^2 + 1(2|x + 1| + 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": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(2sin(1\u03c0(2|(4x^2 + 2) - 1| + 3) + 2) + 1) and g(x) = -3cos(2\u03c0((3(3|x| - 2))/(-3(3|x| - 2) - 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": "18", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(-3sin(2\u03c0(-3(3x^2 + 2x - 2) + 3)) - 3) - 2 and g(x) = 1cos(2\u03c0(-4(3|x - 2| + 3) - 2) + 2) - 2 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": "Determine the number of intersection points between the functions f(x) = 2|(-2e^(1((-2(-3|x - 1| - 3) - 1)/(3(-3|x - 1| - 3) - 1))) - 2) + 3| and g(x) = 5((-2(3x^2 - 4x - 2) + 2)/(-2(3x^2 - 4x - 2) - 1))^2 + 2((-2(3x^2 - 4x - 2) + 2)/(-2(3x^2 - 4x - 2) - 1)) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(-2|(-3sin(1\u03c0(-3x^2 + 3x) + 2) - 3) - 1| + 2) and g(x) = (-3(-5(-4x^2 + 5x - 2) + 1) + 1)/(-3(-5(-4x^2 + 5x - 2) + 1) + 4) 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 solutions does the equation 1sin(2\u03c0(1(2(-3|x - 2| + 3)^2 + 4(-3|x - 2| + 3) + 5)^2 - 1(2(-3|x - 2| + 3)^2 + 4(-3|x - 2| + 3) + 5) + 5)) + 2 = -2((-2(-1e^(1x)))/(1(-1e^(1x)))) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "765", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|((3(-2(-2|x + 2| + 2)^2 + 4(-2|x + 2| + 2) - 2) + 2)/(2(-2(-2|x + 2| + 2)^2 + 4(-2|x + 2| + 2) - 2) + 3)) + 1| and g(x) = 2(-4((-2x)/(-2x))^2 + 3((-2x)/(-2x)) + 5)^2 - 1(-4((-2x)/(-2x))^2 + 3((-2x)/(-2x)) + 5) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(-2|(4((-2x - 2)/(-2x + 1)) + 1) - 1| + 1))/(2(-2|(4((-2x - 2)/(-2x + 1)) + 1) - 1| + 1) - 1) and g(x) = 2(3sin(1\u03c0(2x - 4) - 2)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(-1(1sin(2\u03c0(-2|x| + 2) - 1) - 3)^2 - 3(1sin(2\u03c0(-2|x| + 2) - 1) - 3) - 3) - 1)/(2(-1(1sin(2\u03c0(-2|x| + 2) - 1) - 3)^2 - 3(1sin(2\u03c0(-2|x| + 2) - 1) - 3) - 3)) and g(x) = (3(2sin(1\u03c0(5x^2 - 1) - 1) + 1))/(-3(2sin(1\u03c0(5x^2 - 1) - 1) + 1) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "230", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4((-1(3e^(-1(-3x^2 + 1))) - 3)/(-1(3e^(-1(-3x^2 + 1))) - 2))^2 - 3((-1(3e^(-1(-3x^2 + 1))) - 3)/(-1(3e^(-1(-3x^2 + 1))) - 2)) - 2 and g(x) = (3((-3(-2sin(2\u03c0x - 1) - 2))/(-1(-2sin(2\u03c0x - 1) - 2) - 1)) + 3)/(1((-3(-2sin(2\u03c0x - 1) - 2))/(-1(-2sin(2\u03c0x - 1) - 2) - 1))) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|((-3(-5(3cos(1\u03c0x) + 2)) - 3)/(-1(-5(3cos(1\u03c0x) + 2)) + 2)) + 2| - 2 and g(x) = 1(5(2x^2 - 5x - 1) - 3) + 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": "Find the number of times the graphs of f(x) = 1(-3|(1|(-1e^(-1x) - 1)| - 3)| - 1) - 2 and g(x) = (2(-1cos(3\u03c0(1x^2 - 1x + 3) + 3) - 1) - 2)/(-2(-1cos(3\u03c0(1x^2 - 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": "174", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(3(1(1cos(1\u03c0x + 3) - 1)^2 + 4) - 2) + 3)/(2(3(1(1cos(1\u03c0x + 3) - 1)^2 + 4) - 2) - 1) and g(x) = 3|(3cos(3\u03c0(2|x + 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": "240", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(2|(-3e^(2(-3|x + 2| - 2)) - 2) + 2| - 3) and g(x) = 5(1cos(1\u03c0(2x + 1) + 1))^2 + 1(1cos(1\u03c0(2x + 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": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|(2(5(3cos(1\u03c0x - 3) + 2) - 5)^2 - 3(5(3cos(1\u03c0x - 3) + 2) - 5) - 5) + 3| + 2 and g(x) = -2((-2(-1cos(1\u03c0x + 2) + 1) + 2)/(2(-1cos(1\u03c0x + 2) + 1) - 1)) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "66", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(-3|(1|((2x)/(-2x - 3)) + 3| + 2) - 3| - 1) - 1)/(2(-3|(1|((2x)/(-2x - 3)) + 3| + 2) - 3| - 1)) and g(x) = 3cos(3\u03c0((3(-2x^2 - 2x + 2) - 3)/(2(-2x^2 - 2x + 2) - 3)) + 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(2sin(2\u03c0(2(-5x^2 + 3x - 4)^2 - 3(-5x^2 + 3x - 4) - 4) - 2) + 3) - 1 and g(x) = -4(-3e^(-2(-2x - 5)) - 1) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "377", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(1(-3sin(3\u03c0(1|x - 2|) + 1) - 1)^2 - 2))/(1(1(-3sin(3\u03c0(1|x - 2|) + 1) - 1)^2 - 2) + 1) and g(x) = -1|(1|(5x^2 + 2x - 5) + 1| - 3) - 2| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(-5(-2cos(2\u03c0((2x - 2)/(3x - 2))) - 2) + 1)^2 - 3(-5(-2cos(2\u03c0((2x - 2)/(3x - 2))) - 2) + 1) + 1 and g(x) = -2cos(1\u03c0(-4(1x^2 - 2x + 2) - 4) - 2) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(2|(-1(-4x + 2) - 1) + 1| - 1) + 4 and g(x) = 3(-1cos(1\u03c0(4x^2 - 2x) + 3) - 1) + 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": "Determine the number of intersection points between the functions f(x) = 1(3|(2(-4x^2 + 1x + 2)^2 - 3(-4x^2 + 1x + 2) + 1) - 1| + 3) - 5 and g(x) = 1|(-1|(1x^2 + 3x) + 3|) - 1| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(1(-2cos(2\u03c0(4x^2 + 5x + 1) - 2) - 2)^2 + 1(-2cos(2\u03c0(4x^2 + 5x + 1) - 2) - 2) - 1)^2 - 1(1(-2cos(2\u03c0(4x^2 + 5x + 1) - 2) - 2)^2 + 1(-2cos(2\u03c0(4x^2 + 5x + 1) - 2) - 2) - 1) - 2 and g(x) = -4(1|(-5x + 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": "989", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3cos(2\u03c0(4(-3((-1x - 2)/(-3x))^2 - 3((-1x - 2)/(-3x)) - 2)^2 + 4)) - 2 and g(x) = (3(5(-2x - 3) + 4) - 3)/(-3(5(-2x - 3) + 4)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "376", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2((-1((-2(-3x^2 + 1x - 5) - 2)/(2(-3x^2 + 1x - 5) - 3)) - 1)/(-3((-2(-3x^2 + 1x - 5) - 2)/(2(-3x^2 + 1x - 5) - 3)) - 1)) + 2)/(3((-1((-2(-3x^2 + 1x - 5) - 2)/(2(-3x^2 + 1x - 5) - 3)) - 1)/(-3((-2(-3x^2 + 1x - 5) - 2)/(2(-3x^2 + 1x - 5) - 3)) - 1))) = 2(-2|(3e^(2x) + 1) - 3| + 3)^2 - 4(-2|(3e^(2x) + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(-3|(-3(2sin(3\u03c0x - 3) - 3)^2 - 1(2sin(3\u03c0x - 3) - 3)) + 3| + 1)^2 - 5(-3|(-3(2sin(3\u03c0x - 3) - 3)^2 - 1(2sin(3\u03c0x - 3) - 3)) + 3| + 1) + 4 and g(x) = -1sin(3\u03c0(-1(2x - 4) + 4) + 3) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "220", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(3(-2|(4x - 1) - 2| + 3)^2 - 4(-2|(4x - 1) - 2| + 3) + 2) - 2)/(2(3(-2|(4x - 1) - 2| + 3)^2 - 4(-2|(4x - 1) - 2| + 3) + 2) - 1) and g(x) = 2|(1cos(2\u03c0(2x^2 + 2x) - 1) + 2) - 3| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "721", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(1cos(1\u03c0(-5(-1|x - 2| - 2)^2 - 4(-1|x - 2| - 2) - 5) - 2) - 3)^2 - 3(1cos(1\u03c0(-5(-1|x - 2| - 2)^2 - 4(-1|x - 2| - 2) - 5) - 2) - 3) - 3 = 2|(2|(1cos(1\u03c0x + 2)) + 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": "674", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1cos(2\u03c0((-3(-4(1x^2 - 2x) - 5) + 3)/(3(-4(1x^2 - 2x) - 5) + 3)) - 2) + 3 and g(x) = 4(1(1x^2 - 3x)^2 + 5(1x^2 - 3x))^2 - 3(1(1x^2 - 3x)^2 + 5(1x^2 - 3x)) - 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((-2(1e^(-1(-1|x - 1| + 2)) + 3))/(1(1e^(-1(-1|x - 1| + 2)) + 3) + 2)) + 3 = 2|((-2(-3sin(2\u03c0x + 2) - 2) - 3)/(-2(-3sin(2\u03c0x + 2) - 2) + 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": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5((-1(-1(-2|x - 2| - 2) - 1) + 1)/(2(-1(-2|x - 2| - 2) - 1) + 1)) - 4 and g(x) = 3sin(2\u03c0(3|(-2x - 4) - 3| + 3) - 2) - 3 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 points of intersection exist between the functions f(x) = 1(-1|((-2(-5x^2 - 1x + 5) + 2)/(-1(-5x^2 - 1x + 5) + 3)) + 1| - 1) - 4 and g(x) = -1(-5((-1x + 2)/(-1x)) - 5)^2 - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(-1e^(-1(1|(-1x + 1)| + 1)) + 1) - 3)/(2(-1e^(-1(1|(-1x + 1)| + 1)) + 1)) and g(x) = (2(4(-5x^2 - 4x + 1)^2 - 2(-5x^2 - 4x + 1) - 1) - 2)/(1(4(-5x^2 - 4x + 1)^2 - 2(-5x^2 - 4x + 1) - 1) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2((-2(-1|(5x^2 + 3x + 5)| + 2) + 3)/(2(-1|(5x^2 + 3x + 5)| + 2) - 1)))/(-1((-2(-1|(5x^2 + 3x + 5)| + 2) + 3)/(2(-1|(5x^2 + 3x + 5)| + 2) - 1)) - 2) = -3|((-1(-2cos(2\u03c0x - 3)) - 3)/(3(-2cos(2\u03c0x - 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": "80", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(-2(-1sin(1\u03c0(-4x^2 + 5x + 3) + 2) - 2)) + 3| - 1 and g(x) = -2|(-2e^(-2((-1x + 2)/(1x))) - 1) - 1| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "304", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3sin(1\u03c0(-2|((-1(1x + 5) + 2)/(-3(1x + 5) + 4))|) - 1) - 3 and g(x) = -4(-3e^(-1((-3x + 3)/(-2x - 2))))^2 + 1(-3e^(-1((-3x + 3)/(-2x - 2)))) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(1(-1e^(-1(-2|x| + 2)) - 3) - 5) + 3| - 3 and g(x) = 3((3(-4x^2 + 2x + 3))/(3(-4x^2 + 2x + 3) + 1))^2 - 5((3(-4x^2 + 2x + 3))/(3(-4x^2 + 2x + 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": "Find the number of times the graphs of f(x) = -2sin(3\u03c0(1|(1|(1x - 4) + 2| + 3) - 1| - 3)) - 1 and g(x) = -2|((3(-1|x - 2| - 1) + 1)/(-3(-1|x - 2| - 1) + 2)) - 1| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(1|(-4(-3x - 5) - 1) + 3| + 3) + 3)/(1(1|(-4(-3x - 5) - 1) + 3| + 3) + 2) and g(x) = -5(1|(-2|x + 2|)| - 1)^2 + 4(1|(-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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(3((-1(-2x^2 - 1x) - 3)/(3(-2x^2 - 1x)))^2 + 3((-1(-2x^2 - 1x) - 3)/(3(-2x^2 - 1x))) - 3)| + 2 and g(x) = -5(3(4x - 2)^2 - 1(4x - 2) - 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(-3cos(1\u03c0(1(4x + 1) + 5) - 3) + 1) - 3)/(2(-3cos(1\u03c0(1(4x + 1) + 5) - 3) + 1) + 3) and g(x) = 3|(-1(1sin(2\u03c0x + 3) - 3) - 2) + 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": "Find the number of times the graphs of f(x) = -2((1(3(-1x^2 - 5x + 1) + 5) - 1)/(-3(3(-1x^2 - 5x + 1) + 5) - 2))^2 - 2((1(3(-1x^2 - 5x + 1) + 5) - 1)/(-3(3(-1x^2 - 5x + 1) + 5) - 2)) + 2 and g(x) = 1sin(1\u03c0(1(-1x^2 - 2x)^2 - 4(-1x^2 - 2x) + 2)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2e^(1(3(3|((-3x)/(-3x - 2))| - 2)^2 + 5(3|((-3x)/(-3x - 2))| - 2))) and g(x) = -1(1|((-2x - 2)/(-2x + 3)) + 1| - 1)^2 + 4(1|((-2x - 2)/(-2x + 3)) + 1| - 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) = 1(2|((3(2|x + 3|) + 2)/(1(2|x + 3|) + 1)) - 2|)^2 + 5(2|((3(2|x + 3|) + 2)/(1(2|x + 3|) + 1)) - 2|) - 4 and g(x) = (3(3(2|x + 2|) - 3) + 3)/(-3(3(2|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": "Find the number of times the graphs of f(x) = 5(1(-1((-2x - 2)/(2x - 1)) - 4) + 5) and g(x) = -4(4(-3cos(3\u03c0x - 1)) - 3)^2 + 3(4(-3cos(3\u03c0x - 1)) - 3) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(5(3sin(3\u03c0(2x - 5) + 1) - 1) + 3)^2 + 1(5(3sin(3\u03c0(2x - 5) + 1) - 1) + 3) + 5 and g(x) = 2|(-3e^(1((-1x - 3)/(2x - 1)))) - 1| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(3\u03c0((-1(-1|((1x - 2)/(3x + 2)) - 3| - 1) - 1)/(-2(-1|((1x - 2)/(3x + 2)) - 3| - 1) + 1))) - 2 and g(x) = 3(-3(-2cos(3\u03c0x - 2) + 2) + 5)^2 + 4(-3(-2cos(3\u03c0x - 2) + 2) + 5) - 1 where -10 \u2264 x \u2264 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) = 1sin(2\u03c0(5(-4(-3x - 1) + 5)^2 - 3) + 2) and g(x) = (1(2cos(1\u03c0((1x - 2)/(-1x + 3)) - 1) + 1) + 3)/(2(2cos(1\u03c0((1x - 2)/(-1x + 3)) - 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": "314", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(3|(-1(-2sin(1\u03c0x - 2) + 1) - 1) + 3| - 3)^2 + 1(3|(-1(-2sin(1\u03c0x - 2) + 1) - 1) + 3| - 3) - 1 and g(x) = -2|((-2(-3cos(2\u03c0x + 3) - 1) - 3)/(-2(-3cos(2\u03c0x + 3) - 1) + 3)) + 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(1\u03c0(-2(-1(-2|x - 3| - 1)^2 + 4(-2|x - 3| - 1) + 5) - 4) + 2) + 3 and g(x) = 5(-2cos(3\u03c0((-2x - 3)/(-1x - 1)) - 2) + 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "414", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(2|((-1((3x)/(-2x + 3)) + 3)/(-2((3x)/(-2x + 3)) - 1)) + 1| - 1) + 2 and g(x) = 3cos(3\u03c0(-4(-1x + 4) + 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(-1|((3(3|x + 1| + 2) + 1)/(-3(3|x + 1| + 2) + 2)) + 2| + 1)^2 - 3(-1|((3(3|x + 1| + 2) + 1)/(-3(3|x + 1| + 2) + 2)) + 2| + 1) - 2 and g(x) = (-1(3(2cos(1\u03c0x + 3) + 1) - 2) + 1)/(-1(3(2cos(1\u03c0x + 3) + 1) - 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((1(-1|(-2cos(1\u03c0x + 3)) + 2| - 3) - 1)/(-1(-1|(-2cos(1\u03c0x + 3)) + 2| - 3) + 3)) + 1)/(3((1(-1|(-2cos(1\u03c0x + 3)) + 2| - 3) - 1)/(-1(-1|(-2cos(1\u03c0x + 3)) + 2| - 3) + 3)) + 3) and g(x) = (-2(4(-2|x| + 1)^2 + 2(-2|x| + 1) - 5) + 1)/(-2(4(-2|x| + 1)^2 + 2(-2|x| + 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) = (1(1sin(1\u03c0(2|(3|x - 1|) + 3| - 2) + 1) + 1) - 2)/(-2(1sin(1\u03c0(2|(3|x - 1|) + 3| - 2) + 1) + 1) - 2) and g(x) = 3e^(-2(2|(3|x + 1|) - 3| + 2)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(4(3sin(3\u03c0(1x^2 + 5x - 4) + 3) + 3)) + 2 = 2cos(3\u03c0(-5(4x - 4) - 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": "108", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1((1(-4(1x^2 - 1x + 4)^2 + 2) + 2)/(3(-4(1x^2 - 1x + 4)^2 + 2) - 1)) + 1 and g(x) = 1(-1(3sin(1\u03c0x - 2) + 3) + 2)^2 - 5(-1(3sin(1\u03c0x - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(-1|(-1|(-4x) - 1| - 1) - 3| + 2) + 3| - 3 = 1(-1e^(-1(-5x^2 + 5x + 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(3sin(3\u03c0(3|(5x^2 - 5x - 2) - 1| + 1) + 3)))/(-1(3sin(3\u03c0(3|(5x^2 - 5x - 2) - 1| + 1) + 3)) - 2) and g(x) = 1(1((-3x + 1)/(1x))^2 + 4((-3x + 1)/(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": "465", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(1|((-3(-2|x + 1| + 2) - 3)/(-1(-2|x + 1| + 2) + 3)) - 3| - 1) - 3| + 1 and g(x) = 3(-4(3|x - 1| - 3)^2 + 2(3|x - 1| - 3) + 4) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(-3e^(1(5(-1|x + 1| + 2) - 2)) + 1) + 2| - 1 and g(x) = 5(-3cos(1\u03c0((-1x + 3)/(-2x - 1)) + 3) + 3)^2 - 4(-3cos(1\u03c0((-1x + 3)/(-2x - 1)) + 3) + 3) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(3\u03c0(3|(-1(2|x + 1| - 2)^2 - 3(2|x + 1| - 2) - 2) - 1| + 2)) - 2 and g(x) = (-3(2sin(3\u03c0(-3x - 4) + 1)))/(3(2sin(3\u03c0(-3x - 4) + 1)) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1109", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|((-2(-1(3x - 4)^2 - 2(3x - 4)) - 1)/(-3(-1(3x - 4)^2 - 2(3x - 4)) + 2)) - 1| - 1 and g(x) = -1(-5((2x - 3)/(1x + 1)) + 2)^2 + 1(-5((2x - 3)/(1x + 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": "Determine the number of intersection points between the functions f(x) = 1cos(2\u03c0(3|((-1(-5x - 1) - 3)/(-2(-5x - 1) - 1)) - 1| + 2) - 2) + 2 and g(x) = (-2((-1(-2sin(3\u03c0x - 1)) + 1)/(3(-2sin(3\u03c0x - 1)) + 3)) - 1)/(3((-1(-2sin(3\u03c0x - 1)) + 1)/(3(-2sin(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": "61", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(4((-1(1|x + 1| - 3))/(-2(1|x + 1| - 3) - 2))^2 - 3((-1(1|x + 1| - 3))/(-2(1|x + 1| - 3) - 2)) - 5)^2 - 5(4((-1(1|x + 1| - 3))/(-2(1|x + 1| - 3) - 2))^2 - 3((-1(1|x + 1| - 3))/(-2(1|x + 1| - 3) - 2)) - 5) + 3 and g(x) = 1|((-1((-3x)/(2x + 2)) + 2)/(-3((-3x)/(2x + 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(-1(-2|(-3sin(3\u03c0x + 2) - 3) + 1| + 3) + 2))/(-3(-1(-2|(-3sin(3\u03c0x + 2) - 3) + 1| + 3) + 2) + 2) and g(x) = -1(-1sin(1\u03c0(1x + 4) - 1) + 3)^2 - 4(-1sin(1\u03c0(1x + 4) - 1) + 3) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(-3sin(1\u03c0((1(-1|x - 3| + 3) - 1)/(3(-1|x - 3| + 3) - 2)) + 3))^2 - 4(-3sin(1\u03c0((1(-1|x - 3| + 3) - 1)/(3(-1|x - 3| + 3) - 2)) + 3)) = -1cos(2\u03c0(2|(-1|x - 3| + 2) - 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": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(2cos(3\u03c0((3(-4x^2 + 2x - 2) - 2)/(-2(-4x^2 + 2x - 2) - 1)) + 1) + 2) - 2 and g(x) = 3sin(3\u03c0(-2(-1|x - 2| + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(-3cos(1\u03c0(3(-2|x| - 3)^2 + 4(-2|x| - 3) - 1) + 1) - 1)^2 + 4(-3cos(1\u03c0(3(-2|x| - 3)^2 + 4(-2|x| - 3) - 1) + 1) - 1) - 4 and g(x) = (2(-1e^(-1(3x)) - 1) + 1)/(-1(-1e^(-1(3x)) - 1) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "977", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(-5((1(-3|x|) + 1)/(-3(-3|x|) + 2)) + 1)| + 3 and g(x) = 1|((-3(2sin(2\u03c0x) - 1) + 3)/(2(2sin(2\u03c0x) - 1) - 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 points of intersection exist between the functions f(x) = (-2(4(1|(-1x^2 - 4x + 2) + 1| - 2)^2 - 5(1|(-1x^2 - 4x + 2) + 1| - 2) + 5) + 2)/(1(4(1|(-1x^2 - 4x + 2) + 1| - 2)^2 - 5(1|(-1x^2 - 4x + 2) + 1| - 2) + 5) + 2) and g(x) = 1sin(1\u03c0((-3(-1x - 4) + 2)/(3(-1x - 4) - 2)) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1sin(1\u03c0(-3(-3((-3x - 1)/(2x + 2))^2 - 2((-3x - 1)/(2x + 2)) - 3)^2 + 2(-3((-3x - 1)/(2x + 2))^2 - 2((-3x - 1)/(2x + 2)) - 3) + 2) + 2) - 3 and g(x) = (1(-3cos(2\u03c0(-5x^2 - 3x + 3) - 1) - 3))/(-2(-3cos(2\u03c0(-5x^2 - 3x + 3) - 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": "271", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-1e^(2(1(4x^2 + 3x + 1) - 5)) - 1) - 1 and g(x) = 1|(-2(-2cos(3\u03c0x - 1) + 1)^2 - 2(-2cos(3\u03c0x - 1) + 1) + 4) + 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": "Determine the number of intersection points between the functions f(x) = 2((2(1(1x^2 - 4x - 1)^2 - 2(1x^2 - 4x - 1) - 5) - 2)/(2(1(1x^2 - 4x - 1)^2 - 2(1x^2 - 4x - 1) - 5) + 3))^2 - 2((2(1(1x^2 - 4x - 1)^2 - 2(1x^2 - 4x - 1) - 5) - 2)/(2(1(1x^2 - 4x - 1)^2 - 2(1x^2 - 4x - 1) - 5) + 3)) - 4 and g(x) = (-3(3sin(2\u03c0(2x + 3) - 2) - 2) - 3)/(-1(3sin(2\u03c0(2x + 3) - 2) - 2) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "82", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-2(2cos(2\u03c0(4x^2 + 1x + 1)) - 2)^2 + 2(2cos(2\u03c0(4x^2 + 1x + 1)) - 2) - 5)^2 - 1(-2(2cos(2\u03c0(4x^2 + 1x + 1)) - 2)^2 + 2(2cos(2\u03c0(4x^2 + 1x + 1)) - 2) - 5) - 5 and g(x) = -1(-2|(3e^(-1x) + 3) + 3| - 1)^2 - 5(-2|(3e^(-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": "316", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(2|(1|(-1|x - 1| + 2)| - 3) + 3| - 3)| + 1 and g(x) = 1|(1((-1x + 1)/(2x + 3)) + 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": "How many solutions does the equation (1((-1(-1cos(2\u03c0(3|x - 2| + 3) + 1)) - 2)/(1(-1cos(2\u03c0(3|x - 2| + 3) + 1)))) - 3)/(1((-1(-1cos(2\u03c0(3|x - 2| + 3) + 1)) - 2)/(1(-1cos(2\u03c0(3|x - 2| + 3) + 1)))) + 3) = 2sin(1\u03c0((-2((1x - 1)/(1x)) + 1)/(2((1x - 1)/(1x)))) + 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "124", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-3sin(2\u03c0(-3(3|x - 1|)^2 - 5(3|x - 1|) - 3) + 3) - 3) - 1)/(3(-3sin(2\u03c0(-3(3|x - 1|)^2 - 5(3|x - 1|) - 3) + 3) - 3) + 1) and g(x) = -1|(1(2|x - 2| + 3)^2) - 3| + 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": "Find the number of times the graphs of f(x) = 1|(-2sin(1\u03c0(3(-3|x + 2| - 2)^2 - 5) + 3) + 1) + 1| and g(x) = -2(-3(2x) - 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-2sin(1\u03c0(-2((-1x)/(1x - 3))^2 - 2((-1x)/(1x - 3)) - 5) - 2) - 2)^2 + 1(-2sin(1\u03c0(-2((-1x)/(1x - 3))^2 - 2((-1x)/(1x - 3)) - 5) - 2) - 2) + 2 and g(x) = 3|((1(2|x| + 3) + 3)/(-3(2|x| + 3)))| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "168", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(-4(-1(-3x^2 - 1x + 5) - 5) - 3) + 4 = (3(-2|(-2x - 5) + 1| + 2) - 2)/(-3(-2|(-2x - 5) + 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": "Find the number of times the graphs of f(x) = -3(3|((-2(-2cos(1\u03c0x + 1)) - 2)/(-3(-2cos(1\u03c0x + 1)) - 1)) + 1| - 2)^2 - 4(3|((-2(-2cos(1\u03c0x + 1)) - 2)/(-3(-2cos(1\u03c0x + 1)) - 1)) + 1| - 2) + 1 and g(x) = (-2(1(2cos(2\u03c0x + 1) + 2) - 2) + 2)/(2(1(2cos(2\u03c0x + 1) + 2) - 2) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "93", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(-1((3(3cos(1\u03c0x - 1) + 1) + 1)/(1(3cos(1\u03c0x - 1) + 1) - 3)) + 2)^2 - 3(-1((3(3cos(1\u03c0x - 1) + 1) + 1)/(1(3cos(1\u03c0x - 1) + 1) - 3)) + 2) - 5 and g(x) = -1(-1(3x + 5) + 4)^2 + 5(-1(3x + 5) + 4) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|((-1(4((2x + 2)/(-2x - 2))) - 1)/(-2(4((2x + 2)/(-2x - 2))))) + 1| + 3 = 4(1sin(3\u03c0(-5x^2 + 1x - 4) - 3) - 2)^2 - 3(1sin(3\u03c0(-5x^2 + 1x - 4) - 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": "1254", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(1|(1sin(1\u03c0(1x + 4) + 2) + 2) + 1| - 3) - 2 and g(x) = -2|(1sin(3\u03c0(-5x - 1) + 3) + 3) - 2| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(3|(-3sin(3\u03c0(1|x + 1| + 2) + 1) + 3) - 3| + 3)^2 - 3(3|(-3sin(3\u03c0(1|x + 1| + 2) + 1) + 3) - 3| + 3) + 1 and g(x) = (-3(-1|(1x + 3) - 3| - 3) - 3)/(-2(-1|(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": "70", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(3sin(1\u03c0(2|(-1|x + 2|) - 1| + 2) + 1)) + 2)/(3(3sin(1\u03c0(2|(-1|x + 2|) - 1| + 2) + 1)) + 2) and g(x) = -1(3|(-2cos(3\u03c0x - 3)) - 2| + 2)^2 - 5(3|(-2cos(3\u03c0x - 3)) - 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": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(3|(1|(-2cos(3\u03c0x)) + 1| - 3)| + 1)^2 + 1(3|(1|(-2cos(3\u03c0x)) + 1| - 3)| + 1) + 2 = 2e^(2(-1((-1x + 2)/(2x))^2 + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(-5(1((-3x + 3)/(-3x - 2))^2 - 3((-3x + 3)/(-3x - 2)) + 2)^2 + 1(1((-3x + 3)/(-3x - 2))^2 - 3((-3x + 3)/(-3x - 2)) + 2) + 1) + 3 and g(x) = -4(3sin(2\u03c0(-1x + 3) - 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": "34", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2cos(2\u03c0(1|((2(-3|x - 1| + 3) - 3)/(-1(-3|x - 1| + 3) - 1)) + 3| - 1) - 1) = 2(1cos(1\u03c0((-1x + 3)/(-2x - 1)) + 3) + 3) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "95", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2((-1(-2(-1x + 4) - 5) - 2)/(-2(-2(-1x + 4) - 5) - 2)) - 3)/(1((-1(-2(-1x + 4) - 5) - 2)/(-2(-2(-1x + 4) - 5) - 2)) - 2) and g(x) = 4(5(2|x + 2| - 2) + 3)^2 - 3(5(2|x + 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": "Find the number of times the graphs of f(x) = 3(-5(2(-1x^2 + 2x + 3) - 2) - 1)^2 + 1(-5(2(-1x^2 + 2x + 3) - 2) - 1) - 3 and g(x) = 2|((1(-1x - 4))/(-1(-1x - 4) + 3)) + 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(2cos(1\u03c0(-3(3x^2 - 4x - 4) - 2) - 2) + 1)^2 + 1 and g(x) = -5(-3sin(2\u03c0((1x)/(3x + 1)) - 1) + 1)^2 - 5(-3sin(2\u03c0((1x)/(3x + 1)) - 1) + 1) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "154", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(1((3(-2e^(1x) - 3) - 2)/(1(-2e^(1x) - 3) + 1)) + 5) - 3)/(-1(1((3(-2e^(1x) - 3) - 2)/(1(-2e^(1x) - 3) + 1)) + 5) - 2) and g(x) = (-3(5(1|x + 1| - 3)^2 + 1(1|x + 1| - 3) - 4) - 3)/(1(5(1|x + 1| - 3)^2 + 1(1|x + 1| - 3) - 4) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2((1((-3((1x - 3)/(-1x - 3)) - 2)/(2((1x - 3)/(-1x - 3)) + 1)) - 3)/(-2((-3((1x - 3)/(-1x - 3)) - 2)/(2((1x - 3)/(-1x - 3)) + 1))))^2 - 5((1((-3((1x - 3)/(-1x - 3)) - 2)/(2((1x - 3)/(-1x - 3)) + 1)) - 3)/(-2((-3((1x - 3)/(-1x - 3)) - 2)/(2((1x - 3)/(-1x - 3)) + 1)))) - 1 and g(x) = (-1(3cos(1\u03c0(1x^2 - 3x + 3) - 1) + 2) + 2)/(-1(3cos(1\u03c0(1x^2 - 3x + 3) - 1) + 2)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2((-2(3(2x^2 + 5x)^2 - 2(2x^2 + 5x) - 2))/(-1(3(2x^2 + 5x)^2 - 2(2x^2 + 5x) - 2) + 3)) - 3)/(3((-2(3(2x^2 + 5x)^2 - 2(2x^2 + 5x) - 2))/(-1(3(2x^2 + 5x)^2 - 2(2x^2 + 5x) - 2) + 3)) - 1) and g(x) = 1|(1sin(1\u03c0(4x - 5) - 1) - 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": "Find the number of times the graphs of f(x) = 2(-3|(2(-5x^2 + 2)^2 + 4(-5x^2 + 2)) + 3| - 3) - 2 and g(x) = -5(1cos(2\u03c0(-5x^2 + 2x - 2) - 3) - 2)^2 + 5(1cos(2\u03c0(-5x^2 + 2x - 2) - 3) - 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-5((-2(3x^2 + 5x + 1) - 2)/(2(3x^2 + 5x + 1) - 3))^2 + 3((-2(3x^2 + 5x + 1) - 2)/(2(3x^2 + 5x + 1) - 3)) + 3)^2 - 1(-5((-2(3x^2 + 5x + 1) - 2)/(2(3x^2 + 5x + 1) - 3))^2 + 3((-2(3x^2 + 5x + 1) - 2)/(2(3x^2 + 5x + 1) - 3)) + 3) + 2 and g(x) = (-1(1|(-2|x - 1| - 3) + 3|) + 1)/(3(1|(-2|x - 1| - 3) + 3|) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(1(1(-3cos(3\u03c0x) + 3)^2 + 1(-3cos(3\u03c0x) + 3))^2 - 3(1(-3cos(3\u03c0x) + 3)^2 + 1(-3cos(3\u03c0x) + 3)) - 3)^2 - 4(1(1(-3cos(3\u03c0x) + 3)^2 + 1(-3cos(3\u03c0x) + 3))^2 - 3(1(-3cos(3\u03c0x) + 3)^2 + 1(-3cos(3\u03c0x) + 3)) - 3) + 4 and g(x) = -2|((3(3x - 1) + 1)/(-1(3x - 1))) - 2| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(2|(-5(-1x^2 + 4x - 3)^2 + 1(-1x^2 + 4x - 3) + 5) + 1| + 3) + 2)/(3(2|(-5(-1x^2 + 4x - 3)^2 + 1(-1x^2 + 4x - 3) + 5) + 1| + 3) - 2) and g(x) = -2((-3(3sin(1\u03c0x + 1) + 1) + 1)/(-2(3sin(1\u03c0x + 1) + 1)))^2 - 5((-3(3sin(1\u03c0x + 1) + 1) + 1)/(-2(3sin(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": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(-3cos(3\u03c0(-2|(-4x - 3) - 3| - 3)) - 3) - 1| + 2 and g(x) = -5((2((-1x - 3)/(2x + 2)) + 3)/(1((-1x - 3)/(2x + 2)))) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1((-2(-1(-2sin(3\u03c0x - 2))^2 - 5(-2sin(3\u03c0x - 2)) + 2) - 3)/(-2(-1(-2sin(3\u03c0x - 2))^2 - 5(-2sin(3\u03c0x - 2)) + 2) + 3)) + 3 = (2((1(3e^(1x) + 3) - 3)/(-3(3e^(1x) + 3) - 2)) - 1)/(2((1(3e^(1x) + 3) - 3)/(-3(3e^(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": "60", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(1(-2sin(2\u03c0(-3x - 4) + 1) + 2)^2 + 3(-2sin(2\u03c0(-3x - 4) + 1) + 2))^2 - 1(1(-2sin(2\u03c0(-3x - 4) + 1) + 2)^2 + 3(-2sin(2\u03c0(-3x - 4) + 1) + 2)) + 1 = 2|(-1(2x^2 + 5x + 5)^2 + 1(2x^2 + 5x + 5) + 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": "42", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(-3sin(2\u03c0(-3(1x - 1)^2 + 1(1x - 1) - 4) - 3) - 2)^2 - 5 = -2(-4((3x - 2)/(2x + 2))^2 + 4((3x - 2)/(2x + 2)) + 3)^2 + 4(-4((3x - 2)/(2x + 2))^2 + 4((3x - 2)/(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": "381", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-4(-1cos(3\u03c0(-2|x + 2| + 1)) + 1)^2)^2 - 3(-4(-1cos(3\u03c0(-2|x + 2| + 1)) + 1)^2) + 1 = (-1(1(-2e^(-1x) + 2)^2 - 5(-2e^(-1x) + 2) - 3) + 3)/(-1(1(-2e^(-1x) + 2)^2 - 5(-2e^(-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": "194", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(-2e^(-2(1(-1x + 3)^2 + 4(-1x + 3) + 3)) - 1)^2 - 4(-2e^(-2(1(-1x + 3)^2 + 4(-1x + 3) + 3)) - 1) + 1 = 1|(2sin(2\u03c0(-2|x + 3| - 2) + 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": "68", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(-1|(2e^(-1(-3x + 2)) + 3) - 1| + 3)^2 - 3(-1|(2e^(-1(-3x + 2)) + 3) - 1| + 3) + 4 and g(x) = -2cos(2\u03c0(4(5x^2 - 5x - 1)^2 + 1(5x^2 - 5x - 1) - 5) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(1|((2(-3x + 3) - 1)/(-1(-3x + 3) - 2))| - 1) - 2| + 2 and g(x) = 3|(1(-5x^2 + 3x + 3) - 1) + 3| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-1sin(1\u03c0(-3|(-2x - 3) + 2| - 2) - 3) - 2) = -2(3sin(3\u03c0(1x) + 2) - 3)^2 + 1(3sin(3\u03c0(1x) + 2) - 3) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-2e^(-2(-1|((-2x - 1)/(3x - 2)) - 1|)) + 3) - 3)/(-1(-2e^(-2(-1|((-2x - 1)/(3x - 2)) - 1|)) + 3) + 2) and g(x) = 1sin(3\u03c0(4(-1x^2 - 4x - 3) - 5)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1562", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-2|(-2(2e^(2x) + 1) + 3) - 3| - 1) + 2 and g(x) = 3cos(1\u03c0(-1|(3x^2 - 1x + 1) - 2| + 1) - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "305", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(1\u03c0(-5(-2(2|x + 3| - 3)^2 + 4(2|x + 3| - 3) - 4)^2 - 4(-2(2|x + 3| - 3)^2 + 4(2|x + 3| - 3) - 4) + 4) - 1) + 3 and g(x) = -2|(-2e^(-2((-2x - 2)/(-1x + 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": "517", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2cos(3\u03c0(-1|(-3|(2x^2 + 2x + 5) + 1|)| - 3)) + 3 = -5(2cos(3\u03c0(1x^2 - 1x - 2) + 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": "601", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(2cos(1\u03c0(2(1x - 4) + 1) + 3) - 2)^2 - 4(2cos(1\u03c0(2(1x - 4) + 1) + 3) - 2) - 2 and g(x) = -1cos(1\u03c0((-1((2x + 3)/(1x)) + 2)/(-3((2x + 3)/(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": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|((3(1(5x^2 + 4x - 5) + 2) - 2)/(1(1(5x^2 + 4x - 5) + 2))) + 1| + 2 and g(x) = -1|((-3(-2e^(1x) + 3) + 3)/(2(-2e^(1x) + 3) + 2)) - 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(2e^(1((-2(2|x - 3| + 1) + 1)/(3(2|x - 3| + 1)))) - 3) + 1 and g(x) = -2sin(3\u03c0(-5(-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": "300", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2e^(-1(4(-3|(-2x^2 + 2x - 5)| - 3)^2 - 3(-3|(-2x^2 + 2x - 5)| - 3) + 1)) and g(x) = -2sin(2\u03c0(-4(1|x - 2| + 3) + 2) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "160", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(-3|((1(2|x - 1| + 3) + 3)/(-1(2|x - 1| + 3) + 2)) + 2| + 3) + 3)/(1(-3|((1(2|x - 1| + 3) + 3)/(-1(2|x - 1| + 3) + 2)) + 2| + 3) + 1) and g(x) = (-2((2(-1sin(3\u03c0x - 1) - 1))/(-1(-1sin(3\u03c0x - 1) - 1) - 2)) - 2)/(-2((2(-1sin(3\u03c0x - 1) - 1))/(-1(-1sin(3\u03c0x - 1) - 1) - 2))) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|(1|((1(1|x + 2| - 3) + 3)/(3(1|x + 2| - 3) - 1)) - 3| - 2) + 1| - 2 and g(x) = -3e^(1(-5(-3x + 1)^2 - 5(-3x + 1) - 5)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3((2((3(-2e^(2x) + 3) + 1)/(3(-2e^(2x) + 3) - 2)) + 3)/(-1((3(-2e^(2x) + 3) + 1)/(3(-2e^(2x) + 3) - 2)))) - 3 and g(x) = (-2(4(-2sin(3\u03c0x - 3)) + 1) + 1)/(1(4(-2sin(3\u03c0x - 3)) + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1((-2(-5(-1|x - 1| + 1)^2 - 1(-1|x - 1| + 1) + 2))/(2(-5(-1|x - 1| + 1)^2 - 1(-1|x - 1| + 1) + 2) + 1))^2 - 3((-2(-5(-1|x - 1| + 1)^2 - 1(-1|x - 1| + 1) + 2))/(2(-5(-1|x - 1| + 1)^2 - 1(-1|x - 1| + 1) + 2) + 1)) - 2 and g(x) = 1|((1(2|x - 3|) - 3)/(3(2|x - 3|) + 3)) + 3| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(-5(-5(3sin(1\u03c0x + 1) + 1)^2 + 1(3sin(1\u03c0x + 1) + 1) + 2) + 3)^2 + 3(-5(-5(3sin(1\u03c0x + 1) + 1)^2 + 1(3sin(1\u03c0x + 1) + 1) + 2) + 3) + 2 and g(x) = 1e^(-1((1(2|x - 2| - 3) + 2)/(1(2|x - 2| - 3) - 3))) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(-2sin(3\u03c0((-1((2x - 3)/(2x - 1)) + 2)/(-1((2x - 3)/(2x - 1)) - 1)) + 2) + 2) + 5 and g(x) = (2(3sin(3\u03c0(-1x^2 + 1x - 4) + 1) - 1) - 2)/(1(3sin(3\u03c0(-1x^2 + 1x - 4) + 1) - 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "246", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(-2cos(2\u03c0(-3|(-5x^2 + 1x - 2)|) + 3) - 3) - 3 and g(x) = -5(3(-1e^(-2x) + 1)^2 + 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": "Determine the number of intersection points between the functions f(x) = 2sin(1\u03c0(-1|((-1(5x^2 - 5x - 3) - 2)/(-3(5x^2 - 5x - 3) + 1)) + 1| - 2) + 3) and g(x) = (3(-4(4x^2 - 2)^2 + 4(4x^2 - 2) + 5) - 2)/(-2(-4(4x^2 - 2)^2 + 4(4x^2 - 2) + 5) - 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 points of intersection exist between the functions f(x) = 1(-2|(2(-3cos(2\u03c0x - 3) - 1) + 5) - 3|)^2 - 5(-2|(2(-3cos(2\u03c0x - 3) - 1) + 5) - 3|) - 3 and g(x) = -3cos(1\u03c0((-3(1|x + 1| - 3) - 1)/(-2(1|x + 1| - 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": "41", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(3((1((-1x - 2)/(2x - 1)) + 2)/(-2((-1x - 2)/(2x - 1)) - 2))) + 2| - 3 and g(x) = (-1(-1sin(3\u03c0(-5x^2 + 4x - 4) - 3) - 3) - 2)/(3(-1sin(3\u03c0(-5x^2 + 4x - 4) - 3) - 3) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-1sin(2\u03c0(3|(2|x - 3| + 3)| + 3) + 2) - 3)^2 + 4(-1sin(2\u03c0(3|(2|x - 3| + 3)| + 3) + 2) - 3) + 2 and g(x) = 4(-3|(5x^2 + 4x - 2) - 1| - 2) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2e^(1(-5(-2(3x)^2 - 1(3x) - 5)^2 - 4(-2(3x)^2 - 1(3x) - 5) - 3)) and g(x) = 2sin(3\u03c0(5(2x^2 - 1x - 3)^2 - 3(2x^2 - 1x - 3)) + 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1209", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(3|(2sin(3\u03c0(-2x - 1) - 3) + 1)| + 2) + 4 and g(x) = 3((3(4x^2 + 2x - 3) - 1)/(-1(4x^2 + 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": "216", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(3sin(1\u03c0((-2((2x + 1)/(-1x - 1)) + 2)/(3((2x + 1)/(-1x - 1)) + 3)) + 3) - 2) - 1)/(1(3sin(1\u03c0((-2((2x + 1)/(-1x - 1)) + 2)/(3((2x + 1)/(-1x - 1)) + 3)) + 3) - 2) + 1) and g(x) = (-3(3|(1cos(1\u03c0x) - 2) + 1| - 2) - 2)/(-3(3|(1cos(1\u03c0x) - 2) + 1| - 2) + 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 solutions does the equation (2(3(5((1x + 1)/(-1x - 3))^2 - 1((1x + 1)/(-1x - 3)) - 4) + 1) - 1)/(2(3(5((1x + 1)/(-1x - 3))^2 - 1((1x + 1)/(-1x - 3)) - 4) + 1) + 1) = 3sin(2\u03c0(5(4x) + 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": "767", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1((-2(-2|(2x^2 + 4x - 3)| + 3) + 1)/(1(-2|(2x^2 + 4x - 3)| + 3) - 2)) - 2)/(-1((-2(-2|(2x^2 + 4x - 3)| + 3) + 1)/(1(-2|(2x^2 + 4x - 3)| + 3) - 2)) + 2) = -3|(-2|(1|x| - 3) - 2| - 1)| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|((3((-3(3|x| - 1) + 2)/(-2(3|x| - 1) + 1)) + 2)/(3((-3(3|x| - 1) + 2)/(-2(3|x| - 1) + 1)) - 2))| - 1 and g(x) = 1sin(3\u03c0(-1|(1x + 5) - 2| - 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": "Determine the number of intersection points between the functions f(x) = 5(1(-2|(2x - 5) + 2| + 3) + 4)^2 - 2(1(-2|(2x - 5) + 2| + 3) + 4) - 2 and g(x) = -5(-2cos(1\u03c0(-2x - 5) - 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": "How many points of intersection exist between the functions f(x) = -3|(-3(-2sin(1\u03c0((-2x + 3)/(2x + 1)) - 2) + 3)^2 - 1(-2sin(1\u03c0((-2x + 3)/(2x + 1)) - 2) + 3) - 5) + 1| - 2 and g(x) = (2(-2sin(3\u03c0(-3|x + 1|) - 3) - 3) + 2)/(1(-2sin(3\u03c0(-3|x + 1|) - 3) - 3) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(-2|(-2|(-2sin(1\u03c0x + 1) - 3) + 2| + 2) + 1| - 2)| - 3 and g(x) = -2sin(2\u03c0((2((-1x + 3)/(3x)) - 1)/(-1((-1x + 3)/(3x)) - 2)) - 2) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "82", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(3(-2|(1x - 5) + 1| + 1) + 1)^2 - 2(3(-2|(1x - 5) + 1| + 1) + 1) - 1 and g(x) = 2e^(1(2(-2x + 1) + 5)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(1((-2(3|x + 2| - 1) - 1)/(2(3|x + 2| - 1) - 1))^2 + 1((-2(3|x + 2| - 1) - 1)/(2(3|x + 2| - 1) - 1))) - 1)/(1(1((-2(3|x + 2| - 1) - 1)/(2(3|x + 2| - 1) - 1))^2 + 1((-2(3|x + 2| - 1) - 1)/(2(3|x + 2| - 1) - 1))) + 3) and g(x) = -5(5(-2cos(2\u03c0x + 3) - 2) + 3)^2 + 4(5(-2cos(2\u03c0x + 3) - 2) + 3) + 1 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) = -3cos(2\u03c0(3((-1(-1x + 4) + 1)/(1(-1x + 4) + 3)) + 5) - 3) - 1 and g(x) = (-2(2e^(-1(-4x^2 + 1x + 3)) + 2) - 3)/(-2(2e^(-1(-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": "154", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(2|(-3cos(2\u03c0(2x^2 - 3x - 5) + 1) + 1) - 1| + 2)^2 + 3(2|(-3cos(2\u03c0(2x^2 - 3x - 5) + 1) + 1) - 1| + 2) - 3 and g(x) = -2((-2(4x^2 - 2x - 4) - 3)/(-1(4x^2 - 2x - 4)))^2 + 4((-2(4x^2 - 2x - 4) - 3)/(-1(4x^2 - 2x - 4))) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1sin(3\u03c0(-2(-5(-3x^2 + 4)^2 - 4(-3x^2 + 4) + 1)^2 + 4(-5(-3x^2 + 4)^2 - 4(-3x^2 + 4) + 1) + 4) + 1) - 3 = 2sin(3\u03c0(-2|(-1x - 1)| + 2) + 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "102", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(-4(3(3sin(2\u03c0x - 3)) - 5)^2 + 4(3(3sin(2\u03c0x - 3)) - 5) + 4)^2 - 2(-4(3(3sin(2\u03c0x - 3)) - 5)^2 + 4(3(3sin(2\u03c0x - 3)) - 5) + 4) - 1 and g(x) = -3sin(1\u03c0(1|((1x + 2)/(-1x - 1)) + 1| - 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": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|((-3((-2(1|x - 2| - 3) - 2)/(1(1|x - 2| - 3))) - 3)/(-1((-2(1|x - 2| - 3) - 2)/(1(1|x - 2| - 3))) + 3)) - 3| + 2 and g(x) = 3cos(1\u03c0(4((-2x + 3)/(2x - 3))^2 + 5((-2x + 3)/(2x - 3)) - 1) - 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) = (2(-1((1((1x - 1)/(1x + 1)) - 2)/(-3((1x - 1)/(1x + 1)) - 3)) + 4) + 2)/(2(-1((1((1x - 1)/(1x + 1)) - 2)/(-3((1x - 1)/(1x + 1)) - 3)) + 4) - 1) and g(x) = 1|(-1|(5x^2 + 1x - 2) + 1| + 3) + 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(1(-2|(1x + 1) - 3| + 2)^2 - 5)| + 3 and g(x) = -3cos(2\u03c0((-1((-3x - 1)/(3x + 2)) - 3)/(3((-3x - 1)/(3x + 2)) - 2)) + 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1cos(1\u03c0(-2|(-5(-2|x + 1| + 2)) + 1| + 3) + 3) + 2 and g(x) = 4((-1(-3sin(3\u03c0x + 2) + 2))/(1(-3sin(3\u03c0x + 2) + 2) - 3)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "135", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1sin(1\u03c0((-2(-1(-1|x + 3| + 1)) - 1)/(-3(-1(-1|x + 3| + 1)) + 1)) + 3) + 3 and g(x) = 3(2|(2e^(1x) - 1) - 2| - 2) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(2e^(-1(3((-2x + 3)/(-1x))^2 + 5((-2x + 3)/(-1x)) - 5))) + 1)/(3(2e^(-1(3((-2x + 3)/(-1x))^2 + 5((-2x + 3)/(-1x)) - 5))) - 2) = 3sin(2\u03c0((1(-1x^2 - 2x + 1) + 1)/(-3(-1x^2 - 2x + 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": "29", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2((-3(-1(-2x^2 - 2x + 2)^2 + 2) - 3)/(1(-1(-2x^2 - 2x + 2)^2 + 2) - 2)) + 1 = 2|(-3sin(1\u03c0(-1x^2 + 4x + 3) - 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": "5", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(-2(-2cos(2\u03c0(-2|x| - 2) + 3) - 3) - 4) + 4 and g(x) = -3e^(1(-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": "80", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3((-2(-2e^(1(-2x^2 + 4)) + 1) + 1)/(-3(-2e^(1(-2x^2 + 4)) + 1) - 3)) + 2)/(-3((-2(-2e^(1(-2x^2 + 4)) + 1) + 1)/(-3(-2e^(1(-2x^2 + 4)) + 1) - 3)) - 2) = 2|(1(-1|x + 2| + 3)^2 + 5(-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": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(1\u03c0(3(-2(2x) + 4) + 4) + 3) - 2 and g(x) = 2sin(3\u03c0((2(1x^2 + 4) + 2)/(1(1x^2 + 4) + 3)) - 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "237", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(4((-1(3cos(1\u03c0x - 3) - 1) - 1)/(-3(3cos(1\u03c0x - 3) - 1)))^2 - 3((-1(3cos(1\u03c0x - 3) - 1) - 1)/(-3(3cos(1\u03c0x - 3) - 1))) + 4))/(1(4((-1(3cos(1\u03c0x - 3) - 1) - 1)/(-3(3cos(1\u03c0x - 3) - 1)))^2 - 3((-1(3cos(1\u03c0x - 3) - 1) - 1)/(-3(3cos(1\u03c0x - 3) - 1))) + 4)) = (-1(3sin(2\u03c0(1x - 3)) + 2) + 1)/(-2(3sin(2\u03c0(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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(2|(3((3x - 2)/(-1x - 3)) - 1) - 2| - 3) - 5 = (2(-2sin(1\u03c0((-3x - 1)/(-1x + 3))) + 2) - 2)/(2(-2sin(1\u03c0((-3x - 1)/(-1x + 3))) + 2) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1cos(3\u03c0(-5(4(4x^2 - 1x + 1) - 3)^2 + 3(4(4x^2 - 1x + 1) - 3)) + 2) + 3 and g(x) = -2sin(2\u03c0(-4((-1x + 1)/(-1x + 3)) + 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": "121", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(3((-2(1|x + 3| - 2))/(1(1|x + 3| - 2) + 1)) + 5) + 2 and g(x) = -3cos(1\u03c0(3(1x + 1) + 1) + 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(3(-2((-3x + 3)/(3x + 3))^2 - 5((-3x + 3)/(3x + 3)) - 3)) - 3 and g(x) = (-1(-1|((3x)/(-1x - 1)) + 3| + 2) + 3)/(2(-1|((3x)/(-1x - 1)) + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-3|(1|(2x + 5)| - 2) - 3| + 1) - 5 and g(x) = (-1(-1(-4x^2 - 5x - 3)^2 - 3(-4x^2 - 5x - 3) - 2) + 2)/(-1(-1(-4x^2 - 5x - 3)^2 - 3(-4x^2 - 5x - 3) - 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(4((2(-2cos(3\u03c0x + 3) + 3))/(-3(-2cos(3\u03c0x + 3) + 3) + 2))^2 + 4((2(-2cos(3\u03c0x + 3) + 3))/(-3(-2cos(3\u03c0x + 3) + 3) + 2)) + 5)^2 - 4(4((2(-2cos(3\u03c0x + 3) + 3))/(-3(-2cos(3\u03c0x + 3) + 3) + 2))^2 + 4((2(-2cos(3\u03c0x + 3) + 3))/(-3(-2cos(3\u03c0x + 3) + 3) + 2)) + 5) + 1 and g(x) = 4((2(-1|x - 3|))/(-1(-1|x - 3|) - 3)) + 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": "Find the number of times the graphs of f(x) = 3(-3(5((-3x + 3)/(-3x + 2)) + 5) - 5) + 1 and g(x) = 1((1(3sin(1\u03c0x - 1) - 3) - 3)/(1(3sin(1\u03c0x - 1) - 3) + 3)) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(5(-2|(-5x^2 + 5x - 2) + 1|) - 2) + 4 and g(x) = 1sin(3\u03c0(4(-3|x - 2| - 3) - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1((2(-5(-1x + 3)^2 - 3(-1x + 3) + 3) - 2)/(-1(-5(-1x + 3)^2 - 3(-1x + 3) + 3) + 2)) - 3)/(-1((2(-5(-1x + 3)^2 - 3(-1x + 3) + 3) - 2)/(-1(-5(-1x + 3)^2 - 3(-1x + 3) + 3) + 2)) + 2) and g(x) = 2|(-3e^(-1(4x - 2)) + 2)| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(-4(1|(5x^2 - 5x - 3) + 2| - 3) - 4) + 2| - 1 = -5((-1(3sin(3\u03c0x - 3)) - 2)/(1(3sin(3\u03c0x - 3)) + 3))^2 + 1((-1(3sin(3\u03c0x - 3)) - 2)/(1(3sin(3\u03c0x - 3)) + 3)) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(-2(2cos(2\u03c0(-1|x| + 2) + 3) + 1) + 5) - 3| - 1 and g(x) = (3(1cos(2\u03c0(2|x + 1| + 2) + 2) - 2))/(3(1cos(2\u03c0(2|x + 1| + 2) + 2) - 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(1(2cos(2\u03c0(-3x - 4) + 3))^2 - 3(2cos(2\u03c0(-3x - 4) + 3)) + 1)^2 - 4(1(2cos(2\u03c0(-3x - 4) + 3))^2 - 3(2cos(2\u03c0(-3x - 4) + 3)) + 1) - 3 and g(x) = -5((3(3x^2 - 4x + 3) + 2)/(2(3x^2 - 4x + 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": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(-1sin(3\u03c0(-4(-1|x - 1| - 1) + 5) - 2) - 3) + 1| + 1 and g(x) = (-3(-3(-3cos(3\u03c0x - 2) + 1) - 2) - 2)/(-1(-3(-3cos(3\u03c0x - 2) + 1) - 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) = 1e^(-2(1|((-2(-2x + 4) - 2)/(-3(-2x + 4) + 1)) + 2|)) + 3 and g(x) = -1(2(2|x|)^2 - 4(2|x|) - 5)^2 + 4(2(2|x|)^2 - 4(2|x|) - 5) + 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) = 3|(-1|(2|(2e^(-2x) - 2) - 3| - 3) + 3| + 1) - 2| + 3 and g(x) = -3(-4(2cos(2\u03c0x - 1) + 2) + 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-4(-1(-3|x + 3| - 3)^2 - 4(-3|x + 3| - 3) - 1) - 5))/(-3(-4(-1(-3|x + 3| - 3)^2 - 4(-3|x + 3| - 3) - 1) - 5)) and g(x) = (-1(5(3cos(1\u03c0x + 3) + 2)^2 + 3(3cos(1\u03c0x + 3) + 2) + 4) - 3)/(-2(5(3cos(1\u03c0x + 3) + 2)^2 + 3(3cos(1\u03c0x + 3) + 2) + 4) - 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) = -4((2((-2(-1x + 3))/(2(-1x + 3) + 1)) - 2)/(-1((-2(-1x + 3))/(2(-1x + 3) + 1)) + 2)) and g(x) = 1e^(-1(-2|(2|x - 2| - 2)| - 1)) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1cos(2\u03c0(3((1(4x^2 - 4x - 4))/(-2(4x^2 - 4x - 4) - 2)) - 3) - 3) + 1 = 2sin(2\u03c0(3|((2x + 2)/(-3x + 4))| - 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": "114", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(-1(-2sin(3\u03c0(-2x^2 + 2x + 5) - 1) + 2)) - 3| - 1 = -4(4(3cos(1\u03c0x + 2) + 1) + 3)^2 - 3(4(3cos(1\u03c0x + 2) + 1) + 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(1|((2(-2|x - 2| - 1) - 3)/(1(-2|x - 2| - 1) + 2)) - 1| + 3) + 2)/(-3(1|((2(-2|x - 2| - 1) - 3)/(1(-2|x - 2| - 1) + 2)) - 1| + 3)) and g(x) = -2(1|(-1x^2 + 1x + 1) + 3| - 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(2cos(2\u03c0(2(2|x + 2| - 3)^2 + 5(2|x + 2| - 3) - 2) - 3) + 2) - 1)/(-2(2cos(2\u03c0(2(2|x + 2| - 3)^2 + 5(2|x + 2| - 3) - 2) - 3) + 2) + 3) and g(x) = 3e^(-2((3((-3x + 2)/(1x + 1)) + 3)/(2((-3x + 2)/(1x + 1)) - 1))) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-3|(1(-4x^2 + 5x + 3) + 5) + 1|) - 5 and g(x) = (-2(4((1x - 2)/(1x + 1))^2 - 4((1x - 2)/(1x + 1)) - 2) - 1)/(-1(4((1x - 2)/(1x + 1))^2 - 4((1x - 2)/(1x + 1)) - 2) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(-4(-4(3x^2 - 3x + 1) + 5) - 5)^2 - 2(-4(-4(3x^2 - 3x + 1) + 5) - 5) and g(x) = -3(-2|(-3x^2 + 4x + 4) + 3| + 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(-1((-1(-1|x + 3|) + 1)/(2(-1|x + 3|) - 1))^2 + 1((-1(-1|x + 3|) + 1)/(2(-1|x + 3|) - 1)) - 2) - 1 and g(x) = 1|(2sin(3\u03c0(2x^2 - 4))) + 1| + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "762", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|(2(-2(-3sin(1\u03c0x - 3) + 3)^2 + 3(-3sin(1\u03c0x - 3) + 3) - 1) - 4)| and g(x) = 3((-1((2x - 1)/(-3x + 2)) - 3)/(1((2x - 1)/(-3x + 2)) + 2)) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(1|(2cos(2\u03c0(-2|x - 3| + 2) - 2) + 3)| - 3)^2 + 3(1|(2cos(2\u03c0(-2|x - 3| + 2) - 2) + 3)| - 3) + 4 = 3|(-1cos(1\u03c0(-1x^2 - 5x - 3) + 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": "82", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(5(1|(1x - 1) - 3| - 3) + 1) + 3)/(-2(5(1|(1x - 1) - 3| - 3) + 1) + 1) and g(x) = (-2(-5(1x + 3)^2 + 5(1x + 3) + 1) - 1)/(1(-5(1x + 3)^2 + 5(1x + 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) = 2sin(2\u03c0(3((1(2|x + 3| + 3))/(-1(2|x + 3| + 3) + 2))^2 - 3((1(2|x + 3| + 3))/(-1(2|x + 3| + 3) + 2)) - 2) + 2) - 1 and g(x) = -2|(-1((2x + 2)/(1x - 2))^2 - 2((2x + 2)/(1x - 2)) + 5)| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(3|(-3sin(1\u03c0(4x - 4) + 2) - 2) + 1| + 1) + 2)/(-1(3|(-3sin(1\u03c0(4x - 4) + 2) - 2) + 1| + 1) + 3) and g(x) = 1|(1|(4x^2 + 4x + 3) - 1| + 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": "66", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1e^(-2(-3|((1(-3x) + 1)/(1(-3x))) - 1| + 2)) - 1 = 1e^(-2(2|(-5x^2 - 3x + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(1|(-4((-1x - 1)/(3x + 1)) + 1) - 3| + 1) - 2 and g(x) = -5(-3sin(3\u03c0(-1x^2 + 3x + 4) + 2) + 3)^2 - 5(-3sin(3\u03c0(-1x^2 + 3x + 4) + 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": "584", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(-2sin(3\u03c0(3(-1|x| + 3) - 3) - 3) + 2))/(-1(-2sin(3\u03c0(3(-1|x| + 3) - 3) - 3) + 2) + 2) = -5(3|(3|x - 1| + 1)| - 2)^2 + 3(3|(3|x - 1| + 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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1((3(-2(-3|x + 1| + 1)^2 - 3(-3|x + 1| + 1) + 5) + 2)/(2(-2(-3|x + 1| + 1)^2 - 3(-3|x + 1| + 1) + 5) + 2)))/(-2((3(-2(-3|x + 1| + 1)^2 - 3(-3|x + 1| + 1) + 5) + 2)/(2(-2(-3|x + 1| + 1)^2 - 3(-3|x + 1| + 1) + 5) + 2)) + 3) = (-3(-3(-3e^(-1x) - 2) - 5) + 2)/(2(-3(-3e^(-1x) - 2) - 5)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-1(-2|(-1x^2 - 2x + 3) + 2| + 1)^2 - 1(-2|(-1x^2 - 2x + 3) + 2| + 1)) + 2)/(3(-1(-2|(-1x^2 - 2x + 3) + 2| + 1)^2 - 1(-2|(-1x^2 - 2x + 3) + 2| + 1)) - 2) and g(x) = (-3(1(-3x^2 + 5x + 3)^2 - 5(-3x^2 + 5x + 3) + 1) - 1)/(1(1(-3x^2 + 5x + 3)^2 - 5(-3x^2 + 5x + 3) + 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) = 5(-4(4(2sin(2\u03c0x - 1))^2 + 4(2sin(2\u03c0x - 1)) - 2)^2 - 2(4(2sin(2\u03c0x - 1))^2 + 4(2sin(2\u03c0x - 1)) - 2) + 1) - 5 and g(x) = -2((-3(-2|x - 3| + 1) - 3)/(-3(-2|x - 3| + 1) + 4))^2 + 3((-3(-2|x - 3| + 1) - 3)/(-3(-2|x - 3| + 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": "17", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|((-3(3|(5x^2 + 2x - 4) - 1| + 1) - 2)/(1(3|(5x^2 + 2x - 4) - 1| + 1) - 3)) + 1| - 3 and g(x) = -1(4(-2x^2 + 3x - 3) - 1)^2 - 4(4(-2x^2 + 3x - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2(3cos(3\u03c0(-5(1x - 2) - 4) - 1) + 1)^2 + 5(3cos(3\u03c0(-5(1x - 2) - 4) - 1) + 1) + 5 and g(x) = (-2(3sin(3\u03c0(-2|x - 1| - 1) + 3) + 2) - 1)/(3(3sin(3\u03c0(-2|x - 1| - 1) + 3) + 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(2|(-1((1x + 2)/(1x + 3))^2 + 1((1x + 2)/(1x + 3)) + 5) + 3| - 3) + 3 and g(x) = 3cos(2\u03c0(1|(-2x - 1) + 3| - 3) + 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-1(-1|(-3cos(2\u03c0x - 1) - 1)| - 3)^2 + 4(-1|(-3cos(2\u03c0x - 1) - 1)| - 3) - 3) - 5 and g(x) = 5((2(1x + 5) - 2)/(-1(1x + 5) - 3)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(2\u03c0(1(-2(1x^2 - 5x + 3)^2 + 2(1x^2 - 5x + 3) - 1) - 5) - 3) - 2 and g(x) = -5(-1(-4x + 2) + 5)^2 - 4(-1(-4x + 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": "13", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(-2cos(2\u03c0(3((2x + 3)/(-1x + 2)) + 4) - 3)) + 2| + 1 and g(x) = -3cos(1\u03c0((-3((-2x + 3)/(-1x + 3)) + 3)/(-2((-2x + 3)/(-1x + 3)) + 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": "217", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(-1|((2((2x - 1)/(1x - 2)) + 3)/(2((2x - 1)/(1x - 2)) - 1))| + 2) - 2| + 1 and g(x) = (-3(-5(-1x + 3)^2 - 2(-1x + 3) + 5) + 2)/(-3(-5(-1x + 3)^2 - 2(-1x + 3) + 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(5((-2(1x^2 - 5x + 3) + 3)/(-1(1x^2 - 5x + 3) - 2))^2 + 1((-2(1x^2 - 5x + 3) + 3)/(-1(1x^2 - 5x + 3) - 2)) - 2)^2 + 4(5((-2(1x^2 - 5x + 3) + 3)/(-1(1x^2 - 5x + 3) - 2))^2 + 1((-2(1x^2 - 5x + 3) + 3)/(-1(1x^2 - 5x + 3) - 2)) - 2) - 1 and g(x) = 3|(2|(-2sin(1\u03c0x + 3) - 2) + 2| - 2) - 3| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-4(-1(3cos(1\u03c0x + 1) + 3) + 2) - 1) + 5 and g(x) = -1sin(1\u03c0(-2|((-1x + 2)/(1x - 2)) + 2| - 1) - 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": "Determine the number of intersection points between the functions f(x) = 1|(3e^(2(-1(2x^2 - 3x - 5)^2 + 3(2x^2 - 3x - 5) + 3))) + 3| + 2 and g(x) = 1|(-2((3x + 1)/(-2x - 3))^2 + 5((3x + 1)/(-2x - 3)) - 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 points of intersection exist between the functions f(x) = -2sin(2\u03c0(4((-2(3x^2 + 4x - 4))/(1(3x^2 + 4x - 4) - 2)) + 1) - 1) - 2 and g(x) = 3cos(2\u03c0(2|((1x - 2)/(2x)) + 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": "116", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2e^(2(-2(-2((-1x - 2)/(-2x - 2)) + 5)^2 - 5(-2((-1x - 2)/(-2x - 2)) + 5) - 3)) + 1 and g(x) = 4(-5(-3cos(2\u03c0x + 3) + 2) + 1)^2 - 1(-5(-3cos(2\u03c0x + 3) + 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": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(-1(-3((-2x - 1)/(2x - 1)) + 4)^2 - 4(-3((-2x - 1)/(2x - 1)) + 4) - 3) + 1| - 2 and g(x) = 3(-1|(-5x^2 + 1x - 1) - 1| + 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 points of intersection exist between the functions f(x) = -4(-2|((1(-1x^2 - 3x - 1) - 1)/(3(-1x^2 - 3x - 1))) - 1| + 3)^2 + 2(-2|((1(-1x^2 - 3x - 1) - 1)/(3(-1x^2 - 3x - 1))) - 1| + 3) + 5 and g(x) = (1((-2((-3x)/(-3x - 2)) + 3)/(-2((-3x)/(-3x - 2)) + 1)) + 1)/(-3((-2((-3x)/(-3x - 2)) + 3)/(-2((-3x)/(-3x - 2)) + 1))) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2e^(2(-2|(-1((2x - 2)/(3x))^2 - 4((2x - 2)/(3x)))| + 2)) and g(x) = 1|(1((3x - 3)/(-2x + 3)) - 5) + 1| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2((1(-2(-2sin(1\u03c0x + 2) + 1) + 1) + 2)/(2(-2(-2sin(1\u03c0x + 2) + 1) + 1) - 3)) + 3)/(-3((1(-2(-2sin(1\u03c0x + 2) + 1) + 1) + 2)/(2(-2(-2sin(1\u03c0x + 2) + 1) + 1) - 3)) + 4) = -4(-2|(1|x + 2| - 2)|) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-1|((1(2x^2 + 4x) - 2)/(1(2x^2 + 4x) - 3)) - 1| + 2)^2 - 4(-1|((1(2x^2 + 4x) - 2)/(1(2x^2 + 4x) - 3)) - 1| + 2) - 4 and g(x) = 2(-2|(-3|x + 1| + 1) + 3| + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3e^(-2(2((2(3x^2 + 5x - 5) - 1)/(-2(3x^2 + 5x - 5) - 1))^2 + 4((2(3x^2 + 5x - 5) - 1)/(-2(3x^2 + 5x - 5) - 1)) + 1)) + 2 and g(x) = (2(2e^(1(1|x + 3|)) - 2) - 2)/(-1(2e^(1(1|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": "Find the number of times the graphs of f(x) = 1e^(1((-2(1|(2x^2 - 1) + 2| + 1) - 1)/(-2(1|(2x^2 - 1) + 2| + 1) + 3))) + 1 and g(x) = (-1(3cos(1\u03c0(5x + 4) + 3) + 2) - 2)/(1(3cos(1\u03c0(5x + 4) + 3) + 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "96", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|((1(1(1|x - 1| - 3) + 3) - 3)/(1(1(1|x - 1| - 3) + 3) + 3)) + 1| and g(x) = 3(1cos(2\u03c0(5x + 4) + 3) + 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "180", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(1\u03c0(-2|(-3(1|x| + 2)^2 - 5(1|x| + 2) - 4) + 1| + 2) - 1) + 3 and g(x) = 1|(-5(-5x + 5)^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": "22", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|((1(2e^(-1(-3|x| + 2))) + 1)/(2(2e^(-1(-3|x| + 2))))) + 1| = 1(-4(2|x + 2| - 1)^2 + 3(2|x + 2| - 1) + 4)^2 - 3(-4(2|x + 2| - 1)^2 + 3(2|x + 2| - 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(3((-3(3x^2 - 5x))/(2(3x^2 - 5x) - 3))^2 - 4((-3(3x^2 - 5x))/(2(3x^2 - 5x) - 3)) - 3) + 4 and g(x) = 4(4(3cos(2\u03c0x) + 2) - 5) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1sin(1\u03c0(-4((2(2|x + 2|))/(2(2|x + 2|) + 1))^2 - 1((2(2|x + 2|))/(2(2|x + 2|) + 1)) - 4) + 2) + 1 and g(x) = (3(-1|(-4x - 5)| + 3) - 3)/(3(-1|(-4x - 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(-2(2|(-1|x - 3| + 1) + 1| - 2)^2 - 2(2|(-1|x - 3| + 1) + 1| - 2) + 5) + 1)/(-1(-2(2|(-1|x - 3| + 1) + 1| - 2)^2 - 2(2|(-1|x - 3| + 1) + 1| - 2) + 5) + 2) and g(x) = (2(-3|(-1x^2 - 2x + 4)| + 3) - 3)/(3(-3|(-1x^2 - 2x + 4)| + 3) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(4(-2|(-5x - 2) + 2| + 3)^2 + 5(-2|(-5x - 2) + 2| + 3) - 4) + 1)/(2(4(-2|(-5x - 2) + 2| + 3)^2 + 5(-2|(-5x - 2) + 2| + 3) - 4) - 1) = 1cos(2\u03c0((-3(5x^2 + 1x + 2) - 2)/(-1(5x^2 + 1x + 2))) + 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(3sin(1\u03c0(2|((3x - 1)/(-1x - 1)) - 1| - 1) + 3) + 2) + 1 and g(x) = -5(3|((3x - 3)/(-3x - 1)) + 2| - 2)^2 + 1(3|((3x - 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": "12", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(5(2cos(2\u03c0(3|x + 1| - 1) + 1) + 2)^2 - 5(2cos(2\u03c0(3|x + 1| - 1) + 1) + 2) + 5) + 3)/(2(5(2cos(2\u03c0(3|x + 1| - 1) + 1) + 2)^2 - 5(2cos(2\u03c0(3|x + 1| - 1) + 1) + 2) + 5) + 1) and g(x) = -2(2sin(3\u03c0(2|x + 2| - 3) - 1)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(-2cos(2\u03c0((-3(-3|x + 2|))/(-2(-3|x + 2|) - 2)) - 3) - 2)^2 + 3(-2cos(2\u03c0((-3(-3|x + 2|))/(-2(-3|x + 2|) - 2)) - 3) - 2) - 2 and g(x) = -2e^(1(5(5x - 3) - 1)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(-2(3cos(2\u03c0(3|x - 1| - 3) + 2))^2 - 4(3cos(2\u03c0(3|x - 1| - 3) + 2)) + 1)^2 + 4(-2(3cos(2\u03c0(3|x - 1| - 3) + 2))^2 - 4(3cos(2\u03c0(3|x - 1| - 3) + 2)) + 1) + 1 = -1(-2(-5x - 2) - 5)^2 - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-3sin(3\u03c0(4(1x - 1)^2 + 2(1x - 1) - 5) + 1) - 1) - 3| + 2 and g(x) = -1(-4((-3x - 1)/(2x + 2))^2 + 2((-3x - 1)/(2x + 2)) + 1)^2 - 3(-4((-3x - 1)/(2x + 2))^2 + 2((-3x - 1)/(2x + 2)) + 1) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1((-2(-1(-1sin(2\u03c0x + 3) - 1) - 4) + 1)/(1(-1(-1sin(2\u03c0x + 3) - 1) - 4) + 3))^2 - 5((-2(-1(-1sin(2\u03c0x + 3) - 1) - 4) + 1)/(1(-1(-1sin(2\u03c0x + 3) - 1) - 4) + 3)) - 4 and g(x) = (1(-2(-3x^2 - 4x - 2)^2 + 1(-3x^2 - 4x - 2) + 3) + 3)/(-2(-2(-3x^2 - 4x - 2)^2 + 1(-3x^2 - 4x - 2) + 3) - 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": "Determine the number of intersection points between the functions f(x) = 1((-3(-2|((-3x - 2)/(-3x - 3))| - 1) - 2)/(1(-2|((-3x - 2)/(-3x - 3))| - 1)))^2 + 2((-3(-2|((-3x - 2)/(-3x - 3))| - 1) - 2)/(1(-2|((-3x - 2)/(-3x - 3))| - 1))) + 1 and g(x) = 1sin(1\u03c0((2(2|x - 1| - 3) - 2)/(-2(2|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": "29", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(1\u03c0((1(-2|((-1x - 1)/(2x - 3))| + 3) - 3)/(2(-2|((-1x - 1)/(2x - 3))| + 3) - 3)) - 1) + 2 and g(x) = 1sin(1\u03c0(-2(-2x + 4)^2 - 1(-2x + 4) - 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "836", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-2sin(3\u03c0((-2((1x + 2)/(-2x + 3)) + 3)/(2((1x + 2)/(-2x + 3)) + 3)) + 2) + 2)^2 + 4(-2sin(3\u03c0((-2((1x + 2)/(-2x + 3)) + 3)/(2((1x + 2)/(-2x + 3)) + 3)) + 2) + 2) - 3 and g(x) = -3|(3|(-4x - 5) + 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|((3((1(3|x + 3| + 3) - 1)/(-1(3|x + 3| + 3))) - 3)/(3((1(3|x + 3| + 3) - 1)/(-1(3|x + 3| + 3))) + 3)) - 1| + 2 and g(x) = (-1(1|(1cos(3\u03c0x) + 1) + 3| - 1) + 1)/(1(1|(1cos(3\u03c0x) + 1) + 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": "56", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(3((-2(-1x^2 - 3x + 4))/(-3(-1x^2 - 3x + 4))) - 1))/(3(3((-2(-1x^2 - 3x + 4))/(-3(-1x^2 - 3x + 4))) - 1) - 2) and g(x) = (-2(-3cos(2\u03c0((-1x)/(-1x + 2)) - 1) + 3) + 1)/(3(-3cos(2\u03c0((-1x)/(-1x + 2)) - 1) + 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) = (1(-1(2|(4x^2 + 3x + 3) + 1| - 1)) - 1)/(1(-1(2|(4x^2 + 3x + 3) + 1| - 1)) - 3) and g(x) = -3cos(3\u03c0(2|((1x - 2)/(2x + 1)) + 2|) - 1) + 1 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) = 2(-2sin(1\u03c0(-1(1x^2 + 3x - 5) - 4) + 2) - 2) + 5 and g(x) = -3(2(-3|x + 2| + 3) + 5)^2 + 5(2(-3|x + 2| + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(-2|(2(-1cos(1\u03c0x + 3))^2 + 3(-1cos(1\u03c0x + 3)) + 1)| + 1) + 2)/(3(-2|(2(-1cos(1\u03c0x + 3))^2 + 3(-1cos(1\u03c0x + 3)) + 1)| + 1) - 1) = 2cos(3\u03c0((-2((2x)/(-3x + 2)) - 2)/(-3((2x)/(-3x + 2)) + 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": "18", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(-3(3cos(3\u03c0(-2|x + 2| + 1) + 1))^2 + 2(3cos(3\u03c0(-2|x + 2| + 1) + 1)) + 1) + 1 and g(x) = (3(-3sin(1\u03c0(-1x - 3) - 2) - 3) - 2)/(1(-3sin(1\u03c0(-1x - 3) - 2) - 3) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "240", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(-4((3(1|x - 3| - 3) - 3)/(1(1|x - 3| - 3))) - 1) - 3)/(1(-4((3(1|x - 3| - 3) - 3)/(1(1|x - 3| - 3))) - 1)) and g(x) = 3sin(2\u03c0(1((-1x - 2)/(3x - 2))^2 + 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": "Determine the number of intersection points between the functions f(x) = 2|((1(-2sin(3\u03c0(4x^2 + 2x - 3) + 3) - 1) + 3)/(2(-2sin(3\u03c0(4x^2 + 2x - 3) + 3) - 1) - 1)) + 1| + 1 and g(x) = 4(-4(-1|x + 1| + 3) - 2) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "174", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4((1(2sin(3\u03c0(3|x - 3| - 2) + 1) - 2) + 2)/(-1(2sin(3\u03c0(3|x - 3| - 2) + 1) - 2)))^2 + 5((1(2sin(3\u03c0(3|x - 3| - 2) + 1) - 2) + 2)/(-1(2sin(3\u03c0(3|x - 3| - 2) + 1) - 2))) - 2 = -1(1sin(2\u03c0(-5x^2 - 1x - 2) - 1) + 1)^2 + 1(1sin(2\u03c0(-5x^2 - 1x - 2) - 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": "182", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2((-3(-2|(-4x + 1)| + 3) - 2)/(-3(-2|(-4x + 1)| + 3)))^2 - 4((-3(-2|(-4x + 1)| + 3) - 2)/(-3(-2|(-4x + 1)| + 3))) + 2 and g(x) = (1((-2(1|x|) - 3)/(-1(1|x|) - 1)) - 1)/(-3((-2(1|x|) - 3)/(-1(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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(2|(1(1x + 4)^2 - 3(1x + 4) - 4) - 3| + 2) + 3)/(-1(2|(1(1x + 4)^2 - 3(1x + 4) - 4) - 3| + 2)) and g(x) = -2|(-4(-5x^2 - 4x + 1)^2 + 5(-5x^2 - 4x + 1) + 2) + 1| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-5(5(4x^2 + 3x - 4) - 2)^2 - 3(5(4x^2 + 3x - 4) - 2) + 1))/(-1(-5(5(4x^2 + 3x - 4) - 2)^2 - 3(5(4x^2 + 3x - 4) - 2) + 1) + 2) and g(x) = (2((-1(-3|x - 3| - 3) + 1)/(1(-3|x - 3| - 3) - 2)) + 2)/(-3((-1(-3|x - 3| - 3) + 1)/(1(-3|x - 3| - 3) - 2)) + 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 5(1e^(1(-4(-2|x + 3|)^2 - 4(-2|x + 3|) - 1)) + 2)^2 - 4(1e^(1(-4(-2|x + 3|)^2 - 4(-2|x + 3|) - 1)) + 2) - 4 = (-2(-1(-1sin(2\u03c0x - 2) + 2) + 2) + 2)/(3(-1(-1sin(2\u03c0x - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(3|((-1(1e^(1x) + 3) + 1)/(1(1e^(1x) + 3) - 3))| + 3) + 5 and g(x) = 5(2(1x^2 - 3x + 1)^2 + 1(1x^2 - 3x + 1))^2 - 1(2(1x^2 - 3x + 1)^2 + 1(1x^2 - 3x + 1)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1((-2((-1(-1x - 3) + 2)/(1(-1x - 3) - 2)) - 3)/(2((-1(-1x - 3) + 2)/(1(-1x - 3) - 2)) + 3)) + 3)/(3((-2((-1(-1x - 3) + 2)/(1(-1x - 3) - 2)) - 3)/(2((-1(-1x - 3) + 2)/(1(-1x - 3) - 2)) + 3)) - 2) and g(x) = -2(-3(4x^2 - 2x - 2)^2 + 2(4x^2 - 2x - 2) + 1)^2 - 2(-3(4x^2 - 2x - 2)^2 + 2(4x^2 - 2x - 2) + 1) - 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) = -2(-3(-3sin(2\u03c0(-1|x| - 1) - 2) + 2)^2 + 5(-3sin(2\u03c0(-1|x| - 1) - 2) + 2) + 2)^2 + 5(-3(-3sin(2\u03c0(-1|x| - 1) - 2) + 2)^2 + 5(-3sin(2\u03c0(-1|x| - 1) - 2) + 2) + 2) + 4 and g(x) = 3(-1(4x - 2) - 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1((-3(-2sin(1\u03c0(-2x + 3) + 1) + 1) - 2)/(-1(-2sin(1\u03c0(-2x + 3) + 1) + 1) + 3))^2 + 1((-3(-2sin(1\u03c0(-2x + 3) + 1) + 1) - 2)/(-1(-2sin(1\u03c0(-2x + 3) + 1) + 1) + 3)) + 4 and g(x) = 1|(-3|((-3x - 1)/(1x + 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) = -2(2|((-1(-2x - 2) - 2)/(3(-2x - 2) + 1)) + 3| - 3)^2 + 3(2|((-1(-2x - 2) - 2)/(3(-2x - 2) + 1)) + 3| - 3) and g(x) = (-2((-1(-5x^2 - 3x + 4) + 3)/(3(-5x^2 - 3x + 4))) + 3)/(1((-1(-5x^2 - 3x + 4) + 3)/(3(-5x^2 - 3x + 4))) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(3\u03c0(-3|(3((-3x + 3)/(2x - 3)) - 4) - 2| + 2) + 2) + 2 and g(x) = (2(-1e^(1(-2x^2 - 5x - 3)) + 1) - 2)/(-3(-1e^(1(-2x^2 - 5x - 3)) + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "68", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3cos(1\u03c0(-1(3(1|x - 3| - 3) - 5)^2 - 1(3(1|x - 3| - 3) - 5) + 1) + 1) + 2 = -1(3(-3e^(-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": "24", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(3|(5((-3x + 3)/(-2x + 1))) - 1| - 3)^2 - 3(3|(5((-3x + 3)/(-2x + 1))) - 1| - 3) + 2 and g(x) = 1|((-2(1x^2 + 4x + 3) + 2)/(1(1x^2 + 4x + 3) + 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 (-2(3(-2|(3|x|) - 3| + 2) + 1) + 3)/(-2(3(-2|(3|x|) - 3| + 2) + 1) - 1) = 2(1(-2cos(1\u03c0x) - 1)^2 - 3(-2cos(1\u03c0x) - 1) + 2) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2sin(1\u03c0(-1(-4(-3x - 2) - 3) - 1) - 1) - 2 = (-3(1(-5x^2 - 5x + 3)^2 + 4(-5x^2 - 5x + 3) + 1))/(-2(1(-5x^2 - 5x + 3)^2 + 4(-5x^2 - 5x + 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": "How many points of intersection exist between the functions f(x) = -1|(2cos(1\u03c0(-2(-2x)^2 - 4(-2x) - 2) + 1) - 1)| - 3 and g(x) = -4((-2(-1x^2 + 5x - 3) - 2)/(-2(-1x^2 + 5x - 3) + 3))^2 - 5((-2(-1x^2 + 5x - 3) - 2)/(-2(-1x^2 + 5x - 3) + 3)) + 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "980", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2((3(3e^(-2(-4x^2 - 1x + 5)) - 1) - 3)/(3(3e^(-2(-4x^2 - 1x + 5)) - 1) + 1))^2 + 3((3(3e^(-2(-4x^2 - 1x + 5)) - 1) - 3)/(3(3e^(-2(-4x^2 - 1x + 5)) - 1) + 1)) - 2 and g(x) = 3(3|((2x + 3)/(2x))| + 1)^2 - 3(3|((2x + 3)/(2x))| + 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(-4((1(-1|x + 1| + 2) - 1)/(-2(-1|x + 1| + 2) - 2))^2 + 4((1(-1|x + 1| + 2) - 1)/(-2(-1|x + 1| + 2) - 2)) + 4)| - 3 and g(x) = -3(-1e^(2(-2|x - 3| - 2))) + 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 solutions does the equation 3|(3(-1e^(-2(2x^2 - 5x - 1)) - 1) - 2)| + 1 = 2|(2(2|x + 1| - 3)^2 - 5(2|x + 1| - 3) - 2)| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(1(-2cos(1\u03c0(4x - 2) - 1) + 2)^2 + 4(-2cos(1\u03c0(4x - 2) - 1) + 2) - 3) + 3)/(-3(1(-2cos(1\u03c0(4x - 2) - 1) + 2)^2 + 4(-2cos(1\u03c0(4x - 2) - 1) + 2) - 3)) = -2(-1(-3|x + 1| + 1)^2 - 4(-3|x + 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": "18", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(1(2cos(2\u03c0(-4x + 3) + 3) + 3)^2 - 4) - 1 and g(x) = -2sin(2\u03c0(5(-2x^2 + 2x + 3) - 4) + 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "168", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(2(1(-1x^2 - 2x + 2)^2 + 3(-1x^2 - 2x + 2) - 5) + 3) - 1)/(-2(2(1(-1x^2 - 2x + 2)^2 + 3(-1x^2 - 2x + 2) - 5) + 3) + 1) and g(x) = 1(-4(-1sin(1\u03c0x + 2) + 1) - 2) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3((-1((2(-2e^(-1x) - 1) - 1)/(2(-2e^(-1x) - 1))) + 2)/(3((2(-2e^(-1x) - 1) - 1)/(2(-2e^(-1x) - 1))))))/(2((-1((2(-2e^(-1x) - 1) - 1)/(2(-2e^(-1x) - 1))) + 2)/(3((2(-2e^(-1x) - 1) - 1)/(2(-2e^(-1x) - 1))))) - 3) and g(x) = -3sin(2\u03c0(1|(-2x + 4)| - 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": "80", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|((-3((-2(-4x) - 2)/(3(-4x) + 2)) - 2)/(1((-2(-4x) - 2)/(3(-4x) + 2)) + 1)) - 1| = 1(3|(-5x + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(3cos(2\u03c0(1(5x^2 + 4x) + 5) - 3) - 2)^2 + 4(3cos(2\u03c0(1(5x^2 + 4x) + 5) - 3) - 2) - 2 and g(x) = -1(4((3x)/(2x - 1))^2 - 5((3x)/(2x - 1)) + 1)^2 - 4(4((3x)/(2x - 1))^2 - 5((3x)/(2x - 1)) + 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(3|((-2(1x) + 3)/(-3(1x) - 1)) + 1| - 3) - 2 = -1|((-3(5x - 5) - 1)/(-1(5x - 5) - 1)) - 1| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(3(1(-4x^2 + 2x - 4)^2 + 5(-4x^2 + 2x - 4) + 2)^2 - 3) + 2)/(2(3(1(-4x^2 + 2x - 4)^2 + 5(-4x^2 + 2x - 4) + 2)^2 - 3) + 1) and g(x) = (3(1(-2|x| + 1) - 5) + 1)/(2(1(-2|x| + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(2(-1|((-1x - 1)/(1x)) + 2|)^2 - 3(-1|((-1x - 1)/(1x)) + 2|) - 5) - 2)/(2(2(-1|((-1x - 1)/(1x)) + 2|)^2 - 3(-1|((-1x - 1)/(1x)) + 2|) - 5) + 2) and g(x) = 3(5(-3sin(1\u03c0x - 3) - 1) + 1)^2 + 2(5(-3sin(1\u03c0x - 3) - 1) + 1) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2|(3cos(2\u03c0(3(-2|x + 3| + 2) - 4)) + 3) - 1| - 2 = 5(3sin(3\u03c0(1|x - 3| - 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": "58", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2((2(3sin(1\u03c0(3x + 5) + 3) - 2) + 3)/(1(3sin(1\u03c0(3x + 5) + 3) - 2) + 2)) - 4 and g(x) = 1(-4(1x + 3) + 2) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(-1cos(2\u03c0(-4(3x - 3)^2 - 4(3x - 3) + 4) + 2) + 2)^2 + 1(-1cos(2\u03c0(-4(3x - 3)^2 - 4(3x - 3) + 4) + 2) + 2) - 1 and g(x) = 1|(5(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": "480", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(-1|(3|(-3e^(1x) + 3) + 3| + 3) - 2| - 1) + 1| and g(x) = -3|(5((3x - 1)/(-2x + 3))^2 - 5((3x - 1)/(-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": "5", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-1(-3(-3sin(2\u03c0x - 2) - 1) - 4)^2 + 5(-3(-3sin(2\u03c0x - 2) - 1) - 4) - 4) - 5 and g(x) = -1|(-2sin(3\u03c0(1|x| - 3) - 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": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(-2cos(2\u03c0(-3(3|x + 2| - 3)) - 2) - 2)^2 + 1(-2cos(2\u03c0(-3(3|x + 2| - 3)) - 2) - 2) + 2 and g(x) = 5((1(-3cos(2\u03c0x + 3) - 2) + 2)/(-2(-3cos(2\u03c0x + 3) - 2) + 3))^2 + 5((1(-3cos(2\u03c0x + 3) - 2) + 2)/(-2(-3cos(2\u03c0x + 3) - 2) + 3)) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "344", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(1sin(2\u03c0(1((3x)/(3x - 2)) - 3) - 3)) + 3)/(2(1sin(2\u03c0(1((3x)/(3x - 2)) - 3) - 3)) + 1) = -2e^(2((2(3|x| + 3) + 3)/(2(3|x| + 3) + 2))) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3sin(2\u03c0((2(1(3x^2 + 4x - 2)^2 + 5(3x^2 + 4x - 2) - 2) + 2)/(-2(1(3x^2 + 4x - 2)^2 + 5(3x^2 + 4x - 2) - 2) + 3)) + 1) + 2 and g(x) = -1|((-1(2x - 5) - 2)/(2(2x - 5) - 1))| + 3 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) = -1|(-3(-2(-1x^2 - 4x + 2) + 2)^2 - 3(-2(-1x^2 - 4x + 2) + 2) - 2) + 2| and g(x) = -3(-2(3|x + 3| + 2) + 5)^2 - 2(-2(3|x + 3| + 2) + 5) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|((-1(-4(-2sin(1\u03c0x - 1) - 1) + 4) - 1)/(-1(-4(-2sin(1\u03c0x - 1) - 1) + 4) + 2))| + 1 and g(x) = 3(-4(3e^(-1x) - 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": "19", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2((2(1(-1x + 5)^2 - 1(-1x + 5) - 5))/(-1(1(-1x + 5)^2 - 1(-1x + 5) - 5) + 2)) + 2)/(-2((2(1(-1x + 5)^2 - 1(-1x + 5) - 5))/(-1(1(-1x + 5)^2 - 1(-1x + 5) - 5) + 2)) + 3) = -5(2sin(3\u03c0(2x - 3) - 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": "118", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(1sin(2\u03c0(1(-1x^2 + 1x + 5)^2 + 3(-1x^2 + 1x + 5) + 4) + 2) - 1) + 3)/(3(1sin(2\u03c0(1(-1x^2 + 1x + 5)^2 + 3(-1x^2 + 1x + 5) + 4) + 2) - 1) + 2) and g(x) = 4(-3sin(2\u03c0(-1x^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": "275", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(2(-2cos(2\u03c0(2x^2 - 5x - 5) - 2) + 3) - 1) - 3 and g(x) = -2((3(-1|x + 3| + 3) + 3)/(2(-1|x + 3| + 3) - 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(2|(3sin(2\u03c0(2|x - 3|) + 2) - 2) - 3| - 1) - 2)/(1(2|(3sin(2\u03c0(2|x - 3|) + 2) - 2) - 3| - 1)) = 1|(-1cos(3\u03c0((-3x + 3)/(-2x + 3)) + 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": "64", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(-3sin(3\u03c0(3|((1x - 1)/(1x - 2)) + 1| + 1) - 2) + 2) + 2)/(-1(-3sin(3\u03c0(3|((1x - 1)/(1x - 2)) + 1| + 1) - 2) + 2) + 2) and g(x) = -2|(-1e^(-2((-1x - 3)/(3x))) + 3) - 1| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(2|(2cos(3\u03c0(-1|x + 2| - 1))) + 1| - 1) - 1)/(1(2|(2cos(3\u03c0(-1|x + 2| - 1))) + 1| - 1)) and g(x) = -3|(1|(-5x - 1) - 3| - 3)| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(2(1(-2cos(3\u03c0x + 2)) - 2)^2 + 2(1(-2cos(3\u03c0x + 2)) - 2) + 4) + 3 = -3|(4(5x^2 + 1x - 5)^2 - 4(5x^2 + 1x - 5) - 3) + 3| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(1|(-1|((1x)/(2x - 1)) - 3| - 1) - 2| + 2)^2 + 2(1|(-1|((1x)/(2x - 1)) - 3| - 1) - 2| + 2) + 3 and g(x) = (-2(2sin(3\u03c0((2x - 3)/(1x + 2)) + 3) - 2) + 1)/(-2(2sin(3\u03c0((2x - 3)/(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": "11", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(1cos(1\u03c0(-3((3x + 2)/(-1x + 2)) + 2) + 1))^2 - 3(1cos(1\u03c0(-3((3x + 2)/(-1x + 2)) + 2) + 1)) - 4 = 2(1(-5x - 3)^2 - 5(-5x - 3) + 2)^2 - 3(1(-5x - 3)^2 - 5(-5x - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(5(-3cos(2\u03c0(-1x + 2) - 3)) + 4) + 1 and g(x) = -1|(4(-1|x| + 3) - 3) - 3| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(3|(-1|(-3cos(1\u03c0x - 1)) + 3| - 2) + 2| - 3) - 1| + 3 = -1((-1(3|x - 1| - 3) - 2)/(-2(3|x - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(2|(2|(1e^(2x) + 3) - 3| - 3)| + 3) + 1 and g(x) = -3|(-2(-2sin(2\u03c0x + 3) - 2)^2 + 2(-2sin(2\u03c0x + 3) - 2) - 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": "21", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(-2(3(-1x^2 - 3x + 2))^2 + 5(3(-1x^2 - 3x + 2)) - 4) + 3| + 2 and g(x) = (-2(5(-2|x - 2| - 1) + 4) - 3)/(2(5(-2|x - 2| - 1) + 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 3|(2(3e^(-2(-4x^2 + 4)) - 3)^2 - 4(3e^(-2(-4x^2 + 4)) - 3) - 2) + 2| + 2 = 3|((2(1cos(2\u03c0x + 3) - 1) + 3)/(-2(1cos(2\u03c0x + 3) - 1) - 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": "12", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(5((2(-2|x + 3| + 1) + 2)/(-1(-2|x + 3| + 1) + 3)) - 3))/(-1(5((2(-2|x + 3| + 1) + 2)/(-1(-2|x + 3| + 1) + 3)) - 3) - 1) and g(x) = (1(-5(2x)^2 - 1(2x) + 4) + 1)/(1(-5(2x)^2 - 1(2x) + 4)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(1(-3((-1x + 2)/(-1x - 2)) - 5)^2 - 4(-3((-1x + 2)/(-1x - 2)) - 5) + 5) = 3(3cos(3\u03c0((1x + 3)/(1x)) - 2) + 2)^2 + 1(3cos(3\u03c0((1x + 3)/(1x)) - 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": "78", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(1|(1(-1|x - 1| + 1) + 3) + 1| - 2) - 1)/(-1(1|(1(-1|x - 1| + 1) + 3) + 1| - 2) - 2) and g(x) = (-1(4((-3x - 1)/(-2x))^2 + 2((-3x - 1)/(-2x)) - 1))/(2(4((-3x - 1)/(-2x))^2 + 2((-3x - 1)/(-2x)) - 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-2(-4(2|x - 1|) + 5)) + 1 and g(x) = 1|((-1(3x + 4) + 2)/(2(3x + 4) + 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) = 1(-4(-1(1x^2 + 3x - 4) + 2)^2 - 3(-1(1x^2 + 3x - 4) + 2) + 1)^2 + 1(-4(-1(1x^2 + 3x - 4) + 2)^2 - 3(-1(1x^2 + 3x - 4) + 2) + 1) - 4 and g(x) = (1((1((1x - 2)/(3x + 3)) + 2)/(3((1x - 2)/(3x + 3)) - 2)) - 1)/(-2((1((1x - 2)/(3x + 3)) + 2)/(3((1x - 2)/(3x + 3)) - 2))) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(-3sin(2\u03c0((-2((-2x - 1)/(3x + 3)) + 2)/(-2((-2x - 1)/(3x + 3)) + 1)) - 2)) - 3| + 2 and g(x) = 1|((-2(-1x^2 + 2x + 2) - 2)/(2(-1x^2 + 2x + 2) + 3)) + 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(2sin(2\u03c0(4(-3x^2 + 2x)^2 - 2(-3x^2 + 2x) + 4) - 1)) + 3| - 1 and g(x) = 5(3sin(3\u03c0(-1|x + 2| - 3)) + 3)^2 + 5(3sin(3\u03c0(-1|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": "94", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(2|(-1sin(1\u03c0((1x - 2)/(1x)) + 3) - 2) - 3| + 1) + 3)/(-2(2|(-1sin(1\u03c0((1x - 2)/(1x)) + 3) - 2) - 3| + 1) - 1) and g(x) = -1|(2sin(3\u03c0((2x - 3)/(-1x - 2)) + 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": "223", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(1e^(2(-3(-2x + 2) - 3)) - 2) - 5 and g(x) = 5(1cos(1\u03c0(-1x^2 + 5x - 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": "155", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(-2|(-1(2cos(2\u03c0x) + 1) + 3)| + 2)^2 + 5(-2|(-1(2cos(2\u03c0x) + 1) + 3)| + 2) + 4 = -5(2|(1|x + 2| + 2) + 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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(-2(2(-4x - 3)^2 - 5(-4x - 3) - 3) - 4) + 1)/(2(-2(2(-4x - 3)^2 - 5(-4x - 3) - 3) - 4) - 1) = 3sin(2\u03c0(-4((1x - 3)/(3x - 2)) + 5)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "95", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|((-3(2|(-4x^2 + 5x + 4) - 3| + 2))/(1(2|(-4x^2 + 5x + 4) - 3| + 2) + 3)) - 3| + 3 = -3|(3cos(3\u03c0(-1|x + 2| + 3) + 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": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(-4(-4((2x + 3)/(-3x - 2)) - 2) - 4) + 1)/(-1(-4(-4((2x + 3)/(-3x - 2)) - 2) - 4) + 2) and g(x) = -4(2(-3x + 1)^2 + 5(-3x + 1) - 5)^2 - 4(2(-3x + 1)^2 + 5(-3x + 1) - 5) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5((1(-1(-1x^2 + 5x - 5)^2 - 4(-1x^2 + 5x - 5) + 4) - 1)/(3(-1(-1x^2 + 5x - 5)^2 - 4(-1x^2 + 5x - 5) + 4) - 2))^2 + 5((1(-1(-1x^2 + 5x - 5)^2 - 4(-1x^2 + 5x - 5) + 4) - 1)/(3(-1(-1x^2 + 5x - 5)^2 - 4(-1x^2 + 5x - 5) + 4) - 2)) and g(x) = -1(-1(-3cos(3\u03c0x - 1) + 1) + 3)^2 + 1(-1(-3cos(3\u03c0x - 1) + 1) + 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "123", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(5(-1(-2|x - 3| + 3)^2 - 5(-2|x - 3| + 3) + 2)^2 + 4(-1(-2|x - 3| + 3)^2 - 5(-2|x - 3| + 3) + 2) - 5)^2 + 5(5(-1(-2|x - 3| + 3)^2 - 5(-2|x - 3| + 3) + 2)^2 + 4(-1(-2|x - 3| + 3)^2 - 5(-2|x - 3| + 3) + 2) - 5) - 1 and g(x) = 1(-3sin(2\u03c0(4x^2 - 5x - 4) + 3) - 1)^2 + 2(-3sin(2\u03c0(4x^2 - 5x - 4) + 3) - 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(1((1((-2x)/(-2x + 1)) - 2)/(-2((-2x)/(-2x + 1)) + 3))^2 + 5((1((-2x)/(-2x + 1)) - 2)/(-2((-2x)/(-2x + 1)) + 3)) + 5) - 3 = (2(2|(1|x - 1|)| - 2) + 1)/(3(2|(1|x - 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": "Determine the number of intersection points between the functions f(x) = 1((-3(-3cos(2\u03c0(-2x - 1) + 2) - 2) + 3)/(-3(-3cos(2\u03c0(-2x - 1) + 2) - 2))) - 2 and g(x) = 2cos(2\u03c0((-3(-1x - 3) - 1)/(-1(-1x - 3) - 1)) - 2) + 3 where -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 (-2(3(-3(2x^2 + 5x - 1)^2 - 4(2x^2 + 5x - 1) + 5)^2 + 3(-3(2x^2 + 5x - 1)^2 - 4(2x^2 + 5x - 1) + 5) - 1) + 3)/(2(3(-3(2x^2 + 5x - 1)^2 - 4(2x^2 + 5x - 1) + 5)^2 + 3(-3(2x^2 + 5x - 1)^2 - 4(2x^2 + 5x - 1) + 5) - 1) + 3) = -4((-3(3x^2 - 4x) - 2)/(1(3x^2 - 4x) - 3)) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(-1((-1(5x - 4) - 3)/(-1(5x - 4) + 3))^2 - 3((-1(5x - 4) - 3)/(-1(5x - 4) + 3)) - 1) + 2)/(-1(-1((-1(5x - 4) - 3)/(-1(5x - 4) + 3))^2 - 3((-1(5x - 4) - 3)/(-1(5x - 4) + 3)) - 1) + 3) = -2|((2(3x^2 + 1x - 4) - 1)/(1(3x^2 + 1x - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(3(1cos(2\u03c0((-3x - 1)/(1x - 2)) + 2) + 2)^2 - 4(1cos(2\u03c0((-3x - 1)/(1x - 2)) + 2) + 2)) - 1 and g(x) = 1e^(1(-5(-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": "35", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(1(4(-3sin(3\u03c0x - 1) + 3)^2 + 5(-3sin(3\u03c0x - 1) + 3) - 2)) = 1(-1cos(1\u03c0((2x + 3)/(2x + 1)) + 3))^2 + 5(-1cos(1\u03c0((2x + 3)/(2x + 1)) + 3)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "66", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3(3(-2((2x)/(-1x + 2))^2 + 5((2x)/(-1x + 2)) - 2)^2 - 5(-2((2x)/(-1x + 2))^2 + 5((2x)/(-1x + 2)) - 2) - 1) + 2)/(1(3(-2((2x)/(-1x + 2))^2 + 5((2x)/(-1x + 2)) - 2)^2 - 5(-2((2x)/(-1x + 2))^2 + 5((2x)/(-1x + 2)) - 2) - 1) - 3) = -2|(-2(-5x - 2) + 5) + 3| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|(-3((3((-2x)/(2x + 3)))/(-3((-2x)/(2x + 3)) + 2)) + 5) - 2| + 1 and g(x) = -3sin(1\u03c0((-3(3x + 3) + 2)/(-2(3x + 3) + 3)) + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4((2(-3sin(2\u03c0(1x^2 - 3x + 1)) - 1) - 3)/(-1(-3sin(2\u03c0(1x^2 - 3x + 1)) - 1) + 2)) - 1 and g(x) = (-2(1(1x^2 - 3x - 4)^2 - 1(1x^2 - 3x - 4) + 3) + 2)/(-1(1(1x^2 - 3x - 4)^2 - 1(1x^2 - 3x - 4) + 3) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "410", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(-3(-2(-3sin(3\u03c0x + 1) + 2)^2 + 1(-3sin(3\u03c0x + 1) + 2) + 2)^2 + 5(-2(-3sin(3\u03c0x + 1) + 2)^2 + 1(-3sin(3\u03c0x + 1) + 2) + 2))^2 - 3(-3(-2(-3sin(3\u03c0x + 1) + 2)^2 + 1(-3sin(3\u03c0x + 1) + 2) + 2)^2 + 5(-2(-3sin(3\u03c0x + 1) + 2)^2 + 1(-3sin(3\u03c0x + 1) + 2) + 2)) + 3 and g(x) = -1|((2(2sin(1\u03c0x)) - 3)/(3(2sin(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": "358", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3((3(-1((3x - 3)/(-3x + 4)) - 5) + 2)/(-3(-1((3x - 3)/(-3x + 4)) - 5) - 2)) + 2)/(-2((3(-1((3x - 3)/(-3x + 4)) - 5) + 2)/(-3(-1((3x - 3)/(-3x + 4)) - 5) - 2))) = 3sin(3\u03c0(-3(-1|x - 3| + 1)^2 + 2(-1|x - 3| + 1) - 1)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1226", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4((2((2(-1cos(1\u03c0x - 2) - 2))/(1(-1cos(1\u03c0x - 2) - 2) - 3)) - 2)/(2((2(-1cos(1\u03c0x - 2) - 2))/(1(-1cos(1\u03c0x - 2) - 2) - 3)) + 2)) - 1 and g(x) = (2(-2(-2sin(2\u03c0x + 2) + 2) + 5) - 3)/(2(-2(-2sin(2\u03c0x + 2) + 2) + 5) - 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 (2(-3e^(1((-3(1|x - 1| - 2) + 2)/(-3(1|x - 1| - 2) - 1)))) - 1)/(1(-3e^(1((-3(1|x - 1| - 2) + 2)/(-3(1|x - 1| - 2) - 1)))) - 3) = 1(1cos(1\u03c0((1x)/(1x - 2)) - 2) + 2)^2 - 4(1cos(1\u03c0((1x)/(1x - 2)) - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(1(1|(-2sin(2\u03c0x + 3) - 1) + 2|)^2 + 2(1|(-2sin(2\u03c0x + 3) - 1) + 2|) - 5) + 1)/(2(1(1|(-2sin(2\u03c0x + 3) - 1) + 2|)^2 + 2(1|(-2sin(2\u03c0x + 3) - 1) + 2|) - 5) + 2) = -5(2(1cos(1\u03c0x + 1)) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(1cos(2\u03c0(-5((-1x)/(3x + 1))^2 + 5((-1x)/(3x + 1)) - 5) + 1) - 3) - 5 and g(x) = 4(-2cos(1\u03c0((3x - 1)/(1x + 3)) - 3) - 3)^2 + 5(-2cos(1\u03c0((3x - 1)/(1x + 3)) - 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": "102", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(2e^(2((1(-4x^2 - 3x))/(1(-4x^2 - 3x) + 1))) - 3) + 2)/(-2(2e^(2((1(-4x^2 - 3x))/(1(-4x^2 - 3x) + 1))) - 3) + 3) and g(x) = 5(3|(2sin(2\u03c0x))| - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(2sin(2\u03c0(1(2x^2 + 1x - 1)^2 + 2(2x^2 + 1x - 1) - 5) - 3) + 2) + 1)/(1(2sin(2\u03c0(1(2x^2 + 1x - 1)^2 + 2(2x^2 + 1x - 1) - 5) - 3) + 2) - 2) and g(x) = -1|(-3sin(1\u03c0((-1x)/(-1x + 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": "187", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|((-3(-3(3|x - 3| + 2)^2 + 1(3|x - 3| + 2) + 3) + 1)/(3(-3(3|x - 3| + 2)^2 + 1(3|x - 3| + 2) + 3) - 2)) - 3| + 1 and g(x) = (2((-2(-3sin(3\u03c0x - 2) + 2) - 2)/(1(-3sin(3\u03c0x - 2) + 2) + 3)) - 3)/(3((-2(-3sin(3\u03c0x - 2) + 2) - 2)/(1(-3sin(3\u03c0x - 2) + 2) + 3)) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3((-1(1(3cos(3\u03c0x - 3) + 1)) - 1)/(-3(1(3cos(3\u03c0x - 3) + 1)) + 4)) + 4 and g(x) = -1|(3cos(1\u03c0(-4x - 1) - 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": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3((-2(-4(1x - 2) + 1) - 2)/(-3(-4(1x - 2) + 1) + 4))^2 - 1((-2(-4(1x - 2) + 1) - 2)/(-3(-4(1x - 2) + 1) + 4)) + 1 and g(x) = 2|(3sin(1\u03c0((2x)/(-2x + 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": "24", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1((-2(-5(-2|x - 3|) - 1) + 2)/(-3(-5(-2|x - 3|) - 1)))^2 + 4((-2(-5(-2|x - 3|) - 1) + 2)/(-3(-5(-2|x - 3|) - 1))) + 5 and g(x) = -1|(-3(3|x - 2|) - 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) = -1(-4(3|((-1x)/(3x - 2)) - 3| - 2) + 1) - 3 and g(x) = -1(-1|(3x^2 + 4x - 2) + 1| + 2) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(1(-5(1x + 3)^2 - 4(1x + 3) - 3) + 4)^2 - 3(1(-5(1x + 3)^2 - 4(1x + 3) - 3) + 4) and g(x) = -3cos(1\u03c0(-1(-2x - 3) + 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": "How many solutions does the equation -4(1(-2|((2x + 1)/(-3x - 2)) + 3| + 1)^2 + 5(-2|((2x + 1)/(-3x - 2)) + 3| + 1) - 1) - 1 = -1|(2|(3|x + 2| + 1) + 3| + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(1|(2cos(2\u03c0(-5x - 2) - 1)) - 2| + 1) - 1)/(1(1|(2cos(2\u03c0(-5x - 2) - 1)) - 2| + 1)) and g(x) = 2|(-1(3|x + 2|)^2 - 1(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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|(-3|(-3sin(1\u03c0(-1x^2 + 2x - 1) - 3) + 1) - 1| - 3)| + 1 = -2(1e^(-1(-1x^2 + 3x + 5)) - 1)^2 + 4(1e^(-1(-1x^2 + 3x + 5)) - 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1cos(3\u03c0(1(2|(4x) + 1| - 2)^2 + 1(2|(4x) + 1| - 2) + 3) - 3) - 2 and g(x) = 3cos(3\u03c0(1|(2x^2 + 3x - 4) + 2| - 1) + 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "810", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(2\u03c0(-2|(2(-1x^2 + 3x)^2 + 3(-1x^2 + 3x) + 1) + 2|) + 2) - 2 and g(x) = 1e^(-1(4(-3x^2 + 2x + 3)^2 + 5(-3x^2 + 2x + 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": "1106", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(2|((2(1|x + 1| - 2))/(-3(1|x + 1| - 2) + 4)) + 3|) + 2| + 2 and g(x) = -3cos(2\u03c0(-1|(1x^2 - 5x - 5) + 3| - 2) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(2(-4(-5x + 2) - 3) + 4) + 2)/(2(2(-4(-5x + 2) - 3) + 4)) and g(x) = -1|(3cos(2\u03c0(-1x + 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": "79", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(1((-2(4x^2 - 3x + 1) - 1)/(-2(4x^2 - 3x + 1) - 3)) + 5) - 1)/(2(1((-2(4x^2 - 3x + 1) - 1)/(-2(4x^2 - 3x + 1) - 3)) + 5) - 1) = -3cos(3\u03c0(2(3|x - 3| + 1)) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "360", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(3(-1(-1|x + 1| + 3))^2 + 5(-1(-1|x + 1| + 3)) - 5) - 4 and g(x) = 1|((-1(-4x - 2))/(3(-4x - 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 solutions does the equation -3|((3(-3(-1x - 3)^2 - 3(-1x - 3) + 5) - 1)/(3(-3(-1x - 3)^2 - 3(-1x - 3) + 5))) - 2| + 2 = 1sin(3\u03c0((1((1x + 2)/(-3x - 2)) + 2)/(-2((1x + 2)/(-3x - 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": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(2\u03c0((2(1(-5x)^2 - 2(-5x)) + 1)/(2(1(-5x)^2 - 2(-5x)) + 3)) + 3) + 3 and g(x) = -2e^(-1(2|(-2x^2 - 4x - 4) - 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(-1((-3(3cos(2\u03c0x - 2) + 1) - 2)/(2(3cos(2\u03c0x - 2) + 1) + 2)) + 1) - 5 and g(x) = -2|((1((1x + 2)/(2x - 1)) - 3)/(3((1x + 2)/(2x - 1)) - 2)) - 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3((1(-1|((1x - 1)/(-3x + 1)) - 2| + 2) - 1)/(-2(-1|((1x - 1)/(-3x + 1)) - 2| + 2) + 3)) - 3 = 1(2sin(2\u03c0(-1|x - 2| + 1) - 3) - 2)^2 - 4(2sin(2\u03c0(-1|x - 2| + 1) - 3) - 2) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(1|(5(-3sin(2\u03c0x + 2) + 2) + 2) + 1|)^2 + 5(1|(5(-3sin(2\u03c0x + 2) + 2) + 2) + 1|) + 2 = 1|((-3(-2e^(1x) + 2) + 3)/(-2(-2e^(1x) + 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": "79", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(-1(-2(3|x - 2| - 3) - 3) + 1) and g(x) = -5(-2(2cos(3\u03c0x)))^2 + 2(-2(2cos(3\u03c0x))) + 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(-3|(1e^(-2(2|x + 3| - 1)) - 3) - 1| + 2) + 2)/(3(-3|(1e^(-2(2|x + 3| - 1)) - 3) - 1| + 2)) and g(x) = (-3(2(3|x + 1| - 1) + 1) + 1)/(-2(2(3|x + 1| - 1) + 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(2(-1((1x - 2)/(-2x - 2)) + 2) - 1) + 1 and g(x) = -4(-1sin(3\u03c0((-2x + 3)/(3x + 2)) - 3) + 1)^2 - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(-1|(1|((-2x - 3)/(3x + 2)) - 1| - 1) + 1| + 3) - 1| and g(x) = -1cos(1\u03c0(2((-2x - 1)/(1x)) + 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": "Determine the number of intersection points between the functions f(x) = 3sin(3\u03c0(4(-1(-2x - 2))^2 + 2(-1(-2x - 2))) + 2) + 1 and g(x) = -5(-3e^(2((1x - 2)/(2x + 2))))^2 + 3(-3e^(2((1x - 2)/(2x + 2)))) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "73", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5((1((3(2x^2 - 2x + 2))/(-2(2x^2 - 2x + 2) - 3)))/(-1((3(2x^2 - 2x + 2))/(-2(2x^2 - 2x + 2) - 3)) + 2)) + 3 and g(x) = 3(4(-3cos(3\u03c0x) + 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": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-1((3(2|x - 1| - 3) - 3)/(2(2|x - 1| - 3) - 3))) + 1)/(2(-1((3(2|x - 1| - 3) - 3)/(2(2|x - 1| - 3) - 3))) + 1) and g(x) = 1(-5(-1cos(2\u03c0x - 3) - 1)^2 + 5(-1cos(2\u03c0x - 3) - 1) + 4) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(-1((-2(-3x^2 + 2) - 3)/(-2(-3x^2 + 2) + 3)) + 1)^2 - 3(-1((-2(-3x^2 + 2) - 3)/(-2(-3x^2 + 2) + 3)) + 1) + 2 = 3|(3|(1x - 3) - 3| - 2) - 3| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2((2(-3|(-1x + 3) - 2| + 1) + 1)/(1(-3|(-1x + 3) - 2| + 1) + 3)))/(-2((2(-3|(-1x + 3) - 2| + 1) + 1)/(1(-3|(-1x + 3) - 2| + 1) + 3)) + 3) = -4(-3(-3sin(2\u03c0x - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(-2cos(3\u03c0(2|(-5x^2 + 5x - 3) - 1| - 2) + 3) - 2) + 1 = (2(-1|(-2|x + 2| - 2)|))/(1(-1|(-2|x + 2| - 2)|)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "929", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(3e^(-1(5(2|x|)^2 + 3(2|x|) - 2)) - 3)^2 - 4(3e^(-1(5(2|x|)^2 + 3(2|x|) - 2)) - 3) + 1 and g(x) = -2(1e^(-2(-4x^2 + 2x + 2)) - 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1(1e^(-2(5(-4x^2 + 5x) - 4)) - 3)^2 + 1(1e^(-2(5(-4x^2 + 5x) - 4)) - 3) - 1 = -2sin(3\u03c0(-4(-2|x - 1| - 3) + 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": "13", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3sin(1\u03c0(-2(-2|(5x^2 - 2x + 4) + 1|) - 1) + 3) - 1 and g(x) = 1(-1|(3sin(2\u03c0x - 3) + 1) + 2| + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1112", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1cos(3\u03c0(1(2(5x - 1)^2 - 1(5x - 1) + 3) + 4) + 1) + 3 and g(x) = -2sin(3\u03c0(-3|(2x^2 - 2x + 3) + 1| - 2) + 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1222", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5((3(-3sin(2\u03c0(1x - 5) + 1) + 2) + 2)/(-2(-3sin(2\u03c0(1x - 5) + 1) + 2) + 1)) - 1 = 3|(1|(-2|x - 2| - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(1|((-3(-1cos(1\u03c0x - 3) - 3) - 1)/(-2(-1cos(1\u03c0x - 3) - 3) - 3)) - 2| - 2) + 1| - 2 and g(x) = -3cos(2\u03c0(4((3x)/(-3x + 2)) - 2) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "73", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2e^(-1((-3(1(3|x - 1| + 3)^2 + 2(3|x - 1| + 3) - 4))/(1(1(3|x - 1| + 3)^2 + 2(3|x - 1| + 3) - 4) + 2))) - 1 and g(x) = -5(-1(2sin(2\u03c0x - 3) + 2) + 4)^2 + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(-2|(-2|(1x + 2) + 3| + 2)| - 3)^2 - 1(-2|(-2|(1x + 2) + 3| + 2)| - 3) - 4 = 1(5(1e^(1x) - 1)^2 - 5(1e^(1x) - 1) + 3)^2 - 5(5(1e^(1x) - 1)^2 - 5(1e^(1x) - 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": "5", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(3\u03c0(2|(1(-3|x| - 3)^2 - 4(-3|x| - 3) - 2) + 3| - 3) - 2) and g(x) = -3e^(1(5(5x + 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": "583", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(3((-2((-2x)/(3x + 1)) - 3)/(2((-2x)/(3x + 1))))) + 3)/(-1(3((-2((-2x)/(3x + 1)) - 3)/(2((-2x)/(3x + 1))))) + 3) and g(x) = -2(-5(3sin(2\u03c0x - 3) - 1) + 4)^2 + 2(-5(3sin(2\u03c0x - 3) - 1) + 4) - 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(2(-3sin(2\u03c0((-3x + 1)/(-1x - 2)) + 1) - 2) - 1) + 1| + 2 = (1(5(-2|x + 1| - 3)) - 2)/(2(5(-2|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": "83", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-4(-2sin(3\u03c0((-3x + 2)/(2x - 3)) - 2) - 3) - 5) - 2)/(-1(-4(-2sin(3\u03c0((-3x + 2)/(2x - 3)) - 2) - 3) - 5) + 3) and g(x) = -3(-1(1sin(2\u03c0x + 1) + 3) - 4) - 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(2sin(2\u03c0(1|(3x^2 + 2x - 2) - 3| - 1) - 3) + 3) + 3)/(2(2sin(2\u03c0(1|(3x^2 + 2x - 2) - 3| - 1) - 3) + 3) + 1) and g(x) = 2(1((-2x)/(-3x + 1))^2 + 3((-2x)/(-3x + 1)) - 2)^2 + 4(1((-2x)/(-3x + 1))^2 + 3((-2x)/(-3x + 1)) - 2) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(-1|(1(-2cos(2\u03c0x - 3) - 1)^2 - 1(-2cos(2\u03c0x - 3) - 1) - 4) - 3| + 3) - 2 and g(x) = 2cos(2\u03c0(-2|(1x^2 + 1x + 5) - 2| + 2) - 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "167", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(2\u03c0(-2|(-1|((2x - 3)/(-3x + 1)) + 1| + 2) + 3| + 1) - 2) + 2 and g(x) = -2(-2sin(2\u03c0(-2x + 2) - 3) + 3)^2 + 2(-2sin(2\u03c0(-2x + 2) - 3) + 3) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "89", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(2\u03c0((-2(-1(-5x - 5)^2 + 3) - 3)/(-1(-1(-5x - 5)^2 + 3) + 3)) - 2) + 3 = (-1(1(-3x - 1) + 3))/(-2(1(-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": "36", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|((-1(2|(3|x + 2| + 1) - 1| - 3) + 2)/(-2(2|(3|x + 2| + 1) - 1| - 3)))| + 1 = -2sin(1\u03c0(-5(-2|x - 2| + 2)^2 + 4(-2|x - 2| + 2)) - 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1427", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1e^(-1(-2|((-1(3x^2 - 1x - 1))/(-1(3x^2 - 1x - 1) + 2)) - 3| + 2)) - 3 and g(x) = 2|((-1(2|x + 1| - 3) + 2)/(2(2|x + 1| - 3) - 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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3sin(1\u03c0(-2|(4(5x^2 + 2x)^2 + 4(5x^2 + 2x) + 5) + 3| - 3) - 1) + 3 = -2cos(3\u03c0(-3|(2|x - 2| + 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": "648", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(-4((-1(-1x + 1) - 1)/(2(-1x + 1) - 3))^2 - 2((-1(-1x + 1) - 1)/(2(-1x + 1) - 3)) + 3) - 2| = (1((2((-3x + 1)/(-3x - 2)) - 3)/(3((-3x + 1)/(-3x - 2)) + 3)) - 2)/(2((2((-3x + 1)/(-3x - 2)) - 3)/(3((-3x + 1)/(-3x - 2)) + 3)) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-3|(2cos(3\u03c0(3|x - 1|) - 1) + 1) - 2| + 3) + 3)/(-1(-3|(2cos(3\u03c0(3|x - 1|) - 1) + 1) - 2| + 3) + 2) = 3|(-1(4x + 5)^2 + 1(4x + 5)) - 3| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(3(-2cos(2\u03c0((1x - 3)/(3x + 1)) - 2) + 3) + 4)| and g(x) = (-3((2(1x - 5) + 1)/(-3(1x - 5) + 1)) + 1)/(-2((2(1x - 5) + 1)/(-3(1x - 5) + 1)) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-1(-1(2(1x^2 + 5x - 2) - 1) + 3) - 1)/(-1(-1(2(1x^2 + 5x - 2) - 1) + 3) + 2) = 1(-2(1x^2 + 5x + 1) - 2)^2 - 3(-2(1x^2 + 5x + 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) = 1|(1e^(-1(2|((2x)/(-2x))| - 2))) - 2| + 1 and g(x) = 1(2|(3|x - 3| - 2) - 2|)^2 - 5(2|(3|x - 3| - 2) - 2|) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-5(3(1cos(1\u03c0x) + 1)^2 - 3(1cos(1\u03c0x) + 1) - 2)^2 + 2(3(1cos(1\u03c0x) + 1)^2 - 3(1cos(1\u03c0x) + 1) - 2) + 5)^2 - 5(-5(3(1cos(1\u03c0x) + 1)^2 - 3(1cos(1\u03c0x) + 1) - 2)^2 + 2(3(1cos(1\u03c0x) + 1)^2 - 3(1cos(1\u03c0x) + 1) - 2) + 5) + 3 and g(x) = 3sin(2\u03c0(-5(3|x - 2| + 1)^2 - 3(3|x - 2| + 1) - 2) - 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": "Find the number of times the graphs of f(x) = 3(3e^(-1((2(-1x^2 - 2x + 2) + 1)/(2(-1x^2 - 2x + 2)))))^2 - 1(3e^(-1((2(-1x^2 - 2x + 2) + 1)/(2(-1x^2 - 2x + 2))))) + 4 and g(x) = 3(1(-2|x + 1| - 3)^2 + 4(-2|x + 1| - 3) + 5)^2 - 4(1(-2|x + 1| - 3)^2 + 4(-2|x + 1| - 3) + 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) = 3(-1(-2(1cos(2\u03c0x + 1) - 2)^2 + 1(1cos(2\u03c0x + 1) - 2) + 2) - 5) + 5 and g(x) = 5(-1sin(2\u03c0((-1x + 1)/(1x + 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": "51", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2((-2(-1cos(2\u03c0((1x - 1)/(3x - 2)) - 1) - 2) - 3)/(-1(-1cos(2\u03c0((1x - 1)/(3x - 2)) - 1) - 2) + 2)) + 1)/(-1((-2(-1cos(2\u03c0((1x - 1)/(3x - 2)) - 1) - 2) - 3)/(-1(-1cos(2\u03c0((1x - 1)/(3x - 2)) - 1) - 2) + 2)) - 3) and g(x) = (-3(-2|(-4x^2 - 2x) + 1|) - 3)/(3(-2|(-4x^2 - 2x) + 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) = -4(3(4(-3|x + 2| + 3) - 4)^2 + 5(4(-3|x + 2| + 3) - 4) - 1)^2 + 3(3(4(-3|x + 2| + 3) - 4)^2 + 5(4(-3|x + 2| + 3) - 4) - 1) - 5 and g(x) = -5((-2(3|x + 1| - 3) - 3)/(2(3|x + 1| - 3) + 3))^2 + 2((-2(3|x + 1| - 3) - 3)/(2(3|x + 1| - 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(2\u03c0(-3(-2|(-1|x| - 2)| - 3)^2 - 1(-2|(-1|x| - 2)| - 3) - 4) - 2) + 1 and g(x) = (-2((1(-2|x - 2| - 1) + 3)/(-3(-2|x - 2| - 1) + 1)))/(1((1(-2|x - 2| - 1) + 3)/(-3(-2|x - 2| - 1) + 1)) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1390", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(1((-2(-1|x - 2| + 2) - 3)/(2(-1|x - 2| + 2) - 1)))^2 + 3(1((-2(-1|x - 2| + 2) - 3)/(2(-1|x - 2| + 2) - 1))) - 4 and g(x) = 1|((-1(-4x - 1) + 3)/(2(-4x - 1) - 3))| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-3(3|((1x - 3)/(-1x + 2)) + 2| - 1)^2 + 3(3|((1x - 3)/(-1x + 2)) + 2| - 1) + 5))/(-1(-3(3|((1x - 3)/(-1x + 2)) + 2| - 1)^2 + 3(3|((1x - 3)/(-1x + 2)) + 2| - 1) + 5) + 2) and g(x) = -4(-2cos(1\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": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(-3(-2cos(3\u03c0((-2x + 2)/(2x)) - 3) - 1) - 4) + 5 and g(x) = -2(4(-1x^2 + 2x + 4)^2 - 1(-1x^2 + 2x + 4) - 1)^2 - 2(4(-1x^2 + 2x + 4)^2 - 1(-1x^2 + 2x + 4) - 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(3(-2|(1x^2 + 5x - 1) + 1| - 1) + 1)^2 + 2(3(-2|(1x^2 + 5x - 1) + 1| - 1) + 1) - 3 and g(x) = 4(-2e^(1(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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-4(2(2e^(-1x) - 3)^2 - 2(2e^(-1x) - 3) - 3)^2 + 1(2(2e^(-1x) - 3)^2 - 2(2e^(-1x) - 3) - 3) + 1) + 4 and g(x) = -4((2(-3sin(3\u03c0x + 1) + 2))/(3(-3sin(3\u03c0x + 1) + 2) - 2))^2 + 2((2(-3sin(3\u03c0x + 1) + 2))/(3(-3sin(3\u03c0x + 1) + 2) - 2)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(-2(-3(-2|x|)^2 - 1(-2|x|) + 5)^2 - 3(-3(-2|x|)^2 - 1(-2|x|) + 5) + 1))/(-2(-2(-3(-2|x|)^2 - 1(-2|x|) + 5)^2 - 3(-3(-2|x|)^2 - 1(-2|x|) + 5) + 1) - 3) and g(x) = (2((-1(-1x^2 + 2x - 5))/(1(-1x^2 + 2x - 5) + 1)) + 2)/(-1((-1(-1x^2 + 2x - 5))/(1(-1x^2 + 2x - 5) + 1)) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|((3((3((2x)/(2x - 3)))/(3((2x)/(2x - 3)) - 2)))/(3((3((2x)/(2x - 3)))/(3((2x)/(2x - 3)) - 2)) - 1)) - 3| + 3 and g(x) = -5((-1((-1x - 2)/(3x + 3)) - 2)/(1((-1x - 2)/(3x + 3)) + 1))^2 + 2((-1((-1x - 2)/(3x + 3)) - 2)/(1((-1x - 2)/(3x + 3)) + 1)) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|((2(4(2sin(3\u03c0x - 2) - 3)^2 + 4(2sin(3\u03c0x - 2) - 3) - 1) - 3)/(1(4(2sin(3\u03c0x - 2) - 3)^2 + 4(2sin(3\u03c0x - 2) - 3) - 1) + 1)) - 1| and g(x) = -2(1|(-4x^2 - 3x - 3) - 3| + 1) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1((2((1(-1x - 2) + 1)/(2(-1x - 2) + 1)))/(-2((1(-1x - 2) + 1)/(2(-1x - 2) + 1)))) + 1)/(-2((2((1(-1x - 2) + 1)/(2(-1x - 2) + 1)))/(-2((1(-1x - 2) + 1)/(2(-1x - 2) + 1)))) + 3) and g(x) = 3sin(1\u03c0(1((-3x - 1)/(-1x))^2 - 5((-3x - 1)/(-1x)) - 4) + 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(-4((3((2x + 2)/(-2x + 3)) + 3)/(-2((2x + 2)/(-2x + 3)) + 3))^2 + 4))/(1(-4((3((2x + 2)/(-2x + 3)) + 3)/(-2((2x + 2)/(-2x + 3)) + 3))^2 + 4) - 3) and g(x) = 4(2sin(1\u03c0(-1x^2 + 3x - 4)) + 3) - 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1((2(5(-1x + 4)^2 + 2(-1x + 4) + 2) + 2)/(1(5(-1x + 4)^2 + 2(-1x + 4) + 2) + 2)) + 2)/(-1((2(5(-1x + 4)^2 + 2(-1x + 4) + 2) + 2)/(1(5(-1x + 4)^2 + 2(-1x + 4) + 2) + 2)) - 1) and g(x) = -3sin(2\u03c0(3((-2x - 3)/(-1x - 3)) + 1) - 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "152", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-2|(-1|(-3sin(3\u03c0x - 1) + 1) - 3| + 2)| + 1) - 2 and g(x) = 1|(2|(-1x - 2) + 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": "117", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-1|(1sin(3\u03c0(3x^2 + 5x) + 1) - 1) - 3| + 3) - 2)/(3(-1|(1sin(3\u03c0(3x^2 + 5x) + 1) - 1) - 3| + 3) + 3) and g(x) = 2(3|(-2x^2 - 3x + 1) - 3| + 2) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(3sin(1\u03c0((-3(-3|x + 3| + 3))/(1(-3|x + 3| + 3) + 2)) + 3) - 1)^2 - 2(3sin(1\u03c0((-3(-3|x + 3| + 3))/(1(-3|x + 3| + 3) + 2)) + 3) - 1) + 1 and g(x) = -2sin(1\u03c0(2(-4x^2 - 2x + 5) - 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": "96", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(-5(-4(1x^2 + 4x + 1)^2 - 2(1x^2 + 4x + 1) + 1)^2 + 2(-4(1x^2 + 4x + 1)^2 - 2(1x^2 + 4x + 1) + 1) + 2)| - 2 and g(x) = -3|(4(-1e^(-2x) + 2) + 3)| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(2|((-1((-2x + 1)/(-3x + 1)) + 3)/(1((-2x + 1)/(-3x + 1)) - 3)) + 3| - 3) - 1| - 1 and g(x) = -1cos(3\u03c0((-3(2x) - 3)/(2(2x) - 1)) - 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "66", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(-1(1((1x - 1)/(3x + 2))^2 + 3((1x - 1)/(3x + 2)) + 5)^2 + 1(1((1x - 1)/(3x + 2))^2 + 3((1x - 1)/(3x + 2)) + 5) - 1) - 3 and g(x) = -4(-2sin(1\u03c0(5x^2 + 4x - 5) - 2) - 3)^2 + 1(-2sin(1\u03c0(5x^2 + 4x - 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2cos(2\u03c0(-4(2(2x^2 - 5)) + 4) + 3) and g(x) = 5(-4((3x - 3)/(-1x)) + 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": "Determine the number of intersection points between the functions f(x) = (3((3(5(-1|x| + 1) + 3) + 3)/(2(5(-1|x| + 1) + 3))) - 2)/(-3((3(5(-1|x| + 1) + 3) + 3)/(2(5(-1|x| + 1) + 3))) - 2) and g(x) = -2(1(-3cos(1\u03c0x + 3) - 2)^2 + 5(-3cos(1\u03c0x + 3) - 2))^2 - 1(1(-3cos(1\u03c0x + 3) - 2)^2 + 5(-3cos(1\u03c0x + 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": "Determine the number of intersection points between the functions f(x) = -1|(2(5(-3x^2 + 2x - 4) + 4) - 1) + 2| - 3 and g(x) = 4(-3|((-1x - 3)/(1x - 3)) + 2| + 2) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(-1(2(1x^2 - 1x - 3)^2 - 3(1x^2 - 1x - 3) + 4) - 2) + 3| + 3 and g(x) = 1(1e^(-2((-1x - 1)/(-3x + 2))) - 1)^2 + 5(1e^(-2((-1x - 1)/(-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": "How many points of intersection exist between the functions f(x) = -2|(-3(-3sin(2\u03c0(1|x + 3|) - 3) - 2)) + 1| - 1 and g(x) = -1|(2e^(-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": "11", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|(2(-4(1x^2 + 5x - 5)^2 + 1(1x^2 + 5x - 5) - 2) + 4) + 1| + 3 and g(x) = -2|(-2|(3x^2 - 4x + 2) - 1| - 2) - 3| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(1(3|(-1|x + 2| + 1) - 1|)^2 + 3(3|(-1|x + 2| + 1) - 1|) - 4)^2 + 5(1(3|(-1|x + 2| + 1) - 1|)^2 + 3(3|(-1|x + 2| + 1) - 1|) - 4) + 1 and g(x) = (-2(-2cos(1\u03c0(4x - 4) + 2) - 1) + 3)/(-2(-2cos(1\u03c0(4x - 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 (1(1cos(3\u03c0((-3(5x - 5) + 1)/(3(5x - 5) + 2)) - 1) - 2) - 2)/(-3(1cos(3\u03c0((-3(5x - 5) + 1)/(3(5x - 5) + 2)) - 1) - 2) - 3) = (-2(-4(2x + 3)^2 + 1(2x + 3)) + 1)/(1(-4(2x + 3)^2 + 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": "28", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3e^(-1(3(3|(-1|x + 1|) - 2| - 2) + 3)) and g(x) = 3cos(2\u03c0(3((2x)/(-3x + 4))^2 + 2((2x)/(-3x + 4)) - 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": "103", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4((-2(-1e^(1((-1x + 1)/(1x + 1))) - 2) + 3)/(2(-1e^(1((-1x + 1)/(1x + 1))) - 2) + 2)) + 2 and g(x) = 5(-2(-2sin(3\u03c0x) + 3)^2 + 4(-2sin(3\u03c0x) + 3) + 5) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(1\u03c0(3(1((3x)/(-1x - 3)) + 4) - 1)) and g(x) = -1e^(-1(1(2|x - 3| + 3)^2 - 4(2|x - 3| + 3) + 3)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "177", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-3cos(1\u03c0(1|(-2|x - 1| + 2) + 1| + 3) + 1)) + 4 and g(x) = (3(-2((3x + 2)/(2x + 1)) + 4) - 2)/(2(-2((3x + 2)/(2x + 1)) + 4) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-2|(3|((-2x - 1)/(-2x - 2)) - 1| + 1) - 3| + 2) - 1| + 2 and g(x) = (2(2|(1x^2 - 5x + 3) - 3| + 3) + 3)/(2(2|(1x^2 - 5x + 3) - 3| + 3) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(-1e^(1((-2((-1x - 2)/(-3x + 2)) - 3)/(3((-1x - 2)/(-3x + 2)) - 2))) + 3) + 4 = (-2(3sin(3\u03c0(-1|x|)) - 2) - 3)/(-1(3sin(3\u03c0(-1|x|)) - 2)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(-3|(-4(5x^2 + 5x)^2 - 5(5x^2 + 5x) - 1) + 2|))/(2(-3|(-4(5x^2 + 5x)^2 - 5(5x^2 + 5x) - 1) + 2|) + 1) and g(x) = (2(-1((2x - 2)/(2x - 1)) + 4) - 2)/(-2(-1((2x - 2)/(2x - 1)) + 4) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(3cos(1\u03c0(-1(4x - 4) + 1) - 2)) - 3 = -5(-4(1sin(3\u03c0x + 1) + 2) + 5) + 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": "Determine the number of intersection points between the functions f(x) = -2(2|((-3(2|x| - 1) - 3)/(-2(2|x| - 1) + 1)) - 2| + 1) + 4 and g(x) = 3|(-1(3cos(3\u03c0x) + 2)^2 + 5(3cos(3\u03c0x) + 2) + 1) + 2| - 3 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": "How many solutions does the equation (3(1(3((-2x)/(2x + 2))^2 + 1((-2x)/(2x + 2)) - 4)^2 - 2(3((-2x)/(2x + 2))^2 + 1((-2x)/(2x + 2)) - 4) - 2) + 1)/(-1(1(3((-2x)/(2x + 2))^2 + 1((-2x)/(2x + 2)) - 4)^2 - 2(3((-2x)/(2x + 2))^2 + 1((-2x)/(2x + 2)) - 4) - 2) + 3) = 3sin(3\u03c0(-3(-3x + 1)^2 - 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": "334", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1sin(1\u03c0(3|(-4(-2x - 2)^2 - 3) + 1| - 3) - 1) + 2 and g(x) = 2|(3(-3x^2 + 2x + 1)^2 + 1(-3x^2 + 2x + 1) - 4) - 3| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(3cos(1\u03c0(2(5x - 5)^2 + 3(5x - 5) - 4) + 3) + 1) + 2)/(1(3cos(1\u03c0(2(5x - 5)^2 + 3(5x - 5) - 4) + 3) + 1) + 3) and g(x) = -4(-1(3sin(1\u03c0x) + 2) + 4)^2 + 3(-1(3sin(1\u03c0x) + 2) + 4) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "186", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(1\u03c0(-1((-2(1x - 1) + 3)/(1(1x - 1) + 3)) - 1) + 1) - 3 and g(x) = (1(-3(-5x + 4)^2 - 2(-5x + 4) - 4) - 1)/(-3(-3(-5x + 4)^2 - 2(-5x + 4) - 4)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "67", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(5(-2sin(2\u03c0(4x^2 - 5x - 5) + 2) + 2)^2 - 2(-2sin(2\u03c0(4x^2 - 5x - 5) + 2) + 2) - 3) - 3 and g(x) = 3(1(-1|x - 2| - 1)^2 + 1(-1|x - 2| - 1) - 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "972", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3e^(-1(-5((2(-1|x + 3| + 3))/(1(-1|x + 3| + 3) - 2))^2 + 3((2(-1|x + 3| + 3))/(1(-1|x + 3| + 3) - 2)) + 1)) + 1 and g(x) = -1(-1|(-2sin(1\u03c0x - 2) + 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": "Determine the number of intersection points between the functions f(x) = 2(1cos(1\u03c0((-3(-1x - 2) - 1)/(1(-1x - 2) + 3)) - 3) + 3)^2 - 5(1cos(1\u03c0((-3(-1x - 2) - 1)/(1(-1x - 2) + 3)) - 3) + 3) - 5 and g(x) = 1(-1(-1sin(2\u03c0x - 1)) + 4)^2 - 1(-1(-1sin(2\u03c0x - 1)) + 4) - 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 points of intersection exist between the functions f(x) = -2((1(-2|(2|x + 3| - 3) - 3| - 1) - 3)/(-1(-2|(2|x + 3| - 3) - 3| - 1) - 1))^2 - 5((1(-2|(2|x + 3| - 3) - 3| - 1) - 3)/(-1(-2|(2|x + 3| - 3) - 3| - 1) - 1)) + 1 and g(x) = 1((-2(-1|x + 3| - 3))/(-3(-1|x + 3| - 3) + 2))^2 + 3((-2(-1|x + 3| - 3))/(-3(-1|x + 3| - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(-1cos(3\u03c0((-2(-3x^2 + 5x + 5) + 2)/(-2(-3x^2 + 5x + 5) - 3)) + 2) - 3) + 1| + 2 and g(x) = 1(-3e^(2(-1|x - 2| + 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) = -4((-2(-2sin(3\u03c0(-1x + 3) + 2) + 2) + 1)/(3(-2sin(3\u03c0(-1x + 3) + 2) + 2) - 1))^2 + 1((-2(-2sin(3\u03c0(-1x + 3) + 2) + 2) + 1)/(3(-2sin(3\u03c0(-1x + 3) + 2) + 2) - 1)) + 5 and g(x) = (-2(-1e^(-1(3x - 1))))/(-1(-1e^(-1(3x - 1))) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "74", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1cos(2\u03c0(-5(-2(1x + 1) - 1)^2 - 1(-2(1x + 1) - 1) - 4) + 3) + 2 and g(x) = (2(-3(1x^2 - 3) - 1) + 1)/(3(-3(1x^2 - 3) - 1) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(3((2(3x^2 - 5x + 2) - 1)/(1(3x^2 - 5x + 2)))^2 - 1((2(3x^2 - 5x + 2) - 1)/(1(3x^2 - 5x + 2))) - 4) - 1| + 2 and g(x) = -2sin(2\u03c0(3(1x^2 - 5x + 2) - 2) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(4(1cos(2\u03c0(-1|x| + 2) - 2) + 3) - 5) - 1| - 3 and g(x) = 3|(2|((1x - 3)/(1x - 3)) + 2| - 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": "Determine the number of intersection points between the functions f(x) = 4(-5(-5(-1sin(2\u03c0x - 2))^2 - 5(-1sin(2\u03c0x - 2)) + 3))^2 + 3(-5(-5(-1sin(2\u03c0x - 2))^2 - 5(-1sin(2\u03c0x - 2)) + 3)) - 5 and g(x) = (-2(-3e^(-2(-3|x - 2| - 2)) - 1))/(1(-3e^(-2(-3|x - 2| - 2)) - 1) - 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": "How many points of intersection exist between the functions f(x) = 3|(3|((-3(-5x - 2))/(-3(-5x - 2) - 3)) - 1| - 3)| + 1 and g(x) = -1sin(1\u03c0(-1|(1x + 5) + 3| - 1) - 1) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-1|(-2(-5x^2 + 4x - 3)^2 - 2(-5x^2 + 4x - 3)) - 3| + 1) - 1)/(-1(-1|(-2(-5x^2 + 4x - 3)^2 - 2(-5x^2 + 4x - 3)) - 3| + 1) + 2) and g(x) = -4(3cos(3\u03c0(5x + 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": "300", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3((1(-3cos(2\u03c0(-4x^2 + 3x + 4) + 2) - 2) - 1)/(-1(-3cos(2\u03c0(-4x^2 + 3x + 4) + 2) - 2) + 3)) - 1 and g(x) = (-2(-3((-3x + 3)/(-3x + 1)) + 1) + 1)/(3(-3((-3x + 3)/(-3x + 1)) + 1) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "173", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-3(1|(-1x + 2) - 1| - 1) + 4) + 2)/(2(-3(1|(-1x + 2) - 1| - 1) + 4) + 1) and g(x) = -1|(-4(-3|x + 1| + 3) - 1) + 3| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(3(2sin(3\u03c0(1|x - 2| + 1) - 3) - 3) - 5) + 1 and g(x) = 3|(1(-1x - 2) - 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": "16", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2((3(1(1sin(2\u03c0x) + 1) + 4) + 3)/(-3(1(1sin(2\u03c0x) + 1) + 4) + 2)))/(1((3(1(1sin(2\u03c0x) + 1) + 4) + 3)/(-3(1(1sin(2\u03c0x) + 1) + 4) + 2)) - 3) and g(x) = 4(2sin(3\u03c0((-1x - 3)/(3x + 1)) - 2) - 1)^2 - 1(2sin(3\u03c0((-1x - 3)/(3x + 1)) - 2) - 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-3((-1(-3e^(1x) + 1) - 2)/(-2(-3e^(1x) + 1) + 3))^2 - 2((-1(-3e^(1x) + 1) - 2)/(-2(-3e^(1x) + 1) + 3)))^2 + 5(-3((-1(-3e^(1x) + 1) - 2)/(-2(-3e^(1x) + 1) + 3))^2 - 2((-1(-3e^(1x) + 1) - 2)/(-2(-3e^(1x) + 1) + 3))) + 2 = (-2(-3sin(2\u03c0(1x - 2) - 3) - 3) + 1)/(-3(-3sin(2\u03c0(1x - 2) - 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": "25", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(1(-5(1e^(2x) - 3)) - 2) + 3)/(3(1(-5(1e^(2x) - 3)) - 2) - 2) = (-3(-3cos(3\u03c0(1x^2 + 5x + 1) + 2) - 1) - 1)/(-1(-3cos(3\u03c0(1x^2 + 5x + 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": "292", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-2sin(2\u03c0(-5(-5x - 5)^2 - 4(-5x - 5) - 2) + 1) + 1) + 1)/(2(-2sin(2\u03c0(-5(-5x - 5)^2 - 4(-5x - 5) - 2) + 1) + 1) + 2) and g(x) = 2|(3cos(2\u03c0((-1x - 3)/(2x + 1)) + 1) - 3) + 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "814", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2e^(1(1(-1|((-2x)/(-3x - 2)) + 3| + 2)^2 + 2(-1|((-2x)/(-3x - 2)) + 3| + 2) - 3)) + 2 and g(x) = 1cos(2\u03c0(5(-1x^2 + 4) + 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-4(3cos(1\u03c0(4x - 1) + 2) + 2)^2 - 1(3cos(1\u03c0(4x - 1) + 2) + 2) - 1)^2 - 3(-4(3cos(1\u03c0(4x - 1) + 2) + 2)^2 - 1(3cos(1\u03c0(4x - 1) + 2) + 2) - 1) - 1 and g(x) = -5((3(4x^2 - 3x - 1) - 1)/(2(4x^2 - 3x - 1) + 2)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "149", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1sin(1\u03c0((-3(5(-5x^2 + 5x - 1) - 2) - 3)/(3(5(-5x^2 + 5x - 1) - 2) - 1)) - 2) - 3 and g(x) = -3|(1(1|x + 1| - 3) + 1) - 3| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(-1sin(1\u03c0(2(-4x^2 + 1x - 3) - 4) + 3) - 1) - 2 = -1(2|(1x - 4) - 3|) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(2(3|(-2x - 4) - 2| - 3) - 2)^2 - 4(2(3|(-2x - 4) - 2| - 3) - 2) - 3 and g(x) = 2|(-1sin(2\u03c0(-1x^2 + 3x + 5) + 3) + 2) + 2| - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(-5(-1(5x^2 + 4x - 3)^2 - 2(5x^2 + 4x - 3) + 1)^2 - 4(-1(5x^2 + 4x - 3)^2 - 2(5x^2 + 4x - 3) + 1) + 5) + 4 and g(x) = 3(3sin(3\u03c0(-1x^2 - 5x - 5) + 1) + 2)^2 - 4(3sin(3\u03c0(-1x^2 - 5x - 5) + 1) + 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4((1((-3(2|x + 2| - 3))/(3(2|x + 2| - 3) - 2)) + 3)/(-1((-3(2|x + 2| - 3))/(3(2|x + 2| - 3) - 2)) + 2))^2 + 5((1((-3(2|x + 2| - 3))/(3(2|x + 2| - 3) - 2)) + 3)/(-1((-3(2|x + 2| - 3))/(3(2|x + 2| - 3) - 2)) + 2)) - 3 and g(x) = -2|(1(-5x^2 + 5x + 3) + 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-1(-1(2|x + 1| - 1)^2 + 4(2|x + 1| - 1) - 5)^2 - 2(-1(2|x + 1| - 1)^2 + 4(2|x + 1| - 1) - 5) + 4) - 2| + 2 and g(x) = 2((-2(1x + 5) - 2)/(2(1x + 5) - 1))^2 + 2((-2(1x + 5) - 2)/(2(1x + 5) - 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) = (3((1(-3sin(3\u03c0(-2|x - 2|) + 3) - 2) + 2)/(-1(-3sin(3\u03c0(-2|x - 2|) + 3) - 2) + 3)))/(-1((1(-3sin(3\u03c0(-2|x - 2|) + 3) - 2) + 2)/(-1(-3sin(3\u03c0(-2|x - 2|) + 3) - 2) + 3)) + 2) and g(x) = 3|((1(-4x + 3) - 1)/(2(-4x + 3) - 1)) - 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(1(1((3x - 2)/(-3x + 4)) + 2)^2 - 5(1((3x - 2)/(-3x + 4)) + 2) + 3))/(-3(1(1((3x - 2)/(-3x + 4)) + 2)^2 - 5(1((3x - 2)/(-3x + 4)) + 2) + 3) - 3) and g(x) = (-2(5(1cos(3\u03c0x) + 1)^2 - 3(1cos(3\u03c0x) + 1) - 1) + 2)/(-1(5(1cos(3\u03c0x) + 1)^2 - 3(1cos(3\u03c0x) + 1) - 1) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(3(3|(-2cos(2\u03c0x - 3)) + 1| - 2)^2 - 3(3|(-2cos(2\u03c0x - 3)) + 1| - 2) - 5) - 2)/(1(3(3|(-2cos(2\u03c0x - 3)) + 1| - 2)^2 - 3(3|(-2cos(2\u03c0x - 3)) + 1| - 2) - 5)) = -1|(-3e^(-2((2x)/(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": "38", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(-1(4(5x^2 - 1x - 4) + 1) - 5)^2 - 5(-1(4(5x^2 - 1x - 4) + 1) - 5) + 4 and g(x) = (-1(2cos(2\u03c0(5x^2 + 4) + 3) - 2) + 2)/(3(2cos(2\u03c0(5x^2 + 4) + 3) - 2) + 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": "Determine the number of intersection points between the functions f(x) = (3((-1(3sin(1\u03c0((3x)/(-3x - 1)) + 1) + 1) + 2)/(-2(3sin(1\u03c0((3x)/(-3x - 1)) + 1) + 1) + 3)) + 1)/(-2((-1(3sin(1\u03c0((3x)/(-3x - 1)) + 1) + 1) + 2)/(-2(3sin(1\u03c0((3x)/(-3x - 1)) + 1) + 1) + 3)) + 3) and g(x) = -3(-1|((-1x + 2)/(2x + 3)) + 2| - 3) + 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": "Find the number of times the graphs of f(x) = 2(2(1(3cos(2\u03c0x - 3) - 3) + 3) - 2) + 3 and g(x) = 2(2cos(2\u03c0((3x)/(-2x - 3)) - 2)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2(2(2cos(2\u03c0(-2|x + 1|) - 3) + 2)^2 - 4(2cos(2\u03c0(-2|x + 1|) - 3) + 2) + 1) - 4 = -3(3(-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": "28", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(1(-2(-5x^2 + 5x + 1) - 3) + 2) - 2)/(1(1(-2(-5x^2 + 5x + 1) - 3) + 2) - 2) and g(x) = -3(-1cos(1\u03c0((-1x - 2)/(3x + 1)) + 1) + 1)^2 - 4(-1cos(1\u03c0((-1x - 2)/(3x + 1)) + 1) + 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2e^(1((1(5(-5x + 5)) - 1)/(-1(5(-5x + 5))))) + 1 = -5(-3|(1e^(-2x) - 2) + 1| + 2)^2 - 5(-3|(1e^(-2x) - 2) + 1| + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2((-3((-1(-3|x - 3| + 3) - 2)/(-2(-3|x - 3| + 3))) - 1)/(1((-1(-3|x - 3| + 3) - 2)/(-2(-3|x - 3| + 3))) - 2)) - 3)/(2((-3((-1(-3|x - 3| + 3) - 2)/(-2(-3|x - 3| + 3))) - 1)/(1((-1(-3|x - 3| + 3) - 2)/(-2(-3|x - 3| + 3))) - 2)) - 3) and g(x) = -3(5(1cos(3\u03c0x + 2) + 2) + 4) + 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(1(3sin(3\u03c0((-3x + 3)/(-3x - 1)) - 2) + 2) - 5))/(-1(1(3sin(3\u03c0((-3x + 3)/(-3x - 1)) - 2) + 2) - 5) - 2) and g(x) = -3|((1(5x - 1))/(-2(5x - 1) - 2)) + 3| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|(-2sin(1\u03c0(-5(-2|x - 3| + 2)) + 3) + 3)| + 2 = 3(-3cos(2\u03c0(-1x^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": "162", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(-2(-2(4x^2 + 3x - 5)^2 + 2(4x^2 + 3x - 5) - 5)^2 + 2(-2(4x^2 + 3x - 5)^2 + 2(4x^2 + 3x - 5) - 5) + 2) - 3| - 3 and g(x) = -5(3(1sin(3\u03c0x + 2) - 1)^2 - 2)^2 + 2(3(1sin(3\u03c0x + 2) - 1)^2 - 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) = -4(-3sin(3\u03c0(-5(5x^2 - 4x - 1)) - 3) + 1)^2 - 2(-3sin(3\u03c0(-5(5x^2 - 4x - 1)) - 3) + 1) + 1 and g(x) = -3cos(3\u03c0(1((-3x - 1)/(1x + 3)) + 5) + 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "152", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-2(-4(-2|x - 1| + 2)^2 + 1(-2|x - 1| + 2) + 3) + 1) - 1)/(2(-2(-4(-2|x - 1| + 2)^2 + 1(-2|x - 1| + 2) + 3) + 1) - 1) and g(x) = -3|((-3(-2sin(2\u03c0x - 1) - 3) + 3)/(-1(-2sin(2\u03c0x - 1) - 3) + 2)) + 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) = 5(1(2cos(2\u03c0((1x - 1)/(1x)) - 2) + 1)^2 - 1(2cos(2\u03c0((1x - 1)/(1x)) - 2) + 1) - 2) and g(x) = 3sin(3\u03c0(-3((-2x - 2)/(1x)) - 1)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3(2|(2(2|x + 2|) - 5) + 2|) + 2)/(-1(2|(2(2|x + 2|) - 5) + 2|) + 2) = 2|((2(4x + 4) + 1)/(-1(4x + 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": "Find the number of times the graphs of f(x) = (-2((-1(1sin(3\u03c0(3|x| + 2) + 1)) + 2)/(2(1sin(3\u03c0(3|x| + 2) + 1)) - 1)) + 1)/(-2((-1(1sin(3\u03c0(3|x| + 2) + 1)) + 2)/(2(1sin(3\u03c0(3|x| + 2) + 1)) - 1)) - 3) and g(x) = 3|(1|(1|x - 3| - 1) + 3| + 2) + 1| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "95", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(2\u03c0(-1|((3((3x + 3)/(3x)) + 3)/(-3((3x + 3)/(3x)) + 2)) + 2| - 2) + 1) + 2 and g(x) = (-3(-3sin(2\u03c0(-2|x - 3| + 1) - 2) - 2))/(-3(-3sin(2\u03c0(-2|x - 3| + 1) - 2) - 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "193", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(3|(-3|(1x + 5) + 3| + 3)| + 1)^2 + 5(3|(-3|(1x + 5) + 3| + 3)| + 1) - 1 = -1|(3((-2x + 2)/(1x + 3)) + 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(1(-2cos(3\u03c0(-4x + 5) + 2) + 1)^2 + 2(-2cos(3\u03c0(-4x + 5) + 2) + 1) - 3) - 3)/(-1(1(-2cos(3\u03c0(-4x + 5) + 2) + 1)^2 + 2(-2cos(3\u03c0(-4x + 5) + 2) + 1) - 3) - 1) and g(x) = (1(2|((-1x + 3)/(3x - 2)) + 1| + 2) - 1)/(1(2|((-1x + 3)/(3x - 2)) + 1| + 2) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "230", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3(-3((1((-2x)/(-1x - 2)) + 2)/(-2((-2x)/(-1x - 2)) + 3))^2 - 5((1((-2x)/(-1x - 2)) + 2)/(-2((-2x)/(-1x - 2)) + 3)) - 3)^2 - 5(-3((1((-2x)/(-1x - 2)) + 2)/(-2((-2x)/(-1x - 2)) + 3))^2 - 5((1((-2x)/(-1x - 2)) + 2)/(-2((-2x)/(-1x - 2)) + 3)) - 3) - 1 = -5((-2(1sin(2\u03c0x - 3)))/(-1(1sin(2\u03c0x - 3)) - 2))^2 + 5((-2(1sin(2\u03c0x - 3)))/(-1(1sin(2\u03c0x - 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1((2(1|(5x^2 + 3) + 1| - 1) - 1)/(-2(1|(5x^2 + 3) + 1| - 1) - 2))^2 - 1((2(1|(5x^2 + 3) + 1| - 1) - 1)/(-2(1|(5x^2 + 3) + 1| - 1) - 2)) + 4 and g(x) = 3(5(-5x - 5)^2 - 3(-5x - 5) - 5)^2 + 4(5(-5x - 5)^2 - 3(-5x - 5) - 5) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2sin(1\u03c0(2(3|(-1|x + 2| - 3) - 2| - 3)^2 + 2(3|(-1|x + 2| - 3) - 2| - 3) - 3) + 1) and g(x) = (-1(1(1|x - 1| - 2) + 2) - 2)/(1(1(1|x - 1| - 2) + 2) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1435", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(2\u03c0(-1(4(-4x + 1) - 5) + 2) - 3) - 3 and g(x) = -1((2(-1e^(-1x) + 1) - 3)/(3(-1e^(-1x) + 1) + 3)) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "607", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|((-1(-2((-3x)/(1x + 1))^2 - 2((-3x)/(1x + 1)) + 5) - 3)/(1(-2((-3x)/(1x + 1))^2 - 2((-3x)/(1x + 1)) + 5) - 2)) - 3| + 3 and g(x) = 1(2cos(3\u03c0(-2|x + 1| + 3) - 2) + 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) = -3((-2(1(2x^2 + 4x)) - 3)/(-3(1(2x^2 + 4x)) - 3))^2 - 3((-2(1(2x^2 + 4x)) - 3)/(-3(1(2x^2 + 4x)) - 3)) + 2 and g(x) = 2(-5(1x + 5) - 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1cos(1\u03c0(-1|(-3(-2|x| + 2)^2 + 2(-2|x| + 2) - 3) - 3| - 1) - 2) and g(x) = 2((-2(-2|x|) + 3)/(-2(-2|x|) + 3)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "823", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(4(-3sin(1\u03c0(-3x)) + 1) - 2) + 3)/(-2(4(-3sin(1\u03c0(-3x)) + 1) - 2) + 1) and g(x) = -4((2(-4x - 4) + 2)/(1(-4x - 4) + 3)) + 1 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": "How many points of intersection exist between the functions f(x) = 3|(-1cos(2\u03c0(1|(4x - 4) + 1|) + 3) - 3)| - 2 and g(x) = 4((2(-4x^2 + 1x + 2) + 3)/(1(-4x^2 + 1x + 2)))^2 + 4((2(-4x^2 + 1x + 2) + 3)/(1(-4x^2 + 1x + 2))) + 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(4(-3sin(1\u03c0(-4x^2 + 2x) + 2) - 2)^2 + 3) + 3 and g(x) = (-1(-4(-2|x + 2|)^2 - 5(-2|x + 2|) - 1) - 2)/(2(-4(-2|x + 2|)^2 - 5(-2|x + 2|) - 1) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-2cos(1\u03c0(-3(1x^2 + 3x + 4) + 5)) + 1)^2 + 4(-2cos(1\u03c0(-3(1x^2 + 3x + 4) + 5)) + 1) + 5 and g(x) = (-2(-2|((-2x + 1)/(-1x - 1)) - 2| + 1) + 3)/(1(-2|((-2x + 1)/(-1x - 1)) - 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": "1045", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(2\u03c0(-3|(2(2|x + 3| - 2)^2 - 1(2|x + 3| - 2) + 2)| + 1) - 2) + 3 and g(x) = 3sin(1\u03c0(4((3x + 1)/(3x + 3)) - 1) + 2) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "942", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(1|(4(2x + 1)^2 + 5(2x + 1) - 5) + 1| - 3)^2 - 3(1|(4(2x + 1)^2 + 5(2x + 1) - 5) + 1| - 3) and g(x) = (1(2|(-2e^(1x) + 2) + 3| - 2) - 1)/(2(2|(-2e^(1x) + 2) + 3| - 2) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(2((3((-1x - 1)/(2x + 3)) - 3)/(1((-1x - 1)/(2x + 3))))^2 + 1((3((-1x - 1)/(2x + 3)) - 3)/(1((-1x - 1)/(2x + 3)))) - 3) + 2| + 1 and g(x) = 5(-1|((-3x - 1)/(-3x + 4)) + 2| + 2) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(1((3(3|x - 3| + 2) - 1)/(1(3|x - 3| + 2) - 2)) - 1) + 2)/(-3(1((3(3|x - 3| + 2) - 1)/(1(3|x - 3| + 2) - 2)) - 1) - 3) and g(x) = 1(-2|(2cos(1\u03c0x + 1)) - 1| + 1)^2 - 3(-2|(2cos(1\u03c0x + 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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(-1(-2|(-2sin(1\u03c0x - 2) + 1)|)^2 - 1(-2|(-2sin(1\u03c0x - 2) + 1)|) + 3) + 3| + 3 and g(x) = -5(1(-1e^(-1x) + 1)^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": "22", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(2(4(-1e^(1x))^2 - 3(-1e^(1x)) - 5)^2 - 1(4(-1e^(1x))^2 - 3(-1e^(1x)) - 5) - 5)^2 + 3(2(4(-1e^(1x))^2 - 3(-1e^(1x)) - 5)^2 - 1(4(-1e^(1x))^2 - 3(-1e^(1x)) - 5) - 5) + 2 = -1(2sin(3\u03c0(-3x - 4)))^2 - 3(2sin(3\u03c0(-3x - 4))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2e^(1(-5(-1|(-2|x + 3| + 2) + 3| + 2) - 2)) - 2 and g(x) = -1sin(1\u03c0(2((-2x + 1)/(-1x + 3))) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(1\u03c0(1|((1(-1x^2 + 1x + 2) + 2)/(3(-1x^2 + 1x + 2) + 1)) - 3| - 1) - 1) + 1 = (1(-3cos(2\u03c0(1x^2 + 3x + 2) + 3) - 2))/(1(-3cos(2\u03c0(1x^2 + 3x + 2) + 3) - 2)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1e^(-2(-3(-1|(1x^2 - 5x + 5) - 1| + 2) + 3)) + 2 and g(x) = 5(-1|(-3x + 2) + 3| + 3) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|(-2cos(1\u03c0(2|(3x^2 + 2x + 5) + 1| - 3) + 1) + 2) + 3| + 1 and g(x) = (-1((3(3cos(3\u03c0x) - 2) - 3)/(-3(3cos(3\u03c0x) - 2) - 3)) + 1)/(-1((3(3cos(3\u03c0x) - 2) - 3)/(-3(3cos(3\u03c0x) - 2) - 3)) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "101", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|((-1(-3(-3e^(-2x) - 3) - 2) + 3)/(-3(-3(-3e^(-2x) - 3) - 2) + 2)) + 3| + 1 and g(x) = (2(3|(4x^2 - 4x)| - 2) - 1)/(1(3|(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": "How many solutions does the equation (-2(-2sin(1\u03c0(1(1|x| + 1) + 1) - 3)) + 2)/(-3(-2sin(1\u03c0(1(1|x| + 1) + 1) - 3))) = 5(-3sin(2\u03c0(-4x^2 - 3x - 1) + 3) + 3)^2 + 1(-3sin(2\u03c0(-4x^2 - 3x - 1) + 3) + 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "252", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(3sin(1\u03c0(-2(3|x - 1| + 2) - 1) - 2) - 1) - 3 and g(x) = -2(-3sin(3\u03c0(-5x^2 + 2x - 1)) - 1)^2 + 2(-3sin(3\u03c0(-5x^2 + 2x - 1)) - 1) - 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "717", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2((2(-1e^(2((2x - 3)/(1x + 1))) + 1) + 2)/(1(-1e^(2((2x - 3)/(1x + 1))) + 1) + 1)) + 2)/(-3((2(-1e^(2((2x - 3)/(1x + 1))) + 1) + 2)/(1(-1e^(2((2x - 3)/(1x + 1))) + 1) + 1)) + 1) and g(x) = (-1((3(-2sin(3\u03c0x - 3)) + 2)/(1(-2sin(3\u03c0x - 3)) - 2)) - 2)/(3((3(-2sin(3\u03c0x - 3)) + 2)/(1(-2sin(3\u03c0x - 3)) - 2)) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(4(3cos(1\u03c0((3x + 2)/(3x + 3)) + 3) - 3)^2 + 5(3cos(1\u03c0((3x + 2)/(3x + 3)) + 3) - 3) - 4)^2 + 2(4(3cos(1\u03c0((3x + 2)/(3x + 3)) + 3) - 3)^2 + 5(3cos(1\u03c0((3x + 2)/(3x + 3)) + 3) - 3) - 4) - 2 and g(x) = 2cos(3\u03c0(2|(-3x - 1) + 2| - 3) + 2) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-4((2(-3x - 5) + 1)/(1(-3x - 5) + 2)))^2 + 3(-4((2(-3x - 5) + 1)/(1(-3x - 5) + 2))) - 2 and g(x) = -4(-1|(1sin(1\u03c0x)) - 3| - 3)^2 + 3(-1|(1sin(1\u03c0x)) - 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": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-1cos(2\u03c0(-2(-1x - 3)^2 + 2(-1x - 3) - 3)))^2 - 3(-1cos(2\u03c0(-2(-1x - 3)^2 + 2(-1x - 3) - 3))) - 3 and g(x) = -1|(-2|((-2x)/(-1x + 2)) + 1| + 3) - 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "79", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(-3sin(3\u03c0(-5(-5x^2 - 2x + 2)^2 + 4(-5x^2 - 2x + 2) + 2) - 1))| + 1 and g(x) = 1(-2(3e^(-1x))^2 + 5(3e^(-1x)) - 4) + 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((-3(2|x + 2| - 1) + 1)/(1(2|x + 2| - 1) + 3)) + 4)^2 - 4(-3((-3(2|x + 2| - 1) + 1)/(1(2|x + 2| - 1) + 3)) + 4) + 1 and g(x) = 3|(-3((-3x + 3)/(-2x + 3))^2 - 1) + 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": "Determine the number of intersection points between the functions f(x) = (-3(3|((2(2x + 2) - 3)/(-2(2x + 2) + 3)) - 2|) - 1)/(-2(3|((2(2x + 2) - 3)/(-2(2x + 2) + 3)) - 2|) - 1) and g(x) = -3(3cos(2\u03c0(5x^2 - 3x + 3) + 3) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1498", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(-5((-3(5x - 1) - 2)/(2(5x - 1) - 1))^2 + 5((-3(5x - 1) - 2)/(2(5x - 1) - 1)) - 1) - 1| and g(x) = -5(3|(-1|x - 1| + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3(-4((2(1x + 3) - 2)/(-1(1x + 3) + 2))^2 - 2((2(1x + 3) - 2)/(-1(1x + 3) + 2)) - 1) - 2)/(3(-4((2(1x + 3) - 2)/(-1(1x + 3) + 2))^2 - 2((2(1x + 3) - 2)/(-1(1x + 3) + 2)) - 1) - 1) = 2sin(2\u03c0(2(1x - 5)^2 - 1(1x - 5) + 4)) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "990", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(3\u03c0(2|(-1(5x + 1) + 3) - 2|)) + 1 and g(x) = (2(3|(3x + 2) + 1| - 1) - 3)/(-2(3|(3x + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(2(-3cos(2\u03c0(-1x - 4) - 1) + 3) - 3)^2 + 4(2(-3cos(2\u03c0(-1x - 4) - 1) + 3) - 3) + 2 and g(x) = 2(-5(-1|x - 2| + 3) + 1)^2 + 5(-5(-1|x - 2| + 3) + 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(2\u03c0(-3|(-5(2|x - 2| - 2) - 2) - 2| - 3) - 1) and g(x) = 3((1(2|x - 2| + 2))/(1(2|x - 2| + 2) + 1))^2 - 1((1(2|x - 2| + 2))/(1(2|x - 2| + 2) + 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "994", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(2(4(-3|x + 3| - 1) - 4) + 4) - 1)/(-2(2(4(-3|x + 3| - 1) - 4) + 4) + 3) and g(x) = 3(3cos(2\u03c0(-2x^2 + 5x + 4) - 3) - 3)^2 + 5(3cos(2\u03c0(-2x^2 + 5x + 4) - 3) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "771", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1((2((3(-5x^2 + 4x - 1) - 1)/(-1(-5x^2 + 4x - 1) + 2)) + 2)/(-1((3(-5x^2 + 4x - 1) - 1)/(-1(-5x^2 + 4x - 1) + 2)) - 3))^2 + 4((2((3(-5x^2 + 4x - 1) - 1)/(-1(-5x^2 + 4x - 1) + 2)) + 2)/(-1((3(-5x^2 + 4x - 1) - 1)/(-1(-5x^2 + 4x - 1) + 2)) - 3)) - 3 and g(x) = 3((2(-3x^2 + 5x - 1))/(-3(-3x^2 + 5x - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(-2e^(2(2|(-2x + 3) - 1| + 1)) + 2))/(1(-2e^(2(2|(-2x + 3) - 1| + 1)) + 2) + 1) = -4(3cos(3\u03c0(2x^2 - 2x + 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": "1066", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(5(-3(-3cos(3\u03c0x))^2 - 3(-3cos(3\u03c0x)) + 1)^2 + 5(-3(-3cos(3\u03c0x))^2 - 3(-3cos(3\u03c0x)) + 1) + 1) + 1)/(-3(5(-3(-3cos(3\u03c0x))^2 - 3(-3cos(3\u03c0x)) + 1)^2 + 5(-3(-3cos(3\u03c0x))^2 - 3(-3cos(3\u03c0x)) + 1) + 1) - 3) and g(x) = (2((3(2x + 2) + 3)/(1(2x + 2) + 3)) + 2)/(2((3(2x + 2) + 3)/(1(2x + 2) + 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 solutions does the equation 1(-4(-1(-1x^2 + 3x + 3) + 4)^2 + 4(-1(-1x^2 + 3x + 3) + 4) + 3)^2 + 1(-4(-1(-1x^2 + 3x + 3) + 4)^2 + 4(-1(-1x^2 + 3x + 3) + 4) + 3) - 4 = (-1(2(-3sin(2\u03c0x - 2) - 1) - 3))/(-1(2(-3sin(2\u03c0x - 2) - 1) - 3) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-2|(-2cos(2\u03c0(1x^2 - 1) + 3) - 1)| + 3) + 1)/(1(-2|(-2cos(2\u03c0(1x^2 - 1) + 3) - 1)| + 3)) and g(x) = -2e^(-1((2(2x^2 + 4x + 4))/(-1(2x^2 + 4x + 4) - 2))) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(-3|((-2(1|x - 1| - 2) - 3)/(-2(1|x - 1| - 2) + 1)) + 1|)^2 + 4 and g(x) = 2cos(1\u03c0(1(5x + 5)^2 + 4(5x + 5)) - 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(1sin(2\u03c0((-1(2|x - 2| + 1) + 2)/(1(2|x - 2| + 1) + 1)) + 1) + 1)^2 + 3(1sin(2\u03c0((-1(2|x - 2| + 1) + 2)/(1(2|x - 2| + 1) + 1)) + 1) + 1) - 1 and g(x) = -3((-1(1|x - 3|) - 1)/(-2(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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(1(-3|(-1x^2 + 3x + 2) + 1|)^2 - 3(-3|(-1x^2 + 3x + 2) + 1|) - 2) - 3| + 3 and g(x) = -4(-1cos(3\u03c0(1|x - 2| - 1)) + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4((3(2e^(-2((2x + 2)/(-1x + 3))) + 3) - 1)/(-3(2e^(-2((2x + 2)/(-1x + 3))) + 3) - 1)) - 5 = (-1(-1|(2x^2 + 1x) - 3| + 3) + 1)/(-3(-1|(2x^2 + 1x) - 3| + 3) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(-2|(1|(-2|x - 1| + 3) + 3| + 2) + 1| + 3) - 4 and g(x) = 3|((1(-3|x + 2|))/(-3(-3|x + 2|) + 4)) - 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1((1(1(-1x - 1)^2 + 1(-1x - 1) - 4) + 2)/(1(1(-1x - 1)^2 + 1(-1x - 1) - 4)))^2 - 4((1(1(-1x - 1)^2 + 1(-1x - 1) - 4) + 2)/(1(1(-1x - 1)^2 + 1(-1x - 1) - 4))) + 2 = 1sin(3\u03c0(1(1x - 3)^2 + 1(1x - 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": "30", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1((1(2|(-2|x| + 3)| - 1) + 1)/(-1(2|(-2|x| + 3)| - 1) + 2)) - 3)/(-2((1(2|(-2|x| + 3)| - 1) + 1)/(-1(2|(-2|x| + 3)| - 1) + 2)) + 3) = 1(-3cos(2\u03c0(-2x + 3) - 2) - 2)^2 + 1(-3cos(2\u03c0(-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": "152", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(3e^(1(2(-3|x + 3| - 2) + 5)) - 2) - 2)/(3(3e^(1(2(-3|x + 3| - 2) + 5)) - 2)) and g(x) = -1sin(1\u03c0(4(-2|x - 3|)^2 - 5) - 2) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "732", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(3\u03c0(-3(-1|(-2x^2 - 2) - 3| - 2) + 1) - 1) - 1 and g(x) = 5(-1sin(2\u03c0((1x)/(-2x + 3)) - 3) - 2)^2 + 5(-1sin(2\u03c0((1x)/(-2x + 3)) - 3) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "295", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(3sin(2\u03c0(2|((-2x + 2)/(-1x + 3)) - 3| - 3) + 1) + 2) + 5 and g(x) = -5(1(-3e^(1x) + 3) - 4) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-3|((2(-2sin(2\u03c0x - 3) - 1) - 3)/(-1(-2sin(2\u03c0x - 3) - 1) - 1)) - 3| + 1) + 1)/(3(-3|((2(-2sin(2\u03c0x - 3) - 1) - 3)/(-1(-2sin(2\u03c0x - 3) - 1) - 1)) - 3| + 1)) and g(x) = -1|(-4((3x + 3)/(-3x + 2)) + 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": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(1\u03c0(5(1|(-1|x - 3| - 3) + 1| - 3) - 4) + 2) - 1 and g(x) = -2|((-1((2x - 2)/(1x + 2)) + 3)/(1((2x - 2)/(1x + 2)) - 3)) + 2| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(-1((-2(-3e^(2x) + 1) - 1)/(3(-3e^(2x) + 1) + 1))^2 + 4((-2(-3e^(2x) + 1) - 1)/(3(-3e^(2x) + 1) + 1)) - 4) and g(x) = (1(-5(-1cos(2\u03c0x - 3)) - 2) - 1)/(1(-5(-1cos(2\u03c0x - 3)) - 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(-4((1(-3x + 3) + 2)/(-2(-3x + 3) - 1)) - 5)^2 + 1(-4((1(-3x + 3) + 2)/(-2(-3x + 3) - 1)) - 5) + 1 = 1|(2e^(1(4x^2 - 2x - 3)) + 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 2cos(3\u03c0(2|(-2(3|x - 2| + 3) - 1) - 2|) + 3) + 2 = 3sin(3\u03c0(-2|(-3x + 2) + 3| + 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": "360", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1((3(5(-1x - 1) - 1) + 1)/(1(5(-1x - 1) - 1))) - 2)/(-3((3(5(-1x - 1) - 1) + 1)/(1(5(-1x - 1) - 1)))) and g(x) = 4(3sin(3\u03c0(-1|x - 2| + 2) + 2) - 1) + 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 5((-1(-2sin(3\u03c0(-4x - 5) + 2) - 3) + 1)/(3(-2sin(3\u03c0(-4x - 5) + 2) - 3) + 3))^2 + 5 and g(x) = 1(-5(1sin(1\u03c0x - 3) + 3)^2 - 4(1sin(1\u03c0x - 3) + 3) - 1)^2 - 5(-5(1sin(1\u03c0x - 3) + 3)^2 - 4(1sin(1\u03c0x - 3) + 3) - 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "173", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(2sin(2\u03c0(-3|((-2x - 3)/(2x + 1))| - 3) + 1) + 2)^2 - 4(2sin(2\u03c0(-3|((-2x - 3)/(2x + 1))| - 3) + 1) + 2) + 2 and g(x) = 2sin(1\u03c0(-5(-4x + 2) + 1) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "237", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(-3cos(3\u03c0(-3|(-4x - 3)| - 3) + 2) + 2)^2 + 5 and g(x) = -2|(2|(-1cos(2\u03c0x - 2) - 1) - 3| - 1) + 1| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "719", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(3\u03c0(5(4((-2x - 1)/(3x + 3))^2 + 5((-2x - 1)/(3x + 3)) + 3)^2 - 1(4((-2x - 1)/(3x + 3))^2 + 5((-2x - 1)/(3x + 3)) + 3) + 3)) - 3 and g(x) = (-2(-2(-5x - 1)^2 - 1(-5x - 1) - 3) + 1)/(1(-2(-5x - 1)^2 - 1(-5x - 1) - 3) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "152", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-4(2|(1cos(3\u03c0x + 2) - 3) - 2| - 1)^2 - 3(2|(1cos(3\u03c0x + 2) - 3) - 2| - 1) - 1) + 3)/(-2(-4(2|(1cos(3\u03c0x + 2) - 3) - 2| - 1)^2 - 3(2|(1cos(3\u03c0x + 2) - 3) - 2| - 1) - 1) + 3) and g(x) = 1(3(-1|x + 3| + 3) + 2)^2 - 5(3(-1|x + 3| + 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(3cos(3\u03c0(1|(-2|x + 1| + 3) - 3| + 2) + 3) - 1) + 3 and g(x) = 1(-2e^(1((-1x - 3)/(-1x))) + 1) + 5 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(1|(-5((-1x)/(-2x - 2))^2 + 3((-1x)/(-2x - 2)) + 5) + 3| - 2) - 1)/(3(1|(-5((-1x)/(-2x - 2))^2 + 3((-1x)/(-2x - 2)) + 5) + 3| - 2) - 1) and g(x) = (-1((2(-2x^2 - 3x - 4) - 3)/(3(-2x^2 - 3x - 4) + 1)) - 1)/(1((2(-2x^2 - 3x - 4) - 3)/(3(-2x^2 - 3x - 4) + 1))) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(4(-3cos(3\u03c0(-5x^2 + 2x + 4) + 1) - 3) + 4) + 5 and g(x) = -4(1|(-2|x| - 3) - 1| - 1) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1280", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(2(2(-5x) - 3)^2 + 1) + 3)/(3(2(2(-5x) - 3)^2 + 1) - 2) and g(x) = 2sin(1\u03c0(-2(-1x + 3) + 4) - 1) - 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(2(1(5x^2 - 1x + 1)^2 - 5(5x^2 - 1x + 1) - 4) - 5)^2 + 4(2(1(5x^2 - 1x + 1)^2 - 5(5x^2 - 1x + 1) - 4) - 5) + 1 = (2((-2(1x - 3))/(-1(1x - 3) - 3)) - 2)/(-1((-2(1x - 3))/(-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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(1(-1|(2x + 3) + 1| - 3)^2 + 3(-1|(2x + 3) + 1| - 3) - 2) + 3| + 3 and g(x) = -3(-3sin(3\u03c0(-4x^2 - 2x + 5) - 3) - 3)^2 - 4(-3sin(3\u03c0(-4x^2 - 2x + 5) - 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": "209", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(-1(5(3e^(1x))^2 + 2(3e^(1x)) - 2)^2 + 5(5(3e^(1x))^2 + 2(3e^(1x)) - 2) + 5) - 3 and g(x) = 3|(2sin(1\u03c0(-2|x + 2| + 2) + 1) + 3) - 2| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1cos(1\u03c0(3|((1(-4x - 2) - 3)/(-1(-4x - 2) - 1)) + 3| + 2) - 2) and g(x) = -3((-3(2sin(2\u03c0x - 3) + 1) + 3)/(-3(2sin(2\u03c0x - 3) + 1) - 2))^2 - 3((-3(2sin(2\u03c0x - 3) + 1) + 3)/(-3(2sin(2\u03c0x - 3) + 1) - 2)) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "93", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(-1|(-3(3x^2 - 1x - 2)^2 + 1(3x^2 - 1x - 2) + 2) - 1| - 1) - 2| - 2 and g(x) = -1sin(1\u03c0((2(-2|x + 2| + 1) + 3)/(-1(-2|x + 2| + 1) + 3))) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(-1((-2(1|x + 1| + 1) - 3)/(1(1|x + 1| + 1)))^2 + 4)| + 1 and g(x) = (-1(2sin(1\u03c0(-4x^2 - 3x + 5) - 2) - 1) + 1)/(1(2sin(1\u03c0(-4x^2 - 3x + 5) - 2) - 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "379", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(3\u03c0(-3(1|(1|x - 2| - 1) - 2|) + 1) - 2) + 2 and g(x) = 1|(-3|(-3x^2 + 2x + 5) - 1| - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(-5(-3|(3x^2 - 1x - 5) + 3| + 3) - 4) + 2 and g(x) = (3((3((1x - 1)/(-2x - 3)))/(-2((1x - 1)/(-2x - 3)) + 1)) - 2)/(-2((3((1x - 1)/(-2x - 3)))/(-2((1x - 1)/(-2x - 3)) + 1)) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(4((-2(-1x^2 + 5x + 5) - 2)/(-3(-1x^2 + 5x + 5))) + 4) - 2)/(-1(4((-2(-1x^2 + 5x + 5) - 2)/(-3(-1x^2 + 5x + 5))) + 4) + 3) and g(x) = -5(-2(-5x - 5)^2 + 3)^2 + 5(-2(-5x - 5)^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) = (-3(2cos(3\u03c0(5(3x^2 - 4x + 1)^2 + 2(3x^2 - 4x + 1) - 4) - 3) - 3) - 2)/(-1(2cos(3\u03c0(5(3x^2 - 4x + 1)^2 + 2(3x^2 - 4x + 1) - 4) - 3) - 3) - 3) and g(x) = (1(1sin(1\u03c0((-2x)/(-1x - 1))) + 2) - 3)/(1(1sin(1\u03c0((-2x)/(-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": "260", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2((3(1(2sin(3\u03c0x + 1) - 3) + 4) + 1)/(3(1(2sin(3\u03c0x + 1) - 3) + 4) - 2))^2 + 2((3(1(2sin(3\u03c0x + 1) - 3) + 4) + 1)/(3(1(2sin(3\u03c0x + 1) - 3) + 4) - 2)) + 5 and g(x) = 1(2(2e^(1x) - 1)^2 - 1(2e^(1x) - 1) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "65", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-4(-1(-2cos(2\u03c0x - 2))^2 + 4(-2cos(2\u03c0x - 2)) - 1)^2 + 1(-1(-2cos(2\u03c0x - 2))^2 + 4(-2cos(2\u03c0x - 2)) - 1) - 3) + 2)/(3(-4(-1(-2cos(2\u03c0x - 2))^2 + 4(-2cos(2\u03c0x - 2)) - 1)^2 + 1(-1(-2cos(2\u03c0x - 2))^2 + 4(-2cos(2\u03c0x - 2)) - 1) - 3) + 2) and g(x) = 2|((1(-1x^2 - 5x + 3))/(-2(-1x^2 - 5x + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(-1sin(3\u03c0(2|(-1x^2 + 1x - 1) - 1| - 3))) - 5 and g(x) = -3cos(2\u03c0((-1(1|x| + 2) + 1)/(1(1|x| + 2) - 3)) - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(-1|(1|(-3|x - 1| - 3) + 2|)| - 3)| + 2 and g(x) = 3(5((3x + 3)/(2x + 3))^2 + 2((3x + 3)/(2x + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2cos(2\u03c0(-1(-1|(-2|x + 1| + 3) - 1|)^2 + 4(-1|(-2|x + 1| + 3) - 1|) - 4) - 2) - 1 and g(x) = 1e^(-1((2(-2x) + 3)/(-3(-2x) + 4))) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(-5((-2(-1e^(-2x) + 3) - 2)/(-2(-1e^(-2x) + 3) - 3))^2 - 1((-2(-1e^(-2x) + 3) - 2)/(-2(-1e^(-2x) + 3) - 3)) + 4) + 4 and g(x) = -5(3|(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": "7", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(-3(-3(-2sin(1\u03c0x - 3) + 2)^2 + 2(-2sin(1\u03c0x - 3) + 2) + 4) + 4) - 1| - 3 = 1cos(3\u03c0(-3(-3|x - 3| + 2)^2 - 3(-3|x - 3| + 2) + 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": "33", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2((3(3(2|x + 3| + 1)^2 - 5(2|x + 3| + 1) - 4) - 3)/(-3(3(2|x + 3| + 1)^2 - 5(2|x + 3| + 1) - 4)))^2 + 3((3(3(2|x + 3| + 1)^2 - 5(2|x + 3| + 1) - 4) - 3)/(-3(3(2|x + 3| + 1)^2 - 5(2|x + 3| + 1) - 4))) - 3 and g(x) = -1|(-1cos(3\u03c0((-2x)/(-1x + 3)) + 3) + 2) + 3| + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(-2|((-1(2x^2 + 4x - 4) - 1)/(3(2x^2 + 4x - 4) - 2)) - 2| - 2) - 1)/(1(-2|((-1(2x^2 + 4x - 4) - 1)/(3(2x^2 + 4x - 4) - 2)) - 2| - 2) + 2) and g(x) = -5(-1(-3|x + 1| + 1)^2 - 5(-3|x + 1| + 1) + 3) + 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2(4(2e^(-2(3x^2 - 4x - 5)) - 3) - 4)^2 + 4(4(2e^(-2(3x^2 - 4x - 5)) - 3) - 4) + 1 and g(x) = 1|(4(3|x - 3| - 1)^2 - 3(3|x - 3| - 1) - 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) = 1(3(4(1x^2 + 3x - 2)^2 - 2(1x^2 + 3x - 2) - 4)^2 + 1) and g(x) = 1((-3(-1|x| + 1) + 3)/(-2(-1|x| + 1) + 3))^2 + 3((-3(-1|x| + 1) + 3)/(-2(-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": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(5(2(3sin(2\u03c0x + 1) - 1) + 4)^2 - 4) + 1 = 1(-3cos(3\u03c0(-4x^2 + 5x - 1) + 1) - 3)^2 + 5(-3cos(3\u03c0(-4x^2 + 5x - 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": "90", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(4(-1|(-3x^2 - 4x + 1) + 2| + 1)^2 + 2(-1|(-3x^2 - 4x + 1) + 2| + 1) - 2) - 1| + 3 and g(x) = 5((2(2cos(2\u03c0x) - 1) - 2)/(3(2cos(2\u03c0x) - 1) - 1)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(1\u03c0(-1|(-5(4x^2 + 4x + 2) + 1) - 3| + 1) - 2) - 1 and g(x) = 3|(-1|(-1|x| - 2) + 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": "185", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3((1((-3(2|x + 2| - 2) + 1)/(2(2|x + 2| - 2) + 3)) + 2)/(1((-3(2|x + 2| - 2) + 1)/(2(2|x + 2| - 2) + 3)))) - 4 and g(x) = 1sin(3\u03c0(3((-1x + 1)/(-2x - 3)) + 4) + 2) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(1e^(1((3((-2x + 2)/(-1x - 3)))/(2((-2x + 2)/(-1x - 3)) + 3))) - 3) - 3| + 3 = 2|(3(2|x - 1| - 2)^2 - 3(2|x - 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 -5(3sin(1\u03c0(-1(2|x| - 3)^2 - 4(2|x| - 3) + 1) + 2))^2 + 2(3sin(1\u03c0(-1(2|x| - 3)^2 - 4(2|x| - 3) + 1) + 2)) + 5 = -5(2cos(3\u03c0(-3|x + 3| + 3) + 3) - 1)^2 - 3(2cos(3\u03c0(-3|x + 3| + 3) + 3) - 1) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "952", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|((3(-1(-2cos(1\u03c0x - 1) + 1)) + 1)/(-3(-1(-2cos(1\u03c0x - 1) + 1)) + 2)) - 3| - 1 = -1|(-1|(1cos(2\u03c0x + 2) - 3) + 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(-1(4(-2|x + 3| + 3) + 4)^2 + 2(4(-2|x + 3| + 3) + 4) + 5) - 4 and g(x) = 2(3(5x^2 + 4x - 3)^2 - 1(5x^2 + 4x - 3) - 3)^2 + 1(3(5x^2 + 4x - 3)^2 - 1(5x^2 + 4x - 3) - 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3sin(3\u03c0(-5(3|(-1x - 1) - 3|) + 3)) - 3 = -1((2(1cos(3\u03c0x - 2)) - 1)/(-2(1cos(3\u03c0x - 2)) - 1))^2 - 3((2(1cos(3\u03c0x - 2)) - 1)/(-2(1cos(3\u03c0x - 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": "120", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(5(-4(2sin(3\u03c0x + 3) - 2)^2 - 1(2sin(3\u03c0x + 3) - 2)))^2 - 1(5(-4(2sin(3\u03c0x + 3) - 2)^2 - 1(2sin(3\u03c0x + 3) - 2))) + 2 = 2(1sin(3\u03c0(-5x^2 - 1x + 3) + 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": "349", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(-1((-1(-2sin(1\u03c0x - 2)) + 1)/(3(-2sin(1\u03c0x - 2)) + 3))^2 + 3((-1(-2sin(1\u03c0x - 2)) + 1)/(3(-2sin(1\u03c0x - 2)) + 3)) - 2) - 1 and g(x) = (-2(-3sin(3\u03c0(-5x^2 + 1x + 5) + 3) + 1) - 3)/(2(-3sin(3\u03c0(-5x^2 + 1x + 5) + 3) + 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "276", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(1((2(1cos(2\u03c0x + 1) + 1) - 3)/(2(1cos(2\u03c0x + 1) + 1) + 2)) - 2) - 2)/(-1(1((2(1cos(2\u03c0x + 1) + 1) - 3)/(2(1cos(2\u03c0x + 1) + 1) + 2)) - 2) - 2) and g(x) = (3(2cos(3\u03c0(-2|x + 2| + 1) + 1) - 1) + 2)/(-3(2cos(3\u03c0(-2|x + 2| + 1) + 1) - 1) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "65", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3((2(1((3x + 2)/(-2x + 3)) + 2))/(1(1((3x + 2)/(-2x + 3)) + 2) + 3))^2 - 5((2(1((3x + 2)/(-2x + 3)) + 2))/(1(1((3x + 2)/(-2x + 3)) + 2) + 3)) - 2 = 1|(2sin(3\u03c0(-2x + 5) - 3) - 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": "36", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(2\u03c0(4(-3(1x^2 - 2x - 3) + 5) - 3) + 1) - 3 and g(x) = -2e^(-2(5(-2x^2 + 4x - 4)^2 - 5(-2x^2 + 4x - 4) + 1)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1236", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2sin(2\u03c0(1(3(1|x + 2| - 3) - 3)^2 - 2(3(1|x + 2| - 3) - 3) + 2) - 3) + 1 and g(x) = -4(-5((-2x - 2)/(-2x - 1)) + 3) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2sin(2\u03c0(2(-3((1x + 3)/(2x - 3)) - 1) + 4) + 3) + 2 and g(x) = -5(-3((-3x - 2)/(-3x))^2 + 4((-3x - 2)/(-3x)) - 1) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(-2|(2|(2x + 5) + 1| - 3) + 2| - 1) - 2)/(2(-2|(2|(2x + 5) + 1| - 3) + 2| - 1) + 3) = 2e^(2(-1|(2x - 3) + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1((-3(-3(-3|x - 2| + 2) + 4) + 1)/(-1(-3(-3|x - 2| + 2) + 4)))^2 + 5((-3(-3(-3|x - 2| + 2) + 4) + 1)/(-1(-3(-3|x - 2| + 2) + 4))) and g(x) = 3(4(-4x^2 + 5x - 1)^2 - 3(-4x^2 + 5x - 1) - 3)^2 - 2(4(-4x^2 + 5x - 1)^2 - 3(-4x^2 + 5x - 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": "Find the number of times the graphs of f(x) = 4((-1(2(-1|x| + 2) + 1) - 3)/(-2(2(-1|x| + 2) + 1) + 3)) - 1 and g(x) = -1cos(1\u03c0((1(-2|x + 2| + 3) - 3)/(-1(-2|x + 2| + 3) - 2))) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(-3((-2(3|x| + 2))/(-2(3|x| + 2) - 1))^2 + 4((-2(3|x| + 2))/(-2(3|x| + 2) - 1)) - 2) - 1| - 3 and g(x) = -1e^(1(2|(2|x + 3| - 3) - 3| - 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) = 2(-1|(4(-2cos(1\u03c0x - 3) + 3)^2 + 1(-2cos(1\u03c0x - 3) + 3)) - 1| + 2) - 5 and g(x) = -3|((2(-1x) + 2)/(1(-1x) + 1)) + 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": "Determine the number of intersection points between the functions f(x) = 5(2sin(2\u03c0(5(3x + 3)^2 + 2(3x + 3) - 5) - 3) - 2)^2 + 1(2sin(2\u03c0(5(3x + 3)^2 + 2(3x + 3) - 5) - 3) - 2) - 5 and g(x) = -1|(-5(-2|x + 3| + 3) - 5)| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "155", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2((-2(2|((3x)/(-1x + 2)) + 2| + 1) + 2)/(3(2|((3x)/(-1x + 2)) + 2| + 1) + 3)))/(3((-2(2|((3x)/(-1x + 2)) + 2| + 1) + 2)/(3(2|((3x)/(-1x + 2)) + 2| + 1) + 3)) + 1) and g(x) = 2cos(1\u03c0(2((-1x + 2)/(1x + 2)) - 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": "Find the number of times the graphs of f(x) = -3|(3(-4(-2|x + 2| + 1)^2 - 1(-2|x + 2| + 1) + 4) + 2) - 2| - 1 and g(x) = -2e^(-1((2(2|x + 1|) + 1)/(-2(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": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-1(-5((2x)/(-3x + 1)) + 2))^2 - 4(-1(-5((2x)/(-3x + 1)) + 2)) - 4 and g(x) = 2|(-1(-4x^2 - 5x + 2) + 4) - 1| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(4(-4(-3|x + 1| + 2)^2 - 4(-3|x + 1| + 2) + 1)^2 + 2(-4(-3|x + 1| + 2)^2 - 4(-3|x + 1| + 2) + 1) + 1)^2 - 5(4(-4(-3|x + 1| + 2)^2 - 4(-3|x + 1| + 2) + 1)^2 + 2(-4(-3|x + 1| + 2)^2 - 4(-3|x + 1| + 2) + 1) + 1) = 3|((1(-3x + 1) + 3)/(-2(-3x + 1) + 3)) + 3| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(-1cos(2\u03c0((2(3x - 1) - 1)/(2(3x - 1) + 1)) + 1) - 1) - 2)/(-2(-1cos(2\u03c0((2(3x - 1) - 1)/(2(3x - 1) + 1)) + 1) - 1) - 2) and g(x) = 3(-3|(1x + 5) + 2| - 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) = (-2(2sin(3\u03c0(-4(-1x^2 - 2x + 2) - 5) + 1) - 1) - 1)/(1(2sin(3\u03c0(-4(-1x^2 - 2x + 2) - 5) + 1) - 1)) and g(x) = 1|(1sin(2\u03c0((-1x - 2)/(2x - 1)) - 2) - 1)| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "305", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-2sin(3\u03c0((1((-3x - 2)/(-2x)) + 2)/(-2((-3x - 2)/(-2x)) - 2)) - 1)) - 3)/(-2(-2sin(3\u03c0((1((-3x - 2)/(-2x)) + 2)/(-2((-3x - 2)/(-2x)) - 2)) - 1))) and g(x) = -3(-1(2cos(3\u03c0x) + 2) - 3)^2 - 4(-1(2cos(3\u03c0x) + 2) - 3) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(-3|((-1(2cos(1\u03c0x) + 3) + 1)/(1(2cos(1\u03c0x) + 3) + 1)) - 1| + 1) - 3| + 2 and g(x) = 1((-2(-1sin(2\u03c0x + 2) - 1) + 2)/(-2(-1sin(2\u03c0x + 2) - 1) + 1))^2 + 2((-2(-1sin(2\u03c0x + 2) - 1) + 2)/(-2(-1sin(2\u03c0x + 2) - 1) + 1)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(-1cos(2\u03c0(-2(-4x^2 - 2x + 3) - 1) - 2) + 3) - 2)/(-2(-1cos(2\u03c0(-2(-4x^2 - 2x + 3) - 1) - 2) + 3) - 1) = -2(3(-4x)^2 - 3(-4x) - 3)^2 - 5(3(-4x)^2 - 3(-4x) - 3) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(1((-2(3sin(3\u03c0x + 1)) + 1)/(2(3sin(3\u03c0x + 1)) + 3)) - 1)^2 - 1(1((-2(3sin(3\u03c0x + 1)) + 1)/(2(3sin(3\u03c0x + 1)) + 3)) - 1) - 4 and g(x) = 3sin(2\u03c0(4((-1x + 1)/(-2x - 1))^2 + 5((-1x + 1)/(-2x - 1)) - 2) - 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "141", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(2(-3((1x + 1)/(-1x - 2))^2 - 1((1x + 1)/(-1x - 2)) + 4)^2 - 2) - 2 and g(x) = 1(-2(-3sin(3\u03c0x + 1) + 1) - 3)^2 - 5(-2(-3sin(3\u03c0x + 1) + 1) - 3) - 2 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) = -1sin(2\u03c0(-1((1(1|x - 2| + 1) + 3)/(-2(1|x - 2| + 1) - 3))^2 + 5((1(1|x - 2| + 1) + 3)/(-2(1|x - 2| + 1) - 3)) + 3) + 2) - 3 and g(x) = 1sin(1\u03c0((2((-1x + 3)/(-1x + 2)) + 2)/(2((-1x + 3)/(-1x + 2)) - 1)) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(3(1(-3|x - 2|) + 5)^2 + 2(1(-3|x - 2|) + 5) - 3) + 1 and g(x) = 1e^(-2(1|(-2|x - 3| + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(1cos(2\u03c0(-5(-1x^2 + 1x + 4)^2 - 5) + 2) + 2)^2 + 5 and g(x) = 2(3cos(1\u03c0(-1x - 3) + 3) - 1)^2 - 2(3cos(1\u03c0(-1x - 3) + 3) - 1) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "386", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(2\u03c0((1(3|(5x + 5) - 2| - 1) + 3)/(-3(3|(5x + 5) - 2| - 1) + 1)) + 2) + 1 and g(x) = (-1(-3(-5x + 1)^2 + 4(-5x + 1) - 3) + 3)/(1(-3(-5x + 1)^2 + 4(-5x + 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": "23", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(2\u03c0((2(3|(3x - 1)| + 3) + 2)/(1(3|(3x - 1)| + 3) + 2)) + 1) - 1 and g(x) = 1sin(1\u03c0(5((1x - 3)/(1x - 2))) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(1\u03c0(2(-5((-3x + 2)/(-1x - 1)) - 5) + 2) + 3) + 2 and g(x) = -2((1(-4x - 5) + 2)/(3(-4x - 5) + 2)) - 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": "Find the number of times the graphs of f(x) = -2(3cos(2\u03c0(5((-3x - 1)/(-1x - 2))^2 - 2((-3x - 1)/(-1x - 2)) + 4) + 3) - 1)^2 - 4(3cos(2\u03c0(5((-3x - 1)/(-1x - 2))^2 - 2((-3x - 1)/(-1x - 2)) + 4) + 3) - 1) + 2 and g(x) = (2(-3(-4x^2 - 3x + 5)^2 - 1(-4x^2 - 3x + 5) - 5) + 2)/(-3(-3(-4x^2 - 3x + 5)^2 - 1(-4x^2 - 3x + 5) - 5) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "605", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-5((-2((3x + 1)/(-3x + 1)) - 1)/(-3((3x + 1)/(-3x + 1)) + 2)) - 3) + 1)/(1(-5((-2((3x + 1)/(-3x + 1)) - 1)/(-3((3x + 1)/(-3x + 1)) + 2)) - 3) - 3) and g(x) = -1(3|((2x + 1)/(2x - 1)) + 2| - 2)^2 + 5(3|((2x + 1)/(2x - 1)) + 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": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(1cos(2\u03c0(3|((2x - 1)/(1x - 2)) - 1| + 2) - 3)) + 2| + 2 and g(x) = 4(4(-1|x| + 2) + 1)^2 - 2(4(-1|x| + 2) + 1) + 4 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(3cos(2\u03c0(-5(5x + 2) - 1) + 3) + 2) + 2)/(2(3cos(2\u03c0(-5(5x + 2) - 1) + 3) + 2) - 1) and g(x) = 1cos(2\u03c0(-1((1x - 1)/(1x - 3)) + 1) - 1) - 1 intersect in the interval [-10, 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(1|((2(-2|x + 2| + 2) - 2)/(-3(-2|x + 2| + 2) - 3)) + 3| + 1) + 2)/(-1(1|((2(-2|x + 2| + 2) - 2)/(-3(-2|x + 2| + 2) - 3)) + 3| + 1) + 2) = (-1(1cos(3\u03c0(-1|x + 2| + 2) + 2) - 2))/(2(1cos(3\u03c0(-1|x + 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": "30", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(3\u03c0(3(-2|(1x + 1) + 1| + 2)^2 - 4(-2|(1x + 1) + 1| + 2) - 1) - 1) and g(x) = 5((-1(3x^2 - 5x + 2) + 1)/(-2(3x^2 - 5x + 2) - 2))^2 + 5((-1(3x^2 - 5x + 2) + 1)/(-2(3x^2 - 5x + 2) - 2)) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1247", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2((3((-1(-2x - 2))/(-1(-2x - 2) + 3)) - 2)/(-1((-1(-2x - 2))/(-1(-2x - 2) + 3)) + 3)))/(-3((3((-1(-2x - 2))/(-1(-2x - 2) + 3)) - 2)/(-1((-1(-2x - 2))/(-1(-2x - 2) + 3)) + 3)) - 2) = (1(2cos(1\u03c0(-3|x| - 3) + 1) - 1) - 2)/(3(2cos(1\u03c0(-3|x| - 3) + 1) - 1) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(-4((-2(1x^2 - 3x + 1) + 3)/(2(1x^2 - 3x + 1) + 3)) + 3) + 4 and g(x) = (-3(-4(-3cos(1\u03c0x - 2)) + 4))/(2(-4(-3cos(1\u03c0x - 2)) + 4) - 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) = -2|(-3|(4(3|x| - 2) - 3) + 3| + 1) - 2| and g(x) = 5(-4(-1cos(2\u03c0x + 2) + 2)^2 - 5(-1cos(2\u03c0x + 2) + 2)) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(4(-1(3cos(1\u03c0x - 3) + 1) + 2)^2 + 5(-1(3cos(1\u03c0x - 3) + 1) + 2) - 1) - 1| - 2 and g(x) = -5(2|(2|x - 3| - 2) - 2| + 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(-1(-4(-1|x| + 2)^2 + 1(-1|x| + 2) + 1)^2 - 5(-4(-1|x| + 2)^2 + 1(-1|x| + 2) + 1) - 3) + 5 and g(x) = 1(-2sin(3\u03c0(-3x - 3) - 1) - 1) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|((2((-3((-3x - 2)/(-2x - 3)) + 2)/(-2((-3x - 2)/(-2x - 3)) + 3)))/(-1((-3((-3x - 2)/(-2x - 3)) + 2)/(-2((-3x - 2)/(-2x - 3)) + 3)) + 3)) + 2| + 1 and g(x) = -3sin(1\u03c0(-3|(-1|x + 2| - 2) - 1| - 1) - 1) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3((1(2(1x^2 + 3x + 3)^2 + 3(1x^2 + 3x + 3)) - 3)/(-2(2(1x^2 + 3x + 3)^2 + 3(1x^2 + 3x + 3)) + 1)) - 2)/(2((1(2(1x^2 + 3x + 3)^2 + 3(1x^2 + 3x + 3)) - 3)/(-2(2(1x^2 + 3x + 3)^2 + 3(1x^2 + 3x + 3)) + 1)) + 3) = -3|(2|(-1x^2 + 5x + 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": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(2e^(-1(-2|(5x - 5) - 1| + 2)) - 2)^2 - 5(2e^(-1(-2|(5x - 5) - 1| + 2)) - 2) - 5 = 3|(-3(2|x - 3| - 2) + 5) - 2| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2(1|(-1(-1x^2 + 2x + 5) + 1) - 3|) - 3 = -4((-2(-4x))/(-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": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 2sin(1\u03c0(-2((2(4x^2 - 3x - 1) - 3)/(1(4x^2 - 3x - 1) - 3))^2 + 5) + 2) - 3 = 4((-2(-3cos(2\u03c0x - 2)) - 1)/(-3(-3cos(2\u03c0x - 2)))) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3cos(1\u03c0(-3(-2|(2x - 1) + 2| - 3) + 5) - 3) + 2 and g(x) = (-1(-1(-1x^2 + 5x + 3)^2 - 3(-1x^2 + 5x + 3) + 5) + 3)/(-2(-1(-1x^2 + 5x + 3)^2 - 3(-1x^2 + 5x + 3) + 5) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "242", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(2cos(1\u03c0(-1(-5x + 4) + 3)) + 1)^2 - 4(2cos(1\u03c0(-1(-5x + 4) + 3)) + 1) + 4 and g(x) = 1|(-1(3e^(-2x) - 3)^2 + 1(3e^(-2x) - 3) - 4)| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2((-2(2(2x^2 - 3x - 1) + 1) - 1)/(3(2(2x^2 - 3x - 1) + 1) + 2))^2 - 4((-2(2(2x^2 - 3x - 1) + 1) - 1)/(3(2(2x^2 - 3x - 1) + 1) + 2)) - 2 and g(x) = -2((2(2x^2 - 1x + 2) + 3)/(2(2x^2 - 1x + 2) + 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": "Determine the number of intersection points between the functions f(x) = -1|(-4(1(2|x - 3| - 3) + 3)^2 - 3(1(2|x - 3| - 3) + 3) + 2)| - 3 and g(x) = -1cos(3\u03c0(1(2x^2 - 2x - 3) + 3) - 3) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(2cos(3\u03c0((-2(-1|x| + 1) + 2)/(3(-1|x| + 1) + 3)) + 2)) - 3 and g(x) = 1(-3cos(3\u03c0((-1x + 3)/(-2x + 1))) + 1) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(-2(-1cos(1\u03c0(1x^2 - 4x - 4) + 1) - 2) + 1) + 1| + 1 = (2(5(-2cos(1\u03c0x - 2) - 1) - 3) + 1)/(2(5(-2cos(1\u03c0x - 2) - 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": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1((2(-1|(-4x)| + 1) - 3)/(-2(-1|(-4x)| + 1) + 1)) + 2)/(-3((2(-1|(-4x)| + 1) - 3)/(-2(-1|(-4x)| + 1) + 1)) - 1) and g(x) = 2((3(1x^2 + 1x - 5) + 3)/(2(1x^2 + 1x - 5) - 3))^2 + 3((3(1x^2 + 1x - 5) + 3)/(2(1x^2 + 1x - 5) - 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 points of intersection exist between the functions f(x) = (-2(3cos(3\u03c0((-1(-3x^2 + 1x - 2) - 3)/(1(-3x^2 + 1x - 2))) - 2) - 1))/(1(3cos(3\u03c0((-1(-3x^2 + 1x - 2) - 3)/(1(-3x^2 + 1x - 2))) - 2) - 1) + 1) and g(x) = -2sin(2\u03c0((2(1x^2 + 2x + 3) + 1)/(-1(1x^2 + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|((2(-3|(3x - 1) + 3| - 3) + 1)/(-1(-3|(3x - 1) + 3| - 3) + 3)) - 1| + 2 and g(x) = 3sin(2\u03c0(-1|(1|x + 2| + 2) + 1| + 3) - 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(1(-3|(-3x - 3) - 1|)^2 + 5(-3|(-3x - 3) - 1|) - 1) - 1| and g(x) = -2e^(-2(-1(1x^2 + 4x - 4) - 2)) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(-2e^(2((-3(2x - 2) - 2)/(1(2x - 2)))) + 1)| + 3 and g(x) = (3(-3|(-3e^(-1x) + 2) + 1| - 2) - 2)/(1(-3|(-3e^(-1x) + 2) + 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) = (-2(-1(3|(3|x - 2| - 3) - 1| - 1) - 1) + 2)/(-2(-1(3|(3|x - 2| - 3) - 1| - 1) - 1) - 2) and g(x) = 1(3(-2|x + 1| + 2)^2 + 5(-2|x + 1| + 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(5((1(-2cos(2\u03c0x - 2) - 2) - 2)/(-1(-2cos(2\u03c0x - 2) - 2) - 3)) + 3) - 4 and g(x) = -2(-2|(-2e^(-2x) + 2) - 1| + 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": "Determine the number of intersection points between the functions f(x) = -1sin(3\u03c0(-4(-3(-4x - 5) + 2)^2 + 4(-3(-4x - 5) + 2) - 5) + 1) - 3 and g(x) = (-2(1(5x - 5)^2 - 3(5x - 5) - 2) - 3)/(1(1(5x - 5)^2 - 3(5x - 5) - 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "250", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3cos(2\u03c0((2(-1((2x + 2)/(-1x - 3))^2 + 1((2x + 2)/(-1x - 3))) + 2)/(-2(-1((2x + 2)/(-1x - 3))^2 + 1((2x + 2)/(-1x - 3))))) - 3) + 2 = -1(2((2x + 2)/(-1x + 3))^2 - 4) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "73", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2((-2(5(-1x^2 + 5x) + 5) + 1)/(1(5(-1x^2 + 5x) + 5) - 2))^2 - 5((-2(5(-1x^2 + 5x) + 5) + 1)/(1(5(-1x^2 + 5x) + 5) - 2)) - 1 and g(x) = (-2(1(4x + 3) + 3) - 1)/(-3(1(4x + 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": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|(2cos(1\u03c0(-1|(2x^2) - 2|) - 3) - 1) - 1| - 1 = -4(-1sin(1\u03c0((-1x)/(1x - 2)) - 2) - 3)^2 - 5(-1sin(1\u03c0((-1x)/(1x - 2)) - 2) - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1((-2(-1|((-2x)/(-1x - 2)) - 2| + 1) - 3)/(1(-1|((-2x)/(-1x - 2)) - 2| + 1)))^2 + 1((-2(-1|((-2x)/(-1x - 2)) - 2| + 1) - 3)/(1(-1|((-2x)/(-1x - 2)) - 2| + 1))) - 2 and g(x) = -2|(1((2x + 3)/(-3x - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(3(-4((-1x - 1)/(-3x - 1)) + 5) - 4) + 2)/(2(3(-4((-1x - 1)/(-3x - 1)) + 5) - 4) + 3) and g(x) = -2cos(2\u03c0(-1(1x - 3) - 1)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(5(5(1|x + 3| - 3) - 3)^2 + 1(5(1|x + 3| - 3) - 3) - 1) - 4 and g(x) = -3(1(-1cos(3\u03c0x + 1) - 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|((3(2sin(2\u03c0(-3x) + 1) + 2) - 1)/(-1(2sin(2\u03c0(-3x) + 1) + 2))) + 2| - 3 and g(x) = -2(-3|((-3x + 3)/(3x + 3)) - 3| + 2)^2 - 3(-3|((-3x + 3)/(3x + 3)) - 3| + 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "118", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-4((3(2x^2 + 3x - 4) - 1)/(2(2x^2 + 3x - 4) + 1))^2 + 1((3(2x^2 + 3x - 4) - 1)/(2(2x^2 + 3x - 4) + 1)) + 1) + 5 and g(x) = -3cos(1\u03c0(-2(-1x^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": "18", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-1(-2|(1|x + 2| - 3) - 1| + 3)^2 + 5(-2|(1|x + 2| - 3) - 1| + 3) + 3)^2 + 4(-1(-2|(1|x + 2| - 3) - 1| + 3)^2 + 5(-2|(1|x + 2| - 3) - 1| + 3) + 3) - 1 and g(x) = 4(-3|(2sin(1\u03c0x + 2) - 1)| + 1)^2 + 3(-3|(2sin(1\u03c0x + 2) - 1)| + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3|(-5(3(-1x + 3) - 2)^2 + 1(3(-1x + 3) - 2) + 3) + 2| + 1 = 2(1|(2x^2 - 5x - 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) = 1cos(3\u03c0((2(2|(-1|x| - 3) - 3| + 2) + 3)/(1(2|(-1|x| - 3) - 3| + 2) - 2)) + 1) + 3 and g(x) = -4(-1(-2|x + 2| - 1) - 4)^2 + 1(-1(-2|x + 2| - 1) - 4) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(-3((2(-2sin(2\u03c0x - 3) + 3) - 2)/(2(-2sin(2\u03c0x - 3) + 3) - 1))^2 - 1((2(-2sin(2\u03c0x - 3) + 3) - 2)/(2(-2sin(2\u03c0x - 3) + 3) - 1)) + 2)^2 + 1(-3((2(-2sin(2\u03c0x - 3) + 3) - 2)/(2(-2sin(2\u03c0x - 3) + 3) - 1))^2 - 1((2(-2sin(2\u03c0x - 3) + 3) - 2)/(2(-2sin(2\u03c0x - 3) + 3) - 1)) + 2) + 5 and g(x) = 2|(2((-2x + 3)/(-2x - 1))^2 - 4((-2x + 3)/(-2x - 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": "68", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(4(5(2x) + 4) + 2) + 1| + 1 and g(x) = 1(1|(2x)| + 2)^2 + 2(1|(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": "Find the number of times the graphs of f(x) = (-1(2cos(3\u03c0(-3(4x^2 - 1x - 3)^2 - 5(4x^2 - 1x - 3) + 5) - 1) + 1) - 3)/(-2(2cos(3\u03c0(-3(4x^2 - 1x - 3)^2 - 5(4x^2 - 1x - 3) + 5) - 1) + 1) + 3) and g(x) = -1|(1sin(3\u03c0(1x^2 + 4x) + 3) + 2) - 1| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "169", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(-2(-1|(-2|x + 3|) - 1| - 3) + 5) + 1| - 2 and g(x) = -3e^(2(3|(4x^2 - 1x - 4)| - 1)) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2cos(2\u03c0(-4((-2(-1|x + 1|))/(-1(-1|x + 1|) + 2)) - 2)) + 1 and g(x) = (3(-2e^(-1((-1x + 3)/(2x - 1))) - 1) - 3)/(1(-2e^(-1((-1x + 3)/(2x - 1))) - 1) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(3(-3((-1x + 1)/(3x + 1))^2 + 5((-1x + 1)/(3x + 1)) + 5)^2 - 1(-3((-1x + 1)/(3x + 1))^2 + 5((-1x + 1)/(3x + 1)) + 5) - 1)^2 + 2(3(-3((-1x + 1)/(3x + 1))^2 + 5((-1x + 1)/(3x + 1)) + 5)^2 - 1(-3((-1x + 1)/(3x + 1))^2 + 5((-1x + 1)/(3x + 1)) + 5) - 1) - 4 and g(x) = -3|(4(1cos(1\u03c0x - 2) + 1)^2 + 4(1cos(1\u03c0x - 2) + 1)) + 2| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(5(2sin(3\u03c0(-2x + 4)) + 1)) + 1)/(3(5(2sin(3\u03c0(-2x + 4)) + 1)) + 3) and g(x) = 1(-1((2x - 1)/(-1x - 2)) - 5)^2 + 3(-1((2x - 1)/(-1x - 2)) - 5) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4((3((1(-1|x + 1| + 2) + 2)/(-1(-1|x + 1| + 2) + 3)) + 3)/(3((1(-1|x + 1| + 2) + 2)/(-1(-1|x + 1| + 2) + 3)) + 1)) - 3 and g(x) = -2|(5(-2|x - 2| - 1)^2 + 5(-2|x - 2| - 1) - 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": "Determine the number of intersection points between the functions f(x) = 2|(2sin(2\u03c0(-3|(5x^2 - 4x + 2) - 1| + 3) - 1) + 3) + 2| - 2 and g(x) = 2(2|((2x - 2)/(-3x - 1))| + 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1101", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(-2(-3cos(1\u03c0(-2x + 5))) + 3) - 2)/(2(-2(-3cos(1\u03c0(-2x + 5))) + 3) + 3) = 2|((-2(1e^(-1x) - 1) + 3)/(-3(1e^(-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": "39", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(-2cos(3\u03c0(-5(-1|x + 1| + 3)^2 + 3(-1|x + 1| + 3) + 4) + 3))^2 - 2(-2cos(3\u03c0(-5(-1|x + 1| + 3)^2 + 3(-1|x + 1| + 3) + 4) + 3)) + 2 and g(x) = (-3(-3|(-5x^2 + 5x + 3)| + 3))/(-3(-3|(-5x^2 + 5x + 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": "863", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 3(1(5(5x + 5) + 5) + 3)^2 + 4(1(5(5x + 5) + 5) + 3) = -1sin(1\u03c0(-2(5x - 2)^2 - 5(5x - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(-3(-1(3cos(2\u03c0x + 2) - 3)^2 - 1(3cos(2\u03c0x + 2) - 3) + 1) - 1) + 3)/(3(-3(-1(3cos(2\u03c0x + 2) - 3)^2 - 1(3cos(2\u03c0x + 2) - 3) + 1) - 1) - 1) and g(x) = (3(5(-2x^2 - 4x + 1) + 1) + 3)/(-1(5(-2x^2 - 4x + 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": "27", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(-3|((-3(5x^2 + 5x + 1) + 1)/(1(5x^2 + 5x + 1) + 2)) + 2| - 1) + 1| + 3 and g(x) = -2sin(3\u03c0((1(5x + 5) - 2)/(3(5x + 5) + 3))) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(5((2(-3x - 2) - 1)/(1(-3x - 2) + 1))^2 + 1((2(-3x - 2) - 1)/(1(-3x - 2) + 1)) - 1)^2 - 1(5((2(-3x - 2) - 1)/(1(-3x - 2) + 1))^2 + 1((2(-3x - 2) - 1)/(1(-3x - 2) + 1)) - 1) - 5 and g(x) = -1(3e^(1(-2|x - 1| - 3)) - 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) = -2(-2(3|((2x + 1)/(1x + 1)) - 3| + 3) + 1)^2 + 5(-2(3|((2x + 1)/(1x + 1)) - 3| + 3) + 1) + 5 and g(x) = -4(5(-1cos(3\u03c0x)) + 3)^2 - 1(5(-1cos(3\u03c0x)) + 3) - 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": "Determine the number of intersection points between the functions f(x) = -2((2((1(-1x^2 + 1x + 2) + 3)/(3(-1x^2 + 1x + 2) - 2)) + 3)/(-2((1(-1x^2 + 1x + 2) + 3)/(3(-1x^2 + 1x + 2) - 2))))^2 - 2((2((1(-1x^2 + 1x + 2) + 3)/(3(-1x^2 + 1x + 2) - 2)) + 3)/(-2((1(-1x^2 + 1x + 2) + 3)/(3(-1x^2 + 1x + 2) - 2)))) - 1 and g(x) = -3|(5(-5x + 3) - 4)| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-2|(3|(-1|x - 1| - 1) + 1| - 2) - 3| + 1)^2 + 4(-2|(3|(-1|x - 1| - 1) + 1| - 2) - 3| + 1) + 5 and g(x) = -1|(1|((-2x + 3)/(2x - 1)) - 1| + 3) - 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 solutions does the equation 1e^(2(-2(4((2x)/(2x - 1)) + 1) - 4)) - 1 = (-1(-3e^(-2(-3x + 1)) - 3) + 2)/(1(-3e^(-2(-3x + 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": "34", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(3(3(2|x + 1| - 3) - 4) - 3) - 2| - 1 and g(x) = 2sin(3\u03c0(3(3x - 3)^2 + 3(3x - 3) - 3) - 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(-1|(1|(1|x + 1| - 3) - 2|) + 2| + 1) + 2| + 1 and g(x) = -5(-2((-3x)/(2x + 1)) - 3)^2 - 5(-2((-3x)/(2x + 1)) - 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(1(-2cos(1\u03c0(-2x^2 - 5x + 5) + 1) - 1)^2 - 5(-2cos(1\u03c0(-2x^2 - 5x + 5) + 1) - 1) - 1)^2 + 4(1(-2cos(1\u03c0(-2x^2 - 5x + 5) + 1) - 1)^2 - 5(-2cos(1\u03c0(-2x^2 - 5x + 5) + 1) - 1) - 1) - 2 and g(x) = -3|((-1(3|x - 3|) - 3)/(-1(3|x - 3|) + 3)) + 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "500", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(-3cos(1\u03c0(-3|(1x^2 + 2x - 5) - 2| - 1) - 1) + 2) - 1| + 1 and g(x) = 5(2|(2sin(2\u03c0x) - 2) - 2| - 3) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "176", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1|(-3((2(3|x + 3| + 2) - 2)/(3(3|x + 3| + 2) + 2))^2 + 3((2(3|x + 3| + 2) - 2)/(3(3|x + 3| + 2) + 2))) + 3| - 1 = -5(2sin(2\u03c0(1x^2 + 1x + 3) - 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": "376", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(-1e^(2(1(-3x^2 - 5x + 5) - 1)) + 1)^2 - 2(-1e^(2(1(-3x^2 - 5x + 5) - 1)) + 1) - 1 and g(x) = -1(1(3x)^2 + 4) + 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) = -4((-3(-2((2x)/(-2x - 1))^2 - 1((2x)/(-2x - 1))) - 2)/(-1(-2((2x)/(-2x - 1))^2 - 1((2x)/(-2x - 1))) + 2))^2 + 1((-3(-2((2x)/(-2x - 1))^2 - 1((2x)/(-2x - 1))) - 2)/(-1(-2((2x)/(-2x - 1))^2 - 1((2x)/(-2x - 1))) + 2)) + 5 and g(x) = -1|(-3|(5x^2 + 5x + 1) - 1| + 3)| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(1sin(1\u03c0(1((-3x + 3)/(-1x + 2)) + 5) + 3) - 1) - 2| + 2 and g(x) = 3(-1(1x^2 - 2x - 5) - 3) - 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": "Find the number of times the graphs of f(x) = 5(3cos(2\u03c0(-2(2x - 2)^2 - 2(2x - 2) + 3)) + 1) - 2 and g(x) = (-1((2(-2|x - 2| + 2) - 3)/(3(-2|x - 2| + 2) - 2)) + 2)/(-3((2(-2|x - 2| + 2) - 3)/(3(-2|x - 2| + 2) - 2)) + 4) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1646", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(3\u03c0(5((2(-3|x - 1| + 1) - 1)/(-1(-3|x - 1| + 1) - 2))) + 3) - 2 and g(x) = 1(5(-3|x| + 2) - 4)^2 - 2(5(-3|x| + 2) - 4) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(1(-1|(-1x - 4) + 1| + 1)^2 + 4(-1|(-1x - 4) + 1| + 1) + 2)^2 + 3(1(-1|(-1x - 4) + 1| + 1)^2 + 4(-1|(-1x - 4) + 1| + 1) + 2) and g(x) = -2((-3(2sin(1\u03c0x) - 2) + 2)/(-1(2sin(1\u03c0x) - 2) + 2))^2 + 3((-3(2sin(1\u03c0x) - 2) + 2)/(-1(2sin(1\u03c0x) - 2) + 2)) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(-1cos(2\u03c0((2((-1x + 3)/(1x - 2)) - 3)/(2((-1x + 3)/(1x - 2)) - 1))) + 2))/(3(-1cos(2\u03c0((2((-1x + 3)/(1x - 2)) - 3)/(2((-1x + 3)/(1x - 2)) - 1))) + 2)) and g(x) = 1(-3sin(3\u03c0(-2|x + 3| + 3) - 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(3|(-3|(2cos(1\u03c0x) - 3)| + 2) + 3| + 3)^2 + 3(3|(-3|(2cos(1\u03c0x) - 3)| + 2) + 3| + 3) and g(x) = 2(-3|(-4x^2 + 2x + 3) - 1| + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(2cos(3\u03c0(-1|(3x + 2) + 3| - 2) + 1)) + 3 and g(x) = (-3(1sin(1\u03c0(-2x^2 - 4x + 4) - 1) + 1) - 2)/(2(1sin(1\u03c0(-2x^2 - 4x + 4) - 1) + 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "253", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-5(-2(-1e^(-1x) + 3)^2 + 3(-1e^(-1x) + 3) + 2)^2) - 1 and g(x) = 3|((-1(1cos(2\u03c0x - 2) - 2) - 3)/(-2(1cos(2\u03c0x - 2) - 2))) - 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": "Determine the number of intersection points between the functions f(x) = 3(-3|(-3sin(3\u03c0(-1x^2 + 5x - 4)) + 3)| + 3) + 2 and g(x) = 1(-5((2x - 2)/(2x))^2 - 5((2x - 2)/(2x)) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "624", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((1(2(-2|x| - 2)^2 - 2(-2|x| - 2) - 3) - 2)/(-1(2(-2|x| - 2)^2 - 2(-2|x| - 2) - 3) - 1))^2 - 1((1(2(-2|x| - 2)^2 - 2(-2|x| - 2) - 3) - 2)/(-1(2(-2|x| - 2)^2 - 2(-2|x| - 2) - 3) - 1)) + 4 and g(x) = -2(-1((-2x - 2)/(1x - 3))^2 + 5((-2x - 2)/(1x - 3)) + 5)^2 + 1(-1((-2x - 2)/(1x - 3))^2 + 5((-2x - 2)/(1x - 3)) + 5) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2sin(3\u03c0(-1|(-1|(2|x - 3| + 1) + 2| + 3) - 2| - 3)) - 3 = (-3(-1(-3|x| + 2) - 5) - 2)/(3(-1(-3|x| + 2) - 5)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "92", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-4(-5((-2x - 1)/(-1x)) + 1)^2 - 5(-5((-2x - 1)/(-1x)) + 1) + 5)^2 - 5(-4(-5((-2x - 1)/(-1x)) + 1)^2 - 5(-5((-2x - 1)/(-1x)) + 1) + 5) + 4 = 2cos(3\u03c0(-2|(-2x + 2) - 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": "How many points of intersection exist between the functions f(x) = -3((-2(4(3|x + 3| - 3)^2 - 2(3|x + 3| - 3) - 3) - 3)/(1(4(3|x + 3| - 3)^2 - 2(3|x + 3| - 3) - 3)))^2 - 5((-2(4(3|x + 3| - 3)^2 - 2(3|x + 3| - 3) - 3) - 3)/(1(4(3|x + 3| - 3)^2 - 2(3|x + 3| - 3) - 3))) + 1 and g(x) = (2(1(4x^2 + 3x - 2)^2 - 4(4x^2 + 3x - 2) + 4) - 1)/(2(1(4x^2 + 3x - 2)^2 - 4(4x^2 + 3x - 2) + 4) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((-3(-1|(1x^2 + 3x - 2)| + 1) - 2)/(-2(-1|(1x^2 + 3x - 2)| + 1) - 3))^2 - 3((-3(-1|(1x^2 + 3x - 2)| + 1) - 2)/(-2(-1|(1x^2 + 3x - 2)| + 1) - 3)) and g(x) = -2cos(1\u03c0(1|(-4x^2 + 5x - 4) - 2| - 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) = -4(2sin(2\u03c0(2|((-2x - 1)/(2x - 1)) - 3| + 3) - 1) - 2)^2 - 5(2sin(2\u03c0(2|((-2x - 1)/(2x - 1)) - 3| + 3) - 1) - 2) - 4 and g(x) = -5(2|(-1cos(1\u03c0x + 3) - 3) + 1| + 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": "How many points of intersection exist between the functions f(x) = (1((1(1cos(1\u03c0(-5x^2 + 4x) + 3) - 1) + 2)/(-2(1cos(1\u03c0(-5x^2 + 4x) + 3) - 1) - 1)) - 3)/(3((1(1cos(1\u03c0(-5x^2 + 4x) + 3) - 1) + 2)/(-2(1cos(1\u03c0(-5x^2 + 4x) + 3) - 1) - 1)) - 2) and g(x) = 1|(-1(1x - 3) + 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": "210", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(1sin(1\u03c0(-1(-2|x - 2| + 2)^2 + 3(-2|x - 2| + 2) + 1) + 3) + 1) + 1| + 1 and g(x) = -4(3(-5x^2 + 2x + 5) + 1)^2 + 1(3(-5x^2 + 2x + 5) + 1) + 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(5((-3(-2x - 4) - 2)/(3(-2x - 4) + 3)) + 5) + 3| + 3 and g(x) = 4(3|((2x + 2)/(1x + 2)) - 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(-5(2|(-3x - 2) - 3| - 2)^2 + 4(2|(-3x - 2) - 3| - 2) - 5)^2 - 2(-5(2|(-3x - 2) - 3| - 2)^2 + 4(2|(-3x - 2) - 3| - 2) - 5) - 5 = -1|(3e^(2(-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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(2(1|((-2x - 3)/(-2x + 3)) - 2| - 1) - 4)^2 - 5 and g(x) = 1cos(1\u03c0((-3(2|x - 2| - 3))/(-1(2|x - 2| - 3) + 2)) + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(5(-3(1|x + 1|) + 2)^2 + 1(-3(1|x + 1|) + 2) - 5) - 3)/(2(5(-3(1|x + 1|) + 2)^2 + 1(-3(1|x + 1|) + 2) - 5) - 1) and g(x) = -1|(1cos(3\u03c0(2x^2 - 4x + 1) + 3) + 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 (2(2e^(1((-3(-1|x| - 3) - 1)/(-2(-1|x| - 3) + 1))) + 2) - 1)/(-2(2e^(1((-3(-1|x| - 3) - 1)/(-2(-1|x| - 3) + 1))) + 2) - 2) = 3|(-3sin(3\u03c0(-3|x| + 1) - 3)) - 1| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "360", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-3(1|(4(2sin(2\u03c0x - 2) - 2)^2 + 4(2sin(2\u03c0x - 2) - 2) + 4) + 2| + 1) + 1)/(1(1|(4(2sin(2\u03c0x - 2) - 2)^2 + 4(2sin(2\u03c0x - 2) - 2) + 4) + 2| + 1) + 1) = -2|((3(-3cos(3\u03c0x + 3) + 2) + 2)/(-1(-3cos(3\u03c0x + 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": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1sin(2\u03c0(4(-2|((2x - 3)/(2x - 3)) - 3|) - 2)) - 2 and g(x) = -3|(2|(-2sin(2\u03c0x + 3) - 3)| - 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": "80", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(-2cos(2\u03c0((2(2|x - 2| - 2) - 1)/(-3(2|x - 2| - 2) - 3)) - 2) + 3)^2 - 3(-2cos(2\u03c0((2(2|x - 2| - 2) - 1)/(-3(2|x - 2| - 2) - 3)) - 2) + 3) - 5 and g(x) = -2sin(2\u03c0(-3(2x^2 - 2x + 3)^2 - 3(2x^2 - 2x + 3) + 5) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "395", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(-3|(-1|((-2x + 1)/(1x + 1))| + 1) + 3| + 1) - 3)/(-1(-3|(-1|((-2x + 1)/(1x + 1))| + 1) + 3| + 1)) and g(x) = -1sin(3\u03c0(3(1x^2 - 5x - 5)^2 + 3(1x^2 - 5x - 5) + 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": "138", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(4(5((-3x)/(3x + 3)) - 1) - 2) + 3)/(-3(4(5((-3x)/(3x + 3)) - 1) - 2)) = (-2(-5(2cos(3\u03c0x) - 2) + 2))/(-2(-5(2cos(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": "57", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(2|(-2cos(2\u03c0(-3x + 3) + 3)) + 1|) + 3)/(-1(2|(-2cos(2\u03c0(-3x + 3) + 3)) + 1|) + 2) and g(x) = -1(-2(3cos(2\u03c0x)) - 1)^2 + 2(-2(3cos(2\u03c0x)) - 1) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1sin(1\u03c0(2|(-2(5x^2 + 1x + 3)^2 - 3) - 2| + 2) - 2) and g(x) = -3(2e^(-1((3x - 3)/(-1x))) - 1) - 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": "How many solutions does the equation -5(3cos(1\u03c0((1(2x + 2) + 1)/(-2(2x + 2) - 3)) - 2) + 2) - 2 = -3cos(1\u03c0((-2(-3|x - 3|) + 2)/(-3(-3|x - 3|) + 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": "10", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(3((3(2|x - 3| - 3) - 3)/(-3(2|x - 3| - 3) + 1)))^2 - 5 and g(x) = -2|(2|(3|x + 1| + 2) + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -5(2|(-3|(-2sin(2\u03c0x + 2) + 1) + 1|)| - 3)^2 = -3e^(1(1|((1x - 2)/(-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": "42", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(3\u03c0(-5(-5(-2|x + 1|) + 1) - 3) + 1) + 2 and g(x) = 1(1(-3sin(2\u03c0x) - 3) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "727", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2e^(1((2(2(5x + 4) + 4) - 1)/(3(2(5x + 4) + 4) + 2))) + 2 = -1|(1(-2sin(1\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": "19", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-5(1cos(2\u03c0(5x) - 2) - 1)^2 - 5(1cos(2\u03c0(5x) - 2) - 1) - 3)^2 + 5(-5(1cos(2\u03c0(5x) - 2) - 1)^2 - 5(1cos(2\u03c0(5x) - 2) - 1) - 3) + 2 and g(x) = 3cos(3\u03c0((1(-2|x - 3|) - 2)/(2(-2|x - 3|) + 1)) - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "219", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(3cos(1\u03c0((-2(3x) + 2)/(-3(3x) - 3)) + 3) + 1)^2 - 5(3cos(1\u03c0((-2(3x) + 2)/(-3(3x) - 3)) + 3) + 1) + 2 and g(x) = 1|(2(1cos(2\u03c0x + 2) + 3) - 4) + 1| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 4(-3sin(1\u03c0(4(-5x + 4) + 4) + 1) + 3) + 3 = -5(-4(-3e^(-1x) + 2) - 4) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(-4(4((1x - 2)/(-2x - 1))^2 - 4)^2 - 5(4((1x - 2)/(-2x - 1))^2 - 4) + 4)| + 2 and g(x) = 4(-4(-1|x - 1| - 3) + 1) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(2(-1(-2cos(3\u03c0x) - 2)^2 + 5)^2 + 1(-1(-2cos(3\u03c0x) - 2)^2 + 5) - 2) and g(x) = -3((-1((-3x - 3)/(3x - 2)) - 1)/(2((-3x - 3)/(3x - 2)) + 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "119", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(3(-3|(-1x^2 - 1x - 2) + 2| - 2) - 4) - 4 and g(x) = -5(2cos(3\u03c0(-2x - 2) - 2) + 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": "Find the number of times the graphs of f(x) = 2|(1(-1((-2x + 3)/(-2x)) - 3) + 5)| - 2 and g(x) = 4(2sin(1\u03c0(3|x + 2| - 1)) + 2)^2 + 5(2sin(1\u03c0(3|x + 2| - 1)) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(-2|((-1(1cos(3\u03c0x - 1) + 2) + 1)/(-2(1cos(3\u03c0x - 1) + 2) + 3)) + 2| + 3) + 2 and g(x) = -4(1(1sin(1\u03c0x + 2) - 3)^2 + 2(1sin(1\u03c0x + 2) - 3) - 1)^2 + 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": "How many solutions does the equation 5(-4((1(-1x^2 - 5x + 5))/(2(-1x^2 - 5x + 5) - 1))^2 + 5) = -3(5(1cos(2\u03c0x + 1))^2 - 2(1cos(2\u03c0x + 1)) + 4) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(3(3(-1x - 5) - 5)^2 + 1(3(-1x - 5) - 5) - 3) + 2)/(1(3(3(-1x - 5) - 5)^2 + 1(3(-1x - 5) - 5) - 3) + 3) and g(x) = 1|(-2|(2cos(2\u03c0x - 3) - 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": "40", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(3|((-2(-1|x + 3| + 3))/(-2(-1|x + 3| + 3) + 3)) - 1| - 3)^2 - 3(3|((-2(-1|x + 3| + 3))/(-2(-1|x + 3| + 3) + 3)) - 1| - 3) - 3 = -3(2|(1sin(3\u03c0x) - 2) - 1| - 3)^2 + 4(2|(1sin(3\u03c0x) - 2) - 1| - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-5(2(-1x^2 + 1x) - 5)^2 + 1) + 1)/(2(-5(2(-1x^2 + 1x) - 5)^2 + 1) - 1) and g(x) = -4(2|(1|x - 2| - 3) - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4((-3(5(1|x - 2| - 1) + 1) - 2)/(-2(5(1|x - 2| - 1) + 1) - 1)) - 5 and g(x) = -3((2(2sin(3\u03c0x + 3) + 1) - 2)/(-3(2sin(3\u03c0x + 3) + 1) + 4)) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(3sin(1\u03c0(3(3|x + 2| - 3) + 2) - 2) - 2)^2 - 3(3sin(1\u03c0(3(3|x + 2| - 3) + 2) - 2) - 2) - 1 and g(x) = -5((1(1x + 5) + 2)/(2(1x + 5) + 2))^2 + 1((1(1x + 5) + 2)/(2(1x + 5) + 2)) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "348", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-4(-2cos(2\u03c0(-3|x - 3| - 1)) - 1)^2 + 5(-2cos(2\u03c0(-3|x - 3| - 1)) - 1) + 1)^2 + 1(-4(-2cos(2\u03c0(-3|x - 3| - 1)) - 1)^2 + 5(-2cos(2\u03c0(-3|x - 3| - 1)) - 1) + 1) - 1 and g(x) = -1e^(-2(4(1x^2 - 5x + 1) + 1)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "128", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(2\u03c0((3(4(-3|x - 3| + 2)^2 + 5(-3|x - 3| + 2) - 3))/(2(4(-3|x - 3| + 2)^2 + 5(-3|x - 3| + 2) - 3) + 1)) - 3) + 1 and g(x) = (-3(1|(-2|x - 3| + 1) + 1| - 1) - 2)/(-3(1|(-2|x - 3| + 1) + 1| - 1) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-5(-3(4x^2 - 1x - 4) - 2)^2 + 5(-3(4x^2 - 1x - 4) - 2) + 2) + 2 and g(x) = -3((1(2x^2 + 2x - 5))/(-1(2x^2 + 2x - 5) - 2))^2 + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-1(3(-3sin(1\u03c0x - 1) - 1)^2 - 1(-3sin(1\u03c0x - 1) - 1) - 5)^2 - 2(3(-3sin(1\u03c0x - 1) - 1)^2 - 1(-3sin(1\u03c0x - 1) - 1) - 5) + 3)^2 - 5(-1(3(-3sin(1\u03c0x - 1) - 1)^2 - 1(-3sin(1\u03c0x - 1) - 1) - 5)^2 - 2(3(-3sin(1\u03c0x - 1) - 1)^2 - 1(-3sin(1\u03c0x - 1) - 1) - 5) + 3) + 3 and g(x) = (-2(-2(-1x) + 4))/(-2(-2(-1x) + 4) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "139", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-1((-1(3e^(1x) + 1) - 1)/(1(3e^(1x) + 1) + 2)) + 1) + 2)/(1(-1((-1(3e^(1x) + 1) - 1)/(1(3e^(1x) + 1) + 2)) + 1) + 3) and g(x) = 1((-3(1sin(3\u03c0x + 1) - 1))/(-2(1sin(3\u03c0x + 1) - 1) - 2))^2 + 4((-3(1sin(3\u03c0x + 1) - 1))/(-2(1sin(3\u03c0x + 1) - 1) - 2)) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|((-1(-2cos(3\u03c0(-2|x + 2|) - 1) + 1) + 1)/(1(-2cos(3\u03c0(-2|x + 2|) - 1) + 1) - 2))| + 1 and g(x) = (-2((-2((1x + 3)/(2x)) - 1)/(-3((1x + 3)/(2x)) - 1)) + 1)/(2((-2((1x + 3)/(2x)) - 1)/(-3((1x + 3)/(2x)) - 1)) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "236", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(-2cos(2\u03c0(1(-3x^2 + 2x + 1))) - 2) + 2)/(-1(-2cos(2\u03c0(1(-3x^2 + 2x + 1))) - 2) - 3) and g(x) = -3|(-4(-3|x - 2| - 2)^2 - 1(-3|x - 2| - 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": "17", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(2cos(2\u03c0(3(-1|x| - 1) + 4) - 1) - 2) - 3 and g(x) = 1|(2|((-2x - 3)/(3x + 1)) + 3| - 1) - 1| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "118", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(5((3((2x + 2)/(-2x - 1)))/(-3((2x + 2)/(-2x - 1)) - 1))^2 - 3((3((2x + 2)/(-2x - 1)))/(-3((2x + 2)/(-2x - 1)) - 1))) + 4 and g(x) = 4(-3|(1|x + 3| - 2) - 3|)^2 + 2(-3|(1|x + 3| - 2) - 3|) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(5((-2((-2x + 3)/(3x - 1)) - 3)/(1((-2x + 3)/(3x - 1)) - 3)) + 3) + 2)/(-1(5((-2((-2x + 3)/(3x - 1)) - 3)/(1((-2x + 3)/(3x - 1)) - 3)) + 3) - 1) and g(x) = 1|(2|(2|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": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(2|(-5(3x - 3)^2 - 4(3x - 3) + 3)|) - 3)/(-2(2|(-5(3x - 3)^2 - 4(3x - 3) + 3)|) + 3) and g(x) = 4(-1(2|x + 3| - 3)^2 + 5(2|x + 3| - 3) - 5) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(1cos(3\u03c0(2((1x)/(-2x + 3)) - 5) + 2) - 2)^2 + 4 and g(x) = -2cos(3\u03c0(3(1|x + 1| - 1) - 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": "104", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(1\u03c0(-1|(-1|(-4x)|)| - 1)) - 2 and g(x) = -1(3sin(3\u03c0((1x)/(-3x - 1))) - 3)^2 - 3(3sin(3\u03c0((1x)/(-3x - 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": "26", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(4(-1sin(1\u03c0(-1|x + 1|) + 3) - 2) + 5)^2 + 2(4(-1sin(1\u03c0(-1|x + 1|) + 3) - 2) + 5) + 5 and g(x) = 4(-2(3cos(3\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": "18", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(1(-5(2|x - 3| - 3)^2 + 1(2|x - 3| - 3) + 3)) - 2)/(-2(1(-5(2|x - 3| - 3)^2 + 1(2|x - 3| - 3) + 3)) - 2) = -2cos(3\u03c0((-2(5x^2 - 1x - 3) - 1)/(-2(5x^2 - 1x - 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": "Find the number of times the graphs of f(x) = -5(-3sin(3\u03c0(3|(-2x^2 + 4x - 5) - 1| + 2) + 1))^2 + 5(-3sin(3\u03c0(3|(-2x^2 + 4x - 5) - 1| + 2) + 1)) - 1 and g(x) = (1((-1((1x - 1)/(-1x + 3)) - 3)/(1((1x - 1)/(-1x + 3)) + 1)) + 1)/(-1((-1((1x - 1)/(-1x + 3)) - 3)/(1((1x - 1)/(-1x + 3)) + 1)) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "600", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(3|(1sin(3\u03c0(-1x - 2))) - 1| - 2)^2 - 3 = -2e^(2(-1|(-1x + 2) + 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": "100", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(-3((-2(2sin(1\u03c0x + 2) - 2) + 2)/(3(2sin(1\u03c0x + 2) - 2) + 2)) + 3) - 1| + 1 and g(x) = -5(5((2x + 2)/(-2x - 3))^2 - 3) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(2e^(1(3|(3|x - 3|) - 1| + 2))) + 2)/(1(2e^(1(3|(3|x - 3|) - 1| + 2))) - 2) and g(x) = -3((-1(-2sin(3\u03c0x + 2)))/(2(-2sin(3\u03c0x + 2)) - 1)) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(-2(3|(2cos(1\u03c0x + 2)) + 1| + 2)^2 + 2(3|(2cos(1\u03c0x + 2)) + 1| + 2) - 5) + 1| - 3 and g(x) = -4(1sin(2\u03c0(-2x^2 + 1x - 3) - 1) - 3)^2 + 1(1sin(2\u03c0(-2x^2 + 1x - 3) - 1) - 3) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "140", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(-2(2(4x - 5)^2 + 4(4x - 5) - 3) - 3) + 1| - 2 and g(x) = 3cos(3\u03c0(1(-2|x + 2| - 3)^2 + 1(-2|x + 2| - 3) - 3) - 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3((-1(-1|(-2sin(1\u03c0x - 2) - 3)| + 2) + 3)/(-2(-1|(-2sin(1\u03c0x - 2) - 3)| + 2) - 3)) - 5 and g(x) = (-3(2|(2x^2 + 4x + 1) + 1| + 1) + 2)/(-2(2|(2x^2 + 4x + 1) + 1| + 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (-2(1|(-2|(4x^2 - 5) - 2|)| + 2) + 2)/(-1(1|(-2|(4x^2 - 5) - 2|)| + 2) + 3) = 1cos(2\u03c0(-2|(-1|x - 2| - 3) - 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": "80", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|((2((3(2sin(3\u03c0x) - 3) - 3)/(-2(2sin(3\u03c0x) - 3) + 3)) + 1)/(-3((3(2sin(3\u03c0x) - 3) - 3)/(-2(2sin(3\u03c0x) - 3) + 3)) + 1)) + 2| - 1 and g(x) = (-2(-3sin(1\u03c0(1|x| - 3) - 3) + 1) - 3)/(-3(-3sin(1\u03c0(1|x| - 3) - 3) + 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|((3(-2(1x^2 + 4x - 3)^2 - 3(1x^2 + 4x - 3) + 1) + 3)/(3(-2(1x^2 + 4x - 3)^2 - 3(1x^2 + 4x - 3) + 1))) + 1| - 2 and g(x) = (2((3(1|x + 3| + 3))/(-1(1|x + 3| + 3) - 2)))/(-1((3(1|x + 3| + 3))/(-1(1|x + 3| + 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": "5", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(-1(-3sin(1\u03c0(4x^2 - 5x - 2) - 1))^2 - 1(-3sin(1\u03c0(4x^2 - 5x - 2) - 1)) - 2) + 3| + 2 = 3|(-2cos(1\u03c0((-2x + 1)/(-1x + 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": "368", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|((-2(2cos(3\u03c0(3x + 1) - 3)) + 3)/(2(2cos(3\u03c0(3x + 1) - 3)) - 1)) + 2| - 3 and g(x) = 3|(1e^(-1(-3|x| + 1)) + 3) + 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 solutions does the equation -1|(-1|((1(2x^2 - 3x - 2))/(-1(2x^2 - 3x - 2) - 2)) - 3|) + 2| + 2 = (1(1(1|x - 3| + 1) - 1) - 3)/(1(1(1|x - 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 solutions does the equation -5(-2(2cos(2\u03c0(1x + 2) - 2) - 3)^2 + 5(2cos(2\u03c0(1x + 2) - 2) - 3) + 1)^2 + 1(-2(2cos(2\u03c0(1x + 2) - 2) - 3)^2 + 5(2cos(2\u03c0(1x + 2) - 2) - 3) + 1) - 3 = -3(-3(-1x^2 + 2x + 5) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1e^(-1((3(-3(-4x^2 - 5x + 3) - 3) + 3)/(1(-3(-4x^2 - 5x + 3) - 3) - 3))) - 2 and g(x) = 2(-2(1cos(3\u03c0x + 2) + 1)^2 - 2(1cos(3\u03c0x + 2) + 1) + 1) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5((1(-1((2x - 3)/(2x)) - 4) - 3)/(-1(-1((2x - 3)/(2x)) - 4)))^2 - 1((1(-1((2x - 3)/(2x)) - 4) - 3)/(-1(-1((2x - 3)/(2x)) - 4))) + 5 and g(x) = -5(3sin(1\u03c0(1x - 2) - 2) + 1)^2 + 5(3sin(1\u03c0(1x - 2) - 2) + 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(1((1(-2sin(3\u03c0x + 3) - 3))/(-1(-2sin(3\u03c0x + 3) - 3) + 2)) + 1) - 3)/(1(1((1(-2sin(3\u03c0x + 3) - 3))/(-1(-2sin(3\u03c0x + 3) - 3) + 2)) + 1)) and g(x) = -2|((3((3x)/(-2x - 2)) + 1)/(3((3x)/(-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": "46", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(2sin(2\u03c0(1|(3x^2 - 1x - 2) + 3| - 3) - 2) + 2) + 4 and g(x) = 4(-2cos(1\u03c0(3|x - 3| - 2) + 1) + 2)^2 + 2(-2cos(1\u03c0(3|x - 3| - 2) + 1) + 2) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "442", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1sin(1\u03c0(1|(5(5x - 5)^2 + 4(5x - 5) - 4) + 3|) - 3) - 1 = (-3(2(-1x) + 5) - 3)/(1(2(-1x) + 5) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "196", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -3|(5(2|((-2x - 3)/(-3x + 2)) + 1| - 1)) - 2| - 3 = -3((1((-3x + 2)/(-1x - 2)) + 1)/(3((-3x + 2)/(-1x - 2)) - 1))^2 - 2((1((-3x + 2)/(-1x - 2)) + 1)/(3((-3x + 2)/(-1x - 2)) - 1)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2((-2(5(-1x^2 + 4x - 3) - 2) + 2)/(3(5(-1x^2 + 4x - 3) - 2) - 2)) + 2)/(-1((-2(5(-1x^2 + 4x - 3) - 2) + 2)/(3(5(-1x^2 + 4x - 3) - 2) - 2)) - 2) and g(x) = 2(-2sin(1\u03c0(-2x^2 - 5x - 2) - 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": "404", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1sin(1\u03c0(-2((-1(3x + 2) - 1)/(-2(3x + 2))) - 5) - 1) + 1 = (-1(-5(1x^2 - 2x - 5)^2 + 4(1x^2 - 2x - 5) + 2) + 3)/(2(-5(1x^2 - 2x - 5)^2 + 4(1x^2 - 2x - 5) + 2) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(1|(-4(2cos(2\u03c0x + 1) + 1) - 2) + 3| - 1)^2 - 5(1|(-4(2cos(2\u03c0x + 1) + 1) - 2) + 3| - 1) - 2 and g(x) = 3|((3(1|x + 1| - 3) - 1)/(-2(1|x + 1| - 3) - 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": "54", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2sin(1\u03c0(1(-1(1x^2 - 1x + 3))^2 - 1(-1(1x^2 - 1x + 3)) + 3) + 3) + 3 = 3((2(-3sin(1\u03c0x + 1) - 1) - 3)/(2(-3sin(1\u03c0x + 1) - 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": "441", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(3(-3|(-3sin(1\u03c0x + 2) + 3)| + 1)^2 - 5(-3|(-3sin(1\u03c0x + 2) + 3)| + 1) + 1) + 3| - 2 and g(x) = 2|(-1|(-2|x - 2| + 2) + 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": "How many solutions does the equation -3cos(1\u03c0(-3|(3|(5x + 4) - 1| - 2) + 1| - 3) - 1) + 2 = (1((2(3cos(3\u03c0x + 3) + 3) + 3)/(-2(3cos(3\u03c0x + 3) + 3) + 3)) - 2)/(-1((2(3cos(3\u03c0x + 3) + 3) + 3)/(-2(3cos(3\u03c0x + 3) + 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": "423", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(2cos(1\u03c0(2|(-1|x + 3|)| + 1) + 3) + 2) + 5 and g(x) = -2cos(3\u03c0(-1|(-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": "40", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(1(4(2|x + 3| - 2) + 2)^2 + 2(4(2|x + 3| - 2) + 2) - 5))/(-1(1(4(2|x + 3| - 2) + 2)^2 + 2(4(2|x + 3| - 2) + 2) - 5) - 1) and g(x) = -2e^(-1(2(2|x - 3| - 1) + 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3((-1(-2|((-1x - 2)/(1x)) + 1|) - 3)/(3(-2|((-1x - 2)/(1x)) + 1|) + 1))^2 + 5((-1(-2|((-1x - 2)/(1x)) + 1|) - 3)/(3(-2|((-1x - 2)/(1x)) + 1|) + 1)) and g(x) = -3((3(-1cos(2\u03c0x - 3) - 2) + 1)/(1(-1cos(2\u03c0x - 3) - 2) + 3)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (3(-5(-2e^(2(-1|x| - 2)) + 3) + 1) - 3)/(2(-5(-2e^(2(-1|x| - 2)) + 3) + 1) + 1) = -1cos(3\u03c0(-2(-5x^2 - 5x + 1)^2 - 4(-5x^2 - 5x + 1) - 5) + 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "802", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(-3cos(3\u03c0(1((-3x + 2)/(-1x + 2))^2 + 3((-3x + 2)/(-1x + 2)) + 2)) + 1) + 2 and g(x) = 4(1sin(1\u03c0(-4x - 5) - 3) - 3) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "170", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-1|((1(1|x + 1| - 3) + 1)/(3(1|x + 1| - 3) + 1))| + 1) + 1)/(1(-1|((1(1|x + 1| - 3) + 1)/(3(1|x + 1| - 3) + 1))| + 1) + 3) and g(x) = (-2(-4(-2e^(1x) + 2)^2 - 5(-2e^(1x) + 2) + 4) - 2)/(-2(-4(-2e^(1x) + 2)^2 - 5(-2e^(1x) + 2) + 4) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2((3(3(-1e^(1x) - 1) + 4) + 3)/(-2(3(-1e^(1x) - 1) + 4))) + 5 and g(x) = -5(2cos(2\u03c0(1|x - 2| + 3) - 1) - 3)^2 - 2(2cos(2\u03c0(1|x - 2| + 3) - 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": "17", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(-2|(3((-1x - 3)/(2x)) - 1) - 3| + 3) - 3)/(-3(-2|(3((-1x - 3)/(2x)) - 1) - 3| + 3) + 4) and g(x) = -1|(2|(2sin(3\u03c0x + 1) - 3)| + 1) - 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(-3(1(4x^2 - 2x + 5)^2 - 5(4x^2 - 2x + 5) - 1)^2 + 4) - 5 and g(x) = (3((1(-3|x - 1| - 2) + 3)/(-1(-3|x - 1| - 2) + 2)) + 1)/(1((1(-3|x - 1| - 2) + 3)/(-1(-3|x - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -1(-3sin(3\u03c0((-2(1|x - 3| - 1) - 3)/(2(1|x - 3| - 1) - 3))) - 2) - 4 = (3(3|(3sin(2\u03c0x - 3) + 2) + 1| - 1) + 3)/(-3(3|(3sin(2\u03c0x - 3) + 2) + 1| - 1) + 4) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "212", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 1|(2e^(1((1(1x^2 - 1x + 5) + 2)/(-3(1x^2 - 1x + 5) + 4))) - 3) - 2| - 2 = 3cos(3\u03c0(-5(1x - 4)^2 - 3(1x - 4) - 4) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1445", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(5((-3(2|x + 1|))/(2(2|x + 1|) + 3)) - 5)^2 + 1(5((-3(2|x + 1|))/(2(2|x + 1|) + 3)) - 5) - 5 and g(x) = 3|(1|(-1x^2 + 2x - 2) - 3| + 3) - 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 solutions does the equation (-1(-1|(-4(-2|x - 1| - 2)^2 - 4(-2|x - 1| - 2) + 3) + 3| + 3) + 3)/(3(-1|(-4(-2|x - 1| - 2)^2 - 4(-2|x - 1| - 2) + 3) + 3| + 3) - 2) = 1(3cos(3\u03c0(-2x^2 - 5x + 4) + 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": "1077", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-2(2(3e^(1x) - 3) - 5) - 1) - 3 and g(x) = (-3(-4(-2cos(2\u03c0x - 1) - 2)^2 + 1(-2cos(2\u03c0x - 1) - 2) + 4) + 1)/(-1(-4(-2cos(2\u03c0x - 1) - 2)^2 + 1(-2cos(2\u03c0x - 1) - 2) + 4) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1cos(2\u03c0(2((1((2x + 1)/(-1x - 2)) - 1)/(-2((2x + 1)/(-1x - 2)) - 3))^2 - 4((1((2x + 1)/(-1x - 2)) - 1)/(-2((2x + 1)/(-1x - 2)) - 3)) + 3)) - 3 and g(x) = -2((-3(-3sin(3\u03c0x - 1) + 1) + 2)/(-2(-3sin(3\u03c0x - 1) + 1) - 2)) - 4 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) = (3((-2(4((3x + 3)/(2x)) + 3) - 2)/(-2(4((3x + 3)/(2x)) + 3) + 1)) + 1)/(-3((-2(4((3x + 3)/(2x)) + 3) - 2)/(-2(4((3x + 3)/(2x)) + 3) + 1)) + 2) and g(x) = 2(3sin(3\u03c0(4x - 4) - 3))^2 - 5(3sin(3\u03c0(4x - 4) - 3)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "472", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(1|(-1(-3sin(1\u03c0x + 3) + 1)^2 + 2(-3sin(1\u03c0x + 3) + 1) - 3) - 1| - 2)^2 - 2(1|(-1(-3sin(1\u03c0x + 3) + 1)^2 + 2(-3sin(1\u03c0x + 3) + 1) - 3) - 1| - 2) - 2 and g(x) = 1|((-3(1|x + 1| - 3) - 2)/(3(1|x + 1| - 3) + 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": "40", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(-4((3(-3sin(2\u03c0x + 1) - 1) - 2)/(-1(-3sin(2\u03c0x + 1) - 1) + 2)) - 4)^2 + 2(-4((3(-3sin(2\u03c0x + 1) - 1) - 2)/(-1(-3sin(2\u03c0x + 1) - 1) + 2)) - 4) - 1 and g(x) = 2sin(1\u03c0(3(3x)^2 - 2(3x) - 4) - 2) + 1 where -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 points of intersection exist between the functions f(x) = (2(2sin(2\u03c0((1((2x + 1)/(2x - 1)) - 2)/(-3((2x + 1)/(2x - 1)) + 2)) + 2) - 1) + 1)/(-1(2sin(2\u03c0((1((2x + 1)/(2x - 1)) - 2)/(-3((2x + 1)/(2x - 1)) + 2)) + 2) - 1) + 2) and g(x) = -1(3|(-1x - 3) + 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": "44", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(2(1cos(1\u03c0((-1x + 1)/(1x)) + 3) - 3)^2 - 3(1cos(1\u03c0((-1x + 1)/(1x)) + 3) - 3) - 3) + 3| - 2 and g(x) = -2|((3(3x^2 - 4x) - 2)/(-2(3x^2 - 4x) - 1)) - 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(2(-3(-3x^2 - 2x + 3) - 5))^2 - 4(2(-3(-3x^2 - 2x + 3) - 5)) - 2 and g(x) = -4(1(-1|x| + 1) + 3) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2((-1(-3(-2x)^2 + 5(-2x) - 4))/(-3(-3(-2x)^2 + 5(-2x) - 4) - 2))^2 + 5((-1(-3(-2x)^2 + 5(-2x) - 4))/(-3(-3(-2x)^2 + 5(-2x) - 4) - 2)) + 5 and g(x) = -1(-4(2cos(3\u03c0x - 2))^2 + 3(2cos(3\u03c0x - 2))) - 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 solutions does the equation -1((1((3(3sin(3\u03c0x) + 3) + 1)/(2(3sin(3\u03c0x) + 3) - 3)))/(1((3(3sin(3\u03c0x) + 3) + 1)/(2(3sin(3\u03c0x) + 3) - 3)) + 2))^2 - 4((1((3(3sin(3\u03c0x) + 3) + 1)/(2(3sin(3\u03c0x) + 3) - 3)))/(1((3(3sin(3\u03c0x) + 3) + 1)/(2(3sin(3\u03c0x) + 3) - 3)) + 2)) + 2 = (2((1(4x^2 - 2x + 1) - 2)/(1(4x^2 - 2x + 1))) + 2)/(1((1(4x^2 - 2x + 1) - 2)/(1(4x^2 - 2x + 1))) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(1e^(-1(-1(4x^2 - 3x - 2) - 1))) - 3| - 3 and g(x) = 3cos(3\u03c0(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": "14", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(-2|(-3|(-1|x - 1| + 3) + 3| + 2) - 2| - 1) - 1 and g(x) = (1(1(3cos(3\u03c0x + 2) + 2)^2 + 5(3cos(3\u03c0x + 2) + 2) + 2) - 3)/(-1(1(3cos(3\u03c0x + 2) + 2)^2 + 5(3cos(3\u03c0x + 2) + 2) + 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": "How many points of intersection exist between the functions f(x) = -3|(1(2|(-3x - 2)| - 1)^2 - 1(2|(-3x - 2)| - 1) - 1) + 1| + 1 and g(x) = -2sin(2\u03c0((1(2|x - 3| - 1) - 3)/(-2(2|x - 3| - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-2sin(2\u03c0(1|((1x)/(3x - 2)) - 3|) + 3) + 3)^2 - 5(-2sin(2\u03c0(1|((1x)/(3x - 2)) - 3|) + 3) + 3) + 1 and g(x) = -2(-3sin(1\u03c0((2x - 2)/(2x + 3)) - 2) - 2)^2 + 5(-3sin(1\u03c0((2x - 2)/(2x + 3)) - 2) - 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(-2|((2((1x + 2)/(-1x - 1)) + 1)/(3((1x + 2)/(-1x - 1)) - 1)) - 1|) + 4 and g(x) = -5(1|((3x)/(1x - 2)) - 3| - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-3e^(-2(5(-3x^2 + 5x - 1)^2 + 4(-3x^2 + 5x - 1))) - 1) + 2)/(2(-3e^(-2(5(-3x^2 + 5x - 1)^2 + 4(-3x^2 + 5x - 1))) - 1) + 2) and g(x) = (1((2(-1sin(3\u03c0x) - 3) - 1)/(2(-1sin(3\u03c0x) - 3) + 3)) + 1)/(-2((2(-1sin(3\u03c0x) - 3) - 1)/(2(-1sin(3\u03c0x) - 3) + 3)) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|((-2(1|(-1cos(1\u03c0x + 2) - 1) + 1| - 1) + 2)/(1(1|(-1cos(1\u03c0x + 2) - 1) + 1| - 1) - 2)) - 1| - 3 and g(x) = 2((3(-3sin(1\u03c0x)) + 2)/(1(-3sin(1\u03c0x)) - 3)) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1cos(1\u03c0((3((-2((-3x + 2)/(2x + 1)) - 2)/(2((-3x + 2)/(2x + 1)))) + 1)/(-2((-2((-3x + 2)/(2x + 1)) - 2)/(2((-3x + 2)/(2x + 1)))) + 1)) - 1) and g(x) = 1cos(2\u03c0(3|(5x + 3)| + 3) + 2) + 1 where -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) = 2|(-1cos(1\u03c0(1(-4x^2 - 3x - 3) + 4) - 1) - 2) + 1| - 1 and g(x) = 1|((-1(3sin(1\u03c0x + 2) + 2) + 3)/(-3(3sin(1\u03c0x + 2) + 2) + 1)) - 3| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "711", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(2sin(3\u03c0(-1|(-1|x + 3| + 1) + 3| - 2) + 1) - 2) + 3)/(-1(2sin(3\u03c0(-1|(-1|x + 3| + 1) + 3| - 2) + 1) - 2) + 3) and g(x) = 4(-3cos(1\u03c0(-4x + 5) + 2) - 3)^2 - 5(-3cos(1\u03c0(-4x + 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": "80", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(-1(1(5x^2 + 3x - 4) - 1) + 5) + 1)/(-3(-1(1(5x^2 + 3x - 4) - 1) + 5) - 3) = -2|(2((-1x - 1)/(-3x + 1))^2 - 3((-1x - 1)/(-3x + 1)) + 1) + 1| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(-3((2(2x + 2) + 2)/(-3(2x + 2) + 1)) - 5) - 1| + 3 and g(x) = -2(1((2x - 1)/(-1x + 3))^2 + 4((2x - 1)/(-1x + 3)) - 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1((-2((-1(-1x^2 + 4x - 2) - 3)/(-3(-1x^2 + 4x - 2) - 2)))/(-1((-1(-1x^2 + 4x - 2) - 3)/(-3(-1x^2 + 4x - 2) - 2)) + 3))^2 - 3((-2((-1(-1x^2 + 4x - 2) - 3)/(-3(-1x^2 + 4x - 2) - 2)))/(-1((-1(-1x^2 + 4x - 2) - 3)/(-3(-1x^2 + 4x - 2) - 2)) + 3)) + 5 and g(x) = 3|(-5((1x)/(-1x + 2)) + 1) - 3| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-5(-1(-1x^2 - 2x + 2)^2 - 4(-1x^2 - 2x + 2) + 1)) - 5 and g(x) = 3(2((2x - 3)/(-2x - 2)) - 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": "Determine the number of intersection points between the functions f(x) = (-1(3((2(-4x + 5))/(-3(-4x + 5) - 3))^2 - 5((2(-4x + 5))/(-3(-4x + 5) - 3)) - 1) - 2)/(-1(3((2(-4x + 5))/(-3(-4x + 5) - 3))^2 - 5((2(-4x + 5))/(-3(-4x + 5) - 3)) - 1) + 2) and g(x) = 2(2cos(1\u03c0(-5x - 5) + 3) + 2) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-2(3sin(3\u03c0(-3|x + 3| + 2)) + 2)^2 - 1(3sin(3\u03c0(-3|x + 3| + 2)) + 2) - 4) + 2 and g(x) = (-2(3(2|x + 3| - 2)^2 - 4(2|x + 3| - 2) - 2) - 3)/(1(3(2|x + 3| - 2)^2 - 4(2|x + 3| - 2) - 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(-1e^(2((-1(1x^2 - 2x))/(1(1x^2 - 2x) + 1))) + 2) + 2)/(1(-1e^(2((-1(1x^2 - 2x))/(1(1x^2 - 2x) + 1))) + 2) + 3) and g(x) = 3|((-1(1x^2 - 3x - 5) + 2)/(2(1x^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": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -2cos(1\u03c0(2(-4(3|x - 3| + 3) + 3)^2 - 3(-4(3|x - 3| + 3) + 3) + 4) - 1) - 1 and g(x) = 5(-2e^(1(5x - 5)))^2 + 4(-2e^(1(5x - 5))) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "873", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(-3|(-2|(2x - 3) + 1|) + 3| + 1)^2 - 2(-3|(-2|(2x - 3) + 1|) + 3| + 1) + 1 and g(x) = 1(3(-1cos(3\u03c0x) + 1)^2 - 3(-1cos(3\u03c0x) + 1) - 2) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|((2(-2((1x - 3)/(-3x - 2))^2 + 1) - 1)/(3(-2((1x - 3)/(-3x - 2))^2 + 1) - 2)) - 3| - 3 and g(x) = -1|(-4(-1x - 3)^2 - 5(-1x - 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 points of intersection exist between the functions f(x) = -5(-1|((-3(1x^2 - 4))/(-3(1x^2 - 4) - 1)) - 3| + 2) - 2 and g(x) = -4(-2(1x + 2) - 4) - 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-4((-3(-3x^2 + 2x + 4) + 2)/(-2(-3x^2 + 2x + 4))))^2 - 5(-4((-3(-3x^2 + 2x + 4) + 2)/(-2(-3x^2 + 2x + 4)))) + 4 and g(x) = -3|((-2(-5x^2 - 2x + 4) - 1)/(1(-5x^2 - 2x + 4) - 3)) + 3| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(3((-1(3x - 5) + 1)/(-1(3x - 5) - 1))^2 - 4((-1(3x - 5) + 1)/(-1(3x - 5) - 1)) + 2) - 3 and g(x) = 3|((2(-4x - 4) - 1)/(2(-4x - 4) + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(-3cos(3\u03c0(1|(3|x + 2|) + 2| - 1) - 3))^2 + 3(-3cos(3\u03c0(1|(3|x + 2|) + 2| - 1) - 3)) and g(x) = -1cos(1\u03c0(2|(5x^2 - 2x) - 3| + 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": "371", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|((3(1(-1x^2) - 2))/(1(1(-1x^2) - 2) + 3)) + 3| + 1 and g(x) = -2|(-5(-4x^2 - 2x - 5) + 1) + 1| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2(1(-3sin(3\u03c0(5x^2 - 2x + 1) + 3) - 3)^2 + 4(-3sin(3\u03c0(5x^2 - 2x + 1) + 3) - 3) + 2) - 1)/(-2(1(-3sin(3\u03c0(5x^2 - 2x + 1) + 3) - 3)^2 + 4(-3sin(3\u03c0(5x^2 - 2x + 1) + 3) - 3) + 2) - 2) = 2((-3((-3x)/(1x + 2)) + 2)/(2((-3x)/(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": "469", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1((1(4(-5x^2 + 1x + 3)^2 - 3(-5x^2 + 1x + 3) - 4) - 1)/(-2(4(-5x^2 + 1x + 3)^2 - 3(-5x^2 + 1x + 3) - 4)))^2 + 4((1(4(-5x^2 + 1x + 3)^2 - 3(-5x^2 + 1x + 3) - 4) - 1)/(-2(4(-5x^2 + 1x + 3)^2 - 3(-5x^2 + 1x + 3) - 4))) + 5 and g(x) = 3((-1(-1|x - 1| + 1) + 2)/(1(-1|x - 1| + 1) + 2)) + 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) = 3(2((-1(1|x - 2|) - 2)/(3(1|x - 2|) - 2))^2 + 2((-1(1|x - 2|) - 2)/(3(1|x - 2|) - 2)) - 1) + 1 and g(x) = 5((-3(-2sin(2\u03c0x + 3) + 1) - 3)/(3(-2sin(2\u03c0x + 3) + 1) + 2))^2 + 5((-3(-2sin(2\u03c0x + 3) + 1) - 3)/(3(-2sin(2\u03c0x + 3) + 1) + 2)) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(2(4(1cos(1\u03c0x + 3) + 2)^2 + 4(1cos(1\u03c0x + 3) + 2) + 1) - 3) - 2)/(3(2(4(1cos(1\u03c0x + 3) + 2)^2 + 4(1cos(1\u03c0x + 3) + 2) + 1) - 3) + 3) and g(x) = -2|(1|((1x + 2)/(3x)) + 3| - 1) - 1| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(-2|(1cos(3\u03c0((-3x - 1)/(-2x - 3)) - 1) - 2) + 3| + 1) - 3| + 3 and g(x) = (1(2|(2sin(1\u03c0x - 3) - 1) - 1| + 2) - 1)/(1(2|(2sin(1\u03c0x - 3) - 1) - 1| + 2) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(5(-2|(-3|x| + 1) + 2| + 3) - 2) - 4 and g(x) = 2|(2|(1|x + 2|) + 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": "Determine the number of intersection points between the functions f(x) = 3e^(-1(3|(2((1x + 3)/(2x)) - 5) + 1|)) - 3 and g(x) = 5(-5(3x^2 - 5x - 5)^2 - 4(3x^2 - 5x - 5) + 3) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|(3e^(2(4(-4x^2 - 4x + 4)^2 - 5(-4x^2 - 4x + 4) + 1)) + 1) - 2| + 2 = 2e^(1(-2|(-3x - 2)| + 2)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (2((2(3|(-2|x| + 2) - 2| + 1) - 2)/(-2(3|(-2|x| + 2) - 2| + 1))) - 1)/(-1((2(3|(-2|x| + 2) - 2| + 1) - 2)/(-2(3|(-2|x| + 2) - 2| + 1))) + 2) = -4(1(-1x^2 + 5x - 4)^2 + 5(-1x^2 + 5x - 4) - 5) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(1((-2(-2cos(1\u03c0x) + 3) + 2)/(-1(-2cos(1\u03c0x) + 3) + 2))^2 + 5((-2(-2cos(1\u03c0x) + 3) + 2)/(-1(-2cos(1\u03c0x) + 3) + 2)) + 4)^2 - 4(1((-2(-2cos(1\u03c0x) + 3) + 2)/(-1(-2cos(1\u03c0x) + 3) + 2))^2 + 5((-2(-2cos(1\u03c0x) + 3) + 2)/(-1(-2cos(1\u03c0x) + 3) + 2)) + 4) - 4 and g(x) = 3(-2|((-1x + 1)/(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": "13", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(-1|((-3(-1x) - 3)/(-1(-1x) - 2))| + 3) + 1)/(2(-1|((-3(-1x) - 3)/(-1(-1x) - 2))| + 3)) and g(x) = 5(-3sin(1\u03c0(-3|x - 3| - 2) - 2) - 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2((-1(-3(3x^2 + 1x - 2)^2 - 1(3x^2 + 1x - 2) + 4) + 3)/(3(-3(3x^2 + 1x - 2)^2 - 1(3x^2 + 1x - 2) + 4) + 1)) + 3 and g(x) = 2cos(2\u03c0(3(-3x + 4)^2 - 1(-3x + 4) + 5) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3((1(3cos(2\u03c0(2x - 3) - 3) - 3) + 1)/(-3(3cos(2\u03c0(2x - 3) - 3) - 3) - 2)) + 3 and g(x) = 1|(-3(2x + 5) + 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": "How many solutions does the equation (-2((-3(3(-3x + 5)^2 + 2(-3x + 5) - 5) - 2)/(-3(3(-3x + 5)^2 + 2(-3x + 5) - 5) + 4)))/(3((-3(3(-3x + 5)^2 + 2(-3x + 5) - 5) - 2)/(-3(3(-3x + 5)^2 + 2(-3x + 5) - 5) + 4)) + 2) = (2(-2sin(3\u03c0(-2x - 3) + 2) + 1))/(3(-2sin(3\u03c0(-2x - 3) + 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": "120", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2sin(1\u03c0((3(1(-3|x + 3| + 1) + 5) - 3)/(-1(1(-3|x + 3| + 1) + 5) - 1)) - 3) - 3 and g(x) = 2cos(3\u03c0(4((3x)/(-1x + 2)) - 1)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "280", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(1|(3(-2sin(2\u03c0x + 2) + 2) - 4) - 3| + 3) + 3 and g(x) = 1(-1((2x)/(-2x - 3)) + 3)^2 + 3(-1((2x)/(-2x - 3)) + 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "66", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(3|(-3cos(1\u03c0(-3|x + 3| + 3) - 1))| - 2) - 3)/(2(3|(-3cos(1\u03c0(-3|x + 3| + 3) - 1))| - 2) + 3) and g(x) = -2(4(1sin(2\u03c0x - 1)) + 5)^2 + 4(4(1sin(2\u03c0x - 1)) + 5) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-1sin(2\u03c0((2(3|x + 1|) + 1)/(1(3|x + 1|) + 1)) - 3) + 3) - 5 and g(x) = (-2(3cos(1\u03c0(4x^2 + 2x + 5)) + 2) + 2)/(-3(3cos(1\u03c0(4x^2 + 2x + 5)) + 2) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "134", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(4(1(-2x - 1) + 2)^2 + 2(1(-2x - 1) + 2) - 3)^2 + 5(4(1(-2x - 1) + 2)^2 + 2(1(-2x - 1) + 2) - 3) - 5 and g(x) = 2cos(3\u03c0(-4((1x - 1)/(3x)) + 1) + 1) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(1((-2(-2sin(2\u03c0x - 1)) - 3)/(-1(-2sin(2\u03c0x - 1)) - 2))^2 + 5((-2(-2sin(2\u03c0x - 1)) - 3)/(-1(-2sin(2\u03c0x - 1)) - 2)) - 5) - 1 and g(x) = 3|((1((-1x + 1)/(1x - 2)))/(2((-1x + 1)/(1x - 2)) + 1)) + 1| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "78", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(-3(-1(-1|x + 1| - 1) - 2) - 2)^2 - 3(-3(-1(-1|x + 1| - 1) - 2) - 2) - 1 = -1cos(3\u03c0(1|((-1x - 2)/(2x - 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": "Determine the number of intersection points between the functions f(x) = (2((3(1((-1x - 3)/(-1x - 3)) + 3) + 1)/(-2(1((-1x - 3)/(-1x - 3)) + 3) + 1)) - 1)/(-2((3(1((-1x - 3)/(-1x - 3)) + 3) + 1)/(-2(1((-1x - 3)/(-1x - 3)) + 3) + 1)) - 1) and g(x) = 1(3e^(-2(-1|x - 1| + 1)) - 3)^2 - 5(3e^(-2(-1|x - 1| + 1)) - 3) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1cos(3\u03c0(-3(3((-2x + 1)/(2x)) - 3) + 4) - 2) - 2 and g(x) = -2(-3|(1|x + 1|)| + 1)^2 - 5(-3|(1|x + 1|)| + 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -4(1(2cos(1\u03c0((3x - 2)/(3x + 1)) - 3))) = 1(-2|(-1cos(1\u03c0x + 2) - 1) - 2|)^2 + 5(-2|(-1cos(1\u03c0x + 2) - 1) - 2|) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(2|((-2(1x^2 - 2x - 5) + 1)/(-2(1x^2 - 2x - 5) + 3)) - 2| - 3)^2 - 4(2|((-2(1x^2 - 2x - 5) + 1)/(-2(1x^2 - 2x - 5) + 3)) - 2| - 3) + 4 and g(x) = -4((1(5x - 4))/(3(5x - 4) + 2))^2 + 1((1(5x - 4))/(3(5x - 4) + 2)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation (1(-2|(2(3|x + 1|) + 5) + 1| - 2) - 2)/(-3(-2|(2(3|x + 1|) + 5) + 1| - 2) + 2) = -4(-4(-2|x - 2|)^2 + 2(-2|x - 2|) + 5)^2 - 2(-4(-2|x - 2|)^2 + 2(-2|x - 2|) + 5) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(3(-1(-3|x + 3|)^2 + 4(-3|x + 3|) + 5)^2 + 3(-1(-3|x + 3|)^2 + 4(-3|x + 3|) + 5) - 3) + 2 and g(x) = (2((-3(5x^2 - 3x + 4) + 3)/(2(5x^2 - 3x + 4) - 1)) + 1)/(2((-3(5x^2 - 3x + 4) + 3)/(2(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": "4", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-3(1|(5x - 1) + 1| - 1) - 5) + 1| + 3 and g(x) = 1cos(1\u03c0(3((-1x - 2)/(2x - 1))^2 + 5((-1x - 2)/(2x - 1)) + 3) + 2) 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) = (-1(-3|(2(-5x + 4) + 4) + 1| + 2) - 2)/(-2(-3|(2(-5x + 4) + 4) + 1| + 2) + 3) and g(x) = (1(-2cos(3\u03c0(-3x^2 + 3x - 2) - 1) + 1) - 1)/(-1(-2cos(3\u03c0(-3x^2 + 3x - 2) - 1) + 1) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1427", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(1sin(2\u03c0(-1(1x^2 + 1x - 5)^2 + 2(1x^2 + 1x - 5) - 3) + 1) - 1) - 2)/(-3(1sin(2\u03c0(-1(1x^2 + 1x - 5)^2 + 2(1x^2 + 1x - 5) - 3) + 1) - 1) + 4) and g(x) = 2sin(1\u03c0(-5(-2|x - 3| - 3)^2 - 4(-2|x - 3| - 3) + 1) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1528", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-3sin(2\u03c0((-1((-1x - 3)/(2x - 1)) + 1)/(2((-1x - 3)/(2x - 1)) + 1)) - 2)) + 2)/(2(-3sin(2\u03c0((-1((-1x - 3)/(2x - 1)) + 1)/(2((-1x - 3)/(2x - 1)) + 1)) - 2)) + 2) and g(x) = -1((1(3e^(1x) - 2) + 2)/(2(3e^(1x) - 2) + 1))^2 + 5((1(3e^(1x) - 2) + 2)/(2(3e^(1x) - 2) + 1)) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation -2|(-3|((-1(-1|x + 3|) - 2)/(3(-1|x + 3|) + 3)) - 1| + 3) - 2| = -3(-1(3sin(3\u03c0x - 2) - 3)^2 + 2)^2 + 1(-1(3sin(3\u03c0x - 2) - 3)^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": "72", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(-5(2(3|x + 1| - 2)^2 + 5)^2 + 4(2(3|x + 1| - 2)^2 + 5) - 5))/(3(-5(2(3|x + 1| - 2)^2 + 5)^2 + 4(2(3|x + 1| - 2)^2 + 5) - 5) + 2) and g(x) = -3cos(2\u03c0(3|(-1x - 1) + 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": "119", "dataset": "MATH"}
{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(1\u03c0(-5(5(5x) - 3) + 3) - 2) - 1 and g(x) = -2(-1e^(2(-4x - 4)))^2 + 3(-1e^(2(-4x - 4))) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1099", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2sin(3\u03c0(-2|(-3((-2x - 1)/(1x)) + 3) - 3| + 1) - 3) - 3 and g(x) = 2sin(2\u03c0(-5(1|x| + 3)^2 - 4(1|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": "852", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = -3cos(2\u03c0(-3(-2|(1x + 2) + 2| + 2) + 4) - 3) - 2 and g(x) = 1e^(-1(1(4x) - 5)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "107", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-1(3cos(3\u03c0(2x^2 - 4x + 3) + 1) - 3)^2 + 2) + 4 and g(x) = 2(2|(-2sin(2\u03c0x + 3) + 1) - 3| - 1) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "298", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 2(-2(-1(-5x^2 + 4x + 2)^2 - 5(-5x^2 + 4x + 2) + 2)^2 - 2(-1(-5x^2 + 4x + 2)^2 - 5(-5x^2 + 4x + 2) + 2) + 4)^2 - 2(-2(-1(-5x^2 + 4x + 2)^2 - 5(-5x^2 + 4x + 2) + 2)^2 - 2(-1(-5x^2 + 4x + 2)^2 - 5(-5x^2 + 4x + 2) + 2) + 4) and g(x) = 2|(-2sin(1\u03c0((3x + 1)/(-2x + 3)) + 1)) - 1| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
{"messages": [{"content": "How many solutions does the equation 5(-2(1(4x - 2)^2 + 1(4x - 2) + 1)^2 + 4(1(4x - 2)^2 + 1(4x - 2) + 1) - 4) + 3 = -4(-1sin(3\u03c0(3x - 5) - 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": "6", "dataset": "MATH"}
{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(2\u03c0(2(3((2x - 2)/(-1x + 3))^2 + 1((2x - 2)/(-1x + 3)) - 5)^2 + 5(3((2x - 2)/(-1x + 3))^2 + 1((2x - 2)/(-1x + 3)) - 5) - 4) - 1) - 1 and g(x) = (2(-2|(2cos(2\u03c0x + 1) + 2) + 1| + 3) + 2)/(1(-2|(2cos(2\u03c0x + 1) + 2) + 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": "382", "dataset": "MATH"}
{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(3(2sin(2\u03c0((3x - 3)/(2x)) - 1) - 3)^2 + 4(2sin(2\u03c0((3x - 3)/(2x)) - 1) - 3) + 4) - 1 and g(x) = (3((3(-3sin(1\u03c0x + 1) + 1) - 3)/(2(-3sin(1\u03c0x + 1) + 1) + 3)) + 1)/(2((3(-3sin(1\u03c0x + 1) + 1) - 3)/(2(-3sin(1\u03c0x + 1) + 1) + 3)) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}