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+{"messages": [{"content": "How many solutions does the equation (2(-1(1(-2sin(2\u03c0x - 2) - 1)^2 - 5(-2sin(2\u03c0x - 2) - 1) - 3) - 2) - 2)/(-3(-1(1(-2sin(2\u03c0x - 2) - 1)^2 - 5(-2sin(2\u03c0x - 2) - 1) - 3) - 2) - 3) = 5(1cos(3\u03c0((1x - 2)/(1x + 3)) + 1) - 2)^2 - 4(1cos(3\u03c0((1x - 2)/(1x + 3)) + 1) - 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(-3sin(1\u03c0(1(3x^2 - 5x + 3) + 5) - 3) + 1) - 3| + 3 and g(x) = -1((-1(4x^2 - 1x + 4) + 1)/(-2(4x^2 - 1x + 4) + 1)) + 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "390", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-1sin(1\u03c0(-1|(-1x + 3) + 2| + 3) - 2) + 1)^2 + 2(-1sin(1\u03c0(-1|(-1x + 3) + 2| + 3) - 2) + 1) - 2 and g(x) = 1(5((1x + 3)/(-2x + 1))^2 + 5((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": "16", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3((-2(2(-1sin(2\u03c0x + 1) - 3)^2 + 5(-1sin(2\u03c0x + 1) - 3) - 5) - 3)/(2(2(-1sin(2\u03c0x + 1) - 3)^2 + 5(-1sin(2\u03c0x + 1) - 3) - 5))) - 3 = (2(3(3cos(3\u03c0x - 2) + 2) + 5) - 3)/(-1(3(3cos(3\u03c0x - 2) + 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": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(2|(1((3x)/(1x - 3)) + 2) + 3| - 1)^2 and g(x) = 3(-1cos(1\u03c0((2x + 3)/(-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": "7", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(3\u03c0(-4((2(1|x|) - 2)/(-1(1|x|) - 1)) + 1) - 3) - 3 and g(x) = (-3(3|(1cos(2\u03c0x - 1) + 2)| - 2) - 2)/(1(3|(1cos(2\u03c0x - 1) + 2)| - 2) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "88", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(-3(1cos(1\u03c0x + 1) + 1)^2 - 1(1cos(1\u03c0x + 1) + 1) - 2) - 5 and g(x) = 3sin(2\u03c0(-3x - 3) + 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 4(2(3e^(-1((2x - 2)/(-2x - 3))))^2 + 1(3e^(-1((2x - 2)/(-2x - 3))))) + 3 = 2((2(-1sin(1\u03c0x - 3)) - 3)/(1(-1sin(1\u03c0x - 3)))) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|(5((1(4x - 3) - 1)/(-2(4x - 3) - 2))^2 + 1((1(4x - 3) - 1)/(-2(4x - 3) - 2)) - 4) - 3| and g(x) = -2cos(3\u03c0(-3(-1x^2 - 2x - 4) + 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": "8", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2((1(2(3|x + 2| - 3) + 2) - 3)/(3(2(3|x + 2| - 3) + 2) - 2))^2 - 5((1(2(3|x + 2| - 3) + 2) - 3)/(3(2(3|x + 2| - 3) + 2) - 2)) + 3 and g(x) = -4(-2|(-1cos(1\u03c0x - 3)) - 1| + 3) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(1|((-1(-1sin(1\u03c0x + 3) - 2) + 3)/(3(-1sin(1\u03c0x + 3) - 2) + 2)) + 1|)^2 - 3(1|((-1(-1sin(1\u03c0x + 3) - 2) + 3)/(3(-1sin(1\u03c0x + 3) - 2) + 2)) + 1|) - 3 and g(x) = 2|((1(-2cos(2\u03c0x - 2) + 3) - 2)/(-3(-2cos(2\u03c0x - 2) + 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": "20", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 5((1(-3x^2 + 2x + 2) + 3)/(2(-3x^2 + 2x + 2))) - 1 and g(x) = -2cos(1\u03c0(1x - 3) + 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": "Find the number of times the graphs of f(x) = (3(-5(1(1e^(-2x) - 3) + 4)^2 + 5(1(1e^(-2x) - 3) + 4) + 5))/(-3(-5(1(1e^(-2x) - 3) + 4)^2 + 5(1(1e^(-2x) - 3) + 4) + 5) + 4) and g(x) = -3cos(3\u03c0((-3(-5x) + 1)/(2(-5x) + 2))) - 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": "Determine the number of intersection points between the functions f(x) = 5(-3cos(1\u03c0(-2|(3x + 1) + 3| - 2)) + 1)^2 + 5(-3cos(1\u03c0(-2|(3x + 1) + 3| - 2)) + 1) + 3 and g(x) = 2(1|((3x - 1)/(-2x - 2)) - 3| - 1)^2 + 4(1|((3x - 1)/(-2x - 2)) - 3| - 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": "Find the number of times the graphs of f(x) = (3(1|(-1(5x + 4)^2 + 1(5x + 4) - 2) - 3|) + 1)/(2(1|(-1(5x + 4)^2 + 1(5x + 4) - 2) - 3|)) and g(x) = -5((3(-3sin(3\u03c0x - 2) - 1) + 2)/(-2(-3sin(3\u03c0x - 2) - 1) - 3))^2 + 5((3(-3sin(3\u03c0x - 2) - 1) + 2)/(-2(-3sin(3\u03c0x - 2) - 1) - 3)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "110", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(-1|((-3(1x^2 - 2x - 4) + 2)/(-1(1x^2 - 2x - 4))) + 1| + 2) - 2 and g(x) = 2sin(1\u03c0(-2|(2x)| + 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": "72", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-3(-1(2|(-1cos(1\u03c0x - 1) + 2) + 3|) - 2) - 3)/(-2(-1(2|(-1cos(1\u03c0x - 1) + 2) + 3|) - 2) - 1) = 3(-3sin(3\u03c0(-3|x + 1| - 2) + 1) - 1)^2 - 5(-3sin(3\u03c0(-3|x + 1| - 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": "180", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-5(3cos(2\u03c0(2|x - 1| - 3) + 3)) + 4) - 2)/(-1(-5(3cos(2\u03c0(2|x - 1| - 3) + 3)) + 4) + 3) and g(x) = 3|(2(1sin(3\u03c0x) - 2)^2 - 4)| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "143", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(5((1(1x^2 + 3x - 3) + 2)/(3(1x^2 + 3x - 3) - 2)) + 5) - 2| + 3 and g(x) = -1|(5(1cos(1\u03c0x + 3) - 1) + 4)| + 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 -2sin(3\u03c0(4(2(-3|x + 2| + 3)^2 - 4(-3|x + 2| + 3) + 2) - 1) - 3) - 3 = -2sin(3\u03c0((-1(4x + 4) + 1)/(-3(4x + 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": "49", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-1|(1sin(3\u03c0(-5x^2 - 4x + 2) - 1) + 1) - 3|) + 1)/(-1(-1|(1sin(3\u03c0(-5x^2 - 4x + 2) - 1) + 1) - 3|) - 2) and g(x) = 5((-1(1|x + 2| + 2) + 2)/(3(1|x + 2| + 2) - 1))^2 - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "629", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-2(-2sin(2\u03c0(-1|x + 2| + 1) - 3) - 1)^2 - 2(-2sin(2\u03c0(-1|x + 2| + 1) - 3) - 1) - 5))/(3(-2(-2sin(2\u03c0(-1|x + 2| + 1) - 3) - 1)^2 - 2(-2sin(2\u03c0(-1|x + 2| + 1) - 3) - 1) - 5) - 1) and g(x) = 1(-5(-2cos(1\u03c0x - 2) + 2) + 4)^2 + 2(-5(-2cos(1\u03c0x - 2) + 2) + 4) + 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(4x - 2)^2 - 3(4x - 2) - 2))/(-1(-1(4x - 2)^2 - 3(4x - 2) - 2) - 3) and g(x) = (-1(3sin(2\u03c0x - 3) + 3) + 2)/(2(3sin(2\u03c0x - 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 points of intersection exist between the functions f(x) = (-1(-1(-4x - 1)^2 - 4(-4x - 1)) + 3)/(-2(-1(-4x - 1)^2 - 4(-4x - 1)) + 3) and g(x) = -2cos(2\u03c0(4x^2 + 4x + 5) - 3) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1080", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3|((1(-4(-1cos(3\u03c0x + 2) - 1) - 1) + 3)/(3(-4(-1cos(3\u03c0x + 2) - 1) - 1) - 2)) + 2| - 3 = 2|(5(2x^2 - 2x + 5) - 5) + 1| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(2|(-1x - 1) + 1| - 1) - 1)/(-1(2|(-1x - 1) + 1| - 1) + 2) and g(x) = -2(-3sin(1\u03c0x - 2) - 2)^2 - 5(-3sin(1\u03c0x - 2) - 2) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(-1|(-4x^2 + 3x) + 3| + 3) + 2 and g(x) = -1|(2cos(3\u03c0x - 2) + 1) - 1| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(1|(-2((-1x + 1)/(-3x + 1))^2 - 4((-1x + 1)/(-3x + 1)) - 5) + 1| - 3) + 1 and g(x) = (3(3sin(3\u03c0(-2|x| + 1) + 2) + 2) - 1)/(-3(3sin(3\u03c0(-2|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": "79", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(-3cos(3\u03c0(3|x + 3|) + 3) - 2) - 2)/(-1(-3cos(3\u03c0(3|x + 3|) + 3) - 2) + 2) and g(x) = 3(2sin(1\u03c0x + 2)) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3cos(1\u03c0(-1|(-5(5x - 4)^2 + 5(5x - 4) + 5) - 1| + 3) + 3) - 1 and g(x) = -5(3|(-2x + 5) - 2| - 2)^2 + 4(3|(-2x + 5) - 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(-1(2x^2 - 1x - 5) + 2) + 3)/(-2(-1(2x^2 - 1x - 5) + 2) - 2) and g(x) = 2cos(2\u03c0((-2x - 3)/(1x + 2)) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-1(3(-1|x + 3| - 2) + 3)^2 - 3))/(-2(-1(3(-1|x + 3| - 2) + 3)^2 - 3) + 1) and g(x) = 1(-3(2sin(2\u03c0x + 3) - 1)^2 + 4(2sin(2\u03c0x + 3) - 1) + 3) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(-1((3(1cos(3\u03c0x + 3) + 2) + 2)/(-3(1cos(3\u03c0x + 3) + 2) + 2)) - 1)| + 1 and g(x) = -2|(-4(-2e^(-1x) + 3) + 5) - 3| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-3(-1(-3sin(2\u03c0(1|x| - 3) + 3) - 1) - 1))/(2(-1(-3sin(2\u03c0(1|x| - 3) + 3) - 1) - 1) - 1) = 4(-2sin(1\u03c0(2x^2 + 1x - 3) + 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": "404", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(3|(3sin(1\u03c0(-5x^2 - 4x + 2) - 2) - 1)| + 2) and g(x) = -5(1(-2sin(2\u03c0x - 1) + 3)^2 + 1(-2sin(2\u03c0x - 1) + 3) + 5) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "429", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(2\u03c0(-2(1x + 1)^2 - 3(1x + 1) + 2)) - 1 and g(x) = -5((2x)/(3x - 1))^2 + 2((2x)/(3x - 1)) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "236", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(1\u03c0(1(1|(4x - 5)| + 3)^2 + 5(1|(4x - 5)| + 3) + 1) - 2) - 3 and g(x) = 3cos(2\u03c0(4(3x + 5)^2 + 3) + 3) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "533", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-2(3cos(1\u03c0(-3|x + 1| + 2) - 3) + 1) - 3)/(-2(3cos(1\u03c0(-3|x + 1| + 2) - 3) + 1) + 3) = -2(2|x - 3| - 1)^2 + 2(2|x - 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": "23", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(-2cos(3\u03c0(-2|(3x^2 + 3x - 4) - 1| + 3) - 1) - 2) and g(x) = 3((-3(2|x + 1| + 3) - 1)/(-2(2|x + 1| + 3)))^2 + 4((-3(2|x + 1| + 3) - 1)/(-2(2|x + 1| + 3))) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1578", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1(5(-2|(1cos(2\u03c0x) + 1) + 1| + 2) + 5)^2 - 4(5(-2|(1cos(2\u03c0x) + 1) + 1| + 2) + 5) = -5((3(1|x - 2| + 2) + 3)/(2(1|x - 2| + 2) - 1))^2 - 3((3(1|x - 2| + 2) + 3)/(2(1|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": "80", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(2sin(2\u03c0(-1x^2 - 1x + 2) - 1) - 2) + 2| + 2 and g(x) = (3(-5x + 5) + 3)/(1(-5x + 5) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1e^(2((1(4x - 5))/(1(4x - 5)))) - 3 and g(x) = 2cos(3\u03c0((-2x - 3)/(-2x + 3)) - 1) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "98", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 5(-1|(-1((-2x - 3)/(-2x + 1))^2 + 2((-2x - 3)/(-2x + 1)) - 4) + 2| + 1) + 5 = (2(2|(-2sin(3\u03c0x) + 1) - 1|) + 2)/(3(2|(-2sin(3\u03c0x) + 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": "104", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3((1(-1cos(3\u03c0(-4x^2 - 4x + 1) + 3) + 1) - 3)/(-1(-1cos(3\u03c0(-4x^2 - 4x + 1) + 3) + 1) + 3)) - 2 = 1(-1|(1sin(3\u03c0x + 1)) - 1| - 1)^2 + 1(-1|(1sin(3\u03c0x + 1)) - 1| - 1) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(4(-1cos(2\u03c0x) + 2)^2 + 2(-1cos(2\u03c0x) + 2) + 1) - 3| + 3 and g(x) = (-3(3cos(2\u03c0x - 1)) + 1)/(3(3cos(2\u03c0x - 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 3|(1(-2sin(1\u03c0x)) - 4) + 1| - 1 = 2|(1x - 3) + 1| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(-5((3(-1cos(3\u03c0x + 3) + 1) + 1)/(1(-1cos(3\u03c0x + 3) + 1) - 3))^2 + 4) - 3| + 2 and g(x) = -4((-2(-3x^2 + 4) + 1)/(1(-3x^2 + 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": "14", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(4(4((-3x - 2)/(2x + 1)) - 3)^2 - 1(4((-3x - 2)/(2x + 1)) - 3) + 1) - 3| - 3 and g(x) = (-2(-5(1cos(2\u03c0x - 1))^2 + 3(1cos(2\u03c0x - 1)) + 5) + 2)/(1(-5(1cos(2\u03c0x - 1))^2 + 3(1cos(2\u03c0x - 1)) + 5) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(3(-2cos(3\u03c0((3x + 2)/(3x - 2)) + 1) + 2) - 4) - 2| - 3 and g(x) = (-3(-2cos(2\u03c0(-2|x - 2| + 1) - 3)) + 3)/(-2(-2cos(2\u03c0(-2|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": "45", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2e^(-1(5((2x + 3)/(1x - 3))^2 - 2((2x + 3)/(1x - 3)) - 2)) - 1 and g(x) = -3cos(3\u03c0(1x^2) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "528", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1sin(3\u03c0(1(-2|(2x^2 - 4x + 1) - 3|) + 4) + 3) and g(x) = (1(-4(2cos(3\u03c0x + 2) - 3)^2 - 3(2cos(3\u03c0x + 2) - 3) - 4) + 1)/(2(-4(2cos(3\u03c0x + 2) - 3)^2 - 3(2cos(3\u03c0x + 2) - 3) - 4) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1493", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-3|(2sin(2\u03c0x + 2) - 1)| + 2))/(1(-3|(2sin(2\u03c0x + 2) - 1)| + 2) + 1) and g(x) = -1sin(1\u03c0((3x + 2)/(-3x + 4))) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(-3sin(3\u03c0((-1(5x + 5) - 2)/(-2(5x + 5))) - 2) + 3)^2 - 5(-3sin(3\u03c0((-1(5x + 5) - 2)/(-2(5x + 5))) - 2) + 3) + 2 and g(x) = -4(1|(2sin(2\u03c0x - 1) - 1) + 3| + 3) - 4 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": "Find the number of times the graphs of f(x) = (2(-5(2sin(1\u03c0x + 2) + 1) + 5) + 1)/(-3(-5(2sin(1\u03c0x + 2) + 1) + 5) - 3) and g(x) = (2(-3|x - 1| + 1) - 1)/(1(-3|x - 1| + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1sin(2\u03c0(2(2|(5x^2 - 1x - 4) - 2| - 3) - 3) + 3) + 1 and g(x) = -4(1sin(1\u03c0((-2x + 3)/(-1x - 1)) - 2) - 2)^2 - 2(1sin(1\u03c0((-2x + 3)/(-1x - 1)) - 2) - 2) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "229", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(-2cos(3\u03c0(3|(1x^2 - 1x - 5)| + 3) - 3)) + 2)/(-2(-2cos(3\u03c0(3|(1x^2 - 1x - 5)| + 3) - 3))) and g(x) = 2(1e^(2(-3|x + 1| + 3)) - 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "204", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(2sin(2\u03c0(-5(-1|x - 1| - 1) + 3) + 3) + 1)^2 - 5(2sin(2\u03c0(-5(-1|x - 1| - 1) + 3) + 3) + 1) + 2 and g(x) = 3|((1(-1x + 3) + 2)/(-1(-1x + 3) - 1)) - 1| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "216", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(1(-2cos(1\u03c0x) + 3) - 3)^2 - 4 and g(x) = -2(-2e^(1x) + 3) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1|(3(2x + 5)^2 + 2(2x + 5) - 5) - 3| - 2 = 1sin(2\u03c0(4x^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": "16", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(1sin(3\u03c0(3|x + 1| - 3) - 1) + 2) - 1 and g(x) = -5(3x^2 - 3x - 1) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-2(-3|(-2x + 3) + 2|) + 1)/(3(-3|(-2x + 3) + 2|) - 2) = -3cos(3\u03c0((-1x - 1)/(1x))) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2(-1(-1|((1x - 3)/(3x - 2))| + 3)^2 - 4(-1|((1x - 3)/(3x - 2))| + 3) + 5) - 4 = 3sin(2\u03c0(-3|((-3x + 2)/(1x + 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": "6", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(5(2cos(2\u03c0(-1|x + 1| + 3) + 2) - 1) - 5))/(-2(5(2cos(2\u03c0(-1|x + 1| + 3) + 2) - 1) - 5) + 1) and g(x) = -2sin(2\u03c0((-1(-4x - 4) - 2)/(1(-4x - 4) - 2)) + 3) - 1 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|(-3((-3(3x + 3) - 1)/(1(3x + 3) - 3))^2 + 3((-3(3x + 3) - 1)/(1(3x + 3) - 3))) - 3| + 3 and g(x) = (1(3cos(2\u03c0(-1x^2 - 5x + 1) + 3) + 1) - 2)/(-2(3cos(2\u03c0(-1x^2 - 5x + 1) + 3) + 1) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(-1|(1e^(2(5x^2 - 5x + 5)) + 2) - 3| - 2))/(-3(-1|(1e^(2(5x^2 - 5x + 5)) + 2) - 3| - 2) + 2) and g(x) = (-2(3(3cos(3\u03c0x - 1)) - 2) + 3)/(-2(3(3cos(3\u03c0x - 1)) - 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(-4(5x + 1)^2 + 2(5x + 1) + 5) + 1)/(3(-4(5x + 1)^2 + 2(5x + 1) + 5) - 2) and g(x) = 4(-3cos(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": "21", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3|(1(-3sin(3\u03c0(1x^2 - 2x + 3) + 3) - 3) - 2) + 1| - 2 = -1sin(2\u03c0(3|((-1x - 1)/(-2x - 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": "546", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((-2(-4x^2 - 1x - 4) + 3)/(-1(-4x^2 - 1x - 4) + 2)) - 4 and g(x) = 3cos(1\u03c0(-5x - 4) + 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2|(-1e^(-1(-2x^2 + 1x + 4)) - 1) + 1| + 2 = 1sin(2\u03c0((1x - 1)/(-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": "8", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2cos(3\u03c0(3((-1x + 2)/(2x - 3))^2 - 3((-1x + 2)/(2x - 3)) + 4) + 2) = (2(3x + 4) - 1)/(-1(3x + 4)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(1\u03c0(-3|((3x)/(-1x + 2)) - 1|)) + 2 and g(x) = 3(2x^2 - 1x - 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(2sin(3\u03c0(-5((2x - 1)/(-3x + 4))^2 + 2((2x - 1)/(-3x + 4)) - 5) - 3) + 1) - 5 and g(x) = 2e^(1(5((-1x + 3)/(-2x + 3)) + 5)) - 3 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) = 3(-1cos(3\u03c0(2|((-1x - 3)/(-1x))|)) + 3)^2 + 1(-1cos(3\u03c0(2|((-1x - 3)/(-1x))|)) + 3) - 4 and g(x) = (3(2cos(3\u03c0(-3|x + 1| - 1) + 1) + 1) + 3)/(-1(2cos(3\u03c0(-3|x + 1| - 1) + 1) + 1)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "109", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 5((1(3cos(3\u03c0(1|x + 3| - 2) - 1) + 1) + 3)/(-3(3cos(3\u03c0(1|x + 3| - 2) - 1) + 1) + 1))^2 - 5((1(3cos(3\u03c0(1|x + 3| - 2) - 1) + 1) + 3)/(-3(3cos(3\u03c0(1|x + 3| - 2) - 1) + 1) + 1)) - 4 = (2(-2|(-2x^2 + 1x) - 1| + 3) + 3)/(1(-2|(-2x^2 + 1x) - 1| + 3)) 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": "How many points of intersection exist between the functions f(x) = 1(3|(-1sin(3\u03c0(-1x + 1) - 1)) - 2| - 1)^2 + 1(3|(-1sin(3\u03c0(-1x + 1) - 1)) - 2| - 1) and g(x) = 3(-2sin(2\u03c0(3x^2 + 4x - 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": "373", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(3\u03c0(1|(3((2x + 2)/(2x)) + 5) - 1| + 2)) - 2 and g(x) = 5((-1(4x^2 + 2x + 4) + 3)/(2(4x^2 + 2x + 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": "34", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1(2|(-2((3x - 1)/(-3x + 2))^2 - 3((3x - 1)/(-3x + 2)) + 3) - 2| + 3) - 3 = 4(3cos(1\u03c0(1|x - 2| - 1) - 1) + 2)^2 + 4(3cos(1\u03c0(1|x - 2| - 1) - 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": "30", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2sin(2\u03c0(4((1x - 1)/(-2x + 3))^2 - 4((1x - 1)/(-2x + 3)) - 2) - 3) + 2 and g(x) = -3e^(-2(1x^2 + 2x + 3)) 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 points of intersection exist between the functions f(x) = 2|((2(-4(3sin(1\u03c0x) + 3)^2 + 3(3sin(1\u03c0x) + 3) + 1) + 3)/(-1(-4(3sin(1\u03c0x) + 3)^2 + 3(3sin(1\u03c0x) + 3) + 1) + 3)) - 2| + 1 and g(x) = (-3(3(-1x^2 - 1) + 1) - 1)/(-1(3(-1x^2 - 1) + 1) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2sin(2\u03c0(-3((-1x + 2)/(-2x - 1)) - 3) + 1) + 2 = -1sin(1\u03c0((-3x - 2)/(-2x)) - 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3(-3cos(1\u03c0(-4(1|x + 1| - 1) + 3) + 1) - 3)^2 - 3(-3cos(1\u03c0(-4(1|x + 1| - 1) + 3) + 1) - 3) + 1 = 3sin(3\u03c0(-1|(-1x - 4) - 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": "36", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1cos(2\u03c0(3|(2(4x + 4)^2 - 2(4x + 4) - 3) + 1| + 1) + 3) + 1 and g(x) = 1(-4(1|x + 1| + 1)^2 + 5) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(2sin(3\u03c0(2x) - 1)) - 2)/(2(2sin(3\u03c0(2x) - 1)) - 1) and g(x) = 2|(3x^2 + 3x + 5) + 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4((-1(-4(-1|x + 3| + 1) - 4) - 2)/(2(-4(-1|x + 3| + 1) - 4) + 2))^2 - 1((-1(-4(-1|x + 3| + 1) - 4) - 2)/(2(-4(-1|x + 3| + 1) - 4) + 2)) - 4 and g(x) = 3(1(2|x - 1| - 1) - 3)^2 - 4(1(2|x - 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": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1((1(3cos(1\u03c0x + 3) + 2) - 1)/(-2(3cos(1\u03c0x + 3) + 2) + 3))^2 - 2((1(3cos(1\u03c0x + 3) + 2) - 1)/(-2(3cos(1\u03c0x + 3) + 2) + 3)) - 5 = -2|(3cos(1\u03c0x) - 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": "20", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(-3(3sin(3\u03c0x - 3) - 3)^2 - 1(3sin(3\u03c0x - 3) - 3) - 3) + 1| - 1 and g(x) = -1|((-3x + 1)/(1x - 2)) - 1| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(2\u03c0((-3(1|x + 1| - 1) + 1)/(-1(1|x + 1| - 1) + 2)) + 3) and g(x) = 3((-1x + 2)/(-2x - 2)) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "135", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1(-4((-1(-1|x - 1| - 3) + 3)/(1(-1|x - 1| - 3) - 2))^2 + 3((-1(-1|x - 1| - 3) + 3)/(1(-1|x - 1| - 3) - 2)) + 5) - 1 = -3(1cos(1\u03c0(-2|x + 1| - 2) - 3) - 1)^2 + 4(1cos(1\u03c0(-2|x + 1| - 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": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 5(-3cos(2\u03c0(1|x + 2| - 1) - 1) + 2) - 3 = -3((-2x - 3)/(3x - 1))^2 - 1((-2x - 3)/(3x - 1)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3((-1(-3|x - 2|) + 3)/(2(-3|x - 2|) - 3))^2 + 5((-1(-3|x - 2|) + 3)/(2(-3|x - 2|) - 3)) + 4 and g(x) = -1sin(3\u03c0(1x^2 + 1x + 4) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "580", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2((1(3|(-1x - 5) + 2| - 2) + 3)/(-1(3|(-1x - 5) + 2| - 2)))^2 - 5((1(3|(-1x - 5) + 2| - 2) + 3)/(-1(3|(-1x - 5) + 2| - 2))) - 4 and g(x) = -2cos(1\u03c0(1(4x^2 + 5x - 5) - 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": "730", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3(2sin(2\u03c0((-2(-1|x + 2| + 3) - 1)/(1(-1|x + 2| + 3) + 3)) + 2) + 2)^2 + 1 = -1cos(2\u03c0(2(3x^2 + 3x - 4)) + 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "252", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(-1|(3cos(3\u03c0x + 2) + 3) - 3| + 2) + 2)/(-3(-1|(3cos(3\u03c0x + 2) + 3) - 3| + 2) - 3) and g(x) = -3(2|x| - 1)^2 + 2(2|x| - 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "115", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (2(1|(5(-2sin(3\u03c0x)) - 4) + 3| - 3) - 1)/(-1(1|(5(-2sin(3\u03c0x)) - 4) + 3| - 3)) = -1(5(3e^(-1x) + 1)^2 + 5(3e^(-1x) + 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": "20", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(-1|(2((-2x - 3)/(-3x + 2))^2 + 5((-2x - 3)/(-3x + 2)) - 4) + 3| + 3) + 3 and g(x) = (1(-3(-2|x + 1| + 1)^2 - 3(-2|x + 1| + 1) + 4) + 1)/(3(-3(-2|x + 1| + 1)^2 - 3(-2|x + 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": "8", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-5(-4(2|x - 3| + 2)^2 + 4)) + 2)/(-2(-5(-4(2|x - 3| + 2)^2 + 4)) + 3) and g(x) = -5(-3|(1cos(2\u03c0x + 3) - 1) + 2| + 2)^2 - 4(-3|(1cos(2\u03c0x + 3) - 1) + 2| + 2) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1sin(1\u03c0(-5((-3(3x) + 3)/(-1(3x)))^2 + 5((-3(3x) + 3)/(-1(3x))) + 2) - 2) + 2 and g(x) = (2(-4(-2|x + 3| - 3) - 1) + 3)/(1(-4(-2|x + 3| - 3) - 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "161", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(-1sin(1\u03c0(-5x - 4) - 1) + 1)^2 - 4(-1sin(1\u03c0(-5x - 4) - 1) + 1) - 1 and g(x) = 2sin(3\u03c0(-1|x - 3| + 1) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 5(5((-1(3sin(1\u03c0x + 3)) - 3)/(-3(3sin(1\u03c0x + 3)) + 1)))^2 + 2(5((-1(3sin(1\u03c0x + 3)) - 3)/(-3(3sin(1\u03c0x + 3)) + 1))) - 3 = -2|(-3((-3x + 3)/(-2x + 1)) + 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": "18", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(3sin(3\u03c0(2(-3x^2 - 5x - 4) - 5)) + 2)| - 3 and g(x) = 5((3(-1|x| + 2) - 2)/(-1(-1|x| + 2) + 2)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-3(2sin(1\u03c0(-5x - 2) + 2) - 1) + 3)/(2(2sin(1\u03c0(-5x - 2) + 2) - 1) - 3) = (-1(-2|x + 3| - 2))/(3(-2|x + 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": "100", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|((2(-1(2cos(1\u03c0x + 3) - 2)^2 - 4(2cos(1\u03c0x + 3) - 2) + 5))/(-2(-1(2cos(1\u03c0x + 3) - 2)^2 - 4(2cos(1\u03c0x + 3) - 2) + 5) - 1)) - 1| - 2 and g(x) = -2sin(2\u03c0((-2(-1x + 1) - 3)/(-3(-1x + 1) - 2)) + 3) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -4(-3sin(2\u03c0(-1|(3x - 4) + 1| - 2) + 1) + 3)^2 - 4 = -5((-2(-3sin(2\u03c0x - 1) + 2) - 1)/(3(-3sin(2\u03c0x - 1) + 2))) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(1cos(1\u03c0(-4x^2 - 2x) + 3) - 3) + 2| - 1 and g(x) = 1cos(3\u03c0(2x^2 - 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": "800", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(3(-2|(4x^2 - 3x - 5) - 3| + 2) + 3) - 3)/(-1(3(-2|(4x^2 - 3x - 5) - 3| + 2) + 3)) and g(x) = 3(2sin(2\u03c0(-5x - 4) - 2))^2 + 4(2sin(2\u03c0(-5x - 4) - 2)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "398", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(-1(1|(-5x^2 - 2x + 4) + 1| - 3) - 5) - 3)/(-3(-1(1|(-5x^2 - 2x + 4) + 1| - 3) - 5) - 1) and g(x) = (-1(-5(-3cos(1\u03c0x - 3) - 3)^2 + 4(-3cos(1\u03c0x - 3) - 3) + 2) + 1)/(-3(-5(-3cos(1\u03c0x - 3) - 3)^2 + 4(-3cos(1\u03c0x - 3) - 3) + 2)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(-1((-3x - 3)/(2x))^2 + 2((-3x - 3)/(2x)) - 2) + 3| - 3 and g(x) = 3(3cos(3\u03c0x))^2 + 2(3cos(3\u03c0x)) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "106", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|(2(-2cos(2\u03c0(1x^2 - 3x - 2) + 3)) + 2) + 1| and g(x) = -3(3cos(2\u03c0(1|x - 1| + 2) + 2) + 1)^2 - 5(3cos(2\u03c0(1|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": "119", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(2cos(1\u03c0(4x^2 - 2x + 5) - 2) - 2) + 2| - 1 and g(x) = 1(3|x + 1| - 1) - 5 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 solutions does the equation 1sin(3\u03c0(4(-1|((-2x - 3)/(2x + 2))|)^2 - 5(-1|((-2x - 3)/(2x + 2))|)) + 1) + 1 = (2(1|(-3sin(2\u03c0x + 1) - 1) + 1| - 1) + 1)/(1(1|(-3sin(2\u03c0x + 1) - 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": "26", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(-3((3(-2|x - 1| - 3) - 2)/(2(-2|x - 1| - 3) + 2)) + 2)^2 - 5(-3((3(-2|x - 1| - 3) - 2)/(2(-2|x - 1| - 3) + 2)) + 2) and g(x) = 4(3sin(2\u03c0((-2x + 1)/(1x - 3)) - 1) - 3) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(1\u03c0(-2(-4(2|x - 3| + 1)^2 - 2(2|x - 3| + 1) + 4) + 3) - 1) + 3 and g(x) = 2sin(2\u03c0((2(-1x + 5) + 3)/(3(-1x + 5) - 1)) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "763", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(1(2sin(2\u03c0(-3|x| + 2) + 1) + 1) - 1) + 3)/(-3(1(2sin(2\u03c0(-3|x| + 2) + 1) + 1) - 1) + 4) and g(x) = -5(-1|(3x^2 - 4x + 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": "15", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2(1(-2|(2sin(2\u03c0x + 3) + 2) - 1| + 2) + 1)^2 + 5(1(-2|(2sin(2\u03c0x + 3) + 2) - 1| + 2) + 1) + 4 = 2(1(1|x + 2| - 3)^2 - 3(1|x + 2| - 3) - 5) 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 solutions does the equation -5((3(-5(1|x - 3| - 3)^2 - 4(1|x - 3| - 3) + 5) - 3)/(3(-5(1|x - 3| - 3)^2 - 4(1|x - 3| - 3) + 5))) + 5 = 2(1cos(3\u03c0(-1x^2 + 4x + 3) - 1) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (3(-1|(-3x^2 - 1x - 1) - 1| + 1))/(3(-1|(-3x^2 - 1x - 1) - 1| + 1) - 1) = 4(-2cos(3\u03c0x + 3) - 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3sin(1\u03c0(5(-1|((2x)/(3x + 1)) + 1| - 2)^2 - 1(-1|((2x)/(3x + 1)) + 1| - 2) + 1)) - 3 = -3((1(-1e^(-2x) + 1) - 1)/(-3(-1e^(-2x) + 1)))^2 - 3((1(-1e^(-2x) + 1) - 1)/(-3(-1e^(-2x) + 1))) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(2cos(1\u03c0(-1(-2|x + 1| - 1)^2 + 5(-2|x + 1| - 1) - 5) - 3) + 2))/(3(2cos(1\u03c0(-1(-2|x + 1| - 1)^2 + 5(-2|x + 1| - 1) - 5) - 3) + 2) - 2) and g(x) = (1(-2(-1sin(1\u03c0x) + 3)) + 2)/(1(-2(-1sin(1\u03c0x) + 3)) + 3) 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": "How many points of intersection exist between the functions f(x) = 1cos(2\u03c0(-2(-2|x + 2| + 3)^2 - 1(-2|x + 2| + 3)) + 2) and g(x) = (-3(-4x - 4) - 3)/(-3(-4x - 4) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "327", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1cos(3\u03c0(5(-3((1x - 1)/(2x - 1)) - 5)^2 - 2(-3((1x - 1)/(2x - 1)) - 5) - 2) + 2) - 1 and g(x) = -2|(5(-3sin(3\u03c0x) + 1) - 1)| - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1sin(3\u03c0(4(5x^2 + 3x + 4) - 1) + 3) and g(x) = -3(2|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": "55", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5((2(2|(-1x^2 - 4x + 2)| - 2) + 1)/(3(2|(-1x^2 - 4x + 2)| - 2) - 2)) - 4 and g(x) = -2(2sin(2\u03c0(2x + 3) - 2) + 1)^2 + 4(2sin(2\u03c0(2x + 3) - 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": "150", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3((3(2(-2sin(3\u03c0x - 1))^2 + 2(-2sin(3\u03c0x - 1)) - 4) - 2)/(2(2(-2sin(3\u03c0x - 1))^2 + 2(-2sin(3\u03c0x - 1)) - 4) + 1)) + 3 and g(x) = (-1(4(3x - 4)^2 + 2(3x - 4) + 1))/(-3(4(3x - 4)^2 + 2(3x - 4) + 1) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(-4(-3sin(2\u03c0((2x + 3)/(1x)) + 2) - 2) - 5)^2 + 4(-4(-3sin(2\u03c0((2x + 3)/(1x)) + 2) - 2) - 5) - 3 and g(x) = 1(2|((3x + 2)/(3x - 2)) - 2| - 3)^2 - 3(2|((3x + 2)/(3x - 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": "46", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3cos(2\u03c0(5(1|x - 1| - 2)^2 - 3(1|x - 1| - 2) + 4) - 2) - 2 and g(x) = 1|(5x + 2) - 2| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1e^(-1(5(-5x - 1)^2 - 5(-5x - 1) + 3)) = -2sin(1\u03c0(1x - 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": "20", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (1(2(2sin(3\u03c0x - 1) + 1) - 3) + 2)/(-3(2(2sin(3\u03c0x - 1) + 1) - 3) + 4) = (-1(1|x + 3| + 2) - 1)/(-1(1|x + 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) = -3(2cos(3\u03c0(-2|x + 2|) - 3) + 1) + 2 and g(x) = -2(2x + 5)^2 - 2(2x + 5) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1e^(-1((1(5(2|x + 3| + 3) + 3) - 2)/(-2(5(2|x + 3| + 3) + 3) + 3))) and g(x) = (2(3cos(1\u03c0(-1|x + 1| + 2) - 2)) - 3)/(1(3cos(1\u03c0(-1|x + 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": "How many solutions does the equation 1|((-3(3x) - 1)/(1(3x) + 3)) - 1| = -2sin(2\u03c0(-5x^2 - 5x - 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": "39", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(-5(-3sin(1\u03c0x - 1) - 1) - 5) + 3| - 1 and g(x) = 1(-1cos(3\u03c0x - 1) - 3)^2 - 2(-1cos(3\u03c0x - 1) - 3) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1(3(1|((1x + 3)/(-3x - 2)) + 1|)^2 - 3(1|((1x + 3)/(-3x - 2)) + 1|) + 1) + 4 = -2sin(2\u03c0(-1|(1x^2 - 5x + 2)| - 3) + 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(4(1|(-3sin(1\u03c0x - 1))| - 1) + 3)^2 + 2(4(1|(-3sin(1\u03c0x - 1))| - 1) + 3) + 5 and g(x) = -4(-4(-2|x - 3| + 1)^2 - 4(-2|x - 3| + 1) - 5) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2((3(3cos(2\u03c0x - 1) - 2) - 2)/(1(3cos(2\u03c0x - 1) - 2) + 2))^2 + 5((3(3cos(2\u03c0x - 1) - 2) - 2)/(1(3cos(2\u03c0x - 1) - 2) + 2)) - 3 = -3cos(1\u03c0(2x^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": "76", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (2(-5(-1x - 5)^2 + 5(-1x - 5) + 5) + 2)/(3(-5(-1x - 5)^2 + 5(-1x - 5) + 5) + 1) = -1sin(3\u03c0(-4x^2 - 3x - 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(-2|x + 2| + 3)^2 - 3(-2|x + 2| + 3) - 3) + 1)/(-3(4(-2|x + 2| + 3)^2 - 3(-2|x + 2| + 3) - 3) - 3) and g(x) = 1sin(2\u03c0(-1|x + 1|) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(4(-3cos(2\u03c0x - 3) - 1) - 3) + 1| + 3 and g(x) = -4(3x^2 + 3x + 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 4((-3(-5(2|x + 1| - 2)^2 + 2(2|x + 1| - 2) + 1) - 3)/(2(-5(2|x + 1| - 2)^2 + 2(2|x + 1| - 2) + 1) + 2)) + 5 and g(x) = (2(-2e^(-1(5x + 5)) - 3) + 2)/(-2(-2e^(-1(5x + 5)) - 3) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "137", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|((1(4(-1x + 2)^2 - 4(-1x + 2) - 5) + 1)/(-3(4(-1x + 2)^2 - 4(-1x + 2) - 5) - 2)) + 2| and g(x) = -2|(1e^(2((3x - 3)/(-2x - 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) = -2|((-1(-2(4x - 1)^2 + 5(4x - 1) - 2) - 2)/(3(-2(4x - 1)^2 + 5(4x - 1) - 2) - 2)) + 1| - 1 and g(x) = -2|((-3(-2sin(1\u03c0x + 1) + 2))/(-2(-2sin(1\u03c0x + 1) + 2) + 3))| + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|((-1(5(-1x^2 + 2x) - 5) - 3)/(3(5(-1x^2 + 2x) - 5) + 2)) + 1| - 3 and g(x) = -2|(-1((1x + 2)/(3x))^2 - 2((1x + 2)/(3x)) + 4) + 2| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(-2cos(1\u03c0(-5x^2 - 1x - 3) - 1) - 3) + 3 and g(x) = -3(1x^2 + 5x + 2) + 2 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": "Find the number of times the graphs of f(x) = 3e^(-1(-3((-3x + 1)/(-2x - 1)) + 5)) - 3 and g(x) = -3sin(2\u03c0(-3|x| - 3)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "67", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (1(2cos(2\u03c0(-4(2|x - 2| + 2)^2 + 2(2|x - 2| + 2) + 1)) - 3) - 3)/(-2(2cos(2\u03c0(-4(2|x - 2| + 2)^2 + 2(2|x - 2| + 2) + 1)) - 3) - 1) = (1(-1(2|x| - 1)^2 + 3(2|x| - 1)) - 1)/(-3(-1(2|x| - 1)^2 + 3(2|x| - 1))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1((3(4(1cos(1\u03c0x - 2) - 2)^2 + 2(1cos(1\u03c0x - 2) - 2) - 4) + 3)/(1(4(1cos(1\u03c0x - 2) - 2)^2 + 2(1cos(1\u03c0x - 2) - 2) - 4) + 1)) + 3 and g(x) = -2(-2cos(2\u03c0((-2x - 1)/(2x + 2)) + 3) - 1) - 3 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) = 2|(-4(1(1cos(1\u03c0x + 3)) - 2)^2 + 3) + 3| + 2 and g(x) = -4(-2|((2x + 2)/(3x + 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": "Determine the number of intersection points between the functions f(x) = 3|(1((-2(1x^2 + 5x - 2) + 2)/(2(1x^2 + 5x - 2) + 2)) - 1) + 2| - 2 and g(x) = 5(-1sin(1\u03c0(3x + 3) - 2) + 1)^2 - 3(-1sin(1\u03c0(3x + 3) - 2) + 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3sin(3\u03c0(-2|((-2x - 3)/(-2x - 1)) + 1| - 2) - 1) + 2 = -5(-3sin(1\u03c0x - 1)) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(3(1cos(1\u03c0(1x + 4) - 2) - 1)^2 + 1(1cos(1\u03c0(1x + 4) - 2) - 1) - 4) + 3)/(1(3(1cos(1\u03c0(1x + 4) - 2) - 1)^2 + 1(1cos(1\u03c0(1x + 4) - 2) - 1) - 4) + 3) and g(x) = -2|((-3(-1sin(1\u03c0x - 3) - 3) + 2)/(-3(-1sin(1\u03c0x - 3) - 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": "20", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|(-2((1(5x^2 + 4x - 2))/(2(5x^2 + 4x - 2) + 1)) - 5) + 3| + 3 and g(x) = -1(3cos(3\u03c0((-2x - 2)/(-2x)) + 1) + 1)^2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1cos(3\u03c0((-1(-1|x + 3| + 3) + 1)/(1(-1|x + 3| + 3) - 2)) - 1) - 2 and g(x) = 1cos(3\u03c0(2|x - 3|)) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "186", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(3sin(3\u03c0(4x - 1) + 3) - 3)^2 - 4(3sin(3\u03c0(4x - 1) + 3) - 3) - 4 and g(x) = -3e^(-2(3|x - 1| - 3)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "218", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1cos(2\u03c0((1(1|(2x + 5)| + 2))/(-2(1|(2x + 5)| + 2) - 2)) + 3) - 3 = 1(1|(-2sin(3\u03c0x + 2) + 3)| + 1)^2 - 3(1|(-2sin(3\u03c0x + 2) + 3)| + 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|((-3(4x - 3) - 3)/(1(4x - 3))) - 3| + 1 and g(x) = -2sin(3\u03c0(-4x^2 - 1x + 2) + 1) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "298", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|((1(-1(5x^2 - 5x - 1) + 1) + 2)/(2(-1(5x^2 - 5x - 1) + 1) + 3)) + 2| + 2 and g(x) = -1((3(-2x + 3) - 2)/(2(-2x + 3) - 1))^2 + 5((3(-2x + 3) - 2)/(2(-2x + 3) - 1)) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(-1e^(-1(-1x^2 + 1x))) + 2| and g(x) = 3sin(2\u03c0((3x + 3)/(1x - 3)) - 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) = 3cos(2\u03c0(-1|(-1((3x)/(2x - 3))^2 - 4((3x)/(2x - 3)) - 5) + 2| - 3) - 1) + 1 and g(x) = -2|(-4((3x - 1)/(1x + 3))^2 + 3((3x - 1)/(1x + 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": "36", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(-3((-3(3cos(3\u03c0x + 1) - 2) + 2)/(-3(3cos(3\u03c0x + 1) - 2) - 1))^2 - 4((-3(3cos(3\u03c0x + 1) - 2) + 2)/(-3(3cos(3\u03c0x + 1) - 2) - 1)) + 2) + 1| + 3 and g(x) = -3sin(3\u03c0(-4(3|x + 2| - 2) - 5) + 2) + 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(-3(1(2sin(2\u03c0x + 1) + 3)^2 - 4(2sin(2\u03c0x + 1) + 3) + 5) + 3) - 2)/(3(-3(1(2sin(2\u03c0x + 1) + 3)^2 - 4(2sin(2\u03c0x + 1) + 3) + 5) + 3) - 2) = -1|((1(3sin(2\u03c0x + 2) + 1) - 3)/(1(3sin(2\u03c0x + 2) + 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": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|((1(3sin(1\u03c0(4x - 1) - 2) - 2) - 2)/(2(3sin(1\u03c0(4x - 1) - 2) - 2) - 1)) - 2| + 3 and g(x) = 3(-1(-3|x + 3| + 2)^2 - 1(-3|x + 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": "10", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(1sin(1\u03c0(4x^2 - 2x - 4) + 1) + 1) + 3 and g(x) = 2sin(1\u03c0(1x + 1) - 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "799", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(-2cos(3\u03c0(-1x + 5) - 3) - 1) + 1)/(-2(-2cos(3\u03c0(-1x + 5) - 3) - 1) - 1) and g(x) = 2cos(1\u03c0((2x - 2)/(2x - 1)) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(2cos(1\u03c0(-1x) + 2) - 3) + 2)/(1(2cos(1\u03c0(-1x) + 2) - 3) + 2) and g(x) = -3(-2|x + 2| + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (2(-2(2|(-1cos(3\u03c0x + 2) + 1) - 3| - 3)^2 - 3(2|(-1cos(3\u03c0x + 2) + 1) - 3| - 3) + 3) + 2)/(-1(-2(2|(-1cos(3\u03c0x + 2) + 1) - 3| - 3)^2 - 3(2|(-1cos(3\u03c0x + 2) + 1) - 3| - 3) + 3) + 3) = -3cos(3\u03c0(-1(5x)^2 + 1(5x) + 5) - 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "121", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3sin(1\u03c0(3((-1x - 3)/(2x - 1))^2 - 4((-1x - 3)/(2x - 1)) + 3) + 1) + 1 and g(x) = 3cos(1\u03c0(-3x - 5) - 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "125", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-3(-1|(4x^2 + 1x - 5) + 2|) + 2)/(2(-1|(4x^2 + 1x - 5) + 2|) + 1) = 2cos(1\u03c0(-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": "198", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -5(1|(-1(1sin(2\u03c0x - 3) + 3)^2 + 2(1sin(2\u03c0x - 3) + 3) - 3) - 2| - 2) - 3 = -4(-2sin(1\u03c0(4x - 3) + 1) - 2)^2 - 2(-2sin(1\u03c0(4x - 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": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2((-1(-3e^(1(-2|x|)) - 1) + 1)/(-2(-3e^(1(-2|x|)) - 1))) + 2 and g(x) = 1sin(1\u03c0(3|((2x + 1)/(1x - 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": "46", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(2|(-1((-1x - 1)/(-1x + 2)) - 2) - 3| - 2)^2 - 5(2|(-1((-1x - 1)/(-1x + 2)) - 2) - 3| - 2) + 5 and g(x) = 2e^(2(1(3x - 4)^2)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(2\u03c0(-3(1|((2x - 1)/(1x + 2)) - 1| - 2)^2 - 1(1|((2x - 1)/(1x + 2)) - 1| - 2) - 1) - 3) - 1 and g(x) = 2|((3(1x^2 + 3x + 1))/(3(1x^2 + 3x + 1) - 2)) - 1| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "270", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-1(-1e^(-1((-2x + 1)/(1x + 3))) + 1))^2 - 3(-1(-1e^(-1((-2x + 1)/(1x + 3))) + 1)) + 1 and g(x) = (-1(-2sin(3\u03c0(1x^2 - 4x + 2) + 1) - 1) - 3)/(3(-2sin(3\u03c0(1x^2 - 4x + 2) + 1) - 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "73", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(1cos(1\u03c0(-1|((-1x - 1)/(2x + 1)) - 3| + 3) - 3) + 1) - 5 and g(x) = 2((2(-3cos(1\u03c0x - 1) + 1) - 2)/(-1(-3cos(1\u03c0x - 1) + 1) + 2)) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-2(1(2|x - 2| + 3) + 3) + 1)/(3(1(2|x - 2| + 3) + 3) + 2) = 3cos(1\u03c0(4x - 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": "80", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2cos(3\u03c0((-1(1x - 5) + 1)/(2(1x - 5) + 2)) - 1) = (1(3x - 2) - 1)/(1(3x - 2) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 4((1(-3sin(1\u03c0x + 3) + 2) + 1)/(1(-3sin(1\u03c0x + 3) + 2) - 3)) - 2 = -5(2|x + 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": "20", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(-1cos(3\u03c0((-3x - 3)/(-3x)) - 2) + 2)^2 + 2(-1cos(3\u03c0((-3x - 3)/(-3x)) - 2) + 2) + 2 and g(x) = -3sin(2\u03c0((1x + 3)/(-3x - 2)) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "65", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3((2(-2cos(1\u03c0(-2|x + 3| + 3))) - 1)/(2(-2cos(1\u03c0(-2|x + 3| + 3))) + 3))^2 - 5 and g(x) = (-3(-3e^(2(-1x^2 + 3x - 2))) - 3)/(1(-3e^(2(-1x^2 + 3x - 2))) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-5((2x)/(2x - 3)) - 2)^2 + 1(-5((2x)/(2x - 3)) - 2) - 2 and g(x) = (-1(2cos(1\u03c0x + 3) - 1) + 2)/(1(2cos(1\u03c0x + 3) - 1) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|((-3(-1(-1x^2 - 3x - 2) - 5))/(2(-1(-1x^2 - 3x - 2) - 5) - 1)) - 1| + 1 and g(x) = -3(-1cos(3\u03c0(5x^2 - 2x) - 3) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3sin(2\u03c0(-5(-2|x + 2| - 2)^2 + 2(-2|x + 2| - 2) - 5) + 1) + 3 = -2(-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": "385", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-3(-4(-1x^2 - 4x + 5) + 4) + 2)/(-2(-4(-1x^2 - 4x + 5) + 4) - 2) = 2sin(2\u03c0(-1x + 2)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(1cos(3\u03c0((-3x - 1)/(-2x + 3))) + 3)^2 + 5(1cos(3\u03c0((-3x - 1)/(-2x + 3))) + 3) + 3 and g(x) = 4(-2cos(1\u03c0x + 2) - 2)^2 - 2(-2cos(1\u03c0x + 2) - 2) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 4(1(-3cos(3\u03c0x - 1) + 1) - 1)^2 - 5(1(-3cos(3\u03c0x - 1) + 1) - 1) + 4 = -4(-1cos(2\u03c0x - 1) - 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(-3sin(3\u03c0(3((-1x + 2)/(-1x + 3))^2 - 4((-1x + 2)/(-1x + 3)) + 4) + 1) + 3) + 1 and g(x) = 2|(-1((2x)/(-2x + 3)) + 4) - 1| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "150", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|((-1(-2cos(3\u03c0x + 3) + 2))/(-2(-2cos(3\u03c0x + 3) + 2) + 3)) + 1| and g(x) = (-2(-5x^2 + 2x) - 2)/(-1(-5x^2 + 2x) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "114", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(-2sin(3\u03c0(-4x^2 + 1x) + 2) - 1) + 1)/(2(-2sin(3\u03c0(-4x^2 + 1x) + 2) - 1) - 1) and g(x) = (2(-2cos(3\u03c0x - 3) + 3) + 2)/(-2(-2cos(3\u03c0x - 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": "167", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2(3(-1|(-1cos(2\u03c0x + 2) + 2) - 1| + 3) - 5)^2 - 4(3(-1|(-1cos(2\u03c0x + 2) + 2) - 1| + 3) - 5) + 1 and g(x) = 1|(2sin(1\u03c0((1x - 2)/(-2x + 3)) + 3) - 3) + 2| + 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": "How many points of intersection exist between the functions f(x) = -4(-2cos(1\u03c0(-1|((-1x - 2)/(1x)) - 3| - 2)) - 1)^2 and g(x) = -1sin(2\u03c0(4(4x^2 - 5x + 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": "221", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(1\u03c0(-1|(-2x - 3) - 3| - 3)) - 2 and g(x) = 1sin(1\u03c0((-2x)/(2x - 1)) - 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|(1e^(-1(-1x - 3)) - 1) - 3| + 2)^2 + 2 and g(x) = 2cos(1\u03c0(-2(2|x - 2| - 1) + 3) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-1(2|(4x + 3) - 3|)^2 - 3(2|(4x + 3) - 3|) - 1) + 1)/(-2(-1(2|(4x + 3) - 3|)^2 - 3(2|(4x + 3) - 3|) - 1) + 3) and g(x) = -2cos(2\u03c0((1(-3x^2 - 1x + 4) + 1)/(1(-3x^2 - 1x + 4) - 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": "47", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(3|((3(-4x^2 + 3x - 1) - 2)/(3(-4x^2 + 3x - 1) - 2)) - 3| + 3) - 3 and g(x) = 2(-3cos(3\u03c0((3x)/(3x + 3)) + 3) - 2)^2 - 1(-3cos(3\u03c0((3x)/(3x + 3)) + 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": "66", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1|(2(1(-3cos(1\u03c0x - 2) + 3) - 2)^2 + 4(1(-3cos(1\u03c0x - 2) + 3) - 2) + 1)| + 1 = -3(1|(2x^2 + 5x + 1) - 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": "8", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|((-1(2(-2x + 2)^2 - 5(-2x + 2) + 1) + 3)/(-1(2(-2x + 2)^2 - 5(-2x + 2) + 1)))| + 3 and g(x) = -3sin(2\u03c0(-2((1x + 2)/(2x + 1))^2 + 4((1x + 2)/(2x + 1)) - 3) - 3) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2sin(2\u03c0(1|(1((1x - 3)/(-2x - 3))^2 - 5((1x - 3)/(-2x - 3)) - 5) + 2| + 1)) + 1 and g(x) = (-1(4(-2sin(1\u03c0x + 2) - 1)^2 - 2(-2sin(1\u03c0x + 2) - 1) - 4) - 2)/(-2(4(-2sin(1\u03c0x + 2) - 1)^2 - 2(-2sin(1\u03c0x + 2) - 1) - 4) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "171", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (2(3|(-1sin(1\u03c0(-2x - 3)) - 1) - 2| + 1) - 3)/(3(3|(-1sin(1\u03c0(-2x - 3)) - 1) - 2| + 1)) = (-1(2(3cos(1\u03c0x + 2) + 1) + 3) - 3)/(-2(2(3cos(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": "How many points of intersection exist between the functions f(x) = -4(3|(-3cos(3\u03c0(-4x^2 - 1x + 4) - 3) + 3)| - 2) + 4 and g(x) = -2|(-3cos(2\u03c0((2x - 3)/(2x - 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": "1355", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1(-1cos(1\u03c0(1|((2x - 3)/(3x + 3)) - 1| - 1) - 3) + 1)^2 + 2(-1cos(1\u03c0(1|((2x - 3)/(3x + 3)) - 1| - 1) - 3) + 1) + 1 = (1(3(-2|x - 2| - 3)^2 - 2(-2|x - 2| - 3) + 1) + 2)/(1(3(-2|x - 2| - 3)^2 - 2(-2|x - 2| - 3) + 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 5(2|(2sin(2\u03c0x - 1) + 2) - 1|)^2 - 1(2|(2sin(2\u03c0x - 1) + 2) - 1|) - 2 = -4((3x - 2)/(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": "11", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-3(-2|(-2x - 3) + 2| - 2) + 1)/(-1(-2|(-2x - 3) + 2| - 2)) = 1(-3cos(3\u03c0x - 2) + 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2cos(1\u03c0((2(-5x^2 - 2x + 5))/(-2(-5x^2 - 2x + 5) - 3)) + 3) - 3 = 2(-2|x - 2| + 2)^2 + 4(-2|x - 2| + 2) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (2(-2|(-2cos(3\u03c0x) + 1) - 1|) + 1)/(-1(-2|(-2cos(3\u03c0x) + 1) - 1|)) = 2(-2e^(-2x) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(-5(-3cos(3\u03c0(-4x) - 1) - 3)^2 - 1(-3cos(3\u03c0(-4x) - 1) - 3) - 1)| - 3 and g(x) = 2|(2(1cos(1\u03c0x + 3)) + 3)| + 1 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 points of intersection exist between the functions f(x) = -4(-3(1sin(3\u03c0x + 3))^2 - 4(1sin(3\u03c0x + 3)) + 1) + 1 and g(x) = -1(1x - 3)^2 + 3(1x - 3) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|(1((-2(-3e^(-1x) + 1) - 3)/(1(-3e^(-1x) + 1))) + 3)| and g(x) = -3((-1(3cos(3\u03c0x + 3) + 2))/(-3(3cos(3\u03c0x + 3) + 2) - 1)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3((2(-3|x| + 1) - 2)/(3(-3|x| + 1) + 3)) + 4 and g(x) = 2|(1sin(1\u03c0x) + 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": "18", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3((3(2cos(1\u03c0x + 3) - 1) - 2)/(-1(2cos(1\u03c0x + 3) - 1) + 2)) + 1 = -1(3cos(1\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": "20", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(1(-1|((-3x - 2)/(-1x - 2)) + 2| + 3)^2 - 1(-1|((-3x - 2)/(-1x - 2)) + 2| + 3) + 1) + 3 and g(x) = (-1(1(-3x^2 - 2x + 5) - 4))/(-3(1(-3x^2 - 2x + 5) - 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 solutions does the equation (1(3sin(1\u03c0(-1(-1x^2 - 1x - 5) + 1) - 3)) + 3)/(1(3sin(1\u03c0(-1(-1x^2 - 1x - 5) + 1) - 3)) + 2) = -1(-2|((-2x - 3)/(-2x - 2)) - 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": "64", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(1\u03c0(3|((1(-2x^2 + 5x - 3) + 2)/(-3(-2x^2 + 5x - 3) + 4)) + 3| - 1) + 3) + 3 and g(x) = (-3(-1sin(3\u03c0(-2x^2 - 1x + 5) - 2) + 1) - 3)/(3(-1sin(3\u03c0(-2x^2 - 1x + 5) - 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": "35", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(1((-3(1x^2 - 5x))/(-2(1x^2 - 5x) + 1))) - 2| - 1 and g(x) = -2|((2(1x^2 + 5x + 3) + 2)/(2(1x^2 + 5x + 3) + 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": "How many solutions does the equation 3cos(1\u03c0(-2((-2(-2x^2 - 2x + 2) + 3)/(-1(-2x^2 - 2x + 2) - 1)) + 1) + 2) = -3(1e^(-2((-3x + 1)/(2x - 3))) - 1)^2 - 3(1e^(-2((-3x + 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": "38", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(1|((1x + 3)/(2x - 1)) + 3| + 3) + 3 and g(x) = 3cos(3\u03c0(3x - 2) - 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "172", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(2|((-1x - 3)/(-3x - 2)) + 2| - 3)^2 - 2(2|((-1x - 3)/(-3x - 2)) + 2| - 3) + 4 and g(x) = 3|(-1cos(2\u03c0x - 2) - 1) - 3| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (1(-2|(-4x^2 - 2x - 1) + 1| + 2) - 3)/(-2(-2|(-4x^2 - 2x - 1) + 1| + 2)) = 2(2cos(1\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": "18", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-2(-5(-3sin(3\u03c0(-3|x - 2| + 3) - 1) - 3) - 2) + 3)/(1(-5(-3sin(3\u03c0(-3|x - 2| + 3) - 1) - 3) - 2) + 1) = -2(2|(-1x + 4) + 1| - 3)^2 + 1(2|(-1x + 4) + 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": "Determine the number of intersection points between the functions f(x) = -2sin(3\u03c0(-1|(4x - 1)| - 2) - 2) - 2 and g(x) = 3(3sin(3\u03c0x + 3) - 3) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(-2|((-2(4x + 1) - 2)/(-2(4x + 1) - 3)) - 1| + 1)^2 - 3(-2|((-2(4x + 1) - 2)/(-2(4x + 1) - 3)) - 1| + 1) + 2 and g(x) = -3|(3(1x^2 + 1x - 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": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(3|(-2cos(3\u03c0x - 2) + 3)| + 1)^2 - 5(3|(-2cos(3\u03c0x - 2) + 3)| + 1) + 1 and g(x) = 2|(-1x^2 + 4x + 1) - 3| + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(-3|(-3sin(2\u03c0(1x + 3) + 1)) - 1|)^2 - 5(-3|(-3sin(2\u03c0(1x + 3) + 1)) - 1|) + 4 and g(x) = 3(-2|(-4x^2 + 4x - 1)|) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(-2|(1(3e^(2x) + 2)^2 - 2(3e^(2x) + 2) - 2) + 2| - 3) - 1)/(3(-2|(1(3e^(2x) + 2)^2 - 2(3e^(2x) + 2) - 2) + 2| - 3) + 2) and g(x) = -5(3|(-3sin(1\u03c0x + 3)) - 2| - 3)^2 - 5(3|(-3sin(1\u03c0x + 3)) - 2| - 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-1(1|(1sin(1\u03c0x - 3) - 3) - 3|) + 2) + 3)/(-2(-1(1|(1sin(1\u03c0x - 3) - 3) - 3|) + 2) - 3) and g(x) = 1(-3(1cos(3\u03c0x) - 2)^2 - 4(1cos(3\u03c0x) - 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": "Determine the number of intersection points between the functions f(x) = (-2(1(-3sin(1\u03c0(5x^2 + 4x - 5) - 3) + 3) - 3) - 3)/(-2(1(-3sin(1\u03c0(5x^2 + 4x - 5) - 3) + 3) - 3) - 1) and g(x) = 4(-1(-3cos(1\u03c0x + 3) + 2) + 4)^2 + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "133", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(2|(1x + 5) - 2| - 3)^2 + 4(2|(1x + 5) - 2| - 3) + 1 and g(x) = 2|((-2x - 3)/(-3x - 3))| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(3\u03c0(-3((-2x + 3)/(-1x))^2 + 1((-2x + 3)/(-1x)) + 4) - 3) - 2 and g(x) = -1(-1e^(-2x) - 1)^2 - 3(-1e^(-2x) - 1) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(1((-1(3|x| + 2) - 3)/(-1(3|x| + 2) - 1))^2 - 4((-1(3|x| + 2) - 3)/(-1(3|x| + 2) - 1)) - 3) - 4 and g(x) = 3(-3(3sin(3\u03c0x + 1) - 2) - 5)^2 - 2(-3(3sin(3\u03c0x + 1) - 2) - 5) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(3\u03c0(-3((-3(1|x + 1|) - 3)/(1(1|x + 1|)))^2 + 3((-3(1|x + 1|) - 3)/(1(1|x + 1|))) + 3) + 2) - 2 and g(x) = -1|((2(2e^(1x) - 1) + 1)/(3(2e^(1x) - 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": "37", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 5(2sin(1\u03c0((-1x + 2)/(-3x - 1)) - 2) - 3)^2 + 3(2sin(1\u03c0((-1x + 2)/(-3x - 1)) - 2) - 3) - 3 = -3e^(2((-1x - 2)/(-3x - 1))) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1|(2e^(1((-2(-5x^2 - 5x - 1) - 2)/(-1(-5x^2 - 5x - 1) + 2))) + 1)| - 3 = 3sin(2\u03c0(5(-3|x + 1| - 3)) - 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3cos(3\u03c0((-2(1|(1x^2 - 2x - 3) + 3|) + 1)/(-1(1|(1x^2 - 2x - 3) + 3|) + 2)) + 3) - 1 and g(x) = -2cos(1\u03c0(4(3|x + 2| - 2)^2 + 2(3|x + 2| - 2) - 5) + 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1070", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(-2sin(1\u03c0(1(1|x - 3| + 3)^2 - 5(1|x - 3| + 3) + 2)) - 2))/(3(-2sin(1\u03c0(1(1|x - 3| + 3)^2 - 5(1|x - 3| + 3) + 2)) - 2) + 2) and g(x) = (-2(-4(-2cos(2\u03c0x - 2) + 2)) - 2)/(2(-4(-2cos(2\u03c0x - 2) + 2)) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "204", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(-1cos(2\u03c0(2x - 4) + 1))^2 - 4(-1cos(2\u03c0(2x - 4) + 1)) - 5 and g(x) = 2sin(3\u03c0(1x - 1)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5(2cos(3\u03c0(3|((-1x + 1)/(-2x - 2)) + 1| + 3) + 1)) + 3 and g(x) = 1((1(-3|x - 1| - 3))/(-1(-3|x - 1| - 3) - 2)) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "64", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3cos(1\u03c0(3|(2(5x - 5)^2 + 1(5x - 5) - 2) - 1| - 2) + 3) and g(x) = (-2(-4(2x - 2)^2 + 1(2x - 2) + 2) - 3)/(-1(-4(2x - 2)^2 + 1(2x - 2) + 2) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1284", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-1(3cos(3\u03c0(-5x^2 - 3x) - 3) + 2) + 2)/(-3(3cos(3\u03c0(-5x^2 - 3x) - 3) + 2) - 1) = -2(-1x - 5)^2 - 1(-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": "42", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -5((-2(1sin(1\u03c0x) + 1) + 2)/(-2(1sin(1\u03c0x) + 1) - 1))^2 + 5((-2(1sin(1\u03c0x) + 1) + 2)/(-2(1sin(1\u03c0x) + 1) - 1)) - 5 and g(x) = -5(1e^(1x) + 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1sin(2\u03c0(-3(1|((-2x + 1)/(-3x - 3)) - 1| - 2) + 1) + 3) and g(x) = -1|((1(2sin(2\u03c0x - 1) - 1) - 2)/(3(2sin(2\u03c0x - 1) - 1) + 1)) + 1| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "115", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-2(3|((-1x + 2)/(1x)) + 3| + 3) + 5)^2 + 5(-2(3|((-1x + 2)/(1x)) + 3| + 3) + 5) - 5 and g(x) = 2sin(1\u03c0(5(-4x^2 + 1x - 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": "10", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(1cos(1\u03c0(-3x - 1) - 2) + 2))/(2(1cos(1\u03c0(-3x - 1) - 2) + 2) + 1) and g(x) = (-1(1x^2 - 5x + 4) - 3)/(3(1x^2 - 5x + 4) + 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) = -3|(3((3(-1x + 2) - 3)/(-2(-1x + 2) + 1))^2 - 5) + 1| and g(x) = -4(3cos(3\u03c0(-1|x - 1| - 2) - 2) + 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(2sin(3\u03c0(3x - 1) + 1) + 1) - 1| + 2 and g(x) = 2|(2x) - 1| + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(2sin(1\u03c0(2((-3x - 3)/(-1x + 2))^2 + 1((-3x - 3)/(-1x + 2)) + 3) + 2) - 2) - 2 and g(x) = -2|(-5((-2x + 3)/(-2x - 1))^2 - 3((-2x + 3)/(-2x - 1)) + 5)| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "144", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(1(-1e^(-1(2|x| + 2)) + 2)^2 + 5(-1e^(-1(2|x| + 2)) + 2) + 3))/(-3(1(-1e^(-1(2|x| + 2)) + 2)^2 + 5(-1e^(-1(2|x| + 2)) + 2) + 3) - 2) and g(x) = (-2(-2(-2sin(1\u03c0x + 1))^2 - 3(-2sin(1\u03c0x + 1)) + 1) + 2)/(1(-2(-2sin(1\u03c0x + 1))^2 - 3(-2sin(1\u03c0x + 1)) + 1)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(3\u03c0(2|(-2(-3x^2 - 5x - 4) - 1)|) - 2) - 3 and g(x) = -3(-1sin(1\u03c0(-4x) - 2) + 1)^2 - 5(-1sin(1\u03c0(-4x) - 2) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "282", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(2sin(3\u03c0((1x + 1)/(1x + 2)) + 2)) - 4 and g(x) = -1(2|x - 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": "33", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3e^(-1(2|(-5x - 3)| + 1)) - 2 and g(x) = -2cos(1\u03c0(-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": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2((3(-2sin(1\u03c0x + 2) + 1) - 1)/(-1(-2sin(1\u03c0x + 2) + 1) + 3))^2 + 5((3(-2sin(1\u03c0x + 2) + 1) - 1)/(-1(-2sin(1\u03c0x + 2) + 1) + 3)) - 1 = (-3(-1x^2 + 3x - 3) + 2)/(-2(-1x^2 + 3x - 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": "Determine the number of intersection points between the functions f(x) = 2|(1((2(-1x + 1) - 3)/(-2(-1x + 1) - 2))^2 + 2((2(-1x + 1) - 3)/(-2(-1x + 1) - 2)) - 5) - 3| + 2 and g(x) = (-2(1cos(3\u03c0(1x + 5) + 1) + 1) + 2)/(1(1cos(3\u03c0(1x + 5) + 1) + 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": "Find the number of times the graphs of f(x) = -1(2|((-3x - 2)/(1x + 1)) + 1| - 3) + 1 and g(x) = 1sin(1\u03c0(1|x - 2| + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(2cos(3\u03c0(2|x - 2| - 2) + 2) - 2) + 2)/(-1(2cos(3\u03c0(2|x - 2| - 2) + 2) - 2) - 1) and g(x) = -2cos(3\u03c0(1x - 2) + 3) - 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": "Find the number of times the graphs of f(x) = -1cos(3\u03c0((-3(-4(4x^2 + 1x - 4) - 1) + 1)/(-3(-4(4x^2 + 1x - 4) - 1) - 3)) + 2) - 1 and g(x) = 2(3cos(1\u03c0(-3|x + 2| + 2) + 3) + 2) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3|(-2(1sin(1\u03c0x - 1) - 2)^2 - 3(1sin(1\u03c0x - 1) - 2) - 3) + 2| + 1 = -2sin(3\u03c0(4x - 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": "20", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(2(-3|(-5x^2 + 5x + 1) + 3|) - 1) - 1)/(-2(2(-3|(-5x^2 + 5x + 1) + 3|) - 1) - 1) and g(x) = (2(-2|(1cos(3\u03c0x + 3) - 2) + 1| + 2))/(3(-2|(1cos(3\u03c0x + 3) - 2) + 1| + 2) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2(1|(-3cos(1\u03c0x - 3) - 2) - 3| - 3) + 3 = 2|((-3x + 2)/(2x + 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": "18", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-5(-3sin(2\u03c0x) + 2) - 2) - 2)/(3(-5(-3sin(2\u03c0x) + 2) - 2) - 2) and g(x) = -2((-3x + 3)/(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": "41", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 4((-2(3|(-3x + 3)| + 1) + 2)/(-2(3|(-3x + 3)| + 1) + 3))^2 + 5((-2(3|(-3x + 3)| + 1) + 2)/(-2(3|(-3x + 3)| + 1) + 3)) + 1 = 2|((-1(-3sin(1\u03c0x + 2) - 2) - 2)/(1(-3sin(1\u03c0x + 2) - 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": "42", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(2|((-2(-2sin(3\u03c0x + 1) - 2) - 1)/(3(-2sin(3\u03c0x + 1) - 2) - 2)) + 3| - 1)^2 + 3(2|((-2(-2sin(3\u03c0x + 1) - 2) - 1)/(3(-2sin(3\u03c0x + 1) - 2) - 2)) + 3| - 1) + 3 and g(x) = (-1(3(1sin(1\u03c0x + 1) + 1)) + 2)/(1(3(1sin(1\u03c0x + 1) + 1))) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(1cos(3\u03c0(-4(-2x - 5)^2 - 2(-2x - 5) + 3) - 2) + 3) + 1)/(2(1cos(3\u03c0(-4(-2x - 5)^2 - 2(-2x - 5) + 3) - 2) + 3) + 1) and g(x) = (-1(4(1|x + 2| - 2) - 4) - 1)/(-1(4(1|x + 2| - 2) - 4)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(3sin(1\u03c0(-3|x - 1| + 3) + 3)) + 2)/(3(3sin(1\u03c0(-3|x - 1| + 3) + 3)) + 2) and g(x) = 1|((1x)/(1x - 3)) + 1| - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "57", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2sin(1\u03c0((-1(5x) + 3)/(3(5x) + 1)) + 2) - 1 = -1e^(1((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": "15", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3sin(2\u03c0(4(-3|x + 3| - 2)^2 - 5(-3|x + 3| - 2)) + 3) - 2 = (-3(2|x - 3| - 2) + 2)/(-1(2|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": "128", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1sin(3\u03c0(1|((-2(1x - 3) - 1)/(-1(1x - 3) + 2)) + 1| + 3) - 1) and g(x) = (-3(3|(4x - 3) - 2|) - 3)/(3(3|(4x - 3) - 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": "Find the number of times the graphs of f(x) = -2(-3|(-3sin(3\u03c0x + 3) - 3) + 2| + 2) and g(x) = (2(3x - 1) + 2)/(3(3x - 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "119", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((2(3cos(1\u03c0x + 3) + 1) - 2)/(1(3cos(1\u03c0x + 3) + 1)))^2 + 2((2(3cos(1\u03c0x + 3) + 1) - 2)/(1(3cos(1\u03c0x + 3) + 1))) + 1 and g(x) = -1(2cos(1\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": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(3\u03c0(-3|(1x^2 - 2x + 5) + 3|) + 3) - 2 and g(x) = 3(-3|x| + 3) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (2(-3(-1|x - 3| - 3) - 4) - 1)/(2(-3(-1|x - 3| - 3) - 4)) = 2(-3cos(1\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": "20", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3e^(-1(-3(-2x^2 + 4x) + 5)) - 2 and g(x) = 1|(1x^2 - 4x - 3) + 2| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(-2sin(1\u03c0(2(-3x + 3)^2) + 1) + 3) + 3| + 3 and g(x) = -5(-2((2x + 3)/(-2x + 3)) + 3)^2 + 3(-2((2x + 3)/(-2x + 3)) + 3) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-2(3(3cos(2\u03c0(3|x + 2| - 1)) - 2)^2 + 3(3cos(2\u03c0(3|x + 2| - 1)) - 2)) - 1)/(-2(3(3cos(2\u03c0(3|x + 2| - 1)) - 2)^2 + 3(3cos(2\u03c0(3|x + 2| - 1)) - 2))) = (1(2|(-2x + 1) + 2| - 3))/(-2(2|(-2x + 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": "167", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3sin(3\u03c0(-1|(-3(4x^2 - 3x + 3) - 5) - 2| - 3)) + 1 = -1(1sin(1\u03c0(-2x^2 + 2) + 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": "1101", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3e^(2(-2|(3x + 3) + 1| + 2)) + 1 and g(x) = 1cos(1\u03c0(-2|x + 3| - 3) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3((3(-4(-1|x + 1| + 3)^2 - 4(-1|x + 1| + 3) + 4))/(-2(-4(-1|x + 1| + 3)^2 - 4(-1|x + 1| + 3) + 4) - 3)) and g(x) = -3cos(1\u03c0(-3|(3x^2 - 1x - 5) - 1| + 2)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "258", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(1sin(1\u03c0(3|((1x + 3)/(3x + 3)) + 1| + 1) + 1) + 2)^2 - 5(1sin(1\u03c0(3|((1x + 3)/(3x + 3)) + 1| + 1) + 1) + 2) - 3 and g(x) = -2(-5(1|x - 2| - 3) + 2)^2 - 1(-5(1|x - 2| - 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-2(3|(-1x) + 1|)^2 - 3))/(2(-2(3|(-1x) + 1|)^2 - 3) - 1) and g(x) = 1(-5(2sin(2\u03c0x - 1) + 2)^2 - 5(2sin(2\u03c0x - 1) + 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": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (2(-1cos(2\u03c0(5(3x^2 + 5x + 5) - 2) + 1)))/(3(-1cos(2\u03c0(5(3x^2 + 5x + 5) - 2) + 1)) - 2) = 1|((-1(3cos(1\u03c0x - 1) - 3) + 3)/(1(3cos(1\u03c0x - 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": "287", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-3sin(3\u03c0((-1x - 3)/(-2x + 3)) - 1)) and g(x) = 1|((2x - 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": "80", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(-1(3sin(2\u03c0x + 3) - 1) - 1) - 1| + 1 and g(x) = 4(-3sin(1\u03c0x + 2) + 3)^2 + 2(-3sin(1\u03c0x + 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": "20", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-1((-2(-3|x + 1| - 2) + 3)/(-2(-3|x + 1| - 2) + 1))^2 - 1) + 3 and g(x) = (3(-1cos(2\u03c0(-4x^2 + 2x + 2) + 2) + 1) + 1)/(3(-1cos(2\u03c0(-4x^2 + 2x + 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": "251", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(-2|(-2((-3x + 3)/(2x - 1)) + 2) - 1| + 2)^2 - 4(-2|(-2((-3x + 3)/(2x - 1)) + 2) - 1| + 2) and g(x) = -1|(1((3x - 1)/(2x))^2 - 2((3x - 1)/(2x)) - 1) + 1| - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-5(-2|x| - 2) + 5) - 2)/(1(-5(-2|x| - 2) + 5) - 3) and g(x) = (1(3cos(2\u03c0x + 3) + 1) + 3)/(-2(3cos(2\u03c0x + 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": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(1cos(2\u03c0(-2|x + 2| + 3) + 2) + 2)^2 - 2(1cos(2\u03c0(-2|x + 2| + 3) + 2) + 2) + 2 and g(x) = 3|(3sin(3\u03c0x + 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": "100", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2cos(2\u03c0(1(-3|((-1x - 3)/(-2x + 3))| - 2) + 4)) - 2 and g(x) = -4(-5(2|x| - 2) + 3)^2 + 4(-5(2|x| - 2) + 3) + 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) = -5(1((1x + 3)/(-3x + 2)) + 3)^2 and g(x) = -4(1sin(3\u03c0x + 1) - 3)^2 + 2(1sin(3\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": "52", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(1|(3((2x)/(-1x - 3)) + 2)| - 1)^2 - 2(1|(3((2x)/(-1x - 3)) + 2)| - 1) - 2 and g(x) = 2cos(1\u03c0(2|(3x + 5) - 2| - 1) + 3) - 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) = 2(-1|(2sin(1\u03c0x + 1) + 1) - 3| + 1) + 3 and g(x) = 2cos(3\u03c0((-3x - 1)/(-1x - 2))) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|((2(2sin(3\u03c0x + 1) + 1) + 1)/(-2(2sin(3\u03c0x + 1) + 1) - 3)) + 1| - 1 and g(x) = 1cos(1\u03c0(-1x + 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": "20", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1|(4(-1cos(2\u03c0x + 2) - 1)^2 - 5(-1cos(2\u03c0x + 2) - 1)) + 2| - 1 = (-2(2sin(2\u03c0x + 1) - 2) + 1)/(3(2sin(2\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": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-2cos(2\u03c0(-1x^2 + 5x)) + 1))/(-1(-2cos(2\u03c0(-1x^2 + 5x)) + 1) + 2) and g(x) = -1(3cos(1\u03c0x + 1) - 2)^2 - 4(3cos(1\u03c0x + 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": "279", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-2(2(-1|x - 1| + 3) + 5) - 1)/(-3(2(-1|x - 1| + 3) + 5) + 2) = 3sin(2\u03c0(2|x|)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1cos(2\u03c0(5(2|x - 3| + 1)^2 - 5(2|x - 3| + 1) + 5) + 2) - 2 and g(x) = (-2(-1x^2 - 2x + 5) + 1)/(3(-1x^2 - 2x + 5) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|(3e^(2(-1x - 1)) - 3) - 2| + 3 and g(x) = 5(1sin(3\u03c0x - 2) + 1)^2 - 5(1sin(3\u03c0x - 2) + 1) - 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(3|(1(-3x^2 - 4x + 5) + 1) + 2|) - 1)/(-2(3|(1(-3x^2 - 4x + 5) + 1) + 2|) + 3) and g(x) = 3|(1(3x^2 + 4x - 4) + 5)| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3e^(2((-1(1x + 1))/(2(1x + 1)))) + 1 and g(x) = 3sin(3\u03c0(-1|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": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-2(-2sin(3\u03c0(2|x + 1| - 1)) - 2) + 2)/(-3(-2sin(3\u03c0(2|x + 1| - 1)) - 2) - 2) = (-3(2|x - 2| + 2) - 1)/(2(2|x - 2| + 2) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(2((1x - 1)/(-1x + 2))^2 - 5((1x - 1)/(-1x + 2)) - 5) + 2| + 1 and g(x) = 1sin(1\u03c0(-3|x + 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": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(3sin(1\u03c0((3(-3x + 5) + 1)/(1(-3x + 5))) + 3) + 2)^2 + 5 and g(x) = 1cos(2\u03c0(1(-1|x - 2| - 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": "36", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-3sin(3\u03c0((-2x)/(1x + 3)) + 1))^2 - 2(-3sin(3\u03c0((-2x)/(1x + 3)) + 1)) - 1 and g(x) = -5(1|x + 2| - 3) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2cos(2\u03c0(-2|(-1((-1x - 1)/(1x))^2 + 1((-1x - 1)/(1x))) - 3| + 1) - 2) - 2 and g(x) = 1(-3|((-3x + 3)/(1x)) - 3| + 3) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3sin(3\u03c0(1(1x^2 + 5x + 1) + 1)) - 2 and g(x) = 3cos(3\u03c0(-4x - 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": "474", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1((1(-1sin(1\u03c0(-1x^2 + 5x + 1)) + 1) - 2)/(2(-1sin(1\u03c0(-1x^2 + 5x + 1)) + 1) - 3)) + 5 and g(x) = -2(-3e^(2((2x + 3)/(2x + 1))) + 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1e^(2((-1(5x^2 + 3x + 2) - 3)/(1(5x^2 + 3x + 2) - 2))) - 2 = 2sin(2\u03c0((-1x - 3)/(-1x + 2)) - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "112", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((2(2|(1x - 3)| - 2) + 1)/(2(2|(1x - 3)| - 2) + 2))^2 + 5((2(2|(1x - 3)| - 2) + 1)/(2(2|(1x - 3)| - 2) + 2)) - 2 and g(x) = -1(-1cos(3\u03c0(1|x + 1| - 3) - 2) + 1)^2 + 4(-1cos(3\u03c0(1|x + 1| - 3) - 2) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2|(-1(1((2x)/(3x + 2)))^2 - 4(1((2x)/(3x + 2))) + 1) + 3| + 1 and g(x) = -4((1(1sin(3\u03c0x + 1) - 1) + 1)/(-3(1sin(3\u03c0x + 1) - 1)))^2 - 2((1(1sin(3\u03c0x + 1) - 1) + 1)/(-3(1sin(3\u03c0x + 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": "60", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(2((-3(3sin(2\u03c0x - 3)) + 3)/(-2(3sin(2\u03c0x - 3)) + 3)) + 2)^2 + 1(2((-3(3sin(2\u03c0x - 3)) + 3)/(-2(3sin(2\u03c0x - 3)) + 3)) + 2) + 4 and g(x) = -3|(1cos(2\u03c0(1x^2 - 3x + 1) - 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": "Find the number of times the graphs of f(x) = -2sin(1\u03c0(-2(-5((-2x - 2)/(-3x - 1))^2 - 1((-2x - 2)/(-3x - 1)) + 2) + 2) + 2) - 1 and g(x) = 1|(5(3x)^2 + 4(3x) - 4) + 3| - 3 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 -1sin(2\u03c0(-1|((-2x + 3)/(-2x - 1)) + 3| + 3) + 1) - 1 = (2(-1|x + 2| - 2) - 1)/(-2(-1|x + 2| - 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "76", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3cos(3\u03c0(-2|(3x^2) + 2| - 2) + 1) and g(x) = 2|(-2x^2 - 2x) + 3| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "83", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3cos(1\u03c0(-2|(-1x + 4) - 2|) + 3) - 2 and g(x) = -2(-3cos(2\u03c0x + 3) + 2)^2 - 5(-3cos(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": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1e^(-2((-1(2x))/(2(2x) - 3))) - 1 = 2cos(3\u03c0((-2x + 2)/(-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": "7", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2|(2((-2(-1x^2 + 4))/(3(-1x^2 + 4) - 2)) + 4) + 3| + 3 = -5(-1sin(1\u03c0(1|x + 3| + 1)) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 5((2(-2(-2|x - 1| + 2) - 5) - 1)/(-3(-2(-2|x - 1| + 2) - 5) + 4))^2 - 4((2(-2(-2|x - 1| + 2) - 5) - 1)/(-3(-2(-2|x - 1| + 2) - 5) + 4)) - 3 and g(x) = 2(-1|(3x + 1) - 1| + 3)^2 - 3(-1|(3x + 1) - 1| + 3) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1cos(3\u03c0(-1|(-3x^2 - 1x - 5) + 3| + 1) + 1) + 2 and g(x) = 2cos(1\u03c0(2|x - 3| + 3) + 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "533", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1sin(1\u03c0(1(-4(-1|x + 3| - 2) - 2)^2 - 5(-4(-1|x + 3| - 2) - 2)) + 2) + 2 and g(x) = 2|(-3e^(1((2x - 1)/(1x))) - 2) + 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(5((-2(-1|x + 2| + 1) - 2)/(-1(-1|x + 2| + 1))) + 1)^2 - 1(5((-2(-1|x + 2| + 1) - 2)/(-1(-1|x + 2| + 1))) + 1) and g(x) = -4(1(-2cos(2\u03c0x + 3) - 1) - 1)^2 - 3(1(-2cos(2\u03c0x + 3) - 1) - 1) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(3\u03c0(4(4(-1|x + 2| + 1)^2 + 1) - 4) - 1) + 3 and g(x) = 2|((-1(-2x^2 - 4x + 1) + 3)/(-2(-2x^2 - 4x + 1) - 3)) + 3| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "129", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(-2(3sin(3\u03c0(-2|x + 2| - 2) + 2) - 1))^2 + 3(-2(3sin(3\u03c0(-2|x + 2| - 2) + 2) - 1)) + 3 and g(x) = -2((-2(2x + 2))/(-1(2x + 2) - 3))^2 + 4((-2(2x + 2))/(-1(2x + 2) - 3)) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "149", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1|((-3(-3sin(2\u03c0(-5x - 3) - 3) - 2) - 1)/(-2(-3sin(2\u03c0(-5x - 3) - 3) - 2))) + 2| - 1 = 1|(-1cos(1\u03c0((-2x + 2)/(-2x + 3)) + 1) + 2) + 1| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "200", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1|(1(3(-3cos(3\u03c0x + 2) + 3)^2 + 3(-3cos(3\u03c0x + 2) + 3) + 5) + 5) - 3| = (2(2(2sin(1\u03c0x + 1) - 3) + 4))/(-3(2(2sin(1\u03c0x + 1) - 3) + 4) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2e^(1(-5(3x - 1)^2 + 2(3x - 1) + 5)) and g(x) = -4(2sin(2\u03c0x + 1)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-3sin(1\u03c0((3x - 1)/(1x + 3)) - 3) + 3)^2 - 1(-3sin(1\u03c0((3x - 1)/(1x + 3)) - 3) + 3) + 1 and g(x) = -1|(1x^2 - 2x + 4) - 1| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "107", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(5(4(3|x - 3|)^2 + 5(3|x - 3|)) + 2) + 1)/(-2(5(4(3|x - 3|)^2 + 5(3|x - 3|)) + 2) + 3) and g(x) = 2|(3sin(3\u03c0(3x^2 - 3x) + 2) - 3) - 2| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(2sin(2\u03c0(2|(-5x^2 - 1x + 1) + 1| + 1)) - 1) + 2 and g(x) = -3cos(1\u03c0(-2(1x + 1)^2 - 3(1x + 1) - 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": "912", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(2sin(2\u03c0(2(1x^2 + 3x + 3) - 1) - 1) + 2) + 3| and g(x) = 2|((-2(-5x + 4) - 2)/(-1(-5x + 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": "10", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2sin(3\u03c0(4(5x^2 - 1x - 3) - 5) + 3) + 2 and g(x) = -3cos(1\u03c0(1|x + 2| - 2) + 1) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "530", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-2(2x^2 - 5x) + 1) + 1)/(-3(-2(2x^2 - 5x) + 1) + 2) and g(x) = 2sin(3\u03c0(-3x^2 + 1x - 3) + 1) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "975", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(1((1(-3sin(3\u03c0x + 1) + 1) - 3)/(2(-3sin(3\u03c0x + 1) + 1) - 3))^2 + 5((1(-3sin(3\u03c0x + 1) + 1) - 3)/(2(-3sin(3\u03c0x + 1) + 1) - 3)) - 4) - 4 and g(x) = -1|(-1sin(1\u03c0(-3x^2 + 3x - 5) - 1) - 1) + 2| + 1 intersect in the interval [-10, 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) = -1(-3|(3cos(1\u03c0(2x^2 + 2x - 4) + 3))| - 2) - 5 and g(x) = 1|(2cos(1\u03c0(1x - 1) + 2) - 2) + 3| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "797", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1|(-3((2(3x^2 + 2x + 4) + 3)/(3(3x^2 + 2x + 4) + 3)) + 5) + 3| + 1 = (-1(2(-2sin(2\u03c0x + 2) - 1)^2 - 5(-2sin(2\u03c0x + 2) - 1) + 2))/(-3(2(-2sin(2\u03c0x + 2) - 1)^2 - 5(-2sin(2\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": "13", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1(2|(-1((1x - 1)/(-1x - 2)) - 4) + 3| + 2)^2 - 2(2|(-1((1x - 1)/(-1x - 2)) - 4) + 3| + 2) - 2 = (-2(-2(2cos(3\u03c0x - 1) - 2)^2 + 1))/(2(-2(2cos(3\u03c0x - 1) - 2)^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": "17", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(1|(-3((1x)/(3x))^2 - 1((1x)/(3x)) + 1) - 1|) - 5 and g(x) = -3(-2sin(2\u03c0((-1x - 1)/(-1x - 2))))^2 - 5(-2sin(2\u03c0((-1x - 1)/(-1x - 2)))) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "69", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3(-2cos(3\u03c0(2x) + 2))^2 + 5(-2cos(3\u03c0(2x) + 2)) - 3 = 1(1|x + 2| - 1)^2 + 5(1|x + 2| - 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-2(2(-3|(2x^2 + 4x - 2) - 1| - 1) - 1) + 3)/(-2(2(-3|(2x^2 + 4x - 2) - 1| - 1) - 1)) = -4(-2(-2sin(2\u03c0x + 1) - 3) - 2)^2 - 4(-2(-2sin(2\u03c0x + 1) - 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": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5((3(-1(2|x - 1| - 3)^2 + 4(2|x - 1| - 3) - 4) + 2)/(-3(-1(2|x - 1| - 3)^2 + 4(2|x - 1| - 3) - 4) + 4)) - 3 and g(x) = -1(2(-2sin(1\u03c0x + 1) - 3)^2 + 3(-2sin(1\u03c0x + 1) - 3) + 4) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(3(3|x - 1| + 2) - 4) + 1)/(2(3(3|x - 1| + 2) - 4) - 1) and g(x) = 2cos(1\u03c0(-5x^2 + 5x - 5) - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "985", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3|((-3(-5x - 4) + 3)/(-3(-5x - 4) + 1))| - 2 = -1(3cos(2\u03c0x - 1))^2 + 1(3cos(2\u03c0x - 1)) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-1|(2e^(1x) + 1) - 1| - 3) + 2)/(1(-1|(2e^(1x) + 1) - 1| - 3) + 1) and g(x) = -1cos(3\u03c0(-1x + 4) - 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(3e^(-1(1x^2 - 5x + 1)) - 2) - 2| + 3 and g(x) = 1(-2cos(3\u03c0x + 1) + 3)^2 + 2(-2cos(3\u03c0x + 1) + 3) - 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-5(3|(-2cos(1\u03c0x + 3)) + 1| - 1) + 3) + 3)/(-1(-5(3|(-2cos(1\u03c0x + 3)) + 1| - 1) + 3) + 3) and g(x) = -2sin(3\u03c0(3|((-1x - 1)/(2x + 3)) + 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": "48", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 4((-1(-2(2sin(1\u03c0x - 1) - 1)^2 - 4(2sin(1\u03c0x - 1) - 1)) + 3)/(-2(-2(2sin(1\u03c0x - 1) - 1)^2 - 4(2sin(1\u03c0x - 1) - 1)) + 1)) - 3 = -2|(-3sin(2\u03c0(1x^2 - 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": "108", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -5(3cos(1\u03c0((-1x - 3)/(-3x - 3)) + 1))^2 + 5(3cos(1\u03c0((-1x - 3)/(-3x - 3)) + 1)) - 1 = -1e^(1((-2x - 3)/(2x - 1))) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(2|(3cos(3\u03c0(2x + 5) - 3) + 2) + 1| - 3) - 1)/(-1(2|(3cos(3\u03c0(2x + 5) - 3) + 2) + 1| - 3) + 2) and g(x) = -3|(2e^(-2(2x - 4)) + 1) - 2| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (1(1e^(-1(5x^2 - 3x - 5)) - 3) - 3)/(2(1e^(-1(5x^2 - 3x - 5)) - 3) - 1) = -3(-3sin(3\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": "62", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2sin(2\u03c0(2|(-5((1x + 1)/(1x - 2))^2 - 1((1x + 1)/(1x - 2)) + 1) + 2| + 1) + 1) - 1 and g(x) = -3((-3(2x + 2) - 3)/(3(2x + 2) - 1))^2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(4((1(1x + 1) - 3)/(1(1x + 1) + 1))^2 + 1((1(1x + 1) - 3)/(1(1x + 1) + 1)) - 3) - 3| - 2 and g(x) = -4(-5(1sin(3\u03c0x + 2) - 1))^2 + 3(-5(1sin(3\u03c0x + 2) - 1)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-5(-1(-1|x - 3| + 1)^2 - 3(-1|x - 3| + 1) + 3) - 2) - 1)/(3(-5(-1(-1|x - 3| + 1)^2 - 3(-1|x - 3| + 1) + 3) - 2) + 3) and g(x) = 2sin(3\u03c0(-1|((-2x - 3)/(1x - 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": "168", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3cos(1\u03c0((3(-1x) + 2)/(-3(-1x) + 4)) - 1) + 2 and g(x) = 3|((-1x + 1)/(3x - 2))| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3((1(-3cos(1\u03c0x + 2) + 3) + 3)/(-3(-3cos(1\u03c0x + 2) + 3)))^2 - 3((1(-3cos(1\u03c0x + 2) + 3) + 3)/(-3(-3cos(1\u03c0x + 2) + 3))) + 5 = -3|(-3cos(3\u03c0x - 1) + 3) - 2| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(1\u03c0(-1|(3x + 4) - 1|) - 2) and g(x) = (3(-1e^(1x) + 1) + 2)/(-1(-1e^(1x) + 1) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1(-2|(-2cos(2\u03c0x + 2) - 1) + 1| + 2)^2 + 5(-2|(-2cos(2\u03c0x + 2) - 1) + 1| + 2) - 3 = -2(2x^2 - 5x - 1) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2((1(3sin(3\u03c0x - 2)) + 1)/(-1(3sin(3\u03c0x - 2)) + 3))^2 - 4((1(3sin(3\u03c0x - 2)) + 1)/(-1(3sin(3\u03c0x - 2)) + 3)) + 3 and g(x) = 2(-2|x - 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": "52", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(1\u03c0(-5(1x^2 - 4x - 3) + 4) + 1) - 3 and g(x) = -3|(2cos(3\u03c0x - 3) - 3) + 1| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "323", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4((-1(2(-1|x + 2| - 1)^2 + 3(-1|x + 2| - 1) + 5) + 2)/(1(2(-1|x + 2| - 1)^2 + 3(-1|x + 2| - 1) + 5) + 3)) + 1 and g(x) = -1(2|(3sin(3\u03c0x) - 2) - 2|) + 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1sin(3\u03c0(5((-1x - 1)/(3x - 1)) - 5) - 1) = (3(1x) - 2)/(-2(1x) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(1|(-3(3sin(3\u03c0x - 2) - 3)^2 - 3(3sin(3\u03c0x - 2) - 3) - 1) - 2| - 2) - 3)/(-1(1|(-3(3sin(3\u03c0x - 2) - 3)^2 - 3(3sin(3\u03c0x - 2) - 3) - 1) - 2| - 2) + 2) and g(x) = 1(-1|((1x - 3)/(1x + 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": "59", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(4(3|((1x - 2)/(3x)) - 3| - 3)^2 + 3(3|((1x - 2)/(3x)) - 3| - 3) - 5) + 4 and g(x) = (2(-1|(2x - 4) - 2| + 2) + 3)/(1(-1|(2x - 4) - 2| + 2)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-1(-3cos(2\u03c0(1(4x + 1)^2 + 4(4x + 1) - 2) + 3)) - 3)/(3(-3cos(2\u03c0(1(4x + 1)^2 + 4(4x + 1) - 2) + 3))) = (2(1|(1x^2 + 5x + 1) - 2| + 2) + 1)/(1(1|(1x^2 + 5x + 1) - 2| + 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "344", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2|((-1(1x^2 + 2x - 4) + 2)/(-1(1x^2 + 2x - 4) + 3)) - 1| + 3 = 5(3sin(3\u03c0x - 1) + 1)^2 - 1(3sin(3\u03c0x - 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": "120", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1((3(-3|x| - 2) + 2)/(-2(-3|x| - 2) - 2))^2 + 5((3(-3|x| - 2) + 2)/(-2(-3|x| - 2) - 2)) + 3 and g(x) = -5(-2cos(2\u03c0x + 2) + 1)^2 + 3(-2cos(2\u03c0x + 2) + 1) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(3\u03c0(-4(3|x - 1| - 2) + 4) + 2) + 1 and g(x) = 2cos(2\u03c0(2x^2 + 5x + 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "755", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(2((1x + 2)/(2x)) - 3)^2 - 1(2((1x + 2)/(2x)) - 3) + 1 and g(x) = -4(3cos(2\u03c0x - 3) - 3) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (3(3|(-1(-3e^(2x) - 1) - 1) + 1| - 1) + 2)/(-3(3|(-1(-3e^(2x) - 1) - 1) + 1| - 1) + 4) = -5(-4(-2cos(3\u03c0x + 1) - 2)^2 - 2(-2cos(3\u03c0x + 1) - 2) + 1) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2cos(1\u03c0((1(-1x^2 - 1x + 4) - 1)/(2(-1x^2 - 1x + 4) - 1)) + 2) + 1 and g(x) = -1(2sin(3\u03c0x - 2) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "69", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 4((-1(-4(2sin(3\u03c0x + 1)) + 3) + 2)/(1(-4(2sin(3\u03c0x + 1)) + 3) + 1))^2 - 5((-1(-4(2sin(3\u03c0x + 1)) + 3) + 2)/(1(-4(2sin(3\u03c0x + 1)) + 3) + 1)) + 3 = (3(-3|(3sin(2\u03c0x - 3) - 2) + 3| + 2) - 3)/(1(-3|(3sin(2\u03c0x - 3) - 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": "80", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-1(-1sin(3\u03c0(4x + 1) - 1) - 3) - 2)/(2(-1sin(3\u03c0(4x + 1) - 1) - 3) + 1) = 4(2cos(2\u03c0x - 1) + 2) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2(-2sin(1\u03c0(3|x + 3| - 3) - 1)) - 1 and g(x) = 2sin(2\u03c0(-3x + 5) + 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-2sin(3\u03c0(-2|(-4x + 3) + 2|) + 3) - 2)^2 - 5 and g(x) = -2cos(1\u03c0(4(-2x + 4)^2 - 4(-2x + 4) - 2) + 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "478", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(3\u03c0((3(2x - 1) + 2)/(1(2x - 1) - 3)) - 2) - 2 and g(x) = -2|(1cos(2\u03c0x - 3) + 1) - 1| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "144", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2e^(1(-3|(3((1x - 3)/(-2x + 3))^2 + 2((1x - 3)/(-2x + 3)) + 4) + 1| + 3)) + 2 and g(x) = (2(-2cos(3\u03c0(1|x - 1| - 1) - 2) + 2) - 2)/(3(-2cos(3\u03c0(1|x - 1| - 1) - 2) + 2) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(3\u03c0(1|(-4((1x + 3)/(-3x + 4)) + 4) + 1| - 2) - 2) and g(x) = (1(1|(1x^2) - 1| - 2) - 3)/(-1(1|(1x^2) - 1| - 2)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|(-1(1((-3x + 2)/(2x))^2 + 4((-3x + 2)/(2x)) - 5) - 2) + 1| + 2 and g(x) = 2cos(2\u03c0(1((1x - 1)/(1x + 1))^2 - 5((1x - 1)/(1x + 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": "9", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2cos(2\u03c0(-2|((2(5x - 3) - 2)/(1(5x - 3) + 3)) + 3| - 1) - 2) and g(x) = -3|(-2(2sin(3\u03c0x - 2) + 1)^2 - 4(2sin(3\u03c0x - 2) + 1)) + 2| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "65", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-3(1sin(1\u03c0x) + 2)^2 + 1(1sin(1\u03c0x) + 2) + 3) - 4 and g(x) = -1|((1x + 1)/(-3x - 1)) + 1| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1cos(1\u03c0(1|(4x - 5) + 3| + 1) + 3) + 1 and g(x) = (1(-2|x + 1|) - 2)/(2(-2|x + 1|) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "74", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1cos(3\u03c0(-4(-2|x| + 2)^2 + 1(-2|x| + 2)) - 1) + 3 and g(x) = 2e^(-1(-2|x + 3| + 1)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-4(-1cos(3\u03c0x + 1)) - 4) + 2)/(-3(-4(-1cos(3\u03c0x + 1)) - 4) - 1) and g(x) = 1cos(1\u03c0((-1x + 1)/(-1x - 1)) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3(-1|(3cos(3\u03c0((-2x - 2)/(-3x + 1)) - 1) - 3) + 1| - 1) - 1 = 1e^(-1(-2(1|x - 3| - 2)^2 - 1(1|x - 3| - 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": "56", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1((-1(1sin(3\u03c0(-3x - 3) - 2) - 3) - 3)/(1(1sin(3\u03c0(-3x - 3) - 2) - 3) + 2))^2 - 4((-1(1sin(3\u03c0(-3x - 3) - 2) - 3) - 3)/(1(1sin(3\u03c0(-3x - 3) - 2) - 3) + 2)) - 4 and g(x) = 2(2cos(2\u03c0(1x^2 + 5x + 3) - 2) + 1) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "450", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1|(3sin(3\u03c0((2x)/(-3x + 4)) + 2) + 1)| - 2 = (2(-5x^2 - 5x + 5) + 1)/(-2(-5x^2 - 5x + 5)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(2(3|(1sin(2\u03c0x - 2) + 1) + 1| + 2) + 5)^2 + 2(2(3|(1sin(2\u03c0x - 2) + 1) + 1| + 2) + 5) + 2 and g(x) = (1(2sin(3\u03c0(2x^2 - 4x - 4) + 2) + 1) + 2)/(-3(2sin(3\u03c0(2x^2 - 4x - 4) + 2) + 1) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(1cos(1\u03c0(-5(5x + 1)^2 + 5(5x + 1) - 3))) - 3)/(2(1cos(1\u03c0(-5(5x + 1)^2 + 5(5x + 1) - 3))) + 3) and g(x) = 1((-2(2sin(2\u03c0x - 2) + 1) + 3)/(3(2sin(2\u03c0x - 2) + 1) - 2))^2 - 1((-2(2sin(2\u03c0x - 2) + 1) + 3)/(3(2sin(2\u03c0x - 2) + 1) - 2)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1131", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2|((-1(4x^2 - 2x + 1) - 2)/(1(4x^2 - 2x + 1) - 3)) - 1| + 1 = 1sin(3\u03c0((-2x)/(2x - 1))) 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) = 3(-1|(3((-2x + 3)/(-1x - 3)) - 2) + 3| + 3)^2 + 2(-1|(3((-2x + 3)/(-1x - 3)) - 2) + 3| + 3) and g(x) = -1((2(-3sin(3\u03c0x + 2) + 3) + 2)/(-3(-3sin(3\u03c0x + 2) + 3)))^2 - 2((2(-3sin(3\u03c0x + 2) + 3) + 2)/(-3(-3sin(3\u03c0x + 2) + 3))) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(1(-1(-2|x - 3|)^2 + 1)) - 1)/(1(1(-1(-2|x - 3|)^2 + 1))) and g(x) = -1((-2(1cos(2\u03c0x) - 2) + 2)/(-1(1cos(2\u03c0x) - 2))) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-2|((1x + 2)/(2x - 1)) + 1|)^2 + 1(-2|((1x + 2)/(2x - 1)) + 1|) + 4 and g(x) = -1sin(2\u03c0(-1x^2 + 3x + 3) - 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|(2(-2cos(1\u03c0((1x - 3)/(-1x)) - 3) - 3)^2 - 5(-2cos(1\u03c0((1x - 3)/(-1x)) - 3) - 3) - 3) + 2| + 3 and g(x) = (-1(-1cos(1\u03c0(4x + 3) - 2) + 2) + 2)/(-1(-1cos(1\u03c0(4x + 3) - 2) + 2) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "82", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(-1(3(-1|x - 2| + 3) - 4)^2 + 4(3(-1|x - 2| + 3) - 4) - 3) - 3)/(2(-1(3(-1|x - 2| + 3) - 4)^2 + 4(3(-1|x - 2| + 3) - 4) - 3)) and g(x) = 1|((-2(1sin(3\u03c0x - 1) + 3) - 2)/(-2(1sin(3\u03c0x - 1) + 3) + 1)) - 1| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(4((2(3x^2 - 3) - 1)/(-1(3x^2 - 3) - 2)) + 4) - 3| + 1 and g(x) = 5(2cos(2\u03c0(-3|x - 3| + 2) + 3) - 2)^2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "116", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3(3sin(3\u03c0((-2x - 3)/(-1x))) - 1) - 4 = -3|(2cos(3\u03c0x + 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": "129", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(3cos(2\u03c0((-1x - 3)/(-2x + 3)))) + 5 and g(x) = -2e^(-2((-2x + 1)/(1x - 2))) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2sin(1\u03c0(-5(-3|x - 2|)^2 + 1(-3|x - 2|) + 2) - 3) - 2 and g(x) = (-3(2sin(1\u03c0x + 3) + 3) - 2)/(2(2sin(1\u03c0x + 3) + 3) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1460", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2|(-3cos(2\u03c0((-1(-3x^2 + 3x) - 1)/(-2(-3x^2 + 3x) + 3)) + 3) - 1) - 2| + 1 = (-3(2|(2e^(1x) + 3)| - 1) + 1)/(-2(2|(2e^(1x) + 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": "8", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(3(2|x - 3| - 3) + 1) - 3)/(1(3(2|x - 3| - 3) + 1) - 3) and g(x) = 1cos(2\u03c0(3x^2 + 4) - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1168", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3cos(2\u03c0(-2(1((-3x - 1)/(2x - 3))^2 - 2) + 4)) - 3 and g(x) = -3(1e^(1((-1x + 2)/(1x))) - 3)^2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(-2(-2cos(3\u03c0((-2x + 2)/(3x - 2)) + 2) - 2)^2 - 5(-2cos(3\u03c0((-2x + 2)/(3x - 2)) + 2) - 2) + 1) + 3| + 1 and g(x) = -2(-1sin(3\u03c0((2x - 1)/(-1x + 2)) + 3) + 1)^2 + 5(-1sin(3\u03c0((2x - 1)/(-1x + 2)) + 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": "16", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(2(2cos(1\u03c0((-2x + 1)/(1x)) + 3) + 1) + 2) + 1| - 3 and g(x) = -1e^(2(2|(-4x - 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": "Determine the number of intersection points between the functions f(x) = 2sin(3\u03c0(-1(2x + 5)^2 - 1(2x + 5) - 1) - 1) + 3 and g(x) = -2cos(1\u03c0((3x - 1)/(2x + 2)) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "449", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|((-2(5x) + 1)/(3(5x) + 1)) - 3| + 3 and g(x) = 3(1sin(2\u03c0x - 3) + 3)^2 - 4(1sin(2\u03c0x - 3) + 3) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1sin(2\u03c0(-1|((-1x - 2)/(3x)) - 3| - 1) + 1) = -1(-1sin(3\u03c0x + 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": "73", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3|(2(2sin(3\u03c0x))^2 + 4(2sin(3\u03c0x)) - 3) + 2| = (-2(-2sin(3\u03c0x + 3) + 3) - 1)/(1(-2sin(3\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": "60", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-2sin(1\u03c0(4x - 3)))^2 + 2(-2sin(1\u03c0(4x - 3))) + 3 and g(x) = -2|(-1sin(2\u03c0x + 1) + 3) - 2| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "160", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1sin(2\u03c0(-5(1|x| - 3) + 2) + 1) + 1 = 1(-1|x - 1| + 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(2cos(3\u03c0(-4(-1|x| + 2) + 3) + 3))^2 - 4(2cos(3\u03c0(-4(-1|x| + 2) + 3) + 3)) - 5 and g(x) = 3(-2((-3x + 1)/(-1x + 3)) - 4)^2 + 5(-2((-3x + 1)/(-1x + 3)) - 4) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 4(-3sin(3\u03c0((-3x + 3)/(2x + 2))) + 1) + 4 and g(x) = 2(-1x - 1)^2 + 5(-1x - 1) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-4(2|(2x^2 - 1x - 2) - 1| - 3)) + 3)/(1(-4(2|(2x^2 - 1x - 2) - 1| - 3)) + 2) and g(x) = -4(3cos(1\u03c0((-1x - 1)/(-1x - 2)) + 3) + 3)^2 + 1(3cos(1\u03c0((-1x - 1)/(-1x - 2)) + 3) + 3) - 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(5(-1|x - 2| - 2) + 2) - 1)/(1(5(-1|x - 2| - 2) + 2) + 3) and g(x) = -2|(1sin(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": "40", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(-1sin(1\u03c0(-2|(4x^2 - 1x + 5) - 2| + 3) - 3) - 2) - 5 and g(x) = -4((-2(-4x^2 - 4x + 4) + 1)/(-3(-4x^2 - 4x + 4) + 2)) + 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 (3(-4(-5x^2 + 1x - 4) - 2) + 2)/(-3(-4(-5x^2 + 1x - 4) - 2) + 4) = 2sin(1\u03c0(-3x^2 - 4x + 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": "604", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -4((3(3sin(3\u03c0x + 1)) + 1)/(-1(3sin(3\u03c0x + 1)) + 2)) - 5 = -1(3|x| - 1)^2 - 3(3|x| - 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2(-2|(-1cos(1\u03c0((-1x - 1)/(2x - 3)) + 1)) - 2| + 2)^2 + 3(-2|(-1cos(1\u03c0((-1x - 1)/(2x - 3)) + 1)) - 2| + 2) + 5 = 2|(-1(1x^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": "23", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2sin(3\u03c0((3(-5x) - 3)/(2(-5x))) + 2) - 2 = 1(-2sin(3\u03c0x + 3) + 1)^2 + 5(-2sin(3\u03c0x + 3) + 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1cos(1\u03c0(5(-3|x - 1| - 2)^2 + 5) - 1) = 2sin(1\u03c0(2x^2 + 1x - 5) - 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "701", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3sin(2\u03c0((-2(3x^2 - 4x + 4) + 2)/(-1(3x^2 - 4x + 4) - 1)) - 1) + 3 and g(x) = -3sin(3\u03c0((-2x + 3)/(1x + 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": "58", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(2\u03c0(-1((-1x + 3)/(-2x - 1))) + 2) - 1 and g(x) = (-1(2e^(1x) - 2) + 2)/(-2(2e^(1x) - 2) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "66", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(2e^(-1(2x - 5)) - 3) - 1| - 3 and g(x) = 3sin(1\u03c0(-2x - 1) + 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3((-2(-1e^(1x) + 1) - 2)/(2(-1e^(1x) + 1) - 1)) + 3 and g(x) = 1|(-2cos(3\u03c0x + 2) + 1) + 1| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(1sin(2\u03c0(-1x^2 + 5x - 1) + 3) - 2)| - 1 and g(x) = -4(2e^(1x) + 1)^2 + 1(2e^(1x) + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "270", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2((-1(-1sin(1\u03c0x + 3) - 3) - 2)/(-1(-1sin(1\u03c0x + 3) - 3)))^2 + 1((-1(-1sin(1\u03c0x + 3) - 3) - 2)/(-1(-1sin(1\u03c0x + 3) - 3))) and g(x) = -2sin(3\u03c0(5x) - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "258", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2((1(2|(-1cos(2\u03c0x - 3) + 3) + 3| - 2) + 3)/(2(2|(-1cos(2\u03c0x - 3) + 3) + 3| - 2) + 1))^2 + 2((1(2|(-1cos(2\u03c0x - 3) + 3) + 3| - 2) + 3)/(2(2|(-1cos(2\u03c0x - 3) + 3) + 3| - 2) + 1)) - 4 = -5(3sin(3\u03c0(5x + 5) + 3) + 2)^2 + 2(3sin(3\u03c0(5x + 5) + 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": "600", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(-1|((2x - 2)/(-2x - 1)) + 3| - 2) - 2 and g(x) = (-2(2cos(1\u03c0x - 3) - 1) - 2)/(2(2cos(1\u03c0x - 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 solutions does the equation 3(-3|((-1(-1x^2 - 5x - 3) - 1)/(2(-1x^2 - 5x - 3)))| - 2) + 5 = 1(1sin(3\u03c0(1x^2 + 1x - 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": "44", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(2sin(1\u03c0(-4x - 3) - 3) + 1) - 3)/(2(2sin(1\u03c0(-4x - 3) - 3) + 1) - 1) and g(x) = 1|(1x^2 + 4x - 1)| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(-3|((2x + 3)/(-3x))| + 1) - 3 and g(x) = 3|(1cos(3\u03c0x - 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": "24", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(2sin(2\u03c0(-3|(-1x^2 - 2x + 4) + 1| - 1) - 3) + 3) and g(x) = -1((1(2cos(2\u03c0x + 3) + 1) - 3)/(-2(2cos(2\u03c0x + 3) + 1) - 1))^2 - 3((1(2cos(2\u03c0x + 3) + 1) - 3)/(-2(2cos(2\u03c0x + 3) + 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": "130", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(2((-2(3sin(2\u03c0x - 2) + 2) - 1)/(3(3sin(2\u03c0x - 2) + 2))) - 1)^2 + 5(2((-2(3sin(2\u03c0x - 2) + 2) - 1)/(3(3sin(2\u03c0x - 2) + 2))) - 1) - 4 and g(x) = -1|(-5(-1sin(3\u03c0x + 3) + 3)^2 - 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) = -3cos(3\u03c0(4(2x^2 + 3x + 4)) - 1) - 1 and g(x) = 4(-2cos(2\u03c0x + 3))^2 + 2(-2cos(2\u03c0x + 3)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "210", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2cos(3\u03c0(3|(2(1x^2 + 2x + 2) + 3) - 1| + 2) - 1) + 3 = -2|(3(1x^2 - 4x + 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": "25", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(3((1x - 2)/(-1x - 1))^2 + 4((1x - 2)/(-1x - 1)) + 2) - 2| + 3 and g(x) = -2sin(2\u03c0(4x - 5) - 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2((-2(2cos(2\u03c0x + 2) - 2) - 2)/(-3(2cos(2\u03c0x + 2) - 2) - 1)) + 4 and g(x) = -3|(-1e^(-1x)) - 3| + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(-3cos(2\u03c0(4x^2 - 5x) - 1) - 1) - 4 and g(x) = (1(1|x + 3| + 1) + 3)/(-3(1|x + 3| + 1)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1131", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(2\u03c0(1((2(2x + 3) - 3)/(-1(2x + 3) - 1))^2 + 1) + 3) and g(x) = (-2(-3|(-4x + 2) + 1| + 2) + 2)/(-3(-3|(-4x + 2) + 1| + 2) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "165", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1((-2(-1x - 4) - 3)/(2(-1x - 4) + 2))^2 + 5((-2(-1x - 4) - 3)/(2(-1x - 4) + 2)) + 5 and g(x) = -2(3sin(2\u03c0x - 2) + 2)^2 + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-2(3cos(3\u03c0(-1x^2 - 1x + 2)) + 3) + 3)/(-1(3cos(3\u03c0(-1x^2 - 1x + 2)) + 3)) = 3sin(1\u03c0(4x - 4) - 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "495", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-1sin(1\u03c0(-2(-4x + 4)^2 - 1(-4x + 4) + 1) + 2) - 3) + 2| - 3 and g(x) = 5(-5((1x - 3)/(1x + 3)) - 3)^2 - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3cos(1\u03c0(5(-3|x + 1| + 1)^2) - 1) + 3 = -1|((2x)/(-2x - 2))| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "399", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(3\u03c0(-5(2|(-4x^2 - 4x - 2)| + 1) + 1) + 1) and g(x) = 1(-4(-2|x - 1| + 1)^2 - 2(-2|x - 1| + 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": "35", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1e^(1(-1(-3|(-5x^2 - 3x + 1) + 1| + 3) - 1)) - 1 and g(x) = 2cos(2\u03c0(1(2|x - 2| + 3) - 2)) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-1(4(2|x + 1| - 3) + 4)^2 + 4(4(2|x + 1| - 3) + 4) + 5) - 1)/(-1(-1(4(2|x + 1| - 3) + 4)^2 + 4(4(2|x + 1| - 3) + 4) + 5) + 2) and g(x) = 1(-2sin(1\u03c0(-3x) + 2))^2 + 2(-2sin(1\u03c0(-3x) + 2)) - 4 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-3sin(1\u03c0(2x^2 - 2) + 1) + 2) - 3)/(-1(-3sin(1\u03c0(2x^2 - 2) + 1) + 2) - 1) and g(x) = (1(2x^2 - 3x + 4) - 2)/(-3(2x^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": "294", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(3\u03c0(-5((1(4x + 4))/(-1(4x + 4) - 3))^2 - 5((1(4x + 4))/(-1(4x + 4) - 3)) + 3) - 3) - 1 and g(x) = (1(2(-1|x - 3| + 1)) - 2)/(2(2(-1|x - 3| + 1)) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "122", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(-3sin(3\u03c0((3(-1|x - 3| - 1) + 1)/(2(-1|x - 3| - 1) - 3)) - 2) - 1) - 4 and g(x) = -2(-2cos(3\u03c0(3|x + 1| + 1) + 2) + 1)^2 - 3(-2cos(3\u03c0(3|x + 1| + 1) + 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": "115", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3(1|(3cos(1\u03c0x + 3) - 1) - 2| - 3)^2 + 2(1|(3cos(1\u03c0x + 3) - 1) - 2| - 3) + 3 = 3e^(-1(-3x + 5)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -5((-1(2cos(3\u03c0x + 1) + 3) + 3)/(-2(2cos(3\u03c0x + 1) + 3)))^2 - 3((-1(2cos(3\u03c0x + 1) + 3) + 3)/(-2(2cos(3\u03c0x + 1) + 3))) - 2 and g(x) = 2cos(3\u03c0(-5x + 1) - 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "300", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -4((-1(1|(-2x + 2) + 1| - 1))/(1(1|(-2x + 2) + 1| - 1)))^2 - 2((-1(1|(-2x + 2) + 1| - 1))/(1(1|(-2x + 2) + 1| - 1))) + 4 = -3cos(1\u03c0(3((2x - 1)/(2x))^2 - 2((2x - 1)/(2x))) - 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "67", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3((-1(4(-1|x - 1| + 1) - 3) - 2)/(2(4(-1|x - 1| + 1) - 3)))^2 + 1((-1(4(-1|x - 1| + 1) - 3) - 2)/(2(4(-1|x - 1| + 1) - 3))) + 3 and g(x) = 2|(-4((1x)/(2x + 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": "How many solutions does the equation 5((1(-1e^(-1x) + 3) + 3)/(-2(-1e^(-1x) + 3) + 3))^2 - 5((1(-1e^(-1x) + 3) + 3)/(-2(-1e^(-1x) + 3) + 3)) = 3sin(1\u03c0(1x - 2) + 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(1|(2cos(2\u03c0x + 3) + 2)|)^2 + 5(1|(2cos(2\u03c0x + 3) + 2)|) - 2 and g(x) = (-2(2cos(3\u03c0x - 1) + 2))/(-1(2cos(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": "40", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(1|(-1(3e^(-1x) - 3) + 1) - 3|)^2 + 1 and g(x) = -1(1sin(2\u03c0(-3|x + 1| - 3)) - 2)^2 + 1(1sin(2\u03c0(-3|x + 1| - 3)) - 2) + 5 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": "Find the number of times the graphs of f(x) = 1|(4(-3sin(3\u03c0(-5x^2 - 3x + 2) + 2) - 3) - 1) - 1| - 2 and g(x) = (1(1cos(1\u03c0(1|x - 1|)) - 3))/(-1(1cos(1\u03c0(1|x - 1|)) - 3) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "583", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(-1(2cos(2\u03c0x - 2) - 1)^2 + 3(2cos(2\u03c0x - 2) - 1) + 1) - 5 and g(x) = -1e^(1((-2x - 3)/(2x - 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": "How many solutions does the equation -5((-1(-1|x + 2|) - 3)/(1(-1|x + 2|) + 2))^2 + 4((-1(-1|x + 2|) - 3)/(1(-1|x + 2|) + 2)) + 5 = -1cos(2\u03c0(3x^2 + 3x + 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": "423", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2cos(2\u03c0(4((1(-1x + 3))/(-3(-1x + 3) + 2))^2 + 3((1(-1x + 3))/(-3(-1x + 3) + 2)) - 2) + 3) - 2 and g(x) = (2(-3|(-3x^2 + 1x + 3) - 1|))/(-1(-3|(-3x^2 + 1x + 3) - 1|) + 2) 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 points of intersection exist between the functions f(x) = 3(-1|((-3(3cos(1\u03c0x - 1) - 3))/(2(3cos(1\u03c0x - 1) - 3) - 3)) + 1| - 2) and g(x) = 4(1sin(2\u03c0(-1|x|) - 3) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-1(2e^(-1(-1|x| + 2)) + 2)^2 + 3(2e^(-1(-1|x| + 2)) + 2) - 5) - 4 and g(x) = -5(3sin(2\u03c0(-1x - 1) + 2) + 3)^2 - 1(3sin(2\u03c0(-1x - 1) + 2) + 3) + 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) = 2(-2|((3(-1x^2 + 5x + 2) + 2)/(1(-1x^2 + 5x + 2) + 3)) + 2| + 2) + 2 and g(x) = (-3(2e^(-1(3x^2 - 5x - 4)) - 2))/(2(2e^(-1(3x^2 - 5x - 4)) - 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 4(-2sin(2\u03c0(1|x + 1|) - 2)) + 4 = (-3(-2x^2 - 5x - 5) - 2)/(-1(-2x^2 - 5x - 5) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(-3cos(2\u03c0(-1|x - 3| - 3) + 2) + 1) - 2)/(-2(-3cos(2\u03c0(-1|x - 3| - 3) + 2) + 1) + 3) and g(x) = 2(-1cos(1\u03c0x - 1) + 2)^2 - 3(-1cos(1\u03c0x - 1) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(-2((3(2sin(1\u03c0x - 3) + 2) - 1)/(-3(2sin(1\u03c0x - 3) + 2) + 1))^2 - 3((3(2sin(1\u03c0x - 3) + 2) - 1)/(-3(2sin(1\u03c0x - 3) + 2) + 1)) - 1) - 2 and g(x) = -4(-3cos(2\u03c0(-5x - 5) - 3) - 3)^2 + 2(-3cos(2\u03c0(-5x - 5) - 3) - 3) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "200", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -5((2(-3cos(1\u03c0(2x - 3) + 1)) + 3)/(3(-3cos(1\u03c0(2x - 3) + 1)) + 2))^2 - 2((2(-3cos(1\u03c0(2x - 3) + 1)) + 3)/(3(-3cos(1\u03c0(2x - 3) + 1)) + 2)) + 5 = -3(2|(1x^2 + 1x) - 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": "77", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(5(-3x^2 + 2x + 3) + 4) - 1)/(-2(5(-3x^2 + 2x + 3) + 4) + 3) and g(x) = -2(3sin(2\u03c0x + 2) + 2) + 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 solutions does the equation -1|((2(3sin(2\u03c0x + 1) + 2) + 3)/(-2(3sin(2\u03c0x + 1) + 2) + 1)) - 3| + 3 = 4(-3sin(1\u03c0x - 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": "80", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-3sin(3\u03c0(-1|x - 1| + 2) - 3) - 2) + 1)/(2(-3sin(3\u03c0(-1|x - 1| + 2) - 3) - 2) - 3) and g(x) = (-3(3sin(3\u03c0x + 2) + 1) - 3)/(2(3sin(3\u03c0x + 2) + 1) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-2(-5(2(-1e^(-2x) + 2) + 5)^2 - 5) + 2)/(-3(-5(2(-1e^(-2x) + 2) + 5)^2 - 5) + 4) = -2sin(1\u03c0(3(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": "180", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(1((-2(-2sin(1\u03c0x + 2) + 3) - 3)/(-3(-2sin(1\u03c0x + 2) + 3))) + 2)^2 - 4(1((-2(-2sin(1\u03c0x + 2) + 3) - 3)/(-3(-2sin(1\u03c0x + 2) + 3))) + 2) - 2 and g(x) = 3(-2cos(2\u03c0((1x + 3)/(-2x + 3)) + 1) - 2)^2 + 4(-2cos(2\u03c0((1x + 3)/(-2x + 3)) + 1) - 2) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "78", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1(-4(-1cos(3\u03c0((1x)/(-2x - 2)) + 3) - 1)^2 - 1(-1cos(3\u03c0((1x)/(-2x - 2)) + 3) - 1) + 2) - 2 and g(x) = (2(-3(5x^2 + 1x - 2) + 4) - 3)/(2(-3(5x^2 + 1x - 2) + 4) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(-3((3x)/(-2x - 3))^2 - 5((3x)/(-2x - 3)) - 1)| and g(x) = 1|(1cos(3\u03c0x - 2) + 3) + 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-2cos(1\u03c0(-3|x| + 1) + 2) + 3)^2 + 4(-2cos(1\u03c0(-3|x| + 1) + 2) + 3) and g(x) = 1(2cos(3\u03c0x + 2) - 2)^2 + 3(2cos(3\u03c0x + 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": "58", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1((-3(2|(-3sin(3\u03c0x - 1)) - 1| - 1) + 2)/(-2(2|(-3sin(3\u03c0x - 1)) - 1| - 1) - 3))^2 - 2((-3(2|(-3sin(3\u03c0x - 1)) - 1| - 1) + 2)/(-2(2|(-3sin(3\u03c0x - 1)) - 1| - 1) - 3)) - 2 and g(x) = (-2(-2|(-1x^2 + 2x) + 2|) - 3)/(3(-2|(-1x^2 + 2x) + 2|) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "124", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(-3sin(2\u03c0(2(1x^2 - 5x - 5) + 1)) - 2) - 2)/(-3(-3sin(2\u03c0(2(1x^2 - 5x - 5) + 1)) - 2) + 1) and g(x) = (-2(3(-1sin(1\u03c0x + 1) + 1) + 4))/(-1(3(-1sin(1\u03c0x + 1) + 1) + 4) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "349", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1((3(-5(3|x + 2| - 2) + 4) - 1)/(1(-5(3|x + 2| - 2) + 4)))^2 - 4((3(-5(3|x + 2| - 2) + 4) - 1)/(1(-5(3|x + 2| - 2) + 4))) - 4 = (2(-3sin(2\u03c0(-2|x + 2|) + 2) - 1) - 3)/(-2(-3sin(2\u03c0(-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": "80", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(-1(-3|(-5x^2 - 1) - 2| - 1) + 4) - 3)/(-2(-1(-3|(-5x^2 - 1) - 2| - 1) + 4)) and g(x) = -1(-2|(-1cos(1\u03c0x) + 2) - 2| + 1)^2 - 4(-2|(-1cos(1\u03c0x) + 2) - 2| + 1) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3((3(2x^2 - 2x + 3) - 1)/(3(2x^2 - 2x + 3) - 2)) + 3 and g(x) = 2cos(1\u03c0(-5x - 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": "100", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1((-2(4x + 2) + 2)/(1(4x + 2) + 1))^2 + 1((-2(4x + 2) + 2)/(1(4x + 2) + 1)) + 3 and g(x) = (-1(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": "40", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(3(-1sin(2\u03c0(2|x - 2| + 2) - 2)) + 3) + 1)/(3(3(-1sin(2\u03c0(2|x - 2| + 2) - 2)) + 3) - 1) and g(x) = 5((-1(-3|x + 1| - 2) - 2)/(-3(-3|x + 1| - 2)))^2 + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(1sin(1\u03c0(-3|((-1x)/(2x - 1))| - 3) - 1) - 1) - 3 and g(x) = (2(-1(-3|x + 3| + 2)^2 + 5(-3|x + 3| + 2) + 4) + 2)/(2(-1(-3|x + 3| + 2)^2 + 5(-3|x + 3| + 2) + 4)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1cos(3\u03c0((-3(-5(5x^2 + 2x + 5) + 2) + 1)/(1(-5(5x^2 + 2x + 5) + 2) + 3)) + 2) - 2 and g(x) = 2sin(1\u03c0(3|(1x - 2) + 2| + 1) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2((2(3sin(3\u03c0(-3|x + 3| + 1) - 1) + 1) - 2)/(-2(3sin(3\u03c0(-3|x + 3| + 1) - 1) + 1) - 3))^2 + 2((2(3sin(3\u03c0(-3|x + 3| + 1) - 1) + 1) - 2)/(-2(3sin(3\u03c0(-3|x + 3| + 1) - 1) + 1) - 3)) + 1 and g(x) = -2(2|(4x^2 - 1x)|) - 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "254", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 5(-2|(1x^2 - 5x - 5) + 2| + 1) + 1 = (1(-3cos(2\u03c0x - 1) + 2) - 2)/(-2(-3cos(2\u03c0x - 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": "7", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-2sin(2\u03c0(-2x^2 - 2x + 1) - 2) + 3) - 1| + 2 and g(x) = -4(2x^2 + 3x - 5) + 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-3(-3cos(2\u03c0(-1|x - 1| + 2) + 3) + 1) - 2)/(1(-3cos(2\u03c0(-1|x - 1| + 2) + 3) + 1)) = (-2(-3|x - 2| + 3) - 2)/(-3(-3|x - 2| + 3) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(-1(1cos(3\u03c0x) - 2)^2 - 2(1cos(3\u03c0x) - 2) - 5) + 3| + 2 and g(x) = 3(3cos(2\u03c0x - 2) + 1)^2 + 5(3cos(2\u03c0x - 2) + 1) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1|(-3cos(2\u03c0(-4(1x)^2 - 3(1x) + 2)) - 1) - 1| + 1 = 4(-2((1x - 1)/(-2x - 1))^2 + 5((1x - 1)/(-2x - 1)) + 2) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "456", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3cos(1\u03c0(-3(-3|(-5x^2 + 4x - 5) + 2| - 3) - 2)) - 2 = -3|(-1(-3x - 4)^2 + 5(-3x - 4) - 3) + 3| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1|((2(-5x^2 - 1x + 5) - 1)/(3(-5x^2 - 1x + 5) - 2))| - 1 = 3cos(1\u03c0(1|x - 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": "22", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(3sin(1\u03c0((-1x - 1)/(-2x + 1))) + 2) - 4 and g(x) = (2(-2x^2 + 2x + 3) - 3)/(1(-2x^2 + 2x + 3) + 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2cos(1\u03c0(1(-5x^2 - 4x) - 4) - 3) - 1 = -1cos(3\u03c0(-2|x + 2| + 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": "982", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(2cos(1\u03c0(3|((1x)/(3x - 2)) + 2| + 2) - 3) - 1)^2 + 5(2cos(1\u03c0(3|((1x)/(3x - 2)) + 2| + 2) - 3) - 1) - 1 and g(x) = -1|(-5(-2cos(2\u03c0x + 1) - 3) + 5)| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1(1|(1((-1x - 2)/(1x + 3)))| + 3)^2 - 1 = 2|(3sin(3\u03c0((3x)/(-1x + 2)) - 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": "155", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1sin(1\u03c0(1|((-1(1x + 5) - 1)/(-3(1x + 5) + 4)) - 3| + 3) + 3) - 1 and g(x) = 1(3|(-2x^2 - 5x + 5) + 1| - 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 points of intersection exist between the functions f(x) = (-3(-1sin(1\u03c0(3|(-1x^2 - 2x + 1) - 1| - 3) + 2) - 1) - 2)/(1(-1sin(1\u03c0(3|(-1x^2 - 2x + 1) - 1| - 3) + 2) - 1)) and g(x) = (2(-1(2x - 5)^2 + 4(2x - 5) + 1) + 2)/(-3(-1(2x - 5)^2 + 4(2x - 5) + 1) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "600", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(1e^(-2(3|(-1x^2 + 3x - 1) + 3| - 3))) - 3)/(1(1e^(-2(3|(-1x^2 + 3x - 1) + 3| - 3))) + 2) and g(x) = -4(2cos(3\u03c0((-1x + 3)/(3x + 2)) + 1) - 1)^2 + 4(2cos(3\u03c0((-1x + 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": "66", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(3|(1x - 3) + 2| + 1) + 3)/(1(3|(1x - 3) + 2| + 1) - 3) and g(x) = (-1(1sin(1\u03c0x) - 2) - 2)/(2(1sin(1\u03c0x) - 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": "Find the number of times the graphs of f(x) = 1cos(2\u03c0(-5(-1|x - 3| - 2)^2 - 2(-1|x - 3| - 2) + 1) + 1) + 3 and g(x) = -5((2x - 2)/(-3x - 2)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "524", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (3(2cos(3\u03c0(2(-1|x| - 1)^2 + 5(-1|x| - 1) - 3) + 1) - 3) + 1)/(-1(2cos(3\u03c0(2(-1|x| - 1)^2 + 5(-1|x| - 1) - 3) + 1) - 3) + 2) = (-2(2sin(1\u03c0(5x^2 + 3x + 3) - 1) - 3) + 3)/(3(2sin(1\u03c0(5x^2 + 3x + 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": "1146", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(3(2sin(3\u03c0((-3x - 2)/(-1x - 1))) + 1)^2 - 2(2sin(3\u03c0((-3x - 2)/(-1x - 1))) + 1) - 1) + 1| and g(x) = -3(2|((-2x - 2)/(3x - 2)) - 2|)^2 - 5(2|((-2x - 2)/(3x - 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": "20", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1|(-1cos(1\u03c0(1(-2x - 5)^2 + 4(-2x - 5) + 2) - 2))| - 2 and g(x) = 3(-3e^(1((3x + 3)/(-2x - 3))))^2 - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3cos(2\u03c0(-2|((-2(-2x^2 - 5x - 3) - 3)/(-1(-2x^2 - 5x - 3) + 3)) - 2| - 2) - 2) - 3 = 2|(-1((1x + 1)/(-1x + 3))^2 + 1((1x + 1)/(-1x + 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": "12", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2e^(1((-3(4(2|x + 2| + 1)^2 + 5(2|x + 2| + 1) - 4) - 2)/(-3(4(2|x + 2| + 1)^2 + 5(2|x + 2| + 1) - 4) + 2))) - 2 and g(x) = 1sin(1\u03c0(-1|(4x + 5) + 3| - 3) + 3) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-1sin(2\u03c0(-2|(1x) + 2| - 1) + 2) - 1) - 2)/(1(-1sin(2\u03c0(-2|(1x) + 2| - 1) + 2) - 1) + 3) and g(x) = -2(-2cos(3\u03c0(-1|x - 1| + 3))) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(3|(3((-3x)/(3x + 2)) - 1) + 3| - 2)^2 + 4(3|(3((-3x)/(3x + 2)) - 1) + 3| - 2) + 2 and g(x) = (-2(-3sin(1\u03c0(2x^2 + 1x + 5) + 2) + 3) + 2)/(3(-3sin(1\u03c0(2x^2 + 1x + 5) + 2) + 3) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-2e^(-1(-3(2|x| + 2)^2 - 4(2|x| + 2) + 1)) + 1) + 2)/(-1(-2e^(-1(-3(2|x| + 2)^2 - 4(2|x| + 2) + 1)) + 1) + 2) and g(x) = -3cos(1\u03c0(-3|((-3x - 1)/(-3x + 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": "51", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2cos(1\u03c0(2((2x)/(3x - 2)) - 5) + 1) = (2(-1e^(1x) + 2))/(3(-1e^(1x) + 2) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2|(1sin(3\u03c0((1x - 1)/(-1x + 3)) - 3) + 2) - 1| + 3 = 2cos(1\u03c0(1x) + 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "77", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2(-1|((2x - 2)/(-3x + 4)) + 2| - 1) - 2 = 1(-2cos(3\u03c0x + 1) + 1)^2 - 2(-2cos(3\u03c0x + 1) + 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "116", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1cos(3\u03c0(-2|(4x^2 + 1)| + 1) - 1) - 1 and g(x) = 1sin(1\u03c0((-3x - 2)/(3x - 1)) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "198", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (1(1|(2(3x + 2)^2 - 2(3x + 2) - 3) - 1|) + 3)/(1(1|(2(3x + 2)^2 - 2(3x + 2) - 3) - 1|) - 3) = -4((2(3sin(1\u03c0x) - 3) + 1)/(3(3sin(1\u03c0x) - 3))) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(3\u03c0((2(2|x - 1| + 3) + 1)/(1(2|x - 1| + 3) + 3)) - 2) and g(x) = -3(-1sin(1\u03c0x + 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": "Find the number of times the graphs of f(x) = 4(-3sin(2\u03c0((3x + 3)/(1x - 2))) + 2)^2 + 3(-3sin(2\u03c0((3x + 3)/(1x - 2))) + 2) and g(x) = -3e^(-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": "12", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(1sin(1\u03c0(-3|(1x^2 - 4x + 4) - 1| + 3) - 3) - 1) - 3)/(-3(1sin(1\u03c0(-3|(1x^2 - 4x + 4) - 1| + 3) - 3) - 1) - 1) and g(x) = -4(-2|(3x + 5) + 3| - 2)^2 - 4(-2|(3x + 5) + 3| - 2) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|((2(1x - 5) - 1)/(2(1x - 5))) - 1| + 2 and g(x) = 3sin(3\u03c0(-4x^2 + 1x - 4) - 2) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1493", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-1(-4(3sin(3\u03c0(-4x^2 - 4x + 2)) - 3) + 2) + 1)/(1(-4(3sin(3\u03c0(-4x^2 - 4x + 2)) - 3) + 2)) = 1((-1(1x^2 + 5x + 2) - 1)/(-2(1x^2 + 5x + 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": "153", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2|(-2sin(1\u03c0(2((1x)/(-3x - 1)) + 3) - 1) - 3) + 2| = 3cos(1\u03c0((3(-1x^2 + 4x + 2) + 2)/(-1(-1x^2 + 4x + 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": "72", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-5(-1cos(3\u03c0x - 3)) + 4) - 1)/(3(-5(-1cos(3\u03c0x - 3)) + 4) + 1) and g(x) = -5(-3|x - 2| + 1)^2 + 5(-3|x - 2| + 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1(-5((2(-3|x| + 1) + 2)/(3(-3|x| + 1) + 1)) + 1)^2 - 1(-5((2(-3|x| + 1) + 2)/(3(-3|x| + 1) + 1)) + 1) - 1 = -4(3sin(3\u03c0(-2|x + 3| + 3)) + 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "116", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|((-2(1x^2 - 4x) + 2)/(3(1x^2 - 4x) + 1)) - 2| and g(x) = -2cos(3\u03c0(4x + 1) - 1) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "217", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(-1(-1sin(2\u03c0x + 1) - 1) - 3) + 2| and g(x) = (-2(-4x^2 + 1x - 4) + 3)/(-2(-4x^2 + 1x - 4) + 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(-2(1|x - 3| - 2) - 3)^2 - 5(-2(1|x - 3| - 2) - 3) - 2 and g(x) = 1cos(3\u03c0(1x^2 - 4x - 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": "12", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1((-3(-1|(-3x) + 1|))/(-3(-1|(-3x) + 1|) + 4))^2 + 3((-3(-1|(-3x) + 1|))/(-3(-1|(-3x) + 1|) + 4)) = (2(3(3cos(1\u03c0x - 2) - 3)^2 - 1(3cos(1\u03c0x - 2) - 3)) + 1)/(1(3(3cos(1\u03c0x - 2) - 3)^2 - 1(3cos(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": "Determine the number of intersection points between the functions f(x) = -1|((2(-2sin(3\u03c0x - 2) + 1))/(-2(-2sin(3\u03c0x - 2) + 1) - 1)) + 1| + 2 and g(x) = 3e^(-2(-2|x + 3| + 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -5((-1(-2x^2 - 1x - 5) + 2)/(-1(-2x^2 - 1x - 5) + 2)) + 3 = -2|(1sin(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": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-1(3sin(2\u03c0(-2|x + 3| - 2) - 3) - 1)^2 - 4(3sin(2\u03c0(-2|x + 3| - 2) - 3) - 1) + 2) + 1)/(-2(-1(3sin(2\u03c0(-2|x + 3| - 2) - 3) - 1)^2 - 4(3sin(2\u03c0(-2|x + 3| - 2) - 3) - 1) + 2) + 3) and g(x) = (3(-4(2sin(2\u03c0x + 2) + 3) + 5) + 1)/(-1(-4(2sin(2\u03c0x + 2) + 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": "108", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(3e^(2(-3|x| - 3)) + 3) + 2)/(2(3e^(2(-3|x| - 3)) + 3) + 3) and g(x) = 1|(3sin(1\u03c0x)) + 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": "Determine the number of intersection points between the functions f(x) = -2|(1sin(3\u03c0(5(-2x + 3)^2 - 5(-2x + 3) - 1) - 1) - 3) + 3| + 3 and g(x) = 4(1sin(3\u03c0((-1x + 1)/(1x - 2)) + 3) - 1)^2 - 3(1sin(3\u03c0((-1x + 1)/(1x - 2)) + 3) - 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "96", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2e^(-1((1(5(-4x^2 - 3x + 2) - 2) - 1)/(-2(5(-4x^2 - 3x + 2) - 2) - 1))) - 1 = 2|((1(-2sin(2\u03c0x + 3) + 2))/(3(-2sin(2\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": "80", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (3(-1|(-4(-3x - 1)^2 + 2(-3x - 1) + 4) - 1| + 2) - 1)/(-2(-1|(-4(-3x - 1)^2 + 2(-3x - 1) + 4) - 1| + 2)) = (-3(1cos(1\u03c0(3|x + 1| - 2) - 3) + 2) + 2)/(-1(1cos(1\u03c0(3|x + 1| - 2) - 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": "64", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3((2(-3(-2|x - 2| + 3) + 5))/(3(-3(-2|x - 2| + 3) + 5) + 1))^2 - 4((2(-3(-2|x - 2| + 3) + 5))/(3(-3(-2|x - 2| + 3) + 5) + 1)) + 5 and g(x) = 2|(1(5x - 1)^2 - 5(5x - 1) + 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": "Determine the number of intersection points between the functions f(x) = (-1(2|(3(3x^2 + 2x - 4) + 2) - 1| - 2) - 1)/(2(2|(3(3x^2 + 2x - 4) + 2) - 1| - 2) - 1) and g(x) = 1(-1|(3x^2 - 1x + 2)| + 2) + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3|((1(3e^(1x) + 2) + 3)/(-3(3e^(1x) + 2) - 1))| - 3 = 2cos(2\u03c0((-2x + 2)/(3x)) + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(1|(3sin(3\u03c0(5x + 5) - 2) + 3)|) + 2)/(2(1|(3sin(3\u03c0(5x + 5) - 2) + 3)|) - 1) and g(x) = (-1(-2e^(1(-1|x - 1| - 2)) + 2) - 2)/(-3(-2e^(1(-1|x - 1| - 2)) + 2) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "300", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(4(3(3|x - 3| - 1) + 4)^2 - 1(3(3|x - 3| - 1) + 4) - 1) - 2)/(-2(4(3(3|x - 3| - 1) + 4)^2 - 1(3(3|x - 3| - 1) + 4) - 1) - 2) and g(x) = -2cos(3\u03c0(1(-2|x - 1| - 2)^2 + 5(-2|x - 1| - 2) - 2) - 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1511", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|((2(-4(-2x)^2 + 3(-2x) - 1) - 1)/(-1(-4(-2x)^2 + 3(-2x) - 1) + 3))| - 1 and g(x) = (-1(-5(-2sin(3\u03c0x + 3) - 2) - 4) - 2)/(-1(-5(-2sin(3\u03c0x + 3) - 2) - 4) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|((2(-4x^2 - 5x - 1) + 3)/(-3(-4x^2 - 5x - 1) - 1)) + 3| - 2 and g(x) = -1(1|x + 2|) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(1|(-2x + 1) + 1|)^2 - 4(1|(-2x + 1) + 1|) + 3 and g(x) = -3cos(3\u03c0(4x) + 1) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3sin(3\u03c0(-2(1x^2 + 4x + 5) - 4) + 2) - 1 and g(x) = (1(1x^2 + 4x - 5) - 1)/(-1(1x^2 + 4x - 5) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1150", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3cos(1\u03c0(-3(-1|x| + 2)^2 + 2(-1|x| + 2) + 2) + 2) - 1 = 3sin(2\u03c0(-4x + 2) + 2) have for x in the interval [-10, 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|(4((2(-1e^(1x) - 3) + 1)/(-3(-1e^(1x) - 3) + 1))^2 - 3((2(-1e^(1x) - 3) + 1)/(-3(-1e^(1x) - 3) + 1)) - 1) - 3| + 2 = -4(-2sin(3\u03c0((-2x - 3)/(-1x - 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": "65", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3|(1cos(3\u03c0(-1(2x + 2)^2 + 5(2x + 2) - 3) + 1) + 3) + 3| + 3 and g(x) = -2|((1(2x - 1) + 3)/(-1(2x - 1) + 3)) + 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": "Determine the number of intersection points between the functions f(x) = -1|((-2(3(-2x - 1)^2 - 5) + 2)/(1(3(-2x - 1)^2 - 5))) + 2| - 2 and g(x) = -3|(3(2sin(3\u03c0x - 2) + 3)^2 - 2(2sin(3\u03c0x - 2) + 3) + 1) - 3| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2e^(1((-2(5(-3|x + 1| - 1) + 3) + 3)/(-1(5(-3|x + 1| - 1) + 3) - 3))) and g(x) = -3(2cos(1\u03c0((2x - 3)/(1x + 2))) + 2)^2 - 5(2cos(1\u03c0((2x - 3)/(1x + 2))) + 2) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "86", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(1\u03c0(5(1x^2 - 3x - 4) - 2) - 2) - 1 and g(x) = (-2(2e^(-1x) + 1) - 1)/(3(2e^(-1x) + 1) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1009", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(-5((2(-2cos(3\u03c0x + 1)) - 1)/(-1(-2cos(3\u03c0x + 1)) - 2)) - 2)^2 - 2 and g(x) = -4(-1e^(-1((2x + 1)/(1x + 1))) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "118", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1((-3(4(-3|x - 3| - 1)^2 + 3(-3|x - 3| - 1) + 2) - 2)/(-1(4(-3|x - 3| - 1)^2 + 3(-3|x - 3| - 1) + 2) - 1)) - 5 and g(x) = 3cos(1\u03c0((-1(-2x^2 - 2x + 5) - 2)/(1(-2x^2 - 2x + 5) + 1)) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(4(3((2x - 1)/(-3x - 2)) + 1)^2 - 5(3((2x - 1)/(-3x - 2)) + 1) - 4) + 3| - 1 and g(x) = (2(5(1cos(3\u03c0x - 1) - 1) + 4) - 2)/(-3(5(1cos(3\u03c0x - 1) - 1) + 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": "Find the number of times the graphs of f(x) = (3(3sin(3\u03c0(5(-1x + 4)^2 - 2) - 2)) + 1)/(2(3sin(3\u03c0(5(-1x + 4)^2 - 2) - 2)) + 2) and g(x) = -2e^(2(-1((-2x + 1)/(1x))^2 - 4((-2x + 1)/(1x)) - 4)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "529", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(-1|(3sin(3\u03c0(-2x^2 - 5x + 5) - 3) - 3) + 3| + 1) - 1)/(1(-1|(3sin(3\u03c0(-2x^2 - 5x + 5) - 3) - 3) + 3| + 1) + 2) and g(x) = (1(2(-2|x + 3| - 1)^2 + 5(-2|x + 3| - 1) - 4) - 1)/(1(2(-2|x + 3| - 1)^2 + 5(-2|x + 3| - 1) - 4) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2(3sin(3\u03c0((3(2|x - 3| - 3) + 1)/(-3(2|x - 3| - 3))) - 3)) + 1 = -3((-1(-2x^2 - 5x + 1) + 2)/(1(-2x^2 - 5x + 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": "51", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(5((-2(2|x + 1| - 2))/(1(2|x + 1| - 2) + 1)) - 3)^2 - 5(5((-2(2|x + 1| - 2))/(1(2|x + 1| - 2) + 1)) - 3) - 4 and g(x) = -4(-1|((1x)/(-3x + 1)) - 1| - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (3(1(-1|(-2sin(3\u03c0x + 2) + 1) + 2| + 3)^2 + 3(-1|(-2sin(3\u03c0x + 2) + 1) + 2| + 3) - 5) + 2)/(-3(1(-1|(-2sin(3\u03c0x + 2) + 1) + 2| + 3)^2 + 3(-1|(-2sin(3\u03c0x + 2) + 1) + 2| + 3) - 5) + 2) = 3(-2|(-1cos(3\u03c0x) + 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": "118", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1|(3(3sin(1\u03c0((3x - 2)/(-1x - 3)) + 3))^2 + 2(3sin(1\u03c0((3x - 2)/(-1x - 3)) + 3)) - 1) - 2| + 3 = -3e^(1(-3((2x + 1)/(3x + 3))^2 + 3((2x + 1)/(3x + 3)) + 2)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((-1(1sin(1\u03c0x - 1)) + 3)/(3(1sin(1\u03c0x - 1)) - 2))^2 + 5((-1(1sin(1\u03c0x - 1)) + 3)/(3(1sin(1\u03c0x - 1)) - 2)) - 2 and g(x) = (-3(1cos(2\u03c0x - 3) + 1) + 1)/(2(1cos(2\u03c0x - 3) + 1) - 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": "Find the number of times the graphs of f(x) = -2sin(1\u03c0((-3(2x) + 3)/(1(2x) - 3)) - 3) + 1 and g(x) = 1cos(1\u03c0(5x^2 - 2x - 1) - 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "327", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(5(2sin(1\u03c0x - 1) - 1) + 4)| - 3 and g(x) = -1|(-2x - 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": "Find the number of times the graphs of f(x) = (-3(1|(1x^2 + 3x + 2)| - 2) - 3)/(2(1|(1x^2 + 3x + 2)| - 2) + 1) and g(x) = 5(-1cos(3\u03c0x - 3) - 1)^2 - 2(-1cos(3\u03c0x - 3) - 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": "How many solutions does the equation -2(-3cos(3\u03c0(1((-1x - 1)/(1x - 2)) + 5) + 3) + 3)^2 - 3(-3cos(3\u03c0(1((-1x - 1)/(1x - 2)) + 5) + 3) + 3) + 5 = 4(-5((3x + 3)/(3x))^2 + 1((3x + 3)/(3x)) - 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "98", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1(-1(1|((3x - 1)/(-3x - 1))| - 2)^2 + 5(1|((3x - 1)/(-3x - 1))| - 2) - 4) - 3 = (-3(2|(-3cos(2\u03c0x - 3) + 2) - 2|) - 1)/(3(2|(-3cos(2\u03c0x - 3) + 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": "54", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2(-2cos(2\u03c0(5x^2 + 1) + 1) + 3) + 3 and g(x) = (-3(-1cos(2\u03c0x - 1) + 2) + 3)/(2(-1cos(2\u03c0x - 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": "1302", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-2sin(2\u03c0(2x^2 - 4x + 1) - 1)) + 3)/(3(-2sin(2\u03c0(2x^2 - 4x + 1) - 1)) - 1) and g(x) = -5(1|x + 3| - 1)^2 - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1cos(3\u03c0((3(-2|x + 1| - 2) + 3)/(1(-2|x + 1| - 2) + 3)) + 2) + 2 and g(x) = (-3(1sin(3\u03c0x - 2) + 1) + 1)/(1(1sin(3\u03c0x - 2) + 1) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "74", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|((-3(4(-5x^2 + 4x + 2) + 2) - 1)/(3(4(-5x^2 + 4x + 2) + 2) - 2))| + 1 and g(x) = -2|((3(1x^2 - 4) + 1)/(2(1x^2 - 4) - 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": "Determine the number of intersection points between the functions f(x) = (-1(-2|(2cos(2\u03c0x - 2) - 2) - 3| + 3))/(-2(-2|(2cos(2\u03c0x - 2) - 2) - 3| + 3) + 1) and g(x) = 3cos(2\u03c0(3x - 2) - 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 -1cos(1\u03c0(-1((2x + 2)/(2x + 3))^2 + 3((2x + 2)/(2x + 3)) + 5) + 3) + 1 = -2cos(2\u03c0(2|x - 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": "89", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(1cos(2\u03c0(-2|(-1x^2 + 4x + 2) + 3| - 3) - 3) - 1))/(-2(1cos(2\u03c0(-2|(-1x^2 + 4x + 2) + 3| - 3) - 3) - 1) - 3) and g(x) = (-1(-1(-1|x + 2| - 3) - 5) + 2)/(3(-1(-1|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": "76", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(3((2(3|x + 3|) - 2)/(2(3|x + 3|) - 3))^2 + 2) - 3 and g(x) = (-3(2sin(3\u03c0(-2|x - 2| - 3) + 2) - 1) - 2)/(2(2sin(3\u03c0(-2|x - 2| - 3) + 2) - 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "117", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2((-1(4x^2 + 3x + 5))/(3(4x^2 + 3x + 5) - 2)) + 4 and g(x) = 2sin(2\u03c0((-3x)/(-2x + 3)) + 1) + 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": "Determine the number of intersection points between the functions f(x) = -3|(2sin(1\u03c0(5(-2x + 2)^2 - 1(-2x + 2) + 4) + 3))| + 3 and g(x) = 3|((-2(2sin(2\u03c0x - 3)) - 1)/(3(2sin(2\u03c0x - 3)) + 1)) + 1| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "612", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(3sin(3\u03c0(5x - 5) + 1) + 1) + 1| - 2 and g(x) = -1|(2x + 3) + 3| + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(1(-1(1|x - 3| + 2)^2 + 2(1|x - 3| + 2) + 4) - 3))/(3(1(-1(1|x - 3| + 2)^2 + 2(1|x - 3| + 2) + 4) - 3) + 1) and g(x) = 4(1sin(3\u03c0((3x - 3)/(-2x + 1)) + 3) + 1) - 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": "Determine the number of intersection points between the functions f(x) = 4((-1(3|x - 2| + 1) - 2)/(-2(3|x - 2| + 1) + 1))^2 - 3((-1(3|x - 2| + 1) - 2)/(-2(3|x - 2| + 1) + 1)) + 3 and g(x) = 3cos(3\u03c0(-3x^2 + 2x - 1) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "801", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 4(2e^(-1(-1x^2 + 5x - 4)) - 2) - 4 = (-1(-2sin(3\u03c0x + 3) - 3) - 2)/(2(-2sin(3\u03c0x + 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": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1e^(1(1(1(-2|x + 2|)^2 + 5(-2|x + 2|) + 4) + 3)) - 2 and g(x) = 3sin(3\u03c0(-1(-2|x - 2| + 2)^2 + 1(-2|x - 2| + 2) + 3)) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-5(2cos(1\u03c0(5x - 5) - 3) + 1)^2 + 1(2cos(1\u03c0(5x - 5) - 3) + 1) - 4) + 1)/(-1(-5(2cos(1\u03c0(5x - 5) - 3) + 1)^2 + 1(2cos(1\u03c0(5x - 5) - 3) + 1) - 4) - 3) and g(x) = -2|(-5(-3sin(1\u03c0x - 3) - 2)^2 - 5(-3sin(1\u03c0x - 3) - 2) + 5)| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(-1((-3(3e^(-1x) - 3) - 2)/(-1(3e^(-1x) - 3) + 2))^2 - 5((-3(3e^(-1x) - 3) - 2)/(-1(3e^(-1x) - 3) + 2)) - 2) - 1| and g(x) = 5(2((1x - 1)/(2x + 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": "6", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(2\u03c0(2((1(-1x^2 + 5x + 4) - 1)/(-1(-1x^2 + 5x + 4) - 1)))) + 1 and g(x) = (1(-5(2cos(2\u03c0x - 3) + 1) + 4) - 1)/(-3(-5(2cos(2\u03c0x - 3) + 1) + 4)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "88", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2sin(3\u03c0(-4(-1|((1x)/(-3x + 2))| - 3)^2 + 5) - 3) = 1|((-1(5x + 1) - 3)/(-2(5x + 1) - 3)) - 2| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "130", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(2\u03c0(5((-1x + 1)/(-2x + 3))^2 - 5((-1x + 1)/(-2x + 3)) + 5)) - 1 and g(x) = -3|(-3sin(3\u03c0x) + 1) - 3| + 3 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": "Determine the number of intersection points between the functions f(x) = -2cos(2\u03c0(-2|(1x^2 - 3x - 2) + 3| - 1) - 1) - 3 and g(x) = (-3(1e^(-2x) + 1) + 2)/(2(1e^(-2x) + 1) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "440", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1|(-4(-3sin(1\u03c0((1x - 2)/(2x - 1)) - 3) + 2) - 1) + 1| = 4(2cos(3\u03c0(-1x^2 - 3x - 5) + 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": "563", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (2(-3|(2(-1x + 4)^2 - 5(-1x + 4)) + 1| + 1) - 1)/(2(-3|(2(-1x + 4)^2 - 5(-1x + 4)) + 1| + 1) + 2) = (1(2sin(1\u03c0(4x^2 + 4x - 1) - 1) + 1) - 1)/(-2(2sin(1\u03c0(4x^2 + 4x - 1) - 1) + 1) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "151", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-2|(-2x + 5)| + 1) - 1)/(1(-2|(-2x + 5)| + 1) - 3) and g(x) = -5(3cos(1\u03c0x + 2) - 2)^2 + 5(3cos(1\u03c0x + 2) - 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3(2cos(3\u03c0((-2x + 1)/(1x - 2))) + 1)^2 - 2(2cos(3\u03c0((-2x + 1)/(1x - 2))) + 1) + 1 = (1(1x^2 - 1x + 1) - 3)/(-3(1x^2 - 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": "114", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(-1|(3x^2 - 5x - 1) - 2| - 1) + 3)/(3(-1|(3x^2 - 5x - 1) - 2| - 1) + 1) and g(x) = -3sin(1\u03c0(-3|x - 2| - 1) + 2) - 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) = -4(1(2sin(1\u03c0x + 2) + 1) - 3)^2 - 5(1(2sin(1\u03c0x + 2) + 1) - 3) - 3 and g(x) = 1((-3x - 3)/(-1x - 1))^2 - 4((-3x - 3)/(-1x - 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 -2sin(2\u03c0(2|(2(-2x^2 - 4x - 4) + 5) + 1| - 3) + 3) + 1 = (-3(4(2x^2 - 3x - 1) + 1))/(-2(4(2x^2 - 3x - 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": "1614", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(2\u03c0(4(-4(-1|x + 2| - 2)^2 - 4(-1|x + 2| - 2) + 5) + 2) + 3) - 2 and g(x) = -3((-3(3x - 4) - 1)/(-2(3x - 4)))^2 + 1((-3(3x - 4) - 1)/(-2(3x - 4))) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "212", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(2cos(1\u03c0(5(3|x - 3|)^2 + 3(3|x - 3|) - 2) + 3) + 2) - 3 and g(x) = -3|(4((-2x + 2)/(-1x + 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": "49", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3cos(1\u03c0(-3(4x^2 - 5x + 1) - 1) - 1) + 1 and g(x) = (1(-3|x + 3|) - 2)/(-1(-3|x + 3|) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1307", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(2\u03c0(3|(3(4x + 4)^2 - 1(4x + 4) + 5) + 2| + 3) + 2) + 1 and g(x) = 1((-2(2e^(-2x) + 1) - 3)/(2(2e^(-2x) + 1) - 1)) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "792", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-1(-2|(-3e^(-2(-2x + 1)) + 1) + 3|))/(3(-2|(-3e^(-2(-2x + 1)) + 1) + 3|) + 2) = (2(-2|(-3x^2 + 3x + 3) + 1| - 2) + 3)/(-1(-2|(-3x^2 + 3x + 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": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (1(5(-3(-2|x - 2| - 1) - 1)^2 + 1(-3(-2|x - 2| - 1) - 1) - 2))/(1(5(-3(-2|x - 2| - 1) - 1)^2 + 1(-3(-2|x - 2| - 1) - 1) - 2)) = -1cos(3\u03c0(-2|((2x - 3)/(-3x + 4)) + 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": "15", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|((3(-4x + 3) + 3)/(-2(-4x + 3) - 2)) + 1| - 1 and g(x) = 3sin(1\u03c0(2x^2 - 4x + 3) + 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "404", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1((3(1|(-3sin(1\u03c0x + 1) + 3) - 2| - 3) + 1)/(-2(1|(-3sin(1\u03c0x + 1) + 3) - 2| - 3) - 3)) + 5 and g(x) = -3e^(-1((2(3x - 4) + 1)/(3(3x - 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": "38", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1(-1(-2|((-1x + 3)/(-1x - 3)) - 1| - 2)^2 - 3(-2|((-1x + 3)/(-1x - 3)) - 1| - 2) - 2) + 2 = 2|(1cos(2\u03c0(5x - 3) + 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": "24", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-1(-2x - 4)^2 + 2(-2x - 4) - 2) + 1)/(1(-1(-2x - 4)^2 + 2(-2x - 4) - 2)) and g(x) = (2(2sin(1\u03c0x - 1) + 1))/(-2(2sin(1\u03c0x - 1) + 1) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(3\u03c0(2(3|(1x^2 - 1x + 4) - 3| + 1) - 3) - 2) - 2 and g(x) = -2(-1e^(-1(1|x| - 2)) + 3)^2 + 4(-1e^(-1(1|x| - 2)) + 3) - 5 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": "Find the number of times the graphs of f(x) = 2sin(1\u03c0(-4(3|((1x + 2)/(-2x)) + 2| + 3)^2 + 2(3|((1x + 2)/(-2x)) + 2| + 3) - 3) + 2) - 3 and g(x) = -3cos(2\u03c0(1|(1x - 1) + 1| - 2) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "177", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(-5(-2cos(1\u03c0((-1x - 1)/(2x)) - 3) - 2)^2 + 5(-2cos(1\u03c0((-1x - 1)/(2x)) - 3) - 2) + 1) + 1| + 1 and g(x) = 1(-3|(2sin(3\u03c0x + 1) + 3) + 3| - 2) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3(2|(5((-2x + 3)/(-1x + 2)) - 2) + 3| - 3)^2 + 5(2|(5((-2x + 3)/(-1x + 2)) - 2) + 3| - 3) - 4 and g(x) = 3sin(1\u03c0(-2(4x + 5)^2 - 5(4x + 5) + 1) - 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-4(-3(-2|x + 2| + 1)^2 - 5(-2|x + 2| + 1) - 4) + 4) + 3)/(1(-4(-3(-2|x + 2| + 1)^2 - 5(-2|x + 2| + 1) - 4) + 4) - 3) and g(x) = 1sin(1\u03c0(-1(2|x - 2| + 1) - 1) - 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1((3(-1|(2x^2 - 4x + 5) + 2| - 2) + 3)/(1(-1|(2x^2 - 4x + 5) + 2| - 2))) - 2 and g(x) = 3sin(1\u03c0(5(-1|x + 3|) - 5) - 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) = 2cos(1\u03c0(-3((1(-2|x - 3|) + 2)/(1(-2|x - 3|)))^2 + 4((1(-2|x - 3|) + 2)/(1(-2|x - 3|))) + 3)) + 1 and g(x) = 2sin(1\u03c0(2(2|x + 3| - 1)^2 + 4(2|x + 3| - 1) - 5) - 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1249", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2e^(1(-3(4(2|x + 1| - 3))^2 + 5(4(2|x + 1| - 3)) - 2)) + 1 and g(x) = (1(1sin(1\u03c0(-3x - 1) - 2)) + 3)/(-1(1sin(1\u03c0(-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(1(-2cos(1\u03c0x + 1) + 3) - 3) + 1)/(1(1(-2cos(1\u03c0x + 1) + 3) - 3) + 2) and g(x) = (-1(-2sin(2\u03c0x - 1) - 2) + 3)/(2(-2sin(2\u03c0x - 1) - 2) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (3(2(1cos(3\u03c0(-2|x + 3| + 2) + 1)) - 1) - 1)/(-3(2(1cos(3\u03c0(-2|x + 3| + 2) + 1)) - 1) - 3) = 1e^(-1(-1((1x - 2)/(-2x + 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": "86", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(3sin(1\u03c0((-1x - 2)/(1x - 3)) - 3) - 1) + 2| - 2 and g(x) = -3sin(3\u03c0(4x - 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": "161", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2sin(1\u03c0(2(2|((1x - 3)/(3x + 2)) - 3| + 2) + 2) - 3) and g(x) = (1(1|(5x + 3) + 1| + 3) + 1)/(-1(1|(5x + 3) + 1| + 3)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "77", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(2cos(3\u03c0(-3((-3x + 3)/(-2x + 3)) - 1) - 3))^2 + 2(2cos(3\u03c0(-3((-3x + 3)/(-2x + 3)) - 1) - 3)) - 2 and g(x) = 3sin(2\u03c0((-3(-2x - 4) - 3)/(-2(-2x - 4) + 3)) - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "64", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(2sin(1\u03c0(-1x^2 - 4x - 2) - 2) - 1) - 1| - 1 and g(x) = (-3(-1x - 5) + 3)/(3(-1x - 5) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "145", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(2\u03c0(-5(3|(3x^2 + 1x - 1) + 2| + 1) + 1) + 2) and g(x) = -1e^(-2(2|(5x + 4) + 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": "1135", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3|(-5((1x - 3)/(3x))^2 - 1((1x - 3)/(3x))) + 2| + 3 and g(x) = 2|(2cos(2\u03c0x) + 1) + 3| + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(1|(4(-4x - 5)^2 + 2(-4x - 5) - 3) - 3| - 1) - 3)/(-2(1|(4(-4x - 5)^2 + 2(-4x - 5) - 3) - 3| - 1) + 1) and g(x) = 2cos(2\u03c0(-3(-4x - 2)^2 + 5(-4x - 2) + 3) - 2) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1556", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3(-3cos(2\u03c0(3|x| - 1) - 3) - 3) - 5 = 1(2|x + 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": "72", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5(1((2x + 3)/(3x + 1))^2 + 3((2x + 3)/(3x + 1)) - 4) + 4 and g(x) = 1cos(1\u03c0(1x^2 + 3x - 1) + 1) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "83", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2|((1(-3cos(1\u03c0(-1x) - 1) - 1) + 3)/(-3(-3cos(1\u03c0(-1x) - 1) - 1) + 4)) + 3| - 1 = 3(3e^(1(1|x + 2|)) - 2)^2 - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1cos(3\u03c0(-2(3|(-4x^2 - 3x + 5) - 1| - 2) + 1)) and g(x) = (1(1cos(3\u03c0(-4x^2 - 1x + 2) + 1) + 1) + 3)/(2(1cos(3\u03c0(-4x^2 - 1x + 2) + 1) + 1) + 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": "How many points of intersection exist between the functions f(x) = 4(-1(-3sin(3\u03c0((-2x)/(3x - 2)) + 3) - 3))^2 - 2(-1(-3sin(3\u03c0((-2x)/(3x - 2)) + 3) - 3)) - 5 and g(x) = 2sin(3\u03c0((-1(3x^2 + 5x + 1) - 3)/(1(3x^2 + 5x + 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": "35", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1((2(1|(-5x - 1) - 3|) + 1)/(1(1|(-5x - 1) - 3|) - 2))^2 + 2((2(1|(-5x - 1) - 3|) + 1)/(1(1|(-5x - 1) - 3|) - 2)) - 3 = (3(3(2sin(3\u03c0x - 3) + 1)^2 - 5(2sin(3\u03c0x - 3) + 1)) - 2)/(-3(3(2sin(3\u03c0x - 3) + 1)^2 - 5(2sin(3\u03c0x - 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": "55", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1(5(-2|x - 2| + 1))^2 + 1(5(-2|x - 2| + 1)) - 4 = -3((-1x - 3)/(-2x))^2 - 5((-1x - 3)/(-2x)) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(2\u03c0((-2(2(3|x + 2| - 3) + 2) - 1)/(-1(2(3|x + 2| - 3) + 2) + 2)) - 3) - 2 and g(x) = (1(-2(4x^2 - 2x + 1) + 1) - 1)/(-1(-2(4x^2 - 2x + 1) + 1) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "84", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-1sin(1\u03c0(3|x - 2| + 1) + 1) + 3) + 2 and g(x) = -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": "20", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(1sin(3\u03c0((-2x - 3)/(1x + 2)) + 1) + 3) - 2| + 2 and g(x) = (-2(4x) + 2)/(-2(4x) - 1) 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": "Find the number of times the graphs of f(x) = 2|(-3sin(3\u03c0(5(-1x + 2)^2 - 3(-1x + 2)) + 1)) - 2| - 1 and g(x) = -2sin(2\u03c0(-5(-1|x - 3| + 2) - 5) + 3) - 2 intersect 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 (2(1(-3|(2cos(2\u03c0x - 3) - 2) + 1| + 2) + 3) + 2)/(3(1(-3|(2cos(2\u03c0x - 3) - 2) + 1| + 2) + 3) + 2) = 2(2|(-1x^2 + 3x - 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": "10", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-1(2(-3|x - 3|)^2 + 3(-3|x - 3|) - 4) - 3) - 2)/(-1(-1(2(-3|x - 3|)^2 + 3(-3|x - 3|) - 4) - 3) - 3) and g(x) = -3cos(1\u03c0(1(-1|x + 2| - 2)^2 - 1(-1|x + 2| - 2) + 3) - 1) - 3 intersect in the interval [-10, 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(-3((1x + 2)/(-3x)) - 1)^2 + 5(-3((1x + 2)/(-3x)) - 1) + 3 and g(x) = 1cos(2\u03c0(-2x - 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": "24", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3(-3sin(3\u03c0((1(-2x - 2) - 1)/(2(-2x - 2) - 1)) + 1) - 1)^2 - 2(-3sin(3\u03c0((1(-2x - 2) - 1)/(2(-2x - 2) - 1)) + 1) - 1) + 3 and g(x) = -1cos(1\u03c0(3|((1x - 3)/(2x)) - 1| - 1) - 2) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(-2|((-2(-5x - 4))/(-3(-5x - 4) - 1)) - 1|)^2 + 4(-2|((-2(-5x - 4))/(-3(-5x - 4) - 1)) - 1|) + 5 and g(x) = 4((-2(-2x - 3) + 1)/(3(-2x - 3) - 2))^2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|((1(2sin(3\u03c0x + 3) + 3))/(-2(2sin(3\u03c0x + 3) + 3) + 3)) - 1| + 2 and g(x) = -1(1|x - 2| + 3)^2 + 4(1|x - 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": "12", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(-3sin(1\u03c0(2(2|x + 3| - 2) + 5) - 2) + 1)^2 + 3(-3sin(1\u03c0(2(2|x + 3| - 2) + 5) - 2) + 1) - 3 and g(x) = -1sin(1\u03c0(5((3x - 3)/(3x - 2)) - 2) - 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "160", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -4(3cos(2\u03c0(-4((2x - 3)/(3x - 1)) + 4) - 2) + 2)^2 - 1(3cos(2\u03c0(-4((2x - 3)/(3x - 1)) + 4) - 2) + 2) + 2 = -1cos(1\u03c0(-3(3x^2 + 1x + 5) - 5)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "126", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(2(3|(5x) - 1|)^2 + 1(3|(5x) - 1|) - 4))/(2(2(3|(5x) - 1|)^2 + 1(3|(5x) - 1|) - 4)) and g(x) = (-1(3sin(1\u03c0(4x^2 - 1x - 4) - 2) - 1))/(-1(3sin(1\u03c0(4x^2 - 1x - 4) - 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": "748", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(1e^(1(1(-3|x - 2| + 1) + 1)))^2 + 4 and g(x) = -3((-1(3cos(3\u03c0x - 2) - 1) + 1)/(-1(3cos(3\u03c0x - 2) - 1) - 3)) - 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 points of intersection exist between the functions f(x) = -2|(-2sin(2\u03c0((2(1x^2 + 1x + 4))/(1(1x^2 + 1x + 4) + 2)) + 2) - 1) + 3| - 1 and g(x) = 2(2cos(1\u03c0(-2|x + 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": "40", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(1sin(1\u03c0(4x^2 + 1x - 4) - 1)))/(-1(1sin(1\u03c0(4x^2 + 1x - 4) - 1))) and g(x) = 3sin(3\u03c0((1x + 3)/(1x)) - 2) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "114", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(2\u03c0(-3(2|x| - 3) - 3) + 1) and g(x) = -3e^(-1(-1|x - 2| + 1)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3|((1(4(-1x - 4)^2 + 5(-1x - 4) - 5) - 3)/(2(4(-1x - 4)^2 + 5(-1x - 4) - 5) + 2))| - 2 = 2cos(2\u03c0(2(-1|x + 3|)^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": "755", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3e^(2((3(3|(1x - 2) - 3| + 1) + 1)/(2(3|(1x - 2) - 3| + 1) + 2))) - 2 and g(x) = -4(-2|(2sin(2\u03c0x + 2) + 3) + 3| - 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(1\u03c0(2|(4x^2 + 5x + 2) + 3| + 3) + 2) - 3 and g(x) = -2|((-3x)/(1x)) + 3| - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1316", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1(-3sin(2\u03c0(-1|x - 3| + 1) - 3) + 3) - 1 = 5((3x + 3)/(3x - 1))^2 - 4((3x + 3)/(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": "31", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3((1(4(3cos(2\u03c0x + 1) + 3)^2 + 1(3cos(2\u03c0x + 1) + 3) + 1) - 1)/(2(4(3cos(2\u03c0x + 1) + 3)^2 + 1(3cos(2\u03c0x + 1) + 3) + 1) + 1)) - 2 = -1((1(-2x + 5) - 1)/(2(-2x + 5) - 1))^2 - 5((1(-2x + 5) - 1)/(2(-2x + 5) - 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(-1cos(2\u03c0((-3x)/(-3x - 3))) - 1) - 1| and g(x) = -4(-1sin(1\u03c0x + 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": "20", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(-4(-2(3|x + 1| - 2)^2 - 1(3|x + 1| - 2) + 3) + 2) + 2)/(-1(-4(-2(3|x + 1| - 2)^2 - 1(3|x + 1| - 2) + 3) + 2) - 1) and g(x) = (3(2(-2|x| + 3)^2 - 3(-2|x| + 3) - 5) - 3)/(2(2(-2|x| + 3)^2 - 3(-2|x| + 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": "8", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(3sin(3\u03c0(1|(-5x - 1) - 1| + 3) + 3) - 1) - 3)/(-1(3sin(3\u03c0(1|(-5x - 1) - 1| + 3) + 3) - 1) - 1) and g(x) = -1|(-2cos(3\u03c0(-3x^2 - 3) - 3) + 3)| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "606", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(-2|(3sin(2\u03c0((2x - 2)/(-1x + 2)) - 3) - 2) - 2| + 1)^2 - 3(-2|(3sin(2\u03c0((2x - 2)/(-1x + 2)) - 3) - 2) - 2| + 1) and g(x) = -3((2(1|x + 3| - 3) + 2)/(2(1|x + 3| - 3)))^2 + 5((2(1|x + 3| - 3) + 2)/(2(1|x + 3| - 3))) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "57", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (2(-2|(1x + 3) - 1| + 3) - 2)/(2(-2|(1x + 3) - 1| + 3) - 1) and g(x) = -1sin(3\u03c0(3|x + 1|) - 3) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "170", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(-2sin(1\u03c0(1x - 1) - 3) + 3) - 1)/(3(-2sin(1\u03c0(1x - 1) - 3) + 3) + 1) and g(x) = 2cos(2\u03c0(-2|x - 1| + 1) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(1cos(2\u03c0(2x - 2) - 3) - 1) - 1)/(-1(1cos(2\u03c0(2x - 2) - 3) - 1)) and g(x) = -3cos(1\u03c0(-3|x| - 2) + 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2cos(3\u03c0(1|((3x + 2)/(3x - 2)) - 1| - 3) + 1) - 1 and g(x) = (-1(-5x^2 - 1x - 1) + 2)/(1(-5x^2 - 1x - 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3sin(3\u03c0(5(2x^2 + 4x + 4) - 1) + 2) + 1 = 1(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": "330", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2((-2(-4(3|x - 3| + 2)^2 - 3(3|x - 3| + 2) + 5) + 1)/(2(-4(3|x - 3| + 2)^2 - 3(3|x - 3| + 2) + 5) - 1)) - 4 and g(x) = (3(3|(3sin(3\u03c0x + 3))| - 2) - 2)/(2(3|(3sin(3\u03c0x + 3))| - 2) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(2|((-1(3x + 5) - 2)/(3(3x + 5) + 3))|)^2 - 2(2|((-1(3x + 5) - 2)/(3(3x + 5) + 3))|) - 1 and g(x) = -2sin(2\u03c0(-1(3x^2 - 4x + 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": "1035", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-2(1x^2 - 3x - 5) - 5) - 3)/(2(-2(1x^2 - 3x - 5) - 5) + 2) and g(x) = -2(3cos(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": "60", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3cos(1\u03c0(-2(-1(2|x - 1| + 2))^2 + 2(-1(2|x - 1| + 2)) - 4) + 2) - 1 and g(x) = 2e^(-1(4(-3|x + 2| + 2))) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "93", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2e^(-2(3((1x - 3)/(2x + 2))^2 - 4((1x - 3)/(2x + 2)) + 1)) - 3 and g(x) = (2(-3cos(2\u03c0x - 2) - 2) - 2)/(-1(-3cos(2\u03c0x - 2) - 2)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3cos(3\u03c0(-2|(3x^2 + 5x - 1)| + 2) - 2) = 3((1x + 3)/(2x - 1)) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "703", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2sin(1\u03c0(2(-2x - 3)^2 - 5(-2x - 3) + 3) - 2) + 1 = (-1(-1|x + 1| - 3) + 3)/(-1(-1|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": "1375", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2(-3e^(-1(5x + 1)))^2 - 5(-3e^(-1(5x + 1))) + 3 = 5(-3cos(1\u03c0x - 3) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2cos(2\u03c0(3(2x^2 + 1x + 5) - 1) + 2) - 1 = -1(3cos(3\u03c0x) + 1) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((3(1(3|x| + 2)^2 + 3(3|x| + 2) + 5) + 3)/(1(1(3|x| + 2)^2 + 3(3|x| + 2) + 5) - 3)) + 4 and g(x) = -2(2cos(2\u03c0((2x + 1)/(2x - 1)) - 3) - 1)^2 + 3(2cos(2\u03c0((2x + 1)/(2x - 1)) - 3) - 1) + 5 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 points of intersection exist between the functions f(x) = 3cos(2\u03c0(1(2|x + 1| - 3)^2 - 4(2|x + 1| - 3)) + 2) and g(x) = 3(-1sin(1\u03c0x + 2) - 1) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "732", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (2(3(-3cos(2\u03c0x - 2) + 1)^2 + 4(-3cos(2\u03c0x - 2) + 1) + 3) + 1)/(-1(3(-3cos(2\u03c0x - 2) + 1)^2 + 4(-3cos(2\u03c0x - 2) + 1) + 3) + 2) = -2(-1cos(3\u03c0x - 1) - 3)^2 - 5(-1cos(3\u03c0x - 1) - 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3sin(2\u03c0(2(-3|x - 1| + 3)^2 - 5(-3|x - 1| + 3) + 4) + 1) + 3 and g(x) = 4((2x - 3)/(2x + 3)) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "722", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-1sin(3\u03c0((-2(-2x - 3) - 3)/(-3(-2x - 3) - 3)) + 2) - 2)^2 + 1 and g(x) = (1(1|(3cos(3\u03c0x - 3) + 3)| - 3) - 1)/(1(1|(3cos(3\u03c0x - 3) + 3)| - 3) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "64", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-2sin(1\u03c0(-4x^2 + 1x + 2)) + 3)| + 3 and g(x) = -3((-1x + 3)/(1x))^2 + 1((-1x + 3)/(1x)) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "266", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(-5((-2(-2x^2 - 5x + 5) - 2)/(-3(-2x^2 - 5x + 5) + 1)) - 3) - 1| - 3 and g(x) = -5(3sin(2\u03c0(3|x + 3|) + 2) + 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1(5((-3(-3|x|) + 2)/(-2(-3|x|)))^2 - 2((-3(-3|x|) + 2)/(-2(-3|x|))) + 1) + 2 = -2|(-3(3x^2 - 2x - 3) + 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": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|((3(-4(1sin(1\u03c0x - 1))) - 2)/(-3(-4(1sin(1\u03c0x - 1))) - 1)) + 3| and g(x) = -2|((3(-3x^2 - 4x - 3) - 3)/(1(-3x^2 - 4x - 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": "26", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4((1(1(-3|x + 1| - 1) - 2) + 2)/(-3(1(-3|x + 1| - 1) - 2) + 4))^2 - 5((1(1(-3|x + 1| - 1) - 2) + 2)/(-3(1(-3|x + 1| - 1) - 2) + 4)) and g(x) = -1|(3(-2sin(1\u03c0x - 1) + 2) - 2) - 3| + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(-2|(-2sin(2\u03c0x - 1) + 2)| + 3)^2 + 4(-2|(-2sin(2\u03c0x - 1) + 2)| + 3) - 1 and g(x) = (-1(-1x^2 + 2x - 5) - 1)/(1(-1x^2 + 2x - 5)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(-1((2(-4x) + 3)/(2(-4x) + 1))^2 - 3((2(-4x) + 3)/(2(-4x) + 1)) + 4) - 2| + 1 and g(x) = (-1(-3cos(1\u03c0(-2x - 3) - 1) + 2) - 1)/(-2(-3cos(1\u03c0(-2x - 3) - 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 -2((1(-3|x - 3| + 2) + 1)/(3(-3|x - 3| + 2) - 2))^2 - 4((1(-3|x - 3| + 2) + 1)/(3(-3|x - 3| + 2) - 2)) + 3 = -2(-2sin(3\u03c0x + 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": "58", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1|(-1(-3sin(3\u03c0(-5x + 3) + 1))^2 + 1(-3sin(3\u03c0(-5x + 3) + 1)) - 4) + 2| + 2 = -2e^(-2(2((2x + 2)/(1x + 3))^2 + 2((2x + 2)/(1x + 3)) + 1)) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "600", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|((1(3(-1x^2 - 5x) - 5) - 1)/(1(3(-1x^2 - 5x) - 5) - 3)) - 3| + 2 and g(x) = 2|((2(4x + 1) + 1)/(-2(4x + 1) + 3)) - 1| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5((3(-1|(-4x + 3) - 3| + 2) - 3)/(2(-1|(-4x + 3) - 3| + 2)))^2 - 1((3(-1|(-4x + 3) - 3| + 2) - 3)/(2(-1|(-4x + 3) - 3| + 2))) - 5 and g(x) = -1|(5(-2sin(1\u03c0x + 1))^2 - 1(-2sin(1\u03c0x + 1)) + 4) - 2| - 3 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 -1(3(1|(-1cos(2\u03c0x - 2) - 3) + 1| - 2)^2 - 3(1|(-1cos(2\u03c0x - 2) - 3) + 1| - 2) - 2) + 5 = -2((3(-1|x| + 1) + 2)/(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": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3|((-2(-3cos(2\u03c0(3x - 3)) + 1) + 1)/(2(-3cos(2\u03c0(3x - 3)) + 1) + 1)) - 1| + 1 = -1((3(1sin(3\u03c0x + 1) - 1) - 3)/(3(1sin(3\u03c0x + 1) - 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": "120", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(3|(1(-2e^(1x) + 3))| + 1) - 1)/(2(3|(1(-2e^(1x) + 3))| + 1) + 3) and g(x) = 2cos(1\u03c0(3|(2x - 4) - 1|) + 3) - 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": "Determine the number of intersection points between the functions f(x) = 1|(1sin(1\u03c0(-3x^2 + 1) - 1) - 3) - 3| - 2 and g(x) = -2sin(1\u03c0(4x^2 + 4x - 3) + 1) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "723", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(1\u03c0(-5(-2|x - 3| - 1)) + 1) + 3 and g(x) = -5(1|x - 2| - 2)^2 + 5(1|x - 2| - 2) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(4(-1(3sin(3\u03c0x + 2) - 3)^2 - 4(3sin(3\u03c0x + 2) - 3) - 1) - 3) + 3| and g(x) = -2|(-3cos(2\u03c0((-2x + 2)/(1x + 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": "6", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|(-3(-2((-3x - 1)/(2x))^2 - 3((-3x - 1)/(2x)) + 4) - 2) + 2| - 1 and g(x) = 3(4((-2x + 3)/(-1x + 2))^2 + 1((-2x + 3)/(-1x + 2)) - 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) = 3cos(2\u03c0(-3(1|x + 2|) - 1) + 3) - 1 and g(x) = 2(-3x^2 + 1x) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|((-2(2(-2x + 1)^2 + 3(-2x + 1) + 4) + 3)/(2(2(-2x + 1)^2 + 3(-2x + 1) + 4) - 1))| + 1 and g(x) = -1|(-2cos(1\u03c0(5x) - 3) - 1) - 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": "Determine the number of intersection points between the functions f(x) = -1sin(3\u03c0(-1|((-1(-2x - 5) - 2)/(-1(-2x - 5) + 2)) + 2| - 1) - 3) and g(x) = 2((-1(1x^2 + 4x) + 3)/(-2(1x^2 + 4x))) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2|((3(-3cos(3\u03c0(-1x^2 - 2) + 1)) - 1)/(-3(-3cos(3\u03c0(-1x^2 - 2) + 1)) - 2)) + 3| + 2 = (2(1(-2|x - 3| + 3)) + 2)/(2(1(-2|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": "156", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2cos(2\u03c0(3|((3(-4x - 1) + 3)/(1(-4x - 1) + 3)) + 3| + 2) - 3) - 1 and g(x) = 3(-2|(-3cos(1\u03c0x - 2)) + 3| + 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4((-2(-1|(1sin(3\u03c0x) + 1) - 3| + 1) - 3)/(-1(-1|(1sin(3\u03c0x) + 1) - 3| + 1) + 3))^2 - 4((-2(-1|(1sin(3\u03c0x) + 1) - 3| + 1) - 3)/(-1(-1|(1sin(3\u03c0x) + 1) - 3| + 1) + 3)) + 2 and g(x) = -5((1(-1|x| + 3) - 1)/(-2(-1|x| + 3) + 3)) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1e^(-2(1|(1((3x + 2)/(1x + 3)) - 4) + 2| + 2)) + 2 = -1sin(1\u03c0(2(3|x + 1| - 3)^2 - 3(3|x + 1| - 3) + 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": "1508", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2|((3(-1x^2 + 5x + 4) + 1)/(3(-1x^2 + 5x + 4) + 3)) - 3| + 2 = (-2(3x^2 - 5x - 5) - 3)/(2(3x^2 - 5x - 5) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|(3cos(1\u03c0(3x^2 + 4x - 1) - 2)) + 2| + 2 and g(x) = (-2(1e^(2x) + 2) + 2)/(-1(1e^(2x) + 2) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "354", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(-2(5x^2 + 2x - 4)) + 1)/(3(-2(5x^2 + 2x - 4)) - 2) and g(x) = -2sin(3\u03c0(2x^2 + 5x + 2) + 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2e^(-1(1|(1(1x + 5)^2 + 2(1x + 5) + 5) + 2|)) + 3 = 3|((3(-2cos(1\u03c0x + 3) - 1))/(1(-2cos(1\u03c0x + 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": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2((1(-2x^2 + 2x - 2) + 2)/(1(-2x^2 + 2x - 2) - 2)) - 5 = -3cos(2\u03c0(-3x - 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": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(3cos(1\u03c0((-3x + 2)/(-2x - 3)) + 3) - 2)^2 + 5(3cos(1\u03c0((-3x + 2)/(-2x - 3)) + 3) - 2) + 2 and g(x) = -1|(2sin(1\u03c0x) + 3) - 2| - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(-2cos(1\u03c0(-2x^2 + 2x + 1))) + 3)/(2(-2cos(1\u03c0(-2x^2 + 2x + 1))) + 2) and g(x) = 5(-3|x - 3| + 1)^2 - 4(-3|x - 3| + 1) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(5(1|(-5x + 5) + 2| - 3)^2 - 3(1|(-5x + 5) + 2| - 3)))/(3(5(1|(-5x + 5) + 2| - 3)^2 - 3(1|(-5x + 5) + 2| - 3)) - 1) and g(x) = (-3(3|(-1x^2 - 5x + 2) - 2| - 3) + 1)/(2(3|(-1x^2 - 5x + 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": "8", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3cos(3\u03c0(-4(-1|x + 2| + 3)^2 + 3(-1|x + 2| + 3) + 2) - 1) - 2 and g(x) = 2sin(1\u03c0(1x^2 + 1x - 5) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1148", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-3sin(1\u03c0(-1|((-3x + 2)/(1x + 3)) + 3| + 1) - 2) + 2) + 4 and g(x) = -3cos(1\u03c0(-3(-1x + 2)^2 + 1(-1x + 2) - 3) - 1) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "465", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1(-1|(-2((1x + 3)/(2x))^2 - 5((1x + 3)/(2x))) - 2| - 3) - 2 = -2|(3cos(1\u03c0((3x - 3)/(3x + 2)))) - 3| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3cos(1\u03c0(-4(-3|x - 2| - 1)^2 - 1(-3|x - 2| - 1) + 5) - 1) - 1 = -1(3x)^2 - 4(3x) + 1 have for x in the interval [-10, 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) = (-2(-5(3sin(1\u03c0(2|x - 2|) + 3) + 1) + 2) - 2)/(-2(-5(3sin(1\u03c0(2|x - 2|) + 3) + 1) + 2)) and g(x) = 2|(-2((-2x + 3)/(2x - 1))^2 - 3((-2x + 3)/(2x - 1))) - 3| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2sin(1\u03c0(5((1x + 1)/(2x)))) + 3 and g(x) = 3e^(-2((3x - 3)/(2x + 1))) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2e^(1(-5((3x)/(2x + 3)) - 2)) + 1 = 3sin(1\u03c0((3x + 3)/(-1x)) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1(5(-2sin(3\u03c0x) - 2) - 1)^2 + 3(5(-2sin(3\u03c0x) - 2) - 1) - 5 = -2sin(1\u03c0(5x^2 + 5x - 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": "Find the number of times the graphs of f(x) = -1(2((-2x + 3)/(-3x + 1)) + 1)^2 - 4(2((-2x + 3)/(-3x + 1)) + 1) + 2 and g(x) = -3cos(2\u03c0(3x^2 - 4x + 5) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1((3(3|(-1x^2 - 3x + 2) + 1|) - 3)/(1(3|(-1x^2 - 3x + 2) + 1|) + 1)) - 1 = 3|((-2(-3sin(2\u03c0x - 3) - 1) - 2)/(-1(-3sin(2\u03c0x - 3) - 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": "80", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(-1|(4(-1x + 1)^2 + 3(-1x + 1) + 2)| - 2) + 1)/(-2(-1|(4(-1x + 1)^2 + 3(-1x + 1) + 2)| - 2) + 1) and g(x) = 3sin(1\u03c0(-1(-2|x - 1| + 1) + 3) - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2cos(2\u03c0(3|(3x^2 + 1) - 2| + 1) + 1) + 2 and g(x) = 1e^(2((1x - 1)/(-2x - 3))) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1594", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(5(3cos(2\u03c0((3x + 2)/(2x - 1)) + 2) + 2) + 4)^2 + 3 and g(x) = 2|(5((1x + 3)/(2x - 1))^2 - 1((1x + 3)/(2x - 1)) - 3)| + 2 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": "Determine the number of intersection points between the functions f(x) = -1(3sin(1\u03c0(2x^2 - 1x + 1) - 1) + 1) and g(x) = 5(-3sin(3\u03c0x + 3) - 2) 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": "Find the number of times the graphs of f(x) = -1(-2cos(1\u03c0((1x + 1)/(1x + 2)) - 3) - 1)^2 + 4(-2cos(1\u03c0((1x + 1)/(1x + 2)) - 3) - 1) + 5 and g(x) = 3(2e^(2x) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(2(-2cos(2\u03c0x - 3)) + 2) - 3| - 2 and g(x) = (3(2x^2 + 2x + 5) - 3)/(-3(2x^2 + 2x + 5) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(-1sin(3\u03c0(-3|(4x^2 - 3x + 3)| - 2) + 3) + 3) - 2)/(-1(-1sin(3\u03c0(-3|(4x^2 - 3x + 3)| - 2) + 3) + 3) + 2) and g(x) = -2cos(3\u03c0(1(-3x + 3)^2 - 1) - 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1541", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(-2(1cos(3\u03c0x - 2) - 1) - 3) - 1)/(-3(-2(1cos(3\u03c0x - 2) - 1) - 3) + 2) and g(x) = (1(3|x - 3| - 1) + 2)/(1(3|x - 3| - 1) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2cos(3\u03c0(-3|((-2x)/(-2x - 2))|)) and g(x) = (-1(-2x + 4) - 1)/(1(-2x + 4) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "64", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-3sin(2\u03c0(-1((-3x + 1)/(2x - 3))^2 - 3((-3x + 1)/(2x - 3)) - 4)) - 2) - 1 and g(x) = -3sin(1\u03c0(2|(2x^2 + 2x + 2) - 2| + 1) + 2) + 2 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) = 4(4((-2x - 1)/(1x - 2)) + 1)^2 + 4(4((-2x - 1)/(1x - 2)) + 1) - 2 and g(x) = 4(-3cos(2\u03c0x) - 2)^2 + 5(-3cos(2\u03c0x) - 2) - 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 solutions does the equation 2sin(3\u03c0(2(2|x - 1| - 1)^2 - 3(2|x - 1| - 1) - 4) - 3) - 1 = 1((2x - 1)/(-1x - 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "994", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(-1|(5((-2x)/(3x + 2)) + 1) + 3| + 2)^2 + 4(-1|(5((-2x)/(3x + 2)) + 1) + 3| + 2) - 2 and g(x) = 3|((2(2sin(3\u03c0x + 2) - 2) - 1)/(-3(2sin(3\u03c0x + 2) - 2) - 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": "45", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1((-1(5(3|x + 1| - 1)^2 + 3(3|x + 1| - 1) - 3))/(1(5(3|x + 1| - 1)^2 + 3(3|x + 1| - 1) - 3) - 2)) + 1 and g(x) = (-1(4(-3|x - 3| + 2)^2 + 4(-3|x - 3| + 2) - 5))/(1(4(-3|x - 3| + 2)^2 + 4(-3|x - 3| + 2) - 5) + 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3cos(2\u03c0(5(-1|x - 1| - 2) - 1) + 1) + 3 and g(x) = (1(-3cos(3\u03c0x + 2) - 1))/(-2(-3cos(3\u03c0x + 2) - 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) = -2(5((-2(2sin(3\u03c0x + 2) + 1) - 2)/(-3(2sin(3\u03c0x + 2) + 1) - 3)) + 2)^2 + 3(5((-2(2sin(3\u03c0x + 2) + 1) - 2)/(-3(2sin(3\u03c0x + 2) + 1) - 3)) + 2) - 5 and g(x) = (1(5(-1cos(1\u03c0x + 2) + 1) - 5) - 2)/(-3(5(-1cos(1\u03c0x + 2) + 1) - 5) - 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 (-3(1|(5(3x^2 - 1x) - 5) + 1| - 1) - 3)/(-3(1|(5(3x^2 - 1x) - 5) + 1| - 1) - 2) = -2sin(3\u03c0(1|(-1x^2 - 5x + 1) + 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": "94", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(1cos(1\u03c0(-3x^2 - 2x - 5) - 3)) + 2| + 1 and g(x) = 1e^(-1(-1|x - 1| + 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": "Determine the number of intersection points between the functions f(x) = (1(-3cos(3\u03c0(5(3|x| - 3)^2 - 2(3|x| - 3) + 5) + 1) + 3) - 2)/(2(-3cos(3\u03c0(5(3|x| - 3)^2 - 2(3|x| - 3) + 5) + 1) + 3) - 1) and g(x) = 1((1(3sin(3\u03c0x - 2) - 3) - 2)/(1(3sin(3\u03c0x - 2) - 3) + 2))^2 - 4((1(3sin(3\u03c0x - 2) - 3) - 2)/(1(3sin(3\u03c0x - 2) - 3) + 2)) - 2 where -10 \u2264 x \u2264 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 -3|(3cos(1\u03c0(5x^2 - 1x - 1) - 1))| - 2 = -3|(4x^2 + 4x) + 2| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-1(1(-3(-1|x - 2| + 2))^2 - 5(-3(-1|x - 2| + 2)) + 4) + 1)/(-2(1(-3(-1|x - 2| + 2))^2 - 5(-3(-1|x - 2| + 2)) + 4) - 3) and g(x) = -2cos(2\u03c0(2|((-2x - 3)/(2x - 3)) + 2|) - 1) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(3(3x^2 + 1x + 5) - 3) + 3)/(2(3(3x^2 + 1x + 5) - 3) + 1) and g(x) = 3cos(1\u03c0(1|x| + 1) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5((-1(3sin(3\u03c0x - 2) - 2) - 1)/(3(3sin(3\u03c0x - 2) - 2) - 1)) - 5 and g(x) = -1(3sin(2\u03c0x - 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": "Find the number of times the graphs of f(x) = -2sin(3\u03c0(5(-1(-1|x + 3| + 1)^2 + 1(-1|x + 3| + 1) - 5)) + 2) + 2 and g(x) = -4(5(-3|x - 1| + 3))^2 - 1(5(-3|x - 1| + 3)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -4((-2(3x^2 - 4) + 3)/(1(3x^2 - 4) + 3)) - 2 and g(x) = 2|(-2cos(1\u03c0x) + 3) - 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1sin(3\u03c0(1((-2x)/(-3x + 1))^2 + 5((-2x)/(-3x + 1))) + 1) + 2 = -2(-3cos(1\u03c0x) + 1)^2 - 2(-3cos(1\u03c0x) + 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3cos(1\u03c0(-3((2x + 2)/(-2x + 3))^2 - 4((2x + 2)/(-2x + 3)) + 2) - 1) + 2 and g(x) = (3(3x^2 - 1x - 5) + 3)/(-3(3x^2 - 1x - 5) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3|(5(3cos(1\u03c0x + 3) + 3) + 1) + 1| + 1 and g(x) = 3(-1x + 2)^2 + 2(-1x + 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": "Find the number of times the graphs of f(x) = -1(1|((-1x - 1)/(2x - 3)) + 2| + 1)^2 + 1(1|((-1x - 1)/(2x - 3)) + 2| + 1) + 4 and g(x) = -1(-1sin(1\u03c0x) - 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1|(-1(5x - 3)^2 + 3(5x - 3) + 1) + 2| - 2 = 3cos(3\u03c0(5x - 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": "12", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(-2sin(1\u03c0((-1(1x - 2) - 3)/(-3(1x - 2) - 2)) + 1))^2 + 4(-2sin(1\u03c0((-1(1x - 2) - 3)/(-3(1x - 2) - 2)) + 1)) - 5 and g(x) = (1(1cos(2\u03c0(1x^2 + 5x - 2) - 3) - 2))/(2(1cos(2\u03c0(1x^2 + 5x - 2) - 3) - 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2((1(3|(1cos(3\u03c0x) - 1) + 1|) - 2)/(2(3|(1cos(3\u03c0x) - 1) + 1|) - 3))^2 - 4((1(3|(1cos(3\u03c0x) - 1) + 1|) - 2)/(2(3|(1cos(3\u03c0x) - 1) + 1|) - 3)) + 5 = 2cos(2\u03c0(5((1x + 3)/(-3x - 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": "216", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(2sin(3\u03c0(-3(-1|x - 1| + 1)^2 - 4(-1|x - 1| + 1) + 5)) - 1) - 5 and g(x) = -1sin(2\u03c0((1(-2|x - 1| + 1))/(2(-2|x - 1| + 1) + 2)) + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "411", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -4(-3e^(1(-4((-2x - 1)/(3x + 2))^2 + 5((-2x - 1)/(3x + 2)) + 1))) + 5 = (-1(-2cos(1\u03c0(1x - 1) + 3) - 1) - 3)/(-2(-2cos(1\u03c0(1x - 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": "20", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(2|(1(1x + 2)^2 - 3(1x + 2) - 1) + 3| - 2) - 2)/(-1(2|(1(1x + 2)^2 - 3(1x + 2) - 1) + 3| - 2) - 3) and g(x) = 5(-1sin(1\u03c0(-5x^2 + 1x - 4) + 2) + 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "769", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-1(-2|(-1sin(1\u03c0x + 3) - 3) - 3| + 2) - 2)/(-3(-2|(-1sin(1\u03c0x + 3) - 3) - 3| + 2) - 2) and g(x) = -2|(3cos(2\u03c0x + 2) + 3) - 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(5(1sin(2\u03c0((-2x - 1)/(-3x + 2)) + 3) - 2) - 2)^2 - 4(5(1sin(2\u03c0((-2x - 1)/(-3x + 2)) + 3) - 2) - 2) + 2 and g(x) = -3|(-3(2sin(3\u03c0x - 2) + 3)^2 + 3(2sin(3\u03c0x - 2) + 3) + 1) - 2| + 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": "How many solutions does the equation -3e^(2(-2|(1x^2 - 4) + 3| - 3)) - 2 = -3cos(2\u03c0(3x + 2) + 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(1((2x)/(1x)))^2 + 2(1((2x)/(1x))) - 1 and g(x) = -4(2cos(2\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": "40", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4((3(1sin(2\u03c0(-3x^2 + 2) + 2)) + 2)/(-2(1sin(2\u03c0(-3x^2 + 2) + 2)) - 3)) - 4 and g(x) = -3(-3(2sin(2\u03c0x - 1) + 1) + 4)^2 + 2(-3(2sin(2\u03c0x - 1) + 1) + 4) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "132", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2|((1(-5(2x - 4)^2 - 2(2x - 4) + 2) - 1)/(-3(-5(2x - 4)^2 - 2(2x - 4) + 2) - 1)) - 1| and g(x) = -2cos(2\u03c0(-2(-2|x + 3|) - 3) - 3) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "160", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(-3(2cos(2\u03c0x - 2) + 1) + 4) + 3)/(-3(-3(2cos(2\u03c0x - 2) + 1) + 4)) and g(x) = 1(1x^2 + 2x) - 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1(-1|((1(-1x^2 + 1x + 2) - 2)/(1(-1x^2 + 1x + 2) + 2)) + 1| - 1) + 4 = 5(-2((2x + 1)/(3x)) - 3)^2 + 2(-2((2x + 1)/(3x)) - 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(-2e^(2(-4(-3|x + 1| - 1)^2 - 1(-3|x + 1| - 1) + 1)) - 2) - 3 and g(x) = -3|(-3sin(2\u03c0(1x^2 - 5x) + 2) - 3)| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "424", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(-4((1x)/(2x + 2))^2 + 1((1x)/(2x + 2)) + 2) and g(x) = 3|(-2cos(2\u03c0x + 2)) + 3| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 5(4(-3sin(2\u03c0x - 2))^2 + 3(-3sin(2\u03c0x - 2)) + 2) + 3 = -3(-2|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": "64", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2cos(3\u03c0(5(4x - 1)^2 + 5) + 3) + 2 = -1(-2|x - 1| - 1)^2 + 1(-2|x - 1| - 1) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-3(2|(2(2cos(1\u03c0x + 3) - 1)^2 + 2(2cos(1\u03c0x + 3) - 1) + 4) - 2| + 1) - 3)/(2(2|(2(2cos(1\u03c0x + 3) - 1)^2 + 2(2cos(1\u03c0x + 3) - 1) + 4) - 2| + 1) - 1) and g(x) = -2|(-2sin(3\u03c0(4x + 1) + 3) + 2) - 1| + 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": "How many solutions does the equation -2|(-2cos(3\u03c0((1x + 2)/(2x - 1)) + 2) + 2) - 2| + 2 = -3sin(3\u03c0(5x^2 + 5x - 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": "1438", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(-1|((1x)/(-1x - 1)) - 1| + 2)^2 - 4(-1|((1x)/(-1x - 1)) - 1| + 2) - 5 and g(x) = 1|(-1sin(3\u03c0x - 3)) - 1| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2sin(3\u03c0(-1(1((-1x + 2)/(3x + 1)))^2 - 5(1((-1x + 2)/(3x + 1))) + 1) - 3) - 2 = -3|((-1(-1x - 1) + 1)/(-1(-1x - 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": "21", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1((-1(-3(-3cos(2\u03c0x + 3) - 2) - 5) + 3)/(2(-3(-3cos(2\u03c0x + 3) - 2) - 5) - 1))^2 + 5((-1(-3(-3cos(2\u03c0x + 3) - 2) - 5) + 3)/(2(-3(-3cos(2\u03c0x + 3) - 2) - 5) - 1)) - 2 and g(x) = 4(1e^(1(-2|x - 3| - 3)))^2 - 2(1e^(1(-2|x - 3| - 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) = -2|(-3cos(3\u03c0((-3x - 1)/(-1x - 3)) - 3) - 1) - 2| + 3 and g(x) = -3cos(3\u03c0((1x - 1)/(-1x + 2)) - 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "106", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -4(2cos(2\u03c0((-3(-1x - 3) + 2)/(-1(-1x - 3) - 3))))^2 - 5 and g(x) = -3e^(1(-1(-2|x - 2| + 2) + 1)) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5((1(3cos(3\u03c0(-1x - 2) + 2) - 2) - 2)/(1(3cos(3\u03c0(-1x - 2) + 2) - 2) + 2))^2 + 3((1(3cos(3\u03c0(-1x - 2) + 2) - 2) - 2)/(1(3cos(3\u03c0(-1x - 2) + 2) - 2) + 2)) + 4 and g(x) = 1((2(3x^2 + 3x - 4) + 2)/(3(3x^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": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3cos(3\u03c0(1((3(-3x + 1) - 1)/(-3(-3x + 1)))^2 - 1) + 3) - 1 and g(x) = -1(5((-1x - 1)/(1x + 3)) + 2)^2 + 3(5((-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": "30", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-3(3cos(2\u03c0(2|x + 3|) - 1) - 3) - 1)/(-1(3cos(2\u03c0(2|x + 3|) - 1) - 3)) and g(x) = 2cos(3\u03c0(3x^2 + 4x - 1) + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "93", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3(-1|(5(-3sin(1\u03c0x - 1))^2 - 1(-3sin(1\u03c0x - 1)) - 4) - 3| + 1) = (-2(-1(4x + 5)^2 + 4(4x + 5) + 4) - 3)/(-3(-1(4x + 5)^2 + 4(4x + 5) + 4) - 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) = 5(1|(2sin(2\u03c0x) + 1) + 2| - 1)^2 + 3(1|(2sin(2\u03c0x) + 1) + 2| - 1) + 4 and g(x) = (-3(2cos(2\u03c0x + 2) - 2) + 2)/(2(2cos(2\u03c0x + 2) - 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(4(2|(-3x + 5) - 3| - 2)^2 - 1(2|(-3x + 5) - 3| - 2) + 3) - 1)/(-1(4(2|(-3x + 5) - 3| - 2)^2 - 1(2|(-3x + 5) - 3| - 2) + 3) - 3) and g(x) = -1(-2sin(1\u03c0(4x - 1) + 1) + 1)^2 + 5 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-2e^(2(1x)) + 1))/(-1(-2e^(2(1x)) + 1) + 2) and g(x) = -3sin(1\u03c0((-1x - 1)/(3x + 1))) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3sin(2\u03c0((-2(4x^2 - 1x - 4) + 3)/(1(4x^2 - 1x - 4))) + 1) - 3 = -4(-2sin(1\u03c0x) - 1) - 4 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": "How many solutions does the equation 2cos(2\u03c0(-1(1(1|x - 1| - 2)^2 + 5(1|x - 1| - 2) - 2) - 5) - 3) - 3 = 1((-2(1sin(3\u03c0x - 1) + 1) - 3)/(-2(1sin(3\u03c0x - 1) + 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": "62", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1sin(3\u03c0(1(-1|x - 2| - 2)^2 + 1(-1|x - 2| - 2) - 1) - 3) + 1 and g(x) = (2(-1x^2 - 1) - 3)/(2(-1x^2 - 1) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "804", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1((-2(-2sin(2\u03c0x) - 3) + 2)/(3(-2sin(2\u03c0x) - 3) + 2)) and g(x) = -1e^(2((-3x - 1)/(3x + 3))) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(-2(-2|(1sin(3\u03c0x) + 2) - 1| + 2)^2 - 3(-2|(1sin(3\u03c0x) + 2) - 1| + 2) + 4) - 4 and g(x) = -1((3(5x - 1) + 1)/(-1(5x - 1) + 2))^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": "Find the number of times the graphs of f(x) = (3(-2(-5(-3|x - 1| + 1)^2 - 1(-3|x - 1| + 1) + 2) - 1) + 3)/(1(-2(-5(-3|x - 1| + 1)^2 - 1(-3|x - 1| + 1) + 2) - 1)) and g(x) = 1(2|(-3x - 1)| - 3)^2 - 2(2|(-3x - 1)| - 3) - 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": "Determine the number of intersection points between the functions f(x) = 3|(5(-3cos(2\u03c0x + 3) - 3) + 1) - 2| - 1 and g(x) = 3|(-1cos(1\u03c0x - 1) - 3) - 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": "Find the number of times the graphs of f(x) = 5(-2sin(3\u03c0(3|x - 3| + 2) + 1) - 1) and g(x) = -1cos(1\u03c0(5x^2 - 2x + 5) - 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "180", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(-2sin(1\u03c0(-2x - 4) - 2))^2 + 4(-2sin(1\u03c0(-2x - 4) - 2)) - 4 and g(x) = 1|(-1x - 4) + 2| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "64", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2|(4((-2(3cos(1\u03c0x) + 2) + 3)/(-3(3cos(1\u03c0x) + 2) - 3)) - 1) + 1| - 1 = -1(-1|(2cos(3\u03c0x) + 3) + 2| - 2)^2 - 1(-1|(2cos(3\u03c0x) + 3) + 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": "20", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 3(3cos(2\u03c0((-3(-4x) - 2)/(-1(-4x) + 3)) - 1) - 2)^2 + 4(3cos(2\u03c0((-3(-4x) - 2)/(-1(-4x) + 3)) - 1) - 2) - 5 and g(x) = 1|(-3(-3sin(1\u03c0x - 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": "97", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1e^(-1(4((3(-3|x - 3| - 2) + 3)/(1(-3|x - 3| - 2))) - 3)) + 1 and g(x) = 3|(-2cos(2\u03c0(-1x + 1) + 1) - 1) - 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2(1e^(2((3(1x + 1) - 3)/(3(1x + 1) - 2))) - 3)^2 - 2(1e^(2((3(1x + 1) - 3)/(3(1x + 1) - 2))) - 3) - 4 = 1sin(2\u03c0(-5(3|x + 3|) + 4) - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2((-2(1(1|x + 2|) - 5) + 1)/(1(1(1|x + 2|) - 5) + 1))^2 - 2((-2(1(1|x + 2|) - 5) + 1)/(1(1(1|x + 2|) - 5) + 1)) - 2 = -3|(1(3sin(3\u03c0x - 3) - 3) + 1) + 3| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-1|(-5(3cos(3\u03c0x))) + 3|))/(-3(-1|(-5(3cos(3\u03c0x))) + 3|) - 3) and g(x) = 5(3sin(1\u03c0((-1x + 3)/(1x + 1)) - 3) - 1)^2 - 4(3sin(1\u03c0((-1x + 3)/(1x + 1)) - 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": "101", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3sin(3\u03c0(-1|((-2(4x - 3) + 1)/(1(4x - 3) + 3)) - 3| + 3)) - 2 and g(x) = (3(2cos(2\u03c0(-1|x - 3|) - 1) + 1) - 1)/(-1(2cos(2\u03c0(-1|x - 3|) - 1) + 1) + 2) 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 points of intersection exist between the functions f(x) = 3(3sin(1\u03c0(-3|(2x + 3) + 1| - 1) - 1) - 2)^2 - 4(3sin(1\u03c0(-3|(2x + 3) + 1| - 1) - 1) - 2) - 1 and g(x) = 1(1|((-3x + 1)/(1x + 3)) - 1| - 1)^2 - 2(1|((-3x + 1)/(1x + 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": "108", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 5((1(2(-2cos(2\u03c0x + 3) - 1)^2 - 5(-2cos(2\u03c0x + 3) - 1) + 5) + 3)/(3(2(-2cos(2\u03c0x + 3) - 1)^2 - 5(-2cos(2\u03c0x + 3) - 1) + 5) - 1)) - 1 = (3(3cos(2\u03c0(2|x + 1| + 3)) + 1) - 3)/(-1(3cos(2\u03c0(2|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": "80", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-2(-1|(-4x + 2) + 1| + 3)^2 + 1(-1|(-4x + 2) + 1| + 3) + 3) + 1)/(-1(-2(-1|(-4x + 2) + 1| + 3)^2 + 1(-1|(-4x + 2) + 1| + 3) + 3) - 2) and g(x) = -1|(-3cos(2\u03c0((3x + 2)/(2x - 1)) - 3) + 1)| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1((-2(-2|(-2sin(1\u03c0x - 2))|) + 2)/(-3(-2|(-2sin(1\u03c0x - 2))|) - 1))^2 + 4((-2(-2|(-2sin(1\u03c0x - 2))|) + 2)/(-3(-2|(-2sin(1\u03c0x - 2))|) - 1)) - 4 = (2(-1|(-2x + 2)|))/(1(-1|(-2x + 2)|) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "78", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|((2(-5(-3x - 5)^2 - 1) + 1)/(-2(-5(-3x - 5)^2 - 1))) + 3| + 2 and g(x) = (3(-3(-3cos(1\u03c0x - 3) + 2) + 1) - 3)/(1(-3(-3cos(1\u03c0x - 3) + 2) + 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2((1(-3|(2cos(3\u03c0x) - 2) + 3| + 3) + 2)/(3(-3|(2cos(3\u03c0x) - 2) + 3| + 3) + 2)) and g(x) = (3(-2sin(3\u03c0(-4x^2 + 2x)) - 2) + 1)/(-2(-2sin(3\u03c0(-4x^2 + 2x)) - 2) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "694", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3sin(2\u03c0(-1(-1x - 1)^2 + 4) - 1) - 1 = -2cos(2\u03c0(-1|x + 1| - 3) - 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "126", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2sin(2\u03c0(-4(1|x + 2|) - 1)) - 1 and g(x) = -3(2sin(1\u03c0x + 3))^2 - 1(2sin(1\u03c0x + 3)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(3cos(3\u03c0(1x - 1) + 1) - 3) + 1)/(2(3cos(3\u03c0(1x - 1) + 1) - 3) + 3) and g(x) = -2|((-2x - 1)/(-1x)) - 2| where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1|(3e^(1(-5(1x + 1)^2 + 1(1x + 1) + 1))) + 1| - 1 and g(x) = 2(-1(3cos(3\u03c0x + 3))^2 + 4(3cos(3\u03c0x + 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": "60", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(-2cos(1\u03c0(1(5x^2 - 3x - 5) - 1) - 3) - 2) + 1| - 3 and g(x) = 1(-2(2|x| - 3)^2 + 2(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": "16", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(-4((-2x + 3)/(-3x - 1))^2 - 5((-2x + 3)/(-3x - 1)) - 2)| + 2 and g(x) = 2(3|x - 3| - 3)^2 - 5(3|x - 3| - 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -4(1|((2(3sin(1\u03c0x) + 2) + 3)/(-1(3sin(1\u03c0x) + 2))) + 3| + 1)^2 + 3 and g(x) = -1cos(3\u03c0(-3|((-1x)/(-3x)) + 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": "40", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(2(-5(1cos(3\u03c0x)) - 2)^2 - 3(-5(1cos(3\u03c0x)) - 2) + 2) + 2)/(-3(2(-5(1cos(3\u03c0x)) - 2)^2 - 3(-5(1cos(3\u03c0x)) - 2) + 2) + 4) and g(x) = -4(-2|((3x + 3)/(1x - 2)) - 1| - 1) - 2 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) = -1sin(1\u03c0(4((-3x - 3)/(-2x + 3))^2) + 1) + 1 and g(x) = 2((2x - 1)/(-1x + 2)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1(-3(-2cos(1\u03c0(-2|x - 1| - 1) + 2) + 3) + 2)^2 - 4(-3(-2cos(1\u03c0(-2|x - 1| - 1) + 2) + 3) + 2) + 3 = 3|(-3(1x + 1)^2 + 5(1x + 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": "18", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(1cos(1\u03c0((-2x - 2)/(1x - 2)) + 1) - 2)^2 - 2(1cos(1\u03c0((-2x - 2)/(1x - 2)) + 1) - 2) - 5 and g(x) = (-3(2x^2 + 3) + 1)/(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": "72", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(3((-3(-3|x - 2| + 2) + 2)/(-2(-3|x - 2| + 2) - 3))^2 + 3((-3(-3|x - 2| + 2) + 2)/(-2(-3|x - 2| + 2) - 3)) + 1) + 2 and g(x) = 2|(-4(1cos(1\u03c0x + 3) - 2) + 5) + 1| - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(3\u03c0(5(1|x|) - 3) + 3) - 1 and g(x) = 1|(3x^2 - 2x - 1) + 2| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3e^(-2(3|(-2(5x^2 - 2x + 4) + 2) + 3|)) and g(x) = -3sin(3\u03c0(-1(-1|x - 3| + 2)^2 + 5(-1|x - 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": "326", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2|(-1(-1cos(1\u03c0(-2x - 3) - 3) + 1)^2 - 2(-1cos(1\u03c0(-2x - 3) - 3) + 1) + 3) + 3| = 1cos(2\u03c0(-4((-2x - 1)/(3x - 1))^2 - 3((-2x - 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": "90", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(-1sin(3\u03c0(3x^2 + 1x - 1) + 3) + 3)| - 2 and g(x) = 3|(1x - 4) + 3| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "212", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-1sin(2\u03c0(-4x^2 + 3x - 5) + 2) - 3) and g(x) = 2(-1cos(3\u03c0x - 1) + 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "384", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -3sin(1\u03c0(3|(-5(-4x^2 - 4x + 5) - 5) - 3| + 1) + 3) + 3 = -2sin(3\u03c0(3((3x + 2)/(-2x - 1))^2 - 3((3x + 2)/(-2x - 1)) + 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": "286", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (2(-2|(2x^2 + 3x)| - 3) - 3)/(-2(-2|(2x^2 + 3x)| - 3) + 1) = -2cos(3\u03c0((2x - 1)/(-2x - 2)) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "68", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(-2sin(2\u03c0(-4x + 2)) + 2) - 1| + 2 and g(x) = 2sin(3\u03c0(1x - 4) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "203", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|((1(-5(1x^2 - 1x - 3) + 5) - 3)/(-2(-5(1x^2 - 1x - 3) + 5) - 1))| + 2 and g(x) = -5(-3sin(2\u03c0(1x^2 - 4x - 4) - 1) - 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "416", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5((-1(1sin(2\u03c0(2|x + 2| + 2) + 2) - 1) - 1)/(3(1sin(2\u03c0(2|x + 2| + 2) + 2) - 1))) - 1 and g(x) = -1(-1((-3x - 1)/(3x)) + 1)^2 - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2|(-2(1e^(1x))^2 + 4) - 2| and g(x) = -5(2sin(3\u03c0x - 3) + 2)^2 - 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": "Find the number of times the graphs of f(x) = -2sin(1\u03c0(1(-1x^2 - 5x + 3) - 1) - 2) - 2 and g(x) = 1|(-2sin(2\u03c0x + 2) + 3) - 3| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "206", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2|(2((2x - 3)/(-2x)))| + 1 = -1sin(1\u03c0(-3|x + 1|) + 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 4((3(1|(2cos(2\u03c0x - 1) - 2) + 2|) + 2)/(-3(1|(2cos(2\u03c0x - 1) - 2) + 2|) + 2))^2 - 4((3(1|(2cos(2\u03c0x - 1) - 2) + 2|) + 2)/(-3(1|(2cos(2\u03c0x - 1) - 2) + 2|) + 2)) - 5 = (-2(-5(3x - 3)^2 - 1(3x - 3) - 2) + 1)/(-2(-5(3x - 3)^2 - 1(3x - 3) - 2) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(3((-1x + 3)/(2x)))| - 3 and g(x) = -3|(2sin(3\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": "79", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(4(-1|(-1sin(3\u03c0x + 3) + 3) - 3| + 1)^2 - 4(-1|(-1sin(3\u03c0x + 3) + 3) - 3| + 1) + 1) + 1)/(3(4(-1|(-1sin(3\u03c0x + 3) + 3) - 3| + 1)^2 - 4(-1|(-1sin(3\u03c0x + 3) + 3) - 3| + 1) + 1) - 1) and g(x) = 2((2(2x^2 + 4x + 1))/(-1(2x^2 + 4x + 1) + 2)) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "231", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1|(-3(3((-3x + 3)/(-1x + 2)) - 4)^2 - 2(3((-3x + 3)/(-1x + 2)) - 4) + 2) - 1| - 3 = (1(5(1cos(3\u03c0x + 2) - 2)^2 + 5(1cos(3\u03c0x + 2) - 2) - 2) - 2)/(-2(5(1cos(3\u03c0x + 2) - 2)^2 + 5(1cos(3\u03c0x + 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": "6", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1sin(1\u03c0((-3(-5x^2 - 5x + 4) - 1)/(-3(-5x^2 - 5x + 4) + 2)) + 2) + 1 and g(x) = -1(-2sin(3\u03c0x) + 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": "Determine the number of intersection points between the functions f(x) = -3sin(1\u03c0(3((-2x + 2)/(-2x + 3))^2 - 2((-2x + 2)/(-2x + 3)) - 1) - 3) + 1 and g(x) = 1sin(3\u03c0(-3x^2 + 1x - 2) + 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "754", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -5((1(1(2|x - 3| - 3)^2 - 4(2|x - 3| - 3) - 5) - 3)/(-2(1(2|x - 3| - 3)^2 - 4(2|x - 3| - 3) - 5) + 3)) + 2 and g(x) = 4((1(-3sin(3\u03c0x + 2)) + 2)/(3(-3sin(3\u03c0x + 2)) + 1))^2 + 1((1(-3sin(3\u03c0x + 2)) + 2)/(3(-3sin(3\u03c0x + 2)) + 1)) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1(-2|((3(5x^2 - 2x - 5) - 2)/(-1(5x^2 - 2x - 5) + 3))|) and g(x) = -1|(-1(-3cos(3\u03c0x - 3) - 2)^2 - 1(-3cos(3\u03c0x - 3) - 2) + 1) - 3| in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(-2(3sin(1\u03c0x + 1)) + 2) + 3)/(1(-2(3sin(1\u03c0x + 1)) + 2) - 3) and g(x) = -3(2x + 4)^2 - 5(2x + 4) + 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(2((-2(-1x))/(-1(-1x) + 3))^2 - 3((-2(-1x))/(-1(-1x) + 3))) - 3| + 2 and g(x) = 5((-3(-1|x + 2| + 1) - 2)/(2(-1|x + 2| + 1) - 3))^2 - 4((-3(-1|x + 2| + 1) - 2)/(2(-1|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": "6", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(-2e^(1(-3|x + 1| - 3)) + 2) - 2)/(2(-2e^(1(-3|x + 1| - 3)) + 2) + 3) and g(x) = 1cos(3\u03c0(-2|x - 2| + 3) + 2) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(-2cos(2\u03c0(3|x + 1| + 2) + 2)) and g(x) = -2(3x^2 + 1x - 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 points of intersection exist between the functions f(x) = -4((3(-3|(3sin(1\u03c0x + 1) - 1) + 3| + 2) - 3)/(-2(-3|(3sin(1\u03c0x + 1) - 1) + 3| + 2) - 3))^2 - 3((3(-3|(3sin(1\u03c0x + 1) - 1) + 3| + 2) - 3)/(-2(-3|(3sin(1\u03c0x + 1) - 1) + 3| + 2) - 3)) - 1 and g(x) = (-3(-2cos(2\u03c0(-1x - 1) + 3)) + 3)/(2(-2cos(2\u03c0(-1x - 1) + 3)) - 3) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(-2|(-1((1x + 2)/(-2x + 3)) + 1) - 3| + 2)^2 - 2(-2|(-1((1x + 2)/(-2x + 3)) + 1) - 3| + 2) + 5 and g(x) = (2(2|(2cos(3\u03c0x - 1) - 3) - 1| - 2) + 2)/(1(2|(2cos(3\u03c0x - 1) - 3) - 1| - 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 solutions does the equation (2(-2|(1(2x)^2 - 1(2x) - 3) + 1|) + 2)/(-2(-2|(1(2x)^2 - 1(2x) - 3) + 1|) - 3) = -5(-2|((-3x + 2)/(2x - 3)) + 2| + 3)^2 + 1(-2|((-3x + 2)/(2x - 3)) + 2| + 3) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1((-1(-2(3|x + 2| + 3)^2 + 5(3|x + 2| + 3) + 5) - 2)/(-2(-2(3|x + 2| + 3)^2 + 5(3|x + 2| + 3) + 5) - 1)) - 5 and g(x) = 2((-1(1cos(1\u03c0x + 1) - 1))/(1(1cos(1\u03c0x + 1) - 1) + 1)) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(5(-1|((-2x - 3)/(-3x - 1)) + 3| + 3) + 2)^2 - 4 and g(x) = -2(-1cos(3\u03c0(-1x^2 - 1x + 2) + 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "231", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2|(1sin(1\u03c0(-5(1x + 5)^2 + 3)) - 1) + 1| and g(x) = (-3(-5(-1x^2 + 3x - 1) - 3) - 1)/(-1(-5(-1x^2 + 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": "48", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1cos(3\u03c0((2(2(-3|x + 3| + 1) + 2) - 2)/(-2(2(-3|x + 3| + 1) + 2) - 2))) + 3 and g(x) = -2cos(3\u03c0((-1(1x - 1) - 3)/(1(1x - 1) + 1)) - 2) + 2 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": "Find the number of times the graphs of f(x) = (-1(1|(-5(3x^2 + 1x - 4) - 4) - 3| - 2) + 1)/(-1(1|(-5(3x^2 + 1x - 4) - 4) - 3| - 2) + 2) and g(x) = -1((-3(-1sin(3\u03c0x - 2)) + 2)/(1(-1sin(3\u03c0x - 2)) - 2)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "52", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1((1(-1sin(1\u03c0(2x + 5) + 2) - 1) + 2)/(2(-1sin(1\u03c0(2x + 5) + 2) - 1) - 3))^2 - 3((1(-1sin(1\u03c0(2x + 5) + 2) - 1) + 2)/(2(-1sin(1\u03c0(2x + 5) + 2) - 1) - 3)) - 1 and g(x) = (1(5(2sin(3\u03c0x - 1) - 1)^2 + 2(2sin(3\u03c0x - 1) - 1)) + 3)/(1(5(2sin(3\u03c0x - 1) - 1)^2 + 2(2sin(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": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 5(2cos(2\u03c0(-3x + 5) - 2) - 1)^2 - 1 = -2(-1x - 5)^2 - 5(-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": "34", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2((-3(-2(-3|x + 1| + 2)^2 + 4) + 2)/(-2(-2(-3|x + 1| + 2)^2 + 4) - 3)) + 3 and g(x) = 2(1(3cos(1\u03c0x - 3) - 1)^2 + 3(3cos(1\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": "26", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-2sin(1\u03c0(-3(-2|x + 2| - 1) - 2))) - 3)/(3(-2sin(1\u03c0(-3(-2|x + 2| - 1) - 2))) - 2) and g(x) = -3(3(1sin(3\u03c0x + 3)) + 5)^2 - 2(3(1sin(3\u03c0x + 3)) + 5) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(4((-1x - 1)/(2x - 1)) + 4) - 1| - 3 and g(x) = -1sin(2\u03c0(-2x^2 - 3x - 3) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "344", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (1(-1|(-4x^2 - 3x - 1) - 3| - 1) + 3)/(-3(-1|(-4x^2 - 3x - 1) - 3| - 1) - 3) = 3sin(1\u03c0(1|x + 3| + 2) - 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(3|(4x^2 + 4x + 3) - 1| + 1) - 1)/(-3(3|(4x^2 + 4x + 3) - 1| + 1)) and g(x) = -3cos(2\u03c0(5x - 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "200", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-1cos(2\u03c0(3x^2 + 2x + 5)) + 2) + 1)/(1(-1cos(2\u03c0(3x^2 + 2x + 5)) + 2) - 3) and g(x) = -1((-2x + 1)/(2x - 3))^2 - 2((-2x + 1)/(2x - 3)) + 5 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": "How many solutions does the equation 3|((2(-2(-3x - 3)^2 + 2(-3x - 3) + 4) + 3)/(-1(-2(-3x - 3)^2 + 2(-3x - 3) + 4) + 2))| - 1 = -1sin(2\u03c0(2(2|x - 1| + 1)^2 - 1(2|x - 1| + 1) + 5) - 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": "Determine the number of intersection points between the functions f(x) = -1cos(3\u03c0(-5((-2x - 3)/(-3x - 1))^2 - 2((-2x - 3)/(-3x - 1)) + 3) - 2) + 1 and g(x) = -1cos(2\u03c0(-1x^2 + 2x - 3) - 1) + 2 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": "How many points of intersection exist between the functions f(x) = -3cos(2\u03c0(1((3(-1x^2 + 2x + 5) + 2)/(-2(-1x^2 + 2x + 5) - 2)) - 3) - 2) - 2 and g(x) = -1|(-1((-2x + 1)/(-2x)) - 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": "8", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2|(1(1x^2 - 3x) + 3) - 2| + 1 = -2cos(2\u03c0((1x + 2)/(-1x))) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-1(3|(-1x^2 + 3)| + 1) + 2)/(-2(3|(-1x^2 + 3)| + 1) + 3) = 3cos(2\u03c0(3|x + 3| + 3) - 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "122", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1(-4((-1(-1|x - 3|) - 2)/(1(-1|x - 3|) + 2)) - 4)^2 + 1(-4((-1(-1|x - 3|) - 2)/(1(-1|x - 3|) + 2)) - 4) + 4 = -3cos(3\u03c0(-2(-1x^2 - 4x - 3) + 4) + 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "526", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(-2e^(1(-3|x| + 1)) + 3) + 1)/(-3(-2e^(1(-3|x| + 1)) + 3) + 2) and g(x) = -4(-2cos(3\u03c0x - 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) = -2sin(3\u03c0(2(-5x - 3)^2 - 5(-5x - 3) + 3) - 3) - 2 and g(x) = (2(2cos(3\u03c0x - 2) - 3) - 3)/(-2(2cos(3\u03c0x - 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": "1325", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(-2|((-2x)/(-1x + 2)) - 3| - 1) - 1 and g(x) = -3sin(1\u03c0(1x - 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": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -4(-2cos(1\u03c0(-2(-2|x - 1| + 3)^2 + 2(-2|x - 1| + 3)) - 3) + 1) - 1 = -2e^(2(-1(-1|x + 3| + 2)^2 + 3(-1|x + 3| + 2) + 3)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "872", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1|(-2(-2x + 4)^2 + 1(-2x + 4) + 4) - 3| - 3 and g(x) = 5(-2cos(3\u03c0x + 3) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2((2(-2|(1x + 2) + 1| + 1) + 1)/(-3(-2|(1x + 2) + 1| + 1) + 2))^2 - 2((2(-2|(1x + 2) + 1| + 1) + 1)/(-3(-2|(1x + 2) + 1| + 1) + 2)) - 4 and g(x) = -3sin(1\u03c0(-2|(-2x^2 + 4x + 5) + 1| + 1)) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "756", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(1\u03c0(-3|((1(5x^2 - 1x - 2) - 1)/(1(5x^2 - 1x - 2) - 3)) + 3|) + 2) - 1 and g(x) = 3|(-1((1x - 2)/(-2x + 1))^2 + 3((1x - 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": "22", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(-2sin(2\u03c0(4x + 5)) - 3)^2 - 5(-2sin(2\u03c0(4x + 5)) - 3) - 2 and g(x) = -2e^(-1(-2|x + 1|)) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3|(2(1x + 5)^2 - 2(1x + 5) - 2)| + 3 = 4(2sin(1\u03c0x + 3) - 3)^2 - 5(2sin(1\u03c0x + 3) - 3) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 4(2sin(3\u03c0((2(-1|x - 3| + 3) + 3)/(-2(-1|x - 3| + 3) + 3))) + 1)^2 - 5 = 5(-1cos(1\u03c0(2x^2 - 1x - 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": "229", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((-1(-5(1|x - 1| + 3)^2 + 1) - 3)/(-3(-5(1|x - 1| + 3)^2 + 1) - 3)) - 3 and g(x) = 3cos(3\u03c0(1((2x + 3)/(3x - 2))^2 - 1((2x + 3)/(3x - 2)) - 4) + 1) - 3 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) = (3(4(1(-2|x + 3| - 3))^2 - 3(1(-2|x + 3| - 3)) + 1) + 3)/(-1(4(1(-2|x + 3| - 3))^2 - 3(1(-2|x + 3| - 3)) + 1) + 2) and g(x) = -1|(-3(3sin(1\u03c0x + 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": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1(-5((-3x)/(3x))^2 + 4) - 1 and g(x) = 3sin(3\u03c0((2x + 1)/(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": "22", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(2|(3cos(3\u03c0(1x^2 - 2x + 2) - 2) - 2) - 1| - 2) - 2)/(2(2|(3cos(3\u03c0(1x^2 - 2x + 2) - 2) - 2) - 1| - 2) + 1) and g(x) = -5(-3cos(1\u03c0(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": "27", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 4((2(3sin(3\u03c0x - 2) + 2) + 3)/(-3(3sin(3\u03c0x - 2) + 2) - 1))^2 + 1((2(3sin(3\u03c0x - 2) + 2) + 3)/(-3(3sin(3\u03c0x - 2) + 2) - 1)) + 3 = 3(-1e^(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": "73", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2(-2(3sin(3\u03c0(1|x + 1| - 1) + 1) + 3) + 2)^2 - 2(-2(3sin(3\u03c0(1|x + 1| - 1) + 1) + 3) + 2) + 2 and g(x) = (1(1|(-4x^2 - 5x - 4) + 3| + 3) - 3)/(2(1|(-4x^2 - 5x - 4) + 3| + 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 5(-3cos(3\u03c0(-1|((-1x + 3)/(-3x))| + 1) + 2) - 2)^2 - 1(-3cos(3\u03c0(-1|((-1x + 3)/(-3x))| + 1) + 2) - 2) and g(x) = (-1(1(-1|x + 3| - 2)^2 + 5(-1|x + 3| - 2) + 3) + 3)/(-1(1(-1|x + 3| - 2)^2 + 5(-1|x + 3| - 2) + 3) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1sin(2\u03c0(5((-1(1|x - 3| + 3) - 1)/(1(1|x - 3| + 3) - 3))) + 1) - 2 = -1|(5((-1x + 2)/(-1x + 3))^2 + 1((-1x + 2)/(-1x + 3)) - 5) - 1| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2((-2(-1cos(3\u03c0(2|x - 1| + 3)) + 2) + 3)/(-1(-1cos(3\u03c0(2|x - 1| + 3)) + 2) - 2)) + 1 and g(x) = -1(-1sin(1\u03c0(-2|x|) + 2) - 2) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(1\u03c0(3|(1(-4x^2 + 2x) + 3) - 1| + 2) - 1) + 2 and g(x) = (1(-2cos(2\u03c0(-4x + 4))) + 2)/(2(-2cos(2\u03c0(-4x + 4))) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "906", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(4(1|(2x^2 - 3x + 4) + 2| + 2) + 3) + 3)/(-1(4(1|(2x^2 - 3x + 4) + 2| + 2) + 3) - 3) and g(x) = (-1(3sin(1\u03c0(-1|x + 1| + 1) + 3)) + 3)/(-1(3sin(1\u03c0(-1|x + 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": "18", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3cos(1\u03c0((-1(-2x^2 - 2x + 5) - 3)/(2(-2x^2 - 2x + 5))) + 2) - 1 and g(x) = (3(2|x| - 1) + 1)/(3(2|x| - 1) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (-2(-2(-2|(-3x^2 - 4x - 1) + 3| + 2) + 5))/(-3(-2(-2|(-3x^2 - 4x - 1) + 3| + 2) + 5) + 4) and g(x) = 3cos(2\u03c0(4(-1|x + 1| + 2) + 4) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "160", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(1\u03c0((-2(2|x - 3| - 2) + 3)/(1(2|x - 3| - 2) - 3)) - 2) + 2 and g(x) = 3(1sin(1\u03c0x + 1))^2 - 1(1sin(1\u03c0x + 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "90", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(3cos(3\u03c0(3(-3|x + 2| + 2)^2 - 2(-3|x + 2| + 2) + 2) - 2)) + 2 and g(x) = 1|(-1((2x + 2)/(-2x))^2 - 4((2x + 2)/(-2x)) - 4)| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1373", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-3((-2(3x^2 + 4x + 1))/(3(3x^2 + 4x + 1) + 3)) + 5) + 1| and g(x) = (-1(3(2sin(1\u03c0x - 3) - 1)^2 - 2) + 3)/(2(3(2sin(1\u03c0x - 3) - 1)^2 - 2) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3(-1(-3sin(3\u03c0x + 2) + 3)^2 + 3(-3sin(3\u03c0x + 2) + 3)) and g(x) = -1(-2e^(2x) - 3) + 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2cos(2\u03c0(2|(2(5x + 1)^2 + 4(5x + 1) - 3) - 1| + 3) - 3) - 1 and g(x) = 1cos(3\u03c0(2(-1x^2 - 3x - 4) - 2) + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1343", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -4(-1|(-1(-1sin(3\u03c0x - 3) + 1) - 5) + 1| - 1)^2 - 2(-1|(-1(-1sin(3\u03c0x - 3) + 1) - 5) + 1| - 1) = (2(3|(1sin(1\u03c0x - 2) + 2) - 2| - 2) + 3)/(2(3|(1sin(1\u03c0x - 2) + 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": "29", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(3cos(2\u03c0((3x + 1)/(1x - 3)) - 2) + 3) - 1| + 2 and g(x) = (-1(3e^(-2x) - 2) - 1)/(-2(3e^(-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": "157", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(3\u03c0(-3(2x - 1)^2 + 5(2x - 1) + 1) + 1) - 2 and g(x) = -3sin(2\u03c0((-2x + 2)/(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": "143", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(2(-4x + 4)^2 + 5(-4x + 4) - 1) - 1)/(2(2(-4x + 4)^2 + 5(-4x + 4) - 1) + 2) and g(x) = -3sin(1\u03c0(-4x^2 + 5x - 4)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "787", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(3sin(3\u03c0(2(-3x + 5)^2 + 1(-3x + 5) + 1)) + 1) + 3)/(-2(3sin(3\u03c0(2(-3x + 5)^2 + 1(-3x + 5) + 1)) + 1) - 2) and g(x) = -3e^(-1(1(-2|x + 2| - 2)^2 - 5(-2|x + 2| - 2) + 2)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "258", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -5(1((1(-2|x - 3| + 1) + 1)/(2(-2|x - 3| + 1) - 3))^2 - 5((1(-2|x - 3| + 1) + 1)/(2(-2|x - 3| + 1) - 3)) + 2) + 4 = 5((3(3cos(2\u03c0x + 1) + 2))/(2(3cos(2\u03c0x + 1) + 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": "12", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 2|(3((-1(2x - 1) - 2)/(-1(2x - 1) + 2))^2 + 3) - 2| and g(x) = 3(-3cos(3\u03c0((3x - 2)/(2x + 3)) - 1) - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "104", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (3(3|(-2(-3x^2 + 2x + 1) + 1) + 1| + 3) + 2)/(-1(3|(-2(-3x^2 + 2x + 1) + 1) + 1| + 3) + 2) = (-2(3cos(3\u03c0(1|x - 2| + 2) - 3) + 2))/(-1(3cos(3\u03c0(1|x - 2| + 2) - 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": "60", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1(2cos(2\u03c0(-4x^2 - 1x + 5) + 2) - 2) - 5 and g(x) = 1(-5x^2 - 1x) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|((2(3cos(2\u03c0(-4x^2 + 2x + 4) - 3)) - 2)/(-2(3cos(2\u03c0(-4x^2 + 2x + 4) - 3)) + 3)) - 3| + 3 and g(x) = 1|(1(-1x^2 + 2x + 4) - 3) + 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(-2|(1x^2 + 4x - 2) - 3| + 3) + 2)/(2(-2|(1x^2 + 4x - 2) - 3| + 3) - 1) and g(x) = -3(3|x + 1| - 3)^2 + 2(3|x + 1| - 3) + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2(3cos(2\u03c0((1(-2|x + 1| - 3))/(-2(-2|x + 1| - 3) + 3)) - 2) + 2)^2 + 3(3cos(2\u03c0((1(-2|x + 1| - 3))/(-2(-2|x + 1| - 3) + 3)) - 2) + 2) - 4 and g(x) = (-3(-3cos(2\u03c0(1x + 4)) + 2) - 2)/(2(-3cos(2\u03c0(1x + 4)) + 2) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3|(2(3((2x + 1)/(-1x + 2))^2 - 4((2x + 1)/(-1x + 2)) - 2) + 3) - 2| + 2 and g(x) = -2|(1((2x + 1)/(-3x - 2)) - 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": "6", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 5((-2(-2sin(3\u03c0(3|x - 2| - 2) + 3) + 1) - 1)/(-2(-2sin(3\u03c0(3|x - 2| - 2) + 3) + 1) - 3))^2 + 3((-2(-2sin(3\u03c0(3|x - 2| - 2) + 3) + 1) - 1)/(-2(-2sin(3\u03c0(3|x - 2| - 2) + 3) + 1) - 3)) + 4 and g(x) = 3((2(3|x + 3| + 3) - 1)/(-1(3|x + 3| + 3) - 2))^2 - 2 intersect in the interval [-10, 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(1(2sin(3\u03c0x - 3) + 3)^2 - 4(2sin(3\u03c0x - 3) + 3) - 5) - 3)/(3(1(2sin(3\u03c0x - 3) + 3)^2 - 4(2sin(3\u03c0x - 3) + 3) - 5) + 3) = (2(-4x + 1))/(-2(-4x + 1) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(-1|((3(4x^2 - 2x - 2) - 2)/(-3(4x^2 - 2x - 2) - 3)) - 2| + 1) + 1 and g(x) = -1((3(-1|x + 3|))/(2(-1|x + 3|) + 2)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1((-3(-1|(-3x + 3)| - 3))/(3(-1|(-3x + 3)| - 3) - 1))^2 + 2((-3(-1|(-3x + 3)| - 3))/(3(-1|(-3x + 3)| - 3) - 1)) - 1 = -3(1sin(3\u03c0((-2x - 3)/(-1x - 2)) - 2) + 1)^2 - 1(1sin(3\u03c0((-2x - 3)/(-1x - 2)) - 2) + 1) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(3\u03c0(3|((2x)/(-2x - 3)) - 1| + 1) + 2) and g(x) = 4(1|x| - 2)^2 - 4(1|x| - 2) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1e^(1((-2(-4x^2 - 4x + 4) + 1)/(-3(-4x^2 - 4x + 4)))) + 2 = -3cos(3\u03c0(-3x^2 + 5x + 1) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1442", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -5(2sin(3\u03c0((3x - 3)/(3x + 3)) + 2) + 2) - 1 and g(x) = 2cos(3\u03c0(-1x - 3) - 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3cos(2\u03c0(1(-4x^2 + 4) + 4) + 3) - 1 and g(x) = -1(-2cos(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": "355", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2((-3(-1|(-3x^2 + 3x + 4) + 1| + 1) - 2)/(-2(-1|(-3x^2 + 3x + 4) + 1| + 1) + 3)) - 3 and g(x) = -3cos(1\u03c0(1((-1x - 1)/(-2x + 3)) + 5) - 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) = 4(-5(-2sin(3\u03c0(1|x - 2| + 3) + 2) - 1)^2 + 4(-2sin(3\u03c0(1|x - 2| + 3) + 2) - 1) + 4) - 5 and g(x) = -5(-1|((-1x - 3)/(1x - 2)) - 3| - 2)^2 + 2(-1|((-1x - 3)/(1x - 2)) - 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": "34", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2cos(1\u03c0(-1(1x^2 + 4x + 1) + 3) + 1) + 2 and g(x) = (1(1|x + 1| + 2) + 1)/(3(1|x + 1| + 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "208", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 3|(2((3(1cos(2\u03c0x + 2) - 2) + 2)/(1(1cos(2\u03c0x + 2) - 2) + 3))^2 + 1((3(1cos(2\u03c0x + 2) - 2) + 2)/(1(1cos(2\u03c0x + 2) - 2) + 3)) - 5) + 3| + 3 and g(x) = 3((-1(-1|x + 3| + 2) - 1)/(3(-1|x + 3| + 2) - 2))^2 - 3((-1(-1|x + 3| + 2) - 1)/(3(-1|x + 3| + 2) - 2)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(3\u03c0(-2|(5(-2x^2 - 4x - 3) - 2)|)) - 1 and g(x) = (1(1(5x^2 - 4x - 5) + 5))/(2(1(5x^2 - 4x - 5) + 5) - 1) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1145", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4(2|(-3sin(1\u03c0x - 2) - 2) + 2| - 2) + 1 and g(x) = -1(2e^(-2x) - 3) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1sin(2\u03c0(-3(1(3|x - 2| + 3) + 1)^2 + 2(1(3|x - 2| + 3) + 1) - 1) - 3) + 1 and g(x) = 1|(-2(-5x^2 + 4x + 2) + 3) + 3| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1sin(3\u03c0((2(-1|(2x^2 + 2x - 4)| + 1) - 2)/(3(-1|(2x^2 + 2x - 4)| + 1) - 2)) - 1) + 3 and g(x) = -1(-1|(-1x^2 + 4x - 4) + 3| - 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|(4((-2(3cos(2\u03c0x - 2) - 3) + 1)/(-1(3cos(2\u03c0x - 2) - 3) - 3))) + 3| - 1 and g(x) = (-2(2(-3e^(-1x) + 2) - 4))/(-1(2(-3e^(-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": "40", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3sin(3\u03c0((2(5(-1x - 3)^2 - 3(-1x - 3) - 3) - 2)/(2(5(-1x - 3)^2 - 3(-1x - 3) - 3) - 1))) - 3 = (-2(2|(5x^2 + 3x + 1)| + 3) - 2)/(2(2|(5x^2 + 3x + 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": "28", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -4(-1|(4(-3sin(2\u03c0x - 3) + 2)^2 - 2(-3sin(2\u03c0x - 3) + 2) - 3) + 3| - 2) - 5 and g(x) = 3(-3|(3x + 5) - 2| + 2)^2 - 1(-3|(3x + 5) - 2| + 2) + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2sin(1\u03c0(-1|(-3x + 3) + 2| + 2) + 3) + 3 = (2(3cos(2\u03c0x + 3) - 1) + 3)/(2(3cos(2\u03c0x + 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": "20", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(3|(2((-2x + 3)/(1x + 2)) + 2)|)^2 + 4(3|(2((-2x + 3)/(1x + 2)) + 2)|) - 3 and g(x) = -3sin(3\u03c0(2(-1x^2 - 2x + 4) - 1) + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "93", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-5(-2|x| + 2)^2 + 1(-2|x| + 2) + 3) + 1)/(-1(-5(-2|x| + 2)^2 + 1(-2|x| + 2) + 3) + 2) and g(x) = 2cos(2\u03c0((-3x + 2)/(3x + 3)) - 3) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2(3((-1(2|x + 3| + 1) - 2)/(-1(2|x + 3| + 1) + 2))^2 - 1((-1(2|x + 3| + 1) - 2)/(-1(2|x + 3| + 1) + 2)) + 2) + 2 and g(x) = 3|(3cos(2\u03c0(-3x^2 - 3x - 4) + 2)) - 3| - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "670", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1((3(-3|(-4x^2 + 4x - 2) + 2| + 3))/(-1(-3|(-4x^2 + 4x - 2) + 2| + 3))) + 2 and g(x) = (-1(-1|(1sin(2\u03c0x - 3) + 3) - 2| + 2) - 3)/(-2(-1|(1sin(2\u03c0x - 3) + 3) - 2| + 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": "How many points of intersection exist between the functions f(x) = (1(2sin(2\u03c0(3x - 3) - 2) - 1) - 2)/(3(2sin(2\u03c0(3x - 3) - 2) - 1) + 3) and g(x) = (-2(1sin(3\u03c0x + 1) + 1) + 2)/(-1(1sin(3\u03c0x + 1) + 1) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "101", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (1(1|(1(-3x^2 + 3x - 4) + 5) - 2| - 2) - 1)/(3(1|(1(-3x^2 + 3x - 4) + 5) - 2| - 2) + 3) and g(x) = 2(-3|(1x^2 + 4x - 4) - 3|) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(1(3sin(2\u03c0x - 1) - 1) + 3) + 1)/(-2(1(3sin(2\u03c0x - 1) - 1) + 3) - 2) and g(x) = 2cos(1\u03c0(5x + 2) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "100", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(3\u03c0((-3(-3|(-4x^2 + 2x - 2) + 3| - 3) + 2)/(-1(-3|(-4x^2 + 2x - 2) + 3| - 3) + 3)) - 2) + 1 and g(x) = (-2(-5(-1x^2 - 5x)))/(-1(-5(-1x^2 - 5x)) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(-1(3e^(-1(-3|x - 2| - 2)) + 2)^2 + 2) - 1)/(-1(-1(3e^(-1(-3|x - 2| - 2)) + 2)^2 + 2) - 2) and g(x) = 1cos(1\u03c0(3|(4x^2 + 1x + 2) - 2| - 2)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "849", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -5(-1cos(3\u03c0((1(-2x^2 - 4x + 5) + 3)/(-3(-2x^2 - 4x + 5) + 1))) + 2) + 1 and g(x) = -1(2|(1sin(1\u03c0x - 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": "42", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2sin(2\u03c0(4(-2x^2 - 5x - 3) - 3) + 2) - 1 and g(x) = 3e^(2(-1x - 2)) - 2 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "858", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(2|((-2(2cos(1\u03c0x - 3) + 3) + 2)/(-1(2cos(1\u03c0x - 3) + 3) + 2)) + 1| - 1) + 4 and g(x) = -4(-2e^(-1((2x + 1)/(2x - 3))) - 1)^2 - 4(-2e^(-1((2x + 1)/(2x - 3))) - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 5(-3|(2sin(3\u03c0x) + 3) - 1| + 2)^2 - 1(-3|(2sin(3\u03c0x) + 3) - 1| + 2) + 4 and g(x) = -4(1e^(2x) - 3) - 4 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2(-2|(3sin(1\u03c0x + 3) + 1) - 2| - 2) - 1 and g(x) = -1(1|x - 3| - 1)^2 - 4(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": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(1\u03c0(3|(2x - 5) - 2| + 1) - 1) - 2 and g(x) = -1sin(3\u03c0(2x + 1) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "82", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-3(2(1|(-4x + 5) + 3|)^2 + 1(1|(-4x + 5) + 3|) + 2) - 1)/(-1(2(1|(-4x + 5) + 3|)^2 + 1(1|(-4x + 5) + 3|) + 2)) and g(x) = 2|(4(3sin(1\u03c0x) - 1)^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": "80", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(-2|(3x + 4) - 3| + 3) - 2)/(-2(-2|(3x + 4) - 3| + 3) - 3) and g(x) = (1(-2sin(3\u03c0x - 3) - 2))/(-3(-2sin(3\u03c0x - 3) - 2) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3|(1(-4((-1x + 3)/(-3x - 1))^2 - 4((-1x + 3)/(-3x - 1)) + 2) + 3) - 1| - 2 and g(x) = -3(2(2sin(2\u03c0x - 3) - 1) - 3)^2 + 5(2(2sin(2\u03c0x - 3) - 1) - 3) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2(-3|((3(2sin(2\u03c0x - 2) + 2) - 1)/(-2(2sin(2\u03c0x - 2) + 2) + 3)) - 1|) - 2 = -1e^(2(-3((2x)/(3x - 1))^2 + 2)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 1e^(-2(-2((2x - 1)/(-2x - 2))^2 - 4((2x - 1)/(-2x - 2)) + 2)) - 3 and g(x) = 3cos(3\u03c0(-5x^2 + 5x - 1) + 1) - 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1497", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 2|((2(2cos(2\u03c0x + 1) - 1))/(-3(2cos(2\u03c0x + 1) - 1) + 2)) + 2| = -3((2x + 2)/(-1x - 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "80", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (3(1|(-1cos(2\u03c0(1x - 3) + 1)) - 2| - 2) - 1)/(-2(1|(-1cos(2\u03c0(1x - 3) + 1)) - 2| - 2) + 1) and g(x) = 1(-4(-1sin(2\u03c0x - 2))^2 - 5(-1sin(2\u03c0x - 2)) - 5) + 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((-2(3cos(3\u03c0(3x^2 + 1x + 5) - 3) - 2) - 3)/(-1(3cos(3\u03c0(3x^2 + 1x + 5) - 3) - 2) - 2)) - 1 and g(x) = 1|(-4((1x - 1)/(3x + 3))^2 - 2((1x - 1)/(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": "350", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-2cos(1\u03c0((3(-4x^2 - 2x + 1) + 2)/(-2(-4x^2 - 2x + 1))) - 3)) + 2| and g(x) = (-1(3|(2x + 4) - 1| + 2) - 2)/(2(3|(2x + 4) - 1| + 2) - 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(2cos(3\u03c0(-3x) + 2) + 2) - 1)/(3(2cos(3\u03c0(-3x) + 2) + 2) - 2) and g(x) = -1(-2|x + 1|)^2 - 5(-2|x + 1|) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-2(2(1e^(1(-5x + 2)) + 3)^2 + 1(1e^(1(-5x + 2)) + 3)))/(-3(2(1e^(1(-5x + 2)) + 3)^2 + 1(1e^(1(-5x + 2)) + 3)) - 1) and g(x) = 1|(2cos(1\u03c0(-3x^2 + 4x - 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": "966", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(-2((-3x + 1)/(-1x - 3)) + 3) - 3| + 3 and g(x) = 3cos(2\u03c0(-2x^2 - 3x - 5) - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "200", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (3(3sin(3\u03c0(4(3x + 4)^2 + 5(3x + 4) + 5) - 1)) + 3)/(-2(3sin(3\u03c0(4(3x + 4)^2 + 5(3x + 4) + 5) - 1)) + 3) and g(x) = 3cos(2\u03c0((1(3|x + 1| + 3) - 3)/(2(3|x + 1| + 3) - 1)) - 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "318", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 4(-2sin(1\u03c0(-3|x - 3| + 3) - 1) + 2)^2 - 1(-2sin(1\u03c0(-3|x - 3| + 3) - 1) + 2) + 3 and g(x) = 1|(-3x + 4) - 3| + 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": "Find the number of times the graphs of f(x) = (2(2(-2e^(-2(-3|x - 3| - 2)) - 3) - 4) - 2)/(-3(2(-2e^(-2(-3|x - 3| - 2)) - 3) - 4) + 1) and g(x) = 2|(-1(2sin(1\u03c0x + 2) + 1) + 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) = (-1(1(-2sin(3\u03c0x + 1) - 2)) + 2)/(-3(1(-2sin(3\u03c0x + 1) - 2)) - 2) and g(x) = -5(-3|x + 2|) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3sin(1\u03c0(2|((-2x - 1)/(3x + 3)) + 1| + 3) - 1) + 1 and g(x) = 4(-5x - 5)^2 - 1(-5x - 5) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 4((1(1(-1|x + 1| + 3)^2 - 1(-1|x + 1| + 3)) - 2)/(-2(1(-1|x + 1| + 3)^2 - 1(-1|x + 1| + 3)) + 3)) and g(x) = -4(-3|(-2sin(3\u03c0x)) - 2| + 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 4(-1sin(1\u03c0((3(1x^2 - 2x - 3) + 3)/(2(1x^2 - 2x - 3) + 3)) + 3) + 2) = 2(3|((-1x + 1)/(1x + 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": "23", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1(-2|((-1(-3sin(3\u03c0x - 3) + 1) + 3)/(-2(-3sin(3\u03c0x - 3) + 1) + 1)) - 1| + 3) - 2 = (-1(1(-1|x - 1|)^2 + 2(-1|x - 1|) - 4) + 2)/(-2(1(-1|x - 1|)^2 + 2(-1|x - 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": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3|(2(3((-1x - 3)/(-1x + 2)) - 2)^2 - 1(3((-1x - 3)/(-1x + 2)) - 2) + 2) + 2| - 3 = 3|((1(-2cos(2\u03c0x) + 3) - 1)/(-3(-2cos(2\u03c0x) + 3) + 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": "30", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(1|(-3x^2 + 4x - 2) - 1| - 3) + 2)/(-2(1|(-3x^2 + 4x - 2) - 1| - 3) - 2) and g(x) = 3(1cos(3\u03c0x - 2))^2 - 2(1cos(3\u03c0x - 2)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1sin(1\u03c0(-1|(1x^2 + 3x + 4) - 3| - 1) + 2) - 3 and g(x) = 3e^(-1(4x + 3)) - 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": "Determine the number of intersection points between the functions f(x) = (-1(2|(-5x^2 + 4x - 5)|) + 1)/(-3(2|(-5x^2 + 4x - 5)|) + 2) and g(x) = 1(3cos(1\u03c0x + 2))^2 - 5(3cos(1\u03c0x + 2)) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -2cos(1\u03c0(1(2|x - 3| + 3)^2 - 5(2|x - 3| + 3) - 4) - 3) + 2 and g(x) = 2|((-3x + 1)/(-2x - 3)) + 3| + 2 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -1(-2e^(1(-1|x - 3| - 1)) - 3) + 3 = 1(3sin(1\u03c0x - 3) + 1)^2 + 4(3sin(1\u03c0x - 3) + 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = (2(-1(3sin(1\u03c0x + 2) + 1) - 5) - 3)/(2(-1(3sin(1\u03c0x + 2) + 1) - 5) + 2) and g(x) = 2sin(2\u03c0(3x - 4) - 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "120", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1sin(1\u03c0(-5(1x - 3)^2 - 2(1x - 3) - 1) + 3) - 2 and g(x) = 2((-1x - 2)/(-3x + 2)) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "68", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 2|((-2(-1(-2x - 5)^2 + 4(-2x - 5) - 2) - 3)/(2(-1(-2x - 5)^2 + 4(-2x - 5) - 2))) - 1| + 3 and g(x) = -5(-2|(-1cos(3\u03c0x - 1) - 2) + 1| - 1) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|((-3(3sin(2\u03c0x - 3) - 3))/(1(3sin(2\u03c0x - 3) - 3) + 2)) - 1| and g(x) = 3cos(1\u03c0(-1x^2 - 5x - 4) - 1) - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "144", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (2(3cos(3\u03c0(-4(-1x - 1)^2 + 4(-1x - 1) + 2) + 2) + 1) - 2)/(3(3cos(3\u03c0(-4(-1x - 1)^2 + 4(-1x - 1) + 2) + 2) + 1) - 1) and g(x) = -2|(-4(1cos(1\u03c0x - 2) + 3)^2 + 5(1cos(1\u03c0x - 2) + 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": "10", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3(3|(2cos(3\u03c0(-5x^2 - 3x + 2) + 3) - 3) + 1| - 1) + 3 and g(x) = 2(1|(-1sin(2\u03c0x - 3) + 1) - 1| - 1)^2 - 4(1|(-1sin(2\u03c0x - 3) + 1) - 1| - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "778", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2e^(-1((-3(3(-1|x - 1| - 3)^2 + 2(-1|x - 1| - 3) + 1) - 1)/(3(3(-1|x - 1| - 3)^2 + 2(-1|x - 1| - 3) + 1) - 1))) + 3 and g(x) = -2(-2sin(3\u03c0(-3x - 1) - 2))^2 - 5(-2sin(3\u03c0(-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": "180", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3sin(3\u03c0(-2((2(-3|x - 1| - 2) - 2)/(3(-3|x - 1| - 2) + 2)) - 2) - 3) - 2 = -3|(2sin(1\u03c0(-5x^2 + 2x) + 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": "10", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2((-3(-2cos(3\u03c0x - 1) + 3) - 3)/(2(-2cos(3\u03c0x - 1) + 3) + 3))^2 - 1((-3(-2cos(3\u03c0x - 1) + 3) - 3)/(2(-2cos(3\u03c0x - 1) + 3) + 3)) + 2 = -3cos(3\u03c0(-2|x + 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": "84", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = 1cos(2\u03c0((2(2|x| - 1) + 1)/(-1(2|x| - 1) + 2)) + 2) + 1 and g(x) = (-3(-1x + 2))/(3(-1x + 2) - 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) = -5(-1sin(3\u03c0(-5(-3|x - 2|)^2 + 2(-3|x - 2|) - 1) - 1) - 1) - 3 and g(x) = -3((-2(-1|x + 3| + 1) + 1)/(2(-1|x + 3| + 1) - 3))^2 + 1((-2(-1|x + 3| + 1) + 1)/(2(-1|x + 3| + 1) - 3)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "249", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (-1(-5(2cos(3\u03c0(-5x^2 - 1x + 3) + 1) + 2) + 2) - 3)/(-3(-5(2cos(3\u03c0(-5x^2 - 1x + 3) + 1) + 2) + 2) - 3) and g(x) = -2|(4(-2sin(2\u03c0x + 2) - 2) + 4) - 3| + 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "67", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 3cos(3\u03c0(-3(3|x - 2| - 1)^2 - 2)) + 1 = (2(1x) + 3)/(1(1x) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1410", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (1(3sin(3\u03c0(-2|x - 2| - 2) + 1) - 2) - 3)/(1(3sin(3\u03c0(-2|x - 2| - 2) + 1) - 2) + 3) and g(x) = (-2(-2e^(-2x) + 3) + 3)/(1(-2e^(-2x) + 3)) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "118", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (3(-1(-2(-2|x| - 3)^2 - 2(-2|x| - 3) + 2) + 1) + 1)/(1(-1(-2(-2|x| - 3)^2 - 2(-2|x| - 3) + 2) + 1) - 2) and g(x) = 2|((1(3cos(1\u03c0x + 3) - 1))/(1(3cos(1\u03c0x + 3) - 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": "8", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2(-3|(-2cos(1\u03c0((1x + 3)/(1x - 3)) - 3) + 1) - 1| - 1) - 3 = 2(-1|((-1x + 3)/(1x + 3)) - 2| + 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "122", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2(3cos(2\u03c0(-1|(-3x^2 + 3) + 3| + 2) - 2) + 3) + 1 = 1(-1((2x + 3)/(-2x - 3)) + 2)^2 - 2(-1((2x + 3)/(-2x - 3)) + 2) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1008", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = (1(-1(3|x| - 2)^2 - 5(3|x| - 2) + 5) + 3)/(2(-1(3|x| - 2)^2 - 5(3|x| - 2) + 5) - 3) and g(x) = 2|(-1x^2 + 4x + 3) - 3| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2sin(3\u03c0((-1(-2|(4x + 5) - 3| - 1) + 1)/(-1(-2|(4x + 5) - 3| - 1) - 3)) - 3) - 3 and g(x) = -1e^(2((2(5x^2 - 5x - 2) + 2)/(-1(5x^2 - 5x - 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": "61", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1sin(3\u03c0(1|(1((1x + 3)/(-3x + 4))^2 + 2((1x + 3)/(-3x + 4)) + 1) + 3| + 1) + 1) - 3 and g(x) = 2|(1((2x + 2)/(1x - 3)) + 5)| - 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = (-2(-1cos(3\u03c0(1|x + 2| + 2) + 2) + 2) + 3)/(-3(-1cos(3\u03c0(1|x + 2| + 2) + 2) + 2) - 2) and g(x) = -2sin(2\u03c0(2|x + 2| + 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) = -4(-2cos(1\u03c0(-1((-2x + 1)/(-3x + 4)) + 1) - 3) - 2)^2 - 4(-2cos(1\u03c0(-1((-2x + 1)/(-3x + 4)) + 1) - 3) - 2) - 1 and g(x) = (2(-3e^(-1(3|x|)) + 2) - 3)/(-2(-3e^(-1(3|x|)) + 2) + 3) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-2(-4(1x + 1)^2 - 1(1x + 1) + 3) - 1)/(-1(-4(1x + 1)^2 - 1(1x + 1) + 3) + 2) = (-2(-3sin(3\u03c0x - 1) - 1) + 3)/(1(-3sin(3\u03c0x - 1) - 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1|(1((1x - 2)/(3x - 2))^2 - 3((1x - 2)/(3x - 2)) - 4) - 2| - 2 = (-1(-3cos(1\u03c0x - 2)) + 1)/(-3(-3cos(1\u03c0x - 2)) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-2(-4(-2(-2sin(3\u03c0x - 2) + 1) - 1)^2 - 4(-2(-2sin(3\u03c0x - 2) + 1) - 1) - 4) - 1)/(2(-4(-2(-2sin(3\u03c0x - 2) + 1) - 1)^2 - 4(-2(-2sin(3\u03c0x - 2) + 1) - 1) - 4) - 1) = -4(-3sin(2\u03c0((-1x + 3)/(3x - 2)) + 3) - 3)^2 + 3(-3sin(2\u03c0((-1x + 3)/(3x - 2)) + 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": "36", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -2|((3(-2cos(1\u03c0(5x)) - 2) + 3)/(-1(-2cos(1\u03c0(5x)) - 2))) - 1| + 2 = -2(3cos(1\u03c0(1x^2 + 1x - 5) + 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": "204", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -3((2(-3|(-1cos(3\u03c0x - 1) - 1)| + 2) + 1)/(3(-3|(-1cos(3\u03c0x - 1) - 1)| + 2)))^2 - 1((2(-3|(-1cos(3\u03c0x - 1) - 1)| + 2) + 1)/(3(-3|(-1cos(3\u03c0x - 1) - 1)| + 2))) - 4 and g(x) = -5(-1e^(-2((2x - 3)/(-1x + 2))) - 1)^2 - 1(-1e^(-2((2x - 3)/(-1x + 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": "How many points of intersection exist between the functions f(x) = 2|(3(3cos(2\u03c0x + 3) - 1) + 3) - 2| - 3 and g(x) = -2cos(2\u03c0((3x - 2)/(-1x - 1)) + 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "86", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -3cos(1\u03c0((3(3|(1x^2 + 1x - 3) + 2|) - 1)/(-1(3|(1x^2 + 1x - 3) + 2|))) - 2) - 3 and g(x) = -1(1sin(3\u03c0((1x - 2)/(-1x + 2)) - 3) + 2)^2 - 1(1sin(3\u03c0((1x - 2)/(-1x + 2)) - 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": "29", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 4((2(-1(-1|x - 2| + 1)^2 + 2(-1|x - 2| + 1) + 2) - 2)/(3(-1(-1|x - 2| + 1)^2 + 2(-1|x - 2| + 1) + 2))) - 5 = 1sin(2\u03c0(-3((-2x - 2)/(2x - 3))^2 + 4((-2x - 2)/(2x - 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": "12", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3e^(-1(5(3|x| - 1)^2 + 3(3|x| - 1))) + 2 and g(x) = -3cos(2\u03c0((3x + 2)/(2x))) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(2\u03c0(5(1|x - 1| + 2) + 1) + 2) and g(x) = 2(-1|x - 3| + 2) - 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) = -2cos(2\u03c0(-3(-3|(-1x + 5) + 3| + 3)^2 + 5(-3|(-1x + 5) + 3| + 3) + 2) + 1) - 1 and g(x) = 1cos(2\u03c0(-4((-3x - 3)/(1x)) - 3) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1454", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1|(-1((-1x - 3)/(-2x + 3))^2 - 2((-1x - 3)/(-2x + 3)) - 4) + 3| and g(x) = -2sin(1\u03c0(-4x - 1) - 2) - 1 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 5((-1(3|(-4x + 2) - 3| - 1) + 1)/(2(3|(-4x + 2) - 3| - 1) - 1))^2 - 1((-1(3|(-4x + 2) - 3| - 1) + 1)/(2(3|(-4x + 2) - 3| - 1) - 1)) + 4 and g(x) = 5(-2|(2x - 5) - 1|)^2 + 1(-2|(2x - 5) - 1|) + 5 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1cos(1\u03c0(2(-2x - 2)^2 + 3(-2x - 2) - 2) + 1) - 3 and g(x) = 3|(-2sin(3\u03c0x - 1) + 3) - 3| - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "129", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3sin(2\u03c0((3(-2|(-1x - 5) - 3| + 3) - 3)/(1(-2|(-1x - 5) - 3| + 3) + 1)) + 2) and g(x) = -4(3cos(3\u03c0(2|x + 2| + 1) - 1) - 2) where -10 \u2264 x \u2264 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) = -1cos(3\u03c0(-2|(5((-1x - 2)/(1x + 3)) - 4) + 3| - 2) - 2) + 2 and g(x) = -1cos(1\u03c0(-1(-2|x + 1| + 1)^2 + 4(-2|x + 1| + 1) + 4) + 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "563", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -3sin(3\u03c0(-3|((-3x)/(-2x + 1)) - 1| + 1)) - 1 and g(x) = -2((1x - 2)/(-3x - 3)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -5(1cos(2\u03c0((-3(-2|x - 3| - 2) + 3)/(1(-2|x - 3| - 2) + 2)) + 3) - 1)^2 - 4(1cos(2\u03c0((-3(-2|x - 3| - 2) + 3)/(1(-2|x - 3| - 2) + 2)) + 3) - 1) + 5 = (-3(1sin(3\u03c0(-2|x - 1| + 3) + 2) - 3) - 2)/(-2(1sin(3\u03c0(-2|x - 1| + 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": "105", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1(-1|(2x^2 + 1x - 2) - 2| - 2) - 5 and g(x) = -4(-3cos(1\u03c0x + 1)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -1|(1((2x + 3)/(2x - 3)) - 2) + 3| + 3 and g(x) = -1cos(1\u03c0((-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": "24", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = 3(-2((1x + 3)/(-1x - 2))^2 - 1((1x + 3)/(-1x - 2)) - 5) + 1 and g(x) = (-2(-2sin(3\u03c0x + 3) + 1) + 3)/(3(-2sin(3\u03c0x + 3) + 1) + 2) where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "57", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 2cos(2\u03c0(-5(-3x + 2)^2 - 1(-3x + 2) - 4)) + 2 and g(x) = -1((3x - 3)/(1x + 1))^2 + 5((3x - 3)/(1x + 1)) - 3 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "919", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -2cos(1\u03c0(3(-5(-2|x - 2| + 2))^2 - 2(-5(-2|x - 2| + 2)) - 5) - 1) and g(x) = (3(-5(3|x - 2| + 3) - 5))/(-3(-5(3|x - 2| + 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": "1399", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation 1sin(1\u03c0(2((3(1x + 5))/(3(1x + 5) + 1))^2 + 2((3(1x + 5))/(3(1x + 5) + 1)) + 2) - 1) + 3 = -4((1(-3cos(3\u03c0x + 1)))/(1(-3cos(3\u03c0x + 1)) - 2)) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = 1|((-2(-2(5x^2 - 2x + 2) + 2) - 3)/(2(-2(5x^2 - 2x + 2) + 2) - 1)) + 3| - 3 and g(x) = 1sin(1\u03c0(1((-3x - 2)/(-2x)) - 1) + 2) - 1 in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33", "dataset": "MATH"}
+{"messages": [{"content": "How many points of intersection exist between the functions f(x) = -1cos(1\u03c0(2|((-2x - 2)/(1x + 3)) - 2| + 3) + 2) and g(x) = (-1(-5x + 4) + 3)/(-1(-5x + 4) - 2) in the range -10 \u2264 x \u2264 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "84", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation (-3(3cos(2\u03c0(-2(-3|x - 1|)^2 + 2(-3|x - 1|) + 1) + 2) - 1) + 3)/(2(3cos(2\u03c0(-2(-3|x - 1|)^2 + 2(-3|x - 1|) + 1) + 2) - 1) + 2) = -2cos(2\u03c0((-3(1x) - 2)/(-1(1x) - 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": "586", "dataset": "MATH"}
+{"messages": [{"content": "How many solutions does the equation -4(-1(2cos(2\u03c0x + 2) + 2) - 5)^2 - 5(-1(2cos(2\u03c0x + 2) + 2) - 5) + 3 = -2(-1x + 5)^2 + 2(-1x + 5) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11", "dataset": "MATH"}
+{"messages": [{"content": "Find the number of times the graphs of f(x) = -2sin(3\u03c0(-5((-3(-2x - 5) + 3)/(3(-2x - 5) + 2))^2 + 2((-3(-2x - 5) + 3)/(3(-2x - 5) + 2)) - 4) - 2) + 1 and g(x) = -1(1cos(2\u03c0((-1x)/(3x + 3))))^2 - 4(1cos(2\u03c0((-1x)/(3x + 3)))) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "116", "dataset": "MATH"}
+{"messages": [{"content": "Determine the number of intersection points between the functions f(x) = -1(-2cos(2\u03c0(1x - 5) - 1) + 1)^2 + 1(-2cos(2\u03c0(1x - 5) - 1) + 1) + 3 and g(x) = 2cos(3\u03c0((3x - 2)/(-1x + 3))) + 3 where -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "136", "dataset": "MATH"}