{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 4((3(-3|(-3x - 5) + 3| - 3) - 2)/(-2(-3|(-3x - 5) + 3| - 3) - 1))^2 - 2((3(-3|(-3x - 5) + 3| - 3) - 2)/(-2(-3|(-3x - 5) + 3| - 3) - 1)) - 2 and g(x) = 1(1|((2x + 1)/(-1x - 1)) - 1| - 2)^2 - 2(1|((2x + 1)/(-1x - 1)) - 1| - 2) + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 2cos(2\u03c0((-1(1|x + 1| + 3) + 1)/(-2(1|x + 1| + 3) - 3)) - 3) - 2 and g(x) = 2(-2|x - 3| + 2) + 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1cos(2\u03c0(3((3(2|x - 1| + 1) - 3)/(-3(2|x - 1| + 1) - 2))^2 - 4((3(2|x - 1| + 1) - 3)/(-3(2|x - 1| + 1) - 2)) + 1) + 3) - 1 and g(x) = -1sin(3\u03c0(2(1x - 1)^2 + 1(1x - 1) - 5) - 3) - 3 intersect 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2|(-1((3(2x + 1) - 2)/(2(2x + 1) + 3))^2 - 1) - 2| + 2 and g(x) = 1(-1|(1x^2 + 4x - 1) + 1| + 1) + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1e^(-2(-3((1x)/(-1x - 3))^2 - 1((1x)/(-1x - 3)))) - 2 and g(x) = (2(2e^(-1x) - 3) - 2)/(3(2e^(-1x) - 3) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 2|(-4(-2(2e^(2x) - 2)^2 - 2(2e^(2x) - 2) + 1) + 5) + 3| + 1 and g(x) = 2e^(1(5(-1x - 5)^2 - 2(-1x - 5) - 3)) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 5(-2(2sin(2\u03c0x - 1) - 2)^2 - 3(2sin(2\u03c0x - 1) - 2) - 3) + 4 and g(x) = (-2(-3|x + 2| + 3) + 3)/(2(-3|x + 2| + 3) + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 5(4(1|((1x - 2)/(3x - 2)) - 2| + 2) - 5)^2 - 4(4(1|((1x - 2)/(3x - 2)) - 2| + 2) - 5) and g(x) = 1(2|(4x^2 + 4) - 2| - 1) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3|(2(-4x^2 + 5x - 1) + 2) - 3| - 2 and g(x) = (3(-1sin(2\u03c0x + 3) - 3) - 1)/(2(-1sin(2\u03c0x + 3) - 3) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 3|(1(2cos(2\u03c0x + 1) - 1)^2 + 5(2cos(2\u03c0x + 1) - 1) - 3)| and g(x) = -2|(5x^2 + 2) - 3| + 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-1(-3(-2x - 3)^2 + 2(-2x - 3) + 1) + 3)/(1(-3(-2x - 3)^2 + 2(-2x - 3) + 1) + 2) and g(x) = 2|(3e^(-1x)) - 3| + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 3|(2(-5(-3e^(-1x)) - 3)^2 - 5(-5(-3e^(-1x)) - 3) - 3)| - 2 and g(x) = 2|(-3e^(2((2x)/(-2x + 3))) + 1) + 2| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1cos(2\u03c0((2(4(-3|x - 2| - 1) + 3) + 2)/(-2(4(-3|x - 2| - 1) + 3) + 1)) + 2) - 3 and g(x) = -1|(-5(4x + 1)^2 + 3(4x + 1) - 1)| + 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -2|((-1(-3sin(1\u03c0x + 1) - 2) - 2)/(-2(-3sin(1\u03c0x + 1) - 2) - 2)) - 1| - 2 and g(x) = -3((-2x - 2)/(-3x - 2))^2 + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2|(3e^(1(5(-3x^2 + 2x + 4) + 3))) - 1| - 3 and g(x) = -1e^(-1((-2(3x - 2) + 2)/(-2(3x - 2) + 1))) + 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2|(2sin(1\u03c0((1x + 3)/(2x + 1)) - 2) + 2) + 2| + 3 and g(x) = (3(3x - 1))/(3(3x - 1) - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (2(-3(3|(-1sin(2\u03c0x + 2) + 1) - 3| - 1)^2 - 2(3|(-1sin(2\u03c0x + 2) + 1) - 3| - 1) + 1) - 1)/(-2(-3(3|(-1sin(2\u03c0x + 2) + 1) - 3| - 1)^2 - 2(3|(-1sin(2\u03c0x + 2) + 1) - 3| - 1) + 1) - 1) and g(x) = (-1(2(2|x + 1| - 1)^2 + 1(2|x + 1| - 1) - 2) + 3)/(-3(2(2|x + 1| - 1)^2 + 1(2|x + 1| - 1) - 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 3|(5(-2e^(-1((-1x + 1)/(-2x - 3))) - 2)^2 + 2(-2e^(-1((-1x + 1)/(-2x - 3))) - 2)) - 3| + 2 and g(x) = 5((3(-1x^2 - 2x + 4) - 2)/(2(-1x^2 - 2x + 4) + 2)) - 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (2(4(1(-1sin(1\u03c0x + 3) - 1) - 5)^2 + 1(1(-1sin(1\u03c0x + 3) - 1) - 5) - 3) - 1)/(-1(4(1(-1sin(1\u03c0x + 3) - 1) - 5)^2 + 1(1(-1sin(1\u03c0x + 3) - 1) - 5) - 3) + 2) and g(x) = 1((-2(3|x + 3| - 1) - 3)/(-2(3|x + 3| - 1))) + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3|(2sin(1\u03c0((1x)/(2x + 3)) + 2) + 3) + 3| - 1 and g(x) = -1(-2e^(2x) - 1)^2 + 4(-2e^(2x) - 1) + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1cos(1\u03c0(5((2x - 1)/(-2x - 3)) - 3) + 3) - 1 and g(x) = -3e^(-1(1x^2 - 4x + 2)) + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -5(-2|(-2((3x + 3)/(1x - 2)) + 1) - 1| - 3)^2 - 4(-2|(-2((3x + 3)/(1x - 2)) + 1) - 1| - 3) - 1 and g(x) = -2|(-2((2x - 1)/(1x)) + 3) + 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2cos(2\u03c0(-1((-3x - 1)/(2x - 1)) + 1) + 2) - 1 and g(x) = -1|(-1x - 5)| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-1(-1(-1x - 3)^2 - 2(-1x - 3) + 1) + 2)/(1(-1(-1x - 3)^2 - 2(-1x - 3) + 1) + 3) and g(x) = (1(-2x^2 + 4x - 4) + 1)/(3(-2x^2 + 4x - 4) - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-2(2(-2|(-3x - 3) + 3| + 3)^2 - 2(-2|(-3x - 3) + 3| + 3) + 4) + 3)/(-2(2(-2|(-3x - 3) + 3| + 3)^2 - 2(-2|(-3x - 3) + 3| + 3) + 4) - 1) and g(x) = -2|(-3((-3x - 3)/(3x + 2)) - 3) - 3| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-1(-3|(-4(2x^2 + 3x + 1) + 2) - 3| + 3) - 3)/(1(-3|(-4(2x^2 + 3x + 1) + 2) - 3| + 3) - 2) and g(x) = -2(-1((1x - 1)/(1x - 2))^2 - 2) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -1(3((3(3|x + 3| - 1) + 1)/(2(3|x + 3| - 1) + 2)) + 4)^2 - 5(3((3(3|x + 3| - 1) + 1)/(2(3|x + 3| - 1) + 2)) + 4) + 4 and g(x) = 4(-1e^(2(2x^2 + 2x + 2))) + 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 2((1(-3(-2|x - 1| - 2)^2 + 4) - 3)/(-1(-3(-2|x - 1| - 2)^2 + 4))) + 3 and g(x) = 3e^(-2((-2(-1x^2 + 5x - 4) + 1)/(-3(-1x^2 + 5x - 4) + 1))) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -2(4(-1|((3x - 3)/(-2x + 3)) - 3| - 2) + 3)^2 - 1(4(-1|((3x - 3)/(-2x + 3)) - 3| - 2) + 3) + 5 and g(x) = -4(3|(1x + 4) + 2| + 3)^2 + 2(3|(1x + 4) + 2| + 3) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2(1|((1x + 2)/(-3x - 3)) - 3| + 3) + 2 and g(x) = -1|(2x^2 + 2x + 1) - 3| + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3|((-3(1(-2e^(-2x)) + 5) + 1)/(3(1(-2e^(-2x)) + 5))) - 1| and g(x) = 5((-2(-1e^(-1x) - 1) + 3)/(-1(-1e^(-1x) - 1) - 3)) + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1|(-4(4((1x - 2)/(1x + 3))^2 - 5) + 5)| + 1 and g(x) = 1sin(3\u03c0(1(-1|x + 1| - 2) + 5) + 2) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -3e^(-1((1(2(-2x^2 + 5x + 2) + 1) - 3)/(3(2(-2x^2 + 5x + 2) + 1) + 2))) + 1 and g(x) = -2(1|((3x + 2)/(-2x + 3)) + 1| - 1)^2 - 2(1|((3x + 2)/(-2x + 3)) + 1| - 1) + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1(-1((3(2|x| - 1) - 1)/(1(2|x| - 1))) + 2)^2 + 2(-1((3(2|x| - 1) - 1)/(1(2|x| - 1))) + 2) - 1 and g(x) = -3e^(2((2(5x + 2))/(3(5x + 2) + 2))) + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1|(5(1((2x + 3)/(1x))^2 + 3((2x + 3)/(1x)) - 2) + 5) - 1| + 2 and g(x) = -4(1(1|x + 1| - 1)^2 + 3(1|x + 1| - 1) - 4) + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 5(1|((-2(3x + 5) - 2)/(1(3x + 5) + 3)) - 3|)^2 - 1(1|((-2(3x + 5) - 2)/(1(3x + 5) + 3)) - 3|) + 5 and g(x) = 5(2|(-3x - 5) - 3| + 2)^2 + 5(2|(-3x - 5) - 3| + 2) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -3(1|((-1x - 1)/(-3x - 2)) + 1| + 2) - 3 and g(x) = 4((2x - 3)/(-3x + 1)) - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1(5(1sin(2\u03c0((2x - 3)/(-2x + 3)) - 3) + 3)^2 + 4(1sin(2\u03c0((2x - 3)/(-2x + 3)) - 3) + 3) - 2) - 1 and g(x) = 5(2|((-3x + 3)/(-3x - 3)) + 2| + 2)^2 - 5(2|((-3x + 3)/(-3x - 3)) + 2| + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 4(-5(2|((-2x - 3)/(-2x + 1)) - 2| + 1) - 2)^2 - 5(-5(2|((-2x - 3)/(-2x + 1)) - 2| + 1) - 2) - 5 and g(x) = 3(-1|(-2x + 5) - 3| - 2)^2 - 3(-1|(-2x + 5) - 3| - 2) + 5 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-9", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 4((-3(3(1|x| + 2) - 3) + 2)/(-2(3(1|x| + 2) - 3) - 2))^2 + 4((-3(3(1|x| + 2) - 3) + 2)/(-2(3(1|x| + 2) - 3) - 2)) and g(x) = 2|(-5(4x + 1)^2 + 1(4x + 1) + 4) - 2| + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2|((-1(2x^2 - 5x + 2) - 1)/(-1(2x^2 - 5x + 2) - 2)) - 1| - 1 and g(x) = -1|((2x - 1)/(1x)) - 2| - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 2((-1(1(3cos(3\u03c0x + 3) + 3)^2 - 2(3cos(3\u03c0x + 3) + 3) + 3) - 2)/(-2(1(3cos(3\u03c0x + 3) + 3)^2 - 2(3cos(3\u03c0x + 3) + 3) + 3) + 3)) + 2 and g(x) = (-3(-5(-1x + 1)^2 - 1(-1x + 1) + 5) - 1)/(3(-5(-1x + 1)^2 - 1(-1x + 1) + 5) - 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 1|(-5((2(-3x^2 + 3x) + 3)/(3(-3x^2 + 3x) - 1)) + 5) + 1| - 3 and g(x) = 5((2(2x + 1))/(3(2x + 1) + 2))^2 + 1((2(2x + 1))/(3(2x + 1) + 2)) + 5 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1|(-4(5x^2 + 1x - 3) + 2) - 3| + 2 and g(x) = (-2(-1|x - 3| - 2) + 2)/(1(-1|x - 3| - 2) + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -2e^(1(3|(-1(2x^2 + 4x - 1) + 5) + 2| + 2)) + 3 and g(x) = -2|((-1(3x) - 2)/(1(3x) - 3))| + 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (2(-3(1e^(-1x) - 1)^2 - 3(1e^(-1x) - 1) + 4) + 1)/(1(-3(1e^(-1x) - 1)^2 - 3(1e^(-1x) - 1) + 4) - 2) and g(x) = 2e^(2(-1x^2 + 4x - 4)) - 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3|(-3((1x - 1)/(1x)) - 2) + 3| and g(x) = -5(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": "4", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -3e^(1(3(3|((-1x + 1)/(-1x)) + 3|)^2 + 5(3|((-1x + 1)/(-1x)) + 3|) - 1)) + 3 and g(x) = 1(-2(2|x - 3| - 3)^2 - 3(2|x - 3| - 3) + 2) - 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (-3(3(3e^(2x) - 2)^2 - 1(3e^(2x) - 2) - 4) + 3)/(2(3(3e^(2x) - 2)^2 - 1(3e^(2x) - 2) - 4) + 2) and g(x) = 2|(-4x^2 + 2x)| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2(-1cos(1\u03c0((1(3x^2 - 1x + 1) + 1)/(1(3x^2 - 1x + 1) + 2)) - 3) - 1) - 5 and g(x) = -2|(3e^(-2((-3x + 2)/(-2x + 3))) - 1)| + 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1|(-2(3sin(1\u03c0(-2x + 2) + 1) + 3)^2 - 2(3sin(1\u03c0(-2x + 2) + 1) + 3) - 5) + 2| + 2 and g(x) = 1(-3|(-4x)| - 1)^2 - 2(-3|(-4x)| - 1) + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1((-1(-3|(3x - 1) - 2| + 1) - 2)/(-1(-3|(3x - 1) - 2| + 1) + 2))^2 + 3((-1(-3|(3x - 1) - 2| + 1) - 2)/(-1(-3|(3x - 1) - 2| + 1) + 2)) - 3 and g(x) = -5((-3(1sin(2\u03c0x + 1) + 1) - 1)/(-1(1sin(2\u03c0x + 1) + 1))) + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1|(3e^(-1(-5x + 3)) - 1) - 2| - 2 and g(x) = (3(-3x^2 + 2x) + 1)/(-1(-3x^2 + 2x)) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (1(-1sin(3\u03c0(4(1|x - 1| + 1) + 1)) + 3) + 2)/(2(-1sin(3\u03c0(4(1|x - 1| + 1) + 1)) + 3) + 1) and g(x) = 3((-2(-1|x + 3| + 1) - 2)/(-1(-1|x + 3| + 1) - 3)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 2(2sin(1\u03c0((-2(-1|x - 2| - 2))/(2(-1|x - 2| - 2) + 2)) + 2) + 3) - 1 and g(x) = -1(4(-2|x + 1| - 1)^2 + 5(-2|x + 1| - 1) - 4) + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1|(2(-2x^2 + 5x + 1) + 3) - 3| + 1 and g(x) = (3(3|x + 2| - 3) + 2)/(1(3|x + 2| - 3) + 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 4(5((2x + 2)/(-3x - 2))^2 + 3((2x + 2)/(-3x - 2)) - 3) + 2 and g(x) = 2|((-2x + 3)/(2x)) - 1| meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1|(-1((-1(-5x^2 + 5x - 2) - 1)/(1(-5x^2 + 5x - 2))) + 3) + 2| - 1 and g(x) = -1(3|(-2e^(1x)) + 1| - 2) + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -5(1((-3(1sin(3\u03c0x + 3) + 1) - 2)/(2(1sin(3\u03c0x + 3) + 1) - 3)) - 3)^2 + 5(1((-3(1sin(3\u03c0x + 3) + 1) - 2)/(2(1sin(3\u03c0x + 3) + 1) - 3)) - 3) - 3 and g(x) = (2(-3(-3x^2 + 2x + 4) - 2) + 2)/(1(-3(-3x^2 + 2x + 4) - 2) - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -5(3|(1x + 1) + 3|)^2 + 3(3|(1x + 1) + 3|) + 1 and g(x) = -3sin(2\u03c0((1x - 1)/(-1x)) - 2) + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-2(4(2|x - 3|) - 4) - 2)/(-3(4(2|x - 3|) - 4)) and g(x) = (-2(-5x - 4) + 2)/(2(-5x - 4) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2|(-2((-2x + 1)/(2x + 3)) - 2) - 3| - 3 and g(x) = 2cos(2\u03c0(3x + 1)) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 4(-5(-1|((-1x + 2)/(-3x + 1)) + 2| + 3)^2 + 5(-1|((-1x + 2)/(-3x + 1)) + 2| + 3) - 5) and g(x) = -3(3|((-2x + 1)/(2x + 2))|) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 5(-2cos(1\u03c0(-2|x + 3| + 2) + 3) + 3) - 4 and g(x) = (1(2e^(-1x) - 1) + 3)/(-2(2e^(-1x) - 1) + 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1|(4(1((-1x - 2)/(-1x - 2)))^2 + 3(1((-1x - 2)/(-1x - 2))) + 4) - 1| and g(x) = (2(-1(-1e^(1x) - 1) - 3) + 3)/(-1(-1(-1e^(1x) - 1) - 3) - 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (3(1|(-5x^2 + 4x - 2) + 1| - 3) - 1)/(2(1|(-5x^2 + 4x - 2) + 1| - 3)) and g(x) = 1sin(1\u03c0(-1x - 2) - 2) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -4(-1cos(3\u03c0(3|(5x + 1) + 1|) + 3) + 2)^2 + 2(-1cos(3\u03c0(3|(5x + 1) + 1|) + 3) + 2) - 2 and g(x) = (1(-3|(4x) + 1| - 1) + 1)/(-1(-3|(4x) + 1| - 1) - 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2(-1((-1x)/(-2x + 3)) + 4)^2 - 5 and g(x) = -1sin(1\u03c0((-3x + 2)/(3x - 2)) - 1) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(1sin(1\u03c0(3|x + 3| - 2) + 2) - 3) - 2)/(3(1sin(1\u03c0(3|x + 3| - 2) + 2) - 3) - 2) and g(x) = -1|(3x - 2) + 1| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1(-3|(-2((1x + 1)/(-2x + 3))^2 + 3((1x + 1)/(-2x + 3)) + 3) - 3| + 3) + 5 and g(x) = -1(-1(-1e^(-1x) - 2)^2 - 1(-1e^(-1x) - 2) - 1) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1e^(-1((1(2x^2 - 2x - 1))/(3(2x^2 - 2x - 1) - 2))) - 3 and g(x) = -2(3e^(-2x) - 2) + 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -4(2|((-2(4x^2 - 4x + 4) + 1)/(2(4x^2 - 4x + 4) + 1)) + 3| - 2) + 2 and g(x) = -4(5(-1e^(-2x) + 2) + 5)^2 + 4(5(-1e^(-2x) + 2) + 5) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 5(-2|((2x + 3)/(1x + 2)) - 2| - 3)^2 - 1(-2|((2x + 3)/(1x + 2)) - 2| - 3) + 3 and g(x) = (2(3x^2 + 3x - 3) - 3)/(-3(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": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -5(-3|((-1(4x^2 - 2x) + 3)/(-3(4x^2 - 2x)))| + 3) + 2 and g(x) = (-1(1(-2x - 4)^2 - 4(-2x - 4) + 3) - 1)/(1(1(-2x - 4)^2 - 4(-2x - 4) + 3) + 2) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 5((1(-3(-2|x - 3| + 1)^2 + 5(-2|x - 3| + 1) + 4))/(-2(-3(-2|x - 3| + 1)^2 + 5(-2|x - 3| + 1) + 4))) - 4 and g(x) = (-2(-3(3x + 4)^2 - 1(3x + 4) + 3) - 3)/(3(-3(3x + 4)^2 - 1(3x + 4) + 3) - 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-3(2|(2cos(1\u03c0(1x - 4) + 1) + 2) + 1| + 2) - 3)/(1(2|(2cos(1\u03c0(1x - 4) + 1) + 2) + 1| + 2) + 1) and g(x) = (1(-1|(-2e^(1x) + 3) + 1| + 1) + 2)/(-2(-1|(-2e^(1x) + 3) + 1| + 1) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 4(-2|(-3((3x - 2)/(3x - 2))^2 - 5((3x - 2)/(3x - 2)) - 2) - 1| + 2) and g(x) = -1(-1((2x)/(-3x - 2)) - 2)^2 - 1(-1((2x)/(-3x - 2)) - 2) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2((1(3|(4x^2 + 5x - 4) + 2| + 2) + 3)/(1(3|(4x^2 + 5x - 4) + 2| + 2) - 3)) and g(x) = (-1(-2|(5x + 2) - 3| + 2) - 1)/(2(-2|(5x + 2) - 3| + 2) + 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3((-1(2|(3x^2 + 1x - 2) + 1| - 3) + 3)/(3(2|(3x^2 + 1x - 2) + 1| - 3) - 1)) + 2 and g(x) = 3(1|(-2x^2 - 3x - 4) - 3| + 3) - 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1|(-4((1(-1x + 3) + 2)/(-1(-1x + 3)))^2 + 1((1(-1x + 3) + 2)/(-1(-1x + 3))) - 1) - 3| + 2 and g(x) = -1sin(3\u03c0(3|((2x - 2)/(1x + 1))| - 1)) - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3e^(-1((3(-3(-2|x - 3| - 1)^2 - 1) - 2)/(3(-3(-2|x - 3| - 1)^2 - 1) + 1))) - 1 and g(x) = 1(1|(-3e^(1x) + 2) - 1| - 1) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (1(3(-3|(1x - 3) + 1| - 1)^2 + 3(-3|(1x - 3) + 1| - 1) + 1) - 1)/(-2(3(-3|(1x - 3) + 1| - 1)^2 + 3(-3|(1x - 3) + 1| - 1) + 1) + 1) and g(x) = 1sin(1\u03c0((2(-2x^2 + 2x - 1) - 1)/(1(-2x^2 + 2x - 1) - 2))) - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 5(-4(2|(3sin(1\u03c0x - 1) + 3) + 3| - 2) + 3)^2 - 1(-4(2|(3sin(1\u03c0x - 1) + 3) + 3| - 2) + 3) and g(x) = 1e^(-1((3(2x^2 - 5x - 4) - 3)/(1(2x^2 - 5x - 4) + 1))) + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1e^(-2(-5((-3x - 2)/(1x + 2))^2 - 5((-3x - 2)/(1x + 2)) - 1)) and g(x) = 2e^(-2(1|x + 1|)) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3|(-5((2(1x^2 - 3x - 4) - 2)/(-3(1x^2 - 3x - 4) + 2)) + 3) + 3| and g(x) = -2sin(3\u03c0(1|(5x - 4) + 1| + 3) + 2) + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 5(1sin(2\u03c0((-3(-1|x - 3| - 3))/(2(-1|x - 3| - 3) - 3)) - 1) - 1) + 2 and g(x) = -2(2e^(1((-2x)/(-2x - 2)))) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 5(3((1(-3|x| - 3) + 3)/(-2(-3|x| - 3) - 2)) - 1)^2 + 1(3((1(-3|x| - 3) + 3)/(-2(-3|x| - 3) - 2)) - 1) + 3 and g(x) = 1|((3(4x - 5) + 2)/(3(4x - 5) - 2)) - 1| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -5((3(3|(-2x - 5)| + 3) + 2)/(3(3|(-2x - 5)| + 3) - 2))^2 + 2((3(3|(-2x - 5)| + 3) + 2)/(3(3|(-2x - 5)| + 3) - 2)) - 1 and g(x) = 5(1sin(2\u03c0(-2|x - 3| - 2)) - 2) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3|(3(1e^(-1(4x)) + 3)^2 - 1(1e^(-1(4x)) + 3) + 1) + 1| - 2 and g(x) = (-3(4(2sin(3\u03c0x - 3) + 1) + 4) + 3)/(-2(4(2sin(3\u03c0x - 3) + 1) + 4) + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1(-3((-3x + 1)/(-1x))^2 - 1((-3x + 1)/(-1x))) - 4 and g(x) = 2cos(3\u03c0((3x + 1)/(3x + 3)) - 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1cos(2\u03c0(1((-2(-4x^2 + 1x - 1) - 1)/(1(-4x^2 + 1x - 1))) - 2) - 2) - 2 and g(x) = -3((3(-3|x - 1| + 1) + 3)/(1(-3|x - 1| + 1) + 3)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2(-1((-2(-1cos(1\u03c0x + 3) - 3) + 3)/(2(-1cos(1\u03c0x + 3) - 3) - 3))^2 - 3((-2(-1cos(1\u03c0x + 3) - 3) + 3)/(2(-1cos(1\u03c0x + 3) - 3) - 3)) - 2) + 5 and g(x) = 2|((-2(4x + 1) + 1)/(-2(4x + 1) - 2)) - 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (3(2|(-2x^2 + 5x) + 1|) - 1)/(1(2|(-2x^2 + 5x) + 1|)) and g(x) = 1|((2x - 3)/(-2x + 1))| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1((-1(1x) + 3)/(-2(1x) + 1))^2 - 2((-1(1x) + 3)/(-2(1x) + 1)) + 4 and g(x) = 2((1x - 2)/(1x))^2 + 4((1x - 2)/(1x)) + 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (-2(2cos(1\u03c0(2x - 1) + 1)) + 1)/(-2(2cos(1\u03c0(2x - 1) + 1)) + 1) and g(x) = (1(2x^2 - 4x + 1))/(-2(2x^2 - 4x + 1) + 3) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 4((3(5x^2 - 3x - 1) - 2)/(1(5x^2 - 3x - 1) + 3)) - 3 and g(x) = -3|((2x - 1)/(1x + 3)) - 1| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2|(4((-3x - 2)/(3x + 2))^2 - 5((-3x - 2)/(3x + 2))) - 2| + 1 and g(x) = 1(2|x + 2| - 1)^2 - 4(2|x + 2| - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-2(4(-1|(-2e^(-2x)) + 3|)^2 - 3(-1|(-2e^(-2x)) + 3|) - 5) - 3)/(-1(4(-1|(-2e^(-2x)) + 3|)^2 - 3(-1|(-2e^(-2x)) + 3|) - 5) - 2) and g(x) = (-1(2sin(3\u03c0(3x + 5) - 3) + 3) + 3)/(1(2sin(3\u03c0(3x + 5) - 3) + 3) - 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (2(-5(1x - 3)^2 + 1(1x - 3) + 2) - 1)/(1(-5(1x - 3)^2 + 1(1x - 3) + 2) - 3) and g(x) = (1(3e^(1x) - 3) - 3)/(3(3e^(1x) - 3) + 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1cos(2\u03c0(4((-3x - 3)/(-2x - 1)) + 3) + 3) and g(x) = -3|(-2e^(-1x) + 3) - 3| + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3e^(1((-2(-1|(-2x)|) - 2)/(-2(-1|(-2x)|) - 2))) - 2 and g(x) = 4(-2(3|x + 1| + 1)^2 + 4(3|x + 1| + 1) - 3) + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(-1((1x + 2)/(2x + 2))^2 - 1((1x + 2)/(2x + 2)) - 3) + 3| and g(x) = -1e^(-1(4x - 2)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (3(1(-1(-2|x - 3| - 1)^2 + 4(-2|x - 3| - 1) + 3) - 1) + 1)/(-1(1(-1(-2|x - 3| - 1)^2 + 4(-2|x - 3| - 1) + 3) - 1) - 3) and g(x) = (-3(4(2|x + 3| - 2) + 2) + 1)/(-2(4(2|x + 3| - 2) + 2)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (1(1(3cos(2\u03c0x + 3) + 2)^2 + 2) - 3)/(-2(1(3cos(2\u03c0x + 3) + 2)^2 + 2) - 3) and g(x) = -1|(1x^2 - 1)| + 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1(-1|((-2x - 1)/(-3x + 4)) + 1| + 1) - 3 and g(x) = (-2(2x^2 - 3x - 5) + 1)/(-3(2x^2 - 3x - 5) + 2) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (1(-4(2|(-4x^2 + 5x + 3) - 1|) - 5) - 2)/(-2(-4(2|(-4x^2 + 5x + 3) - 1|) - 5) + 1) and g(x) = 2(2|((1x)/(3x + 1)) - 2| - 3) - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 3e^(-1(1|((-1x + 2)/(1x - 3)) + 2| - 3)) - 1 and g(x) = 5(2|x - 3| + 3) - 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-1(-3(3|x| - 2) + 2))/(1(-3(3|x| - 2) + 2) + 3) and g(x) = 3e^(-2((2x + 3)/(1x))) + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1|(1((-1(5x - 4) - 1)/(-1(5x - 4) + 2))^2 + 2) - 1| + 1 and g(x) = -5((-2(2e^(-1x) + 1) + 1)/(-1(2e^(-1x) + 1) + 3)) + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2e^(2((3(1x - 4) - 1)/(-2(1x - 4)))) + 2 and g(x) = 1(2|x - 3|)^2 - 4(2|x - 3|) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2e^(-1((1(-3|(1x^2 + 3x + 3) + 2| + 1) + 2)/(1(-3|(1x^2 + 3x + 3) + 2| + 1)))) - 1 and g(x) = -5(-4((-2x + 3)/(-3x))^2 - 5((-2x + 3)/(-3x)) + 2) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3|(-2((-1(3x^2 - 2x - 2))/(-3(3x^2 - 2x - 2) - 2)) - 5)| + 3 and g(x) = -2cos(2\u03c0(-1|(3x + 2) + 2| - 2) - 2) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 3(1|(-3((-3x)/(1x))^2 + 1((-3x)/(1x))) - 1| - 2) - 2 and g(x) = 1(1((2x + 2)/(3x - 2))^2 + 3((2x + 2)/(3x - 2)) - 5) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -5(-2|((-2(3x^2 - 1x + 2) - 1)/(1(3x^2 - 1x + 2))) - 2| - 1) - 1 and g(x) = 5(1(2|x - 2| - 1)^2 + 4(2|x - 2| - 1) + 5) - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (3(3(-3|(1x^2 - 2x)| - 3) - 3) - 2)/(1(3(-3|(1x^2 - 2x)| - 3) - 3) + 2) and g(x) = 1|((2(3x - 2) - 3)/(-1(3x - 2) + 2)) - 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -4(-4(2|((-2x + 2)/(1x)) - 1| + 1)^2 - 4(2|((-2x + 2)/(1x)) - 1| + 1) - 1) - 2 and g(x) = 5(2e^(-1(3x^2 - 1x - 5)) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-3(-3|(-3(1x^2) - 4) + 2| + 1) - 1)/(-2(-3|(-3(1x^2) - 4) + 2| + 1)) and g(x) = (3(-2|(-1e^(-1x) + 1) + 1| + 1) + 2)/(3(-2|(-1e^(-1x) + 1) + 1| + 1) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -4((2(3|x + 3| - 3) + 3)/(2(3|x + 3| - 3) + 1))^2 + 4((2(3|x + 3| - 3) + 3)/(2(3|x + 3| - 3) + 1)) - 2 and g(x) = -2e^(-1((3x + 2)/(1x + 2))) + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3sin(2\u03c0((1(-2|x - 2| - 2) - 1)/(-3(-2|x - 2| - 2) + 2)) - 3) and g(x) = 1((2x - 3)/(1x + 2))^2 - 5((2x - 3)/(1x + 2)) - 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-2(-1(-1|(4x^2 + 2x - 1) - 3|) - 5) + 3)/(2(-1(-1|(4x^2 + 2x - 1) - 3|) - 5) + 1) and g(x) = 4(2(1|x + 2| - 3)^2 + 1(1|x + 2| - 3) + 3) - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (2(1|(1(3x - 1)^2 + 4(3x - 1) - 4)| - 3) + 3)/(-3(1|(1(3x - 1)^2 + 4(3x - 1) - 4)| - 3) + 1) and g(x) = -1(-2((3x + 3)/(1x)) + 5)^2 - 5(-2((3x + 3)/(1x)) + 5) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1(4(-2|(-2cos(3\u03c0x + 1) - 1)| - 2)^2 - 2(-2|(-2cos(3\u03c0x + 1) - 1)| - 2) + 5) and g(x) = -1|(-5((-1x + 2)/(2x + 1)) + 5) + 3| + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -3|(4((-3(5x^2 + 4x - 2) - 1)/(2(5x^2 + 4x - 2)))) + 2| - 2 and g(x) = -1|(-3cos(1\u03c0(1x^2 + 4) - 1) + 2) + 1| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1(2(1sin(1\u03c0x + 2) + 3) - 2) - 3)/(2(2(1sin(1\u03c0x + 2) + 3) - 2) + 2) and g(x) = (1(3e^(-1x)) - 3)/(1(3e^(-1x)) + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (1(-3e^(1(1(-3x^2 + 3x - 2) + 4)) + 2) + 2)/(-3(-3e^(1(1(-3x^2 + 3x - 2) + 4)) + 2) - 2) and g(x) = -5(2|(-1e^(-1x) + 2) - 2| - 3)^2 - 2(2|(-1e^(-1x) + 2) - 2| - 3) - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1((1(-3e^(1(-1x^2 + 4x + 2)) + 2))/(-1(-3e^(1(-1x^2 + 4x + 2)) + 2) - 3)) + 1 and g(x) = -1|(1e^(2(-2x^2 + 4x - 1)) - 2) + 2| + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (3(2(5x^2 - 3x + 2)) - 3)/(3(2(5x^2 - 3x + 2))) and g(x) = -2|(-1e^(-1x)) + 2| + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1cos(3\u03c0(-1((1x)/(1x - 3)) - 2) - 3) + 2 and g(x) = -5((3x)/(-3x + 1)) - 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-14", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 5((-3(1x^2 + 3x))/(-3(1x^2 + 3x) + 1)) + 4 and g(x) = -1(-1cos(2\u03c0x + 1) + 1)^2 + 5(-1cos(2\u03c0x + 1) + 1) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (2(-2|(1cos(3\u03c0(3x + 4)) - 3)| - 2) - 3)/(-1(-2|(1cos(3\u03c0(3x + 4)) - 3)| - 2) + 3) and g(x) = -2e^(-2((3(3x) - 3)/(-3(3x) - 3))) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 4(-1|(2sin(1\u03c0(-1x^2 + 1x + 5) - 3)) - 2|) + 3 and g(x) = 2|(-2(1x^2 - 2x - 4) + 2) - 2| + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-2(-3|(-5x^2 + 5x - 2)| + 1) - 3)/(1(-3|(-5x^2 + 5x - 2)| + 1) - 3) and g(x) = -1|((-1x + 1)/(-2x - 2)) + 3| + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -4(2|((-3(-1cos(2\u03c0x + 1) - 2) + 1)/(3(-1cos(2\u03c0x + 1) - 2))) + 3| + 3) + 4 and g(x) = -3|(1((2x + 2)/(3x + 2))^2 - 4((2x + 2)/(3x + 2)) + 1)| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -3|((1(-1(-3e^(-1x) - 3) + 4) + 3)/(2(-1(-3e^(-1x) - 3) + 4) + 2))| + 2 and g(x) = -2((-1(4x + 1) + 1)/(-2(4x + 1)))^2 + 5((-1(4x + 1) + 1)/(-2(4x + 1))) - 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1(-1e^(2((1(2|x + 3| - 3) - 2)/(3(2|x + 3| - 3) - 2))) - 2)^2 + 4(-1e^(2((1(2|x + 3| - 3) - 2)/(3(2|x + 3| - 3) - 2))) - 2) + 5 and g(x) = -2(3e^(-2((-2x + 1)/(-3x - 2))) + 3) + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-1(1(-2|(-2x - 5) + 1| - 2)^2 + 3(-2|(-2x - 5) + 1| - 2) - 3) - 3)/(2(1(-2|(-2x - 5) + 1| - 2)^2 + 3(-2|(-2x - 5) + 1| - 2) - 3)) and g(x) = 2|((-1(-2cos(1\u03c0x - 2) - 3) - 2)/(3(-2cos(1\u03c0x - 2) - 3))) - 2| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1|(-4(-3(1cos(1\u03c0x)) - 3)^2 + 3(-3(1cos(1\u03c0x)) - 3) - 4) - 3| + 2 and g(x) = 1|(-2e^(-1(-5x - 4)) - 1) + 2| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (2(3|(5(1x + 1)^2 - 3(1x + 1) + 4) + 2| + 1) - 3)/(1(3|(5(1x + 1)^2 - 3(1x + 1) + 4) + 2| + 1) + 2) and g(x) = -1((-1(4x - 1))/(1(4x - 1) - 3))^2 + 4((-1(4x - 1))/(1(4x - 1) - 3)) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2(3(-1|((-2x + 1)/(2x))| - 2) + 3)^2 + 2(3(-1|((-2x + 1)/(2x))| - 2) + 3) + 5 and g(x) = 2|(3((-3x)/(-1x - 3))^2 + 2((-3x)/(-1x - 3)) - 1) + 2| intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -4((2(-2|(1x^2 + 4x - 4)| + 1) - 3)/(-2(-2|(1x^2 + 4x - 4)| + 1) - 2)) - 1 and g(x) = 5((-1(-2|x + 2| + 1) - 3)/(3(-2|x + 2| + 1) + 2))^2 - 4((-1(-2|x + 2| + 1) - 3)/(3(-2|x + 2| + 1) + 2)) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(2sin(1\u03c0(2x + 1) + 3) + 3) + 1)/(-3(2sin(1\u03c0(2x + 1) + 3) + 3) - 1) and g(x) = -1(2|x - 1| - 2)^2 - 5(2|x - 1| - 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1|(4((1x + 3)/(-1x + 3))^2 - 4((1x + 3)/(-1x + 3)) - 2) - 3| - 2 and g(x) = -2((-3x + 3)/(2x - 1))^2 - 1((-3x + 3)/(2x - 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1(-1|((-3(-2x^2 - 3x + 5) + 2)/(2(-2x^2 - 3x + 5) - 3)) + 3| - 2) - 2 and g(x) = -4(-2(-2e^(-1x) + 3) + 3)^2 + 2(-2(-2e^(-1x) + 3) + 3) + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 2(-2|(-2sin(2\u03c0((3x + 3)/(-3x)))) + 2| + 2)^2 + 2(-2|(-2sin(2\u03c0((3x + 3)/(-3x)))) + 2| + 2) + 3 and g(x) = (-2(2|(-3x + 5) - 1| - 2) + 2)/(-3(2|(-3x + 5) - 1| - 2) + 4) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -4((-1(3|x + 2|))/(-1(3|x + 2|) - 3)) + 3 and g(x) = 5(3|x + 1| + 1)^2 - 5(3|x + 1| + 1) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1(2|(1(-3e^(2x) - 3)^2 - 3(-3e^(2x) - 3)) + 1| + 2) - 1 and g(x) = (-2(1(2x - 3)^2 + 3(2x - 3)) - 2)/(-2(1(2x - 3)^2 + 3(2x - 3)) + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (3(-1|(3x^2 + 5x + 2) + 2| - 3) - 3)/(-1(-1|(3x^2 + 5x + 2) + 2| - 3) - 1) and g(x) = -1(3|x - 2| - 3)^2 - 2(3|x - 2| - 3) + 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2((-2(5(3e^(1x) - 2) + 3) - 1)/(1(5(3e^(1x) - 2) + 3) + 2))^2 - 4((-2(5(3e^(1x) - 2) + 3) - 1)/(1(5(3e^(1x) - 2) + 3) + 2)) + 1 and g(x) = -4(5(-3sin(1\u03c0x - 2) + 3) + 3)^2 - 1(5(-3sin(1\u03c0x - 2) + 3) + 3) - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 1|(2e^(-2((-2x + 3)/(1x + 2))) - 3)| + 3 and g(x) = 2e^(-1(2x^2 - 4x - 1)) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3e^(-1(-1(-5((1x + 3)/(1x))^2 - 1((1x + 3)/(1x)) + 4) + 4)) and g(x) = 4(-3e^(-2(3|x + 3| + 2)) + 1) - 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 3|((-1(3x - 1) - 1)/(1(3x - 1))) + 1| - 1 and g(x) = (-2(1cos(2\u03c0x + 1) + 2) - 1)/(-1(1cos(2\u03c0x + 1) + 2)) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(-3|(1x - 1) + 2| + 1) + 2)/(-2(-3|(1x - 1) + 2| + 1) + 1) and g(x) = -2|((3x)/(1x))| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (3(-1(-1x^2 + 2x - 1) - 4) - 1)/(2(-1(-1x^2 + 2x - 1) - 4) - 1) and g(x) = (-2(3e^(-1x) + 2) + 1)/(2(3e^(-1x) + 2) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1(-4((-1(2|x + 2|) + 1)/(1(2|x + 2|))) + 4)^2 - 3(-4((-1(2|x + 2|) + 1)/(1(2|x + 2|))) + 4) + 4 and g(x) = 1|((2(-4x - 4) + 3)/(1(-4x - 4) - 2)) - 1| - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-2(3|(3sin(3\u03c0(3x + 5) + 1) - 2) + 2| + 3))/(-3(3|(3sin(3\u03c0(3x + 5) + 1) - 2) + 2| + 3) - 3) and g(x) = 3(-2e^(2((3x)/(2x - 1))))^2 + 1(-2e^(2((3x)/(2x - 1)))) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2(-2e^(-1(5(3|x + 3| - 2)^2 + 2(3|x + 3| - 2) + 4)) - 2) - 3 and g(x) = -1|(-1(5x + 4)^2 - 2(5x + 4) - 5) + 2| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 2(-1|(-4((2x - 1)/(-2x - 3))^2 + 1((2x - 1)/(-2x - 3)) - 1) - 2| - 3) + 1 and g(x) = -2(3((3x + 2)/(-3x + 4)) + 2)^2 - 2(3((3x + 2)/(-3x + 4)) + 2) - 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1|((-1(-1(-1x^2 + 1x + 5) - 2))/(-3(-1(-1x^2 + 1x + 5) - 2) - 3)) - 3| + 3 and g(x) = 5(-1|(-1x^2 - 2x - 5) - 1| - 1) + 4 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -3|(-1cos(2\u03c0((-3x - 2)/(1x)) + 2) - 1) - 3| + 2 and g(x) = (1(-1x + 2) + 2)/(3(-1x + 2) - 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2(2e^(1(-4(-1|x| + 1)^2 - 4(-1|x| + 1) - 3)) - 1) and g(x) = 1|(-1(5x - 2)^2 + 1(5x - 2) + 1) + 1| - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1|(-1((-1(2x + 3) + 3)/(-2(2x + 3) - 2))^2 + 4((-1(2x + 3) + 3)/(-2(2x + 3) - 2)) + 4) - 1| + 3 and g(x) = 1|(-2e^(-1(-5x + 2)) + 2) + 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1(-3((-3(-2|x| - 1) + 1)/(-3(-2|x| - 1) + 1))^2 - 2((-3(-2|x| - 1) + 1)/(-3(-2|x| - 1) + 1)) + 3) - 4 and g(x) = (3(-5(3|x + 1| - 3) + 5) - 2)/(-2(-5(3|x + 1| - 3) + 5)) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1|(2((-3(4x^2 + 3x) - 1)/(2(4x^2 + 3x))) - 1) + 1| - 3 and g(x) = 3|(-4(-3cos(1\u03c0x - 1))^2 - 2(-3cos(1\u03c0x - 1)) - 3) + 1| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1((3(-1|x + 3| + 2) - 3)/(3(-1|x + 3| + 2) - 2)) + 1 and g(x) = (-1(3e^(-2x)) + 2)/(1(3e^(-2x)) + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1|((-3(3x - 4) + 3)/(3(3x - 4) - 1)) + 3| - 1 and g(x) = 2|(3sin(3\u03c0x + 3) + 2) + 3| + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1(1(3|((-2x - 1)/(2x - 1)) - 3| + 2)^2 - 5(3|((-2x - 1)/(2x - 1)) - 3| + 2)) - 2 and g(x) = 5(-1|(3e^(-1x) - 3) - 3| - 2) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (2(5(3(-3|x + 3|)^2 - 5(-3|x + 3|) - 1) + 1) - 3)/(3(5(3(-3|x + 3|)^2 - 5(-3|x + 3|) - 1) + 1) + 3) and g(x) = (-1(-1sin(1\u03c0(-4x - 3) - 3) - 1) + 2)/(1(-1sin(1\u03c0(-4x - 3) - 3) - 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2|((-3(-2(4x - 2)^2 - 1(4x - 2) + 3))/(3(-2(4x - 2)^2 - 1(4x - 2) + 3) - 1)) + 1| - 2 and g(x) = -1|(-4(3x - 1)^2 + 4) + 1| + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3(2e^(-1(4((-3x)/(-1x + 2)) + 4)) + 1)^2 + 3(2e^(-1(4((-3x)/(-1x + 2)) + 4)) + 1) + 4 and g(x) = 3e^(1(3|((3x + 2)/(3x - 2)) - 2| - 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 2cos(2\u03c0(2|((-3(2x - 3) + 1)/(3(2x - 3) - 1)) - 1| + 2) + 3) - 2 and g(x) = -2e^(2((2(5x^2 + 4x) + 2)/(1(5x^2 + 4x) + 2))) + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -4(2((-3(3|x - 2| - 1) - 2)/(3(3|x - 2| - 1) - 2)))^2 + 5(2((-3(3|x - 2| - 1) - 2)/(3(3|x - 2| - 1) - 2))) + 5 and g(x) = -3|(2e^(-2(-2x - 4)) + 2) - 1| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1((1(-2|(3x + 1) - 1| - 2) + 1)/(1(-2|(3x + 1) - 1| - 2) + 2))^2 + 3 and g(x) = 3(3((-2x + 2)/(2x + 2)))^2 + 1(3((-2x + 2)/(2x + 2))) - 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (2(1|(1(-5x + 4)^2 - 1(-5x + 4) + 2) - 3| - 3) + 3)/(-3(1|(1(-5x + 4)^2 - 1(-5x + 4) + 2) - 3| - 3)) and g(x) = -2e^(-1((-3(3|x - 1| + 2) - 2)/(2(3|x - 1| + 2) - 1))) + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -2(-3sin(2\u03c0((-2(-1|x + 3| - 2) - 1)/(-3(-1|x + 3| - 2))) - 1))^2 + 5(-3sin(2\u03c0((-2(-1|x + 3| - 2) - 1)/(-3(-1|x + 3| - 2))) - 1)) + 4 and g(x) = -3(-1|(-1x^2 - 2x + 2) - 2| + 2) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -4((-3(1(-2|x - 1| - 3)^2 + 3) + 2)/(-1(1(-2|x - 1| - 3)^2 + 3) - 2)) - 5 and g(x) = -5(-1((1x + 3)/(-1x + 2)) - 1)^2 - 4(-1((1x + 3)/(-1x + 2)) - 1) + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2|(-1(-1e^(-2x) - 3)^2 - 2)| - 1 and g(x) = (-2(1|x + 3|))/(-1(1|x + 3|) + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 3((2(-5(3|x - 2| + 1) + 5) - 1)/(-1(-5(3|x - 2| + 1) + 5) + 2))^2 + 2((2(-5(3|x - 2| + 1) + 5) - 1)/(-1(-5(3|x - 2| + 1) + 5) + 2)) + 2 and g(x) = (2(-3(1|x + 1| + 1) - 5) + 1)/(1(-3(1|x + 1| + 1) - 5) + 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1|(-4(-2((2x + 2)/(-3x + 2))^2 + 5((2x + 2)/(-3x + 2)) - 5) + 3) - 3| and g(x) = (1(-1(2x^2 - 3x - 2) + 2) + 2)/(2(-1(2x^2 - 3x - 2) + 2) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -1(-1|(-1((-1x + 2)/(-2x - 1))^2 + 2) - 2| + 1) - 4 and g(x) = (-2(2(-3x - 5)^2 - 4(-3x - 5)) + 1)/(1(2(-3x - 5)^2 - 4(-3x - 5)) + 1) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-1(-1|(4(-4x + 1)^2 - 5(-4x + 1) - 4) + 2| - 3) + 1)/(-2(-1|(4(-4x + 1)^2 - 5(-4x + 1) - 4) + 2| - 3) + 3) and g(x) = -3e^(2(1(1x^2 + 4x - 3) - 1)) + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2e^(-1(-2((1(-1|x - 1|) - 1)/(3(-1|x - 1|) + 3))^2 + 2((1(-1|x - 1|) - 1)/(3(-1|x - 1|) + 3)))) + 2 and g(x) = (-3(2|(-2x) - 3| - 2))/(3(2|(-2x) - 3| - 2) + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (3(3|(-5(-2x^2 - 1x - 4) - 1) - 3| + 3) - 3)/(-2(3|(-5(-2x^2 - 1x - 4) - 1) - 3| + 3)) and g(x) = -4((1(5x - 4))/(1(5x - 4) + 1))^2 + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1(1((-2(2|x + 2| - 2) - 3)/(-1(2|x + 2| - 2) + 2))^2 - 5((-2(2|x + 2| - 2) - 3)/(-1(2|x + 2| - 2) + 2)) - 5) + 3 and g(x) = -2(2e^(-1(-5x^2 - 5x + 2)) - 3) + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -3|(5((2(3x + 2) + 2)/(1(3x + 2) + 3))^2 + 1((2(3x + 2) + 2)/(1(3x + 2) + 3)) + 5) - 1| + 2 and g(x) = (-2(-2|(-3x - 4) - 2| + 3) + 1)/(-2(-2|(-3x - 4) - 2| + 3) - 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 2|(2(-2((1x - 2)/(3x + 3)) - 5)^2 + 4(-2((1x - 2)/(3x + 3)) - 5) - 2) - 2| + 2 and g(x) = 5(1(-1|x - 2| + 1) + 4)^2 + 3(1(-1|x - 2| + 1) + 4) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -4((-1(3(-1|x|)^2 + 2(-1|x|) + 3) + 3)/(-3(3(-1|x|)^2 + 2(-1|x|) + 3) + 4)) + 2 and g(x) = 2((-3(4x^2 - 5x) - 1)/(-1(4x^2 - 5x) + 2)) + 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (2(1(2e^(2x) - 3)^2 - 3(2e^(2x) - 3) + 5) - 2)/(1(1(2e^(2x) - 3)^2 - 3(2e^(2x) - 3) + 5)) and g(x) = (-3(5x^2 - 2x - 2) - 1)/(-1(5x^2 - 2x - 2) - 1) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (1(-3|(-3(1x^2 + 4x - 4) - 2) - 1| + 3) - 2)/(2(-3|(-3(1x^2 + 4x - 4) - 2) - 1| + 3) - 1) and g(x) = 3|(-3cos(3\u03c0(4x - 1) + 2) + 1)| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (3(-3e^(1(-3(3|x - 2| - 2) - 1)) - 2) + 3)/(-1(-3e^(1(-3(3|x - 2| - 2) - 1)) - 2) + 2) and g(x) = 3(-2((-2x - 3)/(3x - 2)) - 3)^2 - 4(-2((-2x - 3)/(3x - 2)) - 3) - 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1((1(1|(-1x + 1) + 3| + 2) + 3)/(2(1|(-1x + 1) + 3| + 2) - 1))^2 + 4((1(1|(-1x + 1) + 3| + 2) + 3)/(2(1|(-1x + 1) + 3| + 2) - 1)) - 5 and g(x) = (2(3(-1|x - 1| + 1) + 2) - 1)/(1(3(-1|x - 1| + 1) + 2) - 3) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3|((2(4(-2x^2 - 2x + 2) - 1) + 3)/(-1(4(-2x^2 - 2x + 2) - 1) - 1)) - 2| and g(x) = 2cos(3\u03c0(2|((-3x - 3)/(-3x + 4)) - 3| - 3) - 3) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1(1(1e^(-1x) - 3)) + 3)/(-2(1(1e^(-1x) - 3)) - 3) and g(x) = -2((3x - 1)/(1x + 2))^2 + 4((3x - 1)/(1x + 2)) - 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-1(-1cos(2\u03c0(2|x - 1| + 3) - 2) + 2))/(1(-1cos(2\u03c0(2|x - 1| + 3) - 2) + 2) + 2) and g(x) = -2|((1x + 3)/(1x)) + 3| + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -5(-1e^(2(-3|x - 2| - 3)) + 1)^2 - 1(-1e^(2(-3|x - 2| - 3)) + 1) + 2 and g(x) = -4((2x + 1)/(-3x + 2))^2 - 2((2x + 1)/(-3x + 2)) + 4 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (3(3(2|x + 3| - 1) - 5))/(2(3(2|x + 3| - 1) - 5) - 3) and g(x) = (2(-3|x - 2| - 2) + 1)/(1(-3|x - 2| - 2) - 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 3|(1((2(-3x^2 + 2x - 4) + 2)/(3(-3x^2 + 2x - 4) + 2)) + 3) + 1| + 1 and g(x) = 3(4(-2|x + 1|)^2 + 3(-2|x + 1|) - 5) + 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(1|(2cos(3\u03c0x + 1) + 3) - 1| + 2) - 1)/(-1(1|(2cos(3\u03c0x + 1) + 3) - 1| + 2) - 3) and g(x) = 5(4x^2 + 1x) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (2(3(-3e^(-2(1|x + 1| + 1)) - 1)^2 + 3(-3e^(-2(1|x + 1| + 1)) - 1) + 5) + 3)/(1(3(-3e^(-2(1|x + 1| + 1)) - 1)^2 + 3(-3e^(-2(1|x + 1| + 1)) - 1) + 5)) and g(x) = 1|(-1((-1x - 2)/(-2x)) + 5) - 2| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1(3(-1e^(-2((1x)/(-2x - 2))) - 2)^2 + 1(-1e^(-2((1x)/(-2x - 2))) - 2) - 2) + 1 and g(x) = -5(-2|(-1x) + 3| - 3)^2 - 1(-2|(-1x) + 3| - 3) - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-2(-3e^(-2(2x^2 + 4x - 5))) - 1)/(-2(-3e^(-2(2x^2 + 4x - 5))) - 1) and g(x) = (1(2x + 3) - 3)/(3(2x + 3) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1|((-1(3x^2 + 5x - 3) + 2)/(1(3x^2 + 5x - 3))) + 2| - 1 and g(x) = 3e^(1(2x - 4)) - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-1(-2|(-2(-3x - 3)^2 + 2(-3x - 3))| + 3))/(-1(-2|(-2(-3x - 3)^2 + 2(-3x - 3))| + 3) - 3) and g(x) = -2e^(1(-2(4x + 3)^2 + 4(4x + 3) + 4)) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 2(4((-2x - 1)/(-2x + 3))^2 + 2((-2x - 1)/(-2x + 3)) - 5) + 2 and g(x) = (-2(2x^2 + 4x - 2) + 2)/(-2(2x^2 + 4x - 2)) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 2((-1(-4(1|x + 1| + 2)^2 - 5(1|x + 1| + 2)))/(-3(-4(1|x + 1| + 2)^2 - 5(1|x + 1| + 2)) + 2)) - 3 and g(x) = (-1(-3|(3e^(1x)) - 3| + 3) + 1)/(-2(-3|(3e^(1x)) - 3| + 3) - 2) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 1|((-1(4(1x^2 + 1x - 3) - 5) - 2)/(2(4(1x^2 + 1x - 3) - 5) + 3)) - 1| - 3 and g(x) = -1(-1|(-2e^(1x) + 1) + 3| + 2)^2 + 4(-1|(-2e^(1x) + 1) + 3| + 2) - 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -2(1|((3(-3x + 2) - 2)/(-2(-3x + 2) - 2)) + 1| - 2)^2 - 1(1|((3(-3x + 2) - 2)/(-2(-3x + 2) - 2)) + 1| - 2) - 4 and g(x) = (2(1|(-5x^2 + 4x - 1) + 2| - 2) - 2)/(2(1|(-5x^2 + 4x - 1) + 2| - 2) - 1) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 4(-5((3(2e^(1x) - 3) - 3)/(-1(2e^(1x) - 3) - 2)) - 5)^2 + 3(-5((3(2e^(1x) - 3) - 3)/(-1(2e^(1x) - 3) - 2)) - 5) + 1 and g(x) = (-2(-3(-3x - 2)^2 + 3(-3x - 2) + 2) + 2)/(-2(-3(-3x - 2)^2 + 3(-3x - 2) + 2) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-3(-2|(5x - 5) - 3| + 3) + 2)/(-3(-2|(5x - 5) - 3| + 3) - 2) and g(x) = -1sin(1\u03c0(-1x - 5) + 3) - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-1(-3(-3|x + 2|)^2 - 5(-3|x + 2|) - 3) - 2)/(2(-3(-3|x + 2|)^2 - 5(-3|x + 2|) - 3) + 3) and g(x) = 1(-2e^(-1x))^2 + 3(-2e^(-1x)) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 4(-1((3(-1|x + 3| + 2) - 2)/(-3(-1|x + 3| + 2) - 2)) - 4)^2 - 3(-1((3(-1|x + 3| + 2) - 2)/(-3(-1|x + 3| + 2) - 2)) - 4) - 3 and g(x) = -5((1(1|x - 2| - 2) - 1)/(-2(1|x - 2| - 2)))^2 + 5((1(1|x - 2| - 2) - 1)/(-2(1|x - 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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -1|(1cos(1\u03c0(-1(3x^2 - 5x + 5) + 4) - 2) + 3) - 2| - 3 and g(x) = 5(1|((1x + 2)/(-2x + 3)) - 1| - 3) - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2(-2((3(3|x + 2|) - 2)/(3(3|x + 2|) + 2))^2 + 3((3(3|x + 2|) - 2)/(3(3|x + 2|) + 2)) - 2) + 5 and g(x) = (-1(-3(-3e^(2x))^2) - 2)/(1(-3(-3e^(2x))^2) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (2(-1(1|x - 2| + 3)^2 + 5(1|x - 2| + 3) - 2) - 2)/(-2(-1(1|x - 2| + 3)^2 + 5(1|x - 2| + 3) - 2) - 2) and g(x) = -3e^(1(4x + 1)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 3|(1((-3(-1x - 3) - 1)/(3(-1x - 3) - 2))^2 - 1((-3(-1x - 3) - 1)/(3(-1x - 3) - 2)) + 5) - 1| - 3 and g(x) = (-3(5(-2x^2 + 4x + 1) - 3) - 2)/(3(5(-2x^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": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (1(2(-4(-2|x + 2| - 2) - 3)^2 - 3(-4(-2|x + 2| - 2) - 3) + 3) - 3)/(-3(2(-4(-2|x + 2| - 2) - 3)^2 - 3(-4(-2|x + 2| - 2) - 3) + 3)) and g(x) = -2(2((3x + 1)/(1x))^2 - 2((3x + 1)/(1x)) - 3) - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 3(-2|(1cos(2\u03c0((1x - 1)/(-3x - 1)) - 1) - 3) - 3|) - 3 and g(x) = (3(-1(2x^2 + 5x - 3)) + 3)/(2(-1(2x^2 + 5x - 3)) - 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 5(-4((3(-1|x + 1| + 2) + 1)/(3(-1|x + 1| + 2) - 2)) + 4)^2 - 4 and g(x) = 5((1(-1cos(1\u03c0x) + 3) + 1)/(-3(-1cos(1\u03c0x) + 3) + 1)) + 1 meet 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 integer value closest to the x-coordinate where f(x) = 2|((-2(-5(-5x - 1)^2 + 5(-5x - 1) + 4) - 1)/(-1(-5(-5x - 1)^2 + 5(-5x - 1) + 4))) - 3| - 1 and g(x) = -3|(-2e^(1(-4x)) - 1) + 1| + 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2cos(2\u03c0(-1(-1(3|x - 2| - 2)^2 - 1(3|x - 2| - 2) - 2) + 4) + 2) + 3 and g(x) = -1(2e^(-2(5x - 2)) - 1)^2 - 4(2e^(-2(5x - 2)) - 1) + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1(5(1|(2e^(2x) - 1) - 2|) + 3)^2 - 1(5(1|(2e^(2x) - 1) - 2|) + 3) - 3 and g(x) = (-2(-3|(-5x - 1) - 3| + 3) + 2)/(-1(-3|(-5x - 1) - 3| + 3) - 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -3cos(1\u03c0((-1(-1(-1|x + 3| + 1) + 3) - 3)/(3(-1(-1|x + 3| + 1) + 3) - 2)) - 2) + 1 and g(x) = 2(-1|((1x - 2)/(-2x - 3))| + 3) + 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2|(-3(2((-1x + 1)/(3x - 2))^2 + 5((-1x + 1)/(3x - 2)) - 1) - 3) + 2| and g(x) = (2(-1|(-3sin(3\u03c0x + 3) + 3) - 1| + 1) + 1)/(-1(-1|(-3sin(3\u03c0x + 3) + 3) - 1| + 1) + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -5(2cos(2\u03c0((2(-2|x - 3|) - 1)/(3(-2|x - 3|) + 2)) + 3) + 2)^2 - 1(2cos(2\u03c0((2(-2|x - 3|) - 1)/(3(-2|x - 3|) + 2)) + 3) + 2) - 2 and g(x) = (-2(3(-1e^(1x)) + 3) - 3)/(-2(3(-1e^(1x)) + 3) - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1|((-3(2cos(3\u03c0x - 2) - 3) + 3)/(-3(2cos(3\u03c0x - 2) - 3) - 3)) + 2| + 1 and g(x) = (2(2e^(-1x) - 3) - 3)/(-3(2e^(-1x) - 3) - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -2|(-3(3((-2x + 2)/(3x - 2))^2 + 5((-2x + 2)/(3x - 2)) - 4) - 4)| + 1 and g(x) = (-1(1e^(-2(-3|x + 2| + 3)) - 2) - 1)/(-3(1e^(-2(-3|x + 2| + 3)) - 2) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (1(1(-1x^2 + 1) + 5) - 3)/(-1(1(-1x^2 + 1) + 5) - 1) and g(x) = (1(-2cos(3\u03c0x - 3) - 2) + 3)/(1(-2cos(3\u03c0x - 3) - 2) + 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2(1(1|(-3e^(1x) + 2)|)^2 + 5(1|(-3e^(1x) + 2)|) - 4) + 2 and g(x) = 1|(-4(3x^2 - 1x + 3) + 1) - 3| + 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -4((-2(1(-1|x - 1|)^2 - 5(-1|x - 1|) + 5) + 1)/(-2(1(-1|x - 1|)^2 - 5(-1|x - 1|) + 5) + 1)) and g(x) = -3(2e^(2(-4x^2 - 1x + 5)) - 3) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (-1(-3|(-1e^(2x) + 3) - 3| - 3) - 3)/(-2(-3|(-1e^(2x) + 3) - 3| - 3)) and g(x) = (-1(1x^2 + 1x + 2) + 3)/(-1(1x^2 + 1x + 2) + 2) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 3e^(1((2(1(-1x + 5)^2 - 2(-1x + 5) + 4) - 2)/(3(1(-1x + 5)^2 - 2(-1x + 5) + 4) + 3))) - 1 and g(x) = -4(-1|((-2x - 3)/(1x + 3))| + 1) - 5 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (3(2|(-1(-1x^2 + 3x) - 2) - 1| + 3) + 1)/(-3(2|(-1(-1x^2 + 3x) - 2) - 1| + 3) + 2) and g(x) = -1(5(-3e^(-1x) + 3)^2 + 2(-3e^(-1x) + 3) - 1) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (1(-5(1|x|) + 3) + 3)/(3(-5(1|x|) + 3)) and g(x) = -4(1sin(3\u03c0x + 3) - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1(2cos(3\u03c0(-4(-2|x - 2| + 3)^2 + 1(-2|x - 2| + 3) - 4) + 3) - 2) - 2 and g(x) = -3(1((-2x + 2)/(-2x + 3)) + 5)^2 + 4(1((-2x + 2)/(-2x + 3)) + 5) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 5(-2|((-1x - 2)/(-2x)) - 1| + 1) + 1 and g(x) = -1|(-2x^2 - 5x + 4) - 1| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2(-2|((2(1cos(3\u03c0x - 1) - 3) + 3)/(3(1cos(3\u03c0x - 1) - 3) + 1)) - 3| - 2) + 4 and g(x) = -1|((1(-5x - 2) + 2)/(2(-5x - 2) - 3)) - 1| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1|(1e^(-1(5x - 1)) - 2)| - 2 and g(x) = -1cos(3\u03c0(3x^2 - 4x + 1) + 1) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1((-1(-2(2|x + 1| + 1)^2 + 1(2|x + 1| + 1) + 5) - 2)/(3(-2(2|x + 1| + 1)^2 + 1(2|x + 1| + 1) + 5) - 1)) + 1 and g(x) = 3|(1(1e^(-1x) + 1) + 1)| - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (1(1(-3|x| + 3) + 1) + 1)/(-1(1(-3|x| + 3) + 1) + 3) and g(x) = 4(1|x - 1| - 3)^2 + 2(1|x - 1| - 3) + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 4(-3(-3|((-1x - 1)/(3x + 3)) - 2| + 2) - 1)^2 - 3(-3(-3|((-1x - 1)/(3x + 3)) - 2| + 2) - 1) + 2 and g(x) = 3(4((2x - 2)/(-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": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(-2|(5x^2 + 2x) + 2| - 1))/(2(-2|(5x^2 + 2x) + 2| - 1) - 1) and g(x) = (-1(-2|x| + 3) - 3)/(-1(-2|x| + 3) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3|((2(-1x))/(3(-1x) + 2)) - 3| + 3 and g(x) = -3sin(1\u03c0(1x - 1) - 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -4((3(-1e^(1x) - 2) - 2)/(3(-1e^(1x) - 2) + 3)) + 5 and g(x) = (-1(2x) - 3)/(-2(2x) - 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (2(-2e^(-1(2|x - 3|)) + 1) - 1)/(-1(-2e^(-1(2|x - 3|)) + 1) - 3) and g(x) = 1|(5x - 2) - 1| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 1sin(2\u03c0(2|(-5((1x - 2)/(-3x))^2 - 1((1x - 2)/(-3x)) + 3) - 2| + 1) + 1) and g(x) = 4(1((-3x - 1)/(3x - 2)) - 4)^2 - 5 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -4(3((-2(-2|x| + 1) + 3)/(-1(-2|x| + 1) - 3)) + 2)^2 - 3(3((-2(-2|x| + 1) + 3)/(-1(-2|x| + 1) - 3)) + 2) - 2 and g(x) = (-2(3(5x - 3)^2 + 5(5x - 3) - 4) - 3)/(-1(3(5x - 3)^2 + 5(5x - 3) - 4) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -3|(-3((-2x - 1)/(3x + 1))^2 - 3((-2x - 1)/(3x + 1)) + 3) + 3| - 2 and g(x) = -1|(3x - 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 4(-1|((3(-1x^2 + 2x - 1) + 3)/(1(-1x^2 + 2x - 1))) - 1| + 3) - 1 and g(x) = -2cos(3\u03c0((-2(-3|x + 1| + 3) - 2)/(1(-3|x + 1| + 3) + 1)) + 2) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -2|(3((-1x - 2)/(-2x + 1))^2 + 4((-1x - 2)/(-2x + 1)) + 1) - 1| - 1 and g(x) = 3cos(2\u03c0(1|x| + 1) - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1((-3(-3|(5x + 4) - 2| - 2) + 1)/(-1(-3|(5x + 4) - 2| - 2) - 1))^2 - 5((-3(-3|(5x + 4) - 2| - 2) + 1)/(-1(-3|(5x + 4) - 2| - 2) - 1)) - 5 and g(x) = (2(-2sin(3\u03c0(4x - 2) - 1) - 2) + 2)/(-3(-2sin(3\u03c0(4x - 2) - 1) - 2) + 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 3((1(3|(3x - 1) - 3| + 1) + 3)/(-2(3|(3x - 1) - 3| + 1) + 3))^2 + 1 and g(x) = 5(-1|(1x) - 3|)^2 + 2(-1|(1x) - 3|) - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (1(3|(-5(4x^2 + 2x + 5) - 5) - 1| - 2) + 1)/(2(3|(-5(4x^2 + 2x + 5) - 5) - 1| - 2) + 1) and g(x) = (-2(-1(-3|x - 1| - 1) - 4))/(2(-1(-3|x - 1| - 1) - 4) + 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -3sin(2\u03c0(-5(1((1x - 1)/(1x + 2)))^2 + 3(1((1x - 1)/(1x + 2))) - 2) + 2) - 2 and g(x) = 2(-5((-1x + 1)/(-1x))^2 - 1((-1x + 1)/(-1x)) - 1) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -4(1((2(1e^(2x) - 3) - 2)/(-1(1e^(2x) - 3) + 2)) - 5)^2 - 5(1((2(1e^(2x) - 3) - 2)/(-1(1e^(2x) - 3) + 2)) - 5) + 2 and g(x) = -1sin(1\u03c0((-1(4x + 4))/(-1(4x + 4) - 2)) + 1) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2(3|((-2x + 1)/(-2x + 1)) - 1| + 3)^2 + 5(3|((-2x + 1)/(-2x + 1)) - 1| + 3) + 5 and g(x) = -2((-2x + 3)/(-2x - 1)) + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -3((-1(-5(-1e^(1x) + 3) + 3) - 1)/(-1(-5(-1e^(1x) + 3) + 3)))^2 + 5((-1(-5(-1e^(1x) + 3) + 3) - 1)/(-1(-5(-1e^(1x) + 3) + 3))) and g(x) = -2|(5((1x + 1)/(1x + 2))^2 - 4((1x + 1)/(1x + 2)))| + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 4(-2e^(1((-1x)/(-2x + 3))) + 1)^2 - 4(-2e^(1((-1x)/(-2x + 3))) + 1) - 3 and g(x) = -1cos(1\u03c0(1|x| + 1) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -2|((3(-2(3x + 5)^2 + 3) - 3)/(-2(-2(3x + 5)^2 + 3) + 1)) + 1| - 1 and g(x) = (-3(1(2|x| - 3) - 4))/(-2(1(2|x| - 3) - 4) - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3(2|(4((2x - 1)/(-3x - 1)) + 2)| - 2)^2 + 1(2|(4((2x - 1)/(-3x - 1)) + 2)| - 2) + 5 and g(x) = 1(-1((-2x)/(-1x + 3))^2 - 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -4((-3(3x^2 - 4x - 5) - 2)/(-2(3x^2 - 4x - 5) - 3)) + 4 and g(x) = 2|(3sin(1\u03c0x + 1) + 2) - 1| + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1|(-4((1x - 3)/(-1x - 1))^2 + 1((1x - 3)/(-1x - 1)) + 4)| + 3 and g(x) = (-3(2x - 1) + 2)/(1(2x - 1) - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -4(3|((3x + 1)/(-2x + 3)) - 3| + 3)^2 - 3(3|((3x + 1)/(-2x + 3)) - 3| + 3) - 1 and g(x) = 2((1x + 3)/(2x + 1)) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 5(-2((-3x - 3)/(1x))^2 + 4((-3x - 3)/(1x)) + 2) - 3 and g(x) = -3((2x - 3)/(1x))^2 + 2((2x - 3)/(1x)) + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (-2(2|(1(5x^2 + 1) - 2) - 3|) + 3)/(-3(2|(1(5x^2 + 1) - 2) - 3|)) and g(x) = -2sin(1\u03c0((3(2|x + 2| + 1) - 1)/(-2(2|x + 2| + 1) - 2)) + 3) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1(-2e^(-2((-3x - 1)/(-3x))) - 1)^2 + 5(-2e^(-2((-3x - 1)/(-3x))) - 1) - 3 and g(x) = 3(-2e^(2x) + 2)^2 - 1(-2e^(2x) + 2) - 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-1(-3|(2(5x - 3)^2 + 4(5x - 3) + 4) - 3|) + 1)/(-1(-3|(2(5x - 3)^2 + 4(5x - 3) + 4) - 3|) - 3) and g(x) = (2(-2|(3x - 4) + 1| - 2) + 1)/(1(-2|(3x - 4) + 1| - 2) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 1|(-2e^(-2(-4(-1x^2 - 4x - 4))) - 3) + 3| + 2 and g(x) = -1(-3((2x - 1)/(-1x - 3)) - 2)^2 + 1(-3((2x - 1)/(-1x - 3)) - 2) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3(-1cos(3\u03c0((3x + 3)/(-1x))) - 3) + 1 and g(x) = (-1(-2e^(1x) + 3) - 2)/(-3(-2e^(1x) + 3) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-3(-1cos(2\u03c0(3x^2 - 4x - 5) - 1) + 1))/(1(-1cos(2\u03c0(3x^2 - 4x - 5) - 1) + 1)) and g(x) = 3((-3x - 3)/(2x + 1)) - 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -3(1|(-4((3x + 1)/(-1x - 1)) + 4) + 1| + 1)^2 + 4(1|(-4((3x + 1)/(-1x - 1)) + 4) + 1| + 1) + 5 and g(x) = (1(1(1|x - 3| + 3) - 1) - 1)/(1(1(1|x - 3| + 3) - 1)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3|(1(2sin(2\u03c0(1x^2 + 3x + 3) + 1) + 3) + 4) + 3| - 1 and g(x) = -4(-1|(-2e^(-2x)) - 3| + 3) - 5 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -3(-4(1|(3e^(-1x) + 1) + 3| - 3)^2 + 3) + 5 and g(x) = 3(1|(5x^2 + 3x - 2) - 3| - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (2(1(-1|(-2x^2 - 5x - 3)| - 1) + 3))/(-3(1(-1|(-2x^2 - 5x - 3)| - 1) + 3) + 1) and g(x) = 1cos(3\u03c0(3|(-3x^2 + 4x + 4) - 2| - 3) + 2) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -3(2((3(1|x - 1| - 2) - 3)/(-3(1|x - 1| - 2) + 4)) + 1)^2 - 3(2((3(1|x - 1| - 2) - 3)/(-3(1|x - 1| - 2) + 4)) + 1) - 3 and g(x) = -2|(-4(-2e^(2x) + 2) + 4) + 2| + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -3|(-1(1((-2x + 2)/(3x + 1)))^2 - 4(1((-2x + 2)/(3x + 1))) + 4) + 2| - 1 and g(x) = 1sin(2\u03c0((3(-1|x - 1| + 2) + 1)/(-1(-1|x - 1| + 2) + 3)) + 3) - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -2|(1e^(1(1(2x^2 - 1x) + 3)) - 2) + 2| + 3 and g(x) = (1(-1(2e^(2x) - 3)^2) - 1)/(3(-1(2e^(2x) - 3)^2) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2(-2sin(1\u03c0((-3(-5x^2 - 2x - 2) + 1)/(3(-5x^2 - 2x - 2) + 1)) - 3) + 1) + 4 and g(x) = 1(-2(-1e^(2x))^2 - 4(-1e^(2x)) + 5) - 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 4((-1(-2|(-3x^2 - 1x - 2) + 1| + 3) - 3)/(1(-2|(-3x^2 - 1x - 2) + 1| + 3) - 2)) - 2 and g(x) = (-2(-2e^(-1(-2x + 5)) + 2) - 3)/(-1(-2e^(-1(-2x + 5)) + 2) - 3) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2|((2(2(-3x + 1)^2 + 3(-3x + 1) - 1) + 3)/(-1(2(-3x + 1)^2 + 3(-3x + 1) - 1) - 3)) - 3| + 3 and g(x) = -4((2(-3x + 2) - 2)/(-2(-3x + 2) - 3))^2 - 1((2(-3x + 2) - 2)/(-2(-3x + 2) - 3)) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(-1(-2|(4x^2 - 3) + 1|) - 5) - 3)/(-3(-1(-2|(4x^2 - 3) + 1|) - 5) - 1) and g(x) = 2sin(2\u03c0((-3(2x + 1) + 3)/(2(2x + 1) - 3)) - 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (2(1|(2x - 1) + 3|) - 1)/(-3(1|(2x - 1) + 3|) + 4) and g(x) = (2(3|x| - 1) + 3)/(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": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2|(4((-1x)/(1x + 3))) + 1| - 1 and g(x) = (-2(-2|x - 2| - 2) - 3)/(-1(-2|x - 2| - 2) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -3e^(-1(-1|((2x - 3)/(-2x + 3)) + 1| + 3)) - 1 and g(x) = -2(-2x^2 - 4x + 4) - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3cos(1\u03c0(-3(1(-1|x + 2| + 2)^2 - 2(-1|x + 2| + 2) + 5) - 3) - 3) - 1 and g(x) = 5((2(-1e^(-1x) + 2) + 2)/(1(-1e^(-1x) + 2))) - 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3(1e^(-2((-2x + 3)/(-1x - 2))) + 3)^2 - 5(1e^(-2((-2x + 3)/(-1x - 2))) + 3) + 4 and g(x) = (-1(-2x - 1) + 2)/(1(-2x - 1) - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the third x-coordinate (rounded to the nearest integer) where f(x) = 2(1|(1e^(-1x))| - 1) - 5 and g(x) = 2|(2sin(3\u03c0x + 2) + 3) - 3| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1(2((2(-3|x| - 3) + 3)/(1(-3|x| - 3))) + 3)^2 - 3(2((2(-3|x| - 3) + 3)/(1(-3|x| - 3))) + 3) + 1 and g(x) = -1|((2(2e^(1x) - 2))/(2(2e^(1x) - 2) + 2)) + 2| + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -3|((-1(1sin(2\u03c0x) + 1) - 2)/(1(1sin(2\u03c0x) + 1) + 1)) + 1| + 2 and g(x) = 1(-3x^2 + 5) - 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -3e^(2(1|(1(-2x^2 - 5x) - 5)| + 2)) + 2 and g(x) = -1((-1(-1|x + 1| + 3))/(1(-1|x + 1| + 3) - 2))^2 + 1((-1(-1|x + 1| + 3))/(1(-1|x + 1| + 3) - 2)) + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (2(-2(-2|(-1x + 2)|)^2 - 1(-2|(-1x + 2)|) + 1) - 2)/(1(-2(-2|(-1x + 2)|)^2 - 1(-2|(-1x + 2)|) + 1) + 1) and g(x) = (1(1(3x - 1)^2 - 2) + 3)/(2(1(3x - 1)^2 - 2)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2((-3(3|(3x) - 3| - 1) - 2)/(3(3|(3x) - 3| - 1) + 3))^2 + 5((-3(3|(3x) - 3| - 1) - 2)/(3(3|(3x) - 3| - 1) + 3)) + 2 and g(x) = 3((1(3x^2 - 5x + 2) - 1)/(2(3x^2 - 5x + 2) - 1)) + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1sin(1\u03c0(-1|(3x^2 - 1x - 3)|) + 1) + 2 and g(x) = -1(-3e^(-1x) + 1)^2 - 4(-3e^(-1x) + 1) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2cos(3\u03c0(1|((-3x - 2)/(3x - 2)) + 2| + 1) + 3) - 2 and g(x) = -3e^(-2(-1x^2 - 3x - 2)) + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2e^(-2(2|((1(-2x - 1) + 3)/(-3(-2x - 1) + 1)) - 3| + 2)) and g(x) = 1|(-2e^(1((-1x - 2)/(2x + 2))) - 1) + 3| - 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-2(-2|(1(1x^2 - 2x + 5)) - 1|) + 2)/(2(-2|(1(1x^2 - 2x + 5)) - 1|) + 2) and g(x) = -2|(5(-2e^(2x) + 1)^2 - 3(-2e^(2x) + 1) - 4) - 2| + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1|((1(1cos(1\u03c0(2x^2 + 1) + 1) - 1) - 3)/(-1(1cos(1\u03c0(2x^2 + 1) + 1) - 1) - 3)) + 1| - 3 and g(x) = -3|(-4((-1x)/(-2x - 1)) + 1) - 3| - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -3sin(1\u03c0((2(5x^2 - 4x + 2) - 3)/(2(5x^2 - 4x + 2) - 1)) - 1) + 1 and g(x) = 1|((-1x + 2)/(2x - 1)) + 3| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 2|(-5(-2((1x - 2)/(-2x + 3)) - 4)^2 + 5(-2((1x - 2)/(-2x + 3)) - 4)) - 1| and g(x) = (1(2sin(1\u03c0(1x + 3) - 3) + 3) + 1)/(3(2sin(1\u03c0(1x + 3) - 3) + 3) - 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1(-1(-1e^(2((2x + 3)/(3x - 2))) - 2)^2 - 2(-1e^(2((2x + 3)/(3x - 2))) - 2)) - 4 and g(x) = -4(1(-2|x - 2| + 3) + 4)^2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (-2(-3(3|x + 1| - 3) - 2) + 2)/(-1(-3(3|x + 1| - 3) - 2) - 2) and g(x) = 1cos(3\u03c0((1x)/(-3x - 2)) + 1) - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1(-1(3|((3x - 3)/(2x - 3))| + 3)^2 - 4(3|((3x - 3)/(2x - 3))| + 3) - 4) - 4 and g(x) = (3(3e^(-1(1x + 3)) - 2) - 3)/(2(3e^(-1(1x + 3)) - 2) + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1((-1(-1|x| + 1) - 2)/(-1(-1|x| + 1))) - 2 and g(x) = -1((-2x)/(2x - 1))^2 + 1((-2x)/(2x - 1)) - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 5(-2((-1(-1|x - 3| - 2) + 1)/(2(-1|x - 3| - 2) + 2)) + 1)^2 + 4(-2((-1(-1|x - 3| - 2) + 1)/(2(-1|x - 3| - 2) + 2)) + 1) - 5 and g(x) = 3(3|(3x - 5) - 3| + 2)^2 + 3(3|(3x - 5) - 3| + 2) - 5 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -5((-2(3e^(1x) + 1) + 2)/(-2(3e^(1x) + 1) + 1))^2 + 4((-2(3e^(1x) + 1) + 2)/(-2(3e^(1x) + 1) + 1)) + 2 and g(x) = (3(4x^2 - 1x))/(-2(4x^2 - 1x) + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2|((1(-3x^2 + 3) - 3)/(-3(-3x^2 + 3)))| - 2 and g(x) = (-2(-2x^2 - 5x - 1) - 3)/(-3(-2x^2 - 5x - 1) - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1((3(5(-2|x - 1| + 1) - 2) - 2)/(1(5(-2|x - 1| + 1) - 2) - 3))^2 + 5((3(5(-2|x - 1| + 1) - 2) - 2)/(1(5(-2|x - 1| + 1) - 2) - 3)) + 5 and g(x) = (-3(-3(3|x - 2| - 2) - 5) + 2)/(2(-3(3|x - 2| - 2) - 5) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-2(1(1(-2|x| + 2) - 3)^2 + 3(1(-2|x| + 2) - 3) + 1))/(3(1(1(-2|x| + 2) - 3)^2 + 3(1(-2|x| + 2) - 3) + 1)) and g(x) = (3(3e^(1(-1x + 4)) - 2) - 3)/(1(3e^(1(-1x + 4)) - 2) + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1(-3((-2x - 2)/(-1x - 3))^2 - 5((-2x - 2)/(-1x - 3)) - 2) + 5 and g(x) = 2|((3x - 1)/(-3x)) + 3| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 5(-1e^(1((1x + 2)/(1x))))^2 - 4(-1e^(1((1x + 2)/(1x)))) - 4 and g(x) = (2(1|x| + 2) + 3)/(3(1|x| + 2) - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -5(-3e^(1((-3(-2|x| - 1) + 3)/(2(-2|x| - 1) + 1)))) and g(x) = (1(-1|(-4x - 1) - 1| - 2) - 2)/(2(-1|(-4x - 1) - 1| - 2) + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (-2(1|(5(-3x^2 + 3x - 4) + 5) + 3|) + 1)/(-1(1|(5(-3x^2 + 3x - 4) + 5) + 3|) - 3) and g(x) = -2(-1((2x + 3)/(1x))^2 - 5((2x + 3)/(1x)) - 4) + 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1(-3|(2e^(1x) + 1) - 1| + 2)^2 + 4(-3|(2e^(1x) + 1) - 1| + 2) + 4 and g(x) = 1(-2|x + 1| - 3)^2 + 3(-2|x + 1| - 3) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (-1(1|(5x^2 + 4x - 4) - 3| + 2))/(1(1|(5x^2 + 4x - 4) - 3| + 2)) and g(x) = 1|(2x^2 + 2x + 4) - 2| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (1(2(1|(-1e^(1x) - 1) - 3|) + 2))/(-1(2(1|(-1e^(1x) - 1) - 3|) + 2) - 3) and g(x) = 1(4((-1x - 3)/(-2x + 1)) - 5)^2 - 5(4((-1x - 3)/(-2x + 1)) - 5) + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -1(3(-1sin(2\u03c0x - 1) + 3) - 3)^2 - 5 and g(x) = -1|((1x - 3)/(3x)) - 1| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-2(-2(-1|(5x + 2)|)^2 + 2(-1|(5x + 2)|) + 5) - 3)/(-1(-2(-1|(5x + 2)|)^2 + 2(-1|(5x + 2)|) + 5) + 3) and g(x) = -3(-1((2x - 1)/(-1x - 3))^2 - 4((2x - 1)/(-1x - 3)) + 1) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2sin(2\u03c0(5(1(1|x - 2| + 3) - 5)^2 + 3(1(1|x - 2| + 3) - 5) + 5) + 2) - 3 and g(x) = -5(1(3|x + 2| - 1)^2 + 5(3|x + 2| - 1) - 5) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1((-1(-3(-2|x - 1| + 1)) + 3)/(2(-3(-2|x - 1| + 1)) - 1))^2 + 1((-1(-3(-2|x - 1| + 1)) + 3)/(2(-3(-2|x - 1| + 1)) - 1)) + 2 and g(x) = 2|(-3(-3e^(1x) + 1) - 1) - 1| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 3(3e^(1(5(-1|x + 1| - 1)^2 + 1(-1|x + 1| - 1) - 4)) - 3) - 1 and g(x) = 1|(-1sin(3\u03c0(-3x^2 - 3x - 5) + 1) + 1) + 3| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1|((1(1x - 1) - 3)/(1(1x - 1) - 2)) + 2| - 2 and g(x) = -1(2cos(2\u03c0x - 2) + 2)^2 + 4(2cos(2\u03c0x - 2) + 2) + 4 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-3(-3|(-1x^2 - 4x - 2) - 2| - 2))/(-2(-3|(-1x^2 - 4x - 2) - 2| - 2) + 3) and g(x) = 2((1x - 1)/(-1x + 2)) + 5 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-1(5(-3|x| + 1) + 5) - 2)/(-3(5(-3|x| + 1) + 5) - 3) and g(x) = -3cos(3\u03c0(1x^2 + 4x + 3)) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -3(-1(-2sin(1\u03c0(2|x + 3| - 3) - 3))^2 - 4(-2sin(1\u03c0(2|x + 3| - 3) - 3)) - 4) + 5 and g(x) = -3(-1e^(1(1x^2 + 3x - 5)) + 2) + 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1|((1(4x^2 + 5x + 2) - 3)/(-1(4x^2 + 5x + 2) - 3)) - 2| + 2 and g(x) = (-1(-1x + 4) - 3)/(3(-1x + 4) + 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3(3(-1|((-3x + 1)/(-1x + 2)) - 3| + 1))^2 + 5(3(-1|((-3x + 1)/(-1x + 2)) - 3| + 1)) - 1 and g(x) = (-1(-1(2x - 4)^2 - 1(2x - 4) + 4) + 1)/(-2(-1(2x - 4)^2 - 1(2x - 4) + 4)) meet 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = -5((3(1cos(1\u03c0(-1|x - 3| - 2) + 1) - 2) + 1)/(-2(1cos(1\u03c0(-1|x - 3| - 2) + 1) - 2))) + 2 and g(x) = (-2(3|(-1x + 1) - 2| - 2) - 1)/(2(3|(-1x + 1) - 2| - 2) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (2(-1(2|x - 1|)^2 + 2(2|x - 1|) - 1) - 2)/(-1(-1(2|x - 1|)^2 + 2(2|x - 1|) - 1) + 2) and g(x) = (-2(-5x - 4) + 3)/(-1(-5x - 4) + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 5(-2((-1x)/(2x - 1))^2 - 3((-1x)/(2x - 1)) + 5) - 2 and g(x) = -2|(-1sin(1\u03c0x - 3) - 3) + 1| - 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 4(-1|((-1(1x^2 + 5x - 3) - 2)/(3(1x^2 + 5x - 3))) - 2| - 2) - 1 and g(x) = -4(-4((1x - 2)/(-1x - 3)) - 1)^2 - 1(-4((1x - 2)/(-1x - 3)) - 1) + 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1e^(1(1((1x + 2)/(3x + 2))^2 + 5((1x + 2)/(3x + 2)) + 3)) + 3 and g(x) = -4(2cos(1\u03c0x - 1) + 3) + 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 3(2|((1(1e^(2x) - 3))/(-3(1e^(2x) - 3) + 2)) - 1| - 3) + 2 and g(x) = 1(1((1x + 1)/(-1x + 3))^2 - 2((1x + 1)/(-1x + 3)) - 1) + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -3sin(1\u03c0((-2(5(2x^2 - 5x + 2) + 3) - 1)/(3(5(2x^2 - 5x + 2) + 3) + 1)) - 1) - 3 and g(x) = (3(1(3|x + 2| - 1)^2 + 3(3|x + 2| - 1) - 4) - 2)/(2(1(3|x + 2| - 1)^2 + 3(3|x + 2| - 1) - 4) - 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -5((2(-2(-3|x|)^2 - 5(-3|x|) + 2) - 3)/(1(-2(-3|x|)^2 - 5(-3|x|) + 2) - 3)) + 5 and g(x) = -2cos(1\u03c0(-3(-2|x - 2| + 1) + 3) + 3) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2|(-1(5(-3sin(1\u03c0x - 2) - 1)^2 + 4(-3sin(1\u03c0x - 2) - 1) + 2) + 1) + 2| + 3 and g(x) = 3|((-3(-4x - 1) + 3)/(-2(-4x - 1) + 3))| meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(-3|(-3e^(2x) - 3) + 3| - 3) - 3)/(-1(-3|(-3e^(2x) - 3) + 3| - 3) + 2) and g(x) = -2(2x + 2)^2 - 3(2x + 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-2(1(3x)^2 - 5(3x) + 1) - 3)/(-1(1(3x)^2 - 5(3x) + 1) + 3) and g(x) = 3e^(-2(2|x| - 2)) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -4(-2|((3(1x^2 - 3) + 1)/(2(1x^2 - 3) - 1)) + 2| - 1) + 2 and g(x) = -2(-1(-1|x + 3| + 2)^2 - 2(-1|x + 3| + 2) + 5) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (3(5(-4(-2|x + 2| - 3) - 3)^2 - 5(-4(-2|x + 2| - 3) - 3) - 4) - 3)/(3(5(-4(-2|x + 2| - 3) - 3)^2 - 5(-4(-2|x + 2| - 3) - 3) - 4) - 2) and g(x) = 1((-2(3x + 2))/(-3(3x + 2) + 1))^2 + 1((-2(3x + 2))/(-3(3x + 2) + 1)) - 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1|((3(4x^2 + 1x + 1) + 1)/(1(4x^2 + 1x + 1) + 3)) - 2| + 3 and g(x) = 2|(-1x + 3) + 2| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3(2|(-2e^(-2((1x - 2)/(-2x + 3))) + 2) - 3| - 2) - 4 and g(x) = -1|(-2cos(1\u03c0(1x + 4) + 2) - 3) + 2| meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1(-1(-3e^(-2(-3|x + 2|)) - 3) + 1)^2 - 3(-1(-3e^(-2(-3|x + 2|)) - 3) + 1) - 5 and g(x) = 1(-3(3sin(3\u03c0x - 1) + 1) + 1)^2 + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3(3|(-1sin(3\u03c0((3x + 1)/(-3x)) - 3) - 3) - 3| - 1)^2 + 2(3|(-1sin(3\u03c0((3x + 1)/(-3x)) - 3) - 3) - 3| - 1) - 1 and g(x) = -2(-1e^(-1(-2|x + 3| + 1)) + 1)^2 + 4(-1e^(-1(-2|x + 3| + 1)) + 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -1(4(2|((-2x + 1)/(2x)) + 2| + 1)^2 - 4(2|((-2x + 1)/(2x)) + 2| + 1) + 5) - 2 and g(x) = -1|(-2((3x + 3)/(3x)) + 5) + 3| - 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3(3(-1|(3sin(1\u03c0x - 3) + 2) + 1| - 1)^2 + 2(-1|(3sin(1\u03c0x - 3) + 2) + 1| - 1) + 2) - 5 and g(x) = (-1(-3|(3x^2 + 4x - 4) - 1| + 2) + 3)/(-3(-3|(3x^2 + 4x - 4) - 1| + 2) + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (2(-3|(-2sin(2\u03c0x + 1) - 2) - 1| + 3))/(2(-3|(-2sin(2\u03c0x + 1) - 2) - 1| + 3) + 3) and g(x) = -1e^(2(-2|x + 2| + 1)) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -5(1((3(2|x - 2| + 3) + 1)/(-1(2|x - 2| + 3) + 2))^2 + 5((3(2|x - 2| + 3) + 1)/(-1(2|x - 2| + 3) + 2)) - 1) - 2 and g(x) = -3e^(2((1(2|x + 3| + 2))/(-1(2|x + 3| + 2) - 1))) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3|(4(5((1x - 2)/(-1x - 1)) + 4)^2 - 2(5((1x - 2)/(-1x - 1)) + 4) + 4) + 1| + 3 and g(x) = 1((-3(-5x + 1) - 1)/(1(-5x + 1) + 2))^2 + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -3|((3(4(-5x^2 + 4x - 5) - 2) - 2)/(-2(4(-5x^2 + 4x - 5) - 2) - 1)) - 1| + 3 and g(x) = (-3(3(2x^2 + 1x - 5) + 1) + 1)/(-2(3(2x^2 + 1x - 5) + 1) + 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1((-2(2sin(3\u03c0x) - 3) + 3)/(-1(2sin(3\u03c0x) - 3)))^2 + 3((-2(2sin(3\u03c0x) - 3) + 3)/(-1(2sin(3\u03c0x) - 3))) - 5 and g(x) = (-1(2x + 1) - 2)/(-1(2x + 1) - 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3|(5(-2((-3x - 1)/(-2x + 3))^2 + 1((-3x - 1)/(-2x + 3)) - 2) - 5) - 2| + 1 and g(x) = (2(-1(-2|x - 2| + 2)^2 - 3(-2|x - 2| + 2) + 4) + 2)/(2(-1(-2|x - 2| + 2)^2 - 3(-2|x - 2| + 2) + 4)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2|(-5(-2x^2 + 2x + 5) + 1) - 3| + 3 and g(x) = -1cos(1\u03c0((2x - 3)/(-3x + 2)) + 1) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2((-1(-3|(-1e^(1x) - 2)| - 3) + 2)/(-2(-3|(-1e^(1x) - 2)| - 3))) - 4 and g(x) = 1(-3((1x + 1)/(-1x + 2)) - 4)^2 + 4(-3((1x + 1)/(-1x + 2)) - 4) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 5(1|((1x + 3)/(-1x)) + 2| + 1)^2 - 5(1|((1x + 3)/(-1x)) + 2| + 1) - 4 and g(x) = 1e^(1(-3|x| + 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 first intersection point (counting from left to right) between the functions f(x) = 1((-1(-1|(-4x^2 - 1x + 1) - 2| + 2) + 2)/(1(-1|(-4x^2 - 1x + 1) - 2| + 2) + 1)) and g(x) = -3|((2(2sin(2\u03c0x) + 3) - 3)/(2(2sin(2\u03c0x) + 3) + 3)) - 3| - 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -5(-3e^(-2(4(-2|x - 1| + 3)^2 - 5(-2|x - 1| + 3) + 4)) + 2) + 3 and g(x) = -1|(-2(-2x - 2)^2 + 3(-2x - 2)) + 2| + 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(1|(-4(-1sin(3\u03c0x) + 3) + 1) + 2| - 2) + 2)/(-2(1|(-4(-1sin(3\u03c0x) + 3) + 1) + 2| - 2) - 3) and g(x) = 1|(4((3x + 3)/(-1x - 3)) - 4) + 1| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1((-3(-2(-2|x - 1| - 1)^2 + 5(-2|x - 1| - 1) + 2) + 2)/(-1(-2(-2|x - 1| - 1)^2 + 5(-2|x - 1| - 1) + 2) + 2)) - 5 and g(x) = -1((-3(3|x + 1| - 3) - 1)/(2(3|x + 1| - 3) + 3)) + 4 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (1(3|(5x^2 - 4x - 1) - 2| - 1) - 1)/(1(3|(5x^2 - 4x - 1) - 2| - 1) + 2) and g(x) = 2|((-2x + 1)/(3x + 2)) - 1| - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -4((3(-3|(2e^(1x) + 2) + 1| - 2) - 2)/(2(-3|(2e^(1x) + 2) + 1| - 2) + 1)) - 2 and g(x) = (-3(4(-4x - 3)^2 - 5(-4x - 3) + 3) + 3)/(-1(4(-4x - 3)^2 - 5(-4x - 3) + 3) + 2) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -2((2(-3|(-2x^2 - 1x + 1) - 2| + 3) + 2)/(2(-3|(-2x^2 - 1x + 1) - 2| + 3) - 1)) - 1 and g(x) = -2(1e^(1((2x - 2)/(-3x + 4))) - 2) + 4 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 4(1|(1((-2x - 3)/(1x + 3)) + 5) + 1| + 3)^2 + 1(1|(1((-2x - 3)/(1x + 3)) + 5) + 1| + 3) - 5 and g(x) = 1(-2|((-1x - 2)/(3x + 1))| + 3)^2 + 4(-2|((-1x - 2)/(3x + 1))| + 3) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -4(-1|(1((2x + 2)/(1x))^2 + 4((2x + 2)/(1x)) + 1) - 2| + 3) + 3 and g(x) = -3|(-1((2x)/(2x + 2))^2 + 3((2x)/(2x + 2)) + 1) - 1| - 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (3(2(-2(-3|x - 1|) + 3)^2 + 2) + 1)/(-2(2(-2(-3|x - 1|) + 3)^2 + 2) + 1) and g(x) = -3|((-1(-5x + 1))/(-3(-5x + 1) - 2)) + 3| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (1(2e^(-2(-2|x| + 3)) + 2) + 1)/(1(2e^(-2(-2|x| + 3)) + 2)) and g(x) = -1((3x - 3)/(3x - 1))^2 - 1((3x - 3)/(3x - 1)) + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 2e^(-2(2|(-1((2x + 1)/(1x - 3)) + 3) - 3| - 1)) and g(x) = (2(2|(-5x + 5) - 2| + 2) - 1)/(3(2|(-5x + 5) - 2| + 2)) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(1(3((-1x - 3)/(-1x + 2)) + 5)^2 - 1(3((-1x - 3)/(-1x + 2)) + 5) + 3) - 2| + 3 and g(x) = -2|(4(2e^(2x) + 3)^2 - 4(2e^(2x) + 3) + 3) + 2| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -5(-2e^(-2(-1|(4x + 5) - 2| + 1)) + 2)^2 - 4(-2e^(-2(-1|(4x + 5) - 2| + 1)) + 2) - 2 and g(x) = -1(4(3|x - 1| + 3)^2 + 1(3|x - 1| + 3) + 4) - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (3(-4(-1|x + 2| - 1)^2 - 2(-1|x + 2| - 1) + 1) + 2)/(-1(-4(-1|x + 2| - 1)^2 - 2(-1|x + 2| - 1) + 1)) and g(x) = (1(3|x - 2| - 2) - 1)/(-2(3|x - 2| - 2) + 3) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1(-1|((-1x + 2)/(-3x + 2)) - 1| + 1)^2 - 1(-1|((-1x + 2)/(-3x + 2)) - 1| + 1) - 3 and g(x) = -4((2x - 1)/(-2x))^2 - 4((2x - 1)/(-2x)) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1|(-3(-4((-1x)/(1x + 3)) + 2)^2 + 5(-4((-1x)/(1x + 3)) + 2) + 3) + 2| + 1 and g(x) = (2(-3|(-4x) - 1| + 2) - 2)/(3(-3|(-4x) - 1| + 2) - 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -5(3((1(-3|x + 2| - 3) - 2)/(-3(-3|x + 2| - 3))) - 3)^2 + 4(3((1(-3|x + 2| - 3) - 2)/(-3(-3|x + 2| - 3))) - 3) - 4 and g(x) = 3(-2|(-1x^2 + 4x - 1) - 1| - 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": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(5(-4(2|x + 1| + 2)^2 - 3) - 4) + 1)/(-2(5(-4(2|x + 1| + 2)^2 - 3) - 4) - 3) and g(x) = (2(5(-1x^2 + 5x + 2) - 5) - 2)/(-2(5(-1x^2 + 5x + 2) - 5) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 2|(1((-2(1x^2 - 4x + 1) - 2)/(3(1x^2 - 4x + 1) + 2)) - 4)| + 2 and g(x) = 2(1|((3x)/(-1x - 2)) + 2| + 1)^2 - 5(1|((3x)/(-1x - 2)) + 2| + 1) - 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -3|(5(5((3x)/(2x - 1))^2 - 4((3x)/(2x - 1)) + 1) - 1) + 2| + 1 and g(x) = 2((2(1x + 1))/(-2(1x + 1)))^2 + 4((2(1x + 1))/(-2(1x + 1))) - 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2|(2(1sin(1\u03c0((-1x)/(-3x + 1))) - 1)^2 + 3(1sin(1\u03c0((-1x)/(-3x + 1))) - 1) - 1) - 1| + 3 and g(x) = 3(-3((-3x + 1)/(-1x - 2))^2 + 3((-3x + 1)/(-1x - 2)) + 3) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1((-2(1(-2|x + 3| + 1)^2 - 2(-2|x + 3| + 1) - 4) + 1)/(-2(1(-2|x + 3| + 1)^2 - 2(-2|x + 3| + 1) - 4) + 3)) - 1 and g(x) = 1|(1((-3x + 2)/(-3x + 1))^2 + 1((-3x + 2)/(-3x + 1)) + 4) - 1| - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -2(1cos(1\u03c0(-1|x - 1|)) - 1)^2 + 2(1cos(1\u03c0(-1|x - 1|)) - 1) + 1 and g(x) = 4(1|x - 1|)^2 - 2(1|x - 1|) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 4((-3(5(3|x + 3| - 1)^2 + 2(3|x + 3| - 1) - 2) + 1)/(2(5(3|x + 3| - 1)^2 + 2(3|x + 3| - 1) - 2) + 3)) + 5 and g(x) = 5((-1(3|x| - 3) + 3)/(-1(3|x| - 3) - 2)) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-2(1(1|(-1x^2 + 4)|) + 5) - 1)/(-2(1(1|(-1x^2 + 4)|) + 5) - 1) and g(x) = 1((2(1|x + 3|) + 3)/(-2(1|x + 3|) + 3)) + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1|(1((-1x + 3)/(3x + 3)) + 2) - 3| + 2 and g(x) = 1|(4x^2 + 4x + 2) + 3| - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -4((2(2|x + 3| - 2) + 1)/(2(2|x + 3| - 2) - 3)) - 5 and g(x) = -1sin(1\u03c0(2|x - 3|) - 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -3(3e^(-1(-3|((-2x + 2)/(-2x - 2)) - 3| + 2)) - 3) - 5 and g(x) = 3((-2(-5x^2 - 4x - 5) + 1)/(-3(-5x^2 - 4x - 5) + 2)) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -2(-2(-3e^(1(-3|x| - 1)) + 2) + 5)^2 + 1(-2(-3e^(1(-3|x| - 1)) + 2) + 5) + 5 and g(x) = 1|(1((-1x - 2)/(-2x + 1)) - 4)| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1((-2(3|(-3x^2 - 1x - 5) - 1| - 1) - 3)/(1(3|(-3x^2 - 1x - 5) - 1| - 1) - 2)) and g(x) = -2|((-1(3e^(1x)) + 3)/(-3(3e^(1x)) + 4))| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 2(-4((-3x + 1)/(-3x - 3)) + 5)^2 + 2(-4((-3x + 1)/(-3x - 3)) + 5) - 1 and g(x) = (-2(-4x^2 + 3x + 2) - 2)/(1(-4x^2 + 3x + 2)) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (1(-2(2x^2 - 5) - 4))/(-3(-2(2x^2 - 5) - 4) - 3) and g(x) = (2(-1cos(2\u03c0x) - 1) - 1)/(2(-1cos(2\u03c0x) - 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 2((-2(-2sin(2\u03c0x + 2) + 1) - 2)/(1(-2sin(2\u03c0x + 2) + 1) + 1)) - 5 and g(x) = -2|(1x + 4) - 1| + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 3e^(2((-3(1|(-2x - 2) - 1| - 3) - 2)/(3(1|(-2x - 2) - 1| - 3) - 1))) + 3 and g(x) = (1(-2|(-2x + 3) + 2| + 1) + 3)/(2(-2|(-2x + 3) + 2| + 1) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (3(-2(-1|x - 1| - 3)^2 + 3(-1|x - 1| - 3) - 4) - 1)/(-1(-2(-1|x - 1| - 3)^2 + 3(-1|x - 1| - 3) - 4) - 2) and g(x) = (1(-2x^2 + 5x + 5))/(-1(-2x^2 + 5x + 5) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3(-1sin(1\u03c0(1(2|x| + 1)^2 + 4(2|x| + 1) - 5) - 2) - 2) - 1 and g(x) = 2((-3(-2|x - 1|) + 1)/(1(-2|x - 1|) + 1))^2 - 3((-3(-2|x - 1|) + 1)/(1(-2|x - 1|) + 1)) - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1|(1(-5((-1x - 3)/(-1x + 2)) + 3)^2 - 5(-5((-1x - 3)/(-1x + 2)) + 3) - 5) - 3| - 1 and g(x) = (-3(4(-1e^(1x) + 1) + 1) + 3)/(-2(4(-1e^(1x) + 1) + 1) + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-3(1e^(2(1(-2|x - 1| + 2) - 4)) + 3))/(2(1e^(2(1(-2|x - 1| + 2) - 4)) + 3) + 1) and g(x) = -3((-1(-3|x| - 2))/(1(-3|x| - 2) + 2)) - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -1|((1(-3(-1sin(3\u03c0x - 1))^2 - 5(-1sin(3\u03c0x - 1)) + 2) + 1)/(3(-3(-1sin(3\u03c0x - 1))^2 - 5(-1sin(3\u03c0x - 1)) + 2) + 3))| - 3 and g(x) = -3(-2((3x - 1)/(2x - 1))^2 - 4((3x - 1)/(2x - 1)) + 1) + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (3(-3|(4(-4x + 2)^2 + 1(-4x + 2) - 2) + 3| + 1) - 2)/(2(-3|(4(-4x + 2)^2 + 1(-4x + 2) - 2) + 3| + 1) + 2) and g(x) = (1(3|(-1e^(-2x) + 1) + 3| + 1) - 3)/(-3(3|(-1e^(-2x) + 1) + 3| + 1) + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-2(-3(1cos(2\u03c0x - 2) + 2) - 4))/(-2(-3(1cos(2\u03c0x - 2) + 2) - 4)) and g(x) = (-1(-5x^2 + 3x + 4) + 2)/(1(-5x^2 + 3x + 4) + 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2|((1(-2x + 5) + 3)/(-1(-2x + 5) - 3)) + 3| and g(x) = (2(5x^2 + 1) - 3)/(-1(5x^2 + 1) + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3cos(1\u03c0((-3(-4(3|x - 3| - 3) + 3) + 3)/(-2(-4(3|x - 3| - 3) + 3) - 3)) + 2) - 2 and g(x) = -2((2(2x + 5) - 3)/(3(2x + 5) + 2))^2 - 4((2(2x + 5) - 3)/(3(2x + 5) + 2)) + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1(-5((-1(-1e^(-1x) - 3) + 2)/(3(-1e^(-1x) - 3) + 2)) + 2)^2 - 5(-5((-1(-1e^(-1x) - 3) + 2)/(3(-1e^(-1x) - 3) + 2)) + 2) - 4 and g(x) = -2((-2(-1x - 4) + 2)/(1(-1x - 4) - 3))^2 + 3((-2(-1x - 4) + 2)/(1(-1x - 4) - 3)) - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2|(-1((3x)/(-3x + 2))^2 - 3((3x)/(-3x + 2)) - 4) - 3| and g(x) = -3(-3e^(1x) + 3)^2 - 1(-3e^(1x) + 3) - 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3|(3(3((3x + 3)/(3x - 2)))^2 - 5(3((3x + 3)/(3x - 2))) + 3) - 2| - 2 and g(x) = -4(-3e^(1(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": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (-1(1sin(1\u03c0(-1x + 5) - 3) - 2) - 2)/(-2(1sin(1\u03c0(-1x + 5) - 3) - 2) + 1) and g(x) = (-1(-1|x - 1| + 1) + 1)/(2(-1|x - 1| + 1) - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2((-2(2|x| - 3) + 1)/(1(2|x| - 3)))^2 + 2 and g(x) = (-1(2e^(1x)) - 1)/(2(2e^(1x)) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1(-2|((3x + 3)/(2x + 3)) - 2| - 1) + 5 and g(x) = 2|((-2x + 1)/(3x + 3))| - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (3(-5(-1x + 3)^2 - 1(-1x + 3) + 3) + 2)/(3(-5(-1x + 3)^2 - 1(-1x + 3) + 3) - 2) and g(x) = (-2(2x^2 - 1x + 2))/(1(2x^2 - 1x + 2) - 3) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2((2(-1(2|x - 1| + 2)^2 + 5(2|x - 1| + 2) + 2) - 2)/(-1(-1(2|x - 1| + 2)^2 + 5(2|x - 1| + 2) + 2) - 3)) - 1 and g(x) = (2(4(3x + 1)^2 + 4(3x + 1) + 2) - 3)/(-2(4(3x + 1)^2 + 4(3x + 1) + 2) + 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-3(5(2|(4x + 2) - 2| - 2)^2 - 5(2|(4x + 2) - 2| - 2) - 4) - 3)/(-2(5(2|(4x + 2) - 2| - 2)^2 - 5(2|(4x + 2) - 2| - 2) - 4) - 2) and g(x) = 1|(5((-2x + 2)/(-2x - 2)) + 3) + 1| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1(-1|((2(1x^2 - 4x + 2) + 1)/(3(1x^2 - 4x + 2) - 2)) - 3|) + 2 and g(x) = -4(3((-1x - 1)/(-1x)) + 1)^2 - 1(3((-1x - 1)/(-1x)) + 1) - 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1((1(1sin(1\u03c0x + 1)) - 3)/(1(1sin(1\u03c0x + 1)) - 3)) + 1 and g(x) = (-1(1|x - 1| - 2) + 2)/(1(1|x - 1| - 2) + 2) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (-1(-3e^(-1(-2|x - 3| + 3)) + 2) + 3)/(2(-3e^(-1(-2|x - 3| + 3)) + 2) + 1) and g(x) = 1(1cos(3\u03c0x) - 1) - 4 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the third intersection point (counting from left to right) between the functions f(x) = 1(4(3|((2x + 1)/(-2x + 3)) + 2|) + 2)^2 - 4(4(3|((2x + 1)/(-2x + 3)) + 2|) + 2) - 4 and g(x) = (-1(-5(2x + 5)^2 - 4(2x + 5) - 2) - 3)/(2(-5(2x + 5)^2 - 4(2x + 5) - 2) + 3) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3(-3|(2e^(1x) + 1) - 3| + 2) + 4 and g(x) = 2|(-2e^(1x) - 2) + 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -1sin(1\u03c0((2(4x^2 + 5x - 1))/(2(4x^2 + 5x - 1) - 1)) - 3) + 1 and g(x) = 1(1x^2 - 3x + 3) - 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-1(1(3(1e^(-1x) - 3) + 2)^2 + 3(3(1e^(-1x) - 3) + 2) - 3) + 2)/(1(1(3(1e^(-1x) - 3) + 2)^2 + 3(3(1e^(-1x) - 3) + 2) - 3)) and g(x) = 3e^(-1((-1(2|x + 1| - 1))/(-3(2|x + 1| - 1)))) + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2(-1(2|((-2x - 2)/(2x + 2)) - 1| - 3) - 1)^2 + 4(-1(2|((-2x - 2)/(2x + 2)) - 1| - 3) - 1) - 5 and g(x) = -4(1e^(1(3x - 3)))^2 + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3(2|(-1((1x + 1)/(3x))^2 - 5((1x + 1)/(3x)) - 3)| - 1) - 5 and g(x) = -2(-2|(1cos(2\u03c0x - 1) - 3) - 3|)^2 - 3(-2|(1cos(2\u03c0x - 1) - 3) - 3|) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (1(2(3|(-1e^(-2x) - 3) - 1|)^2 + 1(3|(-1e^(-2x) - 3) - 1|) - 1) - 1)/(-1(2(3|(-1e^(-2x) - 3) - 1|)^2 + 1(3|(-1e^(-2x) - 3) - 1|) - 1)) and g(x) = 2(-3((-1x)/(-3x + 1)))^2 + 4(-3((-1x)/(-3x + 1))) - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1((1(5(1|x| + 2) + 4) - 1)/(2(5(1|x| + 2) + 4) + 1))^2 + 3((1(5(1|x| + 2) + 4) - 1)/(2(5(1|x| + 2) + 4) + 1)) - 4 and g(x) = 1(1e^(-1((3x - 2)/(2x - 1))) - 3) - 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2((-1(-3|(-5x^2 - 4x - 3) - 3| - 3) + 2)/(-1(-3|(-5x^2 - 4x - 3) - 3| - 3) - 3)) - 1 and g(x) = 2|((-3(-5x - 2) + 2)/(-2(-5x - 2) + 3))| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -4((3(2|(1x - 4) + 1| + 2))/(3(2|(1x - 4) + 1| + 2) + 1))^2 + 5((3(2|(1x - 4) + 1| + 2))/(3(2|(1x - 4) + 1| + 2) + 1)) - 5 and g(x) = -1((-1(-3|x - 3| + 3) - 3)/(-2(-3|x - 3| + 3))) + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -5(1|(-1e^(-2((-3x + 1)/(-3x + 4))) + 1) - 1|)^2 - 3(1|(-1e^(-2((-3x + 1)/(-3x + 4))) + 1) - 1|) + 4 and g(x) = -4(-3cos(1\u03c0(-3|x| + 3)) + 3) - 4 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 2|(-1sin(3\u03c0(4x - 3) - 3) + 3) + 3| - 3 and g(x) = (-1(-1x^2 - 3x + 4) + 3)/(-1(-1x^2 - 3x + 4) - 1) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1cos(3\u03c0(-2((-1x)/(-1x + 3))^2 - 2((-1x)/(-1x + 3)) - 5) - 2) - 1 and g(x) = 4(5x - 5)^2 - 4(5x - 5) - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -5(1|((-1x + 1)/(-3x + 2))| - 3) + 3 and g(x) = -1cos(1\u03c0((-2x + 3)/(-2x - 3)) - 3) + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 5((-3(-3|x|) + 1)/(2(-3|x|) - 1)) and g(x) = -5((-1x - 3)/(2x))^2 - 3((-1x - 3)/(2x)) - 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -3|((-2(-3(3x^2 + 3x) - 5) - 1)/(-2(-3(3x^2 + 3x) - 5) + 3)) - 3| - 3 and g(x) = -1|(-4(2x^2 - 2x + 1) - 2)| meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2(1|(-1(-3e^(-1x) + 2) - 4) - 3|)^2 + 1(1|(-1(-3e^(-1x) + 2) - 4) - 3|) + 4 and g(x) = -2e^(-1(-2((-1x - 3)/(-2x - 3)) + 4)) - 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1e^(-1((3(3(-2|x + 3| - 2) - 2) + 1)/(1(3(-2|x + 3| - 2) - 2) + 2))) - 3 and g(x) = -1(3|(1x + 3) + 1| + 3)^2 + 5(3|(1x + 3) + 1| + 3) + 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-8", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (-2(3(3|(5x) + 2| + 1)^2 - 5(3|(5x) + 2| + 1) + 2) + 1)/(-1(3(3|(5x) + 2| + 1)^2 - 5(3|(5x) + 2| + 1) + 2) + 2) and g(x) = 3((-2(-3|x| + 3))/(-2(-3|x| + 3)))^2 + 2((-2(-3|x| + 3))/(-2(-3|x| + 3))) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (2(3(-2|(-4x^2 - 2x - 3) - 3| + 2) + 5) + 2)/(-3(3(-2|(-4x^2 - 2x - 3) - 3| + 2) + 5) - 1) and g(x) = 1(-3|((3x - 2)/(-1x + 3)) - 1| - 1) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 2|((1(-3x^2 - 2))/(2(-3x^2 - 2) + 1)) - 2| - 1 and g(x) = (1(-2|x| + 2) + 3)/(-2(-2|x| + 2) + 1) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2|(-1e^(-2((1x + 1)/(1x + 2))) - 3)| and g(x) = 3|(-4x^2 + 1x - 4) - 1| - 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (1(4(-1|x - 1| + 3) + 4) - 2)/(3(4(-1|x - 1| + 3) + 4) - 2) and g(x) = 3(3cos(1\u03c0x + 2) + 3)^2 - 5(3cos(1\u03c0x + 2) + 3) + 4 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2|(-4(-2((-1x - 2)/(-1x - 1))^2 + 5((-1x - 2)/(-1x - 1)) - 3) + 1)| and g(x) = 2|(-5(-1x^2 - 1x + 2) + 3) + 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 3(-2|((-3(3x^2 - 2x + 1))/(-2(3x^2 - 2x + 1) + 1)) + 1| - 1) + 1 and g(x) = (2(1|(2x - 3) - 1| - 2) + 3)/(3(1|(2x - 3) - 1| - 2) + 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3|((-1(-3sin(2\u03c0(-5x^2 - 3x - 1) + 3) + 2) - 2)/(3(-3sin(2\u03c0(-5x^2 - 3x - 1) + 3) + 2))) - 1| - 2 and g(x) = -3e^(2(-3((2x - 2)/(-1x - 2)) + 5)) + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1((1(-2|x| - 3) + 1)/(-2(-2|x| - 3) - 3))^2 - 2((1(-2|x| - 3) + 1)/(-2(-2|x| - 3) - 3)) + 2 and g(x) = (-2(-1x - 5) + 3)/(-3(-1x - 5) + 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -1cos(1\u03c0(1|((1(2x^2 + 3x + 5) - 3)/(3(2x^2 + 3x + 5) + 2)) - 1| - 2) - 2) - 1 and g(x) = (1(5(2|x + 1|)^2 - 5) - 3)/(3(5(2|x + 1|)^2 - 5) + 2) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2((3(-5x + 1) - 2)/(3(-5x + 1) - 2))^2 - 4((3(-5x + 1) - 2)/(3(-5x + 1) - 2)) and g(x) = -1|((-3x + 3)/(1x - 3)) - 3| meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 1|((-1(5x + 4) + 3)/(3(5x + 4) + 3))| + 3 and g(x) = (-3(1|x - 3| - 3) - 3)/(1(1|x - 3| - 3) + 3) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5((2(-3|x| - 1) + 1)/(-1(-3|x| - 1) + 3)) + 1 and g(x) = -1|(1e^(-1x) - 3) + 1| + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3(-5((3(-2|x - 2| - 3) + 2)/(-3(-2|x - 2| - 3)))^2 - 4((3(-2|x - 2| - 3) + 2)/(-3(-2|x - 2| - 3)))) - 2 and g(x) = -2|((1(3x^2 + 4x - 2) - 3)/(-1(3x^2 + 4x - 2))) - 3| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1(3sin(2\u03c0((3(2|x + 3|))/(2(2|x + 3|) + 3)) - 1) - 3)^2 + 2(3sin(2\u03c0((3(2|x + 3|))/(2(2|x + 3|) + 3)) - 1) - 3) + 2 and g(x) = 3|(-3e^(2(-4x - 5)) - 1) + 3| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2e^(-1((-1(-2(2x - 4)^2 - 2(2x - 4) - 2) - 2)/(-2(-2(2x - 4)^2 - 2(2x - 4) - 2) - 2))) - 1 and g(x) = (-1(5(2x^2 - 4x) - 2) - 1)/(-1(5(2x^2 - 4x) - 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 4((2(-4(-2|x + 2| - 2)^2 + 4(-2|x + 2| - 2) - 4) - 2)/(-2(-4(-2|x + 2| - 2)^2 + 4(-2|x + 2| - 2) - 4) + 3)) and g(x) = 2((-2(2x^2 + 3x - 3))/(-1(2x^2 + 3x - 3) - 3)) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (1(-5(1e^(-1(3x^2 - 5x + 1)) + 1)) - 2)/(-1(-5(1e^(-1(3x^2 - 5x + 1)) + 1)) - 1) and g(x) = -4(-2e^(1((2x + 3)/(-2x - 2))) + 2) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -5(-3(-1|((-1x + 2)/(-1x + 2)) - 1| + 2) - 2)^2 - 3(-3(-1|((-1x + 2)/(-1x + 2)) - 1| + 2) - 2) - 2 and g(x) = -5(3(-3|x + 1| + 3)^2 - 3(-3|x + 1| + 3) - 4) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1|((-3(5(-3x^2 + 3x + 3) - 1) + 2)/(3(5(-3x^2 + 3x + 3) - 1) + 2)) + 3| - 3 and g(x) = (-3(1(-2|x - 1| - 2)^2 + 1(-2|x - 1| - 2) - 5) + 1)/(-1(1(-2|x - 1| - 2)^2 + 1(-2|x - 1| - 2) - 5) + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(1sin(1\u03c0(1((2x - 2)/(-2x + 1)) - 2)) - 1) + 1| - 3 and g(x) = (-1(1(2x^2 - 4x - 2) - 5))/(-1(1(2x^2 - 4x - 2) - 5) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2(-3(-3|((-3x - 1)/(1x - 2)) - 2| + 1) + 3)^2 + 1(-3(-3|((-3x - 1)/(1x - 2)) - 2| + 1) + 3) + 5 and g(x) = 3|((2(2cos(1\u03c0x + 1) - 2) + 1)/(-3(2cos(1\u03c0x + 1) - 2) + 2)) - 2| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -2((-2(-1x - 5) + 3)/(1(-1x - 5) + 3))^2 - 5((-2(-1x - 5) + 3)/(1(-1x - 5) + 3)) - 1 and g(x) = -3sin(1\u03c0((-3x - 1)/(1x + 3)) - 2) + 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 5((-2(-2cos(2\u03c0x + 3) + 3) - 2)/(-3(-2cos(2\u03c0x + 3) + 3) + 4))^2 + 5((-2(-2cos(2\u03c0x + 3) + 3) - 2)/(-3(-2cos(2\u03c0x + 3) + 3) + 4)) + 3 and g(x) = -5((1x + 3)/(3x + 3)) - 4 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2|((3(1x - 2) - 2)/(-2(1x - 2)))| + 2 and g(x) = 3(-3|x - 2| - 1)^2 - 1(-3|x - 2| - 1) - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -4(2|(-1x - 5) + 2| - 3)^2 + 1(2|(-1x - 5) + 2| - 3) - 1 and g(x) = (-1(3|x| + 2) + 3)/(-1(3|x| + 2) + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(-2e^(1(-1(-1x + 3)^2 + 3(-1x + 3))) + 2) - 2)/(1(-2e^(1(-1(-1x + 3)^2 + 3(-1x + 3))) + 2) + 3) and g(x) = 2|(-4(4x^2 + 4x - 4) - 3) - 1| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-3(-1(-1|(3x - 1) - 2|)^2 - 1(-1|(3x - 1) - 2|) + 3) - 1)/(-2(-1(-1|(3x - 1) - 2|)^2 - 1(-1|(3x - 1) - 2|) + 3) + 3) and g(x) = 2e^(1(-4(-1x^2 - 3x) + 4)) - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -1((3(-1|(-1x + 5) - 3| - 2) - 2)/(2(-1|(-1x + 5) - 3| - 2) + 2))^2 + 3((3(-1|(-1x + 5) - 3| - 2) - 2)/(2(-1|(-1x + 5) - 3| - 2) + 2)) - 2 and g(x) = 1|((-1(2x^2 - 3x - 1))/(1(2x^2 - 3x - 1) - 2)) - 1| - 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 4((-2(-1|(-1x^2 + 3x - 2)| + 1))/(2(-1|(-1x^2 + 3x - 2)| + 1) - 1)) - 2 and g(x) = (-1(-2e^(-1(1x)) - 2))/(2(-2e^(-1(1x)) - 2)) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -3|(2sin(1\u03c0((-1(2x - 4) + 1)/(1(2x - 4))) + 2) - 2) - 2| - 3 and g(x) = -3(5(1e^(2x) - 3) + 5)^2 - 1(5(1e^(2x) - 3) + 5) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1sin(2\u03c0(1|(-1x + 2) + 3| + 3) - 1) + 3 and g(x) = -1((-2x)/(3x + 2))^2 + 5((-2x)/(3x + 2)) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -4(-4(1|((3x - 3)/(-2x + 1)) - 1|) - 1)^2 - 5 and g(x) = (-3(-3(-2|x - 2| - 1) - 5) + 3)/(-1(-3(-2|x - 2| - 1) - 5) + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (2(4(4(-2|x - 3| - 2)^2 + 3(-2|x - 3| - 2) - 5) + 4))/(2(4(4(-2|x - 3| - 2)^2 + 3(-2|x - 3| - 2) - 5) + 4) - 1) and g(x) = 3(-2|((1x - 1)/(2x - 1)) - 3| + 3) + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 4((-2(1cos(3\u03c0x + 1) + 3) + 3)/(2(1cos(3\u03c0x + 1) + 3)))^2 + 3((-2(1cos(3\u03c0x + 1) + 3) + 3)/(2(1cos(3\u03c0x + 1) + 3))) and g(x) = (1(-4x) - 3)/(2(-4x) - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -3(4((2x - 2)/(-3x + 2))^2 - 5((2x - 2)/(-3x + 2)) - 1) + 4 and g(x) = (-3(-1x^2 + 3x - 2) + 2)/(1(-1x^2 + 3x - 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 2e^(-2(-2|((3(-5x + 3) - 1)/(-3(-5x + 3) + 4))| - 1)) + 3 and g(x) = -3(-2|(3x^2 - 3x + 2) - 1|) + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1(3|(-2e^(-1(-4x^2 + 1x + 4)) - 2) + 3| + 2) + 2 and g(x) = -2((-3(-1|x + 1| + 1))/(-1(-1|x + 1| + 1) - 2)) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (1(-5(-2|(2sin(1\u03c0x + 3) + 2) + 1|)^2 - 1(-2|(2sin(1\u03c0x + 3) + 2) + 1|)) - 3)/(-1(-5(-2|(2sin(1\u03c0x + 3) + 2) + 1|)^2 - 1(-2|(2sin(1\u03c0x + 3) + 2) + 1|)) - 3) and g(x) = (1(5(-1x + 4)^2 - 4(-1x + 4) + 3) + 3)/(-2(5(-1x + 4)^2 - 4(-1x + 4) + 3)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1(1|((2(-3x^2 - 2x + 3) - 2)/(-3(-3x^2 - 2x + 3))) - 1| + 3) + 1 and g(x) = (2(1|(3x - 5) - 1|))/(1(1|(3x - 5) - 1|) - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (1(-2(-1|(-3x + 4) - 2| + 3)^2 + 4(-1|(-3x + 4) - 2| + 3) + 4) - 2)/(-2(-2(-1|(-3x + 4) - 2| + 3)^2 + 4(-1|(-3x + 4) - 2| + 3) + 4) + 3) and g(x) = -1(-2|(4x + 5) - 1| - 2)^2 - 1(-2|(4x + 5) - 1| - 2) + 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2cos(2\u03c0((-2(-5(-1|x - 2|) + 4))/(-2(-5(-1|x - 2|) + 4) + 3)) + 1) + 2 and g(x) = 3(1e^(2((-2x)/(-2x - 3))))^2 - 4(1e^(2((-2x)/(-2x - 3)))) - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 2|(-1(1((-2x + 3)/(-1x + 3))^2 - 1((-2x + 3)/(-1x + 3)) + 2) + 2) - 3| - 1 and g(x) = (3(2|(-1x^2 + 5x - 4) + 3| + 1) + 3)/(3(2|(-1x^2 + 5x - 4) + 3| + 1) - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 3e^(-2(-2|((2(1x^2 - 4x - 2) + 2)/(2(1x^2 - 4x - 2) + 3)) - 1| + 2)) and g(x) = (-2(-3(-5x + 3)^2 + 1(-5x + 3)) - 1)/(3(-3(-5x + 3)^2 + 1(-5x + 3)) - 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 1(-3|(1(-2e^(-1x) + 3)^2 - 4(-2e^(-1x) + 3) - 2) + 1| + 3) - 4 and g(x) = -3sin(3\u03c0((2(-2x - 4) - 3)/(-2(-2x - 4))) - 2) - 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 3|((2(1x - 5) + 2)/(-3(1x - 5) + 2)) + 2| - 3 and g(x) = (-3(2e^(-1x) - 3) + 1)/(2(2e^(-1x) - 3) - 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 2|(4(-5x + 5)^2 + 3(-5x + 5) + 2) + 3| - 2 and g(x) = 3|(-3sin(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": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 2(-2sin(1\u03c0(1|((1x - 2)/(1x - 2))| + 3))) + 2 and g(x) = (-1(1|(2x^2 + 2x + 1) - 3| - 2) + 1)/(2(1|(2x^2 + 2x + 1) - 3| - 2) - 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1(-4(2|((-2x + 1)/(3x + 1)) - 1| - 1)^2 - 3(2|((-2x + 1)/(3x + 1)) - 1| - 1) + 4) - 4 and g(x) = 2sin(3\u03c0(1|(2x^2 - 5x) + 2|) - 1) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 4((-1(-3|x - 2|) + 2)/(2(-3|x - 2|) + 2)) + 1 and g(x) = 1|(-1sin(1\u03c0x) + 3) - 3| - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1|(-2e^(2(-3((-1x + 1)/(2x - 3)) + 3)) + 3) + 1| + 3 and g(x) = 1|(1(1sin(3\u03c0x - 1) + 3)^2 - 5(1sin(3\u03c0x - 1) + 3) - 2) + 3| + 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (3(-1(5(-2e^(-1x) - 1)^2 - 4(-2e^(-1x) - 1) - 5) - 2))/(-2(-1(5(-2e^(-1x) - 1)^2 - 4(-2e^(-1x) - 1) - 5) - 2)) and g(x) = (2(1|(2x + 1)| - 3) - 1)/(-3(1|(2x + 1)| - 3) + 4) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (-2(4(3(-2|x|)^2 - 2) + 1) + 3)/(-2(4(3(-2|x|)^2 - 2) + 1) - 2) and g(x) = (2(-3|(3x - 1) - 3| - 2) - 3)/(1(-3|(3x - 1) - 3| - 2) - 2) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 4(3((-3(-2|x - 2| + 1) - 1)/(2(-2|x - 2| + 1) + 1))^2 + 5((-3(-2|x - 2| + 1) - 1)/(2(-2|x - 2| + 1) + 1)) + 2) + 3 and g(x) = (-1(3(4x^2 - 3x - 4) + 1) + 1)/(-2(3(4x^2 - 3x - 4) + 1) + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1(-3|((-3x - 1)/(-2x - 1))| + 1)^2 - 3(-3|((-3x - 1)/(-2x - 1))| + 1) + 4 and g(x) = (3(3x - 5) + 3)/(-2(3x - 5) + 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1|((-2(1(-1e^(-2x) + 1)^2 - 5(-1e^(-2x) + 1) + 1) + 3)/(3(1(-1e^(-2x) + 1)^2 - 5(-1e^(-2x) + 1) + 1) - 2))| and g(x) = 3|((3(-1sin(1\u03c0x + 3) - 1) + 3)/(-1(-1sin(1\u03c0x + 3) - 1) + 2)) - 3| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1(2((-3x + 3)/(-1x - 3))^2 - 1((-3x + 3)/(-1x - 3))) - 5 and g(x) = (1(5x^2 + 1x + 2))/(2(5x^2 + 1x + 2) + 1) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (2(-3(5(1|x - 2| - 1)^2 + 4) - 5) - 2)/(-2(-3(5(1|x - 2| - 1)^2 + 4) - 5) + 1) and g(x) = -2|((2(2x + 4))/(-1(2x + 4) - 2)) + 3| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -4((-2(1|(-2x + 5) + 1|) - 1)/(-1(1|(-2x + 5) + 1|) + 3))^2 - 3((-2(1|(-2x + 5) + 1|) - 1)/(-1(1|(-2x + 5) + 1|) + 3)) + 3 and g(x) = (1(-1(3cos(3\u03c0x) + 3)^2 - 4(3cos(3\u03c0x) + 3) - 1) + 2)/(2(-1(3cos(3\u03c0x) + 3)^2 - 4(3cos(3\u03c0x) + 3) - 1) - 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -2((3(2|x + 2| - 2) - 3)/(1(2|x + 2| - 2) - 2)) - 4 and g(x) = (3(3x - 5) - 3)/(-1(3x - 5) - 3) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2cos(2\u03c0(-2|((-2x + 3)/(-2x + 1)) + 1| + 3) - 3) - 1 and g(x) = 1(3|x + 3|)^2 - 3(3|x + 3|) - 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2e^(-2(2(2|((-3x - 3)/(-1x - 3)) - 1| - 2) - 3)) and g(x) = 1((-1(-2x^2 + 2) + 3)/(-1(-2x^2 + 2) + 2)) - 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2|(-2(2e^(-1(5x + 4)) - 2)^2 - 5(2e^(-1(5x + 4)) - 2) + 1)| + 1 and g(x) = -3((1(-1|x + 1| + 3))/(1(-1|x + 1| + 3) - 2)) - 4 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-3(1(-1x + 3)^2 + 2(-1x + 3) - 1) + 3)/(2(1(-1x + 3)^2 + 2(-1x + 3) - 1) + 1) and g(x) = (1(-2cos(1\u03c0x - 1) - 1) + 2)/(-1(-2cos(1\u03c0x - 1) - 1) + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 5(2|((1(-3x) - 2)/(-1(-3x) + 3))| + 1)^2 - 4(2|((1(-3x) - 2)/(-1(-3x) + 3))| + 1) - 3 and g(x) = (2(-1(1sin(1\u03c0x + 2)) - 2) - 3)/(2(-1(1sin(1\u03c0x + 2)) - 2)) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (1(3(-2|x + 3| - 2)^2 - 4(-2|x + 3| - 2) + 2) + 1)/(3(3(-2|x + 3| - 2)^2 - 4(-2|x + 3| - 2) + 2) + 3) and g(x) = 2|(2x - 4) - 3| meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 4(-4(1|((3x + 1)/(-1x)) - 3|)^2 + 2(1|((3x + 1)/(-1x)) - 3|) + 3) - 4 and g(x) = -1cos(2\u03c0(4((-2x + 1)/(2x))) - 1) - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -4(1|((1(-1x + 4) + 3)/(-2(-1x + 4))) - 1| - 2)^2 + 1(1|((1(-1x + 4) + 3)/(-2(-1x + 4))) - 1| - 2) + 2 and g(x) = -2(5(-1|x + 1| - 2)^2 - 4(-1|x + 1| - 2) + 4) + 4 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 1sin(2\u03c0(-3|((-3(1x + 4) + 2)/(-2(1x + 4) + 1)) + 1| + 3) + 2) - 2 and g(x) = -1(1((-3x)/(1x + 2)) - 5)^2 + 4(1((-3x)/(1x + 2)) - 5) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(-1(-5((-1x - 3)/(3x - 2))^2 - 5((-1x - 3)/(3x - 2)) - 5) + 2)| + 1 and g(x) = (-3(-2|(1x + 4) + 2| - 1))/(1(-2|(1x + 4) + 2| - 1) + 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -3cos(2\u03c0((-1(-2x^2 - 3x + 2) + 1)/(-1(-2x^2 - 3x + 2) - 3)) - 3) + 2 and g(x) = -4(-1x^2 + 3x + 5) - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1(-1|(1e^(2(-4x^2 + 1x + 4)) - 1) - 3| - 3) and g(x) = (3(3(3e^(-1x) - 2)^2 + 5(3e^(-1x) - 2) - 1) + 3)/(-2(3(3e^(-1x) - 2)^2 + 5(3e^(-1x) - 2) - 1) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-3(-5(1(-1|x + 3| - 2) - 5)^2 - 3(1(-1|x + 3| - 2) - 5)) - 3)/(1(-5(1(-1|x + 3| - 2) - 5)^2 - 3(1(-1|x + 3| - 2) - 5))) and g(x) = -1((1(4x^2 - 5x) + 3)/(-1(4x^2 - 5x) + 2)) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3|(-1((3(-2x^2 + 4x - 4) + 3)/(-1(-2x^2 + 4x - 4) + 3)) + 3) + 3| and g(x) = -3(5(2e^(1x) - 3) + 2)^2 - 2(5(2e^(1x) - 3) + 2) + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (3(4(1(-3|x + 2| - 2) + 1)^2 - 5) + 1)/(-2(4(1(-3|x + 2| - 2) + 1)^2 - 5)) and g(x) = 3|(-1sin(2\u03c0(-1x - 4) - 3) - 3) - 1| intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1(5(2e^(1((2x + 2)/(1x))) + 3)^2 - 4) + 5 and g(x) = 3(-2|(-1e^(2x) + 3)| - 2)^2 + 2(-2|(-1e^(2x) + 3)| - 2) - 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 5(1((-2x + 2)/(2x - 1)) - 5)^2 + 5(1((-2x + 2)/(2x - 1)) - 5) - 4 and g(x) = 1|(1sin(3\u03c0x - 3) - 3) - 1| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2((3(-1|x - 1| - 1) - 3)/(3(-1|x - 1| - 1) - 2))^2 - 5((3(-1|x - 1| - 1) - 3)/(3(-1|x - 1| - 1) - 2)) + 4 and g(x) = 4((2x + 1)/(1x))^2 - 2((2x + 1)/(1x)) - 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 4((1(-5(-3|x - 3| + 3)^2 - 5) + 1)/(-2(-5(-3|x - 3| + 3)^2 - 5) - 3)) - 3 and g(x) = -1e^(-1(-3(-1x + 4)^2 - 4(-1x + 4) - 1)) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1((2(2e^(-1x) + 2))/(1(2e^(-1x) + 2) + 3))^2 + 2((2(2e^(-1x) + 2))/(1(2e^(-1x) + 2) + 3)) and g(x) = (-1(2|x - 1|) + 1)/(-2(2|x - 1|)) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3(-2|(3e^(1((-1x - 3)/(1x))) + 1) - 3| - 2) - 5 and g(x) = (-2(-3|(-1x + 1) + 1|) - 3)/(2(-3|(-1x + 1) + 1|)) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1e^(-1(-2((3x + 1)/(-2x + 1)) - 1)) + 3 and g(x) = (-1(-2x^2 + 2x - 5) - 3)/(3(-2x^2 + 2x - 5) + 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -3((1(2(1|x + 3| + 1) + 3) - 1)/(-2(2(1|x + 3| + 1) + 3) - 3))^2 - 1((1(2(1|x + 3| + 1) + 3) - 1)/(-2(2(1|x + 3| + 1) + 3) - 3)) + 2 and g(x) = -5((1(4x^2 - 4) + 1)/(3(4x^2 - 4) + 1)) - 4 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 1(3|(-3((1x + 2)/(1x + 3))^2 + 5((1x + 2)/(1x + 3)) - 1) + 1| + 1) + 1 and g(x) = (-1(-3|(-5x - 3) - 2| + 1) + 1)/(-1(-3|(-5x - 3) - 2| + 1) - 3) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (1(-5(-1|x| + 3) + 5) - 3)/(2(-5(-1|x| + 3) + 5)) and g(x) = (2(-3x - 2) - 1)/(-1(-3x - 2) - 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 3|((2(5x^2 + 1x - 2) - 1)/(-3(5x^2 + 1x - 2) - 3))| and g(x) = 3e^(-2(-5x^2 + 1x + 3)) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -3e^(2((-3(-1x^2 - 2x + 4) + 3)/(2(-1x^2 - 2x + 4) - 1))) + 3 and g(x) = 1sin(2\u03c0((-1x)/(-2x + 1))) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2((-2(-2|(-2x^2 + 3x + 2) + 3| - 1) + 1)/(1(-2|(-2x^2 + 3x + 2) + 3| - 1))) + 1 and g(x) = 4((-1(-2x - 1) + 2)/(2(-2x - 1) + 2))^2 - 5((-1(-2x - 1) + 2)/(2(-2x - 1) + 2)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3(4(-1e^(2(-2|x - 3| + 2)) - 3)^2 - 4(-1e^(2(-2|x - 3| + 2)) - 3)) - 1 and g(x) = -1|(5(1x + 3)^2 + 4(1x + 3) - 1)| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1(3|(1e^(-1((3x + 3)/(-1x - 3))) - 1)| - 3) + 4 and g(x) = (2(-4(-2x^2 - 5) - 5) + 1)/(1(-4(-2x^2 - 5) - 5) + 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 2|((-3(3(-2cos(1\u03c0x - 1) + 2) - 1) + 1)/(-3(3(-2cos(1\u03c0x - 1) + 2) - 1) + 1))| and g(x) = (-1(2|(2x^2 - 4x - 1) + 2| - 2) + 2)/(1(2|(2x^2 - 4x - 1) + 2| - 2) - 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1(-2sin(1\u03c0(1|((-1x)/(1x)) - 2| + 3) - 2) - 3)^2 + 2(-2sin(1\u03c0(1|((-1x)/(1x)) - 2| + 3) - 2) - 3) - 5 and g(x) = -2e^(-2(-1|(2x + 2) - 2| + 3)) + 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2(3e^(-2((1x + 1)/(2x + 2))) - 1) + 2 and g(x) = 2|((-3x - 1)/(1x - 3)) + 2| + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (1(-5(5x + 1)^2 + 3(5x + 1) + 2) - 1)/(-2(-5(5x + 1)^2 + 3(5x + 1) + 2) + 3) and g(x) = -1sin(1\u03c0(5x^2 - 5x - 3) + 1) - 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 4(1|((-3x - 3)/(1x - 2)) + 1| + 3)^2 + 3(1|((-3x - 3)/(1x - 2)) + 1| + 3) + 4 and g(x) = 3e^(-1(-5x^2 - 3)) - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-3(-5(-2|(-2e^(-1x))| + 2)) + 2)/(1(-5(-2|(-2e^(-1x))| + 2)) - 2) and g(x) = 2|(-4(-2x^2 + 5x - 1) - 5) - 3| + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-1(-3(-3|(-5x + 5)| - 3)^2 + 1(-3|(-5x + 5)| - 3) - 5) - 3)/(-3(-3(-3|(-5x + 5)| - 3)^2 + 1(-3|(-5x + 5)| - 3) - 5) + 4) and g(x) = -2|(4((1x - 1)/(1x + 3)) - 3) + 2| + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1e^(1((-1(-1(-2|x + 1| + 2)^2 + 5(-2|x + 1| + 2) - 3) - 1)/(1(-1(-2|x + 1| + 2)^2 + 5(-2|x + 1| + 2) - 3) - 2))) + 1 and g(x) = -3|((-1(3x^2 + 3x - 1) - 2)/(1(3x^2 + 3x - 1) + 1)) - 1| - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (1(-1(-2|(-2cos(1\u03c0x - 2) + 3) - 3| - 1)^2 + 3(-2|(-2cos(1\u03c0x - 2) + 3) - 3| - 1) - 4) + 3)/(-2(-1(-2|(-2cos(1\u03c0x - 2) + 3) - 3| - 1)^2 + 3(-2|(-2cos(1\u03c0x - 2) + 3) - 3| - 1) - 4) - 3) and g(x) = 5((1(-5x^2 - 3x + 2) - 2)/(-1(-5x^2 - 3x + 2) - 3)) - 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 3((2(1|x + 3|))/(-1(1|x + 3|) + 2))^2 - 3((2(1|x + 3|))/(-1(1|x + 3|) + 2)) - 4 and g(x) = 3e^(-1(-3x + 5)) - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -1(-1|(1((3x + 2)/(-1x + 3)) + 1) - 2| + 1)^2 - 3(-1|(1((3x + 2)/(-1x + 3)) + 1) - 2| + 1) + 2 and g(x) = (-1(-1(1|x - 3|) + 3) + 2)/(-3(-1(1|x - 3|) + 3)) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -1(-2|(-2sin(2\u03c0((3x - 1)/(-3x)) + 2) + 1) + 2| - 2)^2 + 2(-2|(-2sin(2\u03c0((3x - 1)/(-3x)) + 2) + 1) + 2| - 2) - 3 and g(x) = -4(2|((2x + 1)/(3x + 3)) + 2| + 3)^2 - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-3(2|(-4x^2 - 3) + 3| - 3) - 1)/(-2(2|(-4x^2 - 3) + 3| - 3) + 1) and g(x) = (1(-2|x + 2| - 3) + 3)/(-3(-2|x + 2| - 3)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1(1|(5((1x)/(-3x))^2 + 1((1x)/(-3x))) + 3| + 2) - 3 and g(x) = -1((-3(-1x^2 + 4x + 5) + 2)/(-1(-1x^2 + 4x + 5) - 3)) + 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2cos(2\u03c0((-2(3|x - 3| - 3) + 2)/(3(3|x - 3| - 3) - 2)) + 1) and g(x) = -2(-3|x + 2| - 2)^2 - 4(-3|x + 2| - 2) + 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3((3(3e^(-2x) + 3) - 1)/(-3(3e^(-2x) + 3) - 1))^2 + 1((3(3e^(-2x) + 3) - 1)/(-3(3e^(-2x) + 3) - 1)) - 1 and g(x) = 3((-2x - 2)/(2x)) - 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -2((1(-3|x + 2| + 1) + 1)/(-2(-3|x + 2| + 1) - 1)) - 4 and g(x) = -1e^(1(-3|x + 1| - 3)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 3sin(2\u03c0(-2((2(-1|x - 2| + 2) - 2)/(-1(-1|x - 2| + 2) - 2))^2 + 4((2(-1|x - 2| + 2) - 2)/(-1(-1|x - 2| + 2) - 2))) + 1) + 2 and g(x) = -5(1|(-5x + 4)| + 1)^2 + 3(1|(-5x + 4)| + 1) + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -4(3(-1cos(1\u03c0(2|x + 2|)) - 1)^2 + 2(-1cos(1\u03c0(2|x + 2|)) - 1)) - 5 and g(x) = (1(3|(-1x + 4) - 1|) + 1)/(2(3|(-1x + 4) - 1|) - 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 3|(2((-3(3x - 3) - 3)/(-1(3x - 3) + 3))^2 - 1((-3(3x - 3) - 3)/(-1(3x - 3) + 3)) + 5) - 1| + 1 and g(x) = 1|(5(2x - 5)^2 - 3(2x - 5) - 4)| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1(1|(1e^(1((2x)/(-1x - 1))) - 3) + 1| + 2) - 5 and g(x) = -3(-1e^(2(-2|x + 1| - 3)))^2 + 5(-1e^(2(-2|x + 1| - 3))) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2e^(2(-1(-3|((-1x - 3)/(-3x + 4)) - 3|)^2 + 4(-3|((-1x - 3)/(-3x + 4)) - 3|) + 5)) - 2 and g(x) = -2sin(3\u03c0((-2(-5x^2 - 3x + 5) - 3)/(-1(-5x^2 - 3x + 5) - 1)) - 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 2(-2((-1x - 1)/(-2x - 2)) + 5)^2 - 2(-2((-1x - 1)/(-2x - 2)) + 5) + 4 and g(x) = (2(-2|x - 2| + 3) - 2)/(2(-2|x - 2| + 3)) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 5(-3|(1e^(1(-1x^2 + 1x + 4)) + 1) - 2| + 2) + 5 and g(x) = 5(-1|(1x^2 - 3x + 3) - 3|) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1((-1(3e^(1(-2x + 2)) + 3) - 3)/(-2(3e^(1(-2x + 2)) + 3) + 1))^2 + 4((-1(3e^(1(-2x + 2)) + 3) - 3)/(-2(3e^(1(-2x + 2)) + 3) + 1)) - 5 and g(x) = 3|(2sin(3\u03c0((-3x + 2)/(-2x - 3))) - 3)| - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3((-1(-3e^(1x) - 1))/(-3(-3e^(1x) - 1) - 1))^2 - 5((-1(-3e^(1x) - 1))/(-3(-3e^(1x) - 1) - 1)) + 3 and g(x) = -5(-1|x + 2| + 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3(3(-1cos(2\u03c0x - 3) - 2)^2 - 1(-1cos(2\u03c0x - 3) - 2) + 3) - 2 and g(x) = 3e^(2((1x - 2)/(-3x + 1))) + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (2(2|(-2(1x + 1)^2 - 1(1x + 1) + 1) + 1| - 3) + 1)/(-1(2|(-2(1x + 1)^2 - 1(1x + 1) + 1) + 1| - 3) - 3) and g(x) = 5(-1e^(-1(-1|x + 2| - 2)) + 3)^2 - 3(-1e^(-1(-1|x + 2| - 2)) + 3) - 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2|((1(1cos(2\u03c0(-1x + 5) + 3) - 1))/(2(1cos(2\u03c0(-1x + 5) + 3) - 1))) + 2| + 2 and g(x) = -3|(5((-1x + 2)/(-2x - 2)) + 1) - 2| + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 1((-1(5(-1|x + 3| + 2)^2 + 4(-1|x + 3| + 2) + 2) - 3)/(-2(5(-1|x + 3| + 2)^2 + 4(-1|x + 3| + 2) + 2) - 1)) and g(x) = (2(-1|(-2x - 3) - 1| - 2) + 3)/(3(-1|(-2x - 3) - 1| - 2) + 1) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (2(-1|(1e^(1(-4x + 3)) - 2)| + 3) - 2)/(-3(-1|(1e^(1(-4x + 3)) - 2)| + 3) + 4) and g(x) = 3((1(-1sin(2\u03c0x - 1) - 2) + 3)/(2(-1sin(2\u03c0x - 1) - 2)))^2 + 3((1(-1sin(2\u03c0x - 1) - 2) + 3)/(2(-1sin(2\u03c0x - 1) - 2))) + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1e^(1(3|(5x^2 + 4x) + 1| - 2)) + 1 and g(x) = (2(-1x + 4) - 3)/(-1(-1x + 4) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2|(-3((-3(2x - 1) + 2)/(-2(2x - 1)))^2 - 3((-3(2x - 1) + 2)/(-2(2x - 1)))) - 1| and g(x) = -2|((-1(-1cos(1\u03c0x + 3) + 3) - 2)/(2(-1cos(1\u03c0x + 3) + 3) + 1)) + 3| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2(3|(-1(1cos(1\u03c0x - 2) - 3)^2 + 3(1cos(1\u03c0x - 2) - 3) - 2) - 1| + 2) + 3 and g(x) = 2((-2(1x^2 - 1x - 1) - 2)/(-3(1x^2 - 1x - 1) - 2)) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (1(-4(1(-1|x + 2| + 3))^2) - 2)/(-3(-4(1(-1|x + 2| + 3))^2) + 2) and g(x) = (-3(2(-1x^2 + 3x + 3) + 1) + 2)/(-1(2(-1x^2 + 3x + 3) + 1) - 3) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -4((-3(-1(-2|x - 3| - 2)^2 - 3(-2|x - 3| - 2) + 3) + 3)/(2(-1(-2|x - 3| - 2)^2 - 3(-2|x - 3| - 2) + 3) + 3)) and g(x) = (1(-4(2e^(-2x) + 2)^2 - 3(2e^(-2x) + 2) - 3) - 2)/(1(-4(2e^(-2x) + 2)^2 - 3(2e^(-2x) + 2) - 3) - 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -4(-5(-1sin(3\u03c0x - 2) - 1)^2 - 3(-1sin(3\u03c0x - 2) - 1) - 2) + 2 and g(x) = (-2(-2x^2 + 2x + 2) + 1)/(1(-2x^2 + 2x + 2) + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -3|(5(-4((3x - 1)/(1x - 2)) - 3)^2 + 1(-4((3x - 1)/(1x - 2)) - 3) + 2) + 2| - 1 and g(x) = -4(-4((2x)/(-1x - 1)) - 4)^2 - 2(-4((2x)/(-1x - 1)) - 4) + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -4((1(-3x) - 3)/(-1(-3x) + 2))^2 - 1((1(-3x) - 3)/(-1(-3x) + 2)) + 4 and g(x) = (-3(2|x - 3| + 1) - 1)/(3(2|x - 3| + 1) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1((-2(2x - 2) + 2)/(-3(2x - 2) + 2))^2 - 3((-2(2x - 2) + 2)/(-3(2x - 2) + 2)) + 4 and g(x) = (-2(-2sin(1\u03c0x - 3) + 1) - 2)/(-1(-2sin(1\u03c0x - 3) + 1)) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-1(-2e^(-2(1|x + 2|)) + 1) + 3)/(2(-2e^(-2(1|x + 2|)) + 1) + 3) and g(x) = 1|(2x^2 + 4x + 2)| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (1(-1sin(2\u03c0(-2x^2 - 3x - 5) - 2) - 2) - 2)/(-2(-1sin(2\u03c0(-2x^2 - 3x - 5) - 2) - 2) + 3) and g(x) = -2|(3x - 2)| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3((-1(-2(-1|x - 3|)^2 + 1(-1|x - 3|) + 1) - 2)/(1(-2(-1|x - 3|)^2 + 1(-1|x - 3|) + 1))) + 1 and g(x) = -3(-5(1e^(1x) - 3))^2 + 3(-5(1e^(1x) - 3)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (1(-1(-5(2|x + 2| + 3)^2 - 1(2|x + 2| + 3) - 1) + 2) - 3)/(2(-1(-5(2|x + 2| + 3)^2 - 1(2|x + 2| + 3) - 1) + 2) + 1) and g(x) = (1(-3(-1x - 2)^2 + 2(-1x - 2) + 3) + 2)/(-3(-3(-1x - 2)^2 + 2(-1x - 2) + 3) - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-1(3cos(1\u03c0(-1(-5x^2 + 5x - 4) + 4) + 3) + 2))/(2(3cos(1\u03c0(-1(-5x^2 + 5x - 4) + 4) + 3) + 2)) and g(x) = 2(-1((-2x + 2)/(-3x - 2))^2 - 1((-2x + 2)/(-3x - 2))) + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1cos(1\u03c0(-2(1|x|)^2 - 3(1|x|) - 1) - 3) - 1 and g(x) = (1(2sin(2\u03c0x - 1) - 3) + 1)/(1(2sin(2\u03c0x - 1) - 3) - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (3(2|(-4(3x)^2 + 4(3x) - 2) + 2| - 1) + 3)/(-1(2|(-4(3x)^2 + 4(3x) - 2) + 2| - 1) - 1) and g(x) = 1(-5((3x - 2)/(1x + 1)) + 4)^2 - 5(-5((3x - 2)/(1x + 1)) + 4) - 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (2(-3|(4(-4x - 4)^2 - 2(-4x - 4) + 2) + 3| + 3) - 1)/(-2(-3|(4(-4x - 4)^2 - 2(-4x - 4) + 2) + 3| + 3) - 1) and g(x) = 2((3(-3x^2 + 4x + 3) - 3)/(1(-3x^2 + 4x + 3))) - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 2(-5((-1(3e^(-1x) - 1) + 1)/(-2(3e^(-1x) - 1) - 3))^2 + 4((-1(3e^(-1x) - 1) + 1)/(-2(3e^(-1x) - 1) - 3)) - 1) - 2 and g(x) = (-1(-3|(-3x^2 + 4x + 3) - 2| - 1) + 3)/(3(-3|(-3x^2 + 4x + 3) - 2| - 1) + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3|(-2(2((-2x - 3)/(-3x - 1)) - 5)^2 - 3(2((-2x - 3)/(-3x - 1)) - 5) + 1)| + 3 and g(x) = (-2(-3(-1e^(1x) + 3) + 1))/(-1(-3(-1e^(1x) + 3) + 1) - 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1|(4((-2(-4x - 2) + 3)/(3(-4x - 2) - 1))^2 - 3) + 1| - 1 and g(x) = (2(-2(2|x + 1| + 2)^2 + 5(2|x + 1| + 2) - 5) - 2)/(2(-2(2|x + 1| + 2)^2 + 5(2|x + 1| + 2) - 5)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2|(-4((-2x - 2)/(3x + 2))^2 - 1((-2x - 2)/(3x + 2)) - 1) - 1| + 3 and g(x) = 1e^(1((1x + 2)/(-2x + 1))) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -4(2(-1|((2x)/(-1x - 3)) - 2| + 3)^2 - 4(-1|((2x)/(-1x - 3)) - 2| + 3) + 1) - 4 and g(x) = 5(-2e^(-1((2x + 2)/(3x))) + 2)^2 - 3(-2e^(-1((2x + 2)/(3x))) + 2) - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2cos(2\u03c0(3|(4((2x + 2)/(-1x - 3)) - 3) + 2| - 2) + 2) + 1 and g(x) = (-3(1(5x^2 + 1x - 2) - 1) + 1)/(2(1(5x^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": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1(1e^(-1((3x - 1)/(-1x + 2)))) - 5 and g(x) = (2(-2sin(3\u03c0x) - 3) + 2)/(1(-2sin(3\u03c0x) - 3) - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1(-3|((-3x + 1)/(-1x + 2)) - 3| + 3)^2 - 5(-3|((-3x + 1)/(-1x + 2)) - 3| + 3) + 2 and g(x) = (1(-1x - 1) + 3)/(2(-1x - 1) - 1) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -1(-2|(-1((-1x)/(2x + 3))^2 + 1((-1x)/(2x + 3)) + 2) - 1| + 1) - 2 and g(x) = 1|(4((1x + 2)/(-2x - 2))^2 - 5((1x + 2)/(-2x - 2)) + 1) - 1| - 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1|(2(-1e^(1((3x - 1)/(3x - 2))) + 1) + 4) + 3| - 2 and g(x) = -3((3(1x^2 + 3) + 1)/(1(1x^2 + 3))) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 1e^(1(2(-2|x + 3| - 2))) + 2 and g(x) = -3((2x + 1)/(-3x + 2))^2 + 5((2x + 1)/(-3x + 2)) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1((2(-1|(5x^2 - 5x + 1) - 2| + 3) + 1)/(-3(-1|(5x^2 - 5x + 1) - 2| + 3) + 1)) - 5 and g(x) = -3sin(2\u03c0(-1(-1|x + 2|) + 5)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -2sin(2\u03c0(-4((1(1|x - 3| - 3))/(-2(1|x - 3| - 3))) + 4)) - 3 and g(x) = -2e^(2(-1((-1x)/(1x - 3))^2 - 2((-1x)/(1x - 3)) + 2)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2|(1((-2(-1x - 4) + 3)/(1(-1x - 4) + 2))^2 - 2((-2(-1x - 4) + 3)/(1(-1x - 4) + 2)) - 5)| + 1 and g(x) = 2cos(3\u03c0(4(1x^2 + 4x - 1) + 1) - 1) + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -3((-2(-1cos(1\u03c0x + 1) + 2) + 2)/(-1(-1cos(1\u03c0x + 1) + 2) - 3))^2 + 5((-2(-1cos(1\u03c0x + 1) + 2) + 2)/(-1(-1cos(1\u03c0x + 1) + 2) - 3)) + 3 and g(x) = (-1(-3|x + 3| + 2) + 3)/(2(-3|x + 3| + 2) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1|(5(3e^(1(-2x^2 + 5x + 2)) + 1) - 4) + 2| + 1 and g(x) = 1|((1(-5x - 4) - 3)/(-1(-5x - 4) - 1)) - 1| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -3(-4(-2|x + 1| + 1) + 3)^2 - 3(-4(-2|x + 1| + 1) + 3) - 2 and g(x) = -3e^(1((3x + 3)/(-2x))) + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the third intersection point (counting from left to right) between the functions f(x) = 3|(-1((-3x + 3)/(-1x)) - 1)| - 1 and g(x) = 1((-3x + 3)/(-1x + 2)) + 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1((2(1(1|x - 2| - 3)^2 + 5(1|x - 2| - 3) - 1) - 1)/(1(1(1|x - 2| - 3)^2 + 5(1|x - 2| - 3) - 1))) and g(x) = -5((-3(-4x + 3) + 3)/(-1(-4x + 3) - 3))^2 - 2((-3(-4x + 3) + 3)/(-1(-4x + 3) - 3)) + 5 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-1(1(3|(2x^2 - 4x + 3) - 2| - 3) - 1) + 2)/(-2(1(3|(2x^2 - 4x + 3) - 2| - 3) - 1) + 1) and g(x) = 1sin(1\u03c0(1|(5x - 3) - 3| - 1) - 1) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2|((-1(3x + 5) - 2)/(-3(3x + 5) - 2)) - 2| + 3 and g(x) = -1|(2e^(1x) + 3) - 2| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-1(-1sin(3\u03c0(2|x - 1| + 3) + 2) - 2) + 1)/(1(-1sin(3\u03c0(2|x - 1| + 3) + 2) - 2) - 2) and g(x) = -2(-5x + 2)^2 + 1(-5x + 2) + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2|(4(1((3x + 3)/(1x - 2)) - 1)^2 + 4(1((3x + 3)/(1x - 2)) - 1) + 5) + 2| and g(x) = (-3(5(-3x - 1)^2 - 1) - 1)/(-2(5(-3x - 1)^2 - 1) + 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 2e^(2(-5(-4x^2 + 1x + 3) + 5)) - 3 and g(x) = 2e^(-2((2x + 3)/(3x + 1))) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-3(-1(-1(-2|x - 3| - 2))^2 - 5(-1(-2|x - 3| - 2)) + 5) + 2)/(1(-1(-1(-2|x - 3| - 2))^2 - 5(-1(-2|x - 3| - 2)) + 5) + 1) and g(x) = -3((1(-2x^2 + 2x + 4) - 2)/(1(-2x^2 + 2x + 4) + 2)) - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(5(-1|(4x^2 - 4x + 2) - 1| + 1) + 2) + 1)/(1(5(-1|(4x^2 - 4x + 2) - 1| + 1) + 2)) and g(x) = -2sin(2\u03c0((2(1x - 5) + 3)/(-1(1x - 5) - 3))) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3|((-3(-3e^(-1(-2x - 1)) - 1) - 2)/(-3(-3e^(-1(-2x - 1)) - 1) - 3))| + 3 and g(x) = 1((-3(1|x + 2| + 3) + 1)/(-1(1|x + 2| + 3) + 3))^2 - 1((-3(1|x + 2| + 3) + 1)/(-1(1|x + 2| + 3) + 3)) - 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1e^(-2((1(3|x + 2| - 3) - 2)/(2(3|x + 2| - 3) - 1))) + 2 and g(x) = 3(-3e^(-2x) - 3) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1|(3((-1x - 1)/(3x + 2)) - 4)| - 3 and g(x) = -3cos(2\u03c0((1x - 3)/(-3x - 3)) - 3) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 3(-2|(1((2x - 3)/(3x))^2 - 1((2x - 3)/(3x)) - 1)|) - 2 and g(x) = (1(5(-1x - 1)^2 + 4(-1x - 1) - 4) + 3)/(2(5(-1x - 1)^2 + 4(-1x - 1) - 4)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -2|(-2(-2e^(-1((3x - 1)/(-2x + 3))) - 2) + 2) + 2| + 2 and g(x) = (3(-2(-1|x - 2| + 3) + 3) + 3)/(3(-2(-1|x - 2| + 3) + 3) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1cos(2\u03c0(2|((-2x + 2)/(1x + 1))| + 1) - 2) + 2 and g(x) = 2e^(2(4x)) - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -1(2((-1x + 2)/(-3x))^2 - 5((-1x + 2)/(-3x)) - 1) + 4 and g(x) = (3(-3sin(1\u03c0x + 1)) - 2)/(1(-3sin(1\u03c0x + 1)) + 3) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-1(-4(-2|(-2sin(2\u03c0x - 3) + 1) + 2| - 2)^2 - 4(-2|(-2sin(2\u03c0x - 3) + 1) + 2| - 2) + 3) + 2)/(2(-4(-2|(-2sin(2\u03c0x - 3) + 1) + 2| - 2)^2 - 4(-2|(-2sin(2\u03c0x - 3) + 1) + 2| - 2) + 3) - 3) and g(x) = -2e^(-2(5(1x - 5)^2 + 5(1x - 5) + 1)) + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1e^(-2(-1(-3|x + 2| + 3) - 2)) - 2 and g(x) = 1cos(1\u03c0(-1x^2 + 1x + 2) + 2) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-1(2|(1sin(2\u03c0(5x^2 - 1x + 5)) + 2) - 2|))/(-2(2|(1sin(2\u03c0(5x^2 - 1x + 5)) + 2) - 2|) - 1) and g(x) = -4((-1(2e^(-2x) - 2) - 3)/(-2(2e^(-2x) - 2))) - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3(4(-1|((-1x - 3)/(-2x - 2)) + 3| + 2) + 3)^2 + 1(4(-1|((-1x - 3)/(-2x - 2)) + 3| + 2) + 3) + 1 and g(x) = (1(-1|(-5x^2 - 3x + 4) - 1| + 1) - 3)/(-2(-1|(-5x^2 - 3x + 4) - 1| + 1) - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1(-3|((-3(-3x^2 - 1x + 4) + 2)/(-2(-3x^2 - 1x + 4) + 1)) - 1| + 2) + 4 and g(x) = -2(5(2sin(1\u03c0x) + 2) + 1)^2 + 1(5(2sin(1\u03c0x) + 2) + 1) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2(-1e^(-2(2(1|x - 3| - 2)^2 - 1(1|x - 3| - 2) - 4)) - 1) and g(x) = 2|((-1(-4x^2 + 4x + 3) - 1)/(-1(-4x^2 + 4x + 3) - 2)) - 2| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -3|(3((-1x)/(1x + 3))^2 - 4((-1x)/(1x + 3)) - 5) + 2| and g(x) = (1(-2x + 2))/(-1(-2x + 2) - 2) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1e^(1((-1(-3|x - 2| + 3))/(3(-3|x - 2| + 3) + 1))) - 3 and g(x) = 1((-3x - 2)/(1x + 3)) + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1(3(-1|((-1x)/(-1x - 2))| + 3) - 2)^2 - 2(3(-1|((-1x)/(-1x - 2))| + 3) - 2) - 2 and g(x) = -4(-1e^(-1(1x)) - 2)^2 - 2(-1e^(-1(1x)) - 2) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 2sin(3\u03c0(4(2(2|x| - 1)^2 + 2(2|x| - 1) - 3) + 2) + 3) + 1 and g(x) = 2sin(1\u03c0(3((2x + 3)/(-1x))^2 + 3((2x + 3)/(-1x)) + 4)) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 5(3|((2x - 1)/(-1x - 1)) + 3| - 1) + 3 and g(x) = -1((1x - 3)/(-2x - 1)) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-1(-2|(4(3e^(1x) - 2) + 3) + 1| - 3))/(2(-2|(4(3e^(1x) - 2) + 3) + 1| - 3) - 1) and g(x) = -1((3(3|x + 2| + 1) - 3)/(-2(3|x + 2| + 1) + 3))^2 - 5((3(3|x + 2| + 1) - 3)/(-2(3|x + 2| + 1) + 3)) - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 2((-3(-1|x + 3| + 2) - 2)/(-2(-1|x + 3| + 2) - 1))^2 + 4((-3(-1|x + 3| + 2) - 2)/(-2(-1|x + 3| + 2) - 1)) + 4 and g(x) = 3((-2x - 1)/(3x - 1))^2 - 5((-2x - 1)/(3x - 1)) - 5 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-1(1|(4(-1x + 1)^2 - 4(-1x + 1)) + 2| - 1))/(-3(1|(4(-1x + 1)^2 - 4(-1x + 1)) + 2| - 1) + 4) and g(x) = -2|(3(-1x^2 - 3x + 1) + 3)| - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(5(3e^(-2((3x + 2)/(2x + 1))) - 2)^2 - 1(3e^(-2((3x + 2)/(2x + 1))) - 2)) + 2| + 3 and g(x) = (2(1sin(3\u03c0(-1x - 1) + 1) - 2) - 3)/(-2(1sin(3\u03c0(-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": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1e^(-1((1(-3|x + 1| + 2))/(3(-3|x + 1| + 2)))) + 3 and g(x) = 3(4x^2 - 3) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 3|(1((-1x + 1)/(3x - 2)) + 4) + 2| - 3 and g(x) = (3(-2|x + 1| - 1))/(-1(-2|x + 1| - 1) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-1(-2|(-3e^(2x)) - 2| - 1) - 2)/(3(-2|(-3e^(2x)) - 2| - 1)) and g(x) = 2(4x^2 + 3x - 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1((-1(-1sin(3\u03c0(1|x| - 3)) - 3) + 2)/(-2(-1sin(3\u03c0(1|x| - 3)) - 3) + 3))^2 - 4((-1(-1sin(3\u03c0(1|x| - 3)) - 3) + 2)/(-2(-1sin(3\u03c0(1|x| - 3)) - 3) + 3)) + 5 and g(x) = 4((3(3e^(1x) + 2) + 3)/(2(3e^(1x) + 2)))^2 - 5((3(3e^(1x) + 2) + 3)/(2(3e^(1x) + 2))) - 4 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -3(2(1|((-3x - 3)/(-2x + 1)) + 2|)^2 - 2(1|((-3x - 3)/(-2x + 1)) + 2|) - 5) + 3 and g(x) = -4(-2(1e^(2x) + 1) + 5)^2 + 1(-2(1e^(2x) + 1) + 5) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2|(-3e^(1((1(5x - 4) - 1)/(2(5x - 4) + 1))) - 1) + 1| - 3 and g(x) = (-3(2(-1|x - 3| + 1) + 3) + 1)/(-1(2(-1|x - 3| + 1) + 3)) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -2e^(2(1(-3((-1x)/(-2x + 3))^2 + 5) + 2)) - 3 and g(x) = -4(3(2|x + 1| - 3)^2 + 1(2|x + 1| - 3) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1((-1(-1(1|x| - 2)^2 - 4(1|x| - 2) + 5))/(-1(-1(1|x| - 2)^2 - 4(1|x| - 2) + 5) + 3)) - 2 and g(x) = 1(-1|((3x - 2)/(-1x + 2)) - 2| - 2)^2 + 4(-1|((3x - 2)/(-1x + 2)) - 2| - 2) + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (1(-1e^(1(-4x^2 - 3x + 2))) + 3)/(-1(-1e^(1(-4x^2 - 3x + 2))) + 3) and g(x) = (-1(-2cos(3\u03c0x - 3) - 3) - 1)/(-2(-2cos(3\u03c0x - 3) - 3) + 3) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (-3(5(1|x - 3| + 3)^2 - 3(1|x - 3| + 3)) + 2)/(-1(5(1|x - 3| + 3)^2 - 3(1|x - 3| + 3)) + 3) and g(x) = (-3(2|x - 2| - 1))/(-3(2|x - 2| - 1) - 2) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -5(5((2(-3|x - 1| - 1) + 1)/(-2(-3|x - 1| - 1) - 1)) - 2)^2 - 3(5((2(-3|x - 1| - 1) + 1)/(-2(-3|x - 1| - 1) - 1)) - 2) - 3 and g(x) = 2(-2|(4x^2 - 2x - 4) - 2| - 1) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (1(-1(3|(-1e^(2x) - 2) + 2|)^2 - 3) + 1)/(-3(-1(3|(-1e^(2x) - 2) + 2|)^2 - 3) + 4) and g(x) = 1(1|(-3x - 2)| + 1)^2 + 1(1|(-3x - 2)| + 1) - 5 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -3e^(-1(2|(-4((1x + 2)/(-3x - 2)) - 1) - 1| - 1)) + 2 and g(x) = (3(-3(-1x^2 + 2x - 3) - 2) - 1)/(-2(-3(-1x^2 + 2x - 3) - 2) + 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (2(-1(3|(-2x) + 2| - 2)^2 - 5(3|(-2x) + 2| - 2)) - 3)/(-1(-1(3|(-2x) + 2| - 2)^2 - 5(3|(-2x) + 2| - 2)) - 1) and g(x) = -2(-3|(3cos(2\u03c0x - 3) - 1) - 1| - 2)^2 + 1(-3|(3cos(2\u03c0x - 3) - 1) - 1| - 2) + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -3(5((1(1|x| - 3) + 2)/(-1(1|x| - 3) - 1)) + 2)^2 + 1(5((1(1|x| - 3) + 2)/(-1(1|x| - 3) - 1)) + 2) + 5 and g(x) = (-2(3(2x + 3)^2 + 5(2x + 3)) + 2)/(1(3(2x + 3)^2 + 5(2x + 3)) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-3(2e^(-1(-3|x| + 1)) - 1) + 3)/(-3(2e^(-1(-3|x| + 1)) - 1) - 1) and g(x) = -1sin(3\u03c0((1x - 3)/(2x - 3)) + 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1(4(-3|((-2x + 3)/(2x + 2)) - 2| + 1) + 2)^2 + 3 and g(x) = -2(-2(3sin(3\u03c0x - 3) + 2)^2 + 3(3sin(3\u03c0x - 3) + 2) - 1) + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -4(-3e^(-2(4((-2x + 2)/(-1x - 1))^2 + 3((-2x + 2)/(-1x - 1)) - 1)) + 2) + 3 and g(x) = 2|(4((2x + 1)/(-1x + 3)) - 5) + 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -4(1|((-3(3x - 3) - 2)/(2(3x - 3) + 1)) - 3| - 2)^2 - 2(1|((-3(3x - 3) - 2)/(2(3x - 3) + 1)) - 3| - 2) + 4 and g(x) = 2cos(1\u03c0((-3(1x + 5) + 3)/(2(1x + 5) - 3)) - 3) - 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2|(2(2e^(1x) + 1) + 3)| + 1 and g(x) = -1((1x - 1)/(-2x + 3))^2 + 2((1x - 1)/(-2x + 3)) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -5(3|(-2((-1x + 3)/(2x + 1)) + 1) + 3|)^2 - 3(3|(-2((-1x + 3)/(2x + 1)) + 1) + 3|) and g(x) = 1|(-1(5x^2 + 2x + 4) + 5) + 2| - 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-2(2(-2|(4x + 4) - 2| + 3)^2 - 5(-2|(4x + 4) - 2| + 3) + 3) + 2)/(-3(2(-2|(4x + 4) - 2| + 3)^2 - 5(-2|(4x + 4) - 2| + 3) + 3) + 1) and g(x) = -2((3(-2|x + 1| - 2) - 3)/(1(-2|x + 1| - 2) + 3)) + 4 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2sin(2\u03c0(1((-2(5x - 3) - 3)/(-1(5x - 3) - 1))^2 + 4((-2(5x - 3) - 3)/(-1(5x - 3) - 1)) - 5) - 3) - 3 and g(x) = -1|((3(1x - 3))/(1(1x - 3) - 2)) - 2| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -3(1|((1x)/(-1x + 2)) + 1| - 2) + 5 and g(x) = 2|(-4x^2 - 1x - 2) + 2| + 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -2|(1e^(-1(1(-5x^2 - 2x + 5) + 2)) + 1)| - 1 and g(x) = 3(-4((-1x - 2)/(1x - 2)) + 4)^2 + 4(-4((-1x - 2)/(1x - 2)) + 4) - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2((-1(-3|(-5x^2 + 5x + 3) - 2| - 2) - 2)/(1(-3|(-5x^2 + 5x + 3) - 2| - 2) + 3)) and g(x) = -2e^(-1(-4((3x + 3)/(2x - 3))^2 + 5((3x + 3)/(2x - 3)) + 1)) + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1(1((3x - 1)/(-3x + 4)) - 4)^2 - 4(1((3x - 1)/(-3x + 4)) - 4) - 3 and g(x) = 1((2x - 1)/(-3x + 4))^2 - 1((2x - 1)/(-3x + 4)) + 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (3(-3|(1x + 1) + 3| + 1) + 1)/(2(-3|(1x + 1) + 3| + 1) + 2) and g(x) = (-2(-1|x - 1| - 3) + 1)/(3(-1|x - 1| - 3) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 3e^(2((1(-1(2x - 1)^2 - 2(2x - 1) + 1) - 3)/(1(-1(2x - 1)^2 - 2(2x - 1) + 1)))) - 3 and g(x) = -4(-3((-2x)/(1x + 1)) - 1)^2 + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -3e^(1(-2|((1(3x^2 - 2) + 3)/(-1(3x^2 - 2) + 2)) + 1| - 1)) - 1 and g(x) = 1|((-3(5x + 2))/(2(5x + 2) - 3)) + 1| - 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3|(-1e^(-1((3(1x^2 + 3) + 3)/(-2(1x^2 + 3) - 3))) + 2) - 2| - 3 and g(x) = (2(1(1|x| - 3) - 3) + 3)/(-1(1(1|x| - 3) - 3) - 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(-1(5(-1|x + 2| - 1) - 1)^2 + 2(5(-1|x + 2| - 1) - 1) + 4) - 3)/(-1(-1(5(-1|x + 2| - 1) - 1)^2 + 2(5(-1|x + 2| - 1) - 1) + 4) + 3) and g(x) = (-2(1(-3x^2 - 1x) + 5) + 2)/(3(1(-3x^2 - 1x) + 5) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-2(-3(-3|x + 3| - 1)^2 - 1(-3|x + 3| - 1) - 2) + 2)/(2(-3(-3|x + 3| - 1)^2 - 1(-3|x + 3| - 1) - 2) - 3) and g(x) = 3|(-1x^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": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2((2(2(-2|x| - 1)^2 - 1(-2|x| - 1) + 3) + 3)/(2(2(-2|x| - 1)^2 - 1(-2|x| - 1) + 3))) - 3 and g(x) = -2(3((-3x + 1)/(2x + 3))^2 - 4((-3x + 1)/(2x + 3)) - 3) + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1(-4((-1x + 1)/(2x)) + 2)^2 - 3 and g(x) = -1(-3cos(2\u03c0x + 1))^2 - 1(-3cos(2\u03c0x + 1)) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1|(-5((-2(1x^2 - 5x) + 3)/(-1(1x^2 - 5x))) + 1) - 2| - 2 and g(x) = 5(-4(2|x - 1|) - 5)^2 - 1(-4(2|x - 1|) - 5) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3((-1(-2|(1x^2 - 2x + 5) - 2| - 3) + 1)/(2(-2|(1x^2 - 2x + 5) - 2| - 3) - 1)) + 2 and g(x) = 1((2(3e^(1x) - 3) + 2)/(-3(3e^(1x) - 3) + 1)) + 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (1(-1(3e^(2x) + 2) + 1) + 2)/(1(-1(3e^(2x) + 2) + 1) - 3) and g(x) = (2(-1x^2 + 2x + 4) + 2)/(2(-1x^2 + 2x + 4) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (2(-3(-1e^(-1x) + 2) - 2) + 2)/(3(-3(-1e^(-1x) + 2) - 2) + 2) and g(x) = (2(1x - 5) - 2)/(1(1x - 5) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1|((-1(5(-1x + 4)^2 - 3(-1x + 4) - 4) - 1)/(3(5(-1x + 4)^2 - 3(-1x + 4) - 4) + 1)) - 3| + 3 and g(x) = -3sin(1\u03c0(2((-3x)/(1x + 3)) - 4) - 1) + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -5((1(-4x^2 + 2x + 3) - 3)/(-1(-4x^2 + 2x + 3) - 3)) + 5 and g(x) = -1|((1x)/(3x + 2)) - 2| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -4(-2(1|((-1x)/(2x + 2)) + 1| + 1)^2 + 3(1|((-1x)/(2x + 2)) + 1| + 1)) + 2 and g(x) = -3(1(2e^(2x) + 3)^2 - 3(2e^(2x) + 3) - 5) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 5(3|((-2(-4x - 5) + 1)/(-1(-4x - 5) - 3)) - 2|)^2 + 1(3|((-2(-4x - 5) + 1)/(-1(-4x - 5) - 3)) - 2|) - 5 and g(x) = 4(-2|((3x + 3)/(3x)) - 1|) meet 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 first intersection point (counting from left to right) between the functions f(x) = 1|(2((-2(-4x^2 + 5x - 5) + 3)/(-1(-4x^2 + 5x - 5) + 2)) + 1) + 1| - 1 and g(x) = -5((2(-1|x + 2| + 3) + 2)/(3(-1|x + 2| + 3) + 1)) - 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1sin(2\u03c0(5(1|(3x^2 + 1x + 3) + 3| - 3) + 2)) - 2 and g(x) = (2(4(-1x^2 - 5x + 2) + 1) + 2)/(2(4(-1x^2 - 5x + 2) + 1) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -5(3|(2sin(2\u03c0x + 3) - 3) + 3| - 3) + 3 and g(x) = 4(3|x - 2| + 1)^2 + 5(3|x - 2| + 1) - 4 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3(-2(-2|((2x + 3)/(-3x + 1)) + 3|)^2 - 5) - 4 and g(x) = (-2(1|(-3x^2 - 3x - 4) + 3| - 1) - 3)/(-1(1|(-3x^2 - 3x - 4) + 3| - 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2|((-1(3x + 4) + 3)/(-3(3x + 4) + 2))| - 1 and g(x) = -3(-1|x - 2| - 3)^2 + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1e^(1((1(5x^2 + 4x + 2) - 1)/(-3(5x^2 + 4x + 2) - 1))) - 1 and g(x) = -1|(-4x - 3) + 3| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(-2(2e^(-2x) - 3) + 4))/(1(-2(2e^(-2x) - 3) + 4)) and g(x) = (2(4x^2 - 2) + 2)/(-3(4x^2 - 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -4(2(-2|x - 1| + 2) - 4)^2 - 2(2(-2|x - 1| + 2) - 4) + 3 and g(x) = (3(3cos(2\u03c0x - 3)) - 2)/(-1(3cos(2\u03c0x - 3)) - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (2(-1(2|x - 3| - 3) - 2) - 2)/(2(-1(2|x - 3| - 3) - 2) - 1) and g(x) = 2e^(1(-4x^2 + 2x + 2)) + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -4(1((-2x + 3)/(1x))^2 + 3((-2x + 3)/(1x)) - 1) and g(x) = 1((-3x)/(-2x + 1))^2 - 5((-3x)/(-2x + 1)) - 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -2|(-4(-4((2x + 2)/(-2x - 3)) + 5)^2 - 1(-4((2x + 2)/(-2x - 3)) + 5) + 4) + 3| + 3 and g(x) = 5(-3sin(2\u03c0(-2x - 3) - 2) + 3)^2 - 5(-3sin(2\u03c0(-2x - 3) - 2) + 3) + 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -4(-1|(-3x + 1) - 1|)^2 - 5(-1|(-3x + 1) - 1|) + 2 and g(x) = (-2(2|x - 1|))/(1(2|x - 1|)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3(1cos(3\u03c0(-3|((-2x - 2)/(1x + 3)) + 3| - 2) - 3) - 3) and g(x) = 1e^(-2(-3|(-1x + 4) + 3| + 1)) - 2 meet 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = (3(-3|(-1x - 3)| + 2) - 2)/(-2(-3|(-1x - 3)| + 2)) and g(x) = 1((-3x - 2)/(-3x - 1))^2 - 4((-3x - 2)/(-3x - 1)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -2(-1|((1(2x^2 + 2x + 5) - 2)/(-3(2x^2 + 2x + 5))) - 3| + 2) + 1 and g(x) = 2((1(1|x - 3| - 2) + 2)/(-2(1|x - 3| - 2) + 3)) - 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 3|((-1(-4(-5x + 2)^2 - 4) + 3)/(2(-4(-5x + 2)^2 - 4) + 3))| - 1 and g(x) = 2|(-4((-1x + 1)/(2x - 3))^2 - 1) + 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 5(5(3|((-3x - 1)/(3x + 2)) - 2| + 3)^2 + 4(3|((-3x - 1)/(3x + 2)) - 2| + 3) - 3) - 2 and g(x) = 5(-2((3x + 1)/(2x - 3)) + 1)^2 + 5(-2((3x + 1)/(2x - 3)) + 1) + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 2|(-1((3(4x + 4) - 3)/(3(4x + 4)))^2 + 4((3(4x + 4) - 3)/(3(4x + 4))) - 4) - 1| + 1 and g(x) = -4((2(-2|x + 3| + 2) - 2)/(2(-2|x + 3| + 2) - 1))^2 + 3((2(-2|x + 3| + 2) - 2)/(2(-2|x + 3| + 2) - 1)) + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-3(-2(1sin(1\u03c0x + 1) + 2) + 1) + 1)/(1(-2(1sin(1\u03c0x + 1) + 2) + 1) - 3) and g(x) = (-3(4x^2 - 3x - 1) - 3)/(-1(4x^2 - 3x - 1)) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1(-2|((1(-1e^(2x) + 1) - 2)/(-1(-1e^(2x) + 1) - 1)) + 3| + 1)^2 + 3(-2|((1(-1e^(2x) + 1) - 2)/(-1(-1e^(2x) + 1) - 1)) + 3| + 1) + 2 and g(x) = -1e^(1(-2((-2x)/(3x - 2)) - 2)) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5(-3|(-1e^(1(1x - 2)) - 3)| + 2)^2 + 4(-3|(-1e^(1(1x - 2)) - 3)| + 2) - 4 and g(x) = -4(-2e^(-1((3x - 1)/(-3x - 1))) - 1) - 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1(-1|((-2(1x^2 - 4x) - 1)/(-1(1x^2 - 4x) + 2)) - 2| - 2) - 1 and g(x) = -5(-4((3x - 3)/(1x)))^2 + 2(-4((3x - 3)/(1x))) - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (2(-1(-2|(1x^2 - 2x - 3) + 3| + 2) - 5) - 3)/(3(-1(-2|(1x^2 - 2x - 3) + 3| + 2) - 5)) and g(x) = 1(-1|(-2cos(2\u03c0x + 2) + 2) + 3| + 1)^2 - 3(-1|(-2cos(2\u03c0x + 2) + 2) + 3| + 1) + 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 2((-1(5(-2|x - 3| - 2) + 5) - 2)/(-2(5(-2|x - 3| - 2) + 5) + 3))^2 + 5 and g(x) = 4((-1(-2|x + 1| + 1) + 1)/(1(-2|x + 1| + 1)))^2 - 5((-1(-2|x + 1| + 1) + 1)/(1(-2|x + 1| + 1))) - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -2(3|(5((-2x - 1)/(-1x + 2)) + 1) + 3| - 3)^2 - 4(3|(5((-2x - 1)/(-1x + 2)) + 1) + 3| - 3) - 3 and g(x) = 5((2(3|x|))/(-1(3|x|) + 2)) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (3(1(2(-1|x + 1| + 1)^2 + 5) + 4) + 2)/(2(1(2(-1|x + 1| + 1)^2 + 5) + 4) - 3) and g(x) = (-2(-1(-3e^(-1x) - 3)^2 - 5(-3e^(-1x) - 3) + 1) + 1)/(3(-1(-3e^(-1x) - 3)^2 - 5(-3e^(-1x) - 3) + 1) - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1(-3|((-1(1x^2 + 2x - 2) + 2)/(-1(1x^2 + 2x - 2) + 2)) + 1| + 3) and g(x) = -1e^(-1((-1(-5x^2 + 3x + 4) - 3)/(-2(-5x^2 + 3x + 4) - 2))) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -5((-2(-1|(1x^2 - 5x - 5)| - 1))/(-1(-1|(1x^2 - 5x - 5)| - 1) + 2)) - 3 and g(x) = -4((1(1e^(-2x) - 1) - 1)/(-3(1e^(-2x) - 1)))^2 - 2((1(1e^(-2x) - 1) - 1)/(-3(1e^(-2x) - 1))) + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 2(3(1|((2x - 2)/(1x + 3)) + 2| + 3) + 5)^2 + 2(3(1|((2x - 2)/(1x + 3)) + 2| + 3) + 5) - 4 and g(x) = -3(-3((-3x)/(2x + 2))^2 + 5((-3x)/(2x + 2)) - 1) - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1cos(2\u03c0(3((-1x - 3)/(1x - 3)) + 4) + 3) and g(x) = -5(-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": "-5", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-1(2|(3(-5x + 2)^2 - 4(-5x + 2) - 5)| - 2) - 3)/(-3(2|(3(-5x + 2)^2 - 4(-5x + 2) - 5)| - 2)) and g(x) = 1|(-4(-3x + 2)^2 + 3(-3x + 2) + 4) + 2| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1(4((-3(-3cos(2\u03c0x - 3) + 1) + 2)/(3(-3cos(2\u03c0x - 3) + 1) + 1))^2 - 1((-3(-3cos(2\u03c0x - 3) + 1) + 2)/(3(-3cos(2\u03c0x - 3) + 1) + 1)) - 5) + 3 and g(x) = -3|(-1(-5x - 1)^2 - 3(-5x - 1) + 4) - 2| + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 1(1|((-2x)/(1x + 2)) + 2| - 1) + 2 and g(x) = (-2(-1x^2 - 3x + 5) - 3)/(3(-1x^2 - 3x + 5) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-1(1e^(1(-3x^2 - 2x + 5)) + 2) + 2)/(2(1e^(1(-3x^2 - 2x + 5)) + 2) - 3) and g(x) = -5(3e^(-1x) - 1)^2 + 1(3e^(-1x) - 1) + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (3(1(-2|x - 3| + 1) + 5) - 2)/(-1(1(-2|x - 3| + 1) + 5) + 3) and g(x) = 1sin(2\u03c0(2|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": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1cos(2\u03c0(2|(4x^2 - 3x + 4) - 3| + 2) - 3) - 3 and g(x) = -1sin(1\u03c0(-5x - 5) - 1) - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (3(-1|(-2(3x + 4)^2 + 2(3x + 4) + 3) - 3| + 2))/(-3(-1|(-2(3x + 4)^2 + 2(3x + 4) + 3) - 3| + 2) + 4) and g(x) = (-1(2(1|x - 3| - 3) - 1) - 2)/(1(2(1|x - 3| - 3) - 1)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -5(1e^(1((-3(2x) - 3)/(-2(2x) + 3))) + 1)^2 - 2(1e^(1((-3(2x) - 3)/(-2(2x) + 3))) + 1) + 2 and g(x) = -1|(1e^(-1((-3x)/(3x + 3))) - 1) - 2| + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -2|(1e^(-2((3(5x^2 + 5x + 4) + 2)/(2(5x^2 + 5x + 4) + 3))) + 3) + 3| + 3 and g(x) = -3((-3(-5x^2 + 4x - 2) + 3)/(-1(-5x^2 + 4x - 2) + 2)) - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 3(-2|(4((-1x + 1)/(2x))^2 + 5((-1x + 1)/(2x)) + 5) - 1| - 3) - 1 and g(x) = 3(-1e^(-1(-5x^2 + 2x)) - 1) + 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -5((3(-1cos(1\u03c0x - 2) + 1) - 1)/(1(-1cos(1\u03c0x - 2) + 1) + 2))^2 + 1((3(-1cos(1\u03c0x - 2) + 1) - 1)/(1(-1cos(1\u03c0x - 2) + 1) + 2)) + 5 and g(x) = 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2|(-1(5((-1x + 1)/(-3x + 1)) + 1)^2 + 4(5((-1x + 1)/(-3x + 1)) + 1) - 4) - 1| - 2 and g(x) = (1(3|(3x + 4) + 1| - 1))/(-2(3|(3x + 4) + 1| - 1) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1(-2(-1e^(1x) + 3) + 1)^2 - 1(-2(-1e^(1x) + 3) + 1) and g(x) = -3sin(1\u03c0((-3x)/(2x + 2)) - 1) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2|(2sin(3\u03c0((2(-1x - 2) + 2)/(-3(-1x - 2) + 1)) + 2) - 2) + 2| + 1 and g(x) = 1|(4(5x + 4)^2 + 2(5x + 4) - 2) - 1| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1(4(1cos(1\u03c0((-1x - 3)/(3x + 2)) - 3) + 3) + 4)^2 + 5(4(1cos(1\u03c0((-1x - 3)/(3x + 2)) - 3) + 3) + 4) - 1 and g(x) = -2(1|((-3x + 1)/(-3x + 2)) + 1| - 1) + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -2(3|(5((-3x)/(3x + 3))^2 + 3((-3x)/(3x + 3)) - 5) - 1| + 1) + 4 and g(x) = 1((3(3|x - 1| - 3) + 2)/(-2(3|x - 1| - 3) + 3)) + 5 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -2|(4((-2(5x - 2) + 2)/(-3(5x - 2) + 1))^2 - 5((-2(5x - 2) + 2)/(-3(5x - 2) + 1)) - 5) - 2| - 2 and g(x) = 3cos(1\u03c0(-2|(-1x^2 - 3x + 4) - 1| + 1) + 3) - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1|(1(2((2x + 2)/(2x - 3))^2 + 2((2x + 2)/(2x - 3)) - 3) - 2) + 1| + 2 and g(x) = 5(2((1x - 1)/(1x + 2)) - 4)^2 - 2(2((1x - 1)/(1x + 2)) - 4) + 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (3(1(1(-1e^(1x) + 1)^2 - 5(-1e^(1x) + 1) + 2) + 5) + 3)/(3(1(1(-1e^(1x) + 1)^2 - 5(-1e^(1x) + 1) + 2) + 5) + 3) and g(x) = -2(-1|((-1x - 3)/(-3x + 2)) - 2| + 2) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1(-3((-1(-2|x| - 1))/(-3(-2|x| - 1) - 1)) + 3)^2 + 3 and g(x) = -1(3|((1x + 3)/(-2x + 1)) + 2| - 1) + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 5((3(2|(-2x^2 + 3x) - 1| + 2) + 3)/(1(2|(-2x^2 + 3x) - 1| + 2) + 2)) + 1 and g(x) = 3|((-3(2x + 2) - 3)/(2(2x + 2) + 3)) - 2| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (1(-1cos(3\u03c0(1|x + 1| + 3) + 3) + 1) + 2)/(-3(-1cos(3\u03c0(1|x + 1| + 3) + 3) + 1) - 2) and g(x) = -1((-1x - 2)/(1x)) - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -3|((1(-2(2x)^2 + 5(2x) - 1) + 2)/(2(-2(2x)^2 + 5(2x) - 1) - 1)) + 1| + 2 and g(x) = -1(5(3|x + 3| + 1)^2 + 4(3|x + 3| + 1) - 5) + 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-2(-1(3|x - 3| + 3)^2 + 2(3|x - 3| + 3) - 2) - 1)/(-2(-1(3|x - 3| + 3)^2 + 2(3|x - 3| + 3) - 2) - 3) and g(x) = 2e^(1((-2x + 1)/(2x - 3))) - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2|(2(-1e^(2((2x - 2)/(2x + 3))) + 3)^2 + 2(-1e^(2((2x - 2)/(2x + 3))) + 3) - 5) - 1| - 1 and g(x) = 3e^(-1(4(-1x + 2)^2 + 2(-1x + 2) + 3)) - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (3(3|(-4(5x^2 - 3x - 2) - 5) + 2| - 2) - 1)/(2(3|(-4(5x^2 - 3x - 2) - 5) + 2| - 2) + 2) and g(x) = -3|(1(4x - 5)^2 + 4(4x - 5) + 4) + 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the third x-coordinate (rounded to the nearest integer) where f(x) = 1(3|((-2x + 3)/(-3x - 3))| - 3)^2 - 4(3|((-2x + 3)/(-3x - 3))| - 3) and g(x) = 1|((-1x - 2)/(2x + 1)) + 3| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1(3(-1|((1x - 1)/(2x + 2)) - 1| + 2) - 1)^2 - 3(3(-1|((1x - 1)/(2x + 2)) - 1| + 2) - 1) + 3 and g(x) = 1(4(3|x + 2| + 2) - 3)^2 + 5(4(3|x + 2| + 2) - 3) + 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3|((3(-3e^(-1(4x^2 - 3x + 1)) + 3) - 1)/(1(-3e^(-1(4x^2 - 3x + 1)) + 3) + 3)) - 3| - 3 and g(x) = 3(-1((-1x + 2)/(-1x - 1)) - 3)^2 - 2(-1((-1x + 2)/(-1x - 1)) - 3) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -2(3cos(1\u03c0((-1x + 3)/(1x + 2)) - 1) - 1) - 3 and g(x) = -1(1|x - 2| + 1) - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2(-2|((-2x - 1)/(-1x - 1)) - 2| - 3)^2 + 3(-2|((-2x - 1)/(-1x - 1)) - 2| - 3) - 3 and g(x) = -2|(-3x + 5) + 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (2(-4(-3(1cos(2\u03c0x - 2)) + 4)^2 + 2(-3(1cos(2\u03c0x - 2)) + 4) - 4) + 1)/(3(-4(-3(1cos(2\u03c0x - 2)) + 4)^2 + 2(-3(1cos(2\u03c0x - 2)) + 4) - 4) + 3) and g(x) = -1(-3|(-5x + 4) - 2| + 2)^2 + 5(-3|(-5x + 4) - 2| + 2) + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2(-3cos(3\u03c0(1x + 4) + 1) - 3)^2 + 3(-3cos(3\u03c0(1x + 4) + 1) - 3) - 3 and g(x) = (3(1|x - 2| - 2) + 2)/(-2(1|x - 2| - 2) - 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (1(4(-2cos(1\u03c0(1|x + 2| - 1) - 3) - 2)^2 - 3(-2cos(1\u03c0(1|x + 2| - 1) - 3) - 2) - 2) - 1)/(-1(4(-2cos(1\u03c0(1|x + 2| - 1) - 3) - 2)^2 - 3(-2cos(1\u03c0(1|x + 2| - 1) - 3) - 2) - 2) + 2) and g(x) = 4(-1|(-1x^2 + 3x + 1) - 3| - 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -1cos(3\u03c0((2(-2x^2 - 3x + 3) + 1)/(1(-2x^2 - 3x + 3)))) + 3 and g(x) = 1|(-1x - 5) - 2| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -3|(2e^(2((2x - 3)/(2x - 1))) - 2)| + 3 and g(x) = 2e^(-2((2x - 1)/(-2x - 2))) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-1(3(2(-1|x - 1| - 3) - 2)^2 - 1(2(-1|x - 1| - 3) - 2) + 4) - 3)/(-1(3(2(-1|x - 1| - 3) - 2)^2 - 1(2(-1|x - 1| - 3) - 2) + 4) + 2) and g(x) = 2(2e^(-2((-2x)/(-1x + 3)))) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -4(2|(4((-3x - 3)/(1x + 2))^2 + 3((-3x - 3)/(1x + 2)) + 5) + 1| + 3) and g(x) = (2(-1|(-4x^2 + 1x) + 1|) + 1)/(-3(-1|(-4x^2 + 1x) + 1|) - 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 3((-2(1|(3x - 3) + 2| - 2) - 2)/(2(1|(3x - 3) + 2| - 2) + 2))^2 - 3((-2(1|(3x - 3) + 2| - 2) - 2)/(2(1|(3x - 3) + 2| - 2) + 2)) - 4 and g(x) = (3(1|(5x - 1) + 3| - 2))/(-1(1|(5x - 1) + 3| - 2) + 3) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1|((1(4(5x - 3)^2 - 4(5x - 3) - 4))/(-1(4(5x - 3)^2 - 4(5x - 3) - 4) + 3)) - 3| - 2 and g(x) = -4(2|((-2x - 3)/(-1x - 3)) + 2| + 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 1(1|((-1(2x^2 - 4x + 3) - 1)/(2(2x^2 - 4x + 3) + 3)) + 1| + 1) - 5 and g(x) = -1(2|(-1x + 4) + 2| - 3)^2 - 4(2|(-1x + 4) + 2| - 3) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 1((3(5x + 5) + 1)/(-2(5x + 5) + 3))^2 + 5((3(5x + 5) + 1)/(-2(5x + 5) + 3)) - 4 and g(x) = -1cos(2\u03c0(5x^2 - 3x - 4) - 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1((-1(-1(1|x + 3| + 1)^2 + 1(1|x + 3| + 1) + 5) - 3)/(-3(-1(1|x + 3| + 1)^2 + 1(1|x + 3| + 1) + 5) + 2)) + 1 and g(x) = (-2(1|(-1x^2 + 2x - 3) - 1| + 3) + 1)/(-1(1|(-1x^2 + 2x - 3) - 1| + 3)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (3(3(1e^(1x))^2 - 2(1e^(1x)) - 4) + 2)/(1(3(1e^(1x))^2 - 2(1e^(1x)) - 4) + 1) and g(x) = 4((3x - 3)/(2x + 3)) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -2((1(1e^(-2x) + 3) - 2)/(-1(1e^(-2x) + 3) - 1))^2 + 1((1(1e^(-2x) + 3) - 2)/(-1(1e^(-2x) + 3) - 1)) + 1 and g(x) = -3((3x - 2)/(1x + 2))^2 + 2((3x - 2)/(1x + 2)) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (2(2(-2|x + 3| - 2) + 1) + 2)/(-1(2(-2|x + 3| - 2) + 1) - 2) and g(x) = -1((3x)/(-1x + 2))^2 - 2((3x)/(-1x + 2)) - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2(1sin(2\u03c0(-3(-2|x - 1| - 3)^2 + 2(-2|x - 1| - 3) - 3) + 3) - 2) - 5 and g(x) = (-3(5(-3x^2 + 4x + 1) + 5))/(3(5(-3x^2 + 4x + 1) + 5) - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(-3(-1((-1x - 3)/(2x + 3)) - 2)^2 + 4(-1((-1x - 3)/(2x + 3)) - 2) - 3) + 3| + 2 and g(x) = -2|((-2(-1cos(1\u03c0x - 2)) + 1)/(2(-1cos(1\u03c0x - 2)) - 3)) - 2| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -5((3(-4(-3|x + 3| + 1)^2 - 5(-3|x + 3| + 1) - 5) + 2)/(-3(-4(-3|x + 3| + 1)^2 - 5(-3|x + 3| + 1) - 5) - 2)) + 3 and g(x) = (3(5(-4x^2 + 5x + 3) + 5) - 3)/(-2(5(-4x^2 + 5x + 3) + 5) + 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-3(-2(-1e^(1x) - 1)^2 - 2(-1e^(1x) - 1) - 5) - 2)/(-2(-2(-1e^(1x) - 1)^2 - 2(-1e^(1x) - 1) - 5) + 3) and g(x) = -1(-2e^(1x) + 2)^2 - 4(-2e^(1x) + 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1(2((3x - 2)/(2x - 1))^2 - 5((3x - 2)/(2x - 1)) + 1) + 1 and g(x) = -3e^(2((-3x + 3)/(-2x))) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 1e^(2((2(-5x + 3) + 2)/(2(-5x + 3) + 1))) - 2 and g(x) = (-3(3x + 3) - 3)/(-3(3x + 3) - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (2(2(2|(-2cos(2\u03c0x - 1) + 2) - 1| + 3)^2 + 3(2|(-2cos(2\u03c0x - 1) + 2) - 1| + 3) - 2))/(3(2(2|(-2cos(2\u03c0x - 1) + 2) - 1| + 3)^2 + 3(2|(-2cos(2\u03c0x - 1) + 2) - 1| + 3) - 2) - 2) and g(x) = -1|(1((-2x - 3)/(2x + 2))^2 - 5((-2x - 3)/(2x + 2)) + 2) - 3| + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -1|(-2e^(-1(-1((1x)/(-1x + 2))^2 - 5((1x)/(-1x + 2)) + 5)) + 3) - 3| and g(x) = 3(-1|((3x + 2)/(-2x + 3)) + 2| + 2) - 4 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3sin(2\u03c0(-3|(-3((3x + 2)/(-1x))^2 - 3((3x + 2)/(-1x)) + 1) + 1| + 3)) - 3 and g(x) = 1((3(-2x^2 + 5x - 3) + 1)/(2(-2x^2 + 5x - 3) + 1)) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-1(-5(1e^(2x) + 1)^2 + 5(1e^(2x) + 1) - 2) - 3)/(1(-5(1e^(2x) + 1)^2 + 5(1e^(2x) + 1) - 2) + 3) and g(x) = 4((-2x)/(-2x + 1)) + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1|(-2cos(2\u03c0((-2x - 2)/(-2x - 3)) + 1) - 3) - 2| + 1 and g(x) = 3|((1x)/(1x + 1)) - 3| - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (1(-1(-1|(-4x^2 + 4x - 4) + 3| + 3) + 4) - 3)/(1(-1(-1|(-4x^2 + 4x - 4) + 3| + 3) + 4)) and g(x) = 4((-3(-4x + 3) + 1)/(-3(-4x + 3) - 3))^2 - 2((-3(-4x + 3) + 1)/(-3(-4x + 3) - 3)) - 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1|(2(1cos(3\u03c0((3x + 3)/(-3x)) + 1) - 1)^2 + 1) - 1| - 2 and g(x) = -3((3(3x - 5) - 3)/(-2(3x - 5) + 3))^2 - 4((3(3x - 5) - 3)/(-2(3x - 5) + 3)) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -4((1(-1x^2 - 2x + 1) - 2)/(-1(-1x^2 - 2x + 1) - 3)) + 3 and g(x) = -2|((2x - 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": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -5(-5((3(-1|x - 2| + 1))/(-1(-1|x - 2| + 1) - 3))^2 - 5((3(-1|x - 2| + 1))/(-1(-1|x - 2| + 1) - 3)) + 5) + 3 and g(x) = 3((-3(3x - 1) + 2)/(-3(3x - 1) + 4))^2 - 2((-3(3x - 1) + 2)/(-3(3x - 1) + 4)) + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -3|(-3(1((3x + 1)/(3x)) - 5)^2 - 2(1((3x + 1)/(3x)) - 5)) + 3| + 2 and g(x) = (2(-4(2|x - 3| + 3)^2 + 4(2|x - 3| + 3) - 2) - 3)/(3(-4(2|x - 3| + 3)^2 + 4(2|x - 3| + 3) - 2) + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -3(5((1(-3|x + 2| + 3) - 2)/(-2(-3|x + 2| + 3) + 3)) + 1)^2 + 1(5((1(-3|x + 2| + 3) - 2)/(-2(-3|x + 2| + 3) + 3)) + 1) + 3 and g(x) = (2(1(-5x^2 + 5x - 1) - 3) + 1)/(2(1(-5x^2 + 5x - 1) - 3) + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1|(1(-4((-1x + 3)/(1x + 2)) + 1)^2 - 1(-4((-1x + 3)/(1x + 2)) + 1) - 2)| + 2 and g(x) = 4(3|((2x - 1)/(-1x + 3))| + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 5(-1|(1e^(-1((-2x + 2)/(-2x))) - 3) - 3| - 1) + 1 and g(x) = -4(2|((1x - 1)/(-3x + 4)) - 1| + 1)^2 + 3(2|((1x - 1)/(-3x + 4)) - 1| + 1) - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 4((3(-4(-1|x + 2| - 1)) - 2)/(-3(-4(-1|x + 2| - 1)) + 2))^2 - 2((3(-4(-1|x + 2| - 1)) - 2)/(-3(-4(-1|x + 2| - 1)) + 2)) - 5 and g(x) = (-1(-2(3x + 2)^2 - 2(3x + 2) + 4) + 2)/(-3(-2(3x + 2)^2 - 2(3x + 2) + 4) - 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2(2|(1((-1x + 3)/(-1x + 3)) + 4) + 2|)^2 - 4(2|(1((-1x + 3)/(-1x + 3)) + 4) + 2|) and g(x) = -1|(-4(-3e^(1x))^2 - 1(-3e^(1x)) + 2) + 1| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2|(4((1x - 1)/(1x + 2)) - 5)| - 1 and g(x) = (-3(-2x^2 - 2x - 1) - 1)/(-2(-2x^2 - 2x - 1)) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -4(1|(-1e^(-2((-1x - 3)/(-1x - 1)))) + 3| - 3) + 3 and g(x) = -2|(-1(3sin(2\u03c0x - 1) - 2)^2 - 1(3sin(2\u03c0x - 1) - 2) - 2) + 1| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 2|((-1(-3(-2x - 2)^2 - 5(-2x - 2) - 1) - 2)/(3(-3(-2x - 2)^2 - 5(-2x - 2) - 1) + 1)) - 1| + 3 and g(x) = 1|(-1e^(1(4x^2 + 1x - 3)) + 2) + 1| - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -4((-2(2(2|x + 1| + 2) - 3) - 3)/(2(2(2|x + 1| + 2) - 3)))^2 - 1 and g(x) = (-2(2(-1x)^2 - 3) - 2)/(-3(2(-1x)^2 - 3) - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1e^(-2((3(3x - 5) - 3)/(1(3x - 5)))) and g(x) = -1sin(3\u03c0(3x^2 + 4x + 4)) + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -3|(2cos(3\u03c0(5((1x - 2)/(-3x)) + 3) + 3)) + 2| + 1 and g(x) = -2(1e^(1(-1x^2 - 2)) + 3) - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-1(-2(-3|(-3x - 2) - 3| + 1)^2 + 5(-3|(-3x - 2) - 3| + 1) - 3) + 1)/(1(-2(-3|(-3x - 2) - 3| + 1)^2 + 5(-3|(-3x - 2) - 3| + 1) - 3) - 3) and g(x) = (3(-1(-1|x + 3| - 1) - 1))/(-1(-1(-1|x + 3| - 1) - 1) - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -4(-3((2(-1|x - 2| - 2) - 1)/(-1(-1|x - 2| - 2) + 2)) - 1)^2 - 2 and g(x) = -3(3|((-2x + 2)/(1x + 2)) + 1| - 2)^2 - 3(3|((-2x + 2)/(1x + 2)) + 1| - 2) + 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -3(-3cos(1\u03c0(-2|((1x + 1)/(-3x + 1))| + 3) + 3) - 3)^2 + 2(-3cos(1\u03c0(-2|((1x + 1)/(-3x + 1))| + 3) + 3) - 3) - 1 and g(x) = -1(2|(4x - 5) - 2| + 2)^2 - 1(2|(4x - 5) - 2| + 2) - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-1(-2e^(1(-1|(-2x^2 - 2x + 2) + 1| + 3))) - 2)/(2(-2e^(1(-1|(-2x^2 - 2x + 2) + 1| + 3))) - 1) and g(x) = 1|(1(2e^(2x) + 1) + 3) - 3| + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -4(-2|(1((3x - 2)/(-1x - 1)) + 3) + 3| - 1)^2 - 3 and g(x) = (3(-4(-1x^2 - 4x + 2) + 2) + 3)/(-1(-4(-1x^2 - 4x + 2) + 2) + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1(1sin(1\u03c0(-3|x|) + 3) - 1) - 1 and g(x) = 3((2x + 1)/(-1x))^2 - 5((2x + 1)/(-1x)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -5((2(1|(4x + 4) + 2| - 3) - 3)/(2(1|(4x + 4) + 2| - 3) + 2))^2 + 2((2(1|(4x + 4) + 2| - 3) - 3)/(2(1|(4x + 4) + 2| - 3) + 2)) + 5 and g(x) = -2cos(2\u03c0(-2|(-2x - 1) + 2| - 1) - 3) - 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2(-3((-3(-3|x| - 3) - 1)/(-1(-3|x| - 3)))^2 + 4((-3(-3|x| - 3) - 1)/(-1(-3|x| - 3))) + 3) - 1 and g(x) = -1(3|(3x - 1) - 2| - 1)^2 + 5(3|(3x - 1) - 2| - 1) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2e^(2((3(-5(-1x + 2)^2 - 4(-1x + 2) + 2) + 2)/(-1(-5(-1x + 2)^2 - 4(-1x + 2) + 2) + 2))) - 3 and g(x) = -1((3(1|x| - 3) - 3)/(-1(1|x| - 3))) + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1(-2e^(-2(2|((-1x + 3)/(1x + 2)) + 1| - 2)) + 2) and g(x) = 2(3|(-5x^2 + 5x + 4) - 3| - 3) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 2|((-2(5(5x^2 + 3x - 3) - 5) + 1)/(3(5(5x^2 + 3x - 3) - 5) + 1)) + 2| - 2 and g(x) = -1|(2((1x)/(-1x + 3)) - 5) - 3| + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -5((2(1|(1x^2 + 4x + 5) - 1| - 3) - 3)/(-2(1|(1x^2 + 4x + 5) - 1| - 3) - 2)) + 1 and g(x) = -2|(-2((2x + 3)/(-2x - 2))^2 + 3((2x + 3)/(-2x - 2)))| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5(3cos(1\u03c0(3|x - 2| + 3) + 1) + 3)^2 + 3(3cos(1\u03c0(3|x - 2| + 3) + 1) + 3) + 4 and g(x) = (-2(5x - 4) + 2)/(-1(5x - 4)) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -4(-1|(2((-1x - 2)/(-1x + 2)) - 4) + 3| + 1)^2 + 4(-1|(2((-1x - 2)/(-1x + 2)) - 4) + 3| + 1) + 2 and g(x) = -3|(-2e^(2(3x^2 + 4x)) + 2) - 3| + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 5(-4((1(3|x + 1| - 1) + 1)/(1(3|x + 1| - 1) - 3)))^2 - 2(-4((1(3|x + 1| - 1) + 1)/(1(3|x + 1| - 1) - 3))) - 5 and g(x) = 2sin(2\u03c0(-3|((-1x + 1)/(2x - 3))| - 2) - 3) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -3|((1(-3e^(1x) - 2) - 1)/(1(-3e^(1x) - 2) + 2)) - 2| - 1 and g(x) = -3e^(-1(-2x^2 + 2x + 2)) - 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (2(-5(-1cos(3\u03c0(-3|x + 2| + 2) + 2) - 3)^2 + 4(-1cos(3\u03c0(-3|x + 2| + 2) + 2) - 3) + 5))/(-2(-5(-1cos(3\u03c0(-3|x + 2| + 2) + 2) - 3)^2 + 4(-1cos(3\u03c0(-3|x + 2| + 2) + 2) - 3) + 5) - 2) and g(x) = 1((3(-5x + 1))/(-1(-5x + 1) - 2))^2 - 2((3(-5x + 1))/(-1(-5x + 1) - 2)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 3(3(-1|((-2x + 3)/(-2x)) - 3| - 3) - 3)^2 - 4(3(-1|((-2x + 3)/(-2x)) - 3| - 3) - 3) + 4 and g(x) = 4(3|(4x^2 + 4x - 3) + 1| - 3) - 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -5(-2|((-3x + 3)/(2x + 3)) - 3| - 1)^2 - 4(-2|((-3x + 3)/(2x + 3)) - 3| - 1) + 5 and g(x) = (1(-3cos(1\u03c0x - 1)) - 1)/(1(-3cos(1\u03c0x - 1))) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -4(1|((-3x)/(1x - 3)) - 3| + 3) and g(x) = -4((-2x)/(3x - 2))^2 + 1((-2x)/(3x - 2)) + 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2(5((-2x + 2)/(3x - 2))^2 + 1((-2x + 2)/(3x - 2)) + 3) + 2 and g(x) = (-3(-4x^2 + 3x + 3) + 1)/(-2(-4x^2 + 3x + 3) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1e^(-1((-3(3x + 4) - 2)/(-2(3x + 4) - 2))) - 2 and g(x) = (1(-2|x + 3| + 2) - 1)/(-1(-2|x + 3| + 2)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2e^(2(1(1x^2 - 5x + 5) - 5)) - 3 and g(x) = 3sin(1\u03c0((1x + 3)/(-3x - 2)) + 1) + 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -4(-3((2x + 2)/(3x - 1))^2 + 4((2x + 2)/(3x - 1)) + 3) - 4 and g(x) = 1sin(2\u03c0(-3x^2 + 4x - 5) + 3) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -5(-2|(-3x^2 - 5x - 5) + 3| - 1) - 1 and g(x) = 2|((2x - 1)/(2x + 2)) + 2| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (-3(-5(2x^2 - 4x - 4) - 2) - 1)/(-3(-5(2x^2 - 4x - 4) - 2) + 2) and g(x) = 2((-2x - 2)/(-2x - 1))^2 - 1((-2x - 2)/(-2x - 1)) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 4((-1(-4(2|x - 3| - 1)^2 - 3(2|x - 3| - 1) - 2) + 2)/(-1(-4(2|x - 3| - 1)^2 - 3(2|x - 3| - 1) - 2) + 2)) and g(x) = -4(1|((-3x - 2)/(2x + 3)) - 2|)^2 - 3(1|((-3x - 2)/(2x + 3)) - 2|) + 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 5(1|((-3(-1x^2 + 4x - 3) + 3)/(3(-1x^2 + 4x - 3))) + 3| + 1) - 3 and g(x) = 5(5(-3e^(2x) + 3) - 5)^2 + 4(5(-3e^(2x) + 3) - 5) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -4(2e^(-1(-1(-1|x| - 1))) + 2)^2 + 1(2e^(-1(-1(-1|x| - 1))) + 2) - 3 and g(x) = (2(-2e^(1(1|x + 3|)) + 1) + 3)/(-1(-2e^(1(1|x + 3|)) + 1) - 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-2(-4(-2|(-1x + 3) + 1| - 3)^2 - 5(-2|(-1x + 3) + 1| - 3) + 4) - 1)/(3(-4(-2|(-1x + 3) + 1| - 3)^2 - 5(-2|(-1x + 3) + 1| - 3) + 4) + 2) and g(x) = 3((-2(3e^(-1x) - 2))/(-2(3e^(-1x) - 2) + 3)) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1|((-1(-2(-3x^2 + 1x - 2) + 3))/(2(-2(-3x^2 + 1x - 2) + 3))) + 1| - 2 and g(x) = 1((3(2|x + 1| - 3) - 3)/(1(2|x + 1| - 3))) - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2|(1((1(2x - 2) - 2)/(-1(2x - 2) - 3))^2 + 1((1(2x - 2) - 2)/(-1(2x - 2) - 3))) + 1| - 3 and g(x) = -4(2|(3e^(1x) - 1) - 1| + 3)^2 + 2(2|(3e^(1x) - 1) - 1| + 3) + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -4(2|(-4((3x)/(3x - 2)) - 4)| - 1)^2 + 4(2|(-4((3x)/(3x - 2)) - 4)| - 1) + 3 and g(x) = -2|(-1((-2x + 1)/(3x - 1))^2 - 4((-2x + 1)/(3x - 1)) + 5) + 3| + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (1(5(-5x - 5)^2 + 5(-5x - 5) - 3) - 2)/(1(5(-5x - 5)^2 + 5(-5x - 5) - 3)) and g(x) = -2e^(-1(-3|x - 2| + 3)) + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 1e^(1(-2(-3|x + 3|)^2 - 1(-3|x + 3|) - 2)) + 3 and g(x) = 1e^(2(1|x - 3|)) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 1(-3|((-1(4x - 3) + 2)/(-3(4x - 3) + 2)) - 3| - 1)^2 - 3(-3|((-1(4x - 3) + 2)/(-3(4x - 3) + 2)) - 3| - 1) - 3 and g(x) = 5((1(1|x - 1| + 2) - 1)/(3(1|x - 1| + 2) - 1))^2 - 2((1(1|x - 1| + 2) - 1)/(3(1|x - 1| + 2) - 1)) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -5((-1(3sin(1\u03c0x + 3) + 3) + 3)/(3(3sin(1\u03c0x + 3) + 3) - 2))^2 + 3((-1(3sin(1\u03c0x + 3) + 3) + 3)/(3(3sin(1\u03c0x + 3) + 3) - 2)) - 4 and g(x) = (2(3|x - 2| - 1) - 2)/(2(3|x - 2| - 1)) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 5((2(3|x + 1|) - 2)/(1(3|x + 1|) - 2))^2 - 3((2(3|x + 1|) - 2)/(1(3|x + 1|) - 2)) - 4 and g(x) = -2cos(3\u03c0((2x - 2)/(2x - 1)) - 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1cos(3\u03c0(-1|((1x + 2)/(2x + 1)) + 1| - 3) + 2) + 1 and g(x) = (2(-1x + 2))/(-2(-1x + 2) + 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -1(2sin(1\u03c0((3x - 1)/(-1x - 1)) - 2) - 2)^2 - 3(2sin(1\u03c0((3x - 1)/(-1x - 1)) - 2) - 2) + 4 and g(x) = 4(-2x^2 + 5x) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 5((-3(1e^(-1(5x^2 + 3x)) + 2) + 1)/(-3(1e^(-1(5x^2 + 3x)) + 2) - 2)) - 5 and g(x) = (1(5(1|x - 3| - 3)) - 1)/(-1(5(1|x - 3| - 3)) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 1e^(-1((-3(1x^2 + 3x - 5) - 2)/(1(1x^2 + 3x - 5) - 3))) - 2 and g(x) = (1(2x - 3))/(-1(2x - 3) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -5((-1(-5(-1|x - 2| + 1) + 2) + 3)/(3(-5(-1|x - 2| + 1) + 2) - 1))^2 + 2((-1(-5(-1|x - 2| + 1) + 2) + 3)/(3(-5(-1|x - 2| + 1) + 2) - 1)) + 2 and g(x) = -1(-2((-1x + 2)/(3x + 2))^2 + 5((-1x + 2)/(3x + 2)) - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the third x-coordinate (rounded to the nearest integer) where f(x) = (2(-1|(5(3x^2 + 3x - 3) - 4) - 1| + 2) + 3)/(3(-1|(5(3x^2 + 3x - 3) - 4) - 1| + 2) + 3) and g(x) = -2|(-2(1e^(-1x) + 1) - 1) + 3| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 4((-2(2|x - 2| - 3) - 3)/(2(2|x - 2| - 3) + 1))^2 - 5 and g(x) = 3((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": "5", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3e^(-2(-3((2x + 2)/(-1x + 3)) + 2)) - 3 and g(x) = -1|(-1cos(1\u03c0x - 1) - 2)| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1|(1((1(-1x^2 + 3x + 3) + 2)/(2(-1x^2 + 3x + 3))) - 2) + 2| + 1 and g(x) = 2|(2(-3x - 4)^2 - 3(-3x - 4) + 4) + 3| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2(-1e^(2((3x + 2)/(-2x + 1))) + 1)^2 + 3(-1e^(2((3x + 2)/(-2x + 1))) + 1) + 5 and g(x) = -2cos(3\u03c0(1x - 3) - 1) + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2(-2(1|((-1x - 1)/(1x + 2)) + 1|)^2 + 5(1|((-1x - 1)/(1x + 2)) + 1|) + 5) + 3 and g(x) = -1|(-1sin(2\u03c0((-2x + 2)/(-1x + 3))) - 2) - 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2(1cos(3\u03c0(3|((1x + 3)/(1x)) + 2|) - 1))^2 - 2 and g(x) = 3cos(2\u03c0(5(-3|x - 2| - 1) - 3) - 1) + 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -3e^(1(5(-1|((1x)/(-2x + 3)) - 3|)^2 - 1(-1|((1x)/(-2x + 3)) - 3|) - 5)) + 3 and g(x) = 3((3(-3x^2 + 5x - 1) + 2)/(-1(-3x^2 + 5x - 1) - 2)) - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 1e^(-2((-2(2|(-5x + 2) - 1| - 3) + 1)/(-3(2|(-5x + 2) - 1| - 3) + 4))) + 1 and g(x) = 2|(-4((-3x + 1)/(2x - 3))^2 - 3((-3x + 1)/(2x - 3)) - 4) - 2| + 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2(1e^(1((-2(2x + 4) - 1)/(-2(2x + 4) + 3))) + 2)^2 - 1(1e^(1((-2(2x + 4) - 1)/(-2(2x + 4) + 3))) + 2) - 1 and g(x) = 2e^(-1(-3|((-3x - 2)/(-2x + 3)) + 2|)) - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -4((-3(1|x|) - 1)/(1(1|x|) + 2))^2 - 3((-3(1|x|) - 1)/(1(1|x|) + 2)) - 3 and g(x) = (1(1x^2 - 3x - 3))/(-1(1x^2 - 3x - 3) + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 3(3((2x + 2)/(-3x - 1)) + 5)^2 - 2(3((2x + 2)/(-3x - 1)) + 5) - 4 and g(x) = -3cos(1\u03c0(-4x^2 - 1x) + 1) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -5((2(-2e^(1x) + 2) + 2)/(-3(-2e^(1x) + 2) + 4))^2 + 3((2(-2e^(1x) + 2) + 2)/(-3(-2e^(1x) + 2) + 4)) + 5 and g(x) = 2e^(2(-1x^2 - 1x - 3)) - 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 1|(3e^(1((-3x + 1)/(-1x + 2))) - 3) - 1| + 1 and g(x) = -1(2cos(2\u03c0x - 3) - 2) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -4(1e^(2(-1|x + 1| - 1))) + 5 and g(x) = 1|((2x + 3)/(-2x + 3)) - 2| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -4(1|((-3(1x + 3) + 2)/(-2(1x + 3) - 1)) + 1| + 3)^2 + 5(1|((-3(1x + 3) + 2)/(-2(1x + 3) - 1)) + 1| + 3) and g(x) = -1|((1(-5x + 4) + 3)/(3(-5x + 4) - 1)) - 2| + 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-1(3|(5(4x^2 + 4x - 4) - 4) - 2|) - 1)/(1(3|(5(4x^2 + 4x - 4) - 4) - 2|)) and g(x) = -2((-3(-4x^2 - 1x + 2) + 3)/(-2(-4x^2 - 1x + 2) - 3)) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1((-3(-3(-2|x| + 2)^2 - 2(-2|x| + 2) - 5) - 1)/(3(-3(-2|x| + 2)^2 - 2(-2|x| + 2) - 5) - 2)) + 5 and g(x) = 1|(4((-3x + 1)/(-3x - 3))^2 - 5) - 1| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1|(-1cos(2\u03c0((2x + 2)/(-1x)) - 1) - 2) + 2| + 3 and g(x) = (-1(-1x - 1) - 1)/(-1(-1x - 1) + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -2((1(2e^(2(2|x + 2| + 2)) - 2) + 3)/(2(2e^(2(2|x + 2| + 2)) - 2) - 1)) + 5 and g(x) = (1(-2(-3x^2 + 1x + 5) + 1) - 3)/(-2(-2(-3x^2 + 1x + 5) + 1) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3|((-1(5(-3x^2 - 4x - 3) - 3) + 2)/(1(5(-3x^2 - 4x - 3) - 3) + 2)) + 3| + 2 and g(x) = (2(-1|(-3x - 5) + 3| + 3) - 1)/(3(-1|(-3x - 5) + 3| + 3) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 4((-1(1(2|x + 2| - 3) + 5) - 2)/(3(1(2|x + 2| - 3) + 5)))^2 + 2((-1(1(2|x + 2| - 3) + 5) - 2)/(3(1(2|x + 2| - 3) + 5))) - 1 and g(x) = -5(2|(-4x + 4) + 2|)^2 + 3(2|(-4x + 4) + 2|) - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -4((-3(-1|x + 1| + 3) - 3)/(2(-1|x + 1| + 3) - 1)) and g(x) = 2|(2e^(-1x) - 1) + 1| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1(3|((1x + 1)/(-2x + 1)) + 1| + 3)^2 - 3(3|((1x + 1)/(-2x + 1)) + 1| + 3) - 3 and g(x) = -3|((2x - 2)/(3x - 2)) + 1| + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(-2|(4(2x^2 - 2x - 5) + 1)| - 1) - 2)/(2(-2|(4(2x^2 - 2x - 5) + 1)| - 1) + 2) and g(x) = (3(2|(-1e^(-1x) - 1) - 1|) - 3)/(2(2|(-1e^(-1x) - 1) - 1|) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1sin(3\u03c0(-3|((1(4x^2 + 3x + 1) + 1)/(1(4x^2 + 3x + 1) + 2)) + 1|) - 3) + 3 and g(x) = 5(2|((2x - 1)/(-3x)) - 1| - 1)^2 + 1(2|((2x - 1)/(-3x)) - 1| - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-3(-3(-2x + 1)^2 + 3(-2x + 1)))/(-1(-3(-2x + 1)^2 + 3(-2x + 1)) + 2) and g(x) = (2(1cos(1\u03c0x + 3) - 1) + 2)/(1(1cos(1\u03c0x + 3) - 1) + 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (1(-3|(4(3x^2 + 2x - 4)) - 2| - 3) - 3)/(3(-3|(4(3x^2 + 2x - 4)) - 2| - 3) - 2) and g(x) = (3(1(-4x^2 + 3x - 3) + 4) - 1)/(-2(1(-4x^2 + 3x - 3) + 4) + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-1(2e^(2(-2x^2 + 3x + 2)) + 1) + 2)/(2(2e^(2(-2x^2 + 3x + 2)) + 1) - 3) and g(x) = (3(-3x - 1) + 3)/(3(-3x - 1)) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2|(-4(-1(-2e^(-1x) + 1) - 1)^2 + 3(-1(-2e^(-1x) + 1) - 1) + 1) + 1| + 2 and g(x) = -5((2(-1x + 3) - 3)/(-2(-1x + 3)))^2 - 5((2(-1x + 3) - 3)/(-2(-1x + 3))) + 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-1(1|(-5(-2x^2 - 3x - 4) - 2) + 1| + 1) + 2)/(2(1|(-5(-2x^2 - 3x - 4) - 2) + 1| + 1) + 3) and g(x) = (2(-1(-5x^2 + 5x) + 1) - 1)/(3(-1(-5x^2 + 5x) + 1) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -3(-1e^(2((-1(4x + 3) - 2)/(2(4x + 3)))) - 3)^2 + 4(-1e^(2((-1(4x + 3) - 2)/(2(4x + 3)))) - 3) and g(x) = -1e^(-1(-3(-1x - 4)^2 + 4(-1x - 4) + 1)) - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-1(1(-2|(-3x^2 + 3x + 4) - 1| - 1) + 1) - 1)/(3(1(-2|(-3x^2 + 3x + 4) - 1| - 1) + 1) + 3) and g(x) = -2e^(-1(1(1x^2 - 2x + 3) - 4)) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (-2(3|(-4(-3cos(1\u03c0x - 2) + 3)^2 + 4(-3cos(1\u03c0x - 2) + 3) + 5) + 2| - 1) - 2)/(2(3|(-4(-3cos(1\u03c0x - 2) + 3)^2 + 4(-3cos(1\u03c0x - 2) + 3) + 5) + 2| - 1) - 1) and g(x) = 2e^(1(-5(-5x^2 + 2x + 1) + 4)) + 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 3|(3e^(2((-2(-2x^2 - 5x + 1) + 3)/(-2(-2x^2 - 5x + 1) + 1))) - 1) + 1| - 2 and g(x) = (-1(3(-3x^2 + 1x - 3)) + 2)/(-2(3(-3x^2 + 1x - 3))) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the third intersection point (counting from left to right) between the functions f(x) = 2(1((1x + 2)/(-2x + 1)) + 2)^2 - 5(1((1x + 2)/(-2x + 1)) + 2) - 5 and g(x) = (-3(-1x^2 - 4x + 4) + 2)/(-1(-1x^2 - 4x + 4) + 2) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1|((2(5(-2cos(3\u03c0x))^2 + 1(-2cos(3\u03c0x)) - 5) - 3)/(2(5(-2cos(3\u03c0x))^2 + 1(-2cos(3\u03c0x)) - 5) - 1)) - 2| - 3 and g(x) = -1(-3e^(-2((2x)/(-3x + 1))) + 3) - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (3(3|(-2e^(2x) - 2) + 3| - 2))/(1(3|(-2e^(2x) - 2) + 3| - 2) - 3) and g(x) = 2(1|x - 3| + 3)^2 + 5(1|x - 3| + 3) - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 3e^(-1(2|((2x - 3)/(3x - 2)) + 1| - 2)) - 3 and g(x) = 1(-3x + 1)^2 + 4(-3x + 1) + 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3|(5(5((3x - 1)/(-2x + 3)) - 4)^2 + 1(5((3x - 1)/(-2x + 3)) - 4) + 3) - 1| and g(x) = 5((2(1cos(2\u03c0x + 1) - 2) - 2)/(3(1cos(2\u03c0x + 1) - 2) - 2))^2 + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (1(-2(2x + 2)^2 + 2(2x + 2) + 5) - 3)/(-2(-2(2x + 2)^2 + 2(2x + 2) + 5) + 1) and g(x) = 1|((2x + 1)/(-1x - 1))| + 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 1|((3(-5x^2 - 5x - 2) + 2)/(2(-5x^2 - 5x - 2))) - 2| - 1 and g(x) = -3sin(2\u03c0(1x - 2) + 1) + 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -2|((1(1sin(2\u03c0(-3x + 2) - 1) + 1) - 3)/(2(1sin(2\u03c0(-3x + 2) - 1) + 1))) + 1| - 1 and g(x) = (-2(-2(3|x + 3| - 3)^2 + 5(3|x + 3| - 3) - 3) - 3)/(-3(-2(3|x + 3| - 3)^2 + 5(3|x + 3| - 3) - 3) - 2) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1(3e^(-1((-1x - 2)/(1x + 3))) - 2) - 1 and g(x) = 1|(-3cos(2\u03c0x + 2) - 3) + 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -3e^(-1((-1(1x + 3) + 3)/(2(1x + 3) - 3))) - 1 and g(x) = -1|(-5x^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": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1|((-2(1e^(-1x) - 3) - 2)/(-2(1e^(-1x) - 3) + 3)) + 3| - 3 and g(x) = -2|((2x + 1)/(2x + 1)) - 3| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3((-2(2|x + 1| - 2) - 3)/(-2(2|x + 1| - 2)))^2 + 5((-2(2|x + 1| - 2) - 3)/(-2(2|x + 1| - 2))) + 2 and g(x) = -1(-2cos(2\u03c0x + 2) - 2) + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1((-2(1(-1|x + 2| - 1)^2 - 4))/(-1(1(-1|x + 2| - 1)^2 - 4) + 2)) - 3 and g(x) = -2((2(-5x^2 - 5x - 1) - 1)/(2(-5x^2 - 5x - 1))) - 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1cos(2\u03c0(4(-2|(-2x - 4) - 2| + 3)^2 + 4(-2|(-2x - 4) - 2| + 3) + 2) - 1) - 1 and g(x) = (2(-5(2e^(2x))^2 + 5) + 3)/(2(-5(2e^(2x))^2 + 5) + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 5(3|(3e^(-1((-3x - 3)/(1x))))| - 3) and g(x) = 2(2|(5x^2 - 2x) + 3| + 2) + 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 3|(-1(-2sin(1\u03c0x + 1)) - 2) - 1| + 3 and g(x) = (-2(4x^2 + 4x - 3) - 1)/(-1(4x^2 + 4x - 3) + 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5(2|(2((-1x - 2)/(-1x - 2))^2 - 2((-1x - 2)/(-1x - 2)) + 5)| - 1) + 3 and g(x) = 3|(-3e^(1((1x - 2)/(-3x - 1))) - 3) + 2| + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3|(-3((1(-1x^2 + 2x + 2))/(-2(-1x^2 + 2x + 2) - 3)) + 4)| + 1 and g(x) = 1(-3(-2e^(2x) - 2)^2 + 4(-2e^(2x) - 2) + 3) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2((2(1(1|x + 3| - 1) - 4) + 1)/(-1(1(1|x + 3| - 1) - 4) + 3))^2 + 5((2(1(1|x + 3| - 1) - 4) + 1)/(-1(1(1|x + 3| - 1) - 4) + 3)) + 5 and g(x) = 2(-2(-2|x + 2| - 2)^2 + 2(-2|x + 2| - 2) + 4) + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (-2(-1(-2x^2 + 4x + 3)) + 2)/(3(-1(-2x^2 + 4x + 3)) + 1) and g(x) = (-1(-3|x - 2| - 3) + 3)/(-1(-3|x - 2| - 3) - 1) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1|(3(-5(2e^(1x) - 3) + 5)^2 - 2(-5(2e^(1x) - 3) + 5) - 2)| + 1 and g(x) = -1cos(3\u03c0(5(1|x + 3| + 2)^2 + 1(1|x + 3| + 2) - 2) + 3) - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 3cos(3\u03c0((1(1x + 1) + 2)/(1(1x + 1) + 1)) + 1) and g(x) = 2|(1x - 3) + 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-3(2(-5(-3|x| - 3)^2 - 5(-3|x| - 3) - 1) - 5))/(-2(2(-5(-3|x| - 3)^2 - 5(-3|x| - 3) - 1) - 5) - 1) and g(x) = -2e^(2((2(2x - 3) + 1)/(-2(2x - 3) + 3))) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -1(4(2|x + 2| + 3) + 1)^2 - 1(4(2|x + 2| + 3) + 1) and g(x) = -3|((-1x + 2)/(1x + 3)) - 3| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1|(-2(-3((-1x - 2)/(-2x - 2))^2 - 3((-1x - 2)/(-2x - 2)) - 1) - 3) + 3| - 3 and g(x) = -1e^(1(-2((-2x - 2)/(3x)) - 3)) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3|(1e^(1((1x + 2)/(-2x + 1))) - 3) - 2| and g(x) = 5(2|x + 2| + 3)^2 + 3(2|x + 2| + 3) + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -3(-2|(5((3x)/(1x - 2))^2 - 2((3x)/(1x - 2)) - 1)| + 2) - 5 and g(x) = 4(2|((-3x)/(-1x - 2))|) + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1cos(3\u03c0((-2(1|(1x^2 + 5x + 5) - 2|) + 2)/(2(1|(1x^2 + 5x + 5) - 2|) + 3)) + 2) - 2 and g(x) = -3(5((3x - 3)/(2x + 2))^2 + 3((3x - 3)/(2x + 2)) - 1) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -3sin(2\u03c0((2(1x^2 + 4x - 2))/(3(1x^2 + 4x - 2))) + 3) - 1 and g(x) = 2(-3x^2 - 1x) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1|((1(-2x^2 + 2) - 3)/(-2(-2x^2 + 2) + 1)) - 2| - 2 and g(x) = 1(-3|x| - 1) + 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 4((3(-1|x + 3| + 2) + 3)/(-3(-1|x + 3| + 2) - 2))^2 + 3((3(-1|x + 3| + 2) + 3)/(-3(-1|x + 3| + 2) - 2)) - 2 and g(x) = (1(1|x + 3| - 1) - 1)/(1(1|x + 3| - 1) + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (3(-2e^(2(-2x + 4)) - 2) - 1)/(-3(-2e^(2(-2x + 4)) - 2)) and g(x) = 1e^(2(-5x^2 - 2x + 2)) - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1sin(2\u03c0(5((2(5x) + 3)/(-3(5x) - 3))^2 - 4((2(5x) + 3)/(-3(5x) - 3)) + 3)) + 3 and g(x) = 1(-3|((-2x - 2)/(1x + 2)) - 1| - 1) + 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2(-1|((-2(4x^2 + 2x - 5) - 1)/(2(4x^2 + 2x - 5)))|) - 4 and g(x) = (1(-1(-5x - 4)^2 + 5(-5x - 4) - 2) - 1)/(-1(-1(-5x - 4)^2 + 5(-5x - 4) - 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2sin(1\u03c0((2(2|x + 3| + 2) + 1)/(-2(2|x + 3| + 2) - 3)) - 2) - 2 and g(x) = -3|((-2x + 3)/(2x - 1)) + 2| + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2(-2((-1(2|x - 1| + 2) + 1)/(-2(2|x - 1| + 2) - 1)) + 3)^2 + 3 and g(x) = -1|(-3(-1e^(1x) - 3) + 2) - 3| + 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (3(2|(4x + 4)| - 2) + 1)/(-2(2|(4x + 4)| - 2) + 3) and g(x) = 2((2x - 2)/(-2x - 3)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1((-3(3|(-5x + 1)|) - 3)/(-2(3|(-5x + 1)|) + 3))^2 - 4((-3(3|(-5x + 1)|) - 3)/(-2(3|(-5x + 1)|) + 3)) + 5 and g(x) = (1(2(2cos(3\u03c0x - 3) - 1) + 4) - 1)/(1(2(2cos(3\u03c0x - 3) - 1) + 4)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 5((-2(-1(1|x + 2| - 1)^2 + 3(1|x + 2| - 1) + 5) - 2)/(2(-1(1|x + 2| - 1)^2 + 3(1|x + 2| - 1) + 5) - 1)) - 1 and g(x) = -3|(1(-4x)^2 - 1(-4x) + 4) - 1| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -4(-1((-1(-2|x - 3| - 3) + 2)/(1(-2|x - 3| - 3) - 3))^2 + 5((-1(-2|x - 3| - 3) + 2)/(1(-2|x - 3| - 3) - 3)) - 3) + 3 and g(x) = 5(-2|(1x - 1) - 2| - 1)^2 - 3(-2|(1x - 1) - 2| - 1) - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (3(-3|(-1cos(3\u03c0x - 3) + 2) + 1| + 2) + 3)/(-2(-3|(-1cos(3\u03c0x - 3) + 2) + 1| + 2) - 3) and g(x) = 2((-3x + 2)/(-3x + 4))^2 - 4((-3x + 2)/(-3x + 4)) - 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2|(4(-2sin(2\u03c0(1x^2 - 4x - 1)) + 3) - 1) + 2| - 3 and g(x) = -5((-3(-2|x + 2| + 2) - 2)/(1(-2|x + 2| + 2))) + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 5(3|(1((3x + 1)/(-3x + 1)) + 4) - 3| - 1)^2 + 4(3|(1((3x + 1)/(-3x + 1)) + 4) - 3| - 1) - 4 and g(x) = -2(-1|(-3x^2 + 3x)| - 3) - 5 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1((-1(2|x - 2| - 3) + 1)/(2(2|x - 2| - 3) - 1)) - 2 and g(x) = 3(-1e^(-1x) + 3)^2 + 2(-1e^(-1x) + 3) + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -4((1(-2x^2 - 5x + 4))/(-1(-2x^2 - 5x + 4) - 1)) + 2 and g(x) = 2sin(1\u03c0((-2x + 3)/(-2x - 3)) - 1) - 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (-1(3(-3|x + 1| + 3) + 1) - 1)/(1(3(-3|x + 1| + 3) + 1) - 3) and g(x) = -2|(-2cos(1\u03c0x - 2) - 3) + 3| - 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1(-3|(-3(3x^2 + 5x) - 4) + 3|) + 3)/(3(-3|(-3(3x^2 + 5x) - 4) + 3|)) and g(x) = -2|((-1(-1x + 1) - 3)/(-3(-1x + 1) + 2)) - 2| + 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (2(2|(4x^2 + 2)| - 1) - 1)/(-3(2|(4x^2 + 2)| - 1) - 3) and g(x) = (3(3e^(-2x) - 2))/(3(3e^(-2x) - 2) + 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(4(-2e^(1(-4x - 1)) + 1)^2 - 4(-2e^(1(-4x - 1)) + 1) + 1) - 3)/(2(4(-2e^(1(-4x - 1)) + 1)^2 - 4(-2e^(1(-4x - 1)) + 1) + 1) - 1) and g(x) = -2|(3(-2x + 1)^2 - 2) + 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (-2(2(3x^2 - 5) - 5) - 3)/(-3(2(3x^2 - 5) - 5) - 2) and g(x) = -1(1cos(2\u03c0x - 1) + 1)^2 + 1(1cos(2\u03c0x - 1) + 1) - 5 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1(-1((2(2cos(1\u03c0x + 1) - 3) - 3)/(2(2cos(1\u03c0x + 1) - 3)))^2 - 3((2(2cos(1\u03c0x + 1) - 3) - 3)/(2(2cos(1\u03c0x + 1) - 3))) - 2) + 5 and g(x) = (-1(-5(-3|x + 3|)^2 - 4(-3|x + 3|) + 2) + 1)/(1(-5(-3|x + 3|)^2 - 4(-3|x + 3|) + 2)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(4((3(-4x + 1))/(1(-4x + 1) - 3))^2 + 1((3(-4x + 1))/(1(-4x + 1) - 3)) + 2) + 3| + 1 and g(x) = 2cos(2\u03c0(-1(1x)^2 + 4(1x) - 2) - 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2|((-3(-3x - 5) - 3)/(-3(-3x - 5) - 1)) - 1| - 3 and g(x) = -1|(-3x^2 - 5) - 2| - 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 3|(-2((-3x - 1)/(-1x + 2))^2 + 5((-3x - 1)/(-1x + 2)) + 5) + 2| - 1 and g(x) = 2(-2|x - 1|)^2 + 1(-2|x - 1|) - 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1e^(-1((3(-5(2x^2 - 1x + 1) + 1) + 1)/(3(-5(2x^2 - 1x + 1) + 1) + 2))) - 2 and g(x) = -1|((1(4x + 1) + 2)/(1(4x + 1) - 3)) + 2| + 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2cos(2\u03c0((-3(5(4x - 5)^2 + 5(4x - 5) + 3) - 3)/(-3(5(4x - 5)^2 + 5(4x - 5) + 3) - 3)) - 2) + 2 and g(x) = 4(-2|((3x - 3)/(3x - 2)) - 2| + 2) + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1(1|((-3x - 1)/(1x + 1)) - 1|)^2 - 1 and g(x) = (-3(-1x - 2) - 2)/(3(-1x - 2)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1e^(-1((-2(-1(-1x + 3)^2 - 1(-1x + 3) + 5) + 3)/(-1(-1(-1x + 3)^2 - 1(-1x + 3) + 5) - 3))) + 2 and g(x) = -1sin(1\u03c0(1((-2x + 3)/(1x)))) - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-2(-4(3|(-1x^2 - 2x + 5) - 1| + 1) + 5) + 1)/(-3(-4(3|(-1x^2 - 2x + 5) - 1| + 1) + 5) + 1) and g(x) = (-2(2|(5x^2 + 2) - 3| - 3) - 1)/(2(2|(5x^2 + 2) - 3| - 3) + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (3(-5(1x - 5)^2 + 3(1x - 5)) - 2)/(-2(-5(1x - 5)^2 + 3(1x - 5))) and g(x) = -1cos(2\u03c0(1|x - 1| + 2) - 1) + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-3(3(-3sin(3\u03c0x + 3) - 3) - 2) + 3)/(3(3(-3sin(3\u03c0x + 3) - 3) - 2) - 2) and g(x) = 5((1x - 3)/(3x + 1))^2 - 2((1x - 3)/(3x + 1)) - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2(1(2|(3e^(-2x) - 1)| + 1)^2 - 2(2|(3e^(-2x) - 1)| + 1) + 3) - 1 and g(x) = (3(-5(1|x + 1| - 1) + 5) - 2)/(-2(-5(1|x + 1| - 1) + 5)) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1e^(-1(1|(1x^2 + 5) - 3| + 1)) - 3 and g(x) = -1|((1x + 1)/(-1x - 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": "At what integer value (rounded) do the functions f(x) = (-1(2(3|x - 3| - 1)^2 - 5(3|x - 3| - 1) + 5) - 2)/(-3(2(3|x - 3| - 1)^2 - 5(3|x - 3| - 1) + 5) + 2) and g(x) = (-1(5x^2 - 1) - 1)/(-3(5x^2 - 1) + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 4(-3((-3x + 1)/(-1x - 1)) + 4)^2 + 3(-3((-3x + 1)/(-1x - 1)) + 4) - 5 and g(x) = -1|(-3cos(1\u03c0x) + 1)| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 4(-1|((-2x + 2)/(-1x - 1)) + 1| + 2) - 1 and g(x) = 2cos(2\u03c0((-2x - 2)/(-2x + 1)) + 3) - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 2|((2(4(1sin(3\u03c0x - 2) - 1) - 5) + 2)/(-3(4(1sin(3\u03c0x - 2) - 1) - 5) - 3)) - 3| - 3 and g(x) = (-1(1|(3x + 5)| + 1) - 3)/(-2(1|(3x + 5)| + 1) + 3) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1(1(-2|x - 3| - 2)^2 - 4(-2|x - 3| - 2)) - 3)/(1(1(-2|x - 3| - 2)^2 - 4(-2|x - 3| - 2)) + 3) and g(x) = (3(5x - 2) + 1)/(1(5x - 2) + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 1((1(1|x| - 3) - 3)/(3(1|x| - 3) - 2)) - 2 and g(x) = 3((-1x + 2)/(3x + 3)) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1|((-3(5x - 2))/(1(5x - 2) + 1)) + 1| - 2 and g(x) = 1(1|x - 1| - 1)^2 - 1(1|x - 1| - 1) + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-2(-1|(-3(1sin(2\u03c0x) + 2)^2 - 1(1sin(2\u03c0x) + 2) + 2) - 3| - 2) - 3)/(2(-1|(-3(1sin(2\u03c0x) + 2)^2 - 1(1sin(2\u03c0x) + 2) + 2) - 3| - 2) + 3) and g(x) = 1((-1(-1e^(1x) + 2) + 2)/(1(-1e^(1x) + 2) + 3))^2 - 5((-1(-1e^(1x) + 2) + 2)/(1(-1e^(1x) + 2) + 3)) - 4 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3(1cos(1\u03c0((1x + 1)/(-1x - 2)) + 3) + 1)^2 - 5(1cos(1\u03c0((1x + 1)/(-1x - 2)) + 3) + 1) + 1 and g(x) = -2|((2x - 2)/(3x - 2)) + 2| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1(-3|(-1((-3x)/(1x + 2)) - 4) + 1|)^2 + 1(-3|(-1((-3x)/(1x + 2)) - 4) + 1|) + 5 and g(x) = -1|(5(-1x^2 - 2x - 1) - 5) - 2| + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3|(3e^(2(-4((-3x + 1)/(-1x)))) + 1) + 1| + 2 and g(x) = -1((2(5x - 5) + 2)/(2(5x - 5)))^2 + 2((2(5x - 5) + 2)/(2(5x - 5))) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 1(2|(2x^2 + 1) - 1| - 1) - 2 and g(x) = (3(2|x + 3| + 3) - 2)/(2(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": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -2|(1sin(1\u03c0(-3x^2 - 3x - 5) - 3) - 2) - 1| - 2 and g(x) = (1(-1|x + 1| + 1) + 1)/(-3(-1|x + 1| + 1) - 1) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -3((2(3|(-1x + 2)| - 1) - 3)/(3(3|(-1x + 2)| - 1) + 2))^2 + 4((2(3|(-1x + 2)| - 1) - 3)/(3(3|(-1x + 2)| - 1) + 2)) - 1 and g(x) = 3e^(2(1|(-4x + 2) + 2| - 3)) - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 2|(4(2x)^2 - 4(2x) + 1) + 1| - 3 and g(x) = -1sin(1\u03c0(-3|x + 2| - 1) + 3) - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(-4(5((3x - 3)/(-3x + 1))^2 - 1((3x - 3)/(-3x + 1)) + 5)) - 1| + 3 and g(x) = -3|(-2((1x - 3)/(1x + 2))^2 - 3((1x - 3)/(1x + 2)) - 3) + 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1cos(1\u03c0(-3(1(-3|x - 1|) - 5)^2 + 3(1(-3|x - 1|) - 5)) - 3) + 3 and g(x) = (-2(-1cos(3\u03c0(1|x - 3| + 2) + 3) + 1) + 2)/(1(-1cos(3\u03c0(1|x - 3| + 2) + 3) + 1) + 1) meet 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(-1((-1x - 3)/(-3x + 2)) - 5) + 3| + 1 and g(x) = 2sin(1\u03c0((-2x - 3)/(-2x + 3)) + 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2(2|((-2(4x - 2) + 3)/(-1(4x - 2) + 2)) - 1| + 1)^2 - 2(2|((-2(4x - 2) + 3)/(-1(4x - 2) + 2)) - 1| + 1) - 1 and g(x) = -2(-1(-2|x - 2| - 1)^2 - 2(-2|x - 2| - 1) + 5) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 3|(3(4((-2x - 1)/(3x - 1))^2 + 2((-2x - 1)/(3x - 1)) - 4) + 2) + 3| + 3 and g(x) = -1((2(-3x - 4) + 1)/(1(-3x - 4) - 3))^2 + 5((2(-3x - 4) + 1)/(1(-3x - 4) - 3)) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (-2(-1(-3(-1|x| + 3)^2 + 2(-1|x| + 3) + 1)) + 1)/(-1(-1(-3(-1|x| + 3)^2 + 2(-1|x| + 3) + 1)) - 3) and g(x) = -1|(5((-1x + 1)/(-3x - 2)) + 3) + 2| + 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1sin(1\u03c0(5((-2x + 3)/(-2x + 3))^2 - 5((-2x + 3)/(-2x + 3)) + 5) - 1) + 1 and g(x) = 2(-1|x - 1| + 2) + 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 3|((-2(-1x) + 2)/(3(-1x) + 1)) + 1| + 2 and g(x) = 1((-3x - 1)/(3x))^2 + 1((-3x - 1)/(3x)) - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1|(-1sin(1\u03c0(3x^2 + 5x + 4)) - 1) - 1| - 3 and g(x) = (3(1x) + 2)/(2(1x) - 1) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1((-3(2|(1x - 4) + 3| + 1) + 2)/(1(2|(1x - 4) + 3| + 1) + 3))^2 + 1 and g(x) = (2(1|(-1x - 5)| - 1) + 2)/(1(1|(-1x - 5)| - 1) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1|((-3(-2(5x - 3)^2 + 4(5x - 3)) + 3)/(1(-2(5x - 3)^2 + 4(5x - 3)))) - 3| + 3 and g(x) = 1(-3((-3x - 2)/(2x - 1)) + 5)^2 - 2(-3((-3x - 2)/(2x - 1)) + 5) - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1(5(-1(1|x|)^2 - 5(1|x|) + 4) + 5))/(3(5(-1(1|x|)^2 - 5(1|x|) + 4) + 5) + 2) and g(x) = -2|(-2(2e^(-1x) - 3)^2 + 2(2e^(-1x) - 3) - 4)| + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1|(-1((-2x + 1)/(-2x)) + 3) + 2| - 2 and g(x) = -1((1x + 1)/(-1x - 3))^2 + 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2(1(-3|((1x - 2)/(-1x + 2)) + 1| - 2)^2 + 4(-3|((1x - 2)/(-1x + 2)) + 1| - 2)) + 3 and g(x) = 2|((2(-1x + 1) + 2)/(3(-1x + 1) - 2)) + 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2e^(1((-2(-4x^2 - 5))/(-2(-4x^2 - 5) + 3))) - 2 and g(x) = -1|(2e^(-2x)) + 1| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2(-1|(1((3x - 1)/(-3x - 2)) - 4)| - 1)^2 + 4(-1|(1((3x - 1)/(-3x - 2)) - 4)| - 1) - 2 and g(x) = (1(5(2|x + 1| - 3)^2 - 4(2|x + 1| - 3) - 4) + 1)/(-1(5(2|x + 1| - 3)^2 - 4(2|x + 1| - 3) - 4) - 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -5(2sin(2\u03c0((-3x - 3)/(3x - 1)) - 3) - 2)^2 + 3(2sin(2\u03c0((-3x - 3)/(3x - 1)) - 3) - 2) - 5 and g(x) = (1(3e^(-1x) - 2) - 1)/(-2(3e^(-1x) - 2) + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (3(-1|(-4x^2) + 1| - 1) - 3)/(-2(-1|(-4x^2) + 1| - 1) - 2) and g(x) = 1((-2x + 1)/(-2x + 3))^2 - 5((-2x + 1)/(-2x + 3)) + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -1((1(-1(-1|x + 2| - 2)^2 + 5(-1|x + 2| - 2) + 5) - 3)/(-1(-1(-1|x + 2| - 2)^2 + 5(-1|x + 2| - 2) + 5) + 2)) + 3 and g(x) = 5(3|((2x - 3)/(-3x)) + 2| - 1) + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1|((-2(-3(3x^2 - 5x - 5)) - 3)/(-1(-3(3x^2 - 5x - 5)))) - 1| + 1 and g(x) = -1|(5(-3x^2 + 4x - 2) + 3)| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -5((-3(-1|(1x^2 + 2x + 5) + 3|) + 2)/(-2(-1|(1x^2 + 2x + 5) + 3|) - 1)) + 1 and g(x) = (2(-3e^(-2(-5x^2 - 5x + 1)) + 2) + 2)/(2(-3e^(-2(-5x^2 - 5x + 1)) + 2) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1e^(2(1|(-5(-3x)^2 - 2(-3x) - 2)| + 1)) and g(x) = 4((-3(2x - 1) - 2)/(-2(2x - 1) - 3))^2 + 5((-3(2x - 1) - 2)/(-2(2x - 1) - 3)) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1(-3|(3e^(-1((2x)/(-1x + 2))) - 2) - 3| - 3)^2 - 1(-3|(3e^(-1((2x)/(-1x + 2))) - 2) - 3| - 3) - 5 and g(x) = -3|(3e^(-2(-1x + 2)) - 1) - 3| - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -4(-1(-1e^(-1((3x - 3)/(2x - 3))) + 1)^2 + 2(-1e^(-1((3x - 3)/(2x - 3))) + 1) + 1) - 2 and g(x) = 4(-1|(2cos(3\u03c0x + 2) - 3) + 2| - 1)^2 + 4(-1|(2cos(3\u03c0x + 2) - 3) + 2| - 1) - 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (1(-2(2|(-2sin(1\u03c0x + 2) - 3) - 1| + 2)^2 - 5(2|(-2sin(1\u03c0x + 2) - 3) - 1| + 2) - 5) + 2)/(-2(-2(2|(-2sin(1\u03c0x + 2) - 3) - 1| + 2)^2 - 5(2|(-2sin(1\u03c0x + 2) - 3) - 1| + 2) - 5) - 1) and g(x) = 1(-3e^(-2((2x + 1)/(1x + 1))) + 2) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1(-3cos(2\u03c0((2x + 3)/(1x + 2)) - 1) - 3)^2 + 3(-3cos(2\u03c0((2x + 3)/(1x + 2)) - 1) - 3) - 5 and g(x) = 1(-1|x - 1| + 2)^2 - 4(-1|x - 1| + 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -5((2(-4(-1|x + 2| + 1) + 1) + 3)/(3(-4(-1|x + 2| + 1) + 1) + 1))^2 + 2((2(-4(-1|x + 2| + 1) + 1) + 3)/(3(-4(-1|x + 2| + 1) + 1) + 1)) - 3 and g(x) = (1(-2(2|x + 2| - 1) - 3) - 1)/(1(-2(2|x + 2| - 1) - 3) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 3(-3((2(-1|x + 3|) - 2)/(-2(-1|x + 3|) - 1))^2 - 5((2(-1|x + 3|) - 2)/(-2(-1|x + 3|) - 1)) + 3) + 4 and g(x) = 3|(3(-4x - 2)^2 - 1(-4x - 2) - 2)| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1sin(1\u03c0((-1(-5(-1x + 5)^2 + 1(-1x + 5) - 2))/(-3(-5(-1x + 5)^2 + 1(-1x + 5) - 2) - 3)) + 3) - 2 and g(x) = -3|((2(5x + 5) - 2)/(2(5x + 5) + 2)) + 1| + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 1sin(3\u03c0(1|(1((-3x - 3)/(1x))^2 - 3) + 3| + 3) - 2) + 2 and g(x) = 1|(1(-2cos(1\u03c0x + 3) - 1)^2 - 2(-2cos(1\u03c0x + 3) - 1) + 4) + 2| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 3|(-5(-1((3x + 3)/(-2x + 1)) - 1)^2 + 3(-1((3x + 3)/(-2x + 1)) - 1) + 4) - 1| - 2 and g(x) = 2|(-1(1sin(1\u03c0x - 3) - 1) + 5) + 2| - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (3(1(2e^(1(3x^2 + 3x + 4)) - 3) + 4) + 2)/(1(1(2e^(1(3x^2 + 3x + 4)) - 3) + 4) + 1) and g(x) = (1(-5(2|x + 3| - 3) - 1) + 2)/(1(-5(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": "-4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-1(1|(4x + 1) - 1| + 1) - 1)/(-3(1|(4x + 1) - 1| + 1) - 3) and g(x) = (3(-1e^(1x) + 2) - 3)/(-1(-1e^(1x) + 2) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1|((-1(1(4x - 4)^2 - 3(4x - 4) + 3) - 2)/(2(1(4x - 4)^2 - 3(4x - 4) + 3) - 1)) - 3| - 2 and g(x) = -1((-1(5x^2 - 1) - 2)/(-3(5x^2 - 1) + 4)) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3(1(3|((2x - 2)/(-2x)) + 1| - 2) + 4)^2 + 1 and g(x) = -2((1(3|x| + 1) - 3)/(-1(3|x| + 1) + 2)) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2(3(-1e^(2x)) + 4)^2 - 4(3(-1e^(2x)) + 4) - 2 and g(x) = -3|(-2sin(2\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": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2|(1((3x - 2)/(1x + 3))^2 + 5((3x - 2)/(1x + 3)) - 5) - 2| 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": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3|((-1(2sin(2\u03c0(4x^2 - 2x - 2) - 2)) - 1)/(-1(2sin(2\u03c0(4x^2 - 2x - 2) - 2)) - 1))| + 3 and g(x) = 5(-1(1e^(1x) - 3) - 3)^2 - 4(-1(1e^(1x) - 3) - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1(-3|(-1((2x + 2)/(-2x))^2 + 3) - 3| + 3) + 3 and g(x) = -1(-2e^(-2((3x)/(1x))) - 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2((3(1|x + 3| + 1) - 1)/(-2(1|x + 3| + 1) - 2))^2 and g(x) = -1e^(2(-4x^2 + 2)) - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 2(1e^(-2(-1|x - 3| + 2)) + 1)^2 + 3(1e^(-2(-1|x - 3| + 2)) + 1) + 3 and g(x) = 2|(1e^(1x) - 2)| - 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 2(1|((2(-4x + 2) - 1)/(-1(-4x + 2) + 2)) + 3|)^2 - 1(1|((2(-4x + 2) - 1)/(-1(-4x + 2) + 2)) + 3|) - 4 and g(x) = 2|(-3(-2sin(1\u03c0x + 2) + 2) - 2) + 1| + 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1((2(3|x + 2| + 2) - 2)/(-2(3|x + 2| + 2)))^2 - 5((2(3|x + 2| + 2) - 2)/(-2(3|x + 2| + 2))) + 1 and g(x) = (-2(-1x^2 + 4x - 5) + 3)/(1(-1x^2 + 4x - 5) + 3) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (1(-3(3|(-1cos(3\u03c0x) + 2) + 3| + 1) + 4) + 3)/(-2(-3(3|(-1cos(3\u03c0x) + 2) + 3| + 1) + 4) - 2) and g(x) = 5((-1(2x^2 - 2x - 3))/(-2(2x^2 - 2x - 3) - 1)) - 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5(-2|((-2(-5x - 4) - 2)/(-1(-5x - 4) - 3)) - 3| - 2)^2 - 4(-2|((-2(-5x - 4) - 2)/(-1(-5x - 4) - 3)) - 3| - 2) - 1 and g(x) = 2((-3(-4x) + 1)/(3(-4x) - 2))^2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1|(5((-3(-3x - 5) + 3)/(1(-3x - 5) + 1))^2 + 3((-3(-3x - 5) + 3)/(1(-3x - 5) + 1)) - 2) - 2| + 1 and g(x) = (2(-2|(-5x + 2)| + 3) + 3)/(-1(-2|(-5x + 2)| + 3) + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (2(3|(5x + 2) + 3| + 3) + 2)/(3(3|(5x + 2) + 3| + 3) - 2) and g(x) = -2((-3x + 3)/(-3x))^2 + 3((-3x + 3)/(-3x)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (2(-2|(1sin(1\u03c0x + 2) + 1) + 3| + 3) + 1)/(-3(-2|(1sin(1\u03c0x + 2) + 1) + 3| + 3)) and g(x) = -3(-4x^2 - 5x) - 5 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 2|((3(-4(-1x^2 + 1x + 4) + 4) + 1)/(-1(-4(-1x^2 + 1x + 4) + 4) + 3)) - 2| and g(x) = -2((2(3|x + 3| - 1) + 2)/(3(3|x + 3| - 1) - 2)) - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(-4((-3(4x^2 - 4x - 3) + 1)/(2(4x^2 - 4x - 3) + 3)) - 1) + 1| + 2 and g(x) = 4(1|((-2x + 3)/(2x + 3)) + 2| - 3) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-3(-2|(2x + 1) - 3| + 3))/(-1(-2|(2x + 1) - 3| + 3) - 1) and g(x) = (-2(5x^2 + 2x) + 2)/(-3(5x^2 + 2x) + 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (2(2|(4(4x + 2)^2 + 4(4x + 2) + 1) - 1| + 3) - 1)/(-3(2|(4(4x + 2)^2 + 4(4x + 2) + 1) - 1| + 3) - 1) and g(x) = (-3(-2(2x^2 - 5x + 1) + 3) + 3)/(-1(-2(2x^2 - 5x + 1) + 3) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 4((1(1x^2 + 1x + 3) - 3)/(-1(1x^2 + 1x + 3) + 2)) + 4 and g(x) = 2|((3x - 1)/(-2x + 3)) + 1| - 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2e^(2(5((1x - 3)/(3x + 3))^2 - 1((1x - 3)/(3x + 3)) - 4)) - 1 and g(x) = -1e^(-1(-1|x - 1| + 2)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1cos(2\u03c0(-1(1|((2x + 1)/(-2x - 2))| + 1)^2 + 3(1|((2x + 1)/(-2x - 2))| + 1) - 4) + 2) + 3 and g(x) = 1(3((-1x + 2)/(-3x + 4))^2 + 5((-1x + 2)/(-3x + 4))) + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (2(-2(-1sin(2\u03c0x - 1))^2 - 5) - 3)/(-2(-2(-1sin(2\u03c0x - 1))^2 - 5) - 1) and g(x) = -1|(-1e^(-2x) - 3) + 1| + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1((2(1(2|x - 3| - 2) + 1) + 2)/(-1(1(2|x - 3| - 2) + 1) - 3))^2 + 3((2(1(2|x - 3| - 2) + 1) + 2)/(-1(1(2|x - 3| - 2) + 1) - 3)) + 5 and g(x) = 1|((-2(-1x - 1) + 2)/(-3(-1x - 1))) + 2| - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -4(-1|(2(-3e^(2x) - 3)^2 + 4(-3e^(2x) - 3) - 4) - 3| + 1) - 1 and g(x) = 5((1(1|x| + 3) + 1)/(1(1|x| + 3))) + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1e^(-2((1(1(2|x - 1| - 1) - 2) + 1)/(2(1(2|x - 1| - 1) - 2) - 1))) and g(x) = 2((2(2|x - 2| - 1) + 3)/(-2(2|x - 2| - 1))) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1|(1((-1(-2x - 3))/(-3(-2x - 3) + 1))^2 + 3((-1(-2x - 3))/(-3(-2x - 3) + 1)) + 2) - 1| - 1 and g(x) = -1e^(-1((-3(4x + 2) + 2)/(-1(4x + 2) + 2))) + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2(2e^(2((-3(-1|x|))/(-3(-1|x|)))) - 1)^2 - 1(2e^(2((-3(-1|x|))/(-3(-1|x|)))) - 1) - 1 and g(x) = 3(-3e^(2((3x)/(-3x - 2))) - 2) - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2cos(1\u03c0(-2|((-3(-4x^2 + 1x) + 1)/(2(-4x^2 + 1x))) + 3| - 1) + 2) - 1 and g(x) = -1|(1((-1x + 2)/(1x))^2 + 1((-1x + 2)/(1x)) - 4)| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 3|((-2(5x + 3) - 3)/(-1(5x + 3) + 3)) - 2| - 2 and g(x) = -3(1cos(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": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (1(5(-5x^2 - 4x + 1) + 4) - 3)/(2(5(-5x^2 - 4x + 1) + 4) - 1) and g(x) = 2|((1x + 3)/(-2x - 3)) - 2| + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1(-1(-1|((1x - 2)/(-2x - 3))| + 1)^2 - 3(-1|((1x - 2)/(-2x - 3))| + 1) + 2) - 3 and g(x) = -1e^(1((-2(-2x + 5) + 1)/(3(-2x + 5)))) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(3(-1x^2 - 2x + 3) + 3) + 3)/(-2(3(-1x^2 - 2x + 3) + 3) - 2) and g(x) = -1cos(3\u03c0((-2x + 3)/(-2x - 2)) - 1) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-3(-2|(-3(-2x^2 + 1x + 2) - 5) - 2|) - 2)/(-3(-2|(-3(-2x^2 + 1x + 2) - 5) - 2|) - 3) and g(x) = -3|(-3cos(2\u03c0(1x^2 + 3x - 5) - 2) - 2) + 3| + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-1(3|(-1(3x - 5)^2 - 3(3x - 5) - 2) - 1| - 3) - 2)/(-1(3|(-1(3x - 5)^2 - 3(3x - 5) - 2) - 1| - 3) + 2) and g(x) = 1(-1|(-2x^2 + 1) - 3| + 3) + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -5(2|((3x - 1)/(-1x - 2)) - 2|)^2 + 4(2|((3x - 1)/(-1x - 2)) - 2|) + 5 and g(x) = -1cos(1\u03c0(3x^2 + 4x + 4) - 1) + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -1e^(2(2(-1((1x - 2)/(2x)) - 5)^2 - 4(-1((1x - 2)/(2x)) - 5) + 1)) + 3 and g(x) = -1|(3((-3x - 2)/(2x))^2 + 4((-3x - 2)/(2x)) + 4) - 1| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -2|(2((-2x - 1)/(2x - 1)) - 4) - 2| + 2 and g(x) = -2(1sin(3\u03c0x + 2) - 1)^2 - 1(1sin(3\u03c0x + 2) - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(3|(1(5x^2 + 2x + 1) + 2) - 2| - 1) + 3)/(-3(3|(1(5x^2 + 2x + 1) + 2) - 2| - 1) + 2) and g(x) = 3|((-1(1x + 5))/(-2(1x + 5) + 3)) - 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (2(-3(-2|(3x + 4) - 3|)^2 + 5(-2|(3x + 4) - 3|)) - 2)/(-1(-3(-2|(3x + 4) - 3|)^2 + 5(-2|(3x + 4) - 3|)) + 2) and g(x) = 2((-1(1x + 5) - 3)/(2(1x + 5) - 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": "Find the integer closest to the largest x-value where the functions f(x) = (1(3|(-1cos(2\u03c0(-2x^2 + 2x - 3) + 2)) - 2| + 3))/(2(3|(-1cos(2\u03c0(-2x^2 + 2x - 3) + 2)) - 2| + 3) - 1) and g(x) = 1((1(2|x - 2| - 2) + 2)/(2(2|x - 2| - 2) + 2)) + 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1((-1(-1(-1|x - 2| - 1)^2 - 1(-1|x - 2| - 1) + 4) + 3)/(-2(-1(-1|x - 2| - 1)^2 - 1(-1|x - 2| - 1) + 4) + 3)) - 3 and g(x) = -4(2cos(3\u03c0((-1x - 1)/(3x - 2)) - 2) - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 3|((3(2(-3x + 5)^2 - 4) + 1)/(2(2(-3x + 5)^2 - 4) + 3)) - 3| + 2 and g(x) = 1((-2(-3e^(2x)) + 1)/(-2(-3e^(2x)) + 3)) + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (-2(-1(2|x - 3| + 2) + 1) - 2)/(-1(-1(2|x - 3| + 2) + 1) + 2) and g(x) = 2|((-2x)/(2x - 1)) + 2| - 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -3|(-2((1x - 3)/(-1x + 3))^2 - 5((1x - 3)/(-1x + 3)) - 3) - 1| + 1 and g(x) = 4((1x - 3)/(-2x + 1)) - 1 intersect 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": "If you add up all the x-coordinates where the functions f(x) = (2(-3|(-1e^(1x) + 1) + 2| + 3) + 2)/(-3(-3|(-1e^(1x) + 1) + 2| + 3) - 1) and g(x) = (-2(1x^2 + 5x + 2) + 1)/(3(1x^2 + 5x + 2) + 2) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1|(3(-2((1x + 1)/(-1x))^2 + 2((1x + 1)/(-1x)) - 2)) - 1| and g(x) = (-2(4(1x^2 + 1x - 5) + 1) + 2)/(1(4(1x^2 + 1x - 5) + 1) - 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-2(-2(-5(3|x|))^2 - 4(-5(3|x|)) + 3) + 1)/(-1(-2(-5(3|x|))^2 - 4(-5(3|x|)) + 3) - 2) and g(x) = 2|(5(-4x)^2 + 5(-4x) + 4) + 1| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 2|(1((-1(-3x + 4) - 3)/(1(-3x + 4)))^2 + 3((-1(-3x + 4) - 3)/(1(-3x + 4))) + 2) - 2| + 1 and g(x) = (-3(-1|(-1x^2 + 4x + 3) - 2| + 2))/(-1(-1|(-1x^2 + 4x + 3) - 2| + 2) - 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-1(3(-3|x + 1| - 1)^2 - 4(-3|x + 1| - 1) + 4) - 1)/(-3(3(-3|x + 1| - 1)^2 - 4(-3|x + 1| - 1) + 4) - 1) and g(x) = 5((-3x + 3)/(-2x - 1)) + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2|(2cos(2\u03c0((-3(-2x - 5) - 1)/(1(-2x - 5) + 1)) - 1) + 2) - 2| + 1 and g(x) = (1(-1e^(1(-1|x + 3| - 2)) + 1) - 2)/(2(-1e^(1(-1|x + 3| - 2)) + 1) - 3) intersect 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 first intersection point (counting from left to right) between the functions f(x) = -1|((-2(2(5x^2 - 5x + 1) - 1) - 3)/(-1(2(5x^2 - 5x + 1) - 1) - 1)) + 2| - 2 and g(x) = (-3(-1|(5x - 3) - 1| + 2) + 2)/(3(-1|(5x - 3) - 1| + 2) + 2) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(-4((3x - 1)/(2x - 3)) - 3) - 3| + 1 and g(x) = -3e^(-2(-5x^2 - 5x + 5)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}