{"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|x + 1| - 1)^2) - 2)/(2(2(-1|x + 1| - 1)^2) + 2) and g(x) = 1|(3x + 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 first intersection point (counting from left to right) between the functions f(x) = (2(2|x + 1| - 2) - 2)/(-1(2|x + 1| - 2) + 3) and g(x) = 2|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": "-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((-2(3x - 5) + 2)/(1(3x - 5)))^2 + 1((-2(3x - 5) + 2)/(1(3x - 5))) and g(x) = 1sin(1\u03c0(-2|x - 2| - 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": "Find the integer closest to the smallest x-value where the functions f(x) = (-2(-1(1x + 4)^2 + 1(1x + 4) + 1) + 1)/(1(-1(1x + 4)^2 + 1(1x + 4) + 1) + 2) and g(x) = 2((-2x + 3)/(-1x - 2))^2 + 1((-2x + 3)/(-1x - 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 value closest to the x-coordinate where f(x) = -2e^(2((-2x - 3)/(3x - 2))) + 3 and g(x) = -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": "Find the integer closest to the largest x-value where the functions f(x) = -1(1x - 4)^2 + 3(1x - 4) - 3 and g(x) = -1x - 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (-1(-5(-3|(-2cos(3\u03c0x - 1) - 1) - 1| + 3)^2 - 5(-3|(-2cos(3\u03c0x - 1) - 1) - 1| + 3) - 4) - 3)/(-2(-5(-3|(-2cos(3\u03c0x - 1) - 1) - 1| + 3)^2 - 5(-3|(-2cos(3\u03c0x - 1) - 1) - 1| + 3) - 4) - 1) and g(x) = 1|(-5(2x + 2)^2 - 4(2x + 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": "-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(2|(4x^2 - 1x - 3) - 1| + 1) - 3)/(-1(2|(4x^2 - 1x - 3) - 1| + 1) + 3) and g(x) = -1(-3|x + 3|)^2 + 1(-3|x + 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": "If you add up all the x-coordinates where the functions f(x) = 1|(2(-2x^2 - 4x) - 2) + 2| + 2 and g(x) = 3(-1x)^2 - 5(-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": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 2(-2e^(-2((3(3|x| + 3) - 1)/(3(3|x| + 3)))) - 1) - 5 and g(x) = -2|(4((-2x)/(2x + 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": "Find the integer value closest to the x-coordinate where f(x) = -1(-2(1sin(3\u03c0x + 1) - 1) + 3)^2 + 1(-2(1sin(3\u03c0x + 1) - 1) + 3) - 2 and g(x) = (-1(-5x + 3) - 1)/(1(-5x + 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) = 4(-1|(-1cos(2\u03c0((-2x - 1)/(2x + 1))) - 3) - 1| + 3)^2 + 1(-1|(-1cos(2\u03c0((-2x - 1)/(2x + 1))) - 3) - 1| + 3) - 1 and g(x) = (-3(5(-2|x - 3| + 3) - 5) + 3)/(3(5(-2|x - 3| + 3) - 5) + 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) = -4(5((1(-1|x - 2| - 1) - 2)/(2(-1|x - 2| - 1) + 1))^2 - 5((1(-1|x - 2| - 1) - 2)/(2(-1|x - 2| - 1) + 1))) + 1 and g(x) = (-2(-5(5x^2 - 4x - 3) + 2) + 1)/(3(-5(5x^2 - 4x - 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) = 1(2e^(-2((3x + 1)/(-1x - 2))) - 2)^2 + 5(2e^(-2((3x + 1)/(-1x - 2))) - 2) - 2 and g(x) = (-1(-2e^(-2x) - 3) + 2)/(1(-2e^(-2x) - 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": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1e^(1(-2|x - 1| - 2)) - 3 and g(x) = (-2x + 1)/(-2x) 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((-2(2|x| + 2))/(-2(2|x| + 2) - 2)) + 2 and g(x) = 1(-2e^(-1x) + 3)^2 + 4(-2e^(-1x) + 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((-3(-1|(-5x + 1) - 1| + 3) + 1)/(-1(-1|(-5x + 1) - 1| + 3) - 2))^2 - 3((-3(-1|(-5x + 1) - 1| + 3) + 1)/(-1(-1|(-5x + 1) - 1| + 3) - 2)) - 3 and g(x) = (1(5(3x + 5)^2 + 2(3x + 5) + 2) - 3)/(-2(5(3x + 5)^2 + 2(3x + 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": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (3(1x^2 + 4x + 2) - 3)/(2(1x^2 + 4x + 2) + 3) and g(x) = -5x + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2(3e^(-1((3x + 2)/(-2x))) + 3) - 5 and g(x) = 1((-1x - 3)/(3x - 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) = 2e^(2(-2|(-1x + 3) + 2| + 3)) - 2 and g(x) = 1cos(2\u03c0((-3x + 1)/(-3x - 1)) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (2(-5(-2x)^2 - 4) - 2)/(2(-5(-2x)^2 - 4) + 3) and g(x) = (-1(-1x + 4) + 1)/(2(-1x + 4) - 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 smallest x-value where the functions f(x) = 5((2(1x^2 - 3x - 4) - 2)/(3(1x^2 - 3x - 4) + 1)) - 4 and g(x) = 4((1x - 3)/(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": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 4((-3x + 1)/(-2x)) + 1 and g(x) = -4x 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^2 - 2x - 3) + 3| + 3) - 2) + 2)/(2(1(2|(-3x^2 - 2x - 3) + 3| + 3) - 2) + 2) and g(x) = -2((-1(2|x - 2|) - 1)/(-1(2|x - 2|) + 2)) + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 1(-3|(-3x^2 + 4x + 4) - 3| - 3) - 3 and g(x) = -4((-2x + 1)/(-3x + 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(-3x^2 - 2) + 1)/(-1(-3x^2 - 2) - 2) and g(x) = -3|x - 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) = (2(2(2cos(2\u03c0x + 3) + 3)^2 - 2(2cos(2\u03c0x + 3) + 3) + 4) + 1)/(-1(2(2cos(2\u03c0x + 3) + 3)^2 - 2(2cos(2\u03c0x + 3) + 3) + 4) - 2) and g(x) = 2(-2x^2 + 1x + 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) = (1(2e^(1(1|(-3x^2 - 1x - 5) + 2| - 3)) - 3) + 1)/(-1(2e^(1(1|(-3x^2 - 1x - 5) + 2| - 3)) - 3) + 2) and g(x) = -3|((-1(-5x) - 3)/(1(-5x) - 3)) + 1| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -2|((-1(1sin(1\u03c0x - 3) - 3) + 1)/(-2(1sin(1\u03c0x - 3) - 3) + 3)) + 2| + 1 and g(x) = -3|((-1x + 1)/(-2x - 2)) + 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2(-3e^(-1(5x + 5)) + 3)^2 + 1(-3e^(-1(5x + 5)) + 3) - 5 and g(x) = -5(2|x - 1| - 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": "Find the integer value closest to the x-coordinate where f(x) = -1cos(2\u03c0((1x + 2)/(-1x + 3)) - 3) + 2 and g(x) = (-2x - 2)/(2x + 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": "If you add up all the x-coordinates where the functions f(x) = 2e^(-1(3((1x + 3)/(-3x - 3))^2 + 2((1x + 3)/(-3x - 3)) + 1)) and g(x) = -2(-3x - 1)^2 + 5(-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": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -3e^(1(-3(-3|x - 3| + 2) + 3)) - 2 and g(x) = -4((2x - 2)/(3x - 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = (2(1(-1e^(1(1|x + 1| - 2))) + 2) + 3)/(-3(1(-1e^(1(1|x + 1| - 2))) + 2) + 1) and g(x) = (2(-4(-3|x + 1| - 2)^2 - 1(-3|x + 1| - 2) - 2) + 2)/(2(-4(-3|x + 1| - 2)^2 - 1(-3|x + 1| - 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-3(3x^2 + 5x + 5) + 2)/(2(3x^2 + 5x + 5) + 1) and g(x) = 2x + 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) = -1e^(1(5x - 5)) - 3 and g(x) = -3x^2 - 1x + 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(-3|(-1e^(2((-1x + 3)/(1x - 2))) + 2) - 1| + 1) and g(x) = (-3(1(2|x + 1| + 2) - 5))/(1(1(2|x + 1| + 2) - 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": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1(1(-3|((-3x - 3)/(3x)) + 3| - 2) - 4)^2 - 5(1(-3|((-3x - 3)/(3x)) + 3| - 2) - 4) and g(x) = -2|(-1((1x - 1)/(2x + 3))) + 1| - 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": "If you add up all the x-coordinates where the functions f(x) = 2(4x - 5)^2 + 1(4x - 5) - 4 and g(x) = -2|x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1sin(1\u03c0(-1(3|x + 1| + 3) + 5) + 2) + 3 and g(x) = (2(-1x^2 + 3x + 2) - 3)/(-2(-1x^2 + 3x + 2) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -2sin(2\u03c0(4((1x)/(1x)) - 1) + 2) + 3 and g(x) = (1(5x^2 - 3x - 4) - 3)/(2(5x^2 - 3x - 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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -1|((-2(5(-1x^2 - 2x + 1) - 2) - 3)/(-2(5(-1x^2 - 2x + 1) - 2) - 3)) - 1| - 2 and g(x) = -4(-2|((2x - 3)/(3x + 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": "Find the integer value closest to the x-coordinate where f(x) = (-1(2e^(2x) + 2))/(2(2e^(2x) + 2) - 1) and g(x) = 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": "Find the second intersection point (counting from left to right) between the functions f(x) = (3(1x^2 + 5x - 1) + 2)/(-3(1x^2 + 5x - 1)) and g(x) = (2x + 3)/(-3x - 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 rightmost x-coordinate where f(x) = -1|(-1x^2 - 3x + 4) + 2| - 1 and g(x) = 3x - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-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|((-1(-5x) - 1)/(-2(-5x) + 1))| - 3 and g(x) = -2|((-2x - 3)/(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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2(-2e^(-2(3|x - 2| - 3)) - 1) - 4 and g(x) = 5(-2|x - 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) = 2sin(1\u03c0((-2(2|(-2x^2 - 5x - 2) + 1| + 3) - 1)/(3(2|(-2x^2 - 5x - 2) + 1| + 3) - 1))) - 1 and g(x) = 1((-2(2x^2 - 2x - 4) + 3)/(3(2x^2 - 2x - 4) - 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -5((2(-1x^2 + 2) + 2)/(-3(-1x^2 + 2))) and g(x) = (2(-2e^(1x) + 1))/(-1(-2e^(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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1(-1|((-1x - 2)/(-2x - 3)) + 1| - 2) + 3 and g(x) = (3(-3x - 4) + 1)/(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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1(1|x + 2| + 2) - 4 and g(x) = 1x^2 - 4x + 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 first x-coordinate (rounded to the nearest integer) where f(x) = -3(-5x - 4)^2 + 2(-5x - 4) and g(x) = 4x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -2(3x + 2)^2 + 2(3x + 2) and g(x) = (-1x - 3)/(-3x + 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": "Find the first intersection point (counting from left to right) between the functions f(x) = 3(5x)^2 + 5(5x) - 2 and g(x) = -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 first intersection point (counting from left to right) between the functions f(x) = -2|((3(-2e^(-1(-4x - 5)) + 2) + 2)/(-2(-2e^(-1(-4x - 5)) + 2) + 3)) - 1| + 1 and g(x) = (-2(-2(1cos(2\u03c0x - 1) - 3) + 5) - 2)/(3(-2(1cos(2\u03c0x - 1) - 3) + 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": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1(1|x - 2| - 3)^2 - 4(1|x - 2| - 3) - 4 and g(x) = (3x)/(2x + 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1((-2x)/(-1x + 2))^2 - 2((-2x)/(-1x + 2)) + 2 and g(x) = -2x^2 - 5x - 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|(3e^(1(-4x + 5)) - 2) + 2| - 2 and g(x) = -1(2|x + 3| + 3) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2(1x^2 + 2x + 4) + 5 and g(x) = -1|x - 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": "At what integer value (rounded) do the functions f(x) = -2|((3x)/(-2x - 3)) - 3| + 3 and g(x) = (1x + 2)/(1x + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 4(-3|((-3(1x - 5))/(3(1x - 5))) + 2| + 3)^2 - 4(-3|((-3(1x - 5))/(3(1x - 5))) + 2| + 3) and g(x) = -5(-2e^(1((-3x + 1)/(-2x - 3))) + 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3((-3(-3(3|x - 3|) + 4) + 2)/(2(-3(3|x - 3|) + 4) - 3))^2 + 3((-3(-3(3|x - 3|) + 4) + 2)/(2(-3(3|x - 3|) + 4) - 3)) - 1 and g(x) = 1e^(-2((1(-1x^2 - 3x + 1) - 1)/(1(-1x^2 - 3x + 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": "At what integer value (rounded) do the functions f(x) = 3sin(1\u03c0((-1(3|x - 3| + 1) - 2)/(-2(3|x - 3| + 1))) + 1) + 3 and g(x) = 1((1x - 2)/(-3x)) + 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) = 2e^(-1(3|x + 2|)) - 3 and g(x) = 2x - 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(2|x + 1| - 3) + 3 and g(x) = 2|x| 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((-3(-3(2|x + 1| + 1)^2 + 1(2|x + 1| + 1) + 1) - 3)/(-3(-3(2|x + 1| + 1)^2 + 1(2|x + 1| + 1) + 1))) + 5 and g(x) = (2(-2(-4x^2 + 2x + 2) - 3) + 2)/(-1(-2(-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": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3((2(-3|x - 3| + 1) + 3)/(-2(-3|x - 3| + 1) + 1))^2 + 5((2(-3|x - 3| + 1) + 3)/(-2(-3|x - 3| + 1) + 1)) - 2 and g(x) = -3((-3x - 3)/(-2x + 3))^2 - 1((-3x - 3)/(-2x + 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": "If you add up all the x-coordinates where the functions f(x) = (2(2(-3e^(2(1|x - 1| - 1)) + 3) + 1) - 3)/(-3(2(-3e^(2(1|x - 1| - 1)) + 3) + 1) + 1) and g(x) = -3((1(-1|x + 3| - 2) + 3)/(2(-1|x + 3| - 2) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 5(-3e^(-1x) + 1) + 5 and g(x) = 1x + 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": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 5(3((3x - 3)/(3x)))^2 - 3(3((3x - 3)/(3x))) + 1 and g(x) = (-3(-2|x + 3| - 1) - 2)/(-1(-2|x + 3| - 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 4(-2|x| + 3) - 4 and g(x) = 3x + 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": "At what integer value (rounded) do the functions f(x) = 3(-1cos(2\u03c0(-1x + 3) + 1) + 1)^2 + 5(-1cos(2\u03c0(-1x + 3) + 1) + 1) + 4 and g(x) = 5((-3x - 3)/(2x)) - 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": "If you add up all the x-coordinates where the functions f(x) = 3|((-1(-5(-3x + 4)^2 + 1(-3x + 4)) + 1)/(-1(-5(-3x + 4)^2 + 1(-3x + 4)) + 2))| + 2 and g(x) = 1e^(2(4(2x - 3)^2 - 2(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": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (1(3x^2))/(-1(3x^2) - 1) and g(x) = 5x + 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((3x + 1)/(1x + 3)) + 3) + 3| - 3 and g(x) = -5((1x)/(1x - 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": "-11", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3cos(3\u03c0((-2x - 3)/(1x + 2))) + 2 and g(x) = 2x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 4(1|((3x - 1)/(2x + 3)) + 1| + 3)^2 - 2 and g(x) = 3|(2x^2 - 3x - 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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -1(1|(-4x + 2) + 2|)^2 + 5 and g(x) = -1(1e^(-1x) + 3)^2 - 2(1e^(-1x) + 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) = -4(-3|((-3x - 3)/(-1x + 2)) - 2| - 2) + 3 and g(x) = 3|(-1sin(1\u03c0x - 1)) - 2| + 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": "At what integer value (rounded) do the functions f(x) = 1(-1|x| + 2)^2 - 2(-1|x| + 2) + 1 and g(x) = (1x + 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3(-1e^(1((1x + 1)/(3x + 3))) + 2) + 2 and g(x) = -5(2x^2 - 2x - 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1(1|(-5((-1x + 2)/(-3x + 2)) + 4) + 1| - 3)^2 + 4(1|(-5((-1x + 2)/(-3x + 2)) + 4) + 1| - 3) - 4 and g(x) = -4((-2(-3|x - 3| - 3))/(3(-3|x - 3| - 3)))^2 - 4((-2(-3|x - 3| - 3))/(3(-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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2(-3|x + 2| - 2) + 5 and g(x) = (-1x - 2)/(2x + 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": "At what integer value (rounded) do the functions f(x) = -1((1x - 2)/(3x)) - 5 and g(x) = 1x^2 + 3x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-2(-2|x + 3| - 2) + 3)/(-2(-2|x + 3| - 2) - 1) and g(x) = 2x intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "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) = -2e^(2(3x^2 + 5x + 1)) + 3 and g(x) = (-3x + 1)/(3x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3(1|x - 1| - 2) + 5 and g(x) = -4x + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "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(-3e^(1((3(1x^2 + 4x + 4) + 3)/(1(1x^2 + 4x + 4) - 3))) - 2) - 3 and g(x) = 1((-3(-3|x + 1| + 3) + 2)/(3(-3|x + 1| + 3) + 1))^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 third intersection point (counting from left to right) between the functions f(x) = (-1(5(1sin(1\u03c0(1x) + 1) - 2)^2 + 3(1sin(1\u03c0(1x) + 1) - 2) - 5) + 2)/(-2(5(1sin(1\u03c0(1x) + 1) - 2)^2 + 3(1sin(1\u03c0(1x) + 1) - 2) - 5) - 2) and g(x) = -5(-5((-2x + 3)/(-1x - 2)) + 1)^2 - 2(-5((-2x + 3)/(-1x - 2)) + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1sin(2\u03c0(3|x + 3|)) - 3 and g(x) = (1x)/(2x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 4(2|(-5x^2 + 4x - 4) + 1| - 1) - 5 and g(x) = -2(4x^2 - 2x - 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) = -3(-3e^(-1((-1x + 1)/(1x + 3))) - 3) - 1 and g(x) = (1(-1x - 4) + 1)/(-1(-1x - 4) - 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) = 2((1(1(-2|x - 3|)^2 + 2) + 2)/(-1(1(-2|x - 3|)^2 + 2) + 2)) + 5 and g(x) = -2|((3(4x^2 + 1x) - 3)/(-1(4x^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": "At what integer value (rounded) do the functions f(x) = (-2(-3sin(1\u03c0(-1x - 3) + 1) + 3) - 3)/(3(-3sin(1\u03c0(-1x - 3) + 1) + 3)) and g(x) = (-3(-5x + 4) - 2)/(-2(-5x + 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": "If you add up all the x-coordinates where the functions f(x) = -5((1(2(1|x - 2| + 3)^2 + 2(1|x - 2| + 3) - 4))/(-1(2(1|x - 2| + 3)^2 + 2(1|x - 2| + 3) - 4))) + 2 and g(x) = 2((-3(3|x - 3| - 3) + 3)/(3(3|x - 3| - 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (2(-1e^(-1x) + 1) + 3)/(3(-1e^(-1x) + 1)) and g(x) = (1x - 1)/(2x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-9", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 2(3|x + 2| - 2)^2 + 1(3|x + 2| - 2) - 4 and g(x) = 1x^2 + 1x 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(-2(-4x^2 - 3x) - 3))/(2(-2(-4x^2 - 3x) - 3)) and g(x) = (-3(-1x^2 + 3x + 2) - 3)/(2(-1x^2 + 3x + 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": "At what integer value (rounded) do the functions f(x) = (-1(-4(1(-1|x - 3| + 3)^2 + 2(-1|x - 3| + 3) + 3) + 2))/(1(-4(1(-1|x - 3| + 3)^2 + 2(-1|x - 3| + 3) + 3) + 2) + 3) and g(x) = -5(3e^(2((3x - 2)/(-2x - 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) = -3(2e^(2((3x - 3)/(1x + 2))) + 1)^2 - 2(2e^(2((3x - 3)/(1x + 2))) + 1) and g(x) = (2(-3x^2 - 5x) + 3)/(3(-3x^2 - 5x)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3e^(-1((-1x + 2)/(-3x - 3))) + 2 and g(x) = (-3x)/(-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 first intersection point (counting from left to right) between the functions f(x) = 1((2(-3|(1cos(3\u03c0x - 1) - 2) - 2| - 1) + 1)/(-1(-3|(1cos(3\u03c0x - 1) - 2) - 2| - 1))) - 5 and g(x) = -4((-2(2x + 5))/(2(2x + 5) + 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": "-5", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3|(-1(5x + 5)^2 - 2(5x + 5) - 5) - 3| + 3 and g(x) = -4(-2cos(3\u03c0x + 2) + 1)^2 - 3(-2cos(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": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 4(1|x - 2|)^2 - 2(1|x - 2|) + 3 and g(x) = -2x^2 + 5x + 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((1x + 2)/(3x - 2)) + 2 and g(x) = (2x + 3)/(-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": "If you add up all the x-coordinates where the functions f(x) = 2x + 5 and g(x) = 2x^2 + 2x - 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) = -2((-3x - 1)/(3x - 2)) + 2 and g(x) = 2|x + 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) = 5x - 4 and g(x) = 5x^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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-2(5x^2 + 4x + 4) - 3)/(-1(5x^2 + 4x + 4) - 2) and g(x) = 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": "Find the second intersection point (counting from left to right) between the functions f(x) = 1|(5(2((-2x + 1)/(1x + 2)) + 2)^2 - 3(2((-2x + 1)/(1x + 2)) + 2) + 2) + 3| - 2 and g(x) = (3(3e^(-1(1|x + 1| - 3))) - 2)/(1(3e^(-1(1|x + 1| - 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": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(1x - 5) - 1)/(1(1x - 5)) and g(x) = 2x 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) = -2x and g(x) = -4x^2 - 4x 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|(-1cos(3\u03c0x + 1)) + 2| - 3 and g(x) = -1x^2 + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (-2(-1x^2 - 4x + 3) - 1)/(3(-1x^2 - 4x + 3)) and g(x) = -2x - 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": "-6", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 3|(-2cos(1\u03c0x - 2) - 1) + 1| - 1 and g(x) = -5x^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": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(-1e^(1((-2x + 2)/(1x + 2))) + 1) + 1| + 1 and g(x) = 3|((-3x + 2)/(-1x - 1))| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "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 + 2)/(-1x))^2 - 2((-2x + 2)/(-1x)) + 3) + 4 and g(x) = (-2(1e^(-1x)))/(-1(1e^(-1x)) + 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) = 4(-1((-3x - 1)/(-1x + 3))^2 + 2((-3x - 1)/(-1x + 3)) - 1) and g(x) = -3((1x + 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = (3(3e^(-2(1(-4x - 3)^2 - 3(-4x - 3) + 1))) + 3)/(-1(3e^(-2(1(-4x - 3)^2 - 3(-4x - 3) + 1))) + 2) and g(x) = -1|(2((3x + 1)/(3x + 2))^2 + 3((3x + 1)/(3x + 2)) + 5) - 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": "At what integer value (rounded) do the functions f(x) = -1e^(-2(2(2|((3x + 3)/(-3x - 3)) + 3| + 2)^2 - 4(2|((3x + 3)/(-3x - 3)) + 3| + 2) - 1)) - 1 and g(x) = -3|((2(4x^2 + 3x - 3))/(2(4x^2 + 3x - 3) + 3)) - 2| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1(3|((2x - 1)/(-1x + 3))| - 3) and g(x) = 2|((2x + 1)/(2x - 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": "-5", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -4(1(-2|((-3x)/(3x - 2))| + 3)^2 - 5(-2|((-3x)/(3x - 2))| + 3) - 5) and g(x) = 2((2(1e^(1x) - 2) - 1)/(-2(1e^(1x) - 2) - 1))^2 + 4((2(1e^(1x) - 2) - 1)/(-2(1e^(1x) - 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": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2|((1(-2x + 4) - 3)/(-2(-2x + 4)))| + 1 and g(x) = 3e^(1(4x + 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 integer closest to the largest x-value where the functions f(x) = 3((-3x + 3)/(2x - 1)) - 5 and g(x) = -1|x - 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) = (-3(-2(4x - 4)^2 + 3(4x - 4) - 3) - 1)/(-3(-2(4x - 4)^2 + 3(4x - 4) - 3) - 1) and g(x) = 3|((3x - 3)/(2x - 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) = -5((-2(-2|x - 2| + 2) - 1)/(1(-2|x - 2| + 2))) and g(x) = 1(-2|x| - 2)^2 - 4(-2|x| - 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1(-5(2|((3x + 3)/(-1x)) + 1|))^2 - 1(-5(2|((3x + 3)/(-1x)) + 1|)) - 1 and g(x) = 1(1sin(2\u03c0(-2|x| - 1) + 3) + 2)^2 + 2(1sin(2\u03c0(-2|x| - 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": "Find the integer value closest to the x-coordinate where f(x) = -1((2x + 1)/(-1x - 2))^2 + 3((2x + 1)/(-1x - 2)) and g(x) = (-1x + 3)/(3x + 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 first intersection point (counting from left to right) between the functions f(x) = 1(-1|(-1((-1x + 1)/(2x - 1))^2 - 2((-1x + 1)/(2x - 1)) - 5) + 3| + 3) + 4 and g(x) = -2e^(-2((-3(-1|x - 3| - 3) - 3)/(-1(-1|x - 3| - 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": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2|(2((1(-5x^2 - 5) - 3)/(-3(-5x^2 - 5) + 2))) - 2| - 2 and g(x) = 1(-5(2|x + 2|))^2 + 5(-5(2|x + 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 value closest to the x-coordinate where f(x) = (-2(4(-1|x + 2| - 1)^2 + 4) + 1)/(-2(4(-1|x + 2| - 1)^2 + 4)) and g(x) = 3e^(1(-5x + 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 second intersection point (counting from left to right) between the functions f(x) = -3(-1x + 2)^2 - 5(-1x + 2) - 2 and g(x) = -3|x - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 4(2x - 1)^2 + 3(2x - 1) + 2 and g(x) = 5x^2 + 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 second intersection point (counting from left to right) between the functions f(x) = -1(-4((1(-2|x - 1| + 1) - 3)/(3(-2|x - 1| + 1) - 1)) - 2)^2 + 5(-4((1(-2|x - 1| + 1) - 3)/(3(-2|x - 1| + 1) - 1)) - 2) - 5 and g(x) = (1(-3|(-3x + 2) + 2| - 1) + 2)/(3(-3|(-3x + 2) + 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 closest to the smallest x-value where the functions f(x) = 3|(1((2x)/(-3x - 3)) - 2) - 1| + 2 and g(x) = -1(-3x^2 + 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -5(-4(2|((3x + 3)/(-3x + 2)) - 1| + 2)^2 - 4(2|((3x + 3)/(-3x + 2)) - 1| + 2)) and g(x) = 3|(-1((-3x - 1)/(-1x))^2 + 5) - 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) = (3(1|(2(-1x)^2 + 5(-1x)) + 3| + 1) - 2)/(-3(1|(2(-1x)^2 + 5(-1x)) + 3| + 1) - 2) and g(x) = 1|(5(-3x^2 - 5x) + 5)| - 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 first intersection point (counting from left to right) between the functions f(x) = 1|((-3(1x + 4) - 1)/(-2(1x + 4) + 3)) - 1| and g(x) = -1((-2x + 3)/(1x))^2 + 5((-2x + 3)/(1x)) + 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (2(2(-1cos(2\u03c0(3|x + 3| - 1) + 3) + 3) + 5))/(3(2(-1cos(2\u03c0(3|x + 3| - 1) + 3) + 3) + 5) + 1) and g(x) = -1(5(3e^(2x) - 2)^2 - 3(3e^(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((-2(-1sin(3\u03c0(4x^2 + 1x + 4) + 3) + 2) - 2)/(-2(-1sin(3\u03c0(4x^2 + 1x + 4) + 3) + 2) + 1)) + 3 and g(x) = (-1(3(1e^(-1x) - 3) - 1) + 1)/(-3(3(1e^(-1x) - 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) = -2(-3e^(1x) - 2)^2 + 1(-3e^(1x) - 2) + 1 and g(x) = -1x^2 + 4x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-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(-3x - 1) - 1)/(-1(-3x - 1) - 1) and g(x) = 2|x - 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1e^(1(-1x - 2)) - 1 and g(x) = -4x^2 + 2x + 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) = 3|(2e^(-2(-3x^2 + 3x + 2))) - 3| + 1 and g(x) = 2e^(2((-3x)/(-2x - 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 second x-coordinate (rounded to the nearest integer) where f(x) = (3(-1x^2 + 1x - 3) - 1)/(2(-1x^2 + 1x - 3)) and g(x) = -3x^2 - 3x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "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(5x - 4) + 1)/(3(5x - 4) + 1) and g(x) = (-2x - 1)/(-1x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 5((1(2e^(1x) + 2) - 1)/(3(2e^(1x) + 2) - 2))^2 + 2((1(2e^(1x) + 2) - 1)/(3(2e^(1x) + 2) - 2)) and g(x) = 1(3|x - 2| + 1)^2 - 1(3|x - 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-3(-1|(4(-2sin(2\u03c0x) + 1)^2 - 5(-2sin(2\u03c0x) + 1)) - 1| - 1) + 1)/(-3(-1|(4(-2sin(2\u03c0x) + 1)^2 - 5(-2sin(2\u03c0x) + 1)) - 1| - 1) + 2) and g(x) = 3|((1(-3x) - 2)/(1(-3x) - 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 integer value closest to the x-coordinate where f(x) = -1e^(1((-1(-2x + 5) + 3)/(2(-2x + 5) + 1))) and g(x) = (-2(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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-2(2(-2e^(-2(1x^2 - 1x + 1)) + 1) - 4) + 1)/(2(2(-2e^(-2(1x^2 - 1x + 1)) + 1) - 4) + 2) and g(x) = (1(5(1x)^2 + 4(1x) - 3) + 1)/(2(5(1x)^2 + 4(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": "Find the integer closest to the smallest x-value where the functions f(x) = 1|((-2(-5(4x^2 + 2x + 1) + 2) - 2)/(2(-5(4x^2 + 2x + 1) + 2) + 3)) - 3| + 1 and g(x) = -3(2(-3e^(-2x) + 2)^2 - 3(-3e^(-2x) + 2) - 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": "At what integer value (rounded) do the functions f(x) = 3|((-2x + 1)/(1x + 3)) - 3| - 1 and g(x) = 5x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3(2|x - 1| + 2) + 5 and g(x) = -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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1(-5((-1x + 1)/(-2x + 3)))^2 - 3(-5((-1x + 1)/(-2x + 3))) and g(x) = (-1(-1x^2 - 5x + 4) - 2)/(-1(-1x^2 - 5x + 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": "At what integer value (rounded) do the functions f(x) = -1((-1(1e^(1(-2x^2 - 3x - 2)) - 1) - 2)/(2(1e^(1(-2x^2 - 3x - 2)) - 1) - 3)) - 1 and g(x) = -2(2((-1x - 2)/(1x - 2)) + 2)^2 - 1(2((-1x - 2)/(1x - 2)) + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2|((-1x - 2)/(3x + 1)) - 2| + 3 and g(x) = (-1x - 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) = 2e^(1(1x^2 + 4x)) + 2 and g(x) = (-2x - 2)/(2x) 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) = -4x and g(x) = -2x^2 + 3x + 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) = (-3(-3x^2 + 5) - 1)/(2(-3x^2 + 5) - 3) and g(x) = -4x^2 + 5x 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) = -2|(-5(2e^(-2(-5x^2 + 1x + 3)) - 2) + 1) - 1| + 3 and g(x) = -1(-2sin(2\u03c0((2x - 3)/(3x + 3)) + 2) + 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) = (1(2|x - 3| + 3) - 1)/(-2(2|x - 3| + 3) - 1) and g(x) = 1x + 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1(-2e^(1(-1x^2 + 4x + 1)) - 2) - 2 and g(x) = -1(4x + 1)^2 + 3(4x + 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 largest x-value where the functions f(x) = (1(-2(-2|(-2x + 4) + 3| + 3)^2 + 5(-2|(-2x + 4) + 3| + 3) - 3) + 3)/(-3(-2(-2|(-2x + 4) + 3| + 3)^2 + 5(-2|(-2x + 4) + 3| + 3) - 3) + 2) and g(x) = (-2(-2(1|x + 3| + 1) + 2) - 2)/(2(-2(1|x + 3| + 1) + 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 x-coordinate where the graphs of f(x) = 3|(-2e^(-1x)) - 3| + 1 and g(x) = 2|x - 3| - 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 2|(-3((-1(1sin(1\u03c0x - 1) + 2) - 2)/(-1(1sin(1\u03c0x - 1) + 2) - 3)) + 5) + 1| - 1 and g(x) = 2(1|((3x - 2)/(-2x + 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 first intersection point (counting from left to right) between the functions f(x) = -3|(-5(-2x + 3)^2 - 1(-2x + 3) + 5)| - 2 and g(x) = -2(5x^2 + 1x + 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2|(5(4((3x)/(3x - 2))^2 + 1((3x)/(3x - 2))) + 1) - 2| + 3 and g(x) = (1(-1|(-5x^2 + 4x + 5) + 2| + 3))/(-2(-1|(-5x^2 + 4x + 5) + 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 x-coordinate where the graphs of f(x) = 1e^(1(-2|x - 2|)) + 2 and g(x) = -2x + 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) = (-3(3(1(2|x + 1| - 2)^2 + 1(2|x + 1| - 2) + 3) + 2) + 1)/(-2(3(1(2|x + 1| - 2)^2 + 1(2|x + 1| - 2) + 3) + 2) - 2) and g(x) = (3(-4(-2e^(-2x) + 1) - 4) + 3)/(1(-4(-2e^(-2x) + 1) - 4) - 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(2|(3(-3x^2 + 4x) + 3) + 3| - 2) - 3)/(-1(2|(3(-3x^2 + 4x) + 3) + 3| - 2) + 3) and g(x) = (-1(3(2|x + 1| - 1) - 2) + 2)/(-3(3(2|x + 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 largest x-value where the functions f(x) = 5((-3(-2|(2x + 2)| + 3) - 1)/(-2(-2|(2x + 2)| + 3) - 2))^2 - 4((-3(-2|(2x + 2)| + 3) - 1)/(-2(-2|(2x + 2)| + 3) - 2)) + 4 and g(x) = (1(2|(-5x + 4) + 3| + 1) - 3)/(-2(2|(-5x + 4) + 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) = (-3(3x^2 - 1))/(-1(3x^2 - 1) - 2) and g(x) = (2x + 3)/(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": "Find the integer value closest to the x-coordinate where f(x) = -3(-3|((-3x - 2)/(1x + 1)) - 1| - 1)^2 - 2(-3|((-3x - 2)/(1x + 1)) - 1| - 1) - 3 and g(x) = -1|(5x^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": "10", "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((2x + 1)/(3x - 1))^2 - 5((2x + 1)/(3x - 1)) - 3)) + 3| + 2 and g(x) = (-1(4(-2x - 5)^2 + 5) + 1)/(-3(4(-2x - 5)^2 + 5) + 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) = -3((1(3|(1x + 1) + 3| + 1) - 3)/(-1(3|(1x + 1) + 3| + 1) + 3))^2 - 4((1(3|(1x + 1) + 3| + 1) - 3)/(-1(3|(1x + 1) + 3| + 1) + 3)) + 4 and g(x) = 5(3((-1x - 3)/(-1x - 3)) - 2)^2 - 1(3((-1x - 3)/(-1x - 3)) - 2) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-3(1x - 1) - 2)/(-1(1x - 1) + 2) and g(x) = (1x - 3)/(1x - 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) = -2|(2x + 1) + 3| - 1 and g(x) = 2x - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (1(3e^(1x) - 3) + 1)/(1(3e^(1x) - 3)) and g(x) = 1|x| + 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|((-1(4x^2 - 2x + 2) - 1)/(-3(4x^2 - 2x + 2) - 2)) + 1| and g(x) = -3e^(1(1x - 4)) + 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1|(2x - 2)| + 3 and g(x) = 3x^2 - 4x - 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) = 1|(-1e^(2x) + 3) - 1| - 3 and g(x) = -3x + 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) = 1(3|((-1x + 2)/(-1x + 3)) - 2| - 1) + 4 and g(x) = (2(-2|x - 2| - 1) - 1)/(1(-2|x - 2| - 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -4x + 2 and g(x) = 4x^2 + 4x + 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) = (1(4(-4(1e^(-2x)) - 2)^2 - 1(-4(1e^(-2x)) - 2) + 2) - 3)/(1(4(-4(1e^(-2x)) - 2)^2 - 1(-4(1e^(-2x)) - 2) + 2)) and g(x) = -1((3(2|x - 2| - 3) - 3)/(-3(2|x - 2| - 3) + 4)) - 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 first intersection point (counting from left to right) between the functions f(x) = 5(-2x^2 + 4x + 2) + 4 and g(x) = -1|x + 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": "At what integer value (rounded) do the functions f(x) = -3(-4x^2 + 3x - 5) + 5 and g(x) = (2x + 3)/(1x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1(-2x - 5))/(-1(-2x - 5) + 2) and g(x) = -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": "-3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1cos(2\u03c0(4((3x)/(3x + 3)) - 5) + 1) - 1 and g(x) = (2(1x - 5) + 2)/(1(1x - 5)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-2(-3x^2 + 1x + 4) + 1)/(-1(-3x^2 + 1x + 4) + 3) and g(x) = 5x - 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(-2e^(2(3(-2|x| + 2) + 3))) + 1)/(3(-2e^(2(3(-2|x| + 2) + 3))) - 2) and g(x) = -4(1(2|x - 1| - 1)^2 + 4(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": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2(-1cos(2\u03c0(1|x - 2| - 2) - 1) + 3)^2 - 3(-1cos(2\u03c0(1|x - 2| - 2) - 1) + 3) - 4 and g(x) = -2|(3e^(1x) + 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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 2|(-1(-1((-3x + 2)/(3x))^2 - 4) + 5) + 3| + 2 and g(x) = 1|(1(3x - 3)^2 + 4(3x - 3) - 4) + 1| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3(2e^(1x) - 1)^2 + 4(2e^(1x) - 1) + 5 and g(x) = 2|x - 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": "-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|(-3(2((2x)/(3x + 3)) - 5)^2 + 3(2((2x)/(3x + 3)) - 5) + 3) - 2| - 1 and g(x) = 5(3|((-3x + 3)/(-1x + 3)) - 1| - 3) - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -3|(-1x^2 - 2x + 2)| + 1 and g(x) = -5x^2 - 2x + 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 rightmost x-coordinate where f(x) = (2(-2(-2|(-4x^2 + 2x - 3) + 3|)) - 1)/(-2(-2(-2|(-4x^2 + 2x - 3) + 3|))) and g(x) = 4(3|((1x - 2)/(3x)) - 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) = (-1(-3(2e^(-2(4x^2 - 5x + 1)) - 3) - 1))/(2(-3(2e^(-2(4x^2 - 5x + 1)) - 3) - 1) - 1) and g(x) = (1(-1|(-4x^2 - 5x - 5) + 1| + 1) - 1)/(3(-1|(-4x^2 - 5x - 5) + 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 integer closest to the smallest x-value where the functions f(x) = 2((-1(1(2|x - 3| - 1)) - 1)/(1(1(2|x - 3| - 1)) + 3))^2 - 2((-1(1(2|x - 3| - 1)) - 1)/(1(1(2|x - 3| - 1)) + 3)) + 4 and g(x) = (2(2(-3x^2 + 5x + 5) + 5) + 2)/(-3(2(-3x^2 + 5x + 5) + 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(2sin(1\u03c0x + 3) + 3) - 3)/(1(2sin(1\u03c0x + 3) + 3) + 1) and g(x) = -2x^2 + 3x + 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) = 4((-2x - 2)/(-2x - 2))^2 - 2((-2x - 2)/(-2x - 2)) - 5 and g(x) = -1|x - 3| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "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) = (-1(2(1x - 5)^2 + 1(1x - 5)) + 2)/(-2(2(1x - 5)^2 + 1(1x - 5)) - 1) and g(x) = (3(3|x - 2| + 3) - 3)/(-2(3|x - 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": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -1|(1x^2 - 1x + 1) - 3| and g(x) = 3|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": "If you add up all the x-coordinates where the functions f(x) = -1((-1(1|(1x^2 + 2x - 3) + 3| - 2) - 2)/(1(1|(1x^2 + 2x - 3) + 3| - 2) + 1)) - 1 and g(x) = 3((-2(4x - 3) - 2)/(-3(4x - 3) + 1))^2 + 5((-2(4x - 3) - 2)/(-3(4x - 3) + 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": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2x - 4 and g(x) = -5x^2 - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2|(-3((1(-4x - 2) - 2)/(-2(-4x - 2)))^2 + 4((1(-4x - 2) - 2)/(-2(-4x - 2))) - 5) + 2| - 1 and g(x) = (-2(3(1x^2 + 2x - 5)) + 2)/(3(3(1x^2 + 2x - 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": "If you add up all the x-coordinates where the functions f(x) = 3(2|((-2x - 1)/(-2x + 3)) - 3| - 3)^2 + 2 and g(x) = 3|(3x^2 + 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": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2x + 1 and g(x) = -4x^2 - 2x 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|(1((1(-5x) - 1)/(2(-5x)))^2 + 4((1(-5x) - 1)/(2(-5x))) - 2) - 1| - 2 and g(x) = 3|(4(-1e^(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": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(-4(-2sin(1\u03c0x + 1) + 2)^2 + 5(-2sin(1\u03c0x + 1) + 2) - 1) - 1| - 2 and g(x) = -4(3x - 4)^2 - 4(3x - 4) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2(2|x| + 3) - 5 and g(x) = (2x)/(-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 value closest to the x-coordinate where f(x) = (-3(-2x^2 + 3x) - 3)/(2(-2x^2 + 3x) + 2) and g(x) = -3x - 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|x - 1| + 1) - 5 and g(x) = 1|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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 3((-2x)/(3x + 2))^2 - 4 and g(x) = (-1x - 2)/(-2x + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2|((2(3(-4x^2 - 4x + 5) + 4) - 1)/(-1(3(-4x^2 - 4x + 5) + 4))) - 1| and g(x) = 1(-3(-1|x + 2|)^2 + 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": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -4(-1|(1(-2sin(2\u03c0x + 2))^2 - 3(-2sin(2\u03c0x + 2)) + 2) + 3| - 2) + 1 and g(x) = (3(5(-3cos(1\u03c0x) - 1)^2 + 4(-3cos(1\u03c0x) - 1) - 5))/(2(5(-3cos(1\u03c0x) - 1)^2 + 4(-3cos(1\u03c0x) - 1) - 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": "Find the integer closest to the largest x-value where the functions f(x) = 2|(-2x - 4) + 2| + 3 and g(x) = 3x^2 - 4x 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|((2x + 3)/(2x - 1)) + 1| - 1)^2 - 4(-2|((2x + 3)/(2x - 1)) + 1| - 1) + 1 and g(x) = 3|(-2cos(2\u03c0x + 2) - 1) - 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1|(5(-1e^(2x) - 2)^2 + 3(-1e^(2x) - 2) - 2)| and g(x) = (1(1x + 4) + 3)/(-2(1x + 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": "At what integer value (rounded) do the functions f(x) = -3|((3(4x^2 + 2x + 1) + 3)/(-1(4x^2 + 2x + 1))) + 2| - 2 and g(x) = -2((1x + 2)/(1x)) - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-2(4x^2 + 3x + 1) + 3)/(-3(4x^2 + 3x + 1) + 1) and g(x) = (-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": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-2(3|(-3x^2 + 2x + 5) - 1| - 1) - 3)/(-3(3|(-3x^2 + 2x + 5) - 1| - 1)) and g(x) = 3e^(-2((-2x + 1)/(3x + 1))) + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-2(-1x + 3) - 3)/(2(-1x + 3) - 1) and g(x) = -1|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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (3(-5(-4x)^2 + 1(-4x) + 4) + 2)/(-2(-5(-4x)^2 + 1(-4x) + 4) - 2) and g(x) = 1e^(1(1x + 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2(1|(3x - 4) + 1| - 3)^2 - 1 and g(x) = 4((1x - 3)/(-2x - 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": "If you add up all the x-coordinates where the functions f(x) = -1e^(2(-2|(-5x - 2) - 2| - 3)) and g(x) = -2(1x^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": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-2(1(2(-1|x + 1| - 1) + 2)^2 + 4(2(-1|x + 1| - 1) + 2)))/(2(1(2(-1|x + 1| - 1) + 2)^2 + 4(2(-1|x + 1| - 1) + 2)) - 3) and g(x) = -1((-2(-1sin(2\u03c0x + 1) + 1))/(3(-1sin(2\u03c0x + 1) + 1) + 2)) - 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 integer value closest to the x-coordinate where f(x) = -3(-1x^2 + 5x - 4) + 4 and g(x) = 2x^2 + 5x - 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": "Find the integer closest to the largest x-value where the functions f(x) = (2(-5(3(1|x| + 2)^2 + 2(1|x| + 2) + 4) + 1) + 2)/(-2(-5(3(1|x| + 2)^2 + 2(1|x| + 2) + 4) + 1) - 3) and g(x) = -1|(2((2x + 3)/(3x + 1)) - 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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -2|(4(-1x^2 + 1x + 2) - 1) + 2| - 2 and g(x) = -3|(5x^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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1|(3e^(1((-2(2x^2 + 2x - 4) + 1)/(1(2x^2 + 2x - 4) + 1))) + 3) + 2| - 2 and g(x) = (2(1(-2x^2 - 3x + 4) + 3) - 1)/(3(1(-2x^2 - 3x + 4) + 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": "If you add up all the x-coordinates where the functions f(x) = (1(-2e^(2(-3|(5x^2 + 1) - 3|)) + 1))/(-2(-2e^(2(-3|(5x^2 + 1) - 3|)) + 1) - 1) and g(x) = -1(2|(2x + 1) - 3|)^2 - 3(2|(2x + 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": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (1(1x) - 3)/(-3(1x) + 1) and g(x) = 2|x + 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (1(-1|x + 1| - 2) + 3)/(1(-1|x + 1| - 2) - 2) and g(x) = (-1x + 3)/(1x + 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) = -3((2(2|x + 2| + 3) + 1)/(3(2|x + 2| + 3) + 1))^2 - 3((2(2|x + 2| + 3) + 1)/(3(2|x + 2| + 3) + 1)) + 3 and g(x) = (1(-1e^(2x) + 3) + 2)/(-1(-1e^(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": "Find the integer closest to the smallest x-value where the functions f(x) = -5((-1x + 3)/(-1x - 1))^2 + 5((-1x + 3)/(-1x - 1)) - 5 and g(x) = -1x^2 + 5x - 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 rightmost x-coordinate where f(x) = -1|((-3(1sin(2\u03c0x - 2) + 2))/(-2(1sin(2\u03c0x - 2) + 2) - 2))| and g(x) = 5(3x - 1)^2 - 5(3x - 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) = -5x - 2 and g(x) = -2x^2 - 2x + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "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 + 5)^2 + 1) + 1)/(2(-2(-5x + 5)^2 + 1)) and g(x) = 1((1x - 1)/(-2x - 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(-1x - 3) - 1)/(-3(-1x - 3) - 3) and g(x) = (2x - 2)/(1x) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (2(-1x + 3) + 2)/(2(-1x + 3) + 3) and g(x) = -3x - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (1(-3(1(-2|x - 1| + 3)^2 + 4(-2|x - 1| + 3) - 5) - 4))/(-3(-3(1(-2|x - 1| + 3)^2 + 4(-2|x - 1| + 3) - 5) - 4)) and g(x) = 5((-1(-5x - 3) - 1)/(-2(-5x - 3) + 3))^2 + 5((-1(-5x - 3) - 1)/(-2(-5x - 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": "If you add up all the x-coordinates where the functions f(x) = 1x + 4 and g(x) = -4x^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": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -2(-1|(-3((2x)/(-2x - 2))^2 + 4((2x)/(-2x - 2)) - 3) - 1| + 3) - 5 and g(x) = (-1(2(-3x^2 + 4x + 3) - 1))/(1(2(-3x^2 + 4x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3(-1|((2x - 2)/(1x + 1))| - 1) + 5 and g(x) = 2(-2e^(-2x)) + 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 first x-coordinate (rounded to the nearest integer) where f(x) = (3(-5(-3e^(1(3|x|))) - 1) - 1)/(2(-5(-3e^(1(3|x|))) - 1) - 3) and g(x) = -4(3|(2x - 4) - 3|)^2 + 5(3|(2x - 4) - 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 integer closest to the smallest x-value where the functions f(x) = -1(2e^(2(4x^2 + 1x - 5)) - 3) - 3 and g(x) = -1e^(1((2x + 3)/(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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-3(1(2e^(-2x) - 2)^2 - 5(2e^(-2x) - 2) + 2) + 3)/(2(1(2e^(-2x) - 2)^2 - 5(2e^(-2x) - 2) + 2) + 2) and g(x) = (-2(1x + 5) + 1)/(-1(1x + 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": "At what integer value (rounded) do the functions f(x) = (1(-1x - 1) + 2)/(-2(-1x - 1) + 3) and g(x) = 2|x + 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": "At what integer value (rounded) do the functions f(x) = -5(-5x^2 - 1x - 1) - 4 and g(x) = (-1x + 3)/(-3x + 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) = 1sin(1\u03c0(1|((2(-5x + 5) - 2)/(1(-5x + 5) - 3)) + 1| + 1) + 3) - 2 and g(x) = -2(2|(-3x - 3) - 1|)^2 - 3(2|(-3x - 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": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -4(2x + 5)^2 + 5(2x + 5) - 5 and g(x) = (-1x + 3)/(-1x) 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|x - 1| + 3) + 3 and g(x) = -2x^2 - 3x intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -4(2((-3x - 2)/(-1x))^2 + 3((-3x - 2)/(-1x))) - 2 and g(x) = -4(-1e^(-1x) - 3)^2 + 4(-1e^(-1x) - 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) = 3(-3((3(3|x - 2| + 1) + 3)/(3(3|x - 2| + 1))) + 5)^2 + 3(-3((3(3|x - 2| + 1) + 3)/(3(3|x - 2| + 1))) + 5) + 3 and g(x) = (-1(2e^(-1(3|x - 3|)) + 1) - 3)/(-1(2e^(-1(3|x - 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -4(-1|(5((2x + 1)/(2x + 3)) + 5) - 1| + 2)^2 + 4(-1|(5((2x + 1)/(2x + 3)) + 5) - 1| + 2) - 3 and g(x) = -3(-2((1x - 3)/(-1x + 2)) + 4)^2 + 5(-2((1x - 3)/(-1x + 2)) + 4) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -3(-2e^(1((1(1|x - 2| - 1) + 1)/(1(1|x - 2| - 1) - 2))) - 1)^2 - 5(-2e^(1((1(1|x - 2| - 1) + 1)/(1(1|x - 2| - 1) - 2))) - 1) - 2 and g(x) = (2(2|(-1x) + 2|) + 2)/(-2(2|(-1x) + 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2(-1e^(2((-1x + 3)/(2x + 1))) + 3) + 1 and g(x) = (1(-1e^(2x) - 1))/(-3(-1e^(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 rightmost x-coordinate where f(x) = -5(2e^(-2(-1(1|x - 1| - 1)^2 + 3)) - 2) - 5 and g(x) = -3e^(2(3(-2|x| - 1) - 4)) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3|(2e^(-2x) + 2) + 2| + 1 and g(x) = 1x^2 + 2x + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 4(1e^(-1x) - 2) - 2 and g(x) = 3x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -4(5(-1|((2x + 2)/(2x - 1)) - 1| - 1)^2 + 1(-1|((2x + 2)/(2x - 1)) - 1| - 1) - 1) - 5 and g(x) = -1((3(-3x + 4) + 1)/(-3(-3x + 4) - 3))^2 - 3((3(-3x + 4) + 1)/(-3(-3x + 4) - 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 integer value closest to the x-coordinate where f(x) = 2(-3|x - 2| - 2)^2 + 4(-3|x - 2| - 2) + 2 and g(x) = (1x + 3)/(-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": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (2(3cos(1\u03c0(4x + 2) + 1) - 1) - 3)/(2(3cos(1\u03c0(4x + 2) + 1) - 1) + 1) and g(x) = -5(2|x - 2| + 3)^2 + 1(2|x - 2| + 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": "If you add up all the x-coordinates where the functions f(x) = -1(1((1x + 3)/(2x - 1))^2 + 1((1x + 3)/(2x - 1)) - 2) - 3 and g(x) = 5(2x^2 + 5x - 4) - 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) = 3x + 5 and g(x) = -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": "Find the integer value closest to the x-coordinate where f(x) = -2(-1|x + 2| - 2)^2 - 2(-1|x + 2| - 2) - 3 and g(x) = (1x - 2)/(-2x - 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 closest to the largest x-value where the functions f(x) = 1(-2|((-1(4x + 3) + 2)/(-3(4x + 3) - 3))| - 3)^2 - 2(-2|((-1(4x + 3) + 2)/(-3(4x + 3) - 3))| - 3) + 2 and g(x) = -3(-3(-2|x - 2| - 2)^2 + 4(-2|x - 2| - 2) + 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 smallest x-value where the functions f(x) = (-3(-1e^(-2x) + 1) - 1)/(2(-1e^(-2x) + 1) - 3) and g(x) = -1x^2 + 4x + 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 largest x-value where the functions f(x) = -3|(4x + 5)| + 1 and g(x) = 1|x + 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 closest to the largest x-value where the functions f(x) = -1((2(-1|x + 2|))/(-2(-1|x + 2|) - 3)) - 1 and g(x) = 2|(-1e^(2x) - 2) - 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) = (-2(-3(2|(-1sin(2\u03c0x - 1) - 3) - 2| + 1)^2 - 1(2|(-1sin(2\u03c0x - 1) - 3) - 2| + 1) + 4) + 1)/(-1(-3(2|(-1sin(2\u03c0x - 1) - 3) - 2| + 1)^2 - 1(2|(-1sin(2\u03c0x - 1) - 3) - 2| + 1) + 4) - 2) and g(x) = 3|((2(-1x + 2) + 1)/(-1(-1x + 2) + 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) = (-2(-5x + 5) + 1)/(-2(-5x + 5) + 1) and g(x) = 1x - 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 leftmost x-coordinate where f(x) = 4(-1|(5((-2x + 3)/(-2x + 1))^2 - 5((-2x + 3)/(-2x + 1)) + 3) - 2| + 3) + 3 and g(x) = (1(2(3e^(-1x) - 3) + 3) + 1)/(-2(2(3e^(-1x) - 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": "If you add up all the x-coordinates where the functions f(x) = 3e^(1(-2|(-5(-5x^2 - 4x + 3) - 2) - 2| - 1)) and g(x) = -3e^(1((-3(4x^2 + 3x - 1))/(-1(4x^2 + 3x - 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": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -2(-4x^2 + 4x + 5) + 3 and g(x) = -2x + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1|(-4(-1x^2 + 1x) - 4) - 3| - 1 and g(x) = -3(-3x - 2)^2 + 5(-3x - 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": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1|(-1((2x + 3)/(1x - 3)) - 4) - 2| + 1 and g(x) = -1|(-1x + 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": "Find the integer value closest to the x-coordinate where f(x) = -1|(1e^(1x) - 1)| and g(x) = (3x + 3)/(3x - 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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -1|(1(2x^2 - 2x + 2) - 1) - 3| + 3 and g(x) = -4(3x^2 - 4x) 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|x| + 3) + 4 and g(x) = (2x - 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 integer closest to the largest x-value where the functions f(x) = 1(-2|((2(-4x - 5) + 3)/(-1(-4x - 5) - 1)) - 2| - 2)^2 - 5(-2|((2(-4x - 5) + 3)/(-1(-4x - 5) - 1)) - 2| - 2) - 1 and g(x) = 3|(-3e^(-1((2x - 3)/(2x))) - 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 value closest to the x-coordinate where f(x) = (-2(-3(3|(-3x^2 + 3x - 2) - 2| - 3) + 1) + 2)/(-3(-3(3|(-3x^2 + 3x - 2) - 2| - 3) + 1) - 2) and g(x) = 1((2(-3e^(-2x)) + 2)/(-2(-3e^(-2x)) - 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 integer closest to the smallest x-value where the functions f(x) = -1x + 1 and g(x) = -3x^2 - 5x + 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2|((-3(-3x - 3) + 3)/(-2(-3x - 3) - 1)) - 2| + 3 and g(x) = 1(-3|x + 1| + 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = (2(-2|(-1(-1x - 3)^2 - 3) + 1| - 3) + 3)/(-2(-2|(-1(-1x - 3)^2 - 3) + 1| - 3)) and g(x) = -1((1(-1x - 2) + 1)/(1(-1x - 2) - 2))^2 - 5((1(-1x - 2) + 1)/(1(-1x - 2) - 2)) - 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": "At what integer value (rounded) do the functions f(x) = 1(-3|(2((-2x - 2)/(3x))^2 + 5((-2x - 2)/(3x)) - 4) + 3| - 2) - 4 and g(x) = -1e^(2((-1(1x - 4) - 1)/(3(1x - 4) - 2))) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 2|(-3cos(3\u03c0(-5((2x - 2)/(1x - 3)) - 1) - 2) + 1) - 3| + 2 and g(x) = -2((-1(3|x - 1| - 3))/(-3(3|x - 1| - 3) - 3)) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (3(1cos(1\u03c0(-1x^2 - 5x - 4) - 3) + 3) + 3)/(-3(1cos(1\u03c0(-1x^2 - 5x - 4) - 3) + 3) - 2) and g(x) = 4(-1e^(-1x) + 2)^2 - 4(-1e^(-1x) + 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) = -5((-2(2x^2 - 1x + 1) + 2)/(-1(2x^2 - 1x + 1) + 2)) - 5 and g(x) = 5(3|x - 1| - 2)^2 - 1(3|x - 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2e^(-1(-4(3|((1x + 1)/(3x + 2)) + 1| + 1) + 4)) + 3 and g(x) = -1(-3((-1x - 1)/(-1x - 1)) - 3)^2 + 2(-3((-1x - 1)/(-1x - 1)) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1|(1(-2cos(1\u03c0x - 2) - 2) + 2) - 2| + 3 and g(x) = -5((-2x - 2)/(1x)) - 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) = 1(1|(-4x^2 - 1x - 5) + 3| - 3) - 3 and g(x) = -2|((1x + 1)/(-1x - 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": "Find the integer closest to the largest x-value where the functions f(x) = 3((2(1(1|x + 3|)^2 + 4(1|x + 3|) - 2) + 2)/(-2(1(1|x + 3|)^2 + 4(1|x + 3|) - 2) - 3)) + 3 and g(x) = -1|(1((1x + 2)/(3x + 3)) + 4) - 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 largest x-value where the functions f(x) = 4(2e^(-2x) - 2) + 4 and g(x) = -5x^2 + 4x + 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) = 1x + 2 and g(x) = -2x^2 - 2x + 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": "If you add up all the x-coordinates where the functions f(x) = -5(3|(1((3x - 2)/(1x - 3)) - 2)| + 1)^2 - 1(3|(1((3x - 2)/(1x - 3)) - 2)| + 1) - 3 and g(x) = -3(1((1x + 3)/(-3x - 3)) - 5)^2 - 1(1((1x + 3)/(-3x - 3)) - 5) - 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 value closest to the x-coordinate where f(x) = -1(-2(-1|((-2x)/(1x)) - 3| + 2) + 2)^2 + 2(-2(-1|((-2x)/(1x)) - 3| + 2) + 2) + 5 and g(x) = -2|(1e^(1((2x + 1)/(-1x - 2))) - 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = -4(3|((2(2x^2 - 5x + 3) - 1)/(2(2x^2 - 5x + 3) - 1)) - 3| + 1) + 5 and g(x) = -2|(-1(5x - 4)^2 - 3(5x - 4) - 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) = 4((3(-5(-1e^(1x))^2 - 5(-1e^(1x)) + 3) - 3)/(2(-5(-1e^(1x))^2 - 5(-1e^(1x)) + 3))) and g(x) = (-1(3|(2e^(-1x) - 3) + 1| + 3) + 3)/(-1(3|(2e^(-1x) - 3) + 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 rightmost x-coordinate where f(x) = 1(1|(-4x - 2) + 1|)^2 + 2(1|(-4x - 2) + 1|) + 4 and g(x) = -2(-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) = 2cos(1\u03c0(1|(-2x + 2) + 2| - 3) + 3) + 3 and g(x) = (-3(4x - 5) - 3)/(3(4x - 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": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(1|(-3x^2 + 5x + 5) + 3| - 2) + 3)/(-3(1|(-3x^2 + 5x + 5) + 3| - 2) + 2) and g(x) = -4(-5x + 4)^2 + 1(-5x + 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": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -2|(-3x)| + 2 and g(x) = 4x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1e^(2(1|(-4x - 2) + 3| - 1)) - 1 and g(x) = 1(-2sin(1\u03c0x) + 1)^2 - 1(-2sin(1\u03c0x) + 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": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1(-1|(-4((1x + 2)/(2x - 1)) + 1) + 1| - 3)^2 + 2(-1|(-4((1x + 2)/(2x - 1)) + 1) + 1| - 3) - 3 and g(x) = -3|(-4((2x - 1)/(1x + 1))^2 - 2((2x - 1)/(1x + 1)) + 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 first x-coordinate (rounded to the nearest integer) where f(x) = (1(1|(-5(1x - 1)^2 - 1(1x - 1) + 2) + 1| + 3) - 1)/(2(1|(-5(1x - 1)^2 - 1(1x - 1) + 2) + 1| + 3) - 1) and g(x) = -2(-1(-2e^(1x) + 1)^2 - 5(-2e^(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 x-coordinate where the graphs of f(x) = 2((-3x - 2)/(-3x + 1)) - 5 and g(x) = 1x^2 - 1x + 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) = -2|((3(-1sin(3\u03c0x + 1) - 3) - 2)/(-2(-1sin(3\u03c0x + 1) - 3) - 2)) - 3| + 2 and g(x) = (-2(-2|x + 3| + 1) - 2)/(-1(-2|x + 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3e^(1(1((-2x - 2)/(1x - 2))^2 - 5)) + 3 and g(x) = 3((-2x - 1)/(-2x - 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) = 2(-3|(-2e^(1((1x)/(2x - 1))) + 1) - 1| + 1)^2 - 5(-3|(-2e^(1((1x)/(2x - 1))) + 1) - 1| + 1) - 1 and g(x) = -2sin(2\u03c0((1(-1x + 4) + 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": "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(4x - 1)^2 - 2(4x - 1) + 1 and g(x) = 2x^2 - 1x intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1e^(1(1x + 4)) - 1 and g(x) = 1|x + 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": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (1(-1(-1|(-1e^(-1x) + 2) + 1| - 2) - 5) - 2)/(-3(-1(-1|(-1e^(-1x) + 2) + 1| - 2) - 5) + 1) and g(x) = (-2(-2(1|x| - 3) + 2) + 3)/(-2(-2(1|x| - 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": "-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) = 1e^(-1(-2|(-1x - 3) + 1| + 3)) - 2 and g(x) = -2e^(-1((-3x + 3)/(-1x + 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": "At what integer value (rounded) do the functions f(x) = -3(2sin(3\u03c0x + 1) + 1)^2 - 3(2sin(3\u03c0x + 1) + 1) - 1 and g(x) = (-3x - 1)/(-2x - 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) = 4(3e^(1(1((-2x - 3)/(-1x + 2)) - 2)) - 2)^2 + 2(3e^(1(1((-2x - 3)/(-1x + 2)) - 2)) - 2) - 3 and g(x) = (1(2e^(1(-1x^2 - 2x)) + 1) + 2)/(2(2e^(1(-1x^2 - 2x)) + 1)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1|(3(5x + 5)^2 - 4(5x + 5) - 2)| - 2 and g(x) = (-1(-1x - 3) + 3)/(3(-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": "-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) = (-3(-3|(1(-1x + 1)^2 - 1(-1x + 1) + 2) + 2| + 2) + 2)/(-2(-3|(1(-1x + 1)^2 - 1(-1x + 1) + 2) + 2| + 2) + 3) and g(x) = (-1(-1(1x^2 + 5x + 2) + 5) + 2)/(-3(-1(1x^2 + 5x + 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": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -4((-1(1|x + 1|) - 3)/(3(1|x + 1|) + 3))^2 + 4((-1(1|x + 1|) - 3)/(3(1|x + 1|) + 3)) - 5 and g(x) = -2|(-1x - 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 third intersection point (counting from left to right) between the functions f(x) = (-2(1x + 4) + 3)/(1(1x + 4) - 2) and g(x) = -3x^2 + 5x + 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": "At what integer value (rounded) do the functions f(x) = 2cos(3\u03c0((-1x)/(-2x + 3)) + 3) - 2 and g(x) = 3x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -2|(-2sin(3\u03c0(-1((-2x - 3)/(2x + 2)) + 5) - 3) + 2) + 1| - 2 and g(x) = (1(-2|(-3e^(-2x) + 3) + 1| + 3) - 1)/(2(-2|(-3e^(-2x) + 3) + 1| + 3) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (-3(-1|(1(-2e^(-2x) + 3) + 1) - 3| + 2) + 2)/(-2(-1|(1(-2e^(-2x) + 3) + 1) - 3| + 2) - 2) and g(x) = 5(4((-3x + 2)/(2x - 1)) + 5)^2 - 2(4((-3x + 2)/(2x - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1sin(2\u03c0(1|x + 3| - 2) + 2) + 2 and g(x) = -4x + 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) = 4(-2e^(2x) + 1) + 3 and g(x) = 1x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1e^(-2((-3(-3|(4x^2 - 4x - 1) - 3| - 1) + 1)/(3(-3|(4x^2 - 4x - 1) - 3| - 1)))) and g(x) = 5((2(1x^2 - 5x - 3) - 3)/(-1(1x^2 - 5x - 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) = 1((2(-5(2e^(-1x) - 3)^2 - 5(2e^(-1x) - 3) + 3) + 1)/(-3(-5(2e^(-1x) - 3)^2 - 5(2e^(-1x) - 3) + 3) - 1)) - 1 and g(x) = 1cos(1\u03c0((-1(4x^2 - 2x + 4) - 1)/(-1(4x^2 - 2x + 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|(3sin(2\u03c0x - 2)) - 2| and g(x) = (1x - 1)/(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": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (2(5x^2 - 4x - 3) + 2)/(1(5x^2 - 4x - 3) + 3) and g(x) = 1|x + 1| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1e^(1(-4x)) + 1 and g(x) = 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2((-2x - 2)/(-2x - 1)) - 4 and g(x) = -1x + 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(1|(-4(3x^2 - 2x - 1) - 3) + 3|) - 1)/(-2(1|(-4(3x^2 - 2x - 1) - 3) + 3|) + 3) and g(x) = -2(3|((-2x + 1)/(-3x - 1)) - 2| + 2)^2 - 5(3|((-2x + 1)/(-3x - 1)) - 2| + 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = 4(-3|(-2((3x)/(-2x - 3))) - 3| + 2)^2 - 4(-3|(-2((3x)/(-2x - 3))) - 3| + 2) + 4 and g(x) = 1(1(-2|x + 2| + 1)^2 + 5(-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": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 3(-1((3x - 3)/(-2x))^2 + 2((3x - 3)/(-2x)) - 5) + 2 and g(x) = -1|(3x - 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 leftmost x-coordinate where f(x) = -4((1(4x^2 + 2x - 5) + 3)/(3(4x^2 + 2x - 5) + 3)) + 2 and g(x) = 1(3e^(-1x) + 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 second intersection point (counting from left to right) between the functions f(x) = -1(3sin(1\u03c0(1|(-5x^2 - 3x - 2) - 1| + 1) - 1) - 2) - 1 and g(x) = 1(1((2x + 2)/(3x - 2)) + 5)^2 + 2(1((2x + 2)/(3x - 2)) + 5) - 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 integer value closest to the x-coordinate where f(x) = 5(-2e^(2x)) - 4 and g(x) = 2x - 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": "At what integer value (rounded) do the functions f(x) = (-1(-1(-2|(2x^2 + 4x - 1) + 2| - 3) - 4))/(1(-1(-2|(2x^2 + 4x - 1) + 2| - 3) - 4)) and g(x) = -3(-2e^(-1((1x - 3)/(-3x - 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) = 3((-2(-1e^(-1x) + 1) + 2)/(-2(-1e^(-1x) + 1) + 3))^2 + 3((-2(-1e^(-1x) + 1) + 2)/(-2(-1e^(-1x) + 1) + 3)) - 2 and g(x) = 2|((-3x - 3)/(1x + 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": "Find the integer value closest to the x-coordinate where f(x) = -4(3e^(2x) - 3) and g(x) = -1|x + 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": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2|(1x^2 + 3x + 3) + 3| - 3 and g(x) = 4x^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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2(4(1|((-1x)/(-3x - 3)) - 1| + 2)^2 - 3(1|((-1x)/(-3x - 3)) - 1| + 2) - 1) + 1 and g(x) = (-3(-1(-1x + 3)^2 - 1(-1x + 3) + 5) - 3)/(-3(-1(-1x + 3)^2 - 1(-1x + 3) + 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1e^(-1(3x - 5)) - 3 and g(x) = -2|x - 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 integer value closest to the x-coordinate where f(x) = 1e^(-1((-2(5x - 4) + 3)/(2(5x - 4) - 1))) + 2 and g(x) = 1|(3x - 2) + 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": "Find the integer value closest to the x-coordinate where f(x) = 4((3x - 3)/(-1x + 2)) + 2 and g(x) = 3x^2 + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -4(5x + 2)^2 + 4(5x + 2) - 4 and g(x) = 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": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -2|(1e^(-1((1(-1x - 5) - 1)/(-3(-1x - 5) + 2))) - 3) - 2| and g(x) = -1(-3|(-5x - 1)|)^2 + 3(-3|(-5x - 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": "-6", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1(2x - 5) + 1)/(-2(2x - 5) - 2) and g(x) = 5x^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(-2x^2 + 3x - 2) - 5 and g(x) = 3|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": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1(-2(-3|((-3x + 1)/(-1x - 3)) + 3|)^2 - 2(-3|((-3x + 1)/(-1x - 3)) + 3|) + 4) - 1 and g(x) = -3|((-3(1e^(-2x) - 2) - 1)/(-3(1e^(-2x) - 2) - 1)) - 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(5(-3|((-2x)/(1x + 3)) - 2| + 3)^2 - 4(-3|((-2x)/(1x + 3)) - 2| + 3) - 2) and g(x) = -5(5((1x - 1)/(-2x + 3))^2 - 4((1x - 1)/(-2x + 3)) - 4) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1|(-3(2e^(1((-2x - 1)/(2x - 3))) - 1) - 2) + 3| + 3 and g(x) = -3|(-1((1x)/(-1x)) - 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 3|(1((-2(1x^2 - 3x + 1) + 3)/(1(1x^2 - 3x + 1) - 3)) - 4) + 1| + 2 and g(x) = 1((2(5x - 5) + 2)/(1(5x - 5) + 2))^2 - 1((2(5x - 5) + 2)/(1(5x - 5) + 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 value closest to the x-coordinate where f(x) = -1|(1(-3(-3e^(-2x) - 2)^2 + 5(-3e^(-2x) - 2) + 5) + 4) + 2| + 2 and g(x) = -2|((1(3e^(1x) + 1) + 1)/(1(3e^(1x) + 1))) + 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": "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(1x^2 - 2x + 2) - 1)/(1(1x^2 - 2x + 2) - 2) and g(x) = 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": "At what integer value (rounded) do the functions f(x) = -3sin(1\u03c0(1((2x + 2)/(-2x - 2)) + 1) - 3) + 1 and g(x) = -3((-1x + 3)/(2x + 2)) + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1e^(-1(-2|((-1x + 1)/(2x))| - 2)) + 3 and g(x) = -5(-1|x + 3|)^2 - 3(-1|x + 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": "At what integer value (rounded) do the functions f(x) = (-2(2cos(2\u03c0x + 3)) + 3)/(-2(2cos(2\u03c0x + 3)) + 3) and g(x) = -1x - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1((1x)/(-2x))^2 - 5((1x)/(-2x)) + 4 and g(x) = 1|x + 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": "-6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 5((-1(1(2e^(-1x))^2 + 5(2e^(-1x)) + 4) - 3)/(-1(1(2e^(-1x))^2 + 5(2e^(-1x)) + 4) - 2)) + 2 and g(x) = 1|(-2(3x^2 - 4x - 1) - 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1((-1x)/(-3x - 1))^2 + 5((-1x)/(-3x - 1)) - 1 and g(x) = 1|x + 1| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1(2x^2 + 1x) + 4 and g(x) = 3|x + 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1(-3(-2sin(1\u03c0x + 2) - 2) + 2) - 1)/(3(-3(-2sin(1\u03c0x + 2) - 2) + 2)) and g(x) = -3e^(1((-1x + 2)/(2x))) + 2 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": "At what integer value (rounded) do the functions f(x) = (1(-1|x + 3| - 1) - 2)/(1(-1|x + 3| - 1) - 3) and g(x) = (-2x + 2)/(-3x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1|((3x - 3)/(2x - 1)) + 2| and g(x) = 4x + 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 value closest to the x-coordinate where f(x) = (-1(1|(4(-1x - 4)^2 + 2(-1x - 4) - 4) + 1|) - 3)/(3(1|(4(-1x - 4)^2 + 2(-1x - 4) - 4) + 1|) - 1) and g(x) = (-2(-3e^(-2(4x + 3)) + 1) - 2)/(-2(-3e^(-2(4x + 3)) + 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) = 1e^(-1((3x + 2)/(1x + 1))) and g(x) = -5x + 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(-3((-3(-3|x - 1| + 2) - 3)/(3(-3|x - 1| + 2) - 2))^2 - 5((-3(-3|x - 1| + 2) - 3)/(3(-3|x - 1| + 2) - 2)) - 2) + 3 and g(x) = (2(-3e^(1(1|x - 2| + 2)) - 1) - 1)/(1(-3e^(1(1|x - 2| + 2)) - 1) - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2e^(2((2x + 3)/(2x - 3))) + 2 and g(x) = -1x^2 - 3x + 5 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 x-coordinate where the graphs of f(x) = -4(-3x^2 + 3x - 1) + 3 and g(x) = (2x + 3)/(1x + 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(-3|x + 3| + 2) + 2 and g(x) = 3|x + 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 rightmost x-coordinate where f(x) = -2sin(3\u03c0(-1x - 4) - 3) + 3 and g(x) = -5x 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|x - 2| + 1) + 1)/(-3(1|x - 2| + 1) + 4) and g(x) = (-3x)/(-3x + 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": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -4x and g(x) = -1x^2 + 1x + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (3(4(-4(3|x + 1|)^2 - 2(3|x + 1|) - 1) - 5))/(-3(4(-4(3|x + 1|)^2 - 2(3|x + 1|) - 1) - 5) - 1) and g(x) = 4(3e^(1(1x - 3)) - 2)^2 + 3(3e^(1(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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = -3(-3|x - 3| + 1)^2 - 5(-3|x - 3| + 1) and g(x) = -5x^2 + 1x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2(2((-3(2|x - 1| - 1) - 1)/(1(2|x - 1| - 1) + 3))^2 + 1((-3(2|x - 1| - 1) - 1)/(1(2|x - 1| - 1) + 3)) + 3) + 4 and g(x) = (-1(1(5x + 2)^2 + 1(5x + 2)) - 3)/(3(1(5x + 2)^2 + 1(5x + 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 value closest to the x-coordinate where f(x) = -2cos(1\u03c0(1|x| + 3) + 1) + 3 and g(x) = 4x + 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": "If you add up all the x-coordinates where the functions f(x) = -4(-4(3|((1x + 1)/(-3x)) - 1|)^2 - 5(3|((1x + 1)/(-3x)) - 1|) + 5) - 1 and g(x) = -4(-2((1x - 2)/(2x + 2)) - 5)^2 - 1(-2((1x - 2)/(2x + 2)) - 5) - 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": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3|(-3((-1x + 2)/(2x - 3))) - 2| - 2 and g(x) = (2(-4x + 3) + 2)/(-1(-4x + 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) = (2(2(2|x + 2|) + 1) - 1)/(1(2(2|x + 2|) + 1) + 3) and g(x) = -5((-1x)/(-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 first intersection point (counting from left to right) between the functions f(x) = 1(2|x + 2| - 1) and g(x) = -2x^2 - 2x + 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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (3(-1(-2|(1x^2 + 2x + 2) + 2| + 2) - 4) + 3)/(1(-1(-2|(1x^2 + 2x + 2) + 2| + 2) - 4)) and g(x) = 1|(3((-3x + 2)/(-2x + 3)) - 1) - 1| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 3x + 4 and g(x) = 1x^2 - 5x + 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": "Find the integer value closest to the x-coordinate where f(x) = -1(-2(1|((1x + 2)/(-3x - 3)) - 1|) - 2)^2 - 4(-2(1|((1x + 2)/(-3x - 3)) - 1|) - 2) + 3 and g(x) = 2|(1(5x^2 - 5x + 4) - 4)| - 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 value closest to the x-coordinate where f(x) = 3sin(2\u03c0(1(1|x|)^2 + 5(1|x|) - 5) + 2) + 2 and g(x) = 1cos(2\u03c0(-2x^2 - 3x + 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": "-10", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-3(-5(-1x - 2)^2 - 3(-1x - 2) - 1) - 3)/(2(-5(-1x - 2)^2 - 3(-1x - 2) - 1)) and g(x) = 1((-3x - 2)/(1x - 3)) + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 2(-1|(5((-3x)/(-2x - 2)) + 2) - 2| - 3)^2 + 3(-1|(5((-3x)/(-2x - 2)) + 2) - 2| - 3) - 1 and g(x) = 2|((-2(-1x - 1) + 2)/(-3(-1x - 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 closest to the largest x-value where the functions f(x) = (-3(4(2|x - 3| - 3) - 5) + 1)/(-2(4(2|x - 3| - 3) - 5) - 3) and g(x) = 3(1x + 4)^2 + 3(1x + 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) = 2|(3(1((3x - 2)/(3x - 1)) - 2)^2 + 2(1((3x - 2)/(3x - 1)) - 2) + 3) + 3| + 1 and g(x) = 2e^(-2((3(1x - 2) - 3)/(3(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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2e^(2(4((3x + 1)/(2x))^2 - 2)) + 2 and g(x) = (2(-5x + 2) + 1)/(1(-5x + 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) = 3|((-1(5(-1x - 5)^2 + 2(-1x - 5) - 5))/(-3(5(-1x - 5)^2 + 2(-1x - 5) - 5) - 3))| + 2 and g(x) = 1|((-2(-1x - 2))/(2(-1x - 2) - 1)) + 1| + 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) = -2|(-1e^(1x) + 1) - 1| - 2 and g(x) = (-1x + 1)/(1x) 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) = 5(1(-3|x| + 1)^2 - 2(-3|x| + 1) - 4) - 3 and g(x) = 1(1x - 2)^2 + 1(1x - 2) - 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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (1(4x - 4) + 2)/(2(4x - 4) + 1) and g(x) = -2|x - 1| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2(2e^(1x))^2 + 2(2e^(1x)) + 5 and g(x) = 3|x - 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) = -1(3|((1x - 1)/(-1x + 3)) - 2| - 2)^2 + 1(3|((1x - 1)/(-1x + 3)) - 2| - 2) + 5 and g(x) = 1((3x + 2)/(-2x + 3)) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 4(3|(-4x - 3)| + 1)^2 + 3(3|(-4x - 3)| + 1) + 5 and g(x) = 3(-2|x + 3| + 2)^2 + 2(-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": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -4(1|(2x^2 - 3) - 1| + 1) and g(x) = -5(2|x + 3| + 2)^2 - 2(2|x + 3| + 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": "Find the integer closest to the largest x-value where the functions f(x) = (2(3(-5x - 2)^2 + 2(-5x - 2) - 2) + 2)/(-2(3(-5x - 2)^2 + 2(-5x - 2) - 2) + 1) and g(x) = -1|((-2x - 3)/(1x + 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": "If you add up all the x-coordinates where the functions f(x) = -4(3((-3x + 2)/(1x + 3))^2 - 3((-3x + 2)/(1x + 3)) - 5) + 4 and g(x) = -5(-3|x + 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": "-8", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-2(-1(1|x + 1| + 1)^2 + 2(1|x + 1| + 1) + 1) - 1)/(-2(-1(1|x + 1| + 1)^2 + 2(1|x + 1| + 1) + 1) - 2) and g(x) = -2e^(2(1|x + 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": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 4((-2x + 3)/(1x)) - 4 and g(x) = 3|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": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1(-2|x - 2| - 2) + 2 and g(x) = -1x^2 - 3x 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(1x + 2)^2 + 1(1x + 2) - 3 and g(x) = (-1x + 2)/(-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": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (3(2x^2 + 3x + 1) + 1)/(1(2x^2 + 3x + 1) + 3) and g(x) = -2x^2 + 2x + 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": "If you add up all the x-coordinates where the functions f(x) = -2|(5(-3x^2 + 5x - 1) - 1) + 3| - 1 and g(x) = 1(-3e^(-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": "-6", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(-2(-2sin(3\u03c0(-2|x + 3| + 2) - 1) - 2)^2 + 3(-2sin(3\u03c0(-2|x + 3| + 2) - 1) - 2) - 5))/(1(-2(-2sin(3\u03c0(-2|x + 3| + 2) - 1) - 2)^2 + 3(-2sin(3\u03c0(-2|x + 3| + 2) - 1) - 2) - 5)) and g(x) = (1(2(-5x^2 + 4x + 2) + 1) - 1)/(1(2(-5x^2 + 4x + 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 first intersection point (counting from left to right) between the functions f(x) = (3(3cos(2\u03c0x + 2) - 3) - 1)/(-2(3cos(2\u03c0x + 2) - 3) + 3) and g(x) = -1x + 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": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 1x - 1 and g(x) = 1x^2 - 2x 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|(3e^(-1(-4(-1x^2 - 2x + 5) + 3)) + 1) - 2| + 1 and g(x) = 4(1(-3cos(1\u03c0x - 1) - 2)^2 + 1(-3cos(1\u03c0x - 1) - 2) + 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) = 3sin(2\u03c0((-1(3(1x^2 + 5x + 1) + 5) - 1)/(-2(3(1x^2 + 5x + 1) + 5) - 2)) - 2) - 1 and g(x) = -3(4(-1|x - 1| - 1) + 5)^2 + 5(4(-1|x - 1| - 1) + 5) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (2(-1cos(1\u03c0x + 2) - 2) - 2)/(3(-1cos(1\u03c0x + 2) - 2) + 2) and g(x) = (1x - 3)/(2x) 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) = 1(-2|((3x + 3)/(-3x))| + 3)^2 + 3(-2|((3x + 3)/(-3x))| + 3) + 1 and g(x) = -3sin(1\u03c0(-2x)) - 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(-1x + 2) - 2)/(1(-1x + 2) - 3) and g(x) = -5x^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) = -2|(1(-1((-1x - 2)/(1x + 2))^2 - 2) - 3) + 2| - 3 and g(x) = -3|(-1(3x + 5)^2 - 4(3x + 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 3(-1|((-2(1x + 3) + 1)/(-1(1x + 3))) + 2|)^2 - 2 and g(x) = (2(-5(-3x + 1)^2 - 1(-3x + 1)) - 3)/(-2(-5(-3x + 1)^2 - 1(-3x + 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 closest to the smallest x-value where the functions f(x) = 1(-4((2(1|x + 1| + 2) + 1)/(-3(1|x + 1| + 2) + 4)) - 2)^2 - 3(-4((2(1|x + 1| + 2) + 1)/(-3(1|x + 1| + 2) + 4)) - 2) and g(x) = -3|(-2((-1x + 1)/(-2x + 3)) - 5) + 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) = -3|(-2(-4x^2 - 1x + 3) + 1) + 3| - 2 and g(x) = -1((-3x + 3)/(1x)) - 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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 2e^(-1(1|(4((2x + 3)/(-2x - 3))^2 + 4((2x + 3)/(-2x - 3)) - 1) + 1| - 1)) - 1 and g(x) = 1|(-1((3x + 3)/(3x + 1)) - 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) = -3(3e^(-2x) + 1)^2 - 5(3e^(-2x) + 1) + 2 and g(x) = -1x^2 + 5x - 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (2(-2|x + 1|) - 3)/(-3(-2|x + 1|) + 4) and g(x) = (1x + 3)/(2x + 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) = -3(-1|((-3(3x - 5) - 3)/(1(3x - 5) - 2))| + 3)^2 + 5(-1|((-3(3x - 5) - 3)/(1(3x - 5) - 2))| + 3) + 2 and g(x) = (3(1|(1sin(3\u03c0x - 2) + 3) - 2| + 1) - 2)/(2(1|(1sin(3\u03c0x - 2) + 3) - 2| + 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (1(-2(5x + 4)^2 - 2) - 2)/(2(-2(5x + 4)^2 - 2) + 2) and g(x) = -3(-3x - 4)^2 + 3(-3x - 4) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1|((2(-4(4x - 2)^2 - 2(4x - 2) - 4))/(2(-4(4x - 2)^2 - 2(4x - 2) - 4) - 1)) - 1| + 3 and g(x) = 5((-3(-2e^(-1x) + 1) + 2)/(2(-2e^(-1x) + 1) + 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2(-2|(1(-2e^(-2x) + 2) - 4) + 3| + 2)^2 - 1(-2|(1(-2e^(-2x) + 2) - 4) + 3| + 2) - 5 and g(x) = 2(2e^(-1((2x - 2)/(2x - 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 leftmost x-coordinate where f(x) = -1((3(1|(3x^2 + 2x) - 3| + 1))/(1(1|(3x^2 + 2x) - 3| + 1) + 1)) - 1 and g(x) = -1|((2(-2x + 4) + 1)/(2(-2x + 4) + 3)) + 2| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1(1(2|((-3x + 2)/(-3x + 4)) - 3| + 3)^2 - 1(2|((-3x + 2)/(-3x + 4)) - 3| + 3) + 4) + 1 and g(x) = 2(-2|((1x - 1)/(2x + 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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -5(-2e^(-1x) + 3)^2 - 2(-2e^(-1x) + 3) + 2 and g(x) = (1x - 1)/(-3x + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 5(3e^(1x))^2 - 1(3e^(1x)) + 2 and g(x) = -1x - 4 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": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1|((-2(-1sin(2\u03c0x) - 3) - 1)/(-1(-1sin(2\u03c0x) - 3) + 2)) + 3| and g(x) = 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": "-10", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1|((3(1(-1cos(3\u03c0x) + 3)^2 - 5(-1cos(3\u03c0x) + 3) + 1) + 1)/(3(1(-1cos(3\u03c0x) + 3)^2 - 5(-1cos(3\u03c0x) + 3) + 1) + 2)) - 2| and g(x) = 1((-2(1x - 2) + 3)/(-3(1x - 2)))^2 + 5((-2(1x - 2) + 3)/(-3(1x - 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = 1|((2(2cos(1\u03c0x - 3) + 2) + 1)/(-3(2cos(1\u03c0x - 3) + 2) - 1)) + 2| - 3 and g(x) = (-1(-5x^2 + 1x + 1) + 2)/(2(-5x^2 + 1x + 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 value closest to the x-coordinate where f(x) = (2(-5(2|x| - 1)^2 - 1) + 3)/(3(-5(2|x| - 1)^2 - 1)) and g(x) = (3(2x) + 3)/(1(2x)) 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) = (-1(-3|x + 1| - 1) + 3)/(2(-3|x + 1| - 1) - 1) and g(x) = -1x + 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 leftmost x-coordinate where f(x) = (2(4(3|(-4x^2 + 5x - 1) - 1| + 3) + 2) + 3)/(3(4(3|(-4x^2 + 5x - 1) - 1| + 3) + 2) + 3) and g(x) = 2((2(1|x + 1|) - 1)/(2(1|x + 1|) + 1))^2 + 5((2(1|x + 1|) - 1)/(2(1|x + 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": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (2(-2(2x^2 + 5x) + 2) - 3)/(-2(-2(2x^2 + 5x) + 2) + 1) and g(x) = -1(-1sin(1\u03c0x) - 3)^2 - 5(-1sin(1\u03c0x) - 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1((1(3|x - 1| + 2) + 1)/(1(3|x - 1| + 2))) and g(x) = 4(-1e^(-1x) + 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": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(2(1cos(1\u03c0x + 3) + 2) + 2) - 3)/(3(2(1cos(1\u03c0x + 3) + 2) + 2) + 2) and g(x) = 2|(-1x - 2) - 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-8", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1|((-2(2e^(-1x)))/(-2(2e^(-1x)) + 1)) + 3| - 1 and g(x) = 2((1x - 2)/(2x - 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) = -2|((-2(-1(-4x - 4)^2 + 1(-4x - 4) - 1) - 2)/(-2(-1(-4x - 4)^2 + 1(-4x - 4) - 1) - 3)) - 1| - 3 and g(x) = -1(-1|((3x - 3)/(-1x + 2)) - 3| + 1)^2 - 5(-1|((3x - 3)/(-1x + 2)) - 3| + 1) + 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = -3|(1x + 2) + 2| - 1 and g(x) = 1x + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1(1(2|x + 3| - 1)^2 + 3(2|x + 3| - 1) + 4) - 1)/(-1(1(2|x + 3| - 1)^2 + 3(2|x + 3| - 1) + 4) - 1) and g(x) = (-1(1x - 1) - 3)/(-2(1x - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1(-1((-2x - 2)/(2x + 1))^2 + 3((-2x - 2)/(2x + 1)) + 5) - 3 and g(x) = 3e^(2((2x - 3)/(-2x - 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) = (1(2|(-2e^(1x) - 1) + 2| - 3) - 2)/(1(2|(-2e^(1x) - 1) + 2| - 3) - 3) and g(x) = -3(5x^2 - 4x + 5) + 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 leftmost x-coordinate where f(x) = (-3(-1|(-3x^2 - 4x - 3) + 2| + 2) - 1)/(2(-1|(-3x^2 - 4x - 3) + 2| + 2) - 3) and g(x) = 2|(3sin(1\u03c0x + 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": "If you add up all the x-coordinates where the functions f(x) = 2|(1((1(4x^2 - 3x + 4) + 1)/(-1(4x^2 - 3x + 4) - 3)) + 2) + 2| - 3 and g(x) = 3|(1(-1x + 1)^2 - 2(-1x + 1) - 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": "At what integer value (rounded) do the functions f(x) = 1(-1e^(1x) + 2) and g(x) = 2x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1e^(2(1x - 1)) - 2 and g(x) = 5x - 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) = 2(-2x - 4)^2 - 4(-2x - 4) + 4 and g(x) = -4x 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((1x - 1)/(1x + 2))^2 - 5 and g(x) = -3x^2 + 1x + 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": "If you add up all the x-coordinates where the functions f(x) = 4((-1x + 3)/(3x + 1)) + 2 and g(x) = (-3x + 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": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2e^(-2(1|((3(1x^2 - 3x - 5) + 3)/(3(1x^2 - 3x - 5) - 2)) + 1| + 1)) - 1 and g(x) = -2|((3(-2x - 1) + 1)/(-1(-2x - 1) - 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": "Find the integer value closest to the x-coordinate where f(x) = (-3(-1(-1|x - 2| - 2) + 4) - 2)/(1(-1(-1|x - 2| - 2) + 4) - 3) and g(x) = -1((2x + 2)/(-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": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3(-3((2x)/(-3x + 2))^2 + 5((2x)/(-3x + 2)) + 5) + 1 and g(x) = 2(2e^(1x)) - 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) = (-1(-2|(4(4x)^2 + 2(4x) - 1) + 1| - 1) + 3)/(2(-2|(4(4x)^2 + 2(4x) - 1) + 1| - 1) - 3) and g(x) = 1(1e^(1(-3x - 4)) - 2)^2 - 2(1e^(1(-3x - 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": "Find the integer closest to the smallest x-value where the functions f(x) = 3cos(2\u03c0(1|x + 1|) + 3) - 3 and g(x) = (-3x + 3)/(-1x - 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) = -3e^(1(1|((-2x - 1)/(3x - 1)) - 3| - 1)) + 2 and g(x) = (2(1x + 5) + 3)/(2(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": "-6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 4(-1|(2e^(-1(-2x^2 - 4x + 2)) + 3)| + 3) + 2 and g(x) = (-1(2sin(3\u03c0(1|x| + 2) + 3) - 1) + 2)/(-1(2sin(3\u03c0(1|x| + 2) + 3) - 1) + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2|(4((-1x + 1)/(-1x)) - 5) + 2| - 1 and g(x) = 4(-2|x + 1|)^2 + 4(-2|x + 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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -5(-3x - 1)^2 - 3(-3x - 1) - 1 and g(x) = -3|x - 2| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3|(1(-3x + 5)^2 - 5(-3x + 5) - 2)| - 2 and g(x) = -3(1e^(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": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2|(-2((-2(1x - 2))/(-2(1x - 2) - 1))^2 - 4((-2(1x - 2))/(-2(1x - 2) - 1)) - 5) + 3| + 2 and g(x) = -1|(-2((2x - 2)/(-1x + 2)) - 5) + 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (1(-1e^(1(1x^2 + 3x - 4)) + 2) + 1)/(-1(-1e^(1(1x^2 + 3x - 4)) + 2)) and g(x) = 1(-1|x + 2| - 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1(-3e^(-1(-3|((3x + 3)/(2x - 1)) + 2| + 1)) + 1)^2 + 5(-3e^(-1(-3|((3x + 3)/(2x - 1)) + 2| + 1)) + 1) and g(x) = -2sin(2\u03c0(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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1|(3((-3x - 2)/(-3x + 1))^2 - 1((-3x - 2)/(-3x + 1)) - 3) + 2| + 3 and g(x) = -2((2x + 3)/(1x)) - 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) = 5(2|((1x + 3)/(3x - 2)) - 1| - 1) + 2 and g(x) = -1|(-1e^(-2x) + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 5(1e^(-2x) - 1) + 5 and g(x) = 5x + 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": "Find the integer value closest to the x-coordinate where f(x) = -4((-1x - 2)/(-1x + 3))^2 - 1((-1x - 2)/(-1x + 3)) - 1 and g(x) = (-1x + 2)/(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": "-7", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1((3(4x - 2) - 2)/(-3(4x - 2) + 4))^2 - 3((3(4x - 2) - 2)/(-3(4x - 2) + 4)) - 2 and g(x) = (3(-3e^(-2x) + 2) + 1)/(-1(-3e^(-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": "Find the integer value closest to the x-coordinate where f(x) = -2(-3e^(-1x) - 3)^2 + 5(-3e^(-1x) - 3) + 2 and g(x) = -1x^2 - 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": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2|(5x - 1)| + 1 and g(x) = (-1x - 3)/(2x + 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 closest to the largest x-value where the functions f(x) = 2sin(2\u03c0((2(-3x + 1) + 1)/(2(-3x + 1) + 1))) - 2 and g(x) = 1|(2x - 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) = -2(-3sin(1\u03c0x + 1) - 2) + 1 and g(x) = -5x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -5((2(2e^(-1x)) + 3)/(1(2e^(-1x)) + 2)) - 1 and g(x) = 4(-2e^(-2x) + 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": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -4(1((1x - 1)/(-2x - 3))^2 - 2((1x - 1)/(-2x - 3)) - 3) + 4 and g(x) = 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1e^(1(1x^2 - 4x - 4)) + 3 and g(x) = 5x - 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 integer value closest to the x-coordinate where f(x) = (-2(-1sin(1\u03c0x - 2) + 2) + 2)/(-2(-1sin(1\u03c0x - 2) + 2) - 3) and g(x) = (3x + 3)/(3x - 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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-2(1e^(-2x)) - 3)/(2(1e^(-2x)) - 3) and g(x) = 5x^2 + 4x - 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) = -1cos(1\u03c0(1(5x^2 - 3x - 3) + 5)) - 3 and g(x) = (3(1x - 2) + 3)/(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": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -3|(3(-3cos(1\u03c0x + 2) + 2) + 4) + 2| + 3 and g(x) = -4((-3x)/(2x + 1))^2 - 5((-3x)/(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 first intersection point (counting from left to right) between the functions f(x) = -1|(4((-1(2sin(1\u03c0x + 2) - 3) - 2)/(2(2sin(1\u03c0x + 2) - 3) - 3)) - 3) + 3| + 2 and g(x) = 3((1(3|x - 3| + 2) + 3)/(2(3|x - 3| + 2) - 3))^2 - 5((1(3|x - 3| + 2) + 3)/(2(3|x - 3| + 2) - 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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-2(4(-1cos(2\u03c0(-2x + 1) + 3) + 3)^2 - 5(-1cos(2\u03c0(-2x + 1) + 3) + 3) + 3) + 2)/(-2(4(-1cos(2\u03c0(-2x + 1) + 3) + 3)^2 - 5(-1cos(2\u03c0(-2x + 1) + 3) + 3) + 3)) and g(x) = 2((2(4x + 5))/(-2(4x + 5) - 3))^2 + 3((2(4x + 5))/(-2(4x + 5) - 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) = 1sin(2\u03c0(-2((-3(-1|x + 3| - 2) - 1)/(3(-1|x + 3| - 2) - 2)) - 1) + 1) + 1 and g(x) = (1(-1|(4x^2 + 2x) + 1|) + 2)/(-1(-1|(4x^2 + 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": "If you add up all the x-coordinates where the functions f(x) = -4(4(1|x - 3| + 1) - 3)^2 + 2(4(1|x - 3| + 1) - 3) - 1 and g(x) = -3|(-1x - 5) + 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": "6", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 2e^(-2((-3x + 2)/(1x - 3))) + 1 and g(x) = (1x + 1)/(2x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-3(1cos(3\u03c0(-2|(2x + 1) - 3| + 2) - 1) - 3) - 1)/(-3(1cos(3\u03c0(-2|(2x + 1) - 3| + 2) - 1) - 3) - 2) and g(x) = -1((1(1|x - 1| - 1) - 2)/(1(1|x - 1| - 1) - 3)) + 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": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1x - 5 and g(x) = -4x^2 - 5x - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (1(-1sin(2\u03c0x + 1) + 2) + 3)/(-1(-1sin(2\u03c0x + 1) + 2) - 1) and g(x) = -3x^2 + 5x - 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) = -2sin(2\u03c0((2(-3x - 1) + 1)/(-2(-3x - 1) - 1)) - 3) + 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": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2|(1e^(2((1x + 2)/(1x + 1)))) + 1| + 1 and g(x) = -3((2x - 1)/(1x + 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1((-2(-1(2|x| + 1)^2 + 3(2|x| + 1) + 3) - 2)/(-3(-1(2|x| + 1)^2 + 3(2|x| + 1) + 3) + 1)) - 3 and g(x) = (-2(-3|(3x - 4) + 1| - 1) + 1)/(3(-3|(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": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 4(3e^(-2(1((3x + 3)/(1x)) + 4)) - 1)^2 + 5(3e^(-2(1((3x + 3)/(1x)) + 4)) - 1) - 4 and g(x) = 2cos(2\u03c0(-3|((-1x + 2)/(-1x + 2)) - 2| - 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) = -4(2|((3(1x^2 - 1x + 3))/(3(1x^2 - 1x + 3) + 3)) + 2| + 2) - 1 and g(x) = 5(-1e^(-2(-1|x| + 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) = -5(3e^(1x) - 3)^2 + 2(3e^(1x) - 3) - 5 and g(x) = -2|x + 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) = -1|((-2x + 1)/(-2x + 3)) + 3| and g(x) = -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 integer closest to the smallest x-value where the functions f(x) = 3|(3e^(1(-3x + 4)) - 2) - 3| - 3 and g(x) = -1cos(3\u03c0(-1x^2 + 2x - 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 integer closest to the smallest x-value where the functions f(x) = -5(-2e^(1x) + 2)^2 - 2(-2e^(1x) + 2) - 3 and g(x) = -2|x + 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 value closest to the x-coordinate where f(x) = 1(-1|x + 2| + 1) + 2 and g(x) = -2x - 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": "If you add up all the x-coordinates where the functions f(x) = -1|(-2(2((-2x + 2)/(1x - 2))^2 + 5((-2x + 2)/(1x - 2)) + 5) + 2) + 2| and g(x) = (3(1e^(1(-4x^2 + 1x + 4))) + 2)/(-2(1e^(1(-4x^2 + 1x + 4))) - 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": "Find the integer closest to the largest x-value where the functions f(x) = (-3(5(-2e^(1(5x^2 - 5x - 5))) + 3) - 2)/(1(5(-2e^(1(5x^2 - 5x - 5))) + 3) + 2) and g(x) = -2(-2cos(1\u03c0((3x)/(1x)))) + 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 integer closest to the smallest x-value where the functions f(x) = 4x + 2 and g(x) = 1x^2 - 2x + 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) = 2e^(1(-2|(5((-2x + 2)/(1x + 3)) - 4) - 2| + 3)) + 3 and g(x) = 1((2(4x - 1) - 1)/(-1(4x - 1) + 2))^2 + 4((2(4x - 1) - 1)/(-1(4x - 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-2(3(-2x^2 - 3x - 4) - 5) + 3)/(-2(3(-2x^2 - 3x - 4) - 5)) and g(x) = (-2(2x^2 - 4x + 1) - 2)/(-2(2x^2 - 4x + 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 third intersection point (counting from left to right) between the functions f(x) = (-2(-2x^2 + 5x + 2) + 3)/(-1(-2x^2 + 5x + 2) + 2) and g(x) = -3x - 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 smallest x-value where the functions f(x) = (1(2cos(1\u03c0(-1|x + 1| + 2)) + 1))/(1(2cos(1\u03c0(-1|x + 1| + 2)) + 1)) and g(x) = -2(4x + 1)^2 + 1(4x + 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) = -2|((-2x + 3)/(-2x)) - 2| + 2 and g(x) = 1x^2 - 5x + 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = 5(-1e^(1x) + 3)^2 + 1(-1e^(1x) + 3) - 1 and g(x) = (1x + 2)/(-1x + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2|(-3sin(3\u03c0((3x - 3)/(-1x + 2)) + 2) + 1) + 3| - 3 and g(x) = 5((2x - 1)/(-1x - 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -3cos(2\u03c0(1((-1(3x - 1))/(-3(3x - 1) + 4))^2 - 4((-1(3x - 1))/(-3(3x - 1) + 4)) - 3) - 1) - 2 and g(x) = 5((-2(-5x^2 - 4x + 3) + 3)/(-1(-5x^2 - 4x + 3) - 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 closest to the largest x-value where the functions f(x) = -3((-1(-1|x + 1| + 2) - 3)/(2(-1|x + 1| + 2) + 1)) + 1 and g(x) = (3(-1cos(1\u03c0x + 2) + 2) + 3)/(-3(-1cos(1\u03c0x + 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2(2((1x + 3)/(3x + 3))^2 + 5((1x + 3)/(3x + 3)) + 1) - 4 and g(x) = -1sin(1\u03c0((-3x)/(-1x)) + 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": "Find the integer closest to the largest x-value where the functions f(x) = 3((1(-3x^2 + 4x - 4) - 3)/(3(-3x^2 + 4x - 4) + 1)) - 3 and g(x) = (-3(5x^2 - 5x) - 3)/(-3(5x^2 - 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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (1(-3x - 4) + 1)/(-3(-3x - 4)) and g(x) = 3x + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 2((-2(3|x - 1|) + 1)/(-2(3|x - 1|) + 3))^2 - 5((-2(3|x - 1|) + 1)/(-2(3|x - 1|) + 3)) + 3 and g(x) = 3(1|x + 2| - 3) - 5 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 integer value closest to the x-coordinate where f(x) = (-3(-3x + 1) + 1)/(-2(-3x + 1) + 3) and g(x) = -2|x| + 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = 3sin(1\u03c0((-2(3|x + 1| + 2) - 2)/(1(3|x + 1| + 2))) + 2) - 2 and g(x) = (1(3|x - 3| + 1) - 1)/(3(3|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": "Find the integer closest to the smallest x-value where the functions f(x) = -2(2|x + 1| - 1)^2 - 3(2|x + 1| - 1) + 1 and g(x) = (-1x - 3)/(2x - 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) = 2(-5x)^2 - 5(-5x) + 5 and g(x) = 5x^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 integer closest to the largest x-value where the functions f(x) = 1x - 3 and g(x) = -1x^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": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-3(-2(-1x^2 - 5x + 3) + 4) + 1)/(1(-2(-1x^2 - 5x + 3) + 4) - 2) and g(x) = -1(-3x - 5)^2 - 4(-3x - 5) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 1(-2|(-3((1x + 3)/(1x - 2))^2 - 3((1x + 3)/(1x - 2)) - 1) + 3|) - 3 and g(x) = -3|(-5((1x + 2)/(2x + 2)) + 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": "Find the integer closest to the largest x-value where the functions f(x) = 1|(-5x^2 + 2x + 3) + 1| + 2 and g(x) = -2x + 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 integer value closest to the x-coordinate where f(x) = -1(-1(-2e^(-1x))^2 - 4(-2e^(-1x)) - 3) - 4 and g(x) = 1(3x^2 + 5x - 4) + 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 x-coordinate where the graphs of f(x) = 1e^(1((-3(-2|x + 1| + 3) - 2)/(2(-2|x + 1| + 3) - 3))) - 3 and g(x) = 5(-3|x| + 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) = 2|(4((1x + 3)/(2x + 2)) - 3) + 1| + 2 and g(x) = (1(-2|x + 3| + 3) - 3)/(1(-2|x + 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": "-6", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 5((-2x + 2)/(-2x - 3)) + 4 and g(x) = 1|x + 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 integer closest to the smallest x-value where the functions f(x) = 5((-1x - 2)/(1x + 3)) and g(x) = 4x + 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 x-coordinate where the graphs of f(x) = (2(-4(2x - 2)^2 - 3(2x - 2) - 3) + 1)/(3(-4(2x - 2)^2 - 3(2x - 2) - 3) + 3) and g(x) = -1((1x + 2)/(2x)) - 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1(4x^2 - 5x - 3) + 5 and g(x) = (-1x + 1)/(3x + 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) = (2(-3|(3(4x^2 - 3x + 4) - 5) + 3| - 3) + 2)/(-1(-3|(3(4x^2 - 3x + 4) - 5) + 3| - 3)) and g(x) = -1((2(-4x^2 + 1x + 5) + 1)/(2(-4x^2 + 1x + 5) - 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": "At what integer value (rounded) do the functions f(x) = 2|((1x + 3)/(-2x + 3))| - 3 and g(x) = -1x - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -5(-1(3|((-1x)/(1x - 3)) + 2| - 3)^2 - 2(3|((-1x)/(1x - 3)) + 2| - 3) + 1) - 2 and g(x) = -1cos(2\u03c0(2|((2x)/(-2x - 2)) + 1| + 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1|(-5x - 2) - 1| + 1 and g(x) = (1x + 1)/(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 average of all x-coordinates (rounded to the nearest integer) where f(x) = -2(3cos(1\u03c0(-1|x - 3| - 3) - 3) + 1) - 2 and g(x) = (-2(3|x - 1|) + 1)/(-1(3|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": "Find the integer closest to the largest x-value where the functions f(x) = -3sin(3\u03c0(1|x + 3| - 1) + 3) - 3 and g(x) = (-2x + 1)/(-1x - 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) = 1e^(1((-3x + 3)/(2x - 3))) and g(x) = 2x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5((2x + 1)/(-3x + 4))^2 - 5((2x + 1)/(-3x + 4)) + 4 and g(x) = (-2x)/(1x - 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(-2|x + 2| + 2) - 3 and g(x) = -1x + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3(2|(5((3x - 3)/(2x + 1)) + 4) - 2| + 1)^2 + 3(2|(5((3x - 3)/(2x + 1)) + 4) - 2| + 1) - 4 and g(x) = -1((3(2x^2 + 2x - 4) + 1)/(3(2x^2 + 2x - 4) - 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 integer closest to the smallest x-value where the functions f(x) = 5x + 4 and g(x) = -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(3(1|x + 1| + 2) - 4) + 2)/(-2(3(1|x + 1| + 2) - 4) - 1) and g(x) = 1(-5x - 3)^2 + 4(-5x - 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 rightmost x-coordinate where f(x) = -1((-1(-1x + 2) + 3)/(1(-1x + 2)))^2 - 3((-1(-1x + 2) + 3)/(1(-1x + 2))) + 4 and g(x) = -1e^(-1(-1|x + 2| + 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2|(2sin(1\u03c0x + 2) - 1) - 2| + 2 and g(x) = 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) = (-3(4(2e^(1x) + 2) - 4) + 2)/(-2(4(2e^(1x) + 2) - 4)) and g(x) = 1e^(1(1x^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 x-coordinate where the graphs of f(x) = 2(3(1|x| - 2)^2 + 5(1|x| - 2) - 4) + 3 and g(x) = 3e^(-1(-1x - 3)) + 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 leftmost x-coordinate where f(x) = -4(-1|(-4x^2 + 5x)| - 1) - 1 and g(x) = 1|(-4x^2 - 4x + 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) = 2((2x - 1)/(1x + 1)) - 3 and g(x) = 1|x - 1| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -4((-2(-3x + 5) + 1)/(3(-3x + 5) + 1))^2 + 1((-2(-3x + 5) + 1)/(3(-3x + 5) + 1)) + 5 and g(x) = -1((1x + 1)/(-2x))^2 - 3((1x + 1)/(-2x)) - 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": "If you add up all the x-coordinates where the functions f(x) = (2(-2|(-2sin(3\u03c0(4x^2 - 2x - 2) + 3) + 1) + 3|) + 1)/(-3(-2|(-2sin(3\u03c0(4x^2 - 2x - 2) + 3) + 1) + 3|) + 2) and g(x) = (1(3(2|x + 3| - 1) - 5) + 3)/(2(3(2|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": "At what integer value (rounded) do the functions f(x) = -3((-2(1|x| + 2))/(2(1|x| + 2) + 1))^2 - 5((-2(1|x| + 2))/(2(1|x| + 2) + 1)) + 3 and g(x) = (-2(2x - 4) - 1)/(-2(2x - 4)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-1(4(1x^2 - 5x - 5) + 3) + 1)/(-1(4(1x^2 - 5x - 5) + 3) + 2) and g(x) = -2(1e^(-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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1|(2((-3(4x - 2) - 2)/(-2(4x - 2) - 1))^2 - 3((-3(4x - 2) - 2)/(-2(4x - 2) - 1)) - 3) - 3| + 1 and g(x) = -3(-1(1|x + 3| - 3) + 1)^2 + 1(-1(1|x + 3| - 3) + 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": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -2|(1(-1((1x - 1)/(3x))^2 + 1)) - 3| - 3 and g(x) = -4((-1(5x + 4) - 3)/(3(5x + 4) - 2))^2 - 2((-1(5x + 4) - 3)/(3(5x + 4) - 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((-3(1x - 5) + 3)/(2(1x - 5) + 3))^2 - 4((-3(1x - 5) + 3)/(2(1x - 5) + 3)) - 5 and g(x) = (2(2x^2 + 3x + 2) + 3)/(2(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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3((2x + 3)/(-2x + 1)) - 4 and g(x) = 5x + 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) = (2(-1|(-1x + 4) - 3| + 2) - 1)/(-2(-1|(-1x + 4) - 3| + 2) - 2) and g(x) = 2e^(1(-1|x| + 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = -2(-3|(-3cos(2\u03c0x + 3) + 3) + 2| + 1)^2 - 3(-3|(-3cos(2\u03c0x + 3) + 3) + 2| + 1) and g(x) = -1(1e^(1x) - 2)^2 + 3(1e^(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": "At what integer value (rounded) do the functions f(x) = 3(-2e^(-2x))^2 - 2(-2e^(-2x)) + 5 and g(x) = 2x^2 + 3x - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -4((-1(5x^2 + 1x - 3) - 2)/(3(5x^2 + 1x - 3) + 2)) + 4 and g(x) = 2e^(-2(1x + 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) = -5(2|((-1(-4x^2 + 2x) - 2)/(1(-4x^2 + 2x))) + 1| + 2) - 3 and g(x) = -5(1e^(-1(1|x - 2| - 1)) - 2)^2 + 1(1e^(-1(1|x - 2| - 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": "If you add up all the x-coordinates where the functions f(x) = (1(-2(-2|x - 1| + 3) + 3) - 1)/(-3(-2(-2|x - 1| + 3) + 3) - 2) and g(x) = 2|((-1x - 1)/(-1x - 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 second x-coordinate (rounded to the nearest integer) where f(x) = (2(4(-1|(1x + 3) - 2| - 1)^2 + 1) + 2)/(3(4(-1|(1x + 3) - 2| - 1)^2 + 1) - 2) and g(x) = -5((1(3x^2 - 5x - 5) + 2)/(1(3x^2 - 5x - 5) - 3)) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -2((-1x + 1)/(3x + 3)) - 4 and g(x) = (-1x)/(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": "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((-3(-1x - 3) - 2)/(-3(-1x - 3) + 4))^2 - 2((-3(-1x - 3) - 2)/(-3(-1x - 3) + 4)) - 1 and g(x) = (1(3x^2 + 4x - 5) - 1)/(1(3x^2 + 4x - 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 x-coordinate where the graphs of f(x) = (2(1(-4(-2|x + 3| + 2)^2 + 4(-2|x + 3| + 2) - 2) - 3) + 1)/(3(1(-4(-2|x + 3| + 2)^2 + 4(-2|x + 3| + 2) - 2) - 3) + 1) and g(x) = 2e^(-2((-1(3x - 4) + 1)/(-3(3x - 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 integer value closest to the x-coordinate where f(x) = 2|(-4(-3((-2x + 1)/(1x + 2)) - 3)^2 + 4(-3((-2x + 1)/(1x + 2)) - 3)) + 2| - 1 and g(x) = (3(-3|(2e^(2x) + 2) - 3|))/(-1(-3|(2e^(2x) + 2) - 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": "At what integer value (rounded) do the functions f(x) = (-2(-2|x - 2| + 2))/(2(-2|x - 2| + 2) + 3) and g(x) = (1x + 1)/(1x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(4(1cos(1\u03c0((2x - 2)/(-2x + 1)) + 2) - 3)^2 - 1(1cos(1\u03c0((2x - 2)/(-2x + 1)) + 2) - 3) + 5) - 1| - 1 and g(x) = -3|((1(2e^(2x) - 3) - 1)/(-3(2e^(2x) - 3) - 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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-1(-5x^2 - 1x - 5) - 1)/(-1(-5x^2 - 1x - 5) - 3) and g(x) = 2|x| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 5(-3|(2x + 3)| + 1)^2 - 2(-3|(2x + 3)| + 1) + 1 and g(x) = -5(2x^2 + 3x) + 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) = -3e^(1(-2x^2 + 4x + 2)) - 3 and g(x) = (-3x)/(-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": "If you add up all the x-coordinates where the functions f(x) = 2|(-1((-2x - 3)/(-2x + 1))^2 - 1((-2x - 3)/(-2x + 1)) - 5) - 2| - 2 and g(x) = 3|((-1x + 3)/(-3x)) - 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -4(1((3x + 2)/(1x)) - 1)^2 - 2(1((3x + 2)/(1x)) - 1) - 4 and g(x) = -1(-3e^(-1x) - 3)^2 + 3(-3e^(-1x) - 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-1(-3|(-3(-5x^2 - 1x - 2) - 1) - 3| - 1))/(-2(-3|(-3(-5x^2 - 1x - 2) - 1) - 3| - 1)) and g(x) = -2((-2(-1x^2 - 5x + 2) + 2)/(3(-1x^2 - 5x + 2) + 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": "If you add up all the x-coordinates where the functions f(x) = -4(3((-2(-3|x + 1| - 1) + 2)/(-2(-3|x + 1| - 1) - 1))^2 - 5((-2(-3|x + 1| - 1) + 2)/(-2(-3|x + 1| - 1) - 1)) - 2) + 1 and g(x) = -1(-3sin(1\u03c0((2x - 1)/(3x + 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) = -1(-3|((2x + 1)/(2x - 3))|)^2 + 4(-3|((2x + 1)/(2x - 3))|) - 5 and g(x) = -3|(-3x - 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": "At what integer value (rounded) do the functions f(x) = (1(-2e^(1x)) + 3)/(2(-2e^(1x)) + 1) and g(x) = 2|x - 1| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -3cos(2\u03c0((-3(2(-2x^2 + 2) - 2) + 1)/(-1(2(-2x^2 + 2) - 2) - 2)) - 3) and g(x) = -3(-3|((2x + 1)/(-1x))|)^2 - 4(-3|((2x + 1)/(-1x))|) - 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) = 1|(-2(5((-3x - 3)/(2x - 3)) - 2)^2 - 1(5((-3x - 3)/(2x - 3)) - 2) - 2) - 2| - 1 and g(x) = 4(1((3x - 1)/(1x))^2 - 5((3x - 1)/(1x)) - 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": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -3|((2(-5x + 3) - 3)/(2(-5x + 3) - 1))| and g(x) = -3cos(1\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": "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(-2|(2e^(1x) - 2) - 2| - 3))/(1(-2|(2e^(1x) - 2) - 2| - 3) + 1) and g(x) = 4(-5x + 4)^2 + 1(-5x + 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2|(2(-2x - 1)^2 + 2(-2x - 1) + 2) + 1| + 2 and g(x) = (1(3sin(1\u03c0x + 1) + 1) + 1)/(3(3sin(1\u03c0x + 1) + 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1(-2x^2 - 3x + 2) - 3 and g(x) = (1x + 2)/(-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 average of all x-coordinates (rounded to the nearest integer) where f(x) = -2((-2x)/(-2x - 3)) - 1 and g(x) = -3x + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1sin(1\u03c0(-3(-1|(3x^2 + 5x + 4)| - 3) - 1) + 1) and g(x) = -2sin(2\u03c0(3(3|x + 1| - 1) - 1) - 2) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1e^(1(1(-5((3x)/(-2x + 1))^2 - 4((3x)/(-2x + 1)) - 1) - 1)) - 1 and g(x) = 1(-3|(3e^(2x) - 1) - 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1(-1e^(2x) - 2)^2 + 4(-1e^(2x) - 2) + 3 and g(x) = 1|x + 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-1(-1sin(1\u03c0(-2(-5x + 5)^2 + 1(-5x + 5) - 5) - 3) - 2) - 1)/(1(-1sin(1\u03c0(-2(-5x + 5)^2 + 1(-5x + 5) - 5) - 3) - 2) - 3) and g(x) = 4(-3|((-2x - 3)/(2x + 1)) + 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": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -3|(4x - 2) + 1| + 3 and g(x) = 1x^2 + 2x + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 2(-3cos(3\u03c0((1x)/(-1x)) + 2) + 1)^2 - 3(-3cos(3\u03c0((1x)/(-1x)) + 2) + 1) + 2 and g(x) = 2e^(2(-5x - 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) = -3((1(2x^2 + 2x - 4) + 1)/(3(2x^2 + 2x - 4) - 1)) and g(x) = 1(-2e^(1x) + 1)^2 - 4(-2e^(1x) + 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) = -2|(1(5(-2e^(-2x))^2 - 3(-2e^(-2x)) + 2) + 3) - 1| - 3 and g(x) = -1|((-2(-3x) - 2)/(3(-3x) - 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": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -5(1x^2 - 3x + 3) + 2 and g(x) = -3x - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "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(4x - 1))/(3(4x - 1) - 1) and g(x) = (1x - 1)/(-2x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -5((1(5(-2|x + 1| + 2)^2 + 4(-2|x + 1| + 2) + 5) - 2)/(2(5(-2|x + 1| + 2)^2 + 4(-2|x + 1| + 2) + 5) + 2)) + 4 and g(x) = 3e^(-2(-3((3x + 2)/(-1x - 1)) - 4)) - 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 integer value closest to the x-coordinate where f(x) = -4((-2x + 1)/(-2x - 3))^2 - 2((-2x + 1)/(-2x - 3)) + 5 and g(x) = -3x 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 second intersection point (counting from left to right) between the functions f(x) = 5((-1(-2e^(-1x) + 2) - 1)/(1(-2e^(-1x) + 2) + 2))^2 + 4((-1(-2e^(-1x) + 2) - 1)/(1(-2e^(-1x) + 2) + 2)) - 5 and g(x) = 1|(2sin(3\u03c0x + 3) - 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": "-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(-2((-1x - 3)/(2x - 3)) - 2)^2 + 2 and g(x) = 1((-3x + 3)/(-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": "Find the integer value closest to the x-coordinate where f(x) = 3e^(1(1|x| - 1)) - 2 and g(x) = (-3x - 3)/(-3x) 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) = -5(-3(-2|((2x)/(-3x - 1))| + 1)^2 - 1) - 5 and g(x) = 4(2|(-3x^2 - 1x + 3) - 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": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1|(3((2(-4x^2 - 5x - 1) - 1)/(-1(-4x^2 - 5x - 1)))) - 1| and g(x) = (1(-2(5x + 4)^2 + 5(5x + 4) + 4) + 2)/(2(-2(5x + 4)^2 + 5(5x + 4) + 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 value closest to the x-coordinate where f(x) = -4(3(-1e^(2x) - 3) + 2)^2 - 4(3(-1e^(2x) - 3) + 2) and g(x) = -1e^(2((-1x + 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": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -1(-1(1|((-3x)/(-1x - 2)) + 3| + 1)^2 - 1(1|((-3x)/(-1x - 2)) + 3| + 1) - 5) - 4 and g(x) = 1|(1sin(2\u03c0((2x)/(1x + 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": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2|(-4((1x + 2)/(-1x - 1))^2 - 1((1x + 2)/(-1x - 1))) - 1| - 3 and g(x) = (-3(1e^(1x) - 3) - 3)/(-2(1e^(1x) - 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": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (1(-4(1(1cos(1\u03c0x) + 2) - 5)^2 - 1(1(1cos(1\u03c0x) + 2) - 5) + 5) + 2)/(1(-4(1(1cos(1\u03c0x) + 2) - 5)^2 - 1(1(1cos(1\u03c0x) + 2) - 5) + 5)) and g(x) = (1(-1|(5x) + 1| + 3) + 1)/(3(-1|(5x) + 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": "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|(5x^2 - 4x + 1)| + 3 and g(x) = -1|x - 3| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-1(-1|(-1sin(2\u03c0(3x - 1) + 3) - 2)| + 3) + 2)/(3(-1|(-1sin(2\u03c0(3x - 1) + 3) - 2)| + 3) + 2) and g(x) = 3(-5((3x - 3)/(-3x))^2 - 2((3x - 3)/(-3x)) + 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 integer closest to the smallest x-value where the functions f(x) = 3(1|(5((-1x - 2)/(-3x - 2)) + 5) + 2| - 1)^2 + 2(1|(5((-1x - 2)/(-3x - 2)) + 5) + 2| - 1) + 4 and g(x) = (1(1(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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -3(3((-2(-3|x + 1| - 2) - 2)/(3(-3|x + 1| - 2) + 1))^2 + 3((-2(-3|x + 1| - 2) - 2)/(3(-3|x + 1| - 2) + 1)) + 4) - 2 and g(x) = (1(-4(1|x + 3| - 2) - 4) + 3)/(2(-4(1|x + 3| - 2) - 4) - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (2(-3x - 1) - 3)/(-3(-3x - 1)) and g(x) = -5x^2 + 5x + 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) = 1e^(2(-4x - 5)) - 3 and g(x) = 5x - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "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((3(2sin(2\u03c0(3|x - 2| + 2) + 3) + 1) - 2)/(3(2sin(2\u03c0(3|x - 2| + 2) + 3) + 1) - 2)) - 4 and g(x) = -2|(5((2x - 3)/(-3x)) + 4) - 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 integer closest to the largest x-value where the functions f(x) = 2|(5(3e^(2((1x - 1)/(1x + 3))) - 3)^2 - 3(3e^(2((1x - 1)/(1x + 3))) - 3) + 2) + 3| and g(x) = 1|(2(1x - 5)^2 - 4(1x - 5) - 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": "Find the third intersection point (counting from left to right) between the functions f(x) = -4(-1(3sin(2\u03c0x + 3) + 1) + 1)^2 - 3(-1(3sin(2\u03c0x + 3) + 1) + 1) - 2 and g(x) = -3((-1x + 3)/(3x - 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": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1|(-5(-3((1x)/(1x - 2)) + 5)^2 + 5(-3((1x)/(1x - 2)) + 5) - 2) + 1| + 3 and g(x) = (-3(2|(1e^(-2x) - 1) - 3| + 2) - 1)/(3(2|(1e^(-2x) - 1) - 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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2(-1((2(1|x + 2| + 1) + 1)/(-3(1|x + 2| + 1) - 2)))^2 + 1(-1((2(1|x + 2| + 1) + 1)/(-3(1|x + 2| + 1) - 2))) - 2 and g(x) = -2e^(1(1(-5x - 3)^2 + 5(-5x - 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": "Find the integer value closest to the x-coordinate where f(x) = -3|((-3(-3(-1x + 5)^2 + 2(-1x + 5) - 4) - 3)/(-1(-3(-1x + 5)^2 + 2(-1x + 5) - 4) - 3)) + 3| - 1 and g(x) = -4(5(2e^(-1x) - 1)^2 + 2(2e^(-1x) - 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": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -3((1x - 1)/(2x - 1)) + 3 and g(x) = (1x)/(-2x - 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = 3|((-2(-4x + 5) - 2)/(3(-4x + 5) - 2)) + 2| - 3 and g(x) = (-2(1x + 2) - 1)/(-1(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(-1x - 3) - 3)/(3(-1x - 3) - 2))^2 - 5((-1(-1x - 3) - 3)/(3(-1x - 3) - 2)) - 4) - 3| - 1 and g(x) = (-1(3e^(-1(3x - 2)) - 3) + 2)/(-3(3e^(-1(3x - 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 integer closest to the largest x-value where the functions f(x) = (3(-1(-1(-2|x - 1| + 1) + 2)^2 + 1(-1(-2|x - 1| + 1) + 2) + 3) + 1)/(-1(-1(-1(-2|x - 1| + 1) + 2)^2 + 1(-1(-2|x - 1| + 1) + 2) + 3) + 3) and g(x) = 1(2|(-3x - 5) - 2| + 1)^2 + 5(2|(-3x - 5) - 2| + 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 closest to the smallest x-value where the functions f(x) = -1((3(3x^2 - 1x - 1) + 3)/(-3(3x^2 - 1x - 1) + 4)) - 2 and g(x) = 1|(2cos(2\u03c0x + 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": "At what integer value (rounded) do the functions f(x) = 1|((-2x + 2)/(-3x - 1)) - 3| + 2 and g(x) = (3x + 1)/(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": "If you add up all the x-coordinates where the functions f(x) = (-1(-3e^(-2x) + 2) + 2)/(-2(-3e^(-2x) + 2) - 1) and g(x) = 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 second intersection point (counting from left to right) between the functions f(x) = 3e^(2(-3(-2|((-2x + 3)/(-2x + 3)) + 1|)^2 - 1(-2|((-2x + 3)/(-2x + 3)) + 1|) + 1)) - 1 and g(x) = 3|(-2e^(1((-2x + 1)/(-3x + 4)))) + 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": "At what integer value (rounded) do the functions f(x) = 2(1e^(-2x) + 3)^2 - 4(1e^(-2x) + 3) + 4 and g(x) = 2|x + 2| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3e^(1(5((2x - 3)/(-2x + 1)) - 4)) - 1 and g(x) = -3|((-1x - 2)/(2x - 1)) + 2| + 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 2(5((-1x + 1)/(-3x - 2))^2 - 2((-1x + 1)/(-3x - 2)) + 4) - 2 and g(x) = 1|((-1x - 2)/(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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-2(1cos(3\u03c0(1|x - 3| + 2) + 2) + 2) - 1)/(-3(1cos(3\u03c0(1|x - 3| + 2) + 2) + 2) - 1) and g(x) = -5((-3x)/(3x + 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 x-coordinate where the graphs of f(x) = 1|(-4((-2x - 2)/(-2x + 3)) + 3) - 3| and g(x) = -3((1x - 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) = (3(-3x - 2) + 3)/(-1(-3x - 2) + 2) and g(x) = (-1x - 1)/(3x) 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((2x + 2)/(3x))^2 - 4((2x + 2)/(3x)) - 1 and g(x) = 3x + 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) = 2|((-2(1(3x)^2 - 2(3x) - 1) + 2)/(-1(1(3x)^2 - 2(3x) - 1))) - 2| - 3 and g(x) = -4(-1e^(1(3x^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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 3|((3(4x^2 + 2) + 2)/(1(4x^2 + 2) + 1)) + 1| - 1 and g(x) = 2(5x - 2)^2 + 1(5x - 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -5(-2((3x + 3)/(-1x + 2))^2 - 5((3x + 3)/(-1x + 2)) + 2) + 1 and g(x) = -2|(-2e^(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": "At what integer value (rounded) do the functions f(x) = 2(-3|(1e^(1x) - 2) + 2|)^2 - 4 and g(x) = 4(1|x + 2| + 1) + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "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(1|x - 1| - 2) + 3)/(-2(1|x - 1| - 2) + 3)) - 3 and g(x) = 1(5x - 4)^2 + 3(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 integer value closest to the x-coordinate where f(x) = -1|(1((-2(-3x))/(3(-3x) + 1))^2 - 1) + 3| - 3 and g(x) = -2sin(1\u03c0(-3((-3x + 1)/(1x - 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": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -4(2|x + 2| + 1) - 5 and g(x) = (-3x - 3)/(1x) 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) = 2(-3|x - 1| + 1) + 5 and g(x) = 2|x + 2| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2(-1((1x - 1)/(-2x - 3)) + 4)^2 + 5(-1((1x - 1)/(-2x - 3)) + 4) + 3 and g(x) = 1(2e^(1x) + 2)^2 + 4(2e^(1x) + 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(1x + 4))/(2(1x + 4)) and g(x) = -2x^2 + 5x + 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 second intersection point (counting from left to right) between the functions f(x) = -5(2|x - 2| + 2) - 2 and g(x) = -2x^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": "6", "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(-5(2|x - 3|)^2 + 1(2|x - 3|) + 2)) + 1 and g(x) = 3|((1x - 2)/(-1x + 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": "At what integer value (rounded) do the functions f(x) = 5((-2x - 3)/(-1x + 2))^2 + 2((-2x - 3)/(-1x + 2)) + 4 and g(x) = -3x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3e^(1(-1((2x - 1)/(-1x + 2)) + 1)) + 2 and g(x) = -2(1|x| - 1) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1((-3(3e^(-2(-1x^2 - 5x - 2)) - 2) - 3)/(3(3e^(-2(-1x^2 - 5x - 2)) - 2))) - 4 and g(x) = 2|(1(-1e^(2x) - 2) - 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": "Find the integer value closest to the x-coordinate where f(x) = 5((-1x - 3)/(-3x + 4)) - 5 and g(x) = 2x^2 - 5x - 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": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(1|x| - 3) + 2)/(-1(1|x| - 3) + 3) and g(x) = -1|x - 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1|((3x + 1)/(-2x - 1))| - 2 and g(x) = 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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -2|(-4(-1((-3x + 2)/(-2x - 3)) + 4)^2 - 3(-1((-3x + 2)/(-2x - 3)) + 4) - 2) + 3| - 2 and g(x) = 3(-5(-3e^(2x) + 2)^2 - 3(-3e^(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 value closest to the x-coordinate where f(x) = -1e^(-1(-2|(2x^2 - 2) + 3| + 1)) + 3 and g(x) = -3((2x + 1)/(-3x + 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": "Find the integer value closest to the x-coordinate where f(x) = -1|(2x + 2) - 2| - 1 and g(x) = (2x - 1)/(2x + 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 integer value closest to the x-coordinate where f(x) = 1e^(2(-1((3x + 1)/(-2x - 2))^2 + 4((3x + 1)/(-2x - 2)))) + 3 and g(x) = 1(2e^(-2x) - 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": "Find the integer closest to the largest x-value where the functions f(x) = 3(-1((-3(-1|x + 1|))/(2(-1|x + 1|) + 3))^2 - 4((-3(-1|x + 1|))/(2(-1|x + 1|) + 3)) - 5) + 1 and g(x) = -2(3(-2e^(-2x)) - 1)^2 - 1(3(-2e^(-2x)) - 1) - 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) = (3(2(2e^(-1x) - 2)^2 - 4) + 1)/(3(2(2e^(-1x) - 2)^2 - 4) - 1) and g(x) = -2e^(-1(-2x^2 - 3x + 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1(3|x + 3| + 3)^2 + 5(3|x + 3| + 3) - 4 and g(x) = 1x^2 + 5x 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) = 3|(2e^(-2(-1(-3x^2 - 1x - 4) - 4)) + 1) + 2| - 3 and g(x) = (-1(1(1|x| + 3)^2 - 2(1|x| + 3) - 5) + 1)/(1(1(1|x| + 3)^2 - 2(1|x| + 3) - 5) + 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) = -1e^(2((-1x + 3)/(2x - 3))) and g(x) = -3|x - 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) = -3(2e^(1x) + 2)^2 - 1(2e^(1x) + 2) - 1 and g(x) = (-1x + 2)/(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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1(-5x + 5)^2 + 5(-5x + 5) - 5 and g(x) = 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|(-2cos(2\u03c0((3(-3x^2 + 3) + 1)/(1(-3x^2 + 3))) + 2) + 1) + 2| - 1 and g(x) = -5((1(-3|x + 3| + 1) - 1)/(3(-3|x + 3| + 1))) + 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) = -2sin(2\u03c0(1((3(-5x^2 + 3x) - 1)/(3(-5x^2 + 3x) + 2)) - 4) + 3) - 1 and g(x) = -3e^(-2(-2((-2x)/(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": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 4x + 5 and g(x) = -1x^2 + 2x + 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 value closest to the x-coordinate where f(x) = -3(-3e^(-1((1x + 1)/(-2x - 2))) - 2)^2 - 3(-3e^(-1((1x + 1)/(-2x - 2))) - 2) - 4 and g(x) = -5(-3e^(-2x) - 1)^2 - 5(-3e^(-2x) - 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": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 2e^(-1(-1(4((-3x + 1)/(-2x - 3))^2 + 4((-3x + 1)/(-2x - 3)) - 5) - 5)) - 2 and g(x) = (1(-2|(3x - 2) - 3| - 2) + 3)/(-1(-2|(3x - 2) - 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-1(-2(3|x - 1| - 3) - 1) + 3)/(3(-2(3|x - 1| - 3) - 1) + 2) and g(x) = -1((1x - 3)/(2x + 2)) - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1|(-4x + 3)| - 3 and g(x) = (-1x)/(-2x) 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(-1|(1(1e^(1x))^2 - 5(1e^(1x)) + 2) - 3| + 3) - 2)/(-3(-1|(1(1e^(1x))^2 - 5(1e^(1x)) + 2) - 3| + 3) + 1) and g(x) = 4(2e^(2(-2|x + 2| - 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 integer value closest to the x-coordinate where f(x) = (-3(-4x^2 + 1x) + 1)/(-2(-4x^2 + 1x) - 1) and g(x) = (1x - 3)/(-3x - 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = 5((-2(-1|x - 3|) + 1)/(-2(-1|x - 3|) - 1)) + 3 and g(x) = 1|((-3x + 3)/(-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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = 2|(2x^2 + 4x - 3) + 1| - 2 and g(x) = (1x + 1)/(2x - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2|(-5((-3x + 3)/(2x + 3)) + 5) + 2| + 2 and g(x) = (-3(-2x^2 + 1x - 2) - 1)/(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": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (2(1|x + 2| - 2) + 3)/(-2(1|x + 2| - 2) + 1) and g(x) = -3x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -4(-5((1(-3sin(3\u03c0x - 3) + 3))/(1(-3sin(3\u03c0x - 3) + 3))) + 1)^2 - 2(-5((1(-3sin(3\u03c0x - 3) + 3))/(1(-3sin(3\u03c0x - 3) + 3))) + 1) - 1 and g(x) = -3|(1((-3x)/(1x + 1)) + 3) + 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": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -5(1((-3x + 2)/(-2x - 3))^2 - 3((-3x + 2)/(-2x - 3)) - 5) + 5 and g(x) = -1(2cos(3\u03c0x - 2) - 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": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1|((2(4x^2 - 4x + 3) - 2)/(-1(4x^2 - 4x + 3) + 2)) + 2| and g(x) = -2((-2x - 2)/(-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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3|(4x) - 3| - 1 and g(x) = 5x + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 2|(4((2(1x - 1) + 3)/(-3(1x - 1) + 2))^2 + 5((2(1x - 1) + 3)/(-3(1x - 1) + 2)) + 5) + 1| + 3 and g(x) = (1(2(-2|x + 1| + 3)^2 - 3(-2|x + 1| + 3)) + 1)/(-3(2(-2|x + 1| + 3)^2 - 3(-2|x + 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": "When ordered from left to right, what is the third x-coordinate (rounded to the nearest integer) where f(x) = -2|((2(1x - 3))/(2(1x - 3) + 2)) - 3| and g(x) = (-3(4x + 5) - 2)/(2(4x + 5) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1e^(2((3(-2|(-5x + 3) - 2| - 1) + 1)/(1(-2|(-5x + 3) - 2| - 1) + 2))) - 1 and g(x) = -2|((-2(1x^2 - 5x - 2) - 1)/(1(1x^2 - 5x - 2) + 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 integer value closest to the x-coordinate where f(x) = (3(2cos(2\u03c0x + 1) + 1) - 2)/(3(2cos(2\u03c0x + 1) + 1) - 2) and g(x) = (1x + 2)/(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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = -3cos(2\u03c0(2|(-2((-2x - 2)/(2x + 2)) - 2) - 2| + 2) + 3) + 2 and g(x) = -2((-3(-2|x + 1| + 1) - 2)/(1(-2|x + 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": "Find the integer closest to the largest x-value where the functions f(x) = 2|((-2x + 2)/(-2x + 1))| + 2 and g(x) = 4x^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": "If you add up all the x-coordinates where the functions f(x) = -4x and g(x) = -2x^2 + 5x + 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": "At what integer value (rounded) do the functions f(x) = -2|((1(-2x^2 + 4x + 4) - 3)/(2(-2x^2 + 4x + 4) + 1)) - 3| - 3 and g(x) = -3e^(2(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": "When ordered from left to right, what is the third x-coordinate (rounded to the nearest integer) where f(x) = (-3(3(-2e^(-2x) + 2) - 4) + 3)/(3(3(-2e^(-2x) + 2) - 4) - 2) and g(x) = -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": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 2|(-2sin(2\u03c0((2x + 2)/(-2x - 2)) + 1)) + 1| + 1 and g(x) = 2e^(2(-3|x| + 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) = 3|(-1sin(3\u03c0x - 1))| + 2 and g(x) = (-2x - 1)/(-3x - 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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 4(1|(-2cos(2\u03c0((1x - 2)/(-1x - 3)) + 2)) - 2| - 3)^2 + 2(1|(-2cos(2\u03c0((1x - 2)/(-1x - 3)) + 2)) - 2| - 3) + 2 and g(x) = -4(-2|((3x)/(-1x + 2))| + 3) - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-8", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-1(-5(4(-2|x + 1| - 1) + 5)^2 + 5) + 1)/(1(-5(4(-2|x + 1| - 1) + 5)^2 + 5) + 2) and g(x) = 3|((-2(3sin(1\u03c0x + 3) + 3) - 1)/(1(3sin(1\u03c0x + 3) + 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 first intersection point (counting from left to right) between the functions f(x) = 3|(-4(3((1x + 1)/(3x - 1))^2 + 2((1x + 1)/(3x - 1))) - 4)| - 1 and g(x) = -2(1((2x - 1)/(-1x - 3))^2 - 4((2x - 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": "-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((1x + 1)/(2x - 3))^2 + 1((1x + 1)/(2x - 3)) - 4 and g(x) = -2x^2 + 4x + 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) = -2|(-2x) + 1| + 2 and g(x) = (-3x - 1)/(1x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "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) = -3|((-2(-4x - 1) - 3)/(-2(-4x - 1))) + 3| and g(x) = -2e^(1(5x^2 + 2x + 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": "At what integer value (rounded) do the functions f(x) = 4(3|((-2x + 1)/(1x + 3)) + 1| + 1)^2 + 1(3|((-2x + 1)/(1x + 3)) + 1| + 1) - 3 and g(x) = (1(1x) - 2)/(3(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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = 2x + 4 and g(x) = 4x^2 - 3x 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(1x - 3) - 1)/(3(1x - 3) + 2) and g(x) = -2|x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3|(-4((2x - 3)/(3x - 2))) + 2| + 2 and g(x) = (1(-2x - 4) - 1)/(-1(-2x - 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) = 1sin(1\u03c0(3|x - 3| - 3)) - 1 and g(x) = 4x^2 + 5x + 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) = (2(-5(3e^(1x) - 3) + 4) + 3)/(2(-5(3e^(1x) - 3) + 4) + 1) and g(x) = (-1(2cos(1\u03c0x + 1) + 2) - 2)/(1(2cos(1\u03c0x + 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 value closest to the x-coordinate where f(x) = (-1(-2cos(1\u03c0x + 2) - 2))/(-2(-2cos(1\u03c0x + 2) - 2) + 1) and g(x) = (2x - 2)/(-1x) 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) = (3(-2cos(3\u03c0(-5x - 3) - 3) + 1) + 2)/(3(-2cos(3\u03c0(-5x - 3) - 3) + 1) + 2) and g(x) = (1(1x^2 - 5x + 2) - 1)/(-3(1x^2 - 5x + 2) + 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": "Find the integer closest to the largest x-value where the functions f(x) = 3(2|x - 2| + 1) - 3 and g(x) = 2x^2 + 3x - 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(-3x^2 - 3x - 5)) + 2| - 3 and g(x) = -3((2x + 3)/(2x - 1)) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2sin(2\u03c0((-2(-2(5x^2 + 2x + 3) - 4) + 3)/(-1(-2(5x^2 + 2x + 3) - 4))) - 2) + 2 and g(x) = (2(1|(-4x - 2) - 1| + 2))/(2(1|(-4x - 2) - 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 x-coordinate where the graphs of f(x) = 3|(-2e^(2x)) + 3| - 3 and g(x) = (3x - 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1(3|(-1e^(-1(3x + 2)) + 2) - 1| - 1)^2 + 1(3|(-1e^(-1(3x + 2)) + 2) - 1| - 1) - 1 and g(x) = (2(1(-1|x + 3| + 2)^2 - 4) - 3)/(-1(1(-1|x + 3| + 2)^2 - 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": "If you add up all the x-coordinates where the functions f(x) = 2|(4(2x + 5)^2 + 4(2x + 5) + 2) - 1| and g(x) = -3(-2sin(2\u03c0x + 1))^2 - 4(-2sin(2\u03c0x + 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": "-5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1x - 5 and g(x) = 2x^2 + 5x - 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 integer closest to the smallest x-value where the functions f(x) = -2|(-1(-2e^(-2(3x^2 + 2x - 4))) - 4) + 2| - 2 and g(x) = -1(3(-3|x - 2| - 3)^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": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1e^(2(-1x^2 - 2x)) + 2 and g(x) = 3x^2 - 5x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-1(1|(-2(2e^(2x) + 1)) + 2| - 1) + 1)/(-3(1|(-2(2e^(2x) + 1)) + 2| - 1) + 1) and g(x) = -2|(1((-2x - 1)/(1x - 2))^2 - 5((-2x - 1)/(1x - 2)) + 5) - 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) = 3(5((1(1|x - 2|) - 1)/(-2(1|x - 2|)))^2 + 3((1(1|x - 2|) - 1)/(-2(1|x - 2|))) - 5) - 3 and g(x) = -5(1|((1x + 1)/(3x - 2)) + 3| + 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = 5((-1(2|(4x^2 + 1x) + 3| + 2) - 1)/(-3(2|(4x^2 + 1x) + 3| + 2) + 2)) and g(x) = -2e^(1((-1(4x^2 - 1x - 1))/(2(4x^2 - 1x - 1) + 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((2x - 1)/(2x + 1))^2 + 1((2x - 1)/(2x + 1)) + 4 and g(x) = -3|x + 3| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (1(-1|x + 3|) - 3)/(-2(-1|x + 3|) + 3) and g(x) = 3x - 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": "If you add up all the x-coordinates where the functions f(x) = 1e^(-1(5x + 4)) - 1 and g(x) = (-1x - 2)/(2x + 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 integer closest to the smallest x-value where the functions f(x) = -5((1(-3x) + 2)/(3(-3x) + 3))^2 + 1((1(-3x) + 2)/(3(-3x) + 3)) - 3 and g(x) = -2(1x^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 x-coordinate where the graphs of f(x) = 4(-3|(-4x^2 + 4x - 5)| + 3) - 3 and g(x) = 1((1x + 2)/(-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": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1((-3(1|(-4x^2 + 2x - 4) - 3| - 2))/(-2(1|(-4x^2 + 2x - 4) - 3| - 2) - 3)) and g(x) = (-1(2e^(-2(-1x^2 - 2x + 1)) - 3) + 2)/(-3(2e^(-2(-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": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3|(-3((-3(-5x^2 - 5x - 2) - 3)/(-1(-5x^2 - 5x - 2) - 1)) - 5) - 3| + 1 and g(x) = -4(-5((-2x - 2)/(-1x - 3))^2 - 5((-2x - 2)/(-1x - 3)) + 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3|((-2(-1x^2 - 5x + 2) - 2)/(-1(-1x^2 - 5x + 2) - 1)) - 3| + 1 and g(x) = -4((-2x + 1)/(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 integer value closest to the x-coordinate where f(x) = (-1(-1(-1e^(-1x) + 1) + 4) - 1)/(3(-1(-1e^(-1x) + 1) + 4) - 2) and g(x) = 1e^(2((2x - 1)/(-3x + 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": "If you add up all the x-coordinates where the functions f(x) = (-1(5(-2x)^2 + 4(-2x) - 2) + 3)/(2(5(-2x)^2 + 4(-2x) - 2) - 3) and g(x) = 1sin(3\u03c0(-2x - 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": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-1(1(-3x^2 + 5x) + 2) + 3)/(2(1(-3x^2 + 5x) + 2)) and g(x) = -2e^(2(-1|x + 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 integer value closest to the x-coordinate where f(x) = 1|(-3(5x^2 + 2x + 5) + 4)| - 2 and g(x) = 4((-1x + 3)/(-1x + 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2|(-2((-3x - 3)/(1x - 3))) - 1| + 2 and g(x) = 4((-1x - 3)/(-2x - 1))^2 + 4((-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": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (2(-3|(5x - 4) - 1| + 1) + 1)/(-2(-3|(5x - 4) - 1| + 1) - 3) and g(x) = -3|((3x + 1)/(3x + 1)) + 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(3e^(2(-5x^2 + 1)) - 2) + 1)/(3(3e^(2(-5x^2 + 1)) - 2) - 2) and g(x) = (-2(-2|x - 2| - 1) + 3)/(-3(-2|x - 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) = -1(5((1x - 2)/(2x - 3))^2 + 2((1x - 2)/(2x - 3)) - 5) - 5 and g(x) = (1(-1x^2 + 2x - 3) - 1)/(1(-1x^2 + 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2e^(-1(4(-3((3x + 2)/(2x + 2)) - 4)^2 - 4(-3((3x + 2)/(2x + 2)) - 4) + 3)) + 3 and g(x) = (2(-5(3|x - 2|) + 5) + 1)/(-3(-5(3|x - 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 first intersection point (counting from left to right) between the functions f(x) = (1(-3|(-4(5x^2 + 3x + 5) + 3) + 1| - 2) - 2)/(2(-3|(-4(5x^2 + 3x + 5) + 3) + 1| - 2) - 1) and g(x) = (2(5(3x)^2 - 5(3x)))/(3(5(3x)^2 - 5(3x)) + 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = (1(3e^(-2x) + 1))/(-1(3e^(-2x) + 1) - 2) and g(x) = 2x^2 + 4x - 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 first x-coordinate (rounded to the nearest integer) where f(x) = 3|(3(2e^(-2x) - 1)^2 + 4(2e^(-2x) - 1) - 1)| + 1 and g(x) = -1|(3x^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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-1(-1|x - 1| + 2) + 3)/(-1(-1|x - 1| + 2) - 1) and g(x) = -1x + 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": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 4(4(1|(-1e^(-1x) + 3) + 1| - 2)^2 - 5(1|(-1e^(-1x) + 3) + 1| - 2) - 2) - 3 and g(x) = -1|(2(-5x^2 + 4x - 2) + 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(-2|((-1(-2x^2 - 3x + 2))/(-2(-2x^2 - 3x + 2) + 3)) - 3| - 2) + 1 and g(x) = (3(-1(1x^2 - 5x - 5)) + 1)/(-2(-1(1x^2 - 5x - 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": "Find the integer value closest to the x-coordinate where f(x) = 1((1x - 1)/(-1x - 2)) + 1 and g(x) = -2x^2 - 4x - 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 x-coordinate where the graphs of f(x) = 3(-1|x - 3| + 3) - 5 and g(x) = 5x + 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": "Find the first intersection point (counting from left to right) between the functions f(x) = 5(5(-2|((-1x + 1)/(3x - 2)) - 2| - 2)^2 - 4(-2|((-1x + 1)/(3x - 2)) - 2| - 2) + 4) - 5 and g(x) = 3(2(3|x| - 1) + 3)^2 + 2(2(3|x| - 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": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -3((1x + 1)/(2x - 1)) and g(x) = -5x^2 - 3x - 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1((-3x - 3)/(-1x + 2))^2 + 1((-3x - 3)/(-1x + 2)) + 4 and g(x) = (1x + 2)/(-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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1|((-1(3x^2 + 5x + 2) - 2)/(2(3x^2 + 5x + 2) - 1)) - 3| and g(x) = -1((2x)/(-3x - 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 x-coordinate where the graphs of f(x) = 2|((1(4x - 2) - 1)/(1(4x - 2) + 1)) - 1| - 1 and g(x) = -2((-3x - 1)/(-1x - 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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (3(5(-2|(1x^2 + 1x)| - 3) + 3) + 2)/(3(5(-2|(1x^2 + 1x)| - 3) + 3) + 2) and g(x) = 1(4((-2x - 3)/(3x - 1))^2 - 5((-2x - 3)/(3x - 1)) + 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": "Find the integer value closest to the x-coordinate where f(x) = 1(-2|x - 2| + 3)^2 - 1(-2|x - 2| + 3) + 2 and g(x) = (-1x - 3)/(-1x - 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": "Find the integer value closest to the x-coordinate where f(x) = -2(-1|(-1x - 5) - 3| - 1)^2 + 2(-1|(-1x - 5) - 3| - 1) + 5 and g(x) = 1(-5x^2 - 5x + 4) + 5 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 integer value closest to the x-coordinate where f(x) = -3(1cos(2\u03c0x + 3) - 3)^2 + 3 and g(x) = (-3x + 3)/(3x - 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = (-3(3(1|x + 2| - 3)^2 - 4(1|x + 2| - 3) - 3) - 2)/(2(3(1|x + 2| - 3)^2 - 4(1|x + 2| - 3) - 3) - 3) and g(x) = -3|(5x^2 - 2x - 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": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1sin(1\u03c0(-3|x|) - 2) and g(x) = 2x - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-3(-1(-4(1|x - 2|) - 5)^2 - 3(-4(1|x - 2|) - 5) - 1))/(2(-1(-4(1|x - 2|) - 5)^2 - 3(-4(1|x - 2|) - 5) - 1) - 3) and g(x) = -3|((3(-4x + 5) + 2)/(2(-4x + 5) + 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 x-coordinate where the graphs of f(x) = -3((-1x + 3)/(-1x))^2 - 2((-1x + 3)/(-1x)) - 2 and g(x) = -4x - 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) = -4((3(-1|x - 1| + 2) - 2)/(-3(-1|x - 1| + 2) - 2)) + 2 and g(x) = 2((-1x + 1)/(1x + 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) = 3(-4((-3x)/(3x - 2)) + 5)^2 - 1 and g(x) = (3(-3x + 3) - 3)/(-1(-3x + 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 value closest to the x-coordinate where f(x) = (3(-3x^2 - 2x - 5) - 2)/(-1(-3x^2 - 2x - 5) - 1) and g(x) = (-2x - 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": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1(-2(3|((2x - 2)/(-1x + 2)) + 1| - 1)^2 - 3(3|((2x - 2)/(-1x + 2)) + 1| - 1) + 3) - 1 and g(x) = (1(1(2x^2 - 5x + 5)) + 1)/(3(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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2x + 5 and g(x) = 1x^2 - 2x + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-3(1|x - 2| - 1) + 2)/(2(1|x - 2| - 1)) and g(x) = 2x - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-3(1x + 3) - 3)/(2(1x + 3) - 3) and g(x) = (1x - 3)/(-1x) 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|(1e^(-1x)) - 3| + 2 and g(x) = (1x - 3)/(2x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3(-3|x + 3| - 1) - 3 and g(x) = (2x - 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 integer closest to the smallest x-value where the functions f(x) = (-1(-1(-2(3|x - 1| + 3)^2 + 5(3|x - 1| + 3) + 2) + 4))/(3(-1(-2(3|x - 1| + 3)^2 + 5(3|x - 1| + 3) + 2) + 4) - 2) and g(x) = 1(-3|((3x - 1)/(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": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-3(5x - 1) - 2)/(1(5x - 1) - 3) and g(x) = 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": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (3(3(-1x^2 + 2x + 1) - 4) - 1)/(-2(3(-1x^2 + 2x + 1) - 4)) and g(x) = 3|(-4x + 1) - 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3|(2((-2x - 1)/(1x + 2)) - 1) - 3| and g(x) = (3(-2x^2 - 2x - 3))/(2(-2x^2 - 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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 4(5((2x)/(2x + 1))^2 + 3((2x)/(2x + 1)) - 3) - 2 and g(x) = -2((1x - 1)/(3x + 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = 5(-3|(5(-1e^(1x) - 3) - 4) - 1| + 1)^2 - 4(-3|(5(-1e^(1x) - 3) - 4) - 1| + 1) + 5 and g(x) = 3e^(-2((-1(-1|x| + 3) + 3)/(-2(-1|x| + 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 value closest to the x-coordinate where f(x) = -3|(2e^(1x)) - 1| - 3 and g(x) = (1x + 1)/(-3x + 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": "At what integer value (rounded) do the functions f(x) = 3|(1cos(2\u03c0x + 3) - 2) + 3| + 3 and g(x) = (-2x + 3)/(2x + 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|((-3x + 3)/(2x + 2)) + 3| + 1 and g(x) = -2x^2 + 4x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 3(-4x)^2 - 3(-4x) - 1 and g(x) = 5x 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 rightmost x-coordinate where f(x) = 5((-1(2|x - 3| + 1) + 3)/(-1(2|x - 3| + 1) - 3))^2 - 4((-1(2|x - 3| + 1) + 3)/(-1(2|x - 3| + 1) - 3)) - 3 and g(x) = 3cos(2\u03c0((2x + 3)/(2x + 1)) + 1) + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -2e^(-1((2x - 3)/(-1x - 3))) + 2 and g(x) = -5x^2 + 4x + 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": "Find the integer value closest to the x-coordinate where f(x) = -2((1x - 2)/(-2x))^2 - 4((1x - 2)/(-2x)) + 3 and g(x) = 2x + 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) = (2(-3(2|x - 2| + 2) + 4) - 2)/(-3(-3(2|x - 2| + 2) + 4) - 1) and g(x) = 1(3x - 1)^2 + 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": "If you add up all the x-coordinates where the functions f(x) = 2x + 1 and g(x) = -3x^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": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2(1sin(1\u03c0x + 1) + 2) - 5 and g(x) = 3x^2 - 4x - 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2(3|(1((-1x + 1)/(-1x + 2)) - 3) - 1| + 3)^2 + 5 and g(x) = (-2(2(-3sin(1\u03c0x + 2) - 1) - 4) + 1)/(-3(2(-3sin(1\u03c0x + 2) - 1) - 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3(-1|((1(2e^(2x) - 2) - 1)/(2(2e^(2x) - 2) + 3)) + 3| - 3)^2 + 4(-1|((1(2e^(2x) - 2) - 1)/(2(2e^(2x) - 2) + 3)) + 3| - 3) and g(x) = 5(2|(1e^(-2x) + 2) - 1| + 3)^2 + 1(2|(1e^(-2x) + 2) - 1| + 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|(-1(5x^2 - 3x + 1) + 5) + 2| - 3 and g(x) = -2(3|x - 1| + 1)^2 - 4(3|x - 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": "If you add up all the x-coordinates where the functions f(x) = 3((3(-2|x - 2| - 2) - 3)/(-2(-2|x - 2| - 2) - 1)) - 2 and g(x) = -3(3|x + 1| + 1)^2 + 2(3|x + 1| + 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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-1(-3e^(1(-1(3|x + 2| - 2) - 1)) - 2))/(-3(-3e^(1(-1(3|x + 2| - 2) - 1)) - 2) - 2) and g(x) = -2cos(3\u03c0((-1(2|x - 2| + 2) - 2)/(2(2|x - 2| + 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": "If you add up all the x-coordinates where the functions f(x) = 4(-3x^2 + 5x - 2) + 2 and g(x) = -4x^2 - 3x - 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": "3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3|(-1e^(2x)) - 3| + 1 and g(x) = 2|x| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3e^(1(-5x)) and g(x) = -3x^2 + 2x - 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 integer closest to the smallest x-value where the functions f(x) = -3(-1|(4((2x + 2)/(-3x))^2 + 2((2x + 2)/(-3x)) + 5) + 3| + 3) and g(x) = 4(1((1x - 3)/(1x + 2))^2 - 4((1x - 3)/(1x + 2)) - 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 integer value closest to the x-coordinate where f(x) = (-1(3(2e^(-1x) + 2)^2 + 3(2e^(-1x) + 2) - 1) - 3)/(1(3(2e^(-1x) + 2)^2 + 3(2e^(-1x) + 2) - 1) + 2) and g(x) = -4((3x + 1)/(2x + 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": "Find the second intersection point (counting from left to right) between the functions f(x) = 2|(3((2x + 2)/(-2x + 3)) + 2) - 1| + 3 and g(x) = (-3(-1x^2 + 5x + 5) - 2)/(-2(-1x^2 + 5x + 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": "6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1e^(-1(4x^2 - 5x + 2)) - 2 and g(x) = (3x - 2)/(-2x - 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 integer closest to the largest x-value where the functions f(x) = (1(5(3(-2|x + 1|)^2 - 4(-2|x + 1|) - 5) - 2) + 2)/(-1(5(3(-2|x + 1|)^2 - 4(-2|x + 1|) - 5) - 2) - 3) and g(x) = 1((-3(-2|x + 2| - 3) - 3)/(-1(-2|x + 2| - 3)))^2 - 4((-3(-2|x + 2| - 3) - 3)/(-1(-2|x + 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = 5(1|(-2e^(-2x) + 3) - 1|)^2 - 5(1|(-2e^(-2x) + 3) - 1|) - 5 and g(x) = (-1(-1sin(3\u03c0x + 2) - 3) - 2)/(2(-1sin(3\u03c0x + 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 first x-coordinate (rounded to the nearest integer) where f(x) = 1|(5(4((2x - 2)/(-2x - 1)) - 3)^2 + 1(4((2x - 2)/(-2x - 1)) - 3) + 4) + 1| + 1 and g(x) = 1e^(-2(1((-1x + 2)/(-1x + 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 value closest to the x-coordinate where f(x) = -1|(4(-3e^(-1x) - 1) + 1) - 3| - 3 and g(x) = -1((-1x - 2)/(-3x - 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": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -3|(4((2x - 2)/(-3x + 4))^2 + 2((2x - 2)/(-3x + 4)) - 4) + 1| + 3 and g(x) = -2|(-1cos(3\u03c0x - 3) - 3) - 3| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (1(5(-1|x - 1|)) + 1)/(1(5(-1|x - 1|)) + 1) and g(x) = 4((-1x)/(1x + 3))^2 - 5((-1x)/(1x + 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": "If you add up all the x-coordinates where the functions f(x) = (-1(3e^(-1x)) + 2)/(-3(3e^(-1x)) + 4) and g(x) = 1x^2 + 5x + 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": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(-4(3(-1cos(1\u03c0x + 1) + 3) + 5)^2 + 3(3(-1cos(1\u03c0x + 1) + 3) + 5) + 5) - 3)/(3(-4(3(-1cos(1\u03c0x + 1) + 3) + 5)^2 + 3(3(-1cos(1\u03c0x + 1) + 3) + 5) + 5)) and g(x) = -3|((2(2x - 3) + 3)/(-3(2x - 3) - 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3|((2x + 3)/(-3x + 2)) - 1| + 3 and g(x) = 2x - 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1(-1|(-4((-2x)/(1x - 3)) + 2)| - 3)^2 - 5(-1|(-4((-2x)/(1x - 3)) + 2)| - 3) - 2 and g(x) = 1|((1(-5x - 2) + 2)/(3(-5x - 2) - 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 value closest to the x-coordinate where f(x) = (-3(3|(1x - 3) + 1| - 3) + 2)/(1(3|(1x - 3) + 1| - 3) + 2) and g(x) = (2(5x - 4) + 3)/(-2(5x - 4)) 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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-1(3|x + 2| + 3))/(-2(3|x + 2| + 3) + 3) and g(x) = 3x^2 - 2x - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -5(1|x + 3| + 2) - 1 and g(x) = (2x - 1)/(-1x - 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 largest x-value where the functions f(x) = -3|((-1(-2cos(1\u03c0(-5x + 5) - 1) + 3) - 3)/(3(-2cos(1\u03c0(-5x + 5) - 1) + 3) + 1))| - 3 and g(x) = -2|(-2(-1e^(2x) + 3)^2 - 2(-1e^(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 value closest to the x-coordinate where f(x) = -4(1|(2x^2 + 1x) - 1| + 2) - 4 and g(x) = -1(3|x - 2| + 1)^2 - 1(3|x - 2| + 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": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 3|((2(-5(4x - 3)^2 + 2(4x - 3) + 1) - 2)/(-1(-5(4x - 3)^2 + 2(4x - 3) + 1) + 2)) + 2| - 1 and g(x) = 3|((2(-5x - 3) + 1)/(2(-5x - 3) - 1)) - 1| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-2(2(-2|(1x^2 + 5x - 5)| + 1) - 3))/(1(2(-2|(1x^2 + 5x - 5)| + 1) - 3) + 1) and g(x) = (-1(5(4x^2 - 1) + 1) + 3)/(3(5(4x^2 - 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": "At what integer value (rounded) do the functions f(x) = (3(3(-1|x - 3| + 3)^2 + 4(-1|x - 3| + 3) + 4) - 2)/(-2(3(-1|x - 3| + 3)^2 + 4(-1|x - 3| + 3) + 4) - 1) and g(x) = -3((-1x - 3)/(-3x - 2))^2 - 5((-1x - 3)/(-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) = -1sin(3\u03c0(-1x^2 + 4)) - 3 and g(x) = -4x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1|(2x^2 + 3x) - 2| - 1 and g(x) = 2|x + 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": "If you add up all the x-coordinates where the functions f(x) = (3(5(-1|x - 3| - 2)^2 - 5(-1|x - 3| - 2) - 2) - 3)/(3(5(-1|x - 3| - 2)^2 - 5(-1|x - 3| - 2) - 2) - 2) and g(x) = 1|(-5x + 4) - 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -5((2x)/(-3x - 1))^2 - 1((2x)/(-3x - 1)) + 1 and g(x) = (-1x - 2)/(2x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "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|(3((1(-4x - 3) - 3)/(-3(-4x - 3) - 2))^2 - 4((1(-4x - 3) - 3)/(-3(-4x - 3) - 2)) + 1) + 1| + 1 and g(x) = 3|(4((2x + 3)/(-3x + 2))^2 - 2((2x + 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 value closest to the x-coordinate where f(x) = 2|(-2(3x + 4)^2 - 5(3x + 4) - 1) - 3| - 1 and g(x) = 1((3x + 1)/(1x + 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": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2|((-1(-2cos(3\u03c0(5x - 2) - 2) + 2) - 3)/(-3(-2cos(3\u03c0(5x - 2) - 2) + 2))) + 2| + 2 and g(x) = 4((-3(2|x| - 3) + 2)/(2(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": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 4(2|x + 2| - 3) + 3 and g(x) = -1x^2 + 2x - 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": "If you add up all the x-coordinates where the functions f(x) = -1sin(3\u03c0(4((1(-3|x + 2| + 2) + 3)/(1(-3|x + 2| + 2) - 2))) + 1) + 1 and g(x) = 1(-5((1x - 3)/(-1x)) - 1)^2 + 5(-5((1x - 3)/(-1x)) - 1) - 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": "7", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2|(3(-2x - 1)^2 - 3)| and g(x) = 1sin(1\u03c0(3x^2 + 4x) - 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": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (2(-3(-3(1|x + 2| + 2)^2 - 1) + 5) - 2)/(-2(-3(-3(1|x + 2| + 2)^2 - 1) + 5)) and g(x) = 2((1(1|x| + 1) - 3)/(-3(1|x| + 1) + 4))^2 + 1((1(1|x| + 1) - 3)/(-3(1|x| + 1) + 4)) - 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) = (-1(-3|x - 3|))/(-1(-3|x - 3|) - 3) and g(x) = (3x - 3)/(-1x - 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (2(3|(5x^2 - 1x - 4) + 3| - 1) - 2)/(1(3|(5x^2 - 1x - 4) + 3| - 1) - 3) and g(x) = 3((-3x - 3)/(-1x - 3))^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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1(-2|(2x + 1) - 3|)^2 - 1(-2|(2x + 1) - 3|) + 1 and g(x) = (-3(1e^(1x) - 3) + 2)/(-3(1e^(1x) - 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 leftmost x-coordinate where f(x) = 1x + 2 and g(x) = 2x^2 + 1x intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 2|(-4(-1x + 4)^2 - 1(-1x + 4) + 1) - 2| - 2 and g(x) = (-2(-4x + 5) - 3)/(-1(-4x + 5)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-1(1(5x + 2)^2 - 2(5x + 2) - 1) - 2)/(-1(1(5x + 2)^2 - 2(5x + 2) - 1) + 2) and g(x) = -3|(1e^(-1x)) + 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 integer closest to the largest x-value where the functions f(x) = (3(-1|(-2cos(2\u03c0x)) + 3| - 2) + 3)/(-2(-1|(-2cos(2\u03c0x)) + 3| - 2) - 2) and g(x) = -1|((1x + 3)/(1x))| + 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) = -2|((1(3x^2 - 2x - 4) + 2)/(3(3x^2 - 2x - 4) - 2)) + 2| + 2 and g(x) = -1((-2x)/(-3x - 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": "Find the integer value closest to the x-coordinate where f(x) = -2(1(-2|x + 3| - 3) + 1)^2 + 4(1(-2|x + 3| - 3) + 1) - 2 and g(x) = -1e^(-1((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": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-1(2(5(-2|x + 3| - 3)^2 - 5(-2|x + 3| - 3) + 4) - 4) + 1)/(2(2(5(-2|x + 3| - 3)^2 - 5(-2|x + 3| - 3) + 4) - 4) - 3) and g(x) = (2(3(-3|x - 3|)^2 + 4(-3|x - 3|) - 3) - 3)/(1(3(-3|x - 3|)^2 + 4(-3|x - 3|) - 3) + 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 x-coordinate where the graphs of f(x) = -3|(-3((2x + 3)/(-2x))^2 - 4) - 2| + 3 and g(x) = (-1(3x - 1) + 2)/(-2(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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 1(-1x^2 + 2x - 5) - 1 and g(x) = -3x^2 - 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1|(-3sin(2\u03c0x + 1) - 3) - 3| + 1 and g(x) = 4x^2 + 4x - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3(5((3x + 2)/(-2x + 3)) - 2)^2 - 4(5((3x + 2)/(-2x + 3)) - 2) - 2 and g(x) = 1|(-1x^2 + 1x + 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) = -3((2(3|(-4x^2 - 1x - 2) + 3| - 3) + 1)/(-3(3|(-4x^2 - 1x - 2) + 3| - 3) + 4)) - 5 and g(x) = -5(-1((3x)/(-1x + 2)) - 4)^2 - 1(-1((3x)/(-1x + 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": "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) = -1(-4((1x)/(3x + 2))^2 + 4((1x)/(3x + 2))) - 5 and g(x) = -3sin(2\u03c0(1x^2 - 5x + 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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -4x - 1 and g(x) = 2x^2 - 5x - 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 x-coordinate where the graphs of f(x) = 3((-2x + 2)/(1x + 3)) - 4 and g(x) = -2|x + 3| - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-9", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -4(-1|x| + 2) and g(x) = 1x + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2(-3x^2 - 5x + 5) - 3 and g(x) = 3x + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5((-3(3e^(-1x) - 2) + 2)/(3(3e^(-1x) - 2)))^2 - 2((-3(3e^(-1x) - 2) + 2)/(3(3e^(-1x) - 2))) + 3 and g(x) = 3((-2x - 2)/(-1x + 2)) + 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1(-4(-3|((3x + 1)/(2x - 3))| - 3)^2 - 1(-3|((3x + 1)/(2x - 3))| - 3) - 1) + 4 and g(x) = -3((-2(-5x + 1))/(2(-5x + 1) + 2))^2 + 5((-2(-5x + 1))/(2(-5x + 1) + 2)) - 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 integer value closest to the x-coordinate where f(x) = -3(-2x + 1)^2 - 2(-2x + 1) and g(x) = (2x + 3)/(-2x + 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = -1(2e^(1((-3(-3x^2 + 4x - 1) + 2)/(1(-3x^2 + 4x - 1)))) + 3) - 2 and g(x) = (-3(-2|(2e^(-1x) - 3)| - 1) + 3)/(-1(-2|(2e^(-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": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 2e^(1(-1|x - 1| + 2)) + 1 and g(x) = 4x^2 - 5x - 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) = 2|(-4(-2((-3x)/(-3x + 1))^2 + 1((-3x)/(-3x + 1)) + 5))| and g(x) = (2(2|(-3x - 4) - 1| + 2) + 1)/(1(2|(-3x - 4) - 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-3(3|x + 2| - 1) + 2)/(2(3|x + 2| - 1) + 1) and g(x) = (2x + 3)/(3x + 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|(-2e^(2x) - 1)| - 3 and g(x) = 3x^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": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 5(3e^(-1(-1x + 5)) + 3)^2 + 2(3e^(-1(-1x + 5)) + 3) - 4 and g(x) = (-1(-2|x - 2| + 2) + 3)/(1(-2|x - 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": "4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1((-2x + 1)/(3x + 3))^2 - 4((-2x + 1)/(3x + 3)) + 2 and g(x) = (-1x - 3)/(-3x - 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) = 2((2x - 3)/(-1x - 3))^2 - 1((2x - 3)/(-1x - 3)) - 5 and g(x) = 1|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": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1|(-4(3e^(2((1x + 2)/(-2x + 1))) + 3)^2 + 3(3e^(2((1x + 2)/(-2x + 1))) + 3) - 1) - 3| + 1 and g(x) = -2(1|(-5x + 1) + 3| + 2)^2 - 3(1|(-5x + 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": "Find the integer closest to the smallest x-value where the functions f(x) = -1|(-4x^2 + 4) + 2| - 1 and g(x) = -4x^2 - 1x + 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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = 2e^(-1(-4((-1(2|x - 2| + 3) - 1)/(-3(2|x - 2| + 3)))^2 + 5((-1(2|x - 2| + 3) - 1)/(-3(2|x - 2| + 3))))) + 2 and g(x) = 2(4((1x + 3)/(2x + 3))^2 + 4((1x + 3)/(2x + 3)) + 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) = 1|(5((3x - 3)/(3x + 3))^2 + 2((3x - 3)/(3x + 3)))| + 3 and g(x) = 3e^(-2(1x - 5)) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-1(-1(-1|(3x^2 + 5) - 3| + 3) - 5) + 2)/(3(-1(-1|(3x^2 + 5) - 3| + 3) - 5) - 2) and g(x) = -1(3((3x + 3)/(3x + 3))^2 - 2((3x + 3)/(3x + 3)) - 5) - 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) = 1(2|x - 2| - 3) - 1 and g(x) = -3|x - 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -4((1x)/(2x + 2)) + 3 and g(x) = (2x)/(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 integer value closest to the x-coordinate where f(x) = (-3(3|x + 1| - 3) + 3)/(-3(3|x + 1| - 3) + 2) and g(x) = (3x + 3)/(-2x + 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 x-coordinate where the graphs of f(x) = -3cos(1\u03c0((1(5x + 4) - 2)/(2(5x + 4) - 1)) - 3) - 2 and g(x) = -5(-1e^(-2x)) - 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 integer value closest to the x-coordinate where f(x) = 3((-2x + 3)/(2x - 1))^2 - 5 and g(x) = -3x - 5 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 rightmost x-coordinate where f(x) = (-3(-5(-2e^(-2x)) - 3) + 3)/(-2(-5(-2e^(-2x)) - 3)) and g(x) = 3((3x)/(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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2(2x^2 + 4x - 3) - 5 and g(x) = (2x - 2)/(-1x) 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((3x + 3)/(-1x - 3))^2 - 2((3x + 3)/(-1x - 3)) + 5 and g(x) = 2|x - 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": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1(1|((3x - 2)/(2x - 3)) + 1| - 2) - 4 and g(x) = -1(1x^2 + 2x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2|(1x + 4)| and g(x) = -4x^2 + 5x - 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) = -1e^(1(1|x - 3| - 2)) and g(x) = 2|x - 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (2(-3|x - 3| - 1) + 1)/(-3(-3|x - 3| - 1)) and g(x) = (-2x + 2)/(1x + 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) = 1(-2|(-2x^2 - 2x + 1) - 1| - 2) + 4 and g(x) = (-1(-4x + 1) - 3)/(-2(-4x + 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|(-2x - 1)| + 3 and g(x) = 5x^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 sum of all x-coordinates (rounded to the nearest integer) where f(x) = (1(5(-5x + 4)^2 - 5(-5x + 4) + 2) - 2)/(3(5(-5x + 4)^2 - 5(-5x + 4) + 2) - 2) and g(x) = (1(-1cos(3\u03c0x - 1) + 3) + 2)/(-3(-1cos(3\u03c0x - 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 second intersection point (counting from left to right) between the functions f(x) = -1((-2(3cos(3\u03c0x - 3) - 2) + 1)/(1(3cos(3\u03c0x - 3) - 2) - 3)) - 4 and g(x) = 4((-1x - 2)/(1x - 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": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (3(-3(-1|x + 1|)^2 - 2(-1|x + 1|)) + 3)/(2(-3(-1|x + 1|)^2 - 2(-1|x + 1|))) and g(x) = (2(-3|x - 1| - 1) - 1)/(-1(-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": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -4((-3x - 2)/(1x - 2))^2 - 1((-3x - 2)/(1x - 2)) and g(x) = 3x 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 value closest to the x-coordinate where f(x) = 4(-2|((-2x)/(1x)) - 2|)^2 + 1(-2|((-2x)/(1x)) - 2|) - 1 and g(x) = 1(2e^(-1x)) - 5 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 integer closest to the largest x-value where the functions f(x) = (-2(-1|(-3(1x - 5)^2 - 2(1x - 5) - 3) + 2|))/(1(-1|(-3(1x - 5)^2 - 2(1x - 5) - 3) + 2|) + 3) and g(x) = (-3(5(-3|x - 1| + 1) + 4) - 2)/(2(5(-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": "Find the integer closest to the smallest x-value where the functions f(x) = 2x + 4 and g(x) = 1x^2 + 4x 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) = (3(1|(-2x - 4) - 2|) - 1)/(-2(1|(-2x - 4) - 2|) - 2) and g(x) = (1(4x^2 - 3x + 4) - 3)/(2(4x^2 - 3x + 4) + 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": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1e^(2(1((3x + 3)/(-1x + 2))^2 - 2((3x + 3)/(-1x + 2)) + 1)) - 1 and g(x) = -1(2|x + 3|)^2 + 2(2|x + 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": "If you add up all the x-coordinates where the functions f(x) = -2x - 4 and g(x) = -4x^2 - 4x + 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": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3|(-3cos(2\u03c0x + 3)) - 3| + 3 and g(x) = 3x^2 - 2x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "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(-3|((-2(-1x^2 - 3x + 5) - 1)/(1(-1x^2 - 3x + 5) + 3)) + 2| + 3) + 5 and g(x) = (2(-1|(-2x - 5) + 2| + 2) + 1)/(2(-1|(-2x - 5) + 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) = 4((3(1(-2|x| + 3)^2 - 5(-2|x| + 3) - 4) + 3)/(-2(1(-2|x| + 3)^2 - 5(-2|x| + 3) - 4) - 1)) + 4 and g(x) = 2(-1((1x + 1)/(-2x + 3))^2 - 4((1x + 1)/(-2x + 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (2(-3|(-4(-2e^(1x) - 1)^2 - 2(-2e^(1x) - 1) - 3) + 2|) + 1)/(3(-3|(-4(-2e^(1x) - 1)^2 - 2(-2e^(1x) - 1) - 3) + 2|) + 3) and g(x) = 1(-1((-1x - 1)/(2x - 3))^2 + 2((-1x - 1)/(2x - 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 3|((-2(-1x^2 - 4x + 2) - 2)/(3(-1x^2 - 4x + 2)))| + 3 and g(x) = -2(-2x^2 + 1x - 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) = 2sin(3\u03c0((-1(2|x + 1| + 2) + 2)/(-1(2|x + 1| + 2) - 3)) + 3) and g(x) = 2((1x + 2)/(3x - 2)) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (1(2x^2 - 5x - 4) + 2)/(-1(2x^2 - 5x - 4) - 1) and g(x) = -5x 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(-2(2|(1x + 2) - 1| - 2)^2 - 3(2|(1x + 2) - 1| - 2) - 1))/(3(-2(2|(1x + 2) - 1| - 2)^2 - 3(2|(1x + 2) - 1| - 2) - 1)) and g(x) = -2|((-2(-2x - 4) - 1)/(2(-2x - 4) + 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": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1(3(-2|((-3x - 3)/(-2x + 3)) - 2| - 2) - 5)^2 - 5(3(-2|((-3x - 3)/(-2x + 3)) - 2| - 2) - 5) and g(x) = -4((-3(-1x - 1) + 1)/(-1(-1x - 1)))^2 + 5((-3(-1x - 1) + 1)/(-1(-1x - 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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -2(-2(1|x + 1| + 2) + 5)^2 + 5(-2(1|x + 1| + 2) + 5) + 2 and g(x) = (-2(-3|x + 1| - 2) - 2)/(2(-3|x + 1| - 2) + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1|(-3x + 2) + 1| + 2 and g(x) = 4x - 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((-3(2|(4x^2 + 1) + 3| + 3) - 1)/(1(2|(4x^2 + 1) + 3| + 3) - 2)) - 2 and g(x) = (1(-5(2e^(-2x) - 2) - 2) + 3)/(-3(-5(2e^(-2x) - 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": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -2(2|(1x - 2) - 2| - 1)^2 - 1(2|(1x - 2) - 2| - 1) + 5 and g(x) = 2cos(2\u03c0((1x - 2)/(-3x + 4)) + 3) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -4(-1x^2 + 5x) - 2 and g(x) = 1x^2 - 3x - 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3|((2(-2x - 1) - 2)/(-2(-2x - 1) - 2)) + 2| and g(x) = (-2(3x) + 3)/(-1(3x) + 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) = 3((-2(-2|(-2x - 3) + 3| - 2) - 2)/(3(-2|(-2x - 3) + 3| - 2) + 3))^2 + 3((-2(-2|(-2x - 3) + 3| - 2) - 2)/(3(-2|(-2x - 3) + 3| - 2) + 3)) + 2 and g(x) = 2(3e^(1(1x + 4)) - 1)^2 + 5(3e^(1(1x + 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": "-5", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1e^(1((1x + 2)/(-3x + 2))) + 3 and g(x) = -3x^2 + 2x - 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) = 1e^(-2((1(-1x + 4) + 3)/(2(-1x + 4)))) - 1 and g(x) = 2|(-3x^2 + 5x - 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": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -5x + 2 and g(x) = -4x^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": "If you add up all the x-coordinates where the functions f(x) = -1(-4x + 4)^2 - 1(-4x + 4) and g(x) = 3|x - 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = -1(3e^(2(2|(-4x - 2) + 2| - 3)) - 2)^2 - 3(3e^(2(2|(-4x - 2) + 2| - 3)) - 2) + 5 and g(x) = 2sin(3\u03c0(2|(2x + 3) + 3| - 3) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "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(-5x^2 - 4x - 1) - 1)/(1(-5x^2 - 4x - 1) + 2) and g(x) = 5x - 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 integer closest to the smallest x-value where the functions f(x) = -3|((-1(5(-2x - 3)^2 - 3(-2x - 3) - 2) - 1)/(-3(5(-2x - 3)^2 - 3(-2x - 3) - 2) - 2)) + 1| + 2 and g(x) = 1(-2e^(-1(-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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1|(1e^(2(1((3x + 3)/(-1x + 2))^2 + 4((3x + 3)/(-1x + 2)) - 5)) - 2) - 1| + 2 and g(x) = -1(2(-1|x| - 3) + 1)^2 - 3(2(-1|x| - 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": "At what integer value (rounded) do the functions f(x) = -2|(-5(-1x^2 + 1x - 3) + 1)| and g(x) = (-2(4x + 5) - 2)/(2(4x + 5) + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1|(-4((-2x + 2)/(1x)) - 4)| - 1 and g(x) = -2|(-2x + 3) + 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 first intersection point (counting from left to right) between the functions f(x) = -1|(-2e^(-1(1x^2 - 5x - 2)) - 1) + 2| - 2 and g(x) = -3(-3x + 5)^2 + 5(-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": "Find the second intersection point (counting from left to right) between the functions f(x) = -2(-4(2|((-1x + 2)/(2x - 1)) - 2|)^2 + 4(2|((-1x + 2)/(2x - 1)) - 2|) + 4) - 1 and g(x) = (2(4(1e^(-1x) - 1) + 2) + 1)/(-3(4(1e^(-1x) - 1) + 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": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1|((-1x - 1)/(-3x - 1)) - 2| + 1 and g(x) = -2x^2 - 5x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2|((2x + 3)/(-2x - 2)) + 2| and g(x) = (-3x - 2)/(2x) 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 value closest to the x-coordinate where f(x) = -1sin(3\u03c0(-3(-1|x + 1| - 2) - 2) - 1) and g(x) = -5(-1e^(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": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 4((-2x + 2)/(-2x + 3)) - 3 and g(x) = 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": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1((3(2|x + 2| + 1) - 2)/(-3(2|x + 2| + 1) + 1)) - 3 and g(x) = -3|((1x - 1)/(2x + 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": "If you add up all the x-coordinates where the functions f(x) = (2(-1(-2|x + 2|)^2 - 5(-2|x + 2|) - 3) + 1)/(-1(-1(-2|x + 2|)^2 - 5(-2|x + 2|) - 3) + 2) and g(x) = (-2(3cos(1\u03c0x - 2) + 3) - 2)/(2(3cos(1\u03c0x - 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": "-4", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -5(3|x + 1| - 3)^2 + 1(3|x + 1| - 3) + 3 and g(x) = -5x - 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": "Find the integer closest to the smallest x-value where the functions f(x) = 4(-2|((-1x + 2)/(2x + 1)) + 3| - 2) - 4 and g(x) = (3(1|x - 3| - 1) + 3)/(3(1|x - 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) = (3(-2|x - 3| - 3) + 3)/(-3(-2|x - 3| - 3) + 4) and g(x) = -5x^2 + 5x + 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) = 3e^(2((-3(1x^2 - 5) - 1)/(-3(1x^2 - 5) + 1))) and g(x) = -2sin(3\u03c0(-1x^2 + 2x + 1) + 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 integer closest to the largest x-value where the functions f(x) = 1e^(1(1|(4((1x + 2)/(-2x - 1))^2 + 3((1x + 2)/(-2x - 1)) - 5) + 3| + 1)) + 2 and g(x) = -4(-3e^(1(2x^2 - 3x + 4)) + 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(-2x^2 + 5x + 5) - 2)/(-2(-2x^2 + 5x + 5) - 3) and g(x) = (-2x + 3)/(3x + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -5(2|(-1((-1x - 2)/(2x + 1))^2 + 2((-1x - 2)/(2x + 1)) - 4) + 1| + 1) - 2 and g(x) = -5((-2(3|x + 3| - 3) + 1)/(1(3|x + 3| - 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": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 2e^(2(-4(-5x^2 - 2x + 3) + 1)) - 3 and g(x) = (3(-2|x - 3| - 3) + 1)/(1(-2|x - 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1|((3(1x^2 + 5x + 3) - 1)/(3(1x^2 + 5x + 3))) - 3| and g(x) = (-1(5x))/(2(5x) - 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(-4(3x - 1)^2 + 3(3x - 1) + 4) - 3)/(-3(-4(3x - 1)^2 + 3(3x - 1) + 4) + 1) and g(x) = 1((-3x - 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": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (2(-4(3|(5x - 5) + 2| - 1)^2 + 3(3|(5x - 5) + 2| - 1) + 4) + 2)/(-2(-4(3|(5x - 5) + 2| - 1)^2 + 3(3|(5x - 5) + 2| - 1) + 4) - 1) and g(x) = -2|(3((-2x + 1)/(3x + 1)) + 4) - 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 largest x-value where the functions f(x) = 2|(4((1x + 2)/(-2x - 2))^2 + 2((1x + 2)/(-2x - 2)) - 2) - 1| + 2 and g(x) = 3e^(-1((3x - 3)/(-2x))) - 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": "Find the integer closest to the largest x-value where the functions f(x) = 1|((3x + 3)/(2x - 3)) - 1| - 1 and g(x) = 5x + 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) = -1|(5x^2 - 1x + 1) + 1| + 3 and g(x) = -1x^2 + 4x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (1(-4(1|x + 2| - 2)^2 - 4) + 1)/(-2(-4(1|x + 2| - 2)^2 - 4) + 3) and g(x) = 4(-3x^2 - 4x + 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 rightmost x-coordinate where f(x) = 1(-2|x + 3| - 2) - 1 and g(x) = -4x^2 - 2x - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-2(4(2|x - 3| - 2)^2 - 5(2|x - 3| - 2) + 4) - 1)/(2(4(2|x - 3| - 2)^2 - 5(2|x - 3| - 2) + 4)) and g(x) = 2|(3x - 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": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 3(-1|((-2(-2x) + 1)/(-2(-2x) + 3)) - 3| - 3)^2 + 4(-1|((-2(-2x) + 1)/(-2(-2x) + 3)) - 3| - 3) + 4 and g(x) = (1(2(-2|x - 2| + 1)^2 - 5) + 2)/(3(2(-2|x - 2| + 1)^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 integer closest to the largest x-value where the functions f(x) = 1(3|((-1x + 1)/(-1x + 2)) - 1|)^2 + 2(3|((-1x + 1)/(-1x + 2)) - 1|) + 4 and g(x) = 1(-1|x - 2| + 2)^2 - 3(-1|x - 2| + 2) + 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": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(5(3|(-5x + 1) - 1| + 3)^2 + 5(3|(-5x + 1) - 1| + 3) - 5) + 3)/(-3(5(3|(-5x + 1) - 1| + 3)^2 + 5(3|(-5x + 1) - 1| + 3) - 5) - 3) and g(x) = -1(3(3|x| - 1)^2 - 1(3|x| - 1) - 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5(3|(-2e^(1x) + 2) - 2| - 2)^2 - 2(3|(-2e^(1x) + 2) - 2| - 2) + 5 and g(x) = (1(-5x - 4) - 1)/(-1(-5x - 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": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 5x and g(x) = -1x^2 - 1x intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2|(1x^2 + 3x) + 3| + 3 and g(x) = (-1x + 1)/(-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": "At what integer value (rounded) do the functions f(x) = 2|((-1(2sin(1\u03c0x) + 2))/(-3(2sin(1\u03c0x) + 2) + 4)) + 2| + 3 and g(x) = -3((-2x)/(-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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3((-2(-1x^2 + 5x - 4))/(2(-1x^2 + 5x - 4) + 1)) + 5 and g(x) = -1(3cos(2\u03c0x - 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) = (1(-3(-4x)^2 + 4(-4x) - 1) + 2)/(2(-3(-4x)^2 + 4(-4x) - 1) + 2) and g(x) = -3sin(2\u03c0(5x^2 - 1x + 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) = -2|(1cos(2\u03c0(1x + 3) + 3)) - 3| + 3 and g(x) = -2e^(1(-4x)) + 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -4((-2(1|(-1x^2 - 4x + 2) + 1| - 3) - 2)/(3(1|(-1x^2 - 4x + 2) + 1| - 3))) + 5 and g(x) = -2|(-2(-2x + 2)^2 + 1(-2x + 2) + 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1(1x + 3)^2 + 1(1x + 3) - 5 and g(x) = (-2x - 2)/(-2x - 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": "If you add up all the x-coordinates where the functions f(x) = 5((1(1e^(1x)) + 3)/(-3(1e^(1x)) - 2))^2 + 2((1(1e^(1x)) + 3)/(-3(1e^(1x)) - 2)) + 2 and g(x) = 4(-3x + 3)^2 + 4(-3x + 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 value closest to the x-coordinate where f(x) = -1((1x + 1)/(1x + 1)) + 1 and g(x) = 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": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -3|(5(2((1x - 3)/(-3x - 1))^2 - 2) - 3) + 2| - 2 and g(x) = 2((1(3|x| - 2) + 3)/(-3(3|x| - 2) - 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) = 1|(-1x) - 3| and g(x) = (1x - 2)/(2x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3|((-2(1sin(3\u03c0x - 1)) + 3)/(-2(1sin(3\u03c0x - 1)) + 3)) - 2| + 2 and g(x) = -2e^(-1(-4x)) 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) = -5(-1|(1((1x + 3)/(-3x - 2))^2 - 2((1x + 3)/(-3x - 2)) - 5) - 1| - 2) + 5 and g(x) = 3((-3(-1x - 4) + 2)/(3(-1x - 4)))^2 - 3((-3(-1x - 4) + 2)/(3(-1x - 4))) + 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 average of all x-coordinates (rounded to the nearest integer) where f(x) = 1((3x + 1)/(1x)) - 3 and g(x) = 1x^2 - 5x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "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) = -5(-2|x - 3|)^2 - 4(-2|x - 3|) - 1 and g(x) = (-3x)/(2x + 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 leftmost x-coordinate where f(x) = (-1(-1e^(1(-2|(-3x - 4) - 1| + 1)) + 2) - 3)/(2(-1e^(1(-2|(-3x - 4) - 1| + 1)) + 2) - 1) and g(x) = -3cos(2\u03c0(-2|((-1x - 1)/(-2x + 3)) - 2| + 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 first intersection point (counting from left to right) between the functions f(x) = -4(1|((-2x + 1)/(3x - 2)) + 1| - 2)^2 - 1(1|((-2x + 1)/(3x - 2)) + 1| - 2) - 2 and g(x) = -4(-1e^(1x) - 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": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-3(3(-3|x - 1| + 2) + 1) + 2)/(2(3(-3|x - 1| + 2) + 1) - 3) and g(x) = -3(2e^(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": "Find the first intersection point (counting from left to right) between the functions f(x) = 1|(-2e^(-1x) + 3) - 2| + 1 and g(x) = 1|x - 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) = (-3(-1(-1cos(2\u03c0x - 2)) - 5) - 2)/(-3(-1(-1cos(2\u03c0x - 2)) - 5)) and g(x) = 1|(-2e^(-2x) + 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 rightmost x-coordinate where f(x) = (1(1e^(-1x) - 2) - 1)/(-2(1e^(-1x) - 2) - 2) and g(x) = (1x + 3)/(2x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1|((1(2x - 3))/(2(2x - 3) - 1)) + 2| + 2 and g(x) = 2(-4x^2 - 1x - 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 smallest x-value where the functions f(x) = -5((2(3x - 5) - 1)/(-3(3x - 5)))^2 - 5((2(3x - 5) - 1)/(-3(3x - 5))) + 1 and g(x) = -3sin(1\u03c0(-1|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": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (3(2e^(1(1(-1|x + 1| + 3) - 3))))/(-3(2e^(1(1(-1|x + 1| + 3) - 3))) + 4) and g(x) = -2((-3(-3|x - 3|) - 3)/(1(-3|x - 3|) - 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -3(1(3|((1x - 3)/(2x + 3)) + 3|) + 3)^2 + 4(1(3|((1x - 3)/(2x + 3)) + 3|) + 3) + 5 and g(x) = -1(1|(1x - 5) - 2| - 3)^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": "If you add up all the x-coordinates where the functions f(x) = -1|(-3x^2 + 4x - 1) + 3| - 1 and g(x) = 1|x + 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": "If you add up all the x-coordinates where the functions f(x) = 3|(-2x + 3)| - 1 and g(x) = 2|x + 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": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-1(-1e^(-1(2|x + 2| - 2)) + 3) - 2)/(-1(-1e^(-1(2|x + 2| - 2)) + 3) - 3) and g(x) = 2cos(3\u03c0(-2x - 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": "At what integer value (rounded) do the functions f(x) = (2(-1|x - 2| - 2) + 1)/(-2(-1|x - 2| - 2) - 1) and g(x) = (-2x + 1)/(-1x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1|(3e^(1x)) + 2| + 2 and g(x) = (-2x + 1)/(-1x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 5(4((-3x - 3)/(3x + 2))^2 + 4((-3x - 3)/(3x + 2)) + 1) - 4 and g(x) = -1((3x + 3)/(-1x + 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 leftmost x-coordinate where f(x) = 2x + 2 and g(x) = 1x^2 - 4x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "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) = 3sin(2\u03c0(2|((1(-1x^2 + 3x + 3))/(-3(-1x^2 + 3x + 3) - 1)) + 3| + 2) + 3) and g(x) = -2|((2(-4x + 3) + 3)/(-1(-4x + 3) + 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 second intersection point (counting from left to right) between the functions f(x) = (3(-3e^(2(1x^2 + 1x + 2)) - 1) - 2)/(-3(-3e^(2(1x^2 + 1x + 2)) - 1) + 2) and g(x) = -2|(-2x - 5) + 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": "Find the second intersection point (counting from left to right) between the functions f(x) = -1|((-1(1cos(2\u03c0(4x^2) - 3) + 3) + 2)/(-2(1cos(2\u03c0(4x^2) - 3) + 3))) + 2| + 3 and g(x) = -2|((-2(5x^2 - 2x + 1) + 3)/(-1(5x^2 - 2x + 1) + 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": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 4((2x + 1)/(-3x)) - 2 and g(x) = 1x^2 + 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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -4(2|((-3(1x^2 - 3x - 2) - 3)/(1(1x^2 - 3x - 2) - 3)) + 2| + 3) + 5 and g(x) = (-2(1(3x^2 + 4x - 4) - 1) - 2)/(3(1(3x^2 + 4x - 4) - 1) - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1|((2x + 1)/(2x)) - 3| and g(x) = (-1x + 1)/(-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": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (1(-1(-2|x| + 2)^2 + 4) - 3)/(-2(-1(-2|x| + 2)^2 + 4) + 3) and g(x) = 2e^(-2(-3x^2 + 1x + 1)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (1(-5(1|x - 1| - 1)^2 - 4(1|x - 1| - 1) + 2) + 1)/(3(-5(1|x - 1| - 1)^2 - 4(1|x - 1| - 1) + 2) + 1) and g(x) = -2|(4x + 3) + 3| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 3e^(2(5x + 2)) - 1 and g(x) = -3x^2 - 1x 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) = (3(3e^(-1(1x^2 - 5x - 4)) - 1) - 3)/(1(3e^(-1(1x^2 - 5x - 4)) - 1) - 2) and g(x) = -1|(-2x^2 + 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|(-2x^2 - 5x + 1) + 2| and g(x) = (2x)/(2x + 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(2e^(-2x) - 1))/(-3(2e^(-2x) - 1) + 4) and g(x) = 2|x| 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|((2(-1x - 4) + 2)/(1(-1x - 4) + 2)) - 3| - 1)^2 + 5(1|((2(-1x - 4) + 2)/(1(-1x - 4) + 2)) - 3| - 1) - 1 and g(x) = (3(-2(2x^2 - 3x) + 1) + 3)/(-3(-2(2x^2 - 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": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1(-4(-2|(-2e^(-2x) + 2) + 3| - 1)^2 - 4(-2|(-2e^(-2x) + 2) + 3| - 1) - 1) + 4 and g(x) = (-2(-1sin(2\u03c0(-4x - 4) + 1) - 3) + 3)/(2(-1sin(2\u03c0(-4x - 4) + 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": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 3|((-2x + 1)/(1x - 2)) - 1| - 2 and g(x) = (3x)/(-3x - 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(2|x - 1| - 3)^2 - 1(2|x - 1| - 3) - 2 and g(x) = -2|x - 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) = 1sin(2\u03c0(1|(-4x^2 - 4x + 3) + 2|) + 3) - 2 and g(x) = -2e^(1(-5x + 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) = 5(1|x + 1| - 2) + 2 and g(x) = -3|x + 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": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2|(3((-1(1x^2 - 2) + 1)/(-1(1x^2 - 2) - 1)) - 5)| + 2 and g(x) = (-1(3(5x^2 + 3x - 5) + 4) + 2)/(-2(3(5x^2 + 3x - 5) + 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": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1((1(3e^(-2x) + 1) + 2)/(-3(3e^(-2x) + 1) - 1))^2 + 3((1(3e^(-2x) + 1) + 2)/(-3(3e^(-2x) + 1) - 1)) - 4 and g(x) = -3|((-3x + 3)/(2x - 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 leftmost x-coordinate where f(x) = 3|((2x - 3)/(-1x + 2)) - 3| - 1 and g(x) = 1|x - 1| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 3(-2((1x + 1)/(1x + 3))^2 + 4((1x + 1)/(1x + 3)) + 2) and g(x) = -4(-2|x + 3|)^2 + 3(-2|x + 3|) + 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) = 2|(1e^(-1(-4x - 2)) + 1) - 2| - 2 and g(x) = (2(4x^2 + 3x + 5) - 2)/(2(4x^2 + 3x + 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": "Find the integer value closest to the x-coordinate where f(x) = 2e^(-1((2x)/(1x - 3))) + 1 and g(x) = (3x + 1)/(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": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 3((-1(1(2|x + 1| - 2) - 4) - 3)/(1(1(2|x + 1| - 2) - 4)))^2 + 2((-1(1(2|x + 1| - 2) - 4) - 3)/(1(1(2|x + 1| - 2) - 4))) - 2 and g(x) = -2e^(2(1(-1x)^2 - 2(-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) = 2|((2(-2(-4x + 4)^2 - 1(-4x + 4) + 1))/(2(-2(-4x + 4)^2 - 1(-4x + 4) + 1) + 3)) - 3| - 2 and g(x) = 5((1(-4x + 2) + 2)/(-1(-4x + 2)))^2 - 2((1(-4x + 2) + 2)/(-1(-4x + 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) = (-1(2sin(2\u03c0(4x^2 - 5x + 1) - 3) + 1) - 2)/(2(2sin(2\u03c0(4x^2 - 5x + 1) - 3) + 1) + 3) and g(x) = 1(5x - 3)^2 + 5(5x - 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 closest to the largest x-value where the functions f(x) = (-2(-1(1|(1x - 2) + 2| - 1)^2 - 3(1|(1x - 2) + 2| - 1) + 3) - 2)/(2(-1(1|(1x - 2) + 2| - 1)^2 - 3(1|(1x - 2) + 2| - 1) + 3) + 3) and g(x) = (1(-2(-4x^2 - 5x - 5) + 3) + 1)/(2(-2(-4x^2 - 5x - 5) + 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 first x-coordinate (rounded to the nearest integer) where f(x) = 4(-3cos(3\u03c0(-4((-2x + 3)/(-3x - 2)) - 4) + 2) + 2)^2 - 5(-3cos(3\u03c0(-4((-2x + 3)/(-3x - 2)) - 4) + 2) + 2) + 3 and g(x) = 4(-2|(3x - 3) - 1| - 3)^2 - 3(-2|(3x - 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": "At what integer value (rounded) do the functions f(x) = 4(1|(3cos(2\u03c0x - 2) - 3) - 2| + 3)^2 + 5(1|(3cos(2\u03c0x - 2) - 3) - 2| + 3) + 5 and g(x) = (2(-5x + 1) - 3)/(-1(-5x + 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(4(2(-2|x| + 1) - 1)^2 + 4(2(-2|x| + 1) - 1) + 4))/(3(4(2(-2|x| + 1) - 1)^2 + 4(2(-2|x| + 1) - 1) + 4) - 1) and g(x) = -4(1e^(-1((-2x + 3)/(-1x + 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": "At what integer value (rounded) do the functions f(x) = -1((-2(-1|x - 2| - 1) - 1)/(-3(-1|x - 2| - 1))) + 4 and g(x) = (-2(1x - 2) + 1)/(1(1x - 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"}