messages listlengths 1 1 | ground_truth stringclasses 23 values | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3((3(1|x + 1| + 3) - 3)/(3(1|x + 1| + 3) + 3)) + 1 and g(x) = -1(-2|x + 2| + 3) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -2 | MATH |
[
{
"content": "At what integer value (rounded) do the functions f(x) = (-1(-2e^(1(1|x - 2| - 2))) + 2)/(2(-2e^(1(1|x - 2| - 2))) + 1) and g(x) = (3(3e^(-2x) - 2) - 3)/(3(3e^(-2x) - 2) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "At what integer value (rounded) do the functions f(x) = 1(-2|(4x) - 2|) + 5 and g(x) = -2(-1x - 3)^2 + 3(-1x - 3) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -1 | MATH |
[
{
"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5((2(5x - 4) + 1)/(3(5x - 4) - 1)) and g(x) = -1(-2cos(2πx - 3) + 2)^2 + 3(-2cos(2πx - 3) + 2) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 1 | MATH |
[
{
"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1((-2(2e^(1x) + 1) - 2)/(-2(2e^(1x) + 1) - 3))^2 + 4((-2(2e^(1x) + 1) - 2)/(-2(2e^(1x) + 1) - 3)) + 3 and g(x) = 2e^(-2(-3|x + 3| + 2)) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\bo... | -4 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = 3((-1((-1x + 3)/(3x)))/(2((-1x + 3)/(3x)) + 2)) + 5 and g(x) = 4((-2x - 1)/(-2x - 3)) + 3 intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -1 | MATH |
[
{
"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1|((1(-4x^2 - 5x + 2) + 1)/(-3(-4x^2 - 5x + 2) - 3)) + 2| + 1 and g(x) = (-1((-1x - 3)/(-2x - 1)) - 2)/(1((-1x - 3)/(-2x - 1)) + 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your ... | -1 | MATH |
[
{
"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (3(5((-1x + 3)/(-3x - 3)) + 1) - 1)/(-3(5((-1x + 3)/(-3x - 3)) + 1) - 3) and g(x) = (2(2|x + 1| - 2))/(1(2|x + 1| - 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your a... | 0 | MATH |
[
{
"content": "Find the integer closest to the smallest x-value where the functions f(x) = -5(-1cos(1π(1x + 2) + 2) + 2) - 4 and g(x) = -2|((-2x + 3)/(2x + 1)) - 1| - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -1 | MATH |
[
{
"content": "When ordered from left to right, what is the third x-coordinate (rounded to the nearest integer) where f(x) = (3(1((-2x - 2)/(3x))^2 + 4((-2x - 2)/(3x)) - 5) + 3)/(1(1((-2x - 2)/(3x))^2 + 4((-2x - 2)/(3x)) - 5)) and g(x) = 3|((-1x - 1)/(3x)) - 3| intersect in the interval [-10, 10]?\n\nPresent the... | 0 | MATH |
[
{
"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (3((1(3cos(2πx - 2) + 1) + 2)/(2(3cos(2πx - 2) + 1) - 3)) + 3)/(-2((1(3cos(2πx - 2) + 1) + 2)/(2(3cos(2πx - 2) + 1) - 3)) - 1) and g(x) = 2e^(-1(-1x^2 + 4x + 5)) - 3 intersect in the inter... | -1 | MATH |
[
{
"content": "Find the integer closest to the largest x-value where the functions f(x) = (1((2(1x^2 - 3x + 3) - 1)/(1(1x^2 - 3x + 3) + 3)) + 3)/(2((2(1x^2 - 3x + 3) - 1)/(1(1x^2 - 3x + 3) + 3))) and g(x) = -1(-2x - 4) + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 2 | MATH |
[
{
"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -4(-1(1x - 1)^2 - 1(1x - 1) - 2)^2 - 2(-1(1x - 1)^2 - 1(1x - 1) - 2) and g(x) = -1e^(2(-2x^2 - 2x + 2)) + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\b... | 0 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = -1(1e^(2(3x^2 + 1x + 2))) + 4 and g(x) = -1e^(-2((3x - 2)/(2x - 1))) - 2 intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1e^(-2(2|(-3|x|) - 1|)) - 2 and g(x) = -4(2|x - 3|)^2 + 5(2|x - 3|) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "If you add up all the x-coordinates where the functions f(x) = (-3(-3((2x - 2)/(-1x + 3)) - 4) - 2)/(3(-3((2x - 2)/(-1x + 3)) - 4) - 1) and g(x) = -1(-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 a... | -4 | MATH |
[
{
"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 4((-3(1|x - 1| - 1) - 3)/(-2(1|x - 1| - 1) + 3))^2 + 4((-3(1|x - 1| - 1) - 3)/(-2(1|x - 1| - 1) + 3)) - 5 and g(x) = 1|((3x + 3)/(-2x)) - 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \... | 2 | MATH |
[
{
"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3(1((-3x)/(-1x))^2 + 5((-3x)/(-1x)) - 4) and g(x) = (3(-1x^2 + 5x + 1) + 3)/(1(-1x^2 + 5x + 1) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 2 | MATH |
[
{
"content": "Find the integer closest to the largest x-value where the functions f(x) = 2|(-5(-4x^2 + 1x + 3)^2 + 5(-4x^2 + 1x + 3) + 3) + 3| - 2 and g(x) = -2(5x^2 + 2x - 5)^2 - 1(5x^2 + 2x - 5) + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 1 | MATH |
[
{
"content": "Find the integer closest to the largest x-value where the functions f(x) = -3(1|(-3|x + 3| + 2) - 3| - 2)^2 - 1(1|(-3|x + 3| + 2) - 3| - 2) + 3 and g(x) = (-3(-3e^(-1x) + 3))/(-3(-3e^(-1x) + 3) + 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | -2 | MATH |
[
{
"content": "Find the integer closest to the largest x-value where the functions f(x) = -2cos(2π((2(2x^2 + 1x + 1) + 3)/(3(2x^2 + 1x + 1) + 2)) - 3) - 2 and g(x) = 1|(2x - 4)| - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "At what integer value (rounded) do the functions f(x) = -4(-1|(-3x^2 - 2x + 3) + 2| - 2) + 4 and g(x) = (-2(2x + 4) - 2)/(1(2x + 4)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -2 | MATH |
[
{
"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (3(1|(-5x + 5) + 3| + 1) + 2)/(3(1|(-5x + 5) + 3| + 1) + 2) and g(x) = -1(3e^(-1x) + 1)^2 + 5(3e^(-1x) + 1) - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 0 | MATH |
[
{
"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2(-3|((2x - 3)/(2x + 2)) + 1| + 3) - 5 and g(x) = 1(-4x^2 - 1x) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "Find the third intersection point (counting from left to right) between the functions f(x) = -2(-5((-2x + 3)/(2x - 1))) + 2 and g(x) = 5(3|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"
}
] | 1 | MATH |
[
{
"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-3(-2|(2x^2 + 3x + 4) - 3| + 2) + 3)/(2(-2|(2x^2 + 3x + 4) - 3| + 2) + 2) and g(x) = 2(-2cos(2πx - 1) + 3)^2 + 1(-2cos(2πx - 1) + 3) + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your ... | -2 | MATH |
[
{
"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3|(4(-5x^2 - 2x + 1) - 5) + 2| - 1 and g(x) = 2|(-1sin(2πx - 3) - 3) + 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (-1((-3(-5x + 1))/(-3(-5x + 1) + 4)) + 1)/(2((-3(-5x + 1))/(-3(-5x + 1) + 4)) + 2) and g(x) = (-3(-4x + 1) - 1)/(-2(-4x + 1) + 3) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the ... | 0 | MATH |
[
{
"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1(-3(-2x - 3) - 4)^2 - 4(-3(-2x - 3) - 4) + 3 and g(x) = 4(3e^(2x) - 2)^2 + 1(3e^(2x) - 2) - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -1 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = (-1(-2e^(1(1|x - 2| + 2)) + 1) + 3)/(2(-2e^(1(1|x - 2| + 2)) + 1) + 3) and g(x) = -1(3x - 2) + 2 intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "At what integer value (rounded) do the functions f(x) = 2cos(1π((-3(3|x + 2| - 2) - 1)/(3(3|x + 2| - 2))) + 3) + 2 and g(x) = -4(4x - 5) - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-1(2(-3e^(1x)) + 1) - 2)/(2(2(-3e^(1x)) + 1) - 1) and g(x) = (-1(1x - 4) + 1)/(-1(1x - 4) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -2 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = (2(1(-1|x| + 1) - 3))/(-1(1(-1|x| + 1) - 3)) and g(x) = 5(1e^(1x) - 1) - 5 intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "At what integer value (rounded) do the functions f(x) = 2(2(-5x - 3) + 1) + 2 and g(x) = 2cos(2π((-1x + 3)/(2x - 1))) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1cos(2π((-1(-4x - 1) - 2)/(-3(-4x - 1) - 3)) + 3) + 1 and g(x) = 1(3|x + 2| + 3)^2 - 2(3|x + 2| + 3) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answ... | -2 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = -1(-2((-3x - 3)/(2x - 1)) + 4) - 5 and g(x) = 2(4x^2 + 5x + 5)^2 - 1(4x^2 + 5x + 5) + 3 intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-1(-1(-2cos(2πx + 1) + 2) + 4) + 1)/(3(-1(-2cos(2πx + 1) + 2) + 4) + 1) and g(x) = -1(2x^2 - 5x) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Y... | 3 | MATH |
[
{
"content": "At what integer value (rounded) do the functions f(x) = 1|(-2(4x + 2)^2 - 2(4x + 2) - 1) + 1| - 2 and g(x) = 2((2x - 2)/(2x + 2)) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -1 | MATH |
[
{
"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -3|(3e^(2(-3x))) - 2| - 2 and g(x) = 2((1x)/(3x - 2)) - 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "If you add up all the x-coordinates where the functions f(x) = -2((-1(2e^(-2x)) + 3)/(-3(2e^(-2x)) + 2))^2 + 4((-1(2e^(-2x)) + 3)/(-3(2e^(-2x)) + 2)) - 4 and g(x) = -4(2cos(1πx + 1) - 3)^2 + 2(2cos(1πx + 1) - 3) - 5 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\... | 1 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = 3((2(-1sin(2πx) + 3) - 3)/(2(-1sin(2πx) + 3) + 2))^2 - 3((2(-1sin(2πx) + 3) - 3)/(2(-1sin(2πx) + 3) + 2)) + 5 and g(x) = (-1(-3x + 3) + 2)/(-2(-3x + 3) + 3) intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \... | 1 | MATH |
[
{
"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (3(-1cos(3π((-2x + 2)/(-3x)) - 2) + 3) + 3)/(-3(-1cos(3π((-2x + 2)/(-3x)) - 2) + 3) - 2) and g(x) = 1(3|x - 2| + 1)^2 - 2(3|x - 2| + 1) - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\bo... | 2 | MATH |
[
{
"content": "At what integer value (rounded) do the functions f(x) = 5(4(-5x + 3) - 3)^2 + 5(4(-5x + 3) - 3) + 1 and g(x) = (1(-1|x - 2| + 2) + 2)/(2(-1|x - 2| + 2) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 2(-1(-2|x - 3| - 2) - 3) - 5 and g(x) = 1(3x^2 + 3x - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -2 | MATH |
[
{
"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-2(2(-2|x - 3| + 2)) - 1)/(-1(2(-2|x - 3| + 2))) and g(x) = 1e^(2((2x - 2)/(-3x - 1))) + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "At what integer value (rounded) do the functions f(x) = (2(-1|(-2x^2 - 2)| + 1) + 2)/(-1(-1|(-2x^2 - 2)| + 1) - 1) and g(x) = (-2((-1x + 2)/(3x - 1)) - 2)/(2((-1x + 2)/(3x - 1)) + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 2 | MATH |
[
{
"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1(-4(5x^2 + 5x - 5)^2 + 1(5x^2 + 5x - 5) + 1) + 3 and g(x) = -4(5x^2 - 5x - 3) + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | -2 | MATH |
[
{
"content": "At what integer value (rounded) do the functions f(x) = 5(4(-3x^2 + 4x - 2) + 1) + 3 and g(x) = (2(-3x^2 - 4x + 3) - 1)/(-3(-3x^2 - 4x + 3) - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 2|((2((-3x - 1)/(-2x)) + 3)/(2((-3x - 1)/(-2x)) - 1))| - 1 and g(x) = (-1(-4x - 5))/(-1(-4x - 5) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | -2 | MATH |
[
{
"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (2((2(-3e^(1x) + 1) + 1)/(1(-3e^(1x) + 1) + 3)))/(1((2(-3e^(1x) + 1) + 1)/(1(-3e^(1x) + 1) + 3)) - 3) and g(x) = (2(-3|x - 3| - 2) + 2)/(-1(-3|x - 3| - 2) - 2) in the interval [-10, 10], rounded to the nea... | 3 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = (2(5(-5x) - 4) + 3)/(-2(5(-5x) - 4) + 3) and g(x) = 1e^(1(3x^2 - 3x)) intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = 1|(2|(-3x^2 + 5x + 5) + 3| + 2) + 2| and g(x) = (-1((-3x + 2)/(-2x - 2)) + 3)/(2((-3x + 2)/(-2x - 2)) + 3) intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "At what integer value (rounded) do the functions f(x) = 2e^(1(-4(-1x + 2) - 1)) + 1 and g(x) = (1(4x + 1) - 3)/(-1(4x + 1) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-3(-3(3x^2 + 5x - 4) + 1) - 1)/(3(-3(3x^2 + 5x - 4) + 1) + 2) and g(x) = -1e^(-1(-2|x + 2| + 2)) + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | -2 | MATH |
[
{
"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-3((-2(2e^(1x)) + 2)/(3(2e^(1x)) + 3)))/(1((-2(2e^(1x)) + 2)/(3(2e^(1x)) + 3)) + 2) and g(x) = (1(2x^2 + 2x) - 3)/(3(2x^2 + 2x) + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{You... | -1 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = -1|(1|(3e^(-2x) + 3)| - 2)| - 1 and g(x) = (1((-1x + 1)/(1x - 2)) + 3)/(1((-1x + 1)/(1x - 2)) + 2) intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -2|(-1(-1x - 2)^2 + 5(-1x - 2) - 4) - 1| + 1 and g(x) = -1(3|x - 3| + 2) - 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | -9 | MATH |
[
{
"content": "Find the integer closest to the largest x-value where the functions f(x) = -5((-2((3x + 3)/(-2x)) - 2)/(-2((3x + 3)/(-2x)) + 1)) and g(x) = -1|(1x^2 + 4x + 1) - 1| + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -2|(3e^(2(-3x - 2)) - 3)| + 3 and g(x) = (-3((-1x - 3)/(1x)) - 1)/(-2((-1x - 3)/(1x)) - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | -1 | MATH |
[
{
"content": "At what integer value (rounded) do the functions f(x) = (-3((-2((1x - 2)/(2x - 3)) + 2)/(-1((1x - 2)/(2x - 3)) - 3)) - 2)/(3((-2((1x - 2)/(2x - 3)) + 2)/(-1((1x - 2)/(2x - 3)) - 3)) + 1) and g(x) = 3(-2|x - 1| - 3)^2 - 3(-2|x - 1| - 3) + 5 intersect in the interval [-10, 10]?\n\nPresent the answer... | 5 | MATH |
[
{
"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (3(-1e^(1(-1|x + 2| - 1)) + 2) + 2)/(-1(-1e^(1(-1|x + 2| - 1)) + 2) - 1) and g(x) = -2(-2x - 4) - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | -2 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = -3(-5(-4x - 5) - 2) - 2 and g(x) = (-3(2|x - 2| + 2) + 1)/(1(2|x - 2| + 2)) intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -1 | MATH |
[
{
"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1cos(2π((-3((2x + 3)/(1x + 3)))/(3((2x + 3)/(1x + 3)) + 2)) + 1) + 3 and g(x) = 1(4x + 3)^2 - 1(4x + 3) - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | -1 | MATH |
[
{
"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -1e^(-1(-1(3x^2 + 2x - 5) - 3)) + 3 and g(x) = -3((-3x - 2)/(1x + 2)) - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | -2 | MATH |
[
{
"content": "If you add up all the x-coordinates where the functions f(x) = 1|((2(-2cos(1πx + 2) - 3) + 1)/(-2(-2cos(1πx + 2) - 3) - 2)) + 1| + 3 and g(x) = 2e^(-2(2x^2 - 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{Yo... | 2 | MATH |
[
{
"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2(5(4x + 5)^2 - 2(4x + 5) - 3) - 4 and g(x) = -1cos(1π((-1x - 1)/(-3x - 3))) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -1 | MATH |
[
{
"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 4(5(2|x + 3|) - 3) + 1 and g(x) = 2sin(3π(-1x^2 - 2x + 1) + 2) + 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | -3 | MATH |
[
{
"content": "At what integer value (rounded) do the functions f(x) = (-3(2((-3x)/(1x + 1)) - 3) - 1)/(-3(2((-3x)/(1x + 1)) - 3)) and g(x) = (-2(-2cos(1πx - 3) - 2) + 2)/(-1(-2cos(1πx - 3) - 2) - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 0 | MATH |
[
{
"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (-3(4(3sin(3πx) + 3)^2 - 2(3sin(3πx) + 3) + 3) + 3)/(2(4(3sin(3πx) + 3)^2 - 2(3sin(3πx) + 3) + 3) + 1) and g(x) = -5(1|x + 3|) + 3 intersect in the interval [-10, 10]?\n\nPresent the answe... | -4 | MATH |
[
{
"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1(-3(-2x - 2) - 5)^2 - 4(-3(-2x - 2) - 5) + 5 and g(x) = (3(5x - 3) + 1)/(-1(5x - 3) - 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 3(2(-1x - 2)^2 - 1(-1x - 2) + 5) + 4 and g(x) = 3|(-5x^2 + 3x + 5) - 3| - 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 5 | MATH |
[
{
"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (1(-2|(-1e^(1x)) - 1| + 2) + 3)/(-2(-2|(-1e^(1x)) - 1| + 2) - 1) and g(x) = -3(2e^(1x) + 3) + 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -2 | MATH |
[
{
"content": "At what integer value (rounded) do the functions f(x) = -2cos(3π(3|((-2x + 2)/(-1x + 2)) - 1| + 3) - 1) + 2 and g(x) = (1(-2sin(3πx + 3) - 1) - 1)/(-1(-2sin(3πx + 3) - 1) + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 5 | MATH |
[
{
"content": "If you add up all the x-coordinates where the functions f(x) = -4(4(-3|x - 3| + 2)^2 + 4(-3|x - 3| + 2) + 4)^2 + 2(4(-3|x - 3| + 2)^2 + 4(-3|x - 3| + 2) + 4) - 2 and g(x) = -3(1e^(1x) - 3)^2 - 3(1e^(1x) - 3) - 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nP... | 8 | MATH |
[
{
"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3|(5(-1x^2 - 5x - 5)^2 - 4(-1x^2 - 5x - 5) - 4) - 1| + 2 and g(x) = 2|(-4x^2 + 3x - 4) - 2| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | -6 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = -5((-3(2x - 5) + 3)/(-2(2x - 5) + 3))^2 - 1((-3(2x - 5) + 3)/(-2(2x - 5) + 3)) and g(x) = -5(-4x - 4) - 5 intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -1 | MATH |
[
{
"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1(1(1cos(2πx + 2) - 3)^2 - 3(1cos(2πx + 2) - 3) - 4) + 1 and g(x) = 2((3x)/(2x + 3)) + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -2 | MATH |
[
{
"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(-1(2|x + 3| - 3) + 5) + 3| + 2 and g(x) = 2|(1e^(2x) + 2) - 1| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -8 | MATH |
[
{
"content": "Find the integer closest to the largest x-value where the functions f(x) = 2|(4(-3x + 3)^2 - 5(-3x + 3) - 5) + 2| + 1 and g(x) = 2(3|x + 2| - 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1 | MATH |
[
{
"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (3(1(3x + 5) + 1) - 2)/(-3(1(3x + 5) + 1)) and g(x) = (1(-4x + 2) - 3)/(-2(-4x + 2) - 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 0 | MATH |
[
{
"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5(1e^(-1(-4x + 4)) + 2) + 2 and g(x) = (3(-5x - 2))/(-1(-5x - 2) + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -1 | MATH |
[
{
"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 1(5((2x)/(-3x - 3))^2 - 2((2x)/(-3x - 3)) - 5) + 5 and g(x) = (-2(-4x - 5))/(-2(-4x - 5) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | -1 | MATH |
[
{
"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(-1|(-1|x| - 1)| + 3) - 2)/(1(-1|(-1|x| - 1)| + 3) + 3) and g(x) = -1(-2x^2 - 2x + 4) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3((-1(2e^(-2x) + 3) + 1)/(2(2e^(-2x) + 3) + 2)) - 1 and g(x) = -2|((1x - 3)/(-3x)) + 3| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -2 | MATH |
[
{
"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-2(-1(3|x - 2| - 2) + 2) + 2)/(-3(-1(3|x - 2| - 2) + 2) - 2) and g(x) = -1e^(-1((-2x)/(-1x + 2))) - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{... | 3 | MATH |
[
{
"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -4(-2|(-1x + 5) + 1| + 1) - 4 and g(x) = 2e^(-1((2x - 3)/(-2x + 3))) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1(-2sin(3π((3x + 1)/(3x + 3)) + 2) + 1) - 3 and g(x) = -2(1|x - 1| - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -4(4(-5x - 5) + 3) + 3 and g(x) = -3cos(1π(1x)) + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -1 | MATH |
[
{
"content": "Find the integer closest to the largest x-value where the functions f(x) = 2((2((2x - 2)/(3x + 3)) - 3)/(2((2x - 2)/(3x + 3)) - 1))^2 - 3((2((2x - 2)/(3x + 3)) - 3)/(2((2x - 2)/(3x + 3)) - 1)) - 4 and g(x) = (-2(3e^(-1x) - 2) + 3)/(-3(3e^(-1x) - 2) + 4) meet in the interval [-10, 10].\n\nPresent t... | 1 | MATH |
[
{
"content": "Find the integer closest to the largest x-value where the functions f(x) = (-1(-3|((1x - 3)/(2x + 2)) + 3| + 1) + 2)/(3(-3|((1x - 3)/(2x + 2)) + 3| + 1) + 1) and g(x) = (-2((3x - 1)/(-2x + 1)))/(-2((3x - 1)/(-2x + 1)) - 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\bo... | 0 | MATH |
[
{
"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3e^(2(1|(-1|x + 3| + 3) + 1| + 1)) - 1 and g(x) = (3((1x)/(3x - 2)) - 1)/(3((1x)/(3x - 2))) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = 1((-1(-2|x + 1| - 3) + 2)/(2(-2|x + 1| - 3) - 1)) and g(x) = 5(-2e^(-1x)) intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 3 | MATH |
[
{
"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 5(-1|(1e^(-2x) + 1)| + 3)^2 - 3(-1|(1e^(-2x) + 1)| + 3) + 2 and g(x) = 2|(-1e^(-2x) - 3) + 2| in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your ... | -1 | MATH |
[
{
"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1|(-2(-2|x - 1| - 1)^2 - 1(-2|x - 1| - 1) + 2) + 1| - 1 and g(x) = -1|(-2cos(2πx + 2) + 3) + 1| + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 1 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = (-1(3|(1|x - 2| - 2) + 3| + 2) + 3)/(-2(3|(1|x - 2| - 2) + 3| + 2) + 1) and g(x) = (3(-2x - 3) + 2)/(-2(-2x - 3)) intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | -1 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = (2(-1sin(3π(1x + 5) - 1) + 1))/(-1(-1sin(3π(1x + 5) - 1) + 1)) and g(x) = -3(1x - 3) - 5 intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 2 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = 3|((-3(-3x^2 - 1x - 5) + 1)/(3(-3x^2 - 1x - 5) + 3)) + 2| - 1 and g(x) = 2((1x - 3)/(2x)) - 1 intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -2 | MATH |
[
{
"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (1(-1|(3|x + 3| + 1) + 2|))/(-1(-1|(3|x + 3| + 1) + 2|) - 2) and g(x) = -4(-2x - 1) + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | -1 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = (-2(-1(-1e^(2x) + 3) + 1) + 3)/(-3(-1(-1e^(2x) + 3) + 1) - 2) and g(x) = -1cos(3π((2x)/(1x)) + 1) intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
[
{
"content": "Find the integer value closest to the x-coordinate where f(x) = 3(-4(5x) - 4) + 5 and g(x) = -2sin(1π(-3x - 5)) - 2 intersect in the range -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 0 | MATH |
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