messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation 2(-4(-5x^2 - 4x + 2)^2 + 2)^2 - 3(-4(-5x^2 - 4x + 2)^2 + 2) - 1 = (3(1|x + 3|) + 2)/(3(1|x + 3|) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(3(2cos(1πx + 1) - 2)^2 - 3(2cos(1πx + 1) - 2) + 2)^2 - 4(3(2cos(1πx + 1) - 2)^2 - 3(2cos(1πx + 1) - 2) + 2) and g(x) = 3|(3e^(1x) + 1) + 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 15 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3e^(-1((2(-1|x| + 2) - 3)/(-3(-1|x| + 2) - 2))) + 1 and g(x) = -1sin(2π(2x + 3) + 1) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "How many solutions does the equation -1(-1(-1sin(3πx + 3) + 2) + 1)^2 + 4(-1(-1sin(3πx + 3) + 2) + 1) - 5 = -1(-1x - 5)^2 - 5(-1x - 5) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(2π(5(1x^2 + 4x + 1)^2 + 1) + 1) - 2 and g(x) = 3cos(2π(-2|x - 1|) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 624 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1sin(1π(5(-4x^2 + 3x - 4) + 2) + 2) - 1 and g(x) = -4(-5x^2 - 5x + 4) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(1|(3x^2 + 1x - 2) - 2| - 3) + 3| - 3 and g(x) = -2|(3sin(2πx - 3)) - 2| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(5(1x - 5)^2 + 5(1x - 5) - 4) - 1| + 3 and g(x) = (2(2x - 1) - 2)/(-2(2x - 1) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-2e^(1((-3x - 3)/(2x - 1))) + 1) - 2)/(2(-2e^(1((-3x - 3)/(2x - 1))) + 1) + 2) and g(x) = 3sin(1π(5x^2 - 5x - 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 868 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1sin(3π((-3((1x + 3)/(1x - 2)) + 3)/(-2((1x + 3)/(1x - 2)))) + 2) and g(x) = (1(5x + 4) - 2)/(-1(5x + 4) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 98 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-3sin(3π(-5x^2 - 5x - 3) + 2) + 1)| + 2 and g(x) = 1|((-2x - 2)/(2x - 1))| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 930 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-2((-1x + 2)/(-1x - 1))) - 3)/(2(-2((-1x + 2)/(-1x - 1))) + 1) and g(x) = 2cos(1π(-3|x - 3|) + 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 51 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-2sin(2π(1x - 2) - 1) - 2) - 3)/(-3(-2sin(2π(1x - 2) - 1) - 2)) and g(x) = 4(2x + 4) - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(5((-2x + 3)/(3x - 1))^2 + 2((-2x + 3)/(3x - 1)) + 5) + 5 and g(x) = (3(1cos(1πx - 1)) - 1)/(2(1cos(1πx - 1)) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many solutions does the equation -1cos(2π(4(5x + 2)^2 - 2(5x + 2) - 2) - 1) + 1 = (-1(-5x^2 - 4x))/(-2(-5x^2 - 4x)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1442 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(1π(-1(4x^2 - 3x + 2)) + 1) + 2 and g(x) = (-1(-1e^(-1x) - 1) + 1)/(-1(-1e^(-1x) - 1) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 409 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2((2(1|x + 2| - 3) + 1)/(1(1|x + 2| - 3))) + 2)/(-1((2(1|x + 2| - 3) + 1)/(1(1|x + 2| - 3)))) and g(x) = (2(4x^2 + 2x) - 3)/(2(4x^2 + 2x)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(4(3sin(3πx - 1) - 3)^2 + 1(3sin(3πx - 1) - 3) - 4)| + 2 and g(x) = 4(-3x^2 + 3x + 2)^2 - 3(-3x^2 + 3x + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many solutions does the equation -3cos(3π((-3((-3x - 3)/(-1x + 2)) + 3)/(-3((-3x - 3)/(-1x + 2)) + 4)) - 2) + 1 = -1(-3sin(1πx - 2) + 3)^2 + 4(-3sin(1πx - 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 55 | MATH |
[
{
"content": "How many solutions does the equation 1(-2|(-1e^(2x) - 2) + 2| + 3)^2 + 1(-2|(-1e^(2x) - 2) + 2| + 3) - 2 = -3cos(3π((3x - 1)/(-3x)) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((2(3sin(2πx - 2) - 1))/(1(3sin(2πx - 2) - 1) + 1))^2 - 5((2(3sin(2πx - 2) - 1))/(1(3sin(2πx - 2) - 1) + 1)) + 1 and g(x) = -5(-1cos(2πx)) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 40 | MATH |
[
{
"content": "How many solutions does the equation 2|(3|(1x^2 - 4x - 5) + 2| - 2) + 1| + 2 = 5(2sin(3πx + 3) - 1)^2 + 4(2sin(3πx + 3) - 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(2cos(2π(2x + 2) + 1) + 1) + 3)/(1(2cos(2π(2x + 2) + 1) + 1) + 1) and g(x) = (-1(-3cos(1πx + 1) - 1) + 2)/(2(-3cos(1πx + 1) - 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-4(-1x^2 + 2x + 1)^2 + 2(-1x^2 + 2x + 1) - 1)^2 - 3(-4(-1x^2 + 2x + 1)^2 + 2(-1x^2 + 2x + 1) - 1) + 5 and g(x) = (1(1|x| - 1) - 3)/(1(1|x| - 1) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(4(-3e^(1x) + 2)^2 - 1(-3e^(1x) + 2)) + 1)/(-3(4(-3e^(1x) + 2)^2 - 1(-3e^(1x) + 2)) + 4) and g(x) = -5(-2sin(1πx + 3)) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 20 | MATH |
[
{
"content": "How many solutions does the equation -2|(-4(-2sin(2πx) - 3) - 3) - 2| + 2 = 1|(2x^2 - 5x + 2) + 2| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation (-2((2(-1x - 4))/(-2(-1x - 4) - 2)) - 2)/(1((2(-1x - 4))/(-2(-1x - 4) - 2))) = -5(-1cos(1πx - 2) + 1)^2 - 5(-1cos(1πx - 2) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(3(-2|x - 3| + 3) - 5) + 3| - 1 and g(x) = 3(1|x + 2| - 3) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2((-1((2x + 1)/(2x + 3)) + 1)/(2((2x + 1)/(2x + 3)))))/(2((-1((2x + 1)/(2x + 3)) + 1)/(2((2x + 1)/(2x + 3))))) and g(x) = -2sin(3π(2x + 2) + 1) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 60 | MATH |
[
{
"content": "How many solutions does the equation (3(1((1x)/(-1x + 3)) + 2) + 2)/(3(1((1x)/(-1x + 3)) + 2) + 1) = -2|(3sin(2πx + 2)) - 3| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 37 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(-2|(-3sin(3πx + 1) - 1)| + 3) and g(x) = (-1(-2|x - 3| + 1))/(-1(-2|x - 3| + 1) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 116 | MATH |
[
{
"content": "How many solutions does the equation -1e^(1(3|(-2|x + 2| - 1) + 3| - 2)) + 2 = -1|(-2sin(1πx + 2) - 3)| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(4(-2cos(1πx - 2) - 2) + 4))/(1(4(-2cos(1πx - 2) - 2) + 4) + 3) and g(x) = 2(-1|x + 3| - 1) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(1π(-1|(2|x + 2| - 3) - 1| + 1) - 2) + 3 and g(x) = -3(-2cos(3πx + 1) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5((1(-3e^(-2x) + 3) + 2)/(-2(-3e^(-2x) + 3) - 3))^2 + 2((1(-3e^(-2x) + 3) + 2)/(-2(-3e^(-2x) + 3) - 3)) - 1 and g(x) = -1(-1x^2 + 4x + 3)^2 - 5(-1x^2 + 4x + 3) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: ... | 6 | MATH |
[
{
"content": "How many solutions does the equation 1|(-1e^(-2(1x^2 + 4x)) - 3) + 2| = -2sin(1π(-1x^2 + 5x) + 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 180 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2((2(-3sin(2πx) + 2) + 1)/(-2(-3sin(2πx) + 2) - 3))^2 + 5((2(-3sin(2πx) + 2) + 1)/(-2(-3sin(2πx) + 2) - 3)) + 1 and g(x) = 2|(-2cos(3πx - 2) + 2) - 1| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(2(-3|x - 2| + 1)^2 + 2(-3|x - 2| + 1)) - 3| - 3 and g(x) = (-2(2sin(2πx - 3)) - 1)/(-3(2sin(2πx - 3)) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(3sin(2π(-1|x - 3| - 3) + 1) + 3)^2 - 4(3sin(2π(-1|x - 3| - 3) + 1) + 3) + 4 and g(x) = -4(5x^2 + 2x + 4) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-3cos(2π(-3x^2 - 5x + 2) + 1) + 2) - 4 and g(x) = 3e^(1(3x^2 - 5x - 1)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "How many solutions does the equation (-1(1(3cos(3πx) - 2)^2 - 3(3cos(3πx) - 2) - 5) - 3)/(-3(1(3cos(3πx) - 2)^2 - 3(3cos(3πx) - 2) - 5) - 1) = -2e^(2(-2|x + 1|)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(3π((-1(-1|x + 2| - 2) + 2)/(-2(-1|x + 2| - 2) - 1)) - 2) - 2 and g(x) = 3sin(3π(-1x^2 - 4x - 5) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 28 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3((3(2|x + 2| - 3) + 1)/(1(2|x + 2| - 3) - 2)) + 1)/(-1((3(2|x + 2| - 3) + 1)/(1(2|x + 2| - 3) - 2)) + 3) and g(x) = 3cos(1π(-5x^2 + 5x - 1) + 3) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your an... | 102 | MATH |
[
{
"content": "How many solutions does the equation 3|(1|(3x + 3) - 2| - 3) + 1| - 2 = -1sin(1π(-1x^2 + 3x - 2) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many solutions does the equation 2|(-3(-1|x + 2| + 1) - 1) + 3| = (2(-2x + 1) + 3)/(3(-2x + 1) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "How many solutions does the equation (-1((-1(2x - 5) + 1)/(-3(2x - 5) - 2)) + 3)/(2((-1(2x - 5) + 1)/(-3(2x - 5) - 2)) - 1) = -2|(-2sin(2πx + 3) - 2) - 3| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-3e^(-2((1x)/(-3x + 1))) + 3) - 2)/(-1(-3e^(-2((1x)/(-3x + 1))) + 3) - 2) and g(x) = -1sin(1π(1x^2 - 5x - 5) + 2) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 106 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(5(-2cos(1πx) - 2) - 5) + 4 and g(x) = 2|(-3x^2 - 1x) - 3| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(2π(5((-1x - 1)/(1x))^2 - 2((-1x - 1)/(1x)) - 1) - 3) - 1 and g(x) = (2(-3x + 2) + 1)/(1(-3x + 2) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 118 | MATH |
[
{
"content": "How many solutions does the equation -3cos(1π(2|(-5x^2 + 4x - 2) - 2| + 3) - 3) - 1 = 1e^(-2(-4x - 2)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 771 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(1sin(2π(-1x + 5) - 2)))/(2(1sin(2π(-1x + 5) - 2)) + 1) and g(x) = 2sin(1π(-2|x - 3| + 2) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2((3((2x + 2)/(3x + 1)))/(3((2x + 2)/(3x + 1)) - 2)) - 1 and g(x) = -1cos(3π(-4x - 4) + 1) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "How many solutions does the equation -3|(3|(2|x| - 1)|) - 1| + 2 = (1((2x - 2)/(2x + 3)))/(-3((2x - 2)/(2x + 3)) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(1(-5x - 4)^2 - 3(-5x - 4) - 5) - 1| + 2 and g(x) = -2((-2x + 2)/(2x))^2 - 5((-2x + 2)/(2x)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(2|(-3sin(3πx + 2) - 1) - 3| - 2) + 1)/(2(2|(-3sin(3πx + 2) - 1) - 3| - 2) - 1) and g(x) = -1((3x)/(1x)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "How many solutions does the equation -3(-2cos(1π(-3|x|) + 1) + 1)^2 + 1(-2cos(1π(-3|x|) + 1) + 1) - 4 = -2|((-2x + 3)/(2x - 1))| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3((2(-3|x + 3| - 2) + 2)/(1(-3|x + 3| - 2) - 3)) + 1)/(2((2(-3|x + 3| - 2) + 2)/(1(-3|x + 3| - 2) - 3)) + 3) and g(x) = (-1(-3sin(1πx + 1) + 1) - 1)/(2(-3sin(1πx + 1) + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \... | 10 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1sin(2π(3|(1x^2 + 3x - 1) - 3| + 3) - 1) and g(x) = -3cos(1π((2x + 1)/(1x + 3)) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 224 | MATH |
[
{
"content": "How many solutions does the equation -3((3(-3|x - 3| - 3) + 3)/(2(-3|x - 3| - 3) - 3)) - 2 = 3cos(1π(5x^2 + 4x - 5) + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 292 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-2(-3x - 5)^2 - 5(-3x - 5) - 2) + 1 and g(x) = (2(-2cos(3πx) + 2) + 3)/(-3(-2cos(3πx) + 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 53 | MATH |
[
{
"content": "How many solutions does the equation -3sin(3π(-3|((-1x - 3)/(1x)) + 1| - 3) - 2) - 2 = -1|(1|x + 2| + 2) - 2| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 142 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(5(-1x + 2)^2 - 1)^2 - 5(5(-1x + 2)^2 - 1) - 3 and g(x) = 3cos(2π(2|x - 2| + 3) + 3) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(1sin(2π(-2x^2 - 5x) - 2) - 3) + 1| - 1 and g(x) = -2|(-2cos(3πx) - 2) - 3| - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 243 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(2sin(1π((-2x + 2)/(1x - 3)) + 2) + 1) + 2 and g(x) = -3sin(1π(-1x^2 - 4x - 2) + 1) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 132 | MATH |
[
{
"content": "How many solutions does the equation -3cos(2π((2((3x + 2)/(2x - 1)) + 2)/(2((3x + 2)/(2x - 1)) - 1)) + 2) = -1cos(3π(-4x - 4) + 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 127 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-4(-1sin(2πx + 2) - 2)^2 - 3(-1sin(2πx + 2) - 2) - 4)^2 + 1(-4(-1sin(2πx + 2) - 2)^2 - 3(-1sin(2πx + 2) - 2) - 4) - 4 and g(x) = -2|(3|x| + 1) - 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 16 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(1(3sin(2πx - 2) - 2) + 2) - 2| and g(x) = 4((3x - 3)/(1x + 3)) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "How many solutions does the equation (-1(-2(-3|x + 1| + 3) + 3) - 1)/(2(-2(-3|x + 1| + 3) + 3) - 1) = (-2(5x^2 - 2x - 5) - 3)/(-3(5x^2 - 2x - 5) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1cos(3π((-3((2x + 3)/(-2x + 1)) + 2)/(1((2x + 3)/(-2x + 1)) + 3)) - 2) - 1 and g(x) = 2sin(3π(-5x^2 + 1x - 5) - 2) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 732 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1e^(-1(-3|(-3|x| + 1) + 3| - 2)) - 3 and g(x) = -2|(3cos(1πx) - 3)| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation 2(3(-2cos(1πx - 1) + 2)^2 + 5(-2cos(1πx - 1) + 2) - 1) + 2 = -3(-2x - 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-4(-1sin(1πx + 3) - 1) - 5) + 2)/(-3(-4(-1sin(1πx + 3) - 1) - 5) + 2) and g(x) = 3|(2x + 4) + 1| - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-3sin(2π(2|x + 3| - 2) - 2) - 1))/(2(-3sin(2π(2|x + 3| - 2) - 2) - 1) + 1) and g(x) = -4(-1|x + 3| + 1)^2 + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(-2(2|x + 2| - 3) - 4) + 3| + 1 and g(x) = (-1(-4x^2 + 3x - 4) - 2)/(1(-4x^2 + 3x - 4) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-1|(-2x - 3) - 2| - 3) - 2)/(-1(-1|(-2x - 3) - 2| - 3)) and g(x) = -4(1sin(1πx - 2) - 1)^2 - 3(1sin(1πx - 2) - 1) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-1(2sin(3πx - 1) + 1)^2 + 5(2sin(3πx - 1) + 1) + 2) and g(x) = 2|(3|x + 1| - 1) + 1| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(3cos(1π(-3x + 1) + 2) - 2))/(-3(3cos(1π(-3x + 1) + 2) - 2) - 2) and g(x) = -2|(-2x^2 - 4x - 2) + 1| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(-2sin(3π(2x + 5) - 1) + 1) - 2| - 2 and g(x) = -5(-1x^2 - 4x - 1) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-2((3x + 1)/(-2x + 3))^2 - 1((3x + 1)/(-2x + 3)) - 3) + 1| - 1 and g(x) = 1(1cos(1πx) + 3)^2 - 4(1cos(1πx) + 3) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 10 | MATH |
[
{
"content": "How many solutions does the equation (2(1|(-3sin(3πx) - 3) + 1| - 3) + 2)/(2(1|(-3sin(3πx) - 3) + 1| - 3) + 1) = -1(-1x^2 + 2x + 2)^2 - 5(-1x^2 + 2x + 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(2(-2x^2 - 1x + 2) + 5) - 2)/(-2(2(-2x^2 - 1x + 2) + 5) + 3) and g(x) = 2cos(3π(-4x^2 + 4x - 5) - 3) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "How many solutions does the equation -4(2cos(3π(-1|x + 2| + 1) - 3) + 2)^2 - 4(2cos(3π(-1|x + 2| + 1) - 3) + 2) - 1 = -1|(3cos(1πx) + 2) + 1| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(2π(-4(2|x - 3|) - 5) - 1) + 1 and g(x) = 4((3x - 3)/(-3x - 1))^2 + 1((3x - 3)/(-3x - 1)) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 43 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-5(3cos(2πx + 3) + 2)^2 - 4(3cos(2πx + 3) + 2) + 3) + 5 and g(x) = 4(1x + 4)^2 + 3(1x + 4) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(2(3sin(2πx + 3) - 2) + 2)^2 + 2(2(3sin(2πx + 3) - 2) + 2) - 5 and g(x) = 3(-1|x| + 2) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|((2(4x - 5) + 2)/(-2(4x - 5))) - 2| - 3 and g(x) = 3(1sin(3πx - 3) + 3) - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(2π(1(1x^2 - 3)^2 + 3(1x^2 - 3) - 2) - 3) - 1 and g(x) = 1((2x - 3)/(-3x + 2)) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1045 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1((-1(-2x + 5) - 2)/(-3(-2x + 5) - 2)) + 3)/(1((-1(-2x + 5) - 2)/(-3(-2x + 5) - 2)) + 3) and g(x) = -2sin(1π(1|x - 1| + 3)) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 9 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(2π(1(-1|x - 2| - 1)^2 + 1(-1|x - 2| - 1) + 2) - 3) - 1 and g(x) = -3|(-1|x - 2| + 3) + 1| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 69 | MATH |
[
{
"content": "How many solutions does the equation 1e^(2(1|((3x + 2)/(-3x + 1)) + 1| - 2)) - 3 = 2cos(1π(1x^2 + 3x - 4) + 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 202 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(3|(3sin(3πx - 2) - 1) + 2| - 1))/(-2(3|(3sin(3πx - 2) - 1) + 2| - 1) + 1) and g(x) = 1sin(1π(3|x + 3| + 3) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 137 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(2|(-3sin(1πx + 2) + 3) + 2| - 3)^2 - 4(2|(-3sin(1πx + 2) + 3) + 2| - 3) + 1 and g(x) = -4(-1x - 4) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(3sin(2π(-3|x - 2| + 1) - 2) + 3) + 3)/(-1(3sin(2π(-3|x - 2| + 1) - 2) + 3) + 3) and g(x) = -1(1|x - 2| - 3) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 74 | MATH |
[
{
"content": "How many solutions does the equation -3|(1cos(2π(-2|x - 2| - 3) - 1) - 3) - 1| = (2(2sin(3πx - 2) - 1) - 3)/(-2(2sin(3πx - 2) - 1) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|((3(2sin(2πx - 1) + 1) - 3)/(-2(2sin(2πx - 1) + 1) + 1)) - 3| and g(x) = 1|(-1x^2 + 3x) + 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 59 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(2sin(1π((1x + 2)/(-2x + 3)) + 3) + 3) + 3| and g(x) = (-3(2sin(3πx - 3) - 1) + 3)/(1(2sin(3πx - 3) - 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(1sin(2π((-1x + 1)/(-2x + 3)) - 2) + 2) + 3)/(3(1sin(2π((-1x + 1)/(-2x + 3)) - 2) + 2) - 2) and g(x) = (-3(3x) - 1)/(-2(3x) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 29 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3sin(1π(1|(-4x^2 - 3x) - 2| - 3) + 3) - 3 and g(x) = (3(-2e^(-1x) - 1) + 2)/(2(-2e^(-1x) - 1) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 387 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2e^(-1((-3(-2x^2 + 2x - 1) - 3)/(1(-2x^2 + 2x - 1) + 2))) and g(x) = 4(-1cos(3πx) + 2)^2 - 5(-1cos(3πx) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((2(2|x + 1| - 1) - 3)/(-2(2|x + 1| - 1) - 1))^2 - 3((2(2|x + 1| - 1) - 3)/(-2(2|x + 1| - 1) - 1)) + 2 and g(x) = (1(3sin(3πx) + 1) + 2)/(2(3sin(3πx) + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 58 | MATH |
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