messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(2(-3sin(1πx - 3))^2 + 4(-3sin(1πx - 3)) + 4) + 5 and g(x) = 3(3cos(3πx - 3) - 3) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-5((1x - 1)/(-2x + 1)) + 5) + 2)/(1(-5((1x - 1)/(-2x + 1)) + 5) + 2) and g(x) = 3sin(1π(-1x) - 3) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "How many solutions does the equation (1(3e^(2(-1x^2 - 5x + 5)) + 1) - 2)/(-3(3e^(2(-1x^2 - 5x + 5)) + 1) - 2) = 1|(2sin(1πx + 3) - 2) + 3| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 28 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(1|(-2sin(3πx - 1) - 1) - 1| - 3) and g(x) = 1(-3x - 4) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(1π(-2|((2x - 2)/(1x - 3))| - 1) + 1) + 2 and g(x) = 2sin(3π(-1|x + 2| + 3) + 3) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 44 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(2π((2((-2x + 3)/(-1x + 3)) - 1)/(2((-2x + 3)/(-1x + 3)) + 1)) - 1) + 2 and g(x) = 2sin(3π(3x^2 + 2x + 4) - 3) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 872 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1e^(-1((1(-4x + 5) + 1)/(2(-4x + 5)))) - 2 and g(x) = -2|(-3x^2 - 1x + 2) + 1| + 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 -2(1((2x + 1)/(3x + 1))^2 - 4((2x + 1)/(3x + 1)) + 1) - 1 = 1sin(3π(1x^2 - 5) - 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 596 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-3sin(2π(-1|x + 2|) - 3) + 2) + 5 and g(x) = -1sin(1π((-2x - 1)/(3x - 1)) + 2) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 38 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(1((-1x + 2)/(-2x + 1))^2 - 3((-1x + 2)/(-2x + 1)) - 5)^2 + 5(1((-1x + 2)/(-2x + 1))^2 - 3((-1x + 2)/(-2x + 1)) - 5) - 3 and g(x) = (-1(-1|x + 2| + 2) + 1)/(-1(-1|x + 2| + 2) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex for... | 4 | MATH |
[
{
"content": "How many solutions does the equation (-2(1|(-2x) + 2| - 2) - 2)/(2(1|(-2x) + 2| - 2) - 1) = 1sin(2π(2|x + 3| + 3) - 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 66 | MATH |
[
{
"content": "How many solutions does the equation (1(-1sin(2π(3x - 5) + 2) + 3) - 3)/(3(-1sin(2π(3x - 5) + 2) + 3) - 2) = -4(-2cos(3πx - 1) + 3)^2 + 5(-2cos(3πx - 1) + 3) + 3 have for x 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 -1(2sin(1π((-2x + 3)/(-1x)) + 1) + 3) + 5 = 1(4x^2 + 2x + 5)^2 - 4(4x^2 + 2x + 5) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(3π(-4(-3x^2 + 5x - 5) + 4)) - 3 and g(x) = -2e^(2(-2|x + 1| - 3)) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1460 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(3(1x^2 - 5x + 4) + 2) + 3| - 2 and g(x) = (3((-1x - 3)/(3x + 1)) - 2)/(-2((-1x - 3)/(3x + 1))) where -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) = 2(2cos(2π((3x + 1)/(-2x + 3)) - 1) + 1)^2 - 1(2cos(2π((3x + 1)/(-2x + 3)) - 1) + 1) - 2 and g(x) = 5(-1|x + 3| + 1) + 5 intersect 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 (-1(-3e^(1(1|x - 1| - 2)) + 1) - 2)/(-1(-3e^(1(1|x - 1| - 2)) + 1) + 2) = -4(1x^2 - 2x - 4)^2 - 2(1x^2 - 2x - 4) + 3 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 -5(-2sin(3π(-2|x - 3|) + 2) - 2)^2 - 2(-2sin(3π(-2|x - 3|) + 2) - 2) - 2 = (-1(-1x^2 - 3x - 1) + 2)/(-3(-1x^2 - 3x - 1) - 3) 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 1|(3|(-5x^2 + 2x + 1) + 1| + 2)| + 2 = 4(-1cos(2πx) - 3)^2 + 4(-1cos(2πx) - 3) - 3 have for x 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) = -3(-2cos(3π(3x + 2)) + 2) and g(x) = 1sin(3π((-1x + 1)/(-3x + 4)) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-2((1x + 2)/(2x - 1))^2 + 2((1x + 2)/(2x - 1)) - 2) - 2)/(-2(-2((1x + 2)/(2x - 1))^2 + 2((1x + 2)/(2x - 1)) - 2) + 3) and g(x) = -3sin(1π(-4x - 4) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Yo... | 80 | MATH |
[
{
"content": "How many solutions does the equation -1|(2cos(2π((-2x - 3)/(-1x)) + 2) + 3) - 1| + 2 = 3|(-2x^2 - 5x + 3) + 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": "Find the number of times the graphs of f(x) = -3(2(-1e^(-1x))^2 + 2(-1e^(-1x)))^2 - 3(2(-1e^(-1x))^2 + 2(-1e^(-1x))) and g(x) = -4(-3sin(2πx - 1) + 1)^2 - 3(-3sin(2πx - 1) + 1) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 41 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(2π(3(-4x^2 - 1)^2 + 5(-4x^2 - 1) - 4) + 1) - 2 and g(x) = (-1(1|x|) - 3)/(-3(1|x|) + 4) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 104 | MATH |
[
{
"content": "How many solutions does the equation 5((3(1x^2 - 1x - 1) - 2)/(3(1x^2 - 1x - 1) + 3)) - 5 = 2cos(1π(4x^2 + 3x + 2) + 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 759 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(4(-2cos(2πx + 1) + 1) + 4) + 4 and g(x) = 2e^(2((1x + 2)/(-2x - 3))) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(-2|(-3cos(3πx + 2) + 2) - 2| - 3) - 5 and g(x) = -1(-3x + 3) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 38 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(3sin(3π(-4x^2 + 5x + 3) + 2) - 2) - 2)/(-2(3sin(3π(-4x^2 + 5x + 3) + 2) - 2) + 1) and g(x) = -5((2x + 3)/(2x + 2))^2 + 1((2x + 3)/(2x + 2)) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 482 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(4(1x^2 + 5x)^2 - 3(1x^2 + 5x) - 5) + 1)/(1(4(1x^2 + 5x)^2 - 3(1x^2 + 5x) - 5) - 3) and g(x) = (-2(-4x + 4))/(1(-4x + 4)) 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(3|(-1sin(1πx + 1) - 1) - 1| - 2)^2 + 3(3|(-1sin(1πx + 1) - 1) - 1| - 2) + 5 and g(x) = 1|(2x^2 + 4x - 5) - 3| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many solutions does the equation -3|((3(1x^2 + 3x + 1) - 1)/(-2(1x^2 + 3x + 1) - 1))| - 2 = 2(-1sin(3πx + 1)) - 3 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 points of intersection exist between the functions f(x) = 3(-1cos(3π(4x^2 + 3) + 2) - 1) and g(x) = -3(-2sin(3πx + 3) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 476 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1((-2(-2x^2 + 5) - 1)/(3(-2x^2 + 5) - 2))^2 + 2((-2(-2x^2 + 5) - 1)/(3(-2x^2 + 5) - 2)) + 3 and g(x) = 1(2x - 4) + 4 where -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(-3e^(-2(-1|x + 1| + 1)) + 2) + 1)/(2(-3e^(-2(-1|x + 1| + 1)) + 2) + 3) and g(x) = 3(-2sin(2πx - 1) - 2)^2 + 1(-2sin(2πx - 1) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 38 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1sin(2π(-2|(4x^2 + 2x - 1) - 2| + 3) - 3) - 2 and g(x) = -1((1x - 3)/(2x + 1))^2 + 3((1x - 3)/(2x + 1)) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 755 | MATH |
[
{
"content": "How many solutions does the equation 3(2sin(1π(3|x - 1| - 3) + 2) - 1) = 5((-1x + 2)/(-1x))^2 - 2((-1x + 2)/(-1x)) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 42 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(-1(1e^(2x))) + 1| - 3 and g(x) = (-1(-2cos(2πx - 3)) - 2)/(-3(-2cos(2πx - 3)) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-5(3sin(3πx + 3) + 3) - 2) + 2| + 1 and g(x) = 1|(-2e^(2x)) - 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "How many solutions does the equation (-3(3sin(1π(2x - 1) - 2) - 1) - 1)/(3(3sin(1π(2x - 1) - 2) - 1) + 2) = (1(-3sin(1πx + 1) + 3) - 3)/(-3(-3sin(1πx + 1) + 3) - 2) 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 solutions does the equation -2|(2((-3x + 2)/(-1x - 3)) - 5) - 1| = -5(-3|x + 2| + 3)^2 - 5(-3|x + 2| + 3) + 4 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(1cos(1π(4x - 4)) - 1) + 3)/(1(1cos(1π(4x - 4)) - 1) - 2) and g(x) = (2(-3cos(1πx) + 1) + 2)/(3(-3cos(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 points of intersection exist between the functions f(x) = 2sin(3π(-2|(3x^2 + 3x + 1) - 3| - 1)) and g(x) = -3e^(2(4x - 5)) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 773 | MATH |
[
{
"content": "How many solutions does the equation (-1(-3sin(1π(-4x^2 - 2x + 2) + 1) - 3) - 3)/(2(-3sin(1π(-4x^2 - 2x + 2) + 1) - 3) + 1) = (-2(-1x^2 + 4x - 1) - 3)/(-2(-1x^2 + 4x - 1) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 530 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(2sin(1π(-1x - 1)) - 1)^2 + 4(2sin(1π(-1x - 1)) - 1) + 4 and g(x) = 2|((1x + 2)/(-1x + 3))| - 2 in the range -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) = 3(-3sin(1π(-1|x - 2| + 2) + 2) + 3) - 4 and g(x) = 3sin(2π(1|x| + 1) - 3) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-1(1sin(1πx) - 1) + 2)^2 + 5 and g(x) = -5(-2cos(1πx - 1))^2 - 5(-2cos(1πx - 1)) - 1 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) = 4(-2|(-1x^2 + 4x) + 1| - 1) - 3 and g(x) = -3|(3cos(2πx + 2) + 3) - 3| - 3 in the range -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) = (3(3|(2|x + 1| + 3) + 3| - 1) - 3)/(3(3|(2|x + 1| + 3) + 3| - 1) - 2) and g(x) = -2sin(3π(-3x^2 - 1) + 2) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 135 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(3π(-1(1x - 5) + 4)) - 3 and g(x) = 2cos(3π(-3|x - 2| + 2)) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 180 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(3sin(2π((1x + 1)/(2x + 3)) + 1) + 2)| - 2 and g(x) = -3|(-2x + 2)| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|((3(2sin(2πx - 2) + 3) - 2)/(-2(2sin(2πx - 2) + 3))) + 2| + 3 and g(x) = 2cos(2π(3|x| + 1) + 1) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(3cos(1π((1x - 1)/(-1x + 2)) - 2) + 1) - 3)/(2(3cos(1π((1x - 1)/(-1x + 2)) - 2) + 1) - 1) and g(x) = -2sin(2π(4x^2 + 4) - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 445 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-1|(5x + 3)| - 2) + 3 and g(x) = -1|(2x^2 + 1x + 1) + 2| - 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 (-1(-2cos(2π(-2x^2 - 4x + 1) + 3) - 3) + 2)/(-3(-2cos(2π(-2x^2 - 4x + 1) + 3) - 3) - 3) = (3(-2|x + 2|))/(-2(-2|x + 2|) - 1) 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|(2sin(1π(1x^2 - 2x) - 1) + 3) - 3| and g(x) = 5(2sin(2πx)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2e^(-1((1(-1x + 3) - 3)/(1(-1x + 3) - 2))) - 1 and g(x) = -2sin(2π(-2|x + 3| - 2) - 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 77 | MATH |
[
{
"content": "How many solutions does the equation 3sin(3π((-2((-1x + 3)/(-2x)) + 1)/(-1((-1x + 3)/(-2x)) - 3)) - 3) - 1 = -2sin(3π(-1x^2 - 1x - 4) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 328 | MATH |
[
{
"content": "How many solutions does the equation 2e^(-2((-3(-1|x - 1| + 3) + 1)/(-2(-1|x - 1| + 3) - 2))) + 2 = -2|(-1sin(1πx + 2))| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "How many solutions does the equation 2|((-1((1x + 3)/(2x + 1)) - 1)/(-1((1x + 3)/(2x + 1)) - 1)) - 2| - 1 = 2sin(2π(1|x - 1| - 2) + 2) + 3 have for x in the interval [-10, 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) = 3sin(1π(-3((-3x - 1)/(1x + 1))^2 + 4((-3x - 1)/(1x + 1)) - 5) + 1) + 2 and g(x) = -5((-2x + 3)/(2x)) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 37 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(3|(-1x - 4)| - 2) - 1)/(-1(3|(-1x - 4)| - 2)) and g(x) = 1|((1x + 3)/(1x)) - 3| - 3 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) = 3((3(-2x + 1) + 3)/(-2(-2x + 1))) + 4 and g(x) = -1(-2cos(1πx + 1))^2 - 4(-2cos(1πx + 1)) + 5 in the range -10 ≤ x ≤ 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(2e^(1(3x)) + 2) - 2)/(-3(2e^(1(3x)) + 2)) and g(x) = 3|(-1x^2 - 5x + 4) + 2| in the range -10 ≤ x ≤ 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) = -2cos(1π(4((2x - 2)/(1x))^2 + 3((2x - 2)/(1x)) - 2)) and g(x) = (2(-5x^2 + 1x + 2) + 3)/(-3(-5x^2 + 1x + 2) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 151 | MATH |
[
{
"content": "How many solutions does the equation 3((-1(1|x| - 2) + 2)/(3(1|x| - 2) - 2)) + 4 = -1(-5x^2 + 5)^2 + 5(-5x^2 + 5) + 5 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) = 3((1(-1|x - 3| + 1) - 2)/(3(-1|x - 3| + 1) + 3))^2 + 4((1(-1|x - 3| + 1) - 2)/(3(-1|x - 3| + 1) + 3)) - 4 and g(x) = 2cos(2π(1|x + 2|) + 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 22 | MATH |
[
{
"content": "How many solutions does the equation -2sin(1π((1((-1x - 2)/(-3x - 2)) - 1)/(1((-1x - 2)/(-3x - 2)))) + 1) = (1(-3x^2 - 1x - 2) - 2)/(3(-3x^2 - 1x - 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 72 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(3cos(1π((2x + 3)/(1x + 1)) + 2) + 1)^2 - 1 and g(x) = 3(3sin(3πx - 1) + 3)^2 + 3(3sin(3πx - 1) + 3) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 63 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-5(5x^2 - 3x - 3) - 1) - 1| + 1 and g(x) = (-2((-2x)/(1x + 1)) - 3)/(2((-2x)/(1x + 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(-1cos(3π(-1|x + 3| - 2) - 2) - 3) - 1 and g(x) = -2(-2sin(3πx + 1) + 2)^2 + 4(-2sin(3πx + 1) + 2) - 4 intersect 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) = 2cos(3π(-1|(-5x - 4) + 1| - 3) + 2) and g(x) = -3|((3x + 1)/(2x + 2)) - 3| - 1 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 4(-3cos(1π(-2x - 1) - 2) + 1)^2 + 2(-3cos(1π(-2x - 1) - 2) + 1) = 4(-2e^(-2x) + 1)^2 + 3(-2e^(-2x) + 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 42 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-3((2x + 3)/(2x + 2))^2 - 1((2x + 3)/(2x + 2)) - 5) + 1)/(2(-3((2x + 3)/(2x + 2))^2 - 1((2x + 3)/(2x + 2)) - 5) + 2) and g(x) = -3sin(3π(-2x^2 - 2)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your ... | 1146 | MATH |
[
{
"content": "How many solutions does the equation 2cos(1π(4(-5x^2 - 3x)^2 + 4(-5x^2 - 3x) + 4) + 1) - 2 = (-2(-2x^2 + 2x - 5) - 1)/(2(-2x^2 + 2x - 5) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 686 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-5(-1x^2 + 3x + 4)^2 - 1(-1x^2 + 3x + 4) + 2)| + 2 and g(x) = (-3(3|x + 2| - 1) - 3)/(2(3|x + 2| - 1) + 3) where -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) = 4(1|(3x + 4)| - 3)^2 - 3(1|(3x + 4)| - 3) - 5 and g(x) = 3e^(-1(3x^2 + 5x - 1)) - 3 in the range -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) = (-3(-2|(-3cos(2πx - 2) - 3) - 2|) + 2)/(1(-2|(-3cos(2πx - 2) - 3) - 2|)) and g(x) = (1(2sin(1πx - 2)))/(-3(2sin(1πx - 2)) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(3sin(3π((1x - 3)/(-2x - 3)) - 2) + 2)^2 - 3(3sin(3π((1x - 3)/(-2x - 3)) - 2) + 2) and g(x) = 2(-1sin(1πx + 2) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 85 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(-4(-2cos(1πx + 3)) + 1) - 2)/(-1(-4(-2cos(1πx + 3)) + 1) + 2) and g(x) = -1(2x - 5) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-1|((3x - 3)/(-2x + 3)) - 3| + 1) and g(x) = (1(-2|x + 1| + 1) - 1)/(1(-2|x + 1| + 1)) intersect 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(1(-4x + 3) - 1) + 1)/(2(1(-4x + 3) - 1) - 1) = 1sin(2π(4x^2 + 5) + 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1373 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-1sin(3π(2|x + 3|) - 2) + 2)^2 + 5(-1sin(3π(2|x + 3|) - 2) + 2) - 4 and g(x) = 2(3x - 4)^2 + 3(3x - 4) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1((-1(1x + 4))/(1(1x + 4) + 1)) + 3)/(-3((-1(1x + 4))/(1(1x + 4) + 1))) and g(x) = -3sin(2π((-1x - 2)/(2x)) + 1) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 53 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(1(1x + 2) - 2) + 1)/(-3(1(1x + 2) - 2) + 4) and g(x) = -1cos(3π(2x^2 + 3x - 4) - 3) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 979 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-5(2sin(2πx + 1)) - 2)^2 + 1(-5(2sin(2πx + 1)) - 2) + 2 and g(x) = 1(3x + 4)^2 + 3(3x + 4) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many solutions does the equation -1((-3(-1|x + 2| + 2) + 1)/(-1(-1|x + 2| + 2) + 2))^2 - 1((-3(-1|x + 2| + 2) + 1)/(-1(-1|x + 2| + 2) + 2)) - 1 = -3|(1cos(3πx - 2) - 3) + 1| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 39 | MATH |
[
{
"content": "How many solutions does the equation (3(-2(3x^2 - 5x - 3)^2 + 5(3x^2 - 5x - 3) - 1) + 3)/(2(-2(3x^2 - 5x - 3)^2 + 5(3x^2 - 5x - 3) - 1) - 1) = 1(1|x + 1| - 1) + 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(-1sin(1πx - 3)) - 2)/(1(-1sin(1πx - 3)))) - 2 = 4(3|x + 3| - 1) + 2 have for x 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 2sin(3π(3((-1x - 3)/(1x - 3))) - 1) + 1 = 2(3e^(1x) - 3)^2 - 2(3e^(1x) - 3) - 1 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) = -3|(1|(1x^2 + 2x) - 2| - 3)| + 3 and g(x) = (-1(-1|x + 2| + 2) - 1)/(3(-1|x + 2| + 2)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many solutions does the equation -2((-1(1|x - 3| - 3) + 1)/(-3(1|x - 3| - 3) - 3))^2 + 3((-1(1|x - 3| - 3) + 1)/(-3(1|x - 3| - 3) - 3)) = -2(-1sin(1πx + 2) + 2)^2 - 1(-1sin(1πx + 2) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-2|(-2|x - 1| + 1) - 1| - 3))/(3(-2|(-2|x - 1| + 1) - 1| - 3) - 2) and g(x) = -4(-1sin(2πx) + 2)^2 + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2sin(1π((2(-1|x + 2| + 3) + 1)/(-2(-1|x + 2| + 3) - 2)) - 1) - 3 and g(x) = 2cos(1π(4x^2 - 3x - 3) - 3) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 116 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(2π(3|(4x^2 - 1x) - 3|) - 1) - 2 and g(x) = 3cos(2π(1|x| - 1) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 716 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(-5(-1sin(2πx - 2) + 2) + 4) - 2)/(2(-5(-1sin(2πx - 2) + 2) + 4) + 2) and g(x) = 5((-1x + 2)/(1x + 1))^2 + 5((-1x + 2)/(1x + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 25 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-3(-2|x + 1| + 2)^2 + 1(-2|x + 1| + 2) + 1) + 3)/(-1(-3(-2|x + 1| + 2)^2 + 1(-2|x + 1| + 2) + 1) + 2) and g(x) = (3(3cos(2πx - 2) - 1) - 2)/(3(3cos(2πx - 2) - 1) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex fo... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-1|(2e^(-1x) - 2) + 2| + 2)^2 + 2(-1|(2e^(-1x) - 2) + 2| + 2) + 4 and g(x) = 2cos(2π((-1x)/(-3x - 2)) + 3) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(1π(-5(-3x + 1)^2 + 2(-3x + 1) - 3) - 3) - 1 and g(x) = 3(1|x - 2| - 3) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 244 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-1(-2|x + 3| - 2) - 4)^2 - 5(-1(-2|x + 3| - 2) - 4) + 2 and g(x) = -2cos(1π(-2x + 3) - 1) + 3 where -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) = -3e^(2(-2|(-4x - 2) - 3| + 2)) + 2 and g(x) = (2(-3x^2 - 1x + 2) + 3)/(-3(-3x^2 - 1x + 2) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
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