messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Find the number of times the graphs of f(x) = -3|(3sin(1π(-1|x - 2| - 3) - 1) + 2) + 2| and g(x) = 2(-2|x - 3| - 2) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many solutions does the equation 4(1|(1|x + 1|) - 3| + 2) - 5 = 4(-3cos(3πx - 3) + 1)^2 + 1(-3cos(3πx - 3) + 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 98 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(1(-3|x + 3| + 2) + 5) + 1)/(2(1(-3|x + 3| + 2) + 5) + 1) and g(x) = 3sin(3π(-3|x + 2| - 1) - 3) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 178 | MATH |
[
{
"content": "How many solutions does the equation 3((1(-1x^2 - 1))/(-1(-1x^2 - 1) - 1))^2 + 2((1(-1x^2 - 1))/(-1(-1x^2 - 1) - 1)) = 4(-3cos(2πx + 1) + 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 36 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-3sin(1π((-3x + 1)/(2x)) - 1) - 1)^2 - 4(-3sin(1π((-3x + 1)/(2x)) - 1) - 1) + 1 and g(x) = (1(-1x + 1))/(-1(-1x + 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 27 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-3|(2|x| + 3) - 2| + 1) + 1)/(-3(-3|(2|x| + 3) - 2| + 1) + 1) and g(x) = -1cos(3π(1x^2 - 1x - 3) + 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 280 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(-1|(3cos(1πx + 1) + 3)| + 3))/(-1(-1|(3cos(1πx + 1) + 3)| + 3) + 2) and g(x) = 3|(3sin(3πx) + 2) + 2| + 3 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(-3(-2sin(2πx + 2)) + 3) + 2)/(2(-3(-2sin(2πx + 2)) + 3) + 3) = (1(1cos(2πx + 2) + 1))/(-3(1cos(2πx + 2) + 1) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(1(2x - 4) - 1) + 3| + 1 and g(x) = -1cos(1π(2|x| + 1) + 3) + 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) = 4(1cos(1π((-1x - 3)/(-3x)) - 1))^2 + 2(1cos(1π((-1x - 3)/(-3x)) - 1)) + 3 and g(x) = 2cos(1π(-2|x + 3|) - 3) + 1 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) = -1(1cos(2π(1|x| + 3) - 1) - 2) - 4 and g(x) = 1|(-2|x + 3| + 2) + 3| - 2 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) = -5(-2cos(1π(3|x + 1| + 3) + 3) + 1) - 5 and g(x) = -1|(-5x - 4) - 2| - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2((-2(1cos(2πx - 3) - 1) - 1)/(-3(1cos(2πx - 3) - 1))) + 1)/(2((-2(1cos(2πx - 3) - 1) - 1)/(-3(1cos(2πx - 3) - 1))) + 1) and g(x) = 1cos(1π((3x - 3)/(2x + 2))) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: ... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2((-1(-5x + 1) + 1)/(3(-5x + 1) + 3)) - 2)/(1((-1(-5x + 1) + 1)/(3(-5x + 1) + 3)) + 1) and g(x) = 3(-2cos(3πx + 3) - 1) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 59 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(1|((-2x)/(2x + 1)) + 3| - 2)^2 + 1(1|((-2x)/(2x + 1)) + 3| - 2) - 1 and g(x) = 5(-2x - 5)^2 - 5(-2x - 5) 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((1(1e^(1x) - 2) + 1)/(3(1e^(1x) - 2) - 2)) - 1 and g(x) = -3(-3cos(2πx - 2) + 1)^2 + 3(-3cos(2πx - 2) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 77 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(4(-1|x - 1|)^2 - 2(-1|x - 1|) - 5)^2 - 3(4(-1|x - 1|)^2 - 2(-1|x - 1|) - 5) + 5 and g(x) = 1((3x - 2)/(1x - 3))^2 - 5((3x - 2)/(1x - 3)) where -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) = -2|((-1((1x - 1)/(-2x + 3)))/(-3((1x - 1)/(-2x + 3)) + 2))| - 1 and g(x) = -1|((-1x)/(1x - 3)) + 2| intersect 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(2cos(2π(4x^2 + 3x) - 1) + 1)^2 + 3(2cos(2π(4x^2 + 3x) - 1) + 1) + 4 and g(x) = 3cos(2π(-2|x + 1|) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 61 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(1|((3x + 2)/(-2x)) + 2| + 2) - 1)/(2(1|((3x + 2)/(-2x)) + 2| + 2) + 3) and g(x) = 2cos(3π(-3x + 1) + 1) 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) = (2(4(-1|x| + 1) - 1) - 1)/(-2(4(-1|x| + 1) - 1) - 2) and g(x) = -1cos(3π((-1x + 2)/(3x - 2)) + 3) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "How many solutions does the equation 2|(2|((3x)/(-3x - 2)) + 1| - 3) + 2| - 1 = (1(-1sin(1πx - 2)) - 2)/(-1(-1sin(1πx - 2)) + 3) 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) = 2sin(1π(5(-1x^2 - 2x - 5) - 2) + 1) and g(x) = -1(-1|x - 2| + 3)^2 - 2(-1|x - 2| + 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 139 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-1cos(3π(-4x^2 - 2x + 4) + 2) - 1) + 1| and g(x) = (-2(4x^2 - 5x + 2) + 2)/(2(4x^2 - 5x + 2) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1318 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2e^(-1(-2|(5x^2 - 4x - 4)| - 1)) - 3 and g(x) = 4(-1cos(2πx + 1) + 1) - 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) = 4(-2(1cos(1πx + 2) + 2)^2 + 4(1cos(1πx + 2) + 2) + 4)^2 + 1(-2(1cos(1πx + 2) + 2)^2 + 4(1cos(1πx + 2) + 2) + 4) - 2 and g(x) = 1(3|x - 1| - 1) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 30 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(2(3cos(3πx + 3) + 2)^2 - 4(3cos(3πx + 3) + 2))^2 - 2 and g(x) = 1sin(1π(5x + 2) - 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 240 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(4(1x - 4)^2 + 1(1x - 4) - 4))/(1(4(1x - 4)^2 + 1(1x - 4) - 4) + 2) and g(x) = -3sin(1π(3|x + 3| + 3) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 54 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(-1|(1|x - 2| - 3) + 2|)| + 1 and g(x) = -2|(2sin(3πx + 3) + 2) - 2| + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "How many solutions does the equation 2(1e^(-1(3x^2 - 1x - 5)) + 3) + 2 = 1(3cos(1πx + 2) + 2) + 3 have for x in the interval [-10, 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) = 2|(2|(2|x + 3| + 1) - 2|)| + 3 and g(x) = (-1(-3sin(3πx - 1) - 2) - 3)/(3(-3sin(3πx - 1) - 2) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 39 | MATH |
[
{
"content": "How many solutions does the equation 2|(-1(-1cos(2πx) + 2)^2 + 2(-1cos(2πx) + 2)) + 1| + 2 = 2((2x - 2)/(1x + 2)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 36 | MATH |
[
{
"content": "How many solutions does the equation -3|(-3|(-5x + 2) - 3| + 3)| = -3cos(2π(-2x^2 + 1x + 2) + 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": "Determine the number of intersection points between the functions f(x) = -4((-3(-3x) + 1)/(1(-3x) - 3))^2 - 5((-3(-3x) + 1)/(1(-3x) - 3)) - 3 and g(x) = -2|((-2x - 1)/(1x + 3)) + 2| where -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) = 1sin(2π(1(-4x - 4)^2 + 2(-4x - 4) + 1) + 3) - 3 and g(x) = 1(-2|x - 1| + 1) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 136 | MATH |
[
{
"content": "How many solutions does the equation 1(3sin(3π(4x^2 + 2)) - 1)^2 - 3(3sin(3π(4x^2 + 2)) - 1) + 3 = -3cos(2π(-2x^2 - 5) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 502 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3cos(2π(-5((3x + 1)/(-1x)) + 3) - 1) + 1 and g(x) = 2e^(-2(1x^2 - 3x - 2)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "How many solutions does the equation 2(1(3x^2 + 3x + 5) - 2) - 4 = 3|(-3cos(3πx - 3) + 3) - 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": "Find the number of times the graphs of f(x) = -3|((-3(3x^2 - 5x + 2) + 1)/(2(3x^2 - 5x + 2))) - 2| and g(x) = (-1(-1e^(-2x) - 3) + 1)/(2(-1e^(-2x) - 3) - 1) intersect 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) = 1cos(2π(-3(-3|x + 1| + 2) + 2) + 2) and g(x) = -2sin(3π(1x + 5) - 3) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 164 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(1|((-3x - 3)/(-2x - 1)) - 2| - 2) + 1 and g(x) = 1(1sin(1πx + 1) + 2)^2 - 1(1sin(1πx + 1) + 2) - 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 (2(1(3|x + 2| - 2) + 2) - 2)/(3(1(3|x + 2| - 2) + 2) + 1) = 3sin(2π(2|x - 1| - 3) - 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(1(1sin(2πx - 1) + 3)^2 + 3(1sin(2πx - 1) + 3) - 1) - 3)/(1(1(1sin(2πx - 1) + 3)^2 + 3(1sin(2πx - 1) + 3) - 1) - 3) and g(x) = -3|(-1sin(2πx - 2) + 1) - 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your ... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(1π(-1((-1x + 1)/(2x)) + 1) - 3) - 2 and g(x) = -3sin(2π(-4x^2 + 2x + 3) + 3) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-1(3|x + 2|) - 4) and g(x) = (-3(-2sin(1πx + 3) - 1))/(-1(-2sin(1πx + 3) - 1) - 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) = -3cos(1π(-2|(-5x^2 + 4x) + 3| - 3)) + 1 and g(x) = -2cos(1π(-2x^2 + 1x + 5) - 2) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1440 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-3sin(2π(-1|x + 1| - 3) + 3) + 3) - 1)/(-2(-3sin(2π(-1|x + 1| - 3) + 3) + 3)) and g(x) = (-1(2sin(1πx + 3) + 2) - 3)/(-2(2sin(1πx + 3) + 2) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1((2(1x - 1) - 2)/(-3(1x - 1) - 3)) - 3)/(3((2(1x - 1) - 2)/(-3(1x - 1) - 3)) + 2) and g(x) = 1sin(1π(-2|x + 3| - 3) + 1) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(2(3|x - 1| - 3))^2 + 2(2(3|x - 1| - 3)) and g(x) = 1e^(-2(1x + 2)) - 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) = 4(1cos(3π((2x - 2)/(2x + 2)) - 1) - 2)^2 + 5(1cos(3π((2x - 2)/(2x + 2)) - 1) - 2) + 1 and g(x) = 3((1x + 2)/(3x + 2)) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 67 | MATH |
[
{
"content": "How many solutions does the equation -2sin(2π(-3|(-1x^2 + 3x + 2) + 3| + 3) + 3) + 2 = -1(-1x^2 + 2x + 1)^2 - 4(-1x^2 + 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": "How many points of intersection exist between the functions f(x) = -2|(1(-1cos(2πx - 2) - 1) - 1) - 3| and g(x) = -3|(-2e^(1x) + 2) + 3| + 3 in the range -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) = (-3(-2|(-4x + 1)| + 1) - 1)/(3(-2|(-4x + 1)| + 1) + 1) and g(x) = -2(1|x + 3|)^2 + 4(1|x + 3|) - 2 intersect 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|((-2(-5x + 3) + 3)/(1(-5x + 3))) - 2| - 1 and g(x) = 3|((-2x + 3)/(-3x + 4))| - 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) = -1cos(1π(-2(4x^2 - 5x)^2 - 5(4x^2 - 5x) + 1) + 2) + 2 and g(x) = -3sin(3π((1x + 1)/(-1x - 3))) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 233 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(-3sin(2π(4x + 5))) - 1)/(2(-3sin(2π(4x + 5))) + 3) and g(x) = -1cos(1π(5x - 2) - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 140 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(1(-1|x + 3| - 1)^2 + 1(-1|x + 3| - 1) - 1)^2 - 2(1(-1|x + 3| - 1)^2 + 1(-1|x + 3| - 1) - 1) - 4 and g(x) = (-2(1cos(1πx)))/(2(1cos(1πx)) + 2) in the range -10 ≤ x ≤ 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) = 2|((-3(-2e^(2x) - 2) - 3)/(-1(-2e^(2x) - 2) + 2)) + 1| and g(x) = 5(-2cos(3πx + 1) - 3)^2 - 1(-2cos(3πx + 1) - 3) - 4 in the range -10 ≤ x ≤ 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|(3(3cos(3πx - 2) + 1) - 3)| - 2 and g(x) = 2(-3|x - 3| + 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 35 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(-3sin(1π(-4x + 3)) - 2) + 4 and g(x) = -2e^(-2((-1x - 1)/(-3x + 2))) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 74 | MATH |
[
{
"content": "How many solutions does the equation 3cos(3π(-3|(-2|x - 1| - 1) - 2|)) = (-2(-1sin(3πx - 3)))/(2(-1sin(3πx - 3)) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 300 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-2sin(1π((-2x + 1)/(-3x)) + 1) + 1)^2 - 2(-2sin(1π((-2x + 1)/(-3x)) + 1) + 1) + 2 and g(x) = (3(2e^(2x) - 2) - 3)/(3(2e^(2x) - 2) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(-1cos(1π(-1|x| + 1) + 1) - 1) + 1 and g(x) = -2sin(3π(-5x + 2) - 1) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 276 | MATH |
[
{
"content": "How many solutions does the equation -1(1(-1x - 4)^2 + 4(-1x - 4) + 3)^2 + 5(1(-1x - 4)^2 + 4(-1x - 4) + 3) - 2 = -2(2|x + 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 points of intersection exist between the functions f(x) = 2sin(1π(-1((3x + 1)/(-2x + 1))^2 + 1((3x + 1)/(-2x + 1)) - 4) - 2) and g(x) = (-1(3|x + 2| + 1) - 3)/(-2(3|x + 2| + 1) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 83 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-2|(-1|x + 3| + 2)| + 1) - 1| - 3 and g(x) = 1|(-1x - 5)| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation (3(1(1x) + 1) - 3)/(-1(1(1x) + 1) - 1) = 3cos(3π(-1|x + 2| - 2) + 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 47 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(3π(-1(3x + 2)^2 - 4)) - 3 and g(x) = 2cos(3π(-3x - 3) + 1) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 890 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-2((-3x + 2)/(-2x - 3))^2 - 1((-3x + 2)/(-2x - 3))) - 3| and g(x) = 3(-2sin(2πx)) + 2 intersect 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(3sin(3π(1x^2 + 2) - 2)) - 3)/(2(3sin(3π(1x^2 + 2) - 2)) + 2) and g(x) = -2((3x - 3)/(-1x - 3)) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 215 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(-2|(-3sin(2πx + 1)) + 3| + 3) + 2)/(-2(-2|(-3sin(2πx + 1)) + 3| + 3) + 1) and g(x) = -2sin(1π(-1x^2 + 5x + 5) + 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 203 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-5(-1x^2 + 4x + 3) + 5) + 3| and g(x) = 4(-3|x + 1|) - 5 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) = 4((1((2x)/(3x - 1)) - 1)/(2((2x)/(3x - 1)) - 1))^2 - 3((1((2x)/(3x - 1)) - 1)/(2((2x)/(3x - 1)) - 1)) - 5 and g(x) = -1|(3cos(2πx - 3) + 2) - 2| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(-2cos(2π(-3|x| + 3) - 2) + 2) + 3)/(-3(-2cos(2π(-3|x| + 3) - 2) + 2) + 1) and g(x) = 2|(1e^(-2x) + 1) - 1| + 2 where -10 ≤ x ≤ 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) = -3sin(2π(-1(3x^2 - 5x - 1) - 2) - 1) + 2 and g(x) = 2((-2x - 3)/(-3x - 2)) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(2sin(2π(3|x - 2| + 3) - 2) - 1) - 4 and g(x) = 2|(2|x + 2| + 3)| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-3cos(3π(-1|x - 2| - 1))) - 3 and g(x) = 1|(2e^(-1x) - 3)| - 1 where -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) = 1|(-4((1x + 3)/(-1x + 3)) + 1) + 2| - 3 and g(x) = -1|((-3x)/(-3x - 2)) - 2| + 2 intersect 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) = -3e^(-1(2|(1x^2 - 1x + 3) + 3| + 2)) and g(x) = 2|(-2|x|) + 3| - 1 where -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) = (-3(-5(-4x^2 + 4x + 3) + 2) + 3)/(3(-5(-4x^2 + 4x + 3) + 2) + 2) and g(x) = 3cos(3π(1|x - 2| - 2) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 62 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3e^(-1((1(-1|x + 1| + 3) + 1)/(-1(-1|x + 1| + 3)))) + 1 and g(x) = 4(2cos(1πx + 2) + 1) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(3cos(3π(1|x + 1| - 1) - 1) + 1) - 1 and g(x) = (-1(-5x^2 + 4x - 3) + 1)/(3(-5x^2 + 4x - 3) + 1) where -10 ≤ x ≤ 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) = 3|(1sin(3π(-1x^2 - 3x - 1) + 2) - 1)| + 2 and g(x) = 2sin(1π(-1|x + 3| + 1)) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 323 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(1(3cos(2πx - 3) + 1) + 2)^2 - 1(1(3cos(2πx - 3) + 1) + 2) + 3 and g(x) = (2((-1x)/(-1x - 3)) + 1)/(-2((-1x)/(-1x - 3)) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(3π(-2(2|x + 2| - 2)^2 - 5(2|x + 2| - 2))) - 1 and g(x) = 3((-2x + 3)/(3x)) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 44 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-3sin(1π(4x^2 + 1x + 5) + 1) + 2) - 2| + 1 and g(x) = (-2(4x^2 - 1x - 3) + 1)/(1(4x^2 - 1x - 3) - 3) intersect 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) = -1cos(3π(3|(3x^2 + 4x - 5) - 1| + 1) - 2) and g(x) = 1cos(2π(5x + 1)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 673 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|((-1(-2x - 5))/(3(-2x - 5) - 2)) - 3| + 1 and g(x) = (3(-3cos(1πx + 2) - 2) + 2)/(-1(-3cos(1πx + 2) - 2) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "How many solutions does the equation -3|(-1sin(2π(2|x + 2| - 1) + 3)) + 3| + 3 = -2(2x - 1)^2 - 4 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) = -4(3|(-2x - 5)| - 3)^2 - 2(3|(-2x - 5)| - 3) + 4 and g(x) = -2|(-1sin(3πx + 1) + 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 points of intersection exist between the functions f(x) = -2cos(1π(3(-5x^2 + 1) + 5) - 1) - 1 and g(x) = 4(-2|x - 1| - 1) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "How many solutions does the equation 1cos(1π(1((-1x - 2)/(-3x - 3))^2 - 2((-1x - 2)/(-3x - 3)) - 4) + 3) - 3 = -2(3|x - 3| - 3)^2 + 1(3|x - 3| - 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) = 1e^(1(-3|(2|x + 2| + 3) - 3| - 2)) - 3 and g(x) = 2sin(1π(-3x^2 + 3x + 1) - 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 602 | MATH |
[
{
"content": "How many solutions does the equation 1((2(-1|x - 1| - 1) + 2)/(1(-1|x - 1| - 1))) + 2 = 3|(-5x^2 + 3x + 4)| 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) = -2e^(-1(1|(-1|x| - 2) + 2| - 2)) + 1 and g(x) = 2sin(1π((-2x + 1)/(-1x - 2)) - 2) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 78 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5((3((2x - 3)/(3x + 1)) + 2)/(-1((2x - 3)/(3x + 1)) + 3)) + 5 and g(x) = -3(-3cos(1πx - 1) - 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "How many solutions does the equation 1|(-1(-1sin(1πx - 1) - 2)^2 - 2(-1sin(1πx - 1) - 2) + 3) - 3| + 1 = -3e^(2(-1|x + 2| + 3)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 29 | MATH |
[
{
"content": "How many solutions does the equation (-1(-3(-3x)^2 + 5(-3x) + 5) + 3)/(3(-3(-3x)^2 + 5(-3x) + 5) + 2) = -3|((3x - 3)/(2x + 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": "Find the number of times the graphs of f(x) = -3cos(3π((-1(-2|x - 3| + 2) - 2)/(3(-2|x - 3| + 2) - 2)) - 3) - 1 and g(x) = (-1((2x - 3)/(1x + 2)) - 2)/(2((2x - 3)/(1x + 2)) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 47 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(1π(2(-1x + 4) - 1)) and g(x) = (2((-1x - 2)/(1x)) + 3)/(-2((-1x - 2)/(1x)) - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
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