messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation 3(2sin(1π(3x^2 + 5x + 4) - 2) - 2)^2 - 1(2sin(1π(3x^2 + 5x + 4) - 2) - 2) - 2 = 5((1x + 1)/(-2x + 1))^2 + 4((1x + 1)/(-2x + 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 584 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3((-1(-3e^(1x) + 3))/(-3(-3e^(1x) + 3) - 3)) - 5 and g(x) = -2sin(1π(3|x + 3| - 2) + 2) - 2 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) = -4(-1cos(2π(-4x - 1) - 2)) and g(x) = 3(-3cos(3πx))^2 + 2(-3cos(3πx)) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many solutions does the equation 3|(-2|(3sin(3πx + 1) + 2) + 3| - 3) + 1| = 3(-2e^(-1x) - 2)^2 - 4(-2e^(-1x) - 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 points of intersection exist between the functions f(x) = 1(3sin(3π(-3x + 3) + 1)) + 1 and g(x) = (2(3|x - 1| + 3) - 1)/(-2(3|x - 1| + 3) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 180 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(2|(-3|x - 2| + 2)| - 1)^2 - 1(2|(-3|x - 2| + 2)| - 1) + 4 and g(x) = 1((-2x + 3)/(1x))^2 - 2((-2x + 3)/(1x)) - 4 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) = -1(3(4x^2 - 2x)^2 - 1(4x^2 - 2x) - 2) - 3 and g(x) = -3(1sin(2πx + 3) - 1) - 4 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) = -3(-1(-1sin(2πx - 3) + 2))^2 - 3(-1(-1sin(2πx - 3) + 2)) + 3 and g(x) = -1sin(3π((-1x + 2)/(-2x + 3)) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "How many solutions does the equation 1(-2sin(1π((2x + 1)/(2x - 1)) + 1) + 3) + 1 = (3((-1x - 1)/(2x - 1)) + 1)/(1((-1x - 1)/(2x - 1))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-3sin(2π((-1x)/(-2x + 3))) + 3) - 3)/(-2(-3sin(2π((-1x)/(-2x + 3))) + 3)) and g(x) = 1cos(1π((1x + 3)/(2x - 3)) + 3) + 1 intersect in the interval [-10, 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) = (-1(-5(2cos(1πx - 1) - 3)^2 + 5(2cos(1πx - 1) - 3) - 1) - 3)/(2(-5(2cos(1πx - 1) - 3)^2 + 5(2cos(1πx - 1) - 3) - 1) + 1) and g(x) = 2sin(1π(-4x^2 - 2x - 4) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{You... | 800 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-4(-3cos(2πx - 3))^2 + 1(-3cos(2πx - 3)) + 5) + 3 and g(x) = 1|(-1|x - 3| + 2) - 2| - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 68 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(2π(3((2x + 1)/(3x - 1)) + 4) + 1) - 1 and g(x) = -3|(-3cos(2πx + 2)) - 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 46 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(2|(1x^2 - 1x - 4)| - 1))/(-2(2|(1x^2 - 1x - 4)| - 1) - 1) and g(x) = (-3(-4x^2 - 4x - 3))/(-3(-4x^2 - 4x - 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) = 2|(-5((3x + 3)/(-3x + 2)) - 4) + 2| - 2 and g(x) = -2sin(2π(-1|x + 3| - 3) + 3) + 3 where -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) = 2cos(1π(1(4x^2 + 1x - 4)^2 - 2(4x^2 + 1x - 4) + 1)) + 2 and g(x) = (3((-2x + 1)/(1x + 2)) - 2)/(2((-2x + 1)/(1x + 2))) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 1323 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(3cos(3π(-1|x + 1| + 3) - 3) + 1) + 3)/(3(3cos(3π(-1|x + 1| + 3) - 3) + 1) + 2) and g(x) = (3(3x^2 + 4x) - 3)/(-3(3x^2 + 4x) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 58 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(2π((2((-1x - 1)/(2x + 2)) - 2)/(3((-1x - 1)/(2x + 2)) - 1)) + 3) - 2 and g(x) = 1cos(3π(-1|x + 2| - 3) + 1) - 3 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) = (3(-1(2|x| + 1)^2 + 1(2|x| + 1) + 1) + 1)/(-1(-1(2|x| + 1)^2 + 1(2|x| + 1) + 1) - 2) and g(x) = -2cos(2π(-1x^2 + 3x - 1) + 2) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-2|(-2|x - 1| - 2) - 1| - 3) - 2)/(2(-2|(-2|x - 1| - 2) - 1| - 3) + 2) and g(x) = 3e^(-1(1x^2 + 1x + 4)) - 1 in the range -10 ≤ x ≤ 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) = -4((-1(-3sin(1πx) + 1) - 3)/(-1(-3sin(1πx) + 1) - 1)) + 1 and g(x) = (-3(-2|x| - 3) + 3)/(1(-2|x| - 3) + 3) 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) = 3|(-2|(5x + 1)|) - 1| + 2 and g(x) = 2(2cos(2πx - 1) + 2)^2 + 5 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 -3sin(3π(3|(1x + 3)| - 2) - 3) - 2 = (3((1x - 1)/(1x + 2)))/(-2((1x - 1)/(1x + 2)) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 118 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1cos(2π(-5(1|x - 1|)^2 + 1(1|x - 1|) + 5) + 1) + 2 and g(x) = 2e^(-1(4x + 1)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 649 | MATH |
[
{
"content": "How many solutions does the equation 2(-1(-1|x - 3|) - 4)^2 - 2(-1(-1|x - 3|) - 4) + 4 = 3(-1cos(1πx + 2))^2 + 3(-1cos(1πx + 2)) + 5 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((2(3sin(1πx + 2) + 2) - 1)/(-2(3sin(1πx + 2) + 2) + 3))^2 - 4((2(3sin(1πx + 2) + 2) - 1)/(-2(3sin(1πx + 2) + 2) + 3)) + 5 = 2|(3|x - 3| + 2) + 1| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(2(-2sin(2πx + 1) - 1)^2 - 4(-2sin(2πx + 1) - 1) - 5))/(-2(2(-2sin(2πx + 1) - 1)^2 - 4(-2sin(2πx + 1) - 1) - 5) + 1) and g(x) = 3sin(3π(-3x) + 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 26 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(1cos(3π((-1x - 3)/(-2x)) + 2)) + 3| - 1 and g(x) = 5(2|x - 1| - 3) - 5 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(-4((1x - 3)/(2x - 1))^2 + 3((1x - 3)/(2x - 1)) - 2) + 2)/(1(-4((1x - 3)/(2x - 1))^2 + 3((1x - 3)/(2x - 1)) - 2) - 2) = -1cos(2π((2x + 2)/(1x))) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 53 | MATH |
[
{
"content": "How many solutions does the equation (2(-2sin(3π(5x^2 + 3x - 4) - 1) + 2) - 1)/(3(-2sin(3π(5x^2 + 3x - 4) - 1) + 2) - 2) = 1(4x^2 - 2x - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5((3(2|x + 2| - 2) + 3)/(-2(2|x + 2| - 2) - 3))^2 + 1((3(2|x + 2| - 2) + 3)/(-2(2|x + 2| - 2) - 3)) and g(x) = -3cos(1π(1x^2 + 4x + 2)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-4(1|x - 1| - 2)^2 + 4(1|x - 1| - 2) - 3)| + 3 and g(x) = (1(-1x^2 - 5x + 2) + 3)/(2(-1x^2 - 5x + 2) + 2) intersect 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) = -3e^(-1((-2(1x + 5))/(-2(1x + 5) - 3))) - 1 and g(x) = -2sin(2π(2x - 2) - 1) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 70 | MATH |
[
{
"content": "How many solutions does the equation -5((-1((2x + 1)/(-3x + 1)) + 2)/(1((2x + 1)/(-3x + 1)) - 3))^2 - 5((-1((2x + 1)/(-3x + 1)) + 2)/(1((2x + 1)/(-3x + 1)) - 3)) - 5 = 2sin(1π(-2x^2 - 5x + 5) + 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 406 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1cos(3π(1|(-4x^2 - 2x + 5) + 2| - 3) + 2) + 1 and g(x) = -1e^(-1(2x^2 + 2x - 3)) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 896 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-1|((-2x + 1)/(2x + 2))| + 1)^2 - 4(-1|((-2x + 1)/(2x + 2))| + 1) and g(x) = -1(-1sin(3πx + 3)) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5((-2(2cos(3πx - 3) + 1) + 2)/(2(2cos(3πx - 3) + 1) + 1))^2 + 3((-2(2cos(3πx - 3) + 1) + 2)/(2(2cos(3πx - 3) + 1) + 1)) - 2 and g(x) = -3e^(2(1x^2 + 1)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your... | 16 | MATH |
[
{
"content": "How many solutions does the equation -3|(2e^(2(-2|x| + 1)) + 2) - 3| + 2 = 2cos(3π(-2x + 5) - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 112 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(3cos(2π(4x^2 + 2x - 1) - 3) + 2) - 1 and g(x) = (2(-4x - 4))/(3(-4x - 4) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1307 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(-2(3|x - 1|) - 1) + 1 and g(x) = 1(4x^2 - 1x - 4)^2 - 3(4x^2 - 1x - 4) + 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) = 3((2((2x - 1)/(-2x - 1)) + 3)/(-3((2x - 1)/(-2x - 1)) + 4)) - 1 and g(x) = 1cos(3π(4x - 3) - 1) + 2 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) = -1cos(2π(3((1x - 2)/(-2x + 3)) + 1) + 2) + 3 and g(x) = 2sin(1π(-1x + 4) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(-3sin(3π((3x - 2)/(3x + 1)) - 2) + 2) + 2| - 1 and g(x) = (2(2|x - 1| - 3))/(-3(2|x - 1| - 3) + 2) where -10 ≤ x ≤ 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) = (-2(-3sin(2π((-1x - 1)/(-1x + 3)) - 2)) + 1)/(-2(-3sin(2π((-1x - 1)/(-1x + 3)) - 2)) - 2) and g(x) = 2|(5x + 5) - 2| intersect in the interval [-10, 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) = (1(-1sin(3π((2x + 3)/(2x - 3)) + 3) - 1))/(-2(-1sin(3π((2x + 3)/(2x - 3)) + 3) - 1) - 1) and g(x) = 3(-2|x + 1| - 2) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2e^(1(-3(5x^2 + 4x - 1) - 2)) and g(x) = 3sin(3π(-3x + 2) - 2) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 172 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3cos(3π((-2(5x + 2) - 3)/(-2(5x + 2))) - 3) and g(x) = -1cos(1π(1x + 3) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 35 | MATH |
[
{
"content": "How many solutions does the equation -1(2sin(2π((-1x + 1)/(-2x - 3)) - 2) - 2)^2 - 5(2sin(2π((-1x + 1)/(-2x - 3)) - 2) - 2) = (2(1cos(3πx + 1) + 2) - 2)/(3(1cos(3πx + 1) + 2) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 33 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(-1e^(-1(-2|x + 3| - 1)) + 3) + 3| + 3 and g(x) = (1(1|x - 2| - 3) - 2)/(2(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) = (1(1((-3x)/(-2x - 2))^2 + 5((-3x)/(-2x - 2)) + 1) + 1)/(-3(1((-3x)/(-2x - 2))^2 + 5((-3x)/(-2x - 2)) + 1) - 1) and g(x) = -3sin(3π(3|x| - 1) - 3) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your an... | 182 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(2π(-2|(1x^2 - 2x + 3) + 2| - 1) - 2) - 2 and g(x) = (-3(-1x^2 + 4x) + 2)/(1(-1x^2 + 4x) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 774 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4((1((1x - 3)/(-3x - 3)))/(1((1x - 3)/(-3x - 3)) + 1))^2 + 4((1((1x - 3)/(-3x - 3)))/(1((1x - 3)/(-3x - 3)) + 1)) + 3 and g(x) = 1sin(1π(-4x^2 + 1x + 2)) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed... | 188 | MATH |
[
{
"content": "How many solutions does the equation 1(2(4x - 4) + 4)^2 - 4 = 1cos(1π((3x + 3)/(-3x + 1)) + 2) 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(3sin(3π(-3|x| + 2) + 2) + 1))/(2(3sin(3π(-3|x| + 2) + 2) + 1) + 2) and g(x) = -3sin(3π(-3|x - 1|) + 3) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 167 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-2(2x^2 + 1x - 3)^2 - 3(2x^2 + 1x - 3))^2 - 4(-2(2x^2 + 1x - 3)^2 - 3(2x^2 + 1x - 3)) + 5 and g(x) = -3|(2sin(1πx - 1) + 3) - 1| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1|(1(-4x + 4)^2 - 4(-4x + 4) - 1) - 3| + 1 = 1(-3sin(3πx + 1) - 3) + 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) = 1(-2cos(3π(-4x) + 3) + 1)^2 - 1(-2cos(3π(-4x) + 3) + 1) and g(x) = -2(1x + 1)^2 + 5(1x + 1) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-2|(-3|x - 1| + 1) - 2| - 2) - 3)/(3(-2|(-3|x - 1| + 1) - 2| - 2) + 2) and g(x) = 2cos(2π((1x - 1)/(-1x + 2)) + 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 49 | MATH |
[
{
"content": "How many solutions does the equation 2(-1(4x^2 + 1x - 3) + 1)^2 - 4(-1(4x^2 + 1x - 3) + 1) - 4 = -2(-3x - 5) - 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) = 4(-2cos(3π(-3x^2 + 2x + 3) - 2) + 2)^2 - 5(-2cos(3π(-3x^2 + 2x + 3) - 2) + 2) - 3 and g(x) = 3(-2|x + 1| - 1) + 3 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) = (-3(5((-1x)/(-1x + 2)) - 4))/(-3(5((-1x)/(-1x + 2)) - 4) + 2) and g(x) = 3(1|x - 2| - 3)^2 - 3(1|x - 2| - 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) = -4(2|(-1|x + 2|) + 3|)^2 - 2(2|(-1|x + 2|) + 3|) - 2 and g(x) = 1cos(3π(1|x + 2|)) - 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) = 2|(2|(-1e^(1x) + 2) + 2| - 1) - 1| + 3 and g(x) = 2(-2sin(1πx + 3))^2 + 2(-2sin(1πx + 3)) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(1((-2x + 1)/(-1x - 1))^2 - 5) + 2| + 2 and g(x) = 1sin(3π(1|x - 2| + 3) + 1) - 1 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) = 1sin(3π(1|((-1x + 1)/(-2x - 3)) - 1| + 1) - 2) - 3 and g(x) = -2e^(-1(-3x - 5)) + 3 where -10 ≤ x ≤ 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((1(-2e^(-2x) - 3) + 3)/(2(-2e^(-2x) - 3) - 1)) and g(x) = -1|(2|x + 2| - 3) + 2| + 2 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) = (1(-2e^(-2(1x^2 - 1x - 2)) - 3) + 1)/(3(-2e^(-2(1x^2 - 1x - 2)) - 3) - 1) and g(x) = 1cos(2π(2x^2 - 4x + 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 808 | MATH |
[
{
"content": "How many solutions does the equation -5((-2(-1|x - 2| + 2) - 2)/(-1(-1|x - 2| + 2) - 3)) - 4 = -2sin(3π(5x^2 + 3x + 3) - 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 33 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2e^(1(-1|(-2|x + 2| + 1) + 1| - 3)) + 1 and g(x) = 2cos(2π(2|x| + 1) + 1) - 1 in the range -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) = -2(1|(-1x - 1) + 3| - 1)^2 + 2(1|(-1x - 1) + 3| - 1) + 5 and g(x) = 3cos(1π(1x^2 - 5x + 3) - 2) + 2 intersect 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) = -4(-3sin(1π(-3x^2 - 3x - 3) + 1) + 1)^2 + 1(-3sin(1π(-3x^2 - 3x - 3) + 1) + 1) + 4 and g(x) = 3|(3sin(3πx + 3) - 2) + 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 86 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(1sin(3π(-5x^2 - 1x + 3) + 2) + 1) - 1 and g(x) = 4(-2|x - 2| - 1)^2 + 5(-2|x - 2| - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(-2(3cos(3πx - 3) + 3)^2 + 4(3cos(3πx - 3) + 3) + 5) + 3| + 3 and g(x) = -1e^(-1(-2|x - 3| - 1)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-1(-4x^2 - 3x + 4)^2 - 1(-4x^2 - 3x + 4) + 3) + 2)/(3(-1(-4x^2 - 3x + 4)^2 - 1(-4x^2 - 3x + 4) + 3) + 2) and g(x) = (-2(2e^(1x) - 2) + 2)/(-2(2e^(1x) - 2) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your ... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-1|(2sin(2πx + 3) - 1) - 2| + 2) + 2)/(-1(-1|(2sin(2πx + 3) - 1) - 2| + 2) + 3) and g(x) = (-1(-2|x| + 1) - 1)/(-1(-2|x| + 1) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 28 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(3cos(3π(1|x| + 2) + 1)) + 2| + 3 and g(x) = (-3(5x) - 2)/(-2(5x) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 119 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2cos(2π(4(-3x - 3) - 4) + 1) - 3 and g(x) = 3cos(2π(-1|x - 2| + 2) - 1) - 1 in the range -10 ≤ x ≤ 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|(-5(2sin(1πx) + 1)^2 + 3(2sin(1πx) + 1) + 3)| + 3 and g(x) = 1(4x^2 + 4x - 4) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(2π(2|(-1x^2 - 1x + 4) + 2| - 3) - 1) - 3 and g(x) = -3e^(1(1|x + 3| - 1)) - 1 where -10 ≤ x ≤ 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) = 5((-2(2cos(3πx - 3) - 3) - 3)/(2(2cos(3πx - 3) - 3) + 1))^2 + 4((-2(2cos(3πx - 3) - 3) - 3)/(2(2cos(3πx - 3) - 3) + 1)) - 4 and g(x) = -1|(-2|x + 1| - 2) - 1| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3((-2(-1sin(2πx + 1) + 2) - 1)/(3(-1sin(2πx + 1) + 2) + 1)) - 1)/(-1((-2(-1sin(2πx + 1) + 2) - 1)/(3(-1sin(2πx + 1) + 2) + 1)) - 1) and g(x) = -5(-2cos(1πx - 3) - 2) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\box... | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(5(-3|x + 2| + 3)^2 + 1(-3|x + 2| + 3) - 5)^2 - 3(5(-3|x + 2| + 3)^2 + 1(-3|x + 2| + 3) - 5) - 4 and g(x) = 1sin(1π((-3x - 1)/(2x + 1)) - 3) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 8 | MATH |
[
{
"content": "How many solutions does the equation 2|((1(-5x - 1) - 2)/(2(-5x - 1) - 3))| = -2cos(3π(-2|x + 1| - 1) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 118 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(-1|(1sin(3πx + 1)) + 1| + 1) - 1 and g(x) = 3(-2sin(2πx)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many solutions does the equation -2|((-1(-4x + 2) + 2)/(-1(-4x + 2) - 2)) - 2| = -5(2e^(1x) - 2)^2 + 1(2e^(1x) - 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 1|(-2(2cos(1πx + 1) + 1) + 1)| = -2e^(-2(1|x + 3| - 1)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 36 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2cos(2π((-3(-3x^2 + 4x + 2) + 3)/(-3(-3x^2 + 4x + 2) + 2)) + 1) - 2 and g(x) = -3(-2cos(3πx + 3) - 1) - 5 where -10 ≤ x ≤ 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) = 1sin(3π(3(-5x^2 - 2x - 1) + 4) - 3) - 1 and g(x) = -3((3x - 2)/(-3x)) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1177 | MATH |
[
{
"content": "How many solutions does the equation 1(-3|((3x + 3)/(-3x + 4)) - 1| + 2)^2 + 2(-3|((3x + 3)/(-3x + 4)) - 1| + 2) - 5 = -5(1e^(-1x) - 1)^2 + 5(1e^(-1x) - 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) = (-2(3sin(2π(1x^2 - 5x + 2) - 1) + 3) + 2)/(1(3sin(2π(1x^2 - 5x + 2) - 1) + 3) + 3) and g(x) = -2(2|x - 1| - 1)^2 + 1(2|x - 1| - 1) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3((2(-2sin(3πx - 1) + 1) - 3)/(3(-2sin(3πx - 1) + 1)))^2 - 5((2(-2sin(3πx - 1) + 1) - 3)/(3(-2sin(3πx - 1) + 1))) + 2 and g(x) = 1(2|x - 1| - 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 42 | MATH |
[
{
"content": "How many solutions does the equation 1(-3|(-2sin(1πx - 3) + 1)| + 3)^2 + 2(-3|(-2sin(1πx - 3) + 1)| + 3) - 3 = 3|(2x^2 + 4x - 2) - 2| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(4(-1sin(3πx) + 2) - 5) + 2| + 1 and g(x) = (-2(-4x - 2) - 1)/(1(-4x - 2)) where -10 ≤ x ≤ 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) = 3(-2(1cos(2πx + 2) + 3)^2 + 1(1cos(2πx + 2) + 3))^2 + 1(-2(1cos(2πx + 2) + 3)^2 + 1(1cos(2πx + 2) + 3)) + 4 and g(x) = 3(4x^2 + 4x + 2)^2 - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(-1((2x - 2)/(-2x + 3)) - 3) + 1)/(3(-1((2x - 2)/(-2x + 3)) - 3) - 2) and g(x) = 1|(-1sin(3πx - 3)) + 2| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 61 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|((-2(-1|x - 2| + 1) - 2)/(2(-1|x - 2| + 1) - 1)) + 3| + 1 and g(x) = 2((-2x - 1)/(1x))^2 + 1((-2x - 1)/(1x)) 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 4(-3sin(2π(-2x^2 - 1x - 1))) + 1 = 1e^(-2((-1x + 2)/(-2x + 3))) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 798 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-3(-2sin(2πx - 3) - 1)^2 - 3(-2sin(2πx - 3) - 1) + 2)^2 - 5(-3(-2sin(2πx - 3) - 1)^2 - 3(-2sin(2πx - 3) - 1) + 2) - 1 and g(x) = (1(-2|x - 3| - 2) + 2)/(-1(-2|x - 3| - 2) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex forma... | 120 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(2((-3x + 2)/(-2x - 1))^2 + 5((-3x + 2)/(-2x - 1)) - 1) + 3| + 1 and g(x) = -2sin(2π(-1x^2 + 5x + 4)) - 3 in the range -10 ≤ x ≤ 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) = -1(3sin(2π(1|x + 2| - 2) + 1))^2 + 4(3sin(2π(1|x + 2| - 2) + 1)) - 2 and g(x) = -3cos(3π(-4x^2 - 2x + 3) - 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 705 | MATH |
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