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) = 1(-1sin(2π(-3x - 4) - 1) - 3)^2 + 4(-1sin(2π(-3x - 4) - 1) - 3) + 1 and g(x) = (2((-2x - 2)/(-2x + 3)) + 1)/(-1((-2x - 2)/(-2x + 3)) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 25 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4((3(-2x^2 + 3x - 1) - 1)/(3(-2x^2 + 3x - 1) + 2)) + 5 and g(x) = -3sin(3π(-2x - 5) + 2) + 1 intersect 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((-3(3e^(2x) + 1) + 3)/(2(3e^(2x) + 1) + 2))^2 + 1((-3(3e^(2x) + 1) + 3)/(2(3e^(2x) + 1) + 2)) + 2 and g(x) = -1sin(3π(-4x - 5) - 1) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 47 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((3(1|x - 2|) - 1)/(-3(1|x - 2|))) - 4 and g(x) = -3sin(3π((-2x)/(-1x + 3)) + 2) intersect 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(3(-2x^2 + 5x + 1) + 1)^2 - 4(3(-2x^2 + 5x + 1) + 1) - 4 and g(x) = -1cos(1π(-2|x + 3|) + 1) + 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) = (-1(-1sin(2π(-3x^2 - 5x - 1) - 2) - 3) + 2)/(2(-1sin(2π(-3x^2 - 5x - 1) - 2) - 3) + 2) and g(x) = -3sin(3π((-2x - 1)/(1x)) + 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 89 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(1cos(1π(-5x^2 + 3x - 3) + 2) - 1) + 1)/(-2(1cos(1π(-5x^2 + 3x - 3) + 2) - 1) + 1) and g(x) = -4((3x + 3)/(-2x + 3)) - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 176 | MATH |
[
{
"content": "How many solutions does the equation 1sin(2π(-1|(-2x + 1) - 3| - 1) - 2) - 2 = 1cos(2π(-5x - 4) + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 102 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(-2|(3|x - 2| - 1)| - 1) - 2)/(1(-2|(3|x - 2| - 1)| - 1) - 3) and g(x) = 3|(3|x - 1| - 3) - 1| - 3 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|(-5x^2 - 5x + 5) + 1| - 1)^2 - 4(1|(-5x^2 - 5x + 5) + 1| - 1) and g(x) = 3|(-1cos(1πx + 1)) - 2| - 2 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 3e^(2(-3|(5x + 3) + 1| + 1)) - 2 = (-1(2|x + 1| - 2) - 1)/(2(2|x + 1| - 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) = (-3((-3(-1x^2 + 4x) + 3)/(-3(-1x^2 + 4x) - 3)) - 1)/(3((-3(-1x^2 + 4x) + 3)/(-3(-1x^2 + 4x) - 3)) - 2) and g(x) = (3(3x^2 + 5x + 1) - 2)/(2(3x^2 + 5x + 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(1sin(3π(3x^2 + 4x)) - 3) + 3)/(-3(1sin(3π(3x^2 + 4x)) - 3) - 1) and g(x) = 2sin(1π(-2x - 2) - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 85 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(-3(2|x - 2| - 3)^2 + 5(2|x - 2| - 3) + 3) - 1| + 1 and g(x) = (-3(-1sin(2πx - 3) + 2) + 2)/(-2(-1sin(2πx - 3) + 2) + 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) = 2sin(2π((-1(-1|x + 2| - 2) - 1)/(2(-1|x + 2| - 2) + 2)) + 3) - 3 and g(x) = -5(-3x^2 - 1x + 4)^2 - 2(-3x^2 - 1x + 4) - 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) = -1cos(2π(-2|(-1x - 4)| - 1) + 1) - 1 and g(x) = (1(-4x^2 - 3x - 4) + 1)/(-2(-4x^2 - 3x - 4) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "How many solutions does the equation -4(-3cos(2π(3|x| - 1) + 2))^2 - 3(-3cos(2π(3|x| - 1) + 2)) - 3 = -3e^(1(2|x - 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": "Determine the number of intersection points between the functions f(x) = 4(1((-2x + 2)/(-3x - 1))^2) and g(x) = -5(-2sin(3πx - 1)) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 56 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2e^(2((3((1x + 2)/(2x + 3)) - 3)/(2((1x + 2)/(2x + 3)) + 1))) and g(x) = -3sin(2π(3|x + 1| - 2) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 116 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-2|((-2x)/(1x + 1)) + 3| - 2) - 3 and g(x) = (-1(-1|x - 2| + 2) - 3)/(3(-1|x - 2| + 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) = 3cos(3π(2((-1x + 2)/(-1x + 2))^2 - 3((-1x + 2)/(-1x + 2)) + 5) - 2) + 2 and g(x) = 1sin(2π(-5x^2 + 5x - 3) - 1) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1494 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(3sin(3π(1|x + 3| - 1) - 2)) - 3)/(3(3sin(3π(1|x + 3| - 1) - 2)) + 3) and g(x) = 2((-2x - 2)/(-1x + 3))^2 + 1((-2x - 2)/(-1x + 3)) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 38 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(1π(1|(2x - 4) - 2|) - 3) - 1 and g(x) = (-3((-1x)/(-1x + 2)) + 3)/(-1((-1x)/(-1x + 2)) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 37 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(-3cos(1π(1|x - 1| - 3) + 3) - 1) - 1| - 1 and g(x) = 5(1|x + 2|) where -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(2(3cos(3πx + 2) - 1) - 2))/(-2(2(3cos(3πx + 2) - 1) - 2) - 2) and g(x) = -5(2cos(3πx + 3) - 3)^2 - 1(2cos(3πx + 3) - 3) intersect in the interval [-10, 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) = (-2(1(1x^2 + 5x - 1)^2 + 5(1x^2 + 5x - 1) + 2) + 1)/(1(1(1x^2 + 5x - 1)^2 + 5(1x^2 + 5x - 1) + 2)) and g(x) = 2|(1cos(3πx - 1) + 2)| 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|(2sin(1π(4x - 4) - 1) + 1) + 1| - 1 = (3(-5x^2 - 5x + 5))/(-2(-5x^2 - 5x + 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) = -3|(2e^(2(2|x + 3| - 2))) - 3| + 3 and g(x) = (-2(3x - 2) - 1)/(-2(3x - 2) - 1) 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) = -5(-1(-1x - 1)^2 - 5(-1x - 1) - 2) + 5 and g(x) = (-2(3cos(3πx - 1)) - 3)/(1(3cos(3πx - 1)) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-2|(1|x + 1| - 3) - 3| + 2) + 3 and g(x) = -1cos(3π(1x + 4) + 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) = -2sin(3π(-3|(-5x^2 - 3x + 4) - 2| + 2) - 3) - 2 and g(x) = (-2(1sin(2πx + 1) - 3) + 1)/(1(1sin(2πx + 1) - 3) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 864 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(3sin(3π(-2|x - 3|)))^2 - 1(3sin(3π(-2|x - 3|))) - 4 and g(x) = -2(-5x + 5) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 53 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(1(3sin(2πx - 3) - 3) + 1) + 3)/(3(1(3sin(2πx - 3) - 3) + 1)) and g(x) = -2((3x + 2)/(-2x + 3))^2 - 2((3x + 2)/(-2x + 3)) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 36 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(2π(3|(3x - 5) + 3| + 1) + 2) - 3 and g(x) = -2|(-2x - 3) + 1| - 2 in the range -10 ≤ x ≤ 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) = (-1(1|(5x) + 1| + 2) - 1)/(-1(1|(5x) + 1| + 2) + 3) and g(x) = 3sin(1π(4x + 1) - 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 74 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(2e^(-2(1x^2 - 4x - 4)) - 3))/(-3(2e^(-2(1x^2 - 4x - 4)) - 3) + 4) and g(x) = -2cos(2π(-3|x - 1| + 1) + 2) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 118 | MATH |
[
{
"content": "How many solutions does the equation 4(-1|((-3x + 3)/(1x - 2)) + 3| + 3)^2 + 4(-1|((-3x + 3)/(1x - 2)) + 3| + 3) - 4 = (-2((-3x + 1)/(-3x - 3)))/(3((-3x + 1)/(-3x - 3)) + 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 points of intersection exist between the functions f(x) = 1((1(-1x - 1) + 2)/(2(-1x - 1) + 1))^2 - 3((1(-1x - 1) + 2)/(2(-1x - 1) + 1)) - 4 and g(x) = (1(3|x| - 2) - 2)/(1(3|x| - 2) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 4 | MATH |
[
{
"content": "How many solutions does the equation 5(-4(2sin(3πx + 1)) + 5) - 5 = -3(-3sin(2πx + 1) - 3)^2 + 5(-3sin(2πx + 1) - 3) + 4 have for x in the interval [-10, 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) = (-2(-5(-3x^2 - 5x + 2)^2 + 4(-3x^2 - 5x + 2) + 3) - 1)/(1(-5(-3x^2 - 5x + 2)^2 + 4(-3x^2 - 5x + 2) + 3)) and g(x) = 1(1e^(2x))^2 + 5(1e^(2x)) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(-2(-1x^2 + 2) - 2) + 1)/(2(-2(-1x^2 + 2) - 2) + 3) and g(x) = -2(3cos(1πx + 2)) in the range -10 ≤ x ≤ 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) = -1sin(3π((-1(1x^2 - 2x - 2) - 2)/(-2(1x^2 - 2x - 2) + 1))) - 2 and g(x) = -2cos(2π((3x + 3)/(3x + 3)) + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 52 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2((2(2cos(3πx + 1)) + 2)/(3(2cos(3πx + 1)) - 2)) - 2)/(-1((2(2cos(3πx + 1)) + 2)/(3(2cos(3πx + 1)) - 2))) and g(x) = 1|((-3x + 3)/(-1x + 3)) - 1| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your an... | 11 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(2((-2x - 3)/(-2x + 1)) - 2) - 3)/(-2(2((-2x - 3)/(-2x + 1)) - 2) + 3) and g(x) = 2sin(2π((2x + 3)/(-1x)) - 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 82 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(-3(5x^2 + 2x - 2) - 3) + 2| - 1 and g(x) = 1((1x - 2)/(1x - 3)) + 5 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) = 1(-3sin(2π(-5x + 4) + 1) + 1)^2 + 1(-3sin(2π(-5x + 4) + 1) + 1) - 5 and g(x) = (-1(-3x + 1) - 1)/(3(-3x + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 199 | MATH |
[
{
"content": "How many solutions does the equation 2sin(2π((1((-3x + 3)/(-2x + 3)) + 3)/(2((-3x + 3)/(-2x + 3)))) - 1) + 1 = (-3(5x^2 + 4x + 5) - 3)/(3(5x^2 + 4x + 5) + 3) have for x 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 2|(3cos(3π((-2x - 2)/(-1x - 2))) + 2) - 2| = -3sin(1π((1x - 1)/(-1x + 2)) - 2) + 1 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) = 1((-2(-4x^2 - 5x - 4) + 3)/(-1(-4x^2 - 5x - 4) + 3)) - 4 and g(x) = 3cos(1π(1x + 5) + 1) - 1 in the range -10 ≤ x ≤ 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) = -1(-3(1cos(3πx - 1)) + 2) + 3 and g(x) = (-1(-3cos(1πx + 2) - 2) + 2)/(-3(-3cos(1πx + 2) - 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(1π(-3|(1x^2 + 4x - 3)| + 3) - 3) + 3 and g(x) = (2(1|x| + 1) + 2)/(1(1|x| + 1) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 614 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1cos(2π(3(1|x - 3| - 3) - 5) + 2) + 3 and g(x) = 2|(1|x - 2| - 3) - 1| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-3|(-2cos(3πx + 2))| - 1) + 2)/(1(-3|(-2cos(3πx + 2))| - 1) + 1) and g(x) = 1(-2sin(3πx + 3))^2 - 4(-2sin(3πx + 3)) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 60 | MATH |
[
{
"content": "How many solutions does the equation (-2(2|(-2e^(-1x) - 1) + 2| - 2) - 2)/(2(2|(-2e^(-1x) - 1) + 2| - 2) + 2) = 2sin(1π((-2x + 1)/(-1x - 1)) + 2) - 3 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 3cos(1π(-1|(3x - 2) - 2| + 3) + 2) + 3 = -2|(1|x - 3| + 1) - 2| + 3 have for x 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) = -2sin(2π(-2|(1x + 3) + 2| - 3) + 1) + 1 and g(x) = 2e^(2((2x - 3)/(-3x + 4))) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 79 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1sin(2π(-2|(-3x^2 - 3x - 3) - 3| + 1) + 1) - 3 and g(x) = -4((-1x)/(-2x + 3)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1140 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(2|(-1x + 2) + 1| - 3)^2 - 5(2|(-1x + 2) + 1| - 3) - 2 and g(x) = 5((1x)/(-3x - 3)) + 2 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) = -3(3sin(3π(-4x - 3) + 2) + 1) - 4 and g(x) = 2(3e^(2x) - 2)^2 + 3(3e^(2x) - 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 119 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|((-1(-1x^2 - 4x + 2) + 1)/(-3(-1x^2 - 4x + 2) + 1)) + 2| + 3 and g(x) = 2(-2e^(1x) + 3)^2 - 3(-2e^(1x) + 3) - 2 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) = -5(1|((3x - 3)/(-1x + 2)) + 3| - 3)^2 + 3(1|((3x - 3)/(-1x + 2)) + 3| - 3) + 1 and g(x) = -5(3cos(2πx + 2) + 1)^2 - 1(3cos(2πx + 2) + 1) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 44 | MATH |
[
{
"content": "How many solutions does the equation -1cos(3π(5(-1x - 2)^2 - 3(-1x - 2) - 5) + 3) = (1(-1x))/(-3(-1x) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1560 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(2(1|x + 2| - 2)^2 + 2(1|x + 2| - 2) - 5) - 3)/(2(2(1|x + 2| - 2)^2 + 2(1|x + 2| - 2) - 5) - 1) and g(x) = (-1(3|x| + 2) + 2)/(3(3|x| + 2) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1|(1((-2x + 3)/(3x - 2))^2 - 4((-2x + 3)/(3x - 2)) - 1) + 2| - 1 = 3(-1|x - 2|)^2 + 4(-1|x - 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) = -1(2cos(2π(3|x - 1| + 2)) + 1)^2 + 3(2cos(2π(3|x - 1| + 2)) + 1) - 2 and g(x) = (1(1sin(2πx) - 2) + 2)/(-1(1sin(2πx) - 2) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 200 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-3|((1x + 1)/(3x - 2)) - 2| - 1) + 2| + 3 and g(x) = -3(-1sin(1πx - 3) + 1)^2 - 2(-1sin(1πx - 3) + 1) 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 -3((-2(-3sin(1πx - 1) - 2) - 3)/(2(-3sin(1πx - 1) - 2) + 1))^2 - 5((-2(-3sin(1πx - 1) - 2) - 3)/(2(-3sin(1πx - 1) - 2) + 1)) + 1 = (-3(1sin(1πx + 3) + 2) + 1)/(-2(1sin(1πx + 3) + 2) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex forma... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(1cos(1π(-2x - 5) + 1)) and g(x) = (-2(2|x - 1|) + 2)/(3(2|x - 1|) - 1) in the range -10 ≤ x ≤ 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) = 1((-1(-1cos(3πx - 2) + 3) - 1)/(-1(-1cos(3πx - 2) + 3) + 2)) + 1 and g(x) = 1(1sin(3πx - 2) - 2)^2 + 1(1sin(3πx - 2) - 2) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(2cos(3π((-3x + 1)/(2x - 1)) + 2) - 2) - 2)/(1(2cos(3π((-3x + 1)/(2x - 1)) + 2) - 2)) and g(x) = -2cos(3π((-2x - 3)/(2x + 2)) + 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 48 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(2cos(2π(-3x^2 - 1x - 3)) - 1) - 1 and g(x) = 2(-1e^(-1x) - 1) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(2|(-1cos(2πx + 1) + 3) + 1| + 1) - 2| + 2 and g(x) = -1|(3|x - 3| - 1) - 1| + 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) = -1cos(2π(1(1x^2 - 4x - 1)^2 - 2) - 2) - 3 and g(x) = 1cos(2π(-4x + 5)) - 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 -3cos(3π(-3|(-2x) - 2| + 1) + 1) - 1 = (-2(-1cos(3πx - 3) - 2) - 1)/(3(-1cos(3πx - 3) - 2) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 360 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(1π(1|((-2x - 3)/(1x - 2)) + 2| + 3)) and g(x) = (-2((-2x - 2)/(-2x - 2)))/(-1((-2x - 2)/(-2x - 2)) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 94 | MATH |
[
{
"content": "How many solutions does the equation 1(1(2x) - 3) + 2 = 1sin(1π((2x + 2)/(-3x - 2)) - 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-1sin(2π(3|x + 2| - 3) + 2) + 3)^2 - 2(-1sin(2π(3|x + 2| - 3) + 2) + 3) + 2 and g(x) = -2(-3sin(2πx + 3) - 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "How many solutions does the equation -4(2sin(2π(-3|x - 2| - 1) + 3) + 1)^2 + 1(2sin(2π(-3|x - 2| - 1) + 3) + 1) + 4 = 1sin(3π(2x^2 + 4x - 2)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 120 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(3π(3(2|x - 2| - 2)^2 - 5(2|x - 2| - 2)) + 2) + 2 and g(x) = (3(2x + 2) - 3)/(-3(2x + 2) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 304 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4((-2(1|x - 3| - 2) - 3)/(-1(1|x - 3| - 2) + 3)) - 2 and g(x) = 4(-2|x - 3| + 2) + 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) = 1(1sin(1π((3x - 3)/(3x - 2)) - 3) + 2) - 1 and g(x) = (-2(-2|x| + 3) + 1)/(-3(-2|x| + 3) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(-3sin(1π((2x - 3)/(-2x + 3)) + 3) + 1)^2 + 3(-3sin(1π((2x - 3)/(-2x + 3)) + 3) + 1) - 2 and g(x) = 2(1cos(1πx + 1) + 3) intersect 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) = 3|(-2cos(3π(1x^2 - 3x - 5) + 2) - 1) - 1| and g(x) = -3(3x^2 - 4x - 4) + 3 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 -2(-4(-1x) + 2)^2 + 4(-4(-1x) + 2) - 3 = 2sin(2π(-4x + 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": "How many solutions does the equation 5(1|(2sin(3πx + 3) + 3) - 3| + 3) + 5 = 4(2cos(3πx) + 3)^2 - 4(2cos(3πx) + 3) + 2 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 -2sin(3π(-1(-2x^2 - 1x + 5)) + 2) - 3 = (1(1e^(-2x) + 3) - 2)/(-1(1e^(-2x) + 3) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 147 | MATH |
[
{
"content": "How many solutions does the equation -1e^(2(2(4x)^2 + 5(4x))) - 1 = -2sin(3π(2|x - 3| + 3) + 1) - 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) = -4(-1(-2cos(3πx + 2) - 1)^2 - 4(-2cos(3πx + 2) - 1) - 5)^2 - 5(-1(-2cos(3πx + 2) - 1)^2 - 4(-2cos(3πx + 2) - 1) - 5) - 1 and g(x) = -1|(-1cos(3πx - 2) - 1) - 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{You... | 120 | MATH |
[
{
"content": "How many solutions does the equation -3sin(3π((-2((2x)/(-3x + 4)) - 1)/(1((2x)/(-3x + 4)))) - 3) + 3 = 1(3|x - 2| - 2) - 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(-2((-1x - 3)/(-2x - 3))^2 - 1) + 4 = -2sin(2π(1|x + 1| - 3) - 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 38 | MATH |
[
{
"content": "How many solutions does the equation (2((-1(-3cos(3πx - 1) + 3))/(2(-3cos(3πx - 1) + 3) - 3)) + 2)/(-3((-1(-3cos(3πx - 1) + 3))/(2(-3cos(3πx - 1) + 3) - 3))) = -1sin(1π((-3x)/(1x + 3))) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 63 | MATH |
[
{
"content": "How many solutions does the equation -1(1cos(1π(5x^2 + 2) + 1) - 3) - 1 = 2sin(3π((3x + 1)/(3x - 2)) - 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 125 | MATH |
[
{
"content": "How many solutions does the equation (1(4(1x) - 1) - 3)/(2(4(1x) - 1) + 1) = -2cos(3π(4x - 4)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 235 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(-2sin(3π(-1x - 3))) - 3)/(3(-2sin(3π(-1x - 3)))) and g(x) = (3(-5x - 1) - 2)/(1(-5x - 1) + 2) 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) = -1cos(2π((3((2x + 2)/(3x + 3)) + 1)/(2((2x + 2)/(3x + 3)) + 3)) + 3) and g(x) = -4(-2cos(2πx - 1) - 1) + 1 intersect in the interval [-10, 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(2cos(1π(1|x - 1| - 1) - 2) + 1)^2 - 4(2cos(1π(1|x - 1| - 1) - 2) + 1) + 3 and g(x) = 3e^(1((-1x + 2)/(2x - 1))) + 1 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) = 5(-2cos(2π(-1|x + 2| + 2)) + 3)^2 - 2(-2cos(2π(-1|x + 2| + 2)) + 3) - 4 and g(x) = (2(-3|x - 3| - 3))/(2(-3|x - 3| - 3) + 1) intersect 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 3e^(1(-2|(-2|x + 1| + 2) + 1| - 3)) + 3 = (1(-1cos(1πx + 2) + 1) + 3)/(-1(-1cos(1πx + 2) + 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": "Find the number of times the graphs of f(x) = (-3(-3sin(3π((-1x - 3)/(2x - 1)) - 3)) - 2)/(1(-3sin(3π((-1x - 3)/(2x - 1)) - 3))) and g(x) = 5(2|x - 1|) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(5(2|x - 3| - 1) - 5) + 5 and g(x) = 4(3cos(2πx + 1) - 1) - 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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