messages listlengths 1 1 | ground_truth stringclasses 134 values | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation 1cos(3π((-2x - 3)/(2x - 1)) - 1) - 3 = (2x)/(1x - 2) have for x in the interval [-10, 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) = -1sin(3π((-1x - 2)/(-3x - 1)) - 1) and g(x) = -1x^2 - 2x 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) = 2cos(3π(4x + 2) + 3) + 3 and g(x) = 4x - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "How many solutions does the equation -4(2cos(2πx + 1) - 2) - 3 = 1x^2 - 1x + 4 have for x in the interval [-10, 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(-3sin(1πx - 2) - 1) - 4 and g(x) = -3|x - 3| + 2 where -10 ≤ x ≤ 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) = -4(-2cos(2πx + 1) - 2) - 5 and g(x) = -3x^2 + 3x + 3 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|(2sin(3πx - 2) - 3) + 3| + 2 and g(x) = 3|x| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3sin(3π(4x)) + 1 and g(x) = -1|x + 1| - 1 intersect in the interval [-10, 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(2x^2 + 5x - 5) - 3)/(1(2x^2 + 5x - 5)) and g(x) = 3x^2 - 5x - 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) = 5(1cos(1πx - 3)) + 1 and g(x) = -3x + 2 in the range -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) = -1cos(2π(3x^2 + 1x + 2) + 1) - 2 and g(x) = 3x + 1 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) = 2cos(3π(-1|x - 3| - 3) + 3) + 1 and g(x) = -1|x - 1| 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) = -3(-2sin(2πx + 2) + 3)^2 + 2(-2sin(2πx + 2) + 3) - 2 and g(x) = -5x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(2π(-3|x - 3| - 3) + 3) - 1 and g(x) = -2|x| - 3 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) = -4(-2sin(3πx + 1) - 1) - 1 and g(x) = 3x^2 + 1x + 4 intersect 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) = 3cos(1π(2x^2 - 3x + 3) - 2) + 1 and g(x) = (3x)/(-2x - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 386 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(2π(-1|x + 3| - 3) + 2) + 3 and g(x) = (-1x + 3)/(-3x - 3) intersect in the interval [-10, 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(2πx - 3) - 2) - 3)/(1(3sin(2πx - 3) - 2) + 3) and g(x) = (-1x + 3)/(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) = 3sin(2π(1x^2 - 3x + 4) + 2) - 3 and g(x) = 2x in the range -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) = (1(-2sin(3πx - 2) - 3) - 2)/(2(-2sin(3πx - 2) - 3)) and g(x) = 1|x| - 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) = -1sin(2π(5x + 3) + 3) - 2 and g(x) = -5x^2 - 2x - 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 -3cos(2π(1x^2 - 5x) - 2) + 2 = (-3x + 1)/(2x - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1sin(2π(4x + 4) - 1) - 3 and g(x) = 3|x - 1| - 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) = -3(3sin(1πx + 1) + 1) - 3 and g(x) = (-1x + 3)/(-2x - 3) 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) = -1(-2sin(2πx - 2) - 2) + 1 and g(x) = 2|x - 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2sin(3π((3x)/(-3x - 3)) - 1) - 1 and g(x) = 1x - 1 intersect in the interval [-10, 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) = -2(3cos(3πx - 1) + 2) + 1 and g(x) = 1x^2 + 3x + 4 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) = 5(-3cos(3πx + 3)) and g(x) = -1x + 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) = -2|(3cos(1πx + 1)) + 1| + 2 and g(x) = (2x + 1)/(2x + 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) = 1cos(2π(3x^2 + 1x - 1) - 2) and g(x) = -3x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "How many solutions does the equation (-3(3sin(3πx) - 2) - 1)/(-1(3sin(3πx) - 2) + 2) = (-1x - 2)/(1x + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 59 | MATH |
[
{
"content": "How many solutions does the equation -2sin(2π(5x - 4)) + 1 = 1x^2 + 5x + 1 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) = -3sin(2π((1x + 2)/(3x + 3)) - 2) and g(x) = 1|x + 2| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 26 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(2sin(1πx - 3) + 1) + 3 and g(x) = (3x - 3)/(-2x + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(3sin(3πx + 1))^2 - 2(3sin(3πx + 1)) + 3 and g(x) = -1x + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 51 | MATH |
[
{
"content": "How many solutions does the equation 1(1sin(3πx - 2) - 3)^2 + 1(1sin(3πx - 2) - 3) + 1 = 2|x + 2| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1cos(2π(-3|x + 3| - 3) - 3) + 2 and g(x) = 1x - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "How many solutions does the equation -1|(-2cos(3πx) + 1)| - 2 = -5x^2 - 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": "Find the number of times the graphs of f(x) = (-3(2sin(1πx + 2) - 2))/(-1(2sin(1πx + 2) - 2) - 3) and g(x) = 1x - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(3sin(3πx + 2) + 3) - 1| + 2 and g(x) = 4x^2 - 2x + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-3sin(1πx - 1) + 2)^2 - 1(-3sin(1πx - 1) + 2) - 1 and g(x) = (3x)/(-1x + 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) = -4(-2cos(3πx + 2) - 1)^2 + 1(-2cos(3πx + 2) - 1) - 1 and g(x) = 4x^2 - 2x - 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) = -1(1cos(2πx + 1) - 3)^2 + 1(1cos(2πx + 1) - 3) - 2 and g(x) = 5x - 4 in the range -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) = -2cos(3π(-3|x + 2| + 2) + 1) - 1 and g(x) = -3x where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3sin(3π(-2|x - 2| - 2) - 3) - 1 and g(x) = -4x^2 + 1x + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3sin(2π((-3x - 3)/(-3x + 2)) + 3) - 1 and g(x) = -2x^2 + 5x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 63 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(-1x^2 + 4x + 3) - 1| + 2 and g(x) = -1x - 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 2sin(3π(2x + 3) + 2) = (-1x + 2)/(1x + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 81 | MATH |
[
{
"content": "How many solutions does the equation 2cos(3π(1x^2 - 4x - 2) - 1) + 1 = 1x^2 - 1x + 2 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) = -1sin(3π(1x^2)) + 2 and g(x) = -1x where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 23 | MATH |
[
{
"content": "How many solutions does the equation -1sin(3π(1x^2 + 3x + 2) + 3) + 3 = (3x + 1)/(3x - 2) 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) = -1cos(1π(2|x + 1| - 1) - 2) - 2 and g(x) = (3x - 1)/(-1x - 3) where -10 ≤ x ≤ 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) = -2cos(1π(-3x + 5) + 2) + 1 and g(x) = (-1x - 2)/(2x + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 59 | MATH |
[
{
"content": "How many solutions does the equation 2sin(2π(1|x - 2|)) - 1 = (1x + 1)/(-1x - 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 points of intersection exist between the functions f(x) = 2(2sin(1πx - 2) + 2)^2 - 4(2sin(1πx - 2) + 2) + 2 and g(x) = (-2x - 2)/(-2x) in the range -10 ≤ x ≤ 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) = -3|(3sin(3πx + 3) + 3) - 1| + 3 and g(x) = -5x^2 + 5x + 2 in the range -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) = (3(-1cos(3πx - 3) + 2) + 3)/(-1(-1cos(3πx - 3) + 2)) and g(x) = -1x^2 + 5x - 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) = -5(2x^2 + 4x - 1)^2 - 2(2x^2 + 4x - 1) + 1 and g(x) = -1|x - 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) = (-1(-3sin(2πx + 3) + 2) - 3)/(1(-3sin(2πx + 3) + 2)) and g(x) = 3x^2 - 2x + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1sin(1π(4x^2 - 5x - 1) - 3) - 1 and g(x) = -1x + 1 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) = 4(-2sin(1πx - 1) - 2) - 4 and g(x) = -3|x + 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(2sin(1πx - 3) - 1) - 1 and g(x) = 2|x - 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(-1sin(1πx + 3) + 2)^2 - 3(-1sin(1πx + 3) + 2) - 3 and g(x) = (3x - 2)/(-1x - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "How many solutions does the equation -3sin(1π((-1x - 2)/(1x)) - 2) - 2 = 1x^2 + 5x - 1 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 solutions does the equation (3(-2sin(3πx) - 1) + 3)/(-1(-2sin(3πx) - 1) + 2) = 1x^2 - 5x - 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 -2(3sin(3πx + 2)) + 5 = 1x^2 - 2x + 5 have for x 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) = -3cos(1π(2x - 4) - 3) and g(x) = 2x^2 - 3x + 1 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 (1(-5x^2 - 4x - 1) + 1)/(2(-5x^2 - 4x - 1) + 1) = 3x^2 - 3x + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(1π(-4x^2 - 1x + 1) + 1) + 3 and g(x) = 3|x - 3| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(3cos(1πx + 2) + 1)^2 - 3(3cos(1πx + 2) + 1) + 1 and g(x) = 1x^2 - 2x - 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) = -2cos(3π((-3x)/(3x + 2)) + 1) - 1 and g(x) = (3x + 3)/(-3x - 2) 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(-2cos(2πx + 2))^2 + 5(-2cos(2πx + 2)) - 2 = 1|x - 3| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(2π(3x + 3) + 3) and g(x) = 1x + 3 in the range -10 ≤ x ≤ 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) = -3cos(1π(-2x^2 + 3x + 1) - 1) + 3 and g(x) = -4x - 4 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 -3sin(1π(-1x - 4) + 3) + 1 = 3|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": "Determine the number of intersection points between the functions f(x) = -3sin(1π(-4x^2 - 1) + 2) - 2 and g(x) = (-1x - 2)/(-1x - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 257 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(3π(3|x - 2| + 1)) - 1 and g(x) = 1x + 4 in the range -10 ≤ x ≤ 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) = -2|(2|x + 3| - 2) + 1| + 1 and g(x) = (-2x - 1)/(-2x + 3) 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(-3cos(1πx + 3) - 2) + 5 = 4x^2 + 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(-1sin(3πx + 1) + 1))/(-2(-1sin(3πx + 1) + 1) - 3) and g(x) = (-1x - 2)/(1x - 2) intersect in the interval [-10, 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) = 3cos(1π(1x^2 + 2x + 3) - 3) - 1 and g(x) = 4x^2 - 3x + 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) = 3cos(2π(1x^2 - 2x - 3) + 3) - 3 and g(x) = -4x - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "How many solutions does the equation -2cos(3π(2x + 4) - 3) - 3 = (2x - 3)/(-2x + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3sin(3π((3x + 1)/(-2x + 1)) + 2) - 1 and g(x) = 3x - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(2cos(2πx - 2) - 1) - 3 and g(x) = -2x + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2cos(3π(-3|x + 1| + 3) - 1) - 2 and g(x) = -5x^2 - 5x - 2 where -10 ≤ x ≤ 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) = 3(1sin(3πx + 2) + 2) + 1 and g(x) = 4x - 2 intersect in the interval [-10, 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|(3x^2 - 3x - 5) + 3| + 3 and g(x) = 5x - 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 -2(-2sin(2πx - 1) - 1)^2 - 2(-2sin(2πx - 1) - 1) + 2 = 5x + 1 have for x in the interval [-10, 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) = -3cos(1π(3|x - 2| + 3) - 3) - 1 and g(x) = 1x - 4 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(-3sin(2πx - 2) - 3)^2 + 1(-3sin(2πx - 2) - 3) + 3 and g(x) = -4x + 5 intersect in the interval [-10, 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) = -3cos(2π(2|x + 1| + 3) + 1) - 2 and g(x) = -2|x + 1| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "How many solutions does the equation 1(-2sin(2πx + 3))^2 + 4(-2sin(2πx + 3)) + 3 = (-1x - 1)/(-3x + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 39 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-2sin(3πx - 1) - 1)^2 - 4(-2sin(3πx - 1) - 1) - 2 and g(x) = 3x^2 + 2x - 2 in the range -10 ≤ x ≤ 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) = 3(-2cos(2πx) - 2)^2 + 1(-2cos(2πx) - 2) + 5 and g(x) = -1x + 5 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) = 2sin(1π(-4x + 2) + 3) + 2 and g(x) = 1x^2 - 3 intersect 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) = 4(3cos(3πx - 2))^2 - 2(3cos(3πx - 2)) - 5 and g(x) = -5x^2 - 5x - 5 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(3sin(3πx - 2)) - 3 and g(x) = -3x + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(1cos(3πx - 2) + 3) - 1| + 1 and g(x) = -3x^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) = -3cos(2π(-3|x - 2| - 1) - 3) + 3 and g(x) = -1|x| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
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