messages listlengths 1 1 | ground_truth stringclasses 134 values | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(3sin(2πx - 1) + 3)^2 + 3(3sin(2πx - 1) + 3) + 3 and g(x) = -4x + 5 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) = -1(-5x^2 - 5x + 5)^2 - 5(-5x^2 - 5x + 5) + 4 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 1(3sin(1πx) + 2)^2 - 3(3sin(1πx) + 2) - 1 = -1|x + 2| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(3|x + 2| - 1) - 1)/(-1(3|x + 2| - 1) - 1) and g(x) = -1|x + 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) = 2cos(3π(5x)) + 2 and g(x) = 1x + 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) = -4(-3cos(3πx - 2) + 1)^2 - 1(-3cos(3πx - 2) + 1) - 2 and g(x) = -2x - 4 intersect in the interval [-10, 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|(-3sin(2πx + 2)) + 1| - 1 and g(x) = -1x^2 - 3x + 5 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) = -2sin(3π(5x^2 + 4x + 2) + 2) - 3 and g(x) = -5x^2 + 1x - 3 in the range -10 ≤ x ≤ 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) = (3(-2cos(3πx)))/(-3(-2cos(3πx)) + 4) and g(x) = -1x + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 52 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1sin(3π(-4x - 2) + 1) - 2 and g(x) = -2x - 3 in the range -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) = (1(-1x^2 + 1))/(-2(-1x^2 + 1) + 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"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-2cos(1πx)) + 2)/(3(-2cos(1πx)) - 2) and g(x) = -3x + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(3π(-2x + 3)) + 2 and g(x) = (3x - 3)/(-2x + 3) intersect in the interval [-10, 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) = -1sin(2π(-1|x - 1| - 2) + 2) - 2 and g(x) = -1x - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(3π(-3|x - 1| + 3) + 3) + 2 and g(x) = (-1x + 2)/(-3x - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 151 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(3π((2x - 1)/(-2x + 3)) - 2) - 3 and g(x) = -2x - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 26 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1sin(2π(2x^2 + 4x + 3) + 1) - 3 and g(x) = -1x^2 + 4x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(-3sin(2πx - 1) + 3) - 3 and g(x) = -1x + 5 in the range -10 ≤ x ≤ 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) = -3cos(3π((2x + 3)/(3x - 2)) - 2) - 1 and g(x) = 1x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 71 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(-3sin(3πx - 1) - 3)^2 + 5(-3sin(3πx - 1) - 3) + 3 and g(x) = 4x^2 + 5x - 5 intersect 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) = 3sin(1π(-3|x - 3| - 3)) + 1 and g(x) = 1|x + 1| - 3 intersect 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(-2cos(3πx))^2 + 2(-2cos(3πx)) - 5 and g(x) = (1x)/(-2x - 3) 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) = -1(-1x^2 - 1x + 5)^2 - 2(-1x^2 - 1x + 5) + 5 and g(x) = (2x + 2)/(2x + 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 -1cos(3π(1x^2 + 2x + 3) - 3) + 1 = 2x^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": "How many solutions does the equation 3(5x^2 - 2x - 1)^2 - 3 = (3x - 2)/(2x + 3) 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 solutions does the equation -2(1cos(3πx + 1) - 2) - 1 = -1x + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "How many solutions does the equation 1|(3sin(2πx - 3) - 3) + 3| + 2 = 3x^2 - 2x + 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 solutions does the equation 4(3cos(2πx + 2) - 2) + 3 = -2|x + 2| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 28 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2cos(1π(-2x^2 - 4x + 3) - 1) and g(x) = (-1x - 2)/(1x + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 402 | MATH |
[
{
"content": "How many solutions does the equation 1(1sin(2πx - 2) - 1)^2 + 3 = -1x + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3cos(2π(-5x - 2) - 1) - 3 and g(x) = 2x + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "How many solutions does the equation (2(-1sin(3πx + 1) - 1) + 3)/(-1(-1sin(3πx + 1) - 1) - 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"
}
] | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(4x - 4) + 3| + 3 and g(x) = -3x^2 + 5x - 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) = 2cos(2π((-1x - 2)/(2x - 3)) - 1) + 2 and g(x) = -3x + 5 where -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) = -1(-3sin(2πx + 3) - 1) + 2 and g(x) = (2x + 2)/(1x) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 37 | MATH |
[
{
"content": "How many solutions does the equation 1cos(2π(5x^2 + 3x + 4)) = (-2x - 3)/(-3x) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1076 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(2sin(1πx) + 2) + 2| + 3 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"
}
] | 11 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(-3|x - 1| + 2)^2 - 1(-3|x - 1| + 2) + 2 and g(x) = -1x - 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) = 3(-2sin(3πx - 1) + 3)^2 + 3(-2sin(3πx - 1) + 3) + 1 and g(x) = 3|x - 3| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 44 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1cos(2π(-5x^2 + 1x + 3)) + 1 and g(x) = 2|x - 3| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 106 | MATH |
[
{
"content": "How many solutions does the equation -1cos(2π(-4x^2 - 2) + 3) + 1 = (2x - 3)/(-1x - 1) have for x 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(3x^2 - 2x - 5))/(1(3x^2 - 2x - 5) + 3) and g(x) = 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": "Determine the number of intersection points between the functions f(x) = 1|(-1sin(2πx + 1) - 1) + 3| - 2 and g(x) = -1|x - 2| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-3sin(1πx + 1) + 3) - 3)/(-3(-3sin(1πx + 1) + 3)) and g(x) = 1|x - 2| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "How many solutions does the equation -2sin(1π(-4x - 1) + 1) = -3x^2 + 2x + 5 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(3cos(3πx - 2) - 1)^2 + 4(3cos(3πx - 2) - 1) + 1 and g(x) = -4x - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 27 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(-4x^2 - 4x + 3) - 3| + 3 and g(x) = (-3x - 2)/(2x - 1) in the range -10 ≤ x ≤ 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) = 2cos(2π(-4x^2 + 4x - 2) + 2) - 2 and g(x) = -3x + 1 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 -2cos(3π((3x + 1)/(3x - 1)) + 3) + 1 = -5x + 1 have for x in the interval [-10, 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) = -4(-3|x + 2| + 2)^2 - 1(-3|x + 2| + 2) - 5 and g(x) = -3x^2 + 3x - 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 -2(-3cos(2πx + 2) - 3)^2 - 5(-3cos(2πx + 2) - 3) = 1|x| - 3 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 points of intersection exist between the functions f(x) = -3|(-2cos(2πx + 3) + 2) - 3| + 3 and g(x) = (1x - 1)/(2x - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 81 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-3sin(3πx - 3) + 1)^2 + 2(-3sin(3πx - 3) + 1) + 4 and g(x) = 5x - 3 in the range -10 ≤ x ≤ 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) = -3cos(1π(4x - 3) + 3) + 1 and g(x) = (1x + 2)/(2x - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 75 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3sin(2π(1|x| + 3) + 1) + 2 and g(x) = (3x)/(1x) intersect in the interval [-10, 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|(-2|x - 3| + 2)| - 2 and g(x) = (-1x - 2)/(2x - 1) 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 3(-1cos(1πx + 2) + 1)^2 - 2 = (-2x - 3)/(-2x - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(-1sin(2πx - 1) - 3) - 1| + 1 and g(x) = -1|x - 3| - 1 in the range -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) = 3|(-3|x|) + 2| - 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 2cos(3π(1x^2 - 3x - 4) + 3) + 1 = 1x - 2 have for x 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) = 3sin(2π(-4x - 1) + 2) - 3 and g(x) = (2x - 2)/(1x - 2) 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) = -4(2cos(3πx + 2) - 1) + 3 and g(x) = 1x + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 45 | MATH |
[
{
"content": "How many solutions does the equation -5(-2sin(3πx) - 2) + 5 = 5x + 5 have for x 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) = 2|(-2sin(3πx - 2) + 3) + 3| - 3 and g(x) = 1|x + 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "How many solutions does the equation -3sin(2π(-2x - 1)) + 2 = (-2x - 2)/(-1x - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 72 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2cos(3π(1|x - 3| + 2) - 3) - 1 and g(x) = 1x + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(1π(-4x^2 + 3x - 5)) + 2 and g(x) = 2|x - 1| + 3 in the range -10 ≤ x ≤ 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) = -3cos(3π((-1x - 3)/(-2x + 3)) - 3) - 3 and g(x) = -1x^2 - 5x in the range -10 ≤ x ≤ 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) = 3sin(2π(-2x^2 - 4x - 1) + 3) and g(x) = (-1x)/(-2x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 795 | MATH |
[
{
"content": "How many solutions does the equation -3cos(2π((-2x)/(-1x - 3))) - 1 = 2x + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 106 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(-1cos(1πx + 3) - 1) + 3)/(1(-1cos(1πx + 3) - 1)) and g(x) = -1x^2 - 2x - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1cos(3π(-5x - 5) + 2) + 3 and g(x) = 2|x + 1| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "How many solutions does the equation (3(2cos(3πx - 2) - 2) + 2)/(3(2cos(3πx - 2) - 2) + 1) = 3x + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 39 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(1π((3x - 1)/(-1x - 3))) - 2 and g(x) = (-3x)/(2x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 96 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(1π(5x - 4) + 2) + 3 and g(x) = (-3x + 3)/(-1x) in the range -10 ≤ x ≤ 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) = -2sin(1π(-2|x + 3|) + 1) + 2 and g(x) = -1x - 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(-1x^2 + 4x + 5) - 2)/(-1(-1x^2 + 4x + 5) + 2) and g(x) = -1|x - 2| - 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) = -2cos(3π(1x^2 + 2x - 5) + 1) and g(x) = -1|x| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 102 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(-2cos(1πx + 2) + 1)^2 + 4(-2cos(1πx + 2) + 1) - 3 and g(x) = 5x + 3 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) = -1(-2sin(3πx) - 1)^2 + 3(-2sin(3πx) - 1) and g(x) = -3x - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "How many solutions does the equation -5(-3|x - 2| + 2)^2 - 5(-3|x - 2| + 2) = (-3x + 1)/(-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) = 2cos(1π(1x^2 - 3x - 2) - 2) + 2 and g(x) = -1x + 2 where -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) = -3(2cos(3πx + 1) + 1) + 1 and g(x) = -2x - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(2sin(2πx + 1)) and g(x) = 2x + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(-5x^2 - 5x + 3) + 1| and g(x) = 5x - 4 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) = 2cos(2π(5x) - 3) + 3 and g(x) = -2x^2 + 3x + 5 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) = (-1(-1cos(3πx + 3) - 1) + 2)/(1(-1cos(3πx + 3) - 1)) and g(x) = (2x + 1)/(-1x + 3) intersect in the interval [-10, 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) = -2cos(3π(-2|x + 3| - 3) + 3) - 2 and g(x) = -1x^2 + 4x - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3cos(1π(-3x^2 - 3x + 1) + 1) - 2 and g(x) = (1x - 2)/(2x) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 549 | MATH |
[
{
"content": "How many solutions does the equation 3(2cos(2πx))^2 + 3(2cos(2πx)) - 1 = 2x - 5 have for x 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) = 3(-2sin(2πx - 3) + 2) + 1 and g(x) = -1|x| + 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(5x^2 + 1x + 1) + 3) + 1 and g(x) = 5x - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3cos(3π(-4x^2 + 4x + 4) + 1) + 2 and g(x) = 3x - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 83 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-3cos(3πx - 2) - 3)^2 - 3(-3cos(3πx - 2) - 3) and g(x) = 1|x| - 2 intersect in the interval [-10, 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) = 1sin(2π(-5x^2 + 5x - 3) - 1) + 1 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"
}
] | 34 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(2π((-1x + 3)/(2x + 3)) + 2) + 1 and g(x) = (-2x + 2)/(2x) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3sin(3π((-1x + 3)/(1x)) + 2) - 2 and g(x) = 5x + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 23 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(3sin(2πx - 1) + 1) + 4 and g(x) = (3x + 1)/(3x - 2) 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 3(2cos(2πx + 1) + 2)^2 - 1(2cos(2πx + 1) + 2) + 2 = 1x - 5 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(-1cos(3πx + 3) - 1) + 3 and g(x) = -3x^2 + 5x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
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