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) = (-3(5x^2 + 4x - 4) + 2)/(2(5x^2 + 4x - 4) - 1) and g(x) = -2x^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) = 2|(1sin(3πx + 1) + 3) - 2| - 2 and g(x) = -2x^2 + 4x - 3 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) = 2sin(2π(2x^2 + 4x + 2) + 3) and g(x) = 1x^2 + 1x - 5 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(2sin(1πx + 1)) + 5 = (-2x - 1)/(2x - 1) have for x 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(2cos(3πx + 2) + 3) - 4 = -3|x - 2| 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(2sin(2πx - 3) + 1)^2 - 5(2sin(2πx - 3) + 1) - 2 and g(x) = 4x + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "How many solutions does the equation 3(1x^2 - 4x + 1)^2 - 4(1x^2 - 4x + 1) - 1 = (-2x - 3)/(2x + 3) 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 -4(2cos(1πx - 1)) + 2 = -1|x| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "How many solutions does the equation -3cos(1π(-3x - 1) + 1) + 1 = (-2x + 1)/(-1x + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 53 | MATH |
[
{
"content": "How many solutions does the equation -2sin(3π(2x^2 + 1x + 1)) + 2 = (-2x + 1)/(2x + 3) have for x 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) = (1(-3sin(3πx - 3) + 2) - 2)/(2(-3sin(3πx - 3) + 2) - 1) and g(x) = -2x - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 44 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(3π(-1|x - 1|)) - 3 and g(x) = (1x + 3)/(-1x + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(2sin(2πx - 3) + 1) and g(x) = 5x^2 - 1x + 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) = -1(-3cos(2πx + 3) + 3)^2 + 5(-3cos(2πx + 3) + 3) - 2 and g(x) = 1|x + 3| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1sin(1π((-3x - 2)/(2x - 3))) + 1 and g(x) = 3|x - 1| - 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) = 1cos(3π(-4x - 2) + 2) and g(x) = -3|x - 2| + 1 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) = 3cos(1π(-4x^2 - 2x + 1)) - 2 and g(x) = 2|x + 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) = 4(5x^2 + 4x - 3)^2 + 3(5x^2 + 4x - 3) - 2 and g(x) = 2|x - 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(3cos(2πx + 2) + 1) + 1)/(3(3cos(2πx + 2) + 1) - 2) 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"
}
] | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(-3sin(3πx - 2)) - 3| - 2 and g(x) = -1x^2 + 1x + 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) = -5(-1cos(1πx - 2) + 1) + 5 and g(x) = (-1x - 2)/(-1x) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "How many solutions does the equation 5(1cos(3πx + 1) - 3)^2 + 1 = 2|x - 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 points of intersection exist between the functions f(x) = (3(-1sin(3πx) - 3) + 3)/(-1(-1sin(3πx) - 3) + 3) and g(x) = 2|x + 2| - 3 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(2cos(3πx) - 3) + 5 and g(x) = (-1x)/(2x + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 57 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(-2sin(3πx + 2) - 1) + 2 and g(x) = 3|x + 1| + 2 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) = 3sin(2π(3x^2 - 1x)) - 1 and g(x) = -4x - 2 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) = -3sin(1π(-4x^2 + 1x - 4) + 3) and g(x) = (-3x - 2)/(-3x - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 797 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(3x^2 + 4x - 2)^2 + 4(3x^2 + 4x - 2) - 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"
}
] | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3cos(1π((3x + 1)/(-3x + 2)) + 1) + 3 and g(x) = -1x^2 + 2x + 5 in the range -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) = -4(1cos(3πx + 2) - 3)^2 - 3(1cos(3πx + 2) - 3) + 1 and g(x) = 4x - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 27 | MATH |
[
{
"content": "How many solutions does the equation -3cos(1π(-1x^2 + 4x + 1) + 3) = -1x^2 + 4x + 4 have for x 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) = (-2(-2cos(2πx + 2)) + 1)/(2(-2cos(2πx + 2))) and g(x) = 2x^2 + 3x - 3 in the range -10 ≤ x ≤ 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) = -3cos(1π(-4x^2 - 3x - 2) + 3) + 3 and g(x) = 2x + 2 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) = (-1(2x^2 + 5x - 2) + 3)/(-2(2x^2 + 5x - 2) - 1) and g(x) = 1x^2 - 5x + 5 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 - 3) + 1) = -3x - 4 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) = -1cos(2π(1x^2 - 3x - 5) - 2) + 1 and g(x) = (-2x - 1)/(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) = 5(-2sin(2πx + 2))^2 - 2(-2sin(2πx + 2)) + 1 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"
}
] | 68 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(2π(-5x - 4) + 2) - 1 and g(x) = -1|x + 3| + 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 1(-2cos(1πx + 3) + 3)^2 - 4(-2cos(1πx + 3) + 3) - 5 = (-2x)/(3x) have for x 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 5(2sin(3πx + 3) - 1)^2 + 4(2sin(3πx + 3) - 1) - 3 = 2|x + 3| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 68 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(2cos(2πx + 1) + 1) - 4 and g(x) = -1x + 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) = -2cos(3π((-3x)/(-3x - 3))) and g(x) = (3x + 2)/(2x + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 57 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(2sin(1πx) - 1) + 1)/(-2(2sin(1πx) - 1) + 1) and g(x) = 4x^2 + 5x + 2 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 3(-3cos(3πx - 2) - 3)^2 - 2(-3cos(3πx - 2) - 3) + 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"
}
] | 42 | MATH |
[
{
"content": "How many solutions does the equation 2cos(3π(1x - 5) - 2) + 1 = 5x^2 + 5x + 3 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(-2sin(2πx + 2))^2 - 4(-2sin(2πx + 2)) - 5 = -1x^2 + 5x + 3 have for x in the interval [-10, 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) = -3sin(2π(1|x - 1| + 2) - 1) - 2 and g(x) = (-1x - 1)/(-1x + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 25 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(2π(-1x - 3) + 3) - 3 and g(x) = -1x + 1 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) = (-2(3x^2 + 1x - 4))/(2(3x^2 + 1x - 4) - 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"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation 1(1cos(2πx + 3) + 2)^2 + 5 = 4x + 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 2sin(1π(-1|x + 3|) + 3) - 2 = (-1x + 1)/(1x + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "How many solutions does the equation -2|(-3sin(2πx + 1) + 2) + 3| + 2 = 3x - 5 have for x 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) = 5(-5x^2 + 3x + 4)^2 + 5(-5x^2 + 3x + 4) - 3 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"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(2π(3x - 4)) - 2 and g(x) = -3|x + 3| - 2 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 - 1| - 1) - 1) - 2 and g(x) = (2x + 3)/(-1x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1sin(2π(-2x^2 + 2x - 2)) and g(x) = 1x + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 56 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(1sin(1πx + 1) - 3)^2 + 3(1sin(1πx + 1) - 3) + 4 and g(x) = 2|x| where -10 ≤ x ≤ 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) = -3cos(1π(3|x + 1| - 2) - 3) - 1 and g(x) = 1|x + 2| - 1 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) = -2|(3cos(2πx + 3))| + 2 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"
}
] | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(1sin(3πx + 3) + 1) + 1)/(3(1sin(3πx + 3) + 1)) and g(x) = (-2x)/(-3x + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 55 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3cos(1π(1x^2 + 5x - 1) - 3) + 2 and g(x) = -3|x - 3| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(1π(-3|x| - 1) - 1) + 2 and g(x) = 2x^2 - 5x 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) = 2cos(2π((2x - 2)/(-1x + 2))) and g(x) = -3|x - 1| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 50 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1sin(1π(5x^2 - 1x - 3) - 2) + 2 and g(x) = 5x^2 + 3x - 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(-2sin(1πx - 3) - 3)^2 - 5(-2sin(1πx - 3) - 3) - 5 and g(x) = -2x + 5 in the range -10 ≤ x ≤ 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) = 1sin(1π((2x - 2)/(-3x)) + 3) - 1 and g(x) = 3x^2 + 5x - 2 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) = 2cos(2π(2x^2 - 2x - 4) + 2) + 1 and g(x) = -5x - 4 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) = -2sin(3π(2|x - 2|) - 2) and g(x) = 1x^2 - 1x - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many solutions does the equation 3sin(2π(-3x^2 - 2x - 1)) + 2 = (3x + 2)/(-2x - 3) have for x 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(3π(-1|x + 3| - 1) - 2) and g(x) = (2x + 3)/(-3x - 3) in the range -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|(5x^2 - 5) - 2| - 1 and g(x) = -2x + 4 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) = -2sin(1π(5x^2 - 4x - 4) - 1) and g(x) = -2|x - 2| - 1 where -10 ≤ x ≤ 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) = -4(2sin(3πx - 2) + 2) + 4 and g(x) = (-2x + 3)/(-1x) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 55 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-3sin(3πx + 3)) + 2| + 3 and g(x) = -2x + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "How many solutions does the equation 1cos(2π(-2x^2 - 1x - 5) + 2) + 1 = -3x^2 + 4x + 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 2cos(2π((3x - 3)/(-2x + 1)) + 1) + 3 = -4x + 3 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) = 1sin(2π(-1|x - 3| - 2) - 3) and g(x) = (-1x - 1)/(-3x - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 39 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(-1cos(2πx - 1)) + 1)/(3(-1cos(2πx - 1)) - 1) and g(x) = (2x - 2)/(-2x - 1) where -10 ≤ x ≤ 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) = -2cos(2π((-3x - 2)/(-1x - 3))) + 2 and g(x) = -3|x + 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 25 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(2π(-1x - 2) - 2) and g(x) = 3x^2 + 4x - 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) = -1(1cos(2πx) + 1)^2 - 2(1cos(2πx) + 1) - 3 and g(x) = -2|x - 1| - 2 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 2sin(3π(-1x - 3) - 3) + 3 = 3|x + 3| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1sin(2π(2|x - 3| - 2)) - 2 and g(x) = -5x^2 + 3x - 1 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) = 3sin(2π((1x - 1)/(-3x + 1)) - 2) - 3 and g(x) = -5x - 2 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 -1|(-3cos(3πx - 1) - 3) + 3| - 3 = -3x^2 + 1x + 3 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) = -3(-3x^2 - 1x + 2)^2 + 3(-3x^2 - 1x + 2) + 1 and g(x) = -2x^2 - 2x - 5 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(-1x - 4) - 2) + 2 and g(x) = (-2x - 2)/(-1x + 3) 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) = -1sin(1π((-1x + 1)/(3x + 3)) - 2) - 1 and g(x) = 1|x + 3| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 29 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(1π(3x - 2) - 2) - 1 and g(x) = 1|x - 1| - 3 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) = -1(-3sin(2πx - 3) - 1)^2 + 1(-3sin(2πx - 3) - 1) + 3 and g(x) = 1x intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(1π(1x + 3) + 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"
}
] | 19 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(2cos(1πx - 1) + 3) + 3 and g(x) = (-1x + 2)/(-1x - 1) 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) = (-1(1sin(1πx - 3) - 2) - 1)/(-2(1sin(1πx - 3) - 2) - 3) and g(x) = (1x + 3)/(-2x) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "How many solutions does the equation 2cos(1π(-2|x + 3| + 2) - 1) - 1 = -2|x + 3| - 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) = -3cos(2π(5x + 4) - 1) - 2 and g(x) = -3|x - 1| + 1 in the range -10 ≤ x ≤ 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) = (3(2sin(3πx + 1) - 2))/(-2(2sin(3πx + 1) - 2) - 3) and g(x) = (-1x + 2)/(-1x + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 57 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-2cos(1πx) + 1)^2 + 3(-2cos(1πx) + 1) and g(x) = 1|x + 1| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many solutions does the equation 1sin(3π(-1x^2 + 2) - 1) + 2 = (-1x - 1)/(-2x + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 35 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(3sin(1πx + 2) - 2))/(-3(3sin(1πx + 2) - 2) - 2) and g(x) = -1x^2 - 4x - 2 where -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) = (1(-2sin(3πx - 2)) - 3)/(-1(-2sin(3πx - 2)) - 2) and g(x) = (1x - 2)/(3x + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 28 | MATH |
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