messages listlengths 1 1 | ground_truth stringclasses 134 values | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1cos(3π((-1x - 1)/(-1x - 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"
}
] | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3cos(1π(2|x - 3| - 3) - 1) + 3 and g(x) = -1x^2 - 1x + 4 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) = 1|(-4x^2 + 4x + 5) - 2| + 2 and g(x) = 2|x + 2| + 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) = -3sin(3π(-5x^2 - 3x + 4) - 3) - 2 and g(x) = -4x^2 + 2x - 5 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) = -2cos(3π(5x^2 - 5x - 3) + 2) - 2 and g(x) = 3x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 79 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(-3sin(1πx) - 2) + 1)/(1(-3sin(1πx) - 2)) and g(x) = -2x^2 - 1x - 1 where -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|(-5x^2 + 1) + 2| - 1 = -2|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": "How many points of intersection exist between the functions f(x) = -5(-3sin(3πx + 1) + 2) and g(x) = 1x + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 33 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(-3cos(1πx - 2) - 3) + 3| - 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"
}
] | 37 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3cos(1π((1x)/(-2x + 3)) - 1) + 1 and g(x) = 2x intersect 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) = (-3(2cos(3πx) - 1) - 2)/(-2(2cos(3πx) - 1) + 3) and g(x) = (-1x + 2)/(1x - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(-2sin(3πx) + 1) - 2| - 1 and g(x) = 3x 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) = 3|((3x + 3)/(1x - 2)) + 3| - 1 and g(x) = 3|x| + 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 -3sin(1π(-1x^2 + 5x - 1) + 3) + 1 = 1x - 5 have for x 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) = 2|(1sin(2πx + 3) + 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"
}
] | 7 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1cos(3π((-3x - 2)/(1x + 1)) + 1) + 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": "Find the number of times the graphs of f(x) = (2(1x^2 - 4x - 4) + 1)/(-2(1x^2 - 4x - 4) + 1) and g(x) = 2|x - 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 solutions does the equation 2sin(2π(1|x - 3| + 1) - 2) = -1x^2 - 4x 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 2sin(3π(1|x - 1| - 1)) = -1|x - 1| + 2 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 points of intersection exist between the functions f(x) = 1cos(2π(-1x - 1) + 2) and g(x) = 3|x| - 1 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 -2cos(3π(1|x + 1| - 1) + 2) - 1 = -2|x - 2| + 3 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) = (-3(1cos(1πx + 2)) - 2)/(1(1cos(1πx + 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"
}
] | 13 | MATH |
[
{
"content": "How many solutions does the equation -1|(2|x + 2| - 3) + 1| + 1 = (3x + 3)/(-2x + 1) have for x 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) = 2sin(1π(3x^2 - 4x + 3) - 2) - 1 and g(x) = -5x^2 - 2x + 5 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(1π(-5x^2 - 3x + 3) - 2) - 2 and g(x) = -4x^2 - 5x + 1 where -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) = -3cos(2π(-3x^2 + 3x + 2)) - 2 and g(x) = -2|x + 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 158 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(1cos(1πx - 3) + 1) and g(x) = -1|x + 1| + 1 in the range -10 ≤ x ≤ 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(2cos(1πx + 1) + 2) + 1)/(-3(2cos(1πx + 1) + 2) + 2) and g(x) = 1x + 5 in the range -10 ≤ x ≤ 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) = -2sin(2π(2|x + 1| - 3) - 3) - 1 and g(x) = 2|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) = 2(1cos(2πx - 3) - 1) + 4 and g(x) = 1|x| - 3 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) = 5(1cos(2πx - 1) + 3) + 3 and g(x) = 5x + 5 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) = -3(3sin(1πx - 3) - 1) and g(x) = -1|x| - 3 intersect 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 -3sin(2π(-1x + 1)) + 2 = 1x - 3 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 points of intersection exist between the functions f(x) = (1(2sin(3πx - 1) - 2) + 3)/(1(2sin(3πx - 1) - 2) + 2) and g(x) = -2|x| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 53 | MATH |
[
{
"content": "How many solutions does the equation 1(-1cos(1πx + 1) + 2)^2 + 1(-1cos(1πx + 1) + 2) = 3|x - 3| - 3 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) = 2sin(1π(3x^2 - 4x - 3) + 2) and g(x) = (-1x - 1)/(-1x + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 462 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-2|x - 1| + 2)^2 + 5(-2|x - 1| + 2) + 1 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"
}
] | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-3|x + 2| + 2) + 1| - 1 and g(x) = (3x - 1)/(2x - 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) = -2cos(3π(-1x^2 - 4x - 5)) and g(x) = 2|x - 1| intersect 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) = -1(-2sin(3πx + 3) - 3)^2 - 2(-2sin(3πx + 3) - 3) + 2 and g(x) = (-2x)/(-2x - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 57 | MATH |
[
{
"content": "How many solutions does the equation 2(2sin(2πx + 2) + 1) - 5 = 2x - 2 have for x in the interval [-10, 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(-1sin(2πx - 1) - 1)^2 + 5(-1sin(2πx - 1) - 1) - 1 and g(x) = 4x - 5 intersect 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) = -2sin(1π((1x + 2)/(-1x)) - 3) + 3 and g(x) = (1x + 1)/(-3x) 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(3π(5x - 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": "Find the number of times the graphs of f(x) = (3(-3sin(2πx - 3) + 2) - 2)/(-2(-3sin(2πx - 3) + 2) + 1) 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"
}
] | 9 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3sin(1π(-1x^2 - 4x + 1)) and g(x) = 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) = -2sin(3π((1x - 2)/(3x - 2)) - 3) + 2 and g(x) = (2x)/(-1x + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 36 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(1π(-2|x - 2| - 3) - 2) and g(x) = -3|x - 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) = (-1(3cos(2πx) + 2) + 3)/(-1(3cos(2πx) + 2) - 2) and g(x) = -1x^2 - 3 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 -2sin(3π(2|x - 1|) - 2) - 1 = 3|x - 2| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(3π((1x - 2)/(3x + 3)) - 3) + 1 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 solutions does the equation (-1(-2sin(2πx + 3) - 2) + 3)/(-2(-2sin(2πx + 3) - 2) - 2) = 4x^2 - 2x + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(3sin(2πx - 2) - 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"
}
] | 22 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1cos(3π(3x + 5)) - 3 and g(x) = -2x^2 + 3 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) = -1sin(3π(-2|x - 1| + 1) + 1) + 2 and g(x) = -1x - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(2π(3|x + 3| - 2)) - 2 and g(x) = 4x where -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) = -3cos(1π((1x - 1)/(-3x - 1)) - 2) + 1 and g(x) = 5x + 1 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(-2cos(3πx - 1) + 1)^2 + 2(-2cos(3πx - 1) + 1) = -3|x - 2| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 70 | MATH |
[
{
"content": "How many solutions does the equation -5(-2cos(2πx)) + 5 = 1x have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(-2|x - 1| + 2)^2 + 5(-2|x - 1| + 2) 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"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(1sin(3πx))| + 2 and g(x) = (-1x - 3)/(-3x - 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 -2(-1cos(1πx - 1) + 2)^2 + 5 = -1x + 3 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 -3cos(3π(-1|x - 1| - 1)) = (-3x)/(-1x + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 35 | MATH |
[
{
"content": "How many solutions does the equation 2cos(1π(2x^2 - 2x - 2) - 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": "Find the number of times the graphs of f(x) = 4(-3sin(2πx)) - 3 and g(x) = (-3x - 1)/(-2x + 3) 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) = -3sin(3π(-1|x + 3| - 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"
}
] | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(-2sin(3πx + 3) - 2) + 3| + 3 and g(x) = 2x - 3 in the range -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) = -2(3|x| - 1)^2 + 3(3|x| - 1) + 5 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"
}
] | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(2cos(2πx - 2) + 1) - 1)/(-1(2cos(2πx - 2) + 1) + 2) and g(x) = -2x + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 35 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-2cos(3πx) + 3) - 4 and g(x) = -3x^2 - 3x + 4 in the range -10 ≤ x ≤ 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) = -3sin(2π(-1x^2 - 3x + 1) - 3) + 1 and g(x) = (-3x + 1)/(2x - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 400 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(1π((3x + 1)/(-2x + 3))) + 1 and g(x) = 1|x + 2| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 64 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(2π(2|x + 3| - 3)) and g(x) = 3x^2 + 4x + 1 intersect 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(-3cos(3πx + 1) - 2) + 2)/(-3(-3cos(3πx + 1) - 2)) = 1x^2 + 4x have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(2x + 1)| and g(x) = 1x^2 - 1x 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(3π(4x^2 + 5x - 3)) and g(x) = -2x^2 - 5x - 2 in the range -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) = 3|(3sin(1πx + 1) - 3)| + 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"
}
] | 16 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(3sin(1πx + 3) - 1) + 3)/(-3(3sin(1πx + 3) - 1) + 1) and g(x) = (-1x - 2)/(-3x) intersect 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) = 2(1sin(2πx) - 2) - 3 and g(x) = -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 3cos(2π((-3x - 1)/(2x + 2)) + 1) + 1 = 1|x| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 45 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(-2sin(1πx + 2) + 1) and g(x) = (2x)/(1x) 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) = -5(2sin(1πx + 3) - 1) + 1 and g(x) = 1x - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(3π(-4x - 5)) and g(x) = 5x^2 - 3x - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many solutions does the equation -3sin(1π(-5x + 4) + 3) - 2 = 2x^2 - 4x - 1 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 solutions does the equation -2sin(1π(4x - 4)) - 2 = 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 solutions does the equation -5(-2sin(2πx - 3))^2 - 4(-2sin(2πx - 3)) - 1 = (-3x + 1)/(3x + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 75 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3sin(2π(3|x + 3| - 2) - 2) - 3 and g(x) = (-1x - 3)/(1x + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 120 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(3cos(1πx - 1) + 3) and g(x) = -1|x - 1| - 1 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) = -4(3cos(3πx + 3) + 1) + 3 and g(x) = 3x^2 + 5x - 1 in the range -10 ≤ x ≤ 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) = 1(-3sin(2πx + 3) - 1)^2 + 3(-3sin(2πx + 3) - 1) - 4 and g(x) = -3x + 4 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 -2(4x^2 + 5x + 1)^2 + 3(4x^2 + 5x + 1) - 1 = (-2x)/(1x) 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) = -2(2x^2 - 3x - 5)^2 + 5(2x^2 - 3x - 5) + 2 and g(x) = -2x^2 + 3x + 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(3cos(1πx + 3) + 2) + 3)/(1(3cos(1πx + 3) + 2) + 1) and g(x) = 2|x| + 1 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 (3(2sin(2πx - 1) + 2) - 1)/(1(2sin(2πx - 1) + 2) - 2) = 5x^2 + 4x + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 27 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(2sin(1πx - 2) + 2)^2 - 2(2sin(1πx - 2) + 2) and g(x) = -5x^2 - 1x - 5 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) = 1sin(2π(4x^2 - 1x - 5) + 3) and g(x) = -5x^2 - 4x + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many solutions does the equation -3cos(1π(5x + 4) + 2) - 2 = 3|x + 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 points of intersection exist between the functions f(x) = 1(1sin(3πx + 3) - 3)^2 + 2(1sin(3πx + 3) - 3) - 5 and g(x) = (-2x - 3)/(1x - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 46 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-2sin(2πx + 1) - 2) + 1| + 2 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"
}
] | 16 | MATH |
[
{
"content": "How many solutions does the equation 5(1|x - 2| - 3)^2 + 2(1|x - 2| - 3) - 5 = (-2x - 2)/(2x + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
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