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) = 2|(1cos(2πx - 2) + 2)| - 1 and g(x) = 3|x + 2| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(-3|x - 2|) + 3| - 2 and g(x) = -1x + 5 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 1e^(2(-2|x - 3| + 3)) = 1x^2 + 4x 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) = 2cos(3π(5x^2 + 5x - 1)) + 1 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"
}
] | 208 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(3sin(3πx + 3) + 1) + 2)/(1(3sin(3πx + 3) + 1) - 3) and g(x) = (-1x)/(3x + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 59 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3cos(3π(2|x - 1| - 1) - 1) - 3 and g(x) = 1x - 3 in the range -10 ≤ x ≤ 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) = 4(-1|x| + 2)^2 - 3(-1|x| + 2) - 5 and g(x) = 2|x - 2| - 2 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 (-1(-3cos(1πx - 1)) - 3)/(-2(-3cos(1πx - 1)) + 3) = -5x^2 - 5x + 3 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(3π((3x - 1)/(-1x)) - 3) + 2 = -5x - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "How many solutions does the equation (3(-4x^2 + 5x + 5) + 3)/(-1(-4x^2 + 5x + 5) + 2) = 1x^2 - 3x - 4 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) = -2cos(1π((2x - 1)/(2x + 1)) + 2) - 2 and g(x) = -2x - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "How many solutions does the equation 3cos(1π(-3|x| + 2) - 2) + 3 = 5x^2 - 5x - 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 (1(-1x^2 + 1) + 1)/(3(-1x^2 + 1) - 2) = -4x^2 + 1x + 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) = (-3(3cos(2πx - 2) + 1) - 2)/(-2(3cos(2πx - 2) + 1) + 3) and g(x) = -1x - 2 in the range -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) = 3sin(1π(4x^2 - 2x - 2) - 3) - 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"
}
] | 5 | MATH |
[
{
"content": "How many solutions does the equation -2cos(3π(2x^2 + 4x + 3) + 1) = 5x - 2 have for x in the interval [-10, 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) = 4(-2cos(2πx) + 1) - 5 and g(x) = 3|x + 3| - 2 where -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) = -2sin(2π(-1x^2 + 3x - 1) - 3) and g(x) = (-1x - 3)/(3x + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 401 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2sin(1π(4x - 5) + 2) + 1 and g(x) = -2x^2 + 5x intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many solutions does the equation (1(2cos(3πx - 2)) - 1)/(-1(2cos(3πx - 2))) = (-3x)/(-2x) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(2π(5x - 4) + 3) + 3 and g(x) = 5x - 2 in the range -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(1π(-2|x - 2| + 3) + 2) - 3 and g(x) = (2x - 2)/(-2x) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3cos(3π(2x^2 + 3x - 5) + 1) and g(x) = 5x^2 + 2x - 4 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) = (-2(-2sin(1πx + 1) + 1) - 3)/(-3(-2sin(1πx + 1) + 1) + 1) 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"
}
] | 18 | MATH |
[
{
"content": "How many solutions does the equation -2(1cos(2πx - 3) + 3) + 5 = -4x^2 + 3x + 1 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) = 2(1sin(3πx - 2) - 1) + 1 and g(x) = (3x - 2)/(2x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3sin(1π(-1x - 2) - 2) + 3 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 points of intersection exist between the functions f(x) = -2cos(1π(4x + 3) - 2) 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": "How many points of intersection exist between the functions f(x) = -1|(3x^2 + 2x - 4) + 3| - 1 and g(x) = (2x)/(-1x) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(2π(3x^2 + 2x - 5) - 2) + 1 and g(x) = (-3x + 3)/(-1x - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 589 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(1sin(3πx - 1) - 2) - 2 and g(x) = 1x + 2 in the range -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) = -1sin(1π(-5x^2 + 4x + 4)) + 2 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"
}
] | 32 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(3cos(2πx - 3) + 1) + 1)/(-1(3cos(2πx - 3) + 1) - 2) and g(x) = 1x + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(1sin(2πx)) + 2 and g(x) = (2x - 3)/(1x + 2) intersect in the interval [-10, 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) = -4(-3sin(3πx - 2) + 3) + 2 and g(x) = (-1x - 1)/(-3x) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 61 | MATH |
[
{
"content": "How many solutions does the equation -1sin(2π(1x - 5) - 1) - 1 = 3|x - 2| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation 2(-2cos(1πx - 3) - 3)^2 - 1 = -1x + 4 have for x in the interval [-10, 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(-1cos(1πx + 1) + 1)^2 + 2(-1cos(1πx + 1) + 1) - 3 and g(x) = 2x^2 - 2x 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(1π(-2|x + 1|) + 2) 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"
}
] | 21 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-3cos(2πx) + 2) + 1)/(-2(-3cos(2πx) + 2) + 3) and g(x) = 1x + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 37 | MATH |
[
{
"content": "How many solutions does the equation (-3(3x^2 - 1x - 3) - 3)/(2(3x^2 - 1x - 3) - 3) = -3|x + 1| + 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) = 3|(-3cos(3πx) - 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"
}
] | 18 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-3|x + 1| + 1) - 1| - 1 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"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(2cos(1πx + 3) - 2)^2 + 3 and g(x) = 1|x - 2| 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) = -1(3sin(3πx + 1) + 3) + 1 and g(x) = (3x + 1)/(3x + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 33 | MATH |
[
{
"content": "How many solutions does the equation -2sin(2π(-2x^2 + 5x - 4) + 2) + 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"
}
] | 32 | MATH |
[
{
"content": "How many solutions does the equation -2|(3cos(2πx - 2) - 1) - 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"
}
] | 22 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(3cos(1πx - 2) - 1) + 2| + 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"
}
] | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(-4x^2 + 1x + 5) + 1| + 1 and g(x) = 4x^2 - 1x 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) = 2cos(3π((-1x + 3)/(-2x + 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"
}
] | 48 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(1|x - 2| - 1) - 3)/(-3(1|x - 2| - 1) + 2) and g(x) = 1x^2 + 2x - 5 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) = 2(-1cos(3πx + 2) - 1)^2 + 2(-1cos(3πx + 2) - 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"
}
] | 30 | MATH |
[
{
"content": "How many solutions does the equation -3(2cos(2πx - 2) + 1)^2 - 3(2cos(2πx - 2) + 1) + 1 = 4x + 5 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) = (-2(3sin(3πx + 3) - 2) - 3)/(1(3sin(3πx + 3) - 2) + 3) and g(x) = (-3x - 2)/(2x) 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|(-4x^2 + 3x) + 2| and g(x) = 3x^2 + 2x + 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) = 3(-2sin(2πx + 3) - 1) - 2 and g(x) = 1x^2 + 2x - 5 where -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(2π(4x^2 + 5x - 5) - 2) + 1 = (-1x)/(-1x + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1314 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(3cos(3πx - 3) + 1) + 2| + 2 and g(x) = -4x^2 - 3x 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) = 1e^(1(-3x^2 - 2x + 3)) + 2 and g(x) = 2|x - 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) = 1cos(3π(-2|x - 2| + 3) - 2) + 2 and g(x) = 2|x + 3| - 2 where -10 ≤ x ≤ 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) = -3sin(2π(3|x - 3| - 3) + 1) and g(x) = 1x^2 + 2x + 1 where -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) = -1(-1sin(2πx - 1) - 3)^2 - 2(-1sin(2πx - 1) - 3) + 5 and g(x) = (-2x)/(-3x) 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) = (2(5x^2 - 5))/(3(5x^2 - 5) + 3) and g(x) = 1|x - 1| - 2 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) = -3cos(3π(3x + 3) - 2) + 3 and g(x) = -1|x - 2| + 2 intersect in the interval [-10, 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) = -1cos(2π(-5x - 2) - 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"
}
] | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(1π((3x + 1)/(-1x))) - 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"
}
] | 12 | MATH |
[
{
"content": "How many solutions does the equation -3(-2cos(2πx) - 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"
}
] | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(3sin(1πx - 1))^2 and g(x) = -4x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3cos(1π(-3x^2 - 1x + 4) + 1) - 1 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"
}
] | 600 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1cos(3π(-4x - 4) + 3) + 1 and g(x) = 3|x - 2| - 1 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) = -5((-1x + 1)/(-2x + 3))^2 + 5((-1x + 1)/(-2x + 3)) - 2 and g(x) = -5x^2 + 2 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) = 3(-1sin(2πx + 2) - 2)^2 - 4(-1sin(2πx + 2) - 2) + 3 and g(x) = 4x^2 - 4x - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many solutions does the equation -5(-3sin(2πx - 1)) - 3 = -4x^2 - 5x + 1 have for x 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) = 2|(-2cos(3πx) + 1) + 2| - 2 and g(x) = -1x^2 - 3x + 2 intersect 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|(2cos(2πx - 1)) + 1| + 1 and g(x) = 3x^2 + 5x - 2 where -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) = -2|(-2sin(3πx + 2) + 2) + 1| + 2 and g(x) = 1x + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 25 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(3cos(1πx - 3) - 1) - 2)/(-3(3cos(1πx - 3) - 1)) and g(x) = (3x)/(1x - 2) 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) = 2cos(3π(-2|x| + 1) + 2) and g(x) = (2x - 2)/(2x - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 117 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(3π((-2x + 3)/(3x + 2))) and g(x) = (2x - 1)/(-1x + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 73 | MATH |
[
{
"content": "How many solutions does the equation 1(3|x| - 2)^2 - 5(3|x| - 2) - 5 = 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) = -3cos(3π(2|x - 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"
}
] | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(3cos(3πx + 1) - 3) + 2| - 3 and g(x) = -4x^2 - 4x + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many solutions does the equation 1(1sin(3πx + 3) - 2)^2 - 3(1sin(3πx + 3) - 2) - 5 = 3|x - 3| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 28 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(2sin(2πx - 1)) + 1| and g(x) = 1x^2 + 1x where -10 ≤ x ≤ 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) = -1sin(1π(-1x^2 + 4x + 4) - 1) - 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": "Determine the number of intersection points between the functions f(x) = -1sin(3π(2x - 3) + 1) and g(x) = -1x^2 - 4x - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(3sin(2πx) - 1)^2 - 4(3sin(2πx) - 1) + 3 and g(x) = 2|x + 1| + 3 intersect 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) = 2cos(2π((1x - 2)/(-1x - 1)) + 3) - 2 and g(x) = 3x^2 + 3x 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) = 4(1sin(2πx - 1)) - 2 and g(x) = (-1x + 2)/(1x) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 37 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(3π((3x + 2)/(-1x + 2)) - 3) - 3 and g(x) = -3x - 2 in the range -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) = (-2(-2sin(3πx - 2) + 1) - 2)/(-1(-2sin(3πx - 2) + 1) - 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"
}
] | 30 | MATH |
[
{
"content": "How many solutions does the equation (1(1sin(3πx + 2) + 2) + 3)/(1(1sin(3πx + 2) + 2) - 2) = 5x^2 + 3x + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(2π(-3|x + 2| + 2) + 3) + 3 and g(x) = 2x + 5 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) = 3sin(2π((-2x - 2)/(-1x))) - 3 and g(x) = -4x - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1sin(3π(1x^2 - 3x + 3)) + 2 and g(x) = 3|x| - 2 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 -2cos(2π(1x^2 - 1x + 2) + 1) + 2 = (3x + 2)/(1x) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 393 | MATH |
[
{
"content": "How many solutions does the equation -1cos(1π(-3x^2 - 2x + 2) + 1) + 2 = (-2x - 1)/(-1x - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 389 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-3sin(2πx + 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"
}
] | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(1π(3|x - 1| - 3) - 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"
}
] | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-2cos(1πx - 1) + 3) + 2)/(-1(-2cos(1πx - 1) + 3) + 2) and g(x) = -1x^2 + 4x + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
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