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|(2cos(1πx + 2) - 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"
}
] | 19 | MATH |
[
{
"content": "How many solutions does the equation 3(1cos(1πx + 2) + 1)^2 + 2(1cos(1πx + 2) + 1) + 5 = 1|x - 2| + 1 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(2cos(2πx + 3) - 2)^2 - 5(2cos(2πx + 3) - 2) and g(x) = 1x where -10 ≤ x ≤ 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) = 3cos(2π(4x^2 - 5) + 2) + 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"
}
] | 345 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(-2cos(2πx + 1) + 3)^2 - 3(-2cos(2πx + 1) + 3) - 4 and g(x) = 4x^2 - 3x - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(1sin(2πx + 2) + 1)^2 + 1(1sin(2πx + 2) + 1) + 3 and g(x) = 2|x + 2| - 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) = 5(-3sin(2πx + 3) + 2)^2 - 3(-3sin(2πx + 3) + 2) - 1 and g(x) = 5x + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 25 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(2|x - 1| - 3)^2 - 5(2|x - 1| - 3) - 1 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"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(3sin(1πx + 1) + 1) + 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"
}
] | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(3sin(2πx - 3))^2 - 1(3sin(2πx - 3)) - 3 and g(x) = -4x^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 -3(-1cos(2πx - 3) + 3) + 1 = -2x^2 - 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(-3sin(3πx - 1) - 1)^2 - 5(-3sin(3πx - 1) - 1) + 4 = -1x^2 - 3x + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 56 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-3cos(2πx) + 1) + 5 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"
}
] | 40 | MATH |
[
{
"content": "How many solutions does the equation 1(1sin(3πx) - 2)^2 + 1(1sin(3πx) - 2) + 3 = 4x^2 - 5x + 5 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) = -4(-1cos(1πx - 3) - 2)^2 + 1(-1cos(1πx - 3) - 2) - 4 and g(x) = -4x^2 + 2x + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-3sin(3πx + 3) + 1) - 2)/(3(-3sin(3πx + 3) + 1)) and g(x) = -2x^2 - 5 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) = -3sin(2π(3x^2 - 1x - 5) + 3) and g(x) = (2x - 3)/(-1x + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 972 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(1π(2x^2 - 2x + 2) + 1) + 2 and g(x) = -4x - 1 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) = -3sin(2π(1|x + 2| - 3) + 3) + 3 and g(x) = -2|x| + 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) = -2(-3cos(1πx + 1) + 2)^2 - 2(-3cos(1πx + 1) + 2) and g(x) = -3|x - 3| 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(2sin(1πx + 2) - 1) + 3)/(-1(2sin(1πx + 2) - 1) - 1) and g(x) = -4x^2 + 2x + 1 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) = 2|(-1x^2 - 4x - 2) + 3| - 2 and g(x) = (3x + 1)/(3x + 3) where -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) = 3|(3sin(3πx + 1) + 1) - 3| and g(x) = (-2x)/(-3x - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 118 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-1cos(2πx)) - 3)/(2(-1cos(2πx)) - 1) and g(x) = -4x^2 + 2x - 1 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) = (-1(-1cos(2πx + 2)) - 1)/(3(-1cos(2πx + 2)) + 1) and g(x) = (-2x - 2)/(3x - 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 (1(-4x^2 - 4x + 1) - 1)/(-2(-4x^2 - 4x + 1) - 2) = -3|x - 2| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(2sin(2πx) - 1)^2 - 4(2sin(2πx) - 1) + 2 and g(x) = (1x - 1)/(-2x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 79 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(1cos(2πx - 3)) and g(x) = -4x - 4 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) = 1cos(2π(5x^2 + 2x + 3) + 2) and g(x) = -3|x - 1| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "How many solutions does the equation 2cos(2π(5x) + 2) - 1 = (-1x + 3)/(-1x - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 107 | MATH |
[
{
"content": "How many solutions does the equation -2|(-2cos(2πx + 3) + 1) + 3| + 2 = 3x^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 points of intersection exist between the functions f(x) = -3|(3sin(2πx - 3) - 1) - 3| + 1 and g(x) = -2|x + 3| + 2 in the range -10 ≤ x ≤ 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) = (1(-3cos(3πx - 1)) - 2)/(1(-3cos(3πx - 1)) - 3) and g(x) = 2x^2 - 3x + 1 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 2cos(1π(4x + 4) - 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"
}
] | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2sin(2π(-4x^2 - 3x) - 2) - 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"
}
] | 19 | MATH |
[
{
"content": "How many solutions does the equation 3sin(2π((-3x + 1)/(-1x + 2)) + 2) + 3 = (-3x + 3)/(3x + 3) 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|(3sin(3πx + 1) - 2)| + 2 and g(x) = 3x - 5 intersect in the interval [-10, 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) = (-1(-3cos(1πx - 1) - 2) - 1)/(2(-3cos(1πx - 1) - 2) + 1) 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"
}
] | 12 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(2cos(1πx + 2) + 2)| - 2 and g(x) = -4x^2 - 4x + 4 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(1π(1x^2 + 2x + 4) - 2) - 3 and g(x) = -2|x - 2| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "How many solutions does the equation -1(-3cos(3πx - 3) + 3)^2 - 1(-3cos(3πx - 3) + 3) + 1 = -1x + 3 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) = -3sin(2π(-3|x + 1| - 3) + 2) and g(x) = 3|x + 1| - 1 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) = 1(-1cos(3πx + 1) - 1)^2 + 5 and g(x) = 1x + 5 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) = -4(2sin(1πx - 3) + 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"
}
] | 9 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3cos(3π((-3x + 2)/(-3x - 3)) - 1) + 1 and g(x) = (-1x)/(-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) = -3|(3sin(3πx + 2) - 3) - 3| + 3 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"
}
] | 36 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3cos(2π(-1x - 1) - 3) - 3 and g(x) = (3x - 3)/(-1x - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1cos(3π(-4x^2 - 4x + 4) + 3) + 2 and g(x) = (-3x + 3)/(-2x) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1216 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(2cos(3πx + 2) + 1) + 3)/(2(2cos(3πx + 2) + 1) + 3) and g(x) = 1x^2 - 4x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(-2sin(1πx + 2) + 3) and g(x) = 2|x - 2| + 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) = 3sin(2π((2x + 3)/(-2x - 1)) + 3) - 2 and g(x) = 1|x + 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) = 3cos(2π(-3x^2 - 2) + 1) and g(x) = (-2x + 3)/(-3x + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1145 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3sin(2π(-1x^2 + 3x) + 3) - 1 and g(x) = -4x^2 - 2x intersect in the interval [-10, 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) = 4(-2sin(3πx + 3) - 1)^2 - 4(-2sin(3πx + 3) - 1) - 3 and g(x) = 3x^2 - 4x + 4 intersect 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) = -3|(2cos(1πx + 3) + 2) - 1| and g(x) = -1|x + 2| + 2 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) = -2sin(1π(-1x^2 - 3x - 2) + 2) + 1 and g(x) = -4x + 5 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 1cos(2π(-2|x - 2| - 2) + 2) + 1 = (-1x)/(1x - 3) have for x 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) = -2|(-3cos(1πx) + 3) + 2| 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"
}
] | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1cos(3π(-2x - 1)) + 2 and g(x) = 2|x - 2| - 2 where -10 ≤ x ≤ 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) = 2sin(3π(-2x - 4) + 2) - 3 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"
}
] | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(-2cos(1πx + 2) + 2) - 2 and g(x) = (-1x + 3)/(-2x - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "How many solutions does the equation 3cos(3π(-5x^2 + 5)) = -3x^2 - 5x + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 55 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(2cos(2πx - 1) - 3) - 3)/(-1(2cos(2πx - 1) - 3)) and g(x) = -1x^2 + 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 points of intersection exist between the functions f(x) = 1sin(3π(-1|x| - 1) + 1) - 3 and g(x) = -2|x| + 2 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) = 5(3cos(2πx + 3)) - 5 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"
}
] | 28 | MATH |
[
{
"content": "How many solutions does the equation 1sin(2π(-4x + 2) + 3) = (2x + 2)/(3x) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 141 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(-2sin(3πx - 2) - 1)^2 + 5(-2sin(3πx - 2) - 1) + 3 and g(x) = 5x^2 - 2x - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(-3sin(1πx + 1) - 1) and g(x) = 2|x - 3| + 1 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 4(2sin(2πx) + 1)^2 + 4(2sin(2πx) + 1) - 4 = 3|x - 3| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(-1|x| + 1)^2 - 1(-1|x| + 1) and g(x) = 2|x| - 2 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) = 3|(-1sin(2πx + 2) + 2) - 3| - 2 and g(x) = 1|x| - 2 in the range -10 ≤ x ≤ 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) = (1(3sin(2πx - 3) - 1) + 2)/(1(3sin(2πx - 3) - 1)) and g(x) = 1|x| + 3 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) = -4(2cos(1πx - 2) - 2)^2 + 5(2cos(1πx - 2) - 2) + 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"
}
] | 20 | MATH |
[
{
"content": "How many solutions does the equation -2sin(2π(-5x^2 - 5x - 5) + 1) = (2x + 1)/(3x + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1473 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(2cos(1πx + 3) - 1) + 2)/(2(2cos(1πx + 3) - 1)) and g(x) = -5x^2 + 5x + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "How many solutions does the equation 2sin(2π(-3x + 1) - 1) = -1x + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 23 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(-2sin(2πx - 2) + 1) - 5 and g(x) = 2|x - 1| - 3 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) = -3cos(2π(-2x^2 + 1x + 2) + 1) - 3 and g(x) = -3|x + 3| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 101 | MATH |
[
{
"content": "How many solutions does the equation (-3(3cos(1πx + 2) + 2) - 3)/(3(3cos(1πx + 2) + 2) - 1) = 2x^2 + 3x - 4 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 3cos(2π(-3|x - 1| + 1) + 2) - 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"
}
] | 12 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(3sin(2πx - 3) + 1) + 1| + 2 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 solutions does the equation 1cos(3π(3x^2 + 1x - 4) + 1) + 1 = -3x^2 - 3x + 5 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 solutions does the equation 1(2sin(2πx + 3) + 2)^2 + 3(2sin(2πx + 3) + 2) - 5 = 3|x - 1| - 2 have for x 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) = 2|((3x)/(3x - 2)) - 2| - 1 and g(x) = 2|x + 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(1cos(1πx + 3) + 2))/(-2(1cos(1πx + 3) + 2) + 3) and g(x) = 5x^2 - 4x + 1 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) = 2sin(3π(5x - 2)) + 3 and g(x) = 2x intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 29 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3cos(2π((1x + 1)/(2x)) + 2) - 2 and g(x) = (1x - 1)/(2x) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(1sin(1πx - 3))^2 - 5(1sin(1πx - 3)) - 1 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 points of intersection exist between the functions f(x) = -1cos(1π((3x - 1)/(-3x - 2)) - 2) + 1 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) = -3sin(2π(4x^2 - 1x - 5)) + 1 and g(x) = -4x^2 - 2x + 1 in the range -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) = 1cos(1π(2|x - 1| - 1) - 2) + 1 and g(x) = (1x - 1)/(3x + 1) 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) = 2|(-3sin(2πx - 2) + 2) + 1| - 3 and g(x) = -1x + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 25 | MATH |
[
{
"content": "How many solutions does the equation -5(-1cos(1πx + 3) + 3)^2 - 2(-1cos(1πx + 3) + 3) + 4 = 5x 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) = 1sin(1π(-5x - 4) + 1) + 1 and g(x) = 1x - 5 in the range -10 ≤ x ≤ 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) = 3|(3cos(1πx + 2) + 2) - 2| and g(x) = (2x + 1)/(1x) 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) = 2sin(2π(4x^2 + 1x + 5)) - 3 and g(x) = -5x^2 + 1x - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "How many solutions does the equation (-2(-5x^2 + 1x + 2) - 3)/(2(-5x^2 + 1x + 2) + 2) = 3|x + 3| - 2 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) = 1|(1x^2 - 5x + 1) + 3| - 1 and g(x) = (1x)/(3x) 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) = -2|(1cos(3πx + 3) - 2) - 1| + 3 and g(x) = -2|x - 2| + 2 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) = 2|(-2|x + 1| + 3)| + 2 and g(x) = 3|x| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
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