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(-2sin(1πx - 2) + 2) + 2)/(2(-2sin(1πx - 2) + 2) + 1) and g(x) = (1x)/(-3x - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "How many solutions does the equation 2sin(3π(3x - 2) + 2) + 1 = -2x + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "How many solutions does the equation -1(-1sin(2πx - 2) - 2)^2 - 4(-1sin(2πx - 2) - 2) + 1 = 1x + 4 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 points of intersection exist between the functions f(x) = -2sin(2π((2x - 2)/(3x + 2))) + 1 and g(x) = (1x - 1)/(1x + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 38 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(-3|x + 3| - 1) + 3)/(-1(-3|x + 3| - 1) - 2) and g(x) = -2x^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 solutions does the equation 1cos(1π(4x^2 + 3x - 2) - 3) - 3 = -2x^2 - 4x - 5 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 solutions does the equation -2sin(3π(2|x - 3| - 2) - 2) + 1 = 4x^2 - 4x + 1 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 -2sin(2π(3x - 1) - 3) - 2 = -1|x + 2| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "How many solutions does the equation 1sin(3π(3x^2 - 5x) - 2) = -5x - 1 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) = -3cos(2π(2|x + 2| + 2) + 3) + 1 and g(x) = 4x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "How many solutions does the equation -2|(2sin(2πx + 2) + 2) - 1| + 1 = 1x^2 + 3x - 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) = -2(-3cos(1πx + 2) - 1) - 2 and g(x) = (-3x - 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 points of intersection exist between the functions f(x) = 4(1x^2 - 2x - 4)^2 + 5(1x^2 - 2x - 4) - 2 and g(x) = -1x^2 + 5x + 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 2|(2cos(2πx - 3) + 1) + 1| - 3 = 2x^2 - 4x + 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) = (-3(2sin(2πx - 1) - 2) - 1)/(3(2sin(2πx - 1) - 2)) and g(x) = (3x - 3)/(2x - 3) in the range -10 ≤ x ≤ 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|((-1x + 3)/(-2x + 3)) - 1| and g(x) = 1x^2 - 5x 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) = -4(-2sin(3πx - 3))^2 + 5(-2sin(3πx - 3)) - 4 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"
}
] | 76 | MATH |
[
{
"content": "How many solutions does the equation -1(1cos(2πx + 2))^2 + 2(1cos(2πx + 2)) - 1 = (2x - 3)/(-2x) have for x in the interval [-10, 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) = -3(-3cos(3πx - 1) + 2) - 4 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"
}
] | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(1π(-3x^2 + 3x - 1) - 3) + 2 and g(x) = -3x^2 - 5x intersect in the interval [-10, 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) = 2cos(3π(4x - 5) + 1) - 1 and g(x) = 5x - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "How many solutions does the equation (-2(-1cos(1πx + 1) + 1) - 2)/(2(-1cos(1πx + 1) + 1)) = -4x^2 + 3 have for x 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) = 2(2cos(2πx) + 1)^2 + 1(2cos(2πx) + 1) - 1 and g(x) = 3x^2 - 4x 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) = -2|(1cos(3πx - 1) - 3)| + 1 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"
}
] | 8 | MATH |
[
{
"content": "How many solutions does the equation 1(-1cos(1πx - 3))^2 - 3(-1cos(1πx - 3)) - 3 = 1|x - 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 -1cos(3π((-2x)/(-3x + 4))) + 1 = 2|x - 2| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 50 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(2cos(3πx + 3) - 2)^2 - 4(2cos(3πx + 3) - 2) - 3 and g(x) = (-2x - 2)/(1x - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 65 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(3x^2 - 3x - 5)^2 - 4(3x^2 - 3x - 5) 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"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1cos(2π(3x + 5) + 2) + 3 and g(x) = -1x^2 + 4x + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3sin(1π(-4x - 1) + 1) + 3 and g(x) = 5x^2 + 4 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(3sin(2πx + 2) + 3) + 3)/(2(3sin(2πx + 2) + 3)) and g(x) = 4x^2 - 5x - 4 in the range -10 ≤ x ≤ 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) = 4(-3cos(3πx - 1) - 2)^2 + 1(-3cos(3πx - 1) - 2) - 3 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"
}
] | 90 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1sin(1π(2x - 5) - 3) + 3 and g(x) = (-3x - 1)/(-1x + 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(-1sin(3πx + 3) + 2)^2 - 3(-1sin(3πx + 3) + 2) - 5 and g(x) = 3|x - 1| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 56 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(3π((2x + 2)/(1x)) + 2) + 1 and g(x) = -3x + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 75 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(2cos(2πx) - 2) + 1 and g(x) = 2x^2 - 4x - 5 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 -3cos(1π((-2x - 1)/(-1x - 2)) - 1) = -2x^2 - 3x have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(2sin(2πx + 1) - 3) + 3| and g(x) = -1x where -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(-2cos(2πx + 1) + 2) + 4 and g(x) = 3x^2 + 3x + 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) = (-3(3sin(2πx + 1) - 1))/(2(3sin(2πx + 1) - 1) - 3) and g(x) = (-2x)/(-1x - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 38 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(1sin(1πx + 2) - 2) + 3| + 3 and g(x) = -1x + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "How many solutions does the equation 4(-2sin(1πx - 2) + 3) + 5 = 2x^2 - 2x 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 2sin(3π(-4x^2 + 2x - 2) + 2) - 2 = -2|x - 1| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 63 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(3cos(3πx - 3) + 3) - 3| + 3 and g(x) = 5x + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|((-1x + 2)/(-2x + 3)) + 3| and g(x) = 4x^2 + 1x + 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) = -2|((-1x - 3)/(-2x - 1)) + 3| + 3 and g(x) = -1|x + 1| - 1 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) = 5(-2cos(2πx - 3) - 1) + 2 and g(x) = (1x)/(1x) 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 -1sin(3π(3|x + 3| + 1) - 1) + 1 = 1x - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-3sin(2πx + 3) + 2) + 2)/(3(-3sin(2πx + 3) + 2) - 2) and g(x) = (-2x + 2)/(-1x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 39 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(-3x^2 + 4x + 5) - 2| and g(x) = 2x - 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) = (2(-1x^2 - 5x + 1) - 1)/(2(-1x^2 - 5x + 1) + 2) and g(x) = 1x^2 + 5x - 4 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(3π(-2|x + 2|) + 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"
}
] | 37 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(2cos(3πx + 1) + 3) + 2)/(-2(2cos(3πx + 1) + 3) + 3) and g(x) = -1x^2 - 4x - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 39 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-2sin(2πx + 2))^2 - 5(-2sin(2πx + 2)) - 2 and g(x) = 1x^2 - 4x + 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) = -2sin(1π((-3x + 3)/(3x - 2))) - 1 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"
}
] | 21 | MATH |
[
{
"content": "How many solutions does the equation 5(-3sin(1πx + 1) - 1) - 5 = (-3x + 3)/(2x) have for x 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) = -2sin(3π(2x^2 + 2x) - 1) + 3 and g(x) = 2x - 1 where -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) = 1(-3sin(1πx + 1)) + 2 and g(x) = 1x + 5 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) = -1cos(1π(-2|x + 1| - 1) + 1) + 1 and g(x) = (2x - 1)/(2x + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 37 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-1sin(2πx - 2) + 2) - 2 and g(x) = 1x 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) = -4(3cos(3πx - 2) - 2) - 2 and g(x) = -1x + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 57 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-3sin(2πx + 1) - 2) + 2| - 1 and g(x) = (3x + 3)/(3x - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 77 | MATH |
[
{
"content": "How many solutions does the equation 3sin(2π(-4x + 3) - 3) - 1 = 1|x + 3| - 1 have for x in the interval [-10, 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) = (1(2sin(1πx)) + 1)/(-2(2sin(1πx)) + 3) and g(x) = 4x + 5 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 1cos(2π(-1x^2 + 5x + 5) - 2) - 3 = -2x^2 - 3x + 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 solutions does the equation 2cos(3π(-1x + 1) - 1) - 3 = -2|x + 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 points of intersection exist between the functions f(x) = -2sin(3π(-3x - 5)) - 2 and g(x) = (3x + 1)/(-1x - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 157 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1cos(1π(-2|x - 1| + 1) - 3) + 1 and g(x) = -1x^2 + 3x 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|(-2sin(2πx + 2)) - 1| + 3 and g(x) = (3x - 3)/(1x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-2cos(3πx + 1) - 3)^2 - 2(-2cos(3πx + 1) - 3) - 2 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"
}
] | 24 | MATH |
[
{
"content": "How many solutions does the equation (2(2cos(3πx + 2) - 1) + 3)/(-1(2cos(3πx + 2) - 1)) = -2x + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 55 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(3π(5x - 4) + 2) and g(x) = -3|x - 3| + 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 -3cos(1π(1|x + 1|) + 3) - 2 = -1x 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) = 2sin(3π(3|x + 2| - 3) + 2) and g(x) = 4x^2 + 1x 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) = -4(-3cos(2πx) - 3)^2 - 1(-3cos(2πx) - 3) + 5 and g(x) = -5x^2 + 5x + 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) = (1(3cos(3πx + 3) - 1) + 1)/(-1(3cos(3πx + 3) - 1) + 2) and g(x) = -5x - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 29 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1cos(2π(3x - 5) + 3) + 1 and g(x) = 3x - 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) = (1(2sin(3πx + 1) - 1))/(-1(2sin(3πx + 1) - 1) - 1) and g(x) = 1x + 2 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|(1cos(3πx - 3) - 3) + 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) = -2|(2cos(3πx - 3)) - 1| - 2 and g(x) = 1x^2 + 3x - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(3x^2 - 2x - 1)^2 - 4(3x^2 - 2x - 1) - 4 and g(x) = 3x + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(3π((1x + 1)/(-3x + 2)) + 2) - 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"
}
] | 11 | MATH |
[
{
"content": "How many solutions does the equation -1(1x^2 + 5x + 1)^2 + 3(1x^2 + 5x + 1) - 1 = (-2x + 1)/(1x + 3) 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) = -1|((1x - 3)/(1x - 2)) + 2| - 3 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": "Determine the number of intersection points between the functions f(x) = 4(-2sin(3πx) + 1)^2 + 5(-2sin(3πx) + 1) - 5 and g(x) = -4x^2 - 2x + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(4x^2 + 4x - 4)^2 - 3(4x^2 + 4x - 4) - 1 and g(x) = 3|x + 2| + 3 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) = -3cos(2π(5x^2 - 5x - 3) + 3) and g(x) = (2x)/(-1x + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 934 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(2sin(3πx + 1) - 1) + 3)/(-1(2sin(3πx + 1) - 1) - 2) and g(x) = 2x^2 + 1x + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 46 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(3sin(1πx + 1))^2 - 2(3sin(1πx + 1)) - 3 and g(x) = 2x^2 - 5x - 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 -2cos(2π(2|x + 2| - 3)) + 3 = 2x^2 - 1 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 (3(-1cos(3πx + 1)) + 2)/(-2(-1cos(3πx + 1))) = -1|x - 3| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(1sin(1πx - 3) + 2)^2 - 1(1sin(1πx - 3) + 2) - 2 and g(x) = (2x - 1)/(-1x - 2) 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 (1(3sin(2πx + 3)) + 1)/(-1(3sin(2πx + 3)) + 2) = (1x + 1)/(2x + 1) 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 solutions does the equation -4(3sin(2πx - 2) - 1) + 1 = (1x + 3)/(3x + 1) 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) = -2sin(2π(-2|x + 3|)) - 1 and g(x) = 1x^2 - 5x in the range -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(3sin(2πx - 3) + 1) - 2)/(3(3sin(2πx - 3) + 1) - 1) and g(x) = -5x - 4 intersect in the interval [-10, 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) = -2(5x^2 - 2x - 4)^2 + 5(5x^2 - 2x - 4) + 1 and g(x) = (-3x - 2)/(3x - 1) 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(5x + 3) - 2) + 1 = 3x^2 - 3x + 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 solutions does the equation -3sin(3π(2x^2 - 2x - 4) + 2) + 3 = 5x + 2 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 solutions does the equation -3cos(1π(4x - 4) + 2) = 2x^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 |
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