messages listlengths 1 1 | ground_truth stringclasses 134 values | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-3x^2 + 3x + 3) + 2)/(3(-3x^2 + 3x + 3) + 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"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-2cos(2πx + 1) - 1) + 1)/(-1(-2cos(2πx + 1) - 1) - 3) and g(x) = 3|x| - 2 intersect in the interval [-10, 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) = 2sin(1π(4x - 4) - 1) + 1 and g(x) = (-1x + 1)/(2x - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 77 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-3cos(3πx + 1) + 1)^2 - 5(-3cos(3πx + 1) + 1) - 4 and g(x) = (2x - 3)/(-2x + 3) intersect 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) = 3sin(2π(-4x - 1)) - 1 and g(x) = -1x^2 - 5x - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "How many solutions does the equation (-2(3cos(3πx - 1)) - 3)/(1(3cos(3πx - 1)) + 2) = -1|x + 3| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 54 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(1cos(2πx + 1) + 1) + 2)/(-3(1cos(2πx + 1) + 1) - 1) and g(x) = 4x^2 - 1x - 4 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) = 1|(-1sin(2πx - 3) - 3) - 1| - 3 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"
}
] | 23 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-2sin(2πx + 3) - 2) - 4 and g(x) = -5x - 3 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) = (-3(-2cos(2πx + 2) + 2))/(-3(-2cos(2πx + 2) + 2) - 3) and g(x) = (2x + 1)/(1x - 3) 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) = -2|(1cos(2πx) - 3) + 1| + 1 and g(x) = 1x in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(3sin(1πx - 3)) - 3)/(-3(3sin(1πx - 3)) - 1) and g(x) = 1x + 4 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) = 1cos(2π(-5x^2 + 1x) - 1) - 1 and g(x) = (1x + 2)/(-2x - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1380 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(1π(-5x - 1)) - 1 and g(x) = -1|x - 1| - 1 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) = -1cos(2π(-4x + 2) - 2) + 1 and g(x) = 2x^2 + 1x - 5 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) = -2sin(1π(4x^2 + 1x + 5)) + 3 and g(x) = -2x + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "How many solutions does the equation 3cos(1π(3x^2 + 5) + 1) - 1 = (-3x + 3)/(1x - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 497 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(1cos(3πx + 3) - 1)^2 - 3(1cos(3πx + 3) - 1) - 2 and g(x) = 5x^2 - 4x + 1 intersect in the interval [-10, 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) = 2cos(3π(-3x + 5) - 1) + 2 and g(x) = (2x - 3)/(2x + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 151 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(1cos(2πx - 3) + 2) - 1)/(-2(1cos(2πx - 3) + 2) + 3) and g(x) = 1x^2 - 3x + 3 intersect 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) = 2cos(2π(5x^2 + 3x + 2) + 3) and g(x) = 4x + 3 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) = 1|(3sin(3πx - 3) + 1) + 2| - 1 and g(x) = 1|x + 2| + 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) = (2(1|x| - 2))/(-1(1|x| - 2) + 2) and g(x) = -4x^2 + 4x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation 1sin(1π(1x^2 - 3x + 1) - 3) - 2 = 1x + 5 have for x in the interval [-10, 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) = -2sin(1π((2x - 2)/(-2x - 1)) + 1) - 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"
}
] | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(-2cos(2πx - 3) + 1)^2 + 5(-2cos(2πx - 3) + 1) + 5 and g(x) = 5x + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-3cos(2πx - 2) + 2) - 2| + 3 and g(x) = -2x^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) = 1sin(2π(-4x + 4) + 3) + 1 and g(x) = 1x - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(3π(2|x| + 2)) + 2 and g(x) = 2|x + 2| - 1 where -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) = 1sin(3π(-3|x - 1| + 3) + 1) + 3 and g(x) = -1|x| + 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 -1|(1cos(2πx) - 3) - 1| + 3 = (-2x)/(1x) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(2sin(2πx - 1) + 1))/(2(2sin(2πx - 1) + 1) + 1) 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"
}
] | 27 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(2π(-4x^2 - 3x - 1) + 3) and g(x) = 3x^2 - 4x 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) = (3(-3cos(3πx - 3) + 2) + 2)/(3(-3cos(3πx - 3) + 2) + 1) and g(x) = 4x^2 + 1x + 3 where -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) = 1sin(2π((-3x + 3)/(-2x - 1)) - 2) - 2 and g(x) = 4x^2 - 1x - 5 in the range -10 ≤ x ≤ 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) = 1cos(2π(2|x + 1| + 1)) - 2 and g(x) = -1x + 3 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(-2cos(1πx - 3) - 3)^2 + 1(-2cos(1πx - 3) - 3) - 5 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"
}
] | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(3|x + 3| - 3)^2 + 3 and g(x) = 4x^2 + 1x - 4 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(1π(-4x - 5) - 2) + 1 and g(x) = -1x - 2 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) = 1sin(1π(-4x + 1) - 2) - 3 and g(x) = -2|x - 1| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(3sin(3πx - 2) - 1) - 1 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"
}
] | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-1cos(1πx + 1) + 2) - 3 and g(x) = (-2x - 3)/(-1x) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(1sin(3πx - 2))^2 - 1(1sin(3πx - 2)) - 2 and g(x) = 1x^2 - 2x + 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 (-1(3cos(2πx - 2)) + 2)/(3(3cos(2πx - 2)) - 2) = (-2x + 3)/(-3x - 1) 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) = -2sin(2π((-2x - 1)/(2x - 1)) + 1) + 3 and g(x) = 4x + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2cos(1π((-2x - 1)/(1x + 2)) + 3) and g(x) = 5x^2 + 3x - 4 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) = 3cos(2π(-3|x + 2| + 3) - 2) + 2 and g(x) = (-1x + 1)/(-1x) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 115 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(2π(1|x - 1| - 3) + 3) 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": "Determine the number of intersection points between the functions f(x) = 2cos(3π(-3|x + 3| + 3)) - 3 and g(x) = (3x + 3)/(3x - 1) where -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) = 2cos(1π((-1x)/(-2x + 1))) and g(x) = -1|x| 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 2(1sin(2πx + 1) + 1)^2 - 3(1sin(2πx + 1) + 1) - 1 = 1|x| - 3 have for x 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 (-2(-2|x - 2| + 3) + 2)/(3(-2|x - 2| + 3) + 3) = -5x^2 + 4x + 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(3cos(2πx + 2))^2 - 1(3cos(2πx + 2)) = 5x^2 + 1x + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1sin(3π(-2x^2 + 1x + 4) + 1) - 3 and g(x) = 3|x - 3| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "How many solutions does the equation -3cos(2π(-5x - 1) - 3) + 2 = (-2x - 2)/(3x + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 193 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(3π(-5x^2 - 3x + 1) - 2) + 3 and g(x) = 5x^2 + 4x + 1 intersect in the interval [-10, 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)/(2(1cos(3πx) - 1) + 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"
}
] | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(3sin(1πx - 3) - 3) + 2| + 2 and g(x) = (-2x)/(3x + 1) in the range -10 ≤ x ≤ 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) = (1(1|x - 1| - 1))/(2(1|x - 1| - 1) - 1) and g(x) = -2|x - 3| + 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) = 3(-1|x + 3| + 1)^2 + 5(-1|x + 3| + 1) + 4 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": "How many points of intersection exist between the functions f(x) = -1sin(2π(2x + 5) - 3) + 3 and g(x) = -3x^2 + 1x + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many solutions does the equation 2|(3cos(1πx + 3) - 2)| - 2 = (-3x - 3)/(-2x + 3) 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) = -1(-1cos(2πx + 1) - 2)^2 and g(x) = (3x - 1)/(-1x - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 27 | MATH |
[
{
"content": "How many solutions does the equation 1(3sin(3πx) + 2) + 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"
}
] | 51 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2cos(2π(-5x + 3) - 2) - 3 and g(x) = -3|x + 1| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 26 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(3cos(3πx - 2) + 1) - 3)/(-3(3cos(3πx - 2) + 1) - 1) and g(x) = -2|x + 3| - 2 where -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) = 5(-1cos(2πx + 1) + 3) and g(x) = 2|x - 2| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "How many solutions does the equation -3cos(1π(-2|x + 2| + 1)) - 3 = (2x + 3)/(-2x - 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 solutions does the equation 2sin(2π(-3x - 4) - 1) + 3 = 3|x| - 3 have for x 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 -2|(2cos(2πx + 2) - 1)| + 2 = -5x^2 + 4x + 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 5(-2cos(2πx) - 3)^2 + 5(-2cos(2πx) - 3) - 3 = 3|x| - 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 solutions does the equation 2sin(2π(-1|x| - 1) - 3) + 1 = (2x - 1)/(-3x) have for x 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) = -4(2cos(2πx + 3) - 3)^2 - 1(2cos(2πx + 3) - 3) + 5 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"
}
] | 27 | MATH |
[
{
"content": "How many solutions does the equation 5(-2cos(3πx + 3) + 1)^2 + 2(-2cos(3πx + 3) + 1) - 1 = (-1x - 3)/(-3x) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 115 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-1sin(2πx + 2) + 1) - 3 and g(x) = 2x - 4 in the range -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) = (1(-2sin(2πx + 3) + 2))/(-1(-2sin(2πx + 3) + 2) + 2) and g(x) = 1x + 4 intersect 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 2sin(3π(3|x - 3| + 3) - 2) - 1 = -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": "Determine the number of intersection points between the functions f(x) = -1sin(2π(1x^2 - 5x - 2) - 1) and g(x) = (2x - 3)/(3x + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 265 | MATH |
[
{
"content": "How many solutions does the equation 4(2sin(2πx + 3)) - 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 2cos(2π(3|x| - 2) - 2) - 1 = (-2x + 3)/(1x) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 96 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(3π(-1|x + 2| - 3) + 3) + 2 and g(x) = 2|x + 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 solutions does the equation (-2(2cos(2πx + 1) - 3) + 2)/(-1(2cos(2πx + 1) - 3) + 2) = (1x - 2)/(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": "How many solutions does the equation 1(-2x^2 - 4x)^2 + 3(-2x^2 - 4x) - 3 = -1x - 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) = -2sin(3π(-1x^2 - 1x - 3)) 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"
}
] | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(3π((-3x + 2)/(2x + 3)) + 2) + 3 and g(x) = -3x^2 - 1x + 2 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) = 3cos(2π(2|x - 2| - 2) + 1) + 2 and g(x) = 1|x + 3| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1sin(3π(2|x| + 2) - 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"
}
] | 12 | MATH |
[
{
"content": "How many solutions does the equation -3sin(3π((3x - 1)/(1x)) + 3) + 3 = 3x - 1 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) = -1sin(3π(1x^2 + 4) + 3) - 3 and g(x) = -2|x - 3| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 36 | MATH |
[
{
"content": "How many solutions does the equation 2e^(-2(2|x - 3| - 1)) - 2 = -1|x| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation 3|(-1sin(2πx + 1) - 2) + 2| + 1 = 2x - 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 solutions does the equation -1cos(1π(2x^2 + 3x + 5) - 3) + 2 = -1|x - 2| + 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 solutions does the equation 3sin(1π(2|x + 3|) + 2) + 2 = (-1x - 3)/(3x + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 39 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(1sin(2πx)) + 3 and g(x) = (-1x - 1)/(-2x) 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)/(-3x + 2))^2 + 4((3x)/(-3x + 2)) + 3 and g(x) = -2|x + 3| + 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 2cos(1π((3x)/(-1x - 2)) - 1) = -1|x + 3| + 3 have for x in the interval [-10, 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) = -2cos(1π((-1x - 3)/(2x + 2)) - 3) - 3 and g(x) = -1x - 4 in the range -10 ≤ x ≤ 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) = 5(3sin(1πx - 2) - 3)^2 - 1(3sin(1πx - 2) - 3) - 1 and g(x) = 2|x - 3| 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 2cos(2π(4x + 5) + 3) - 2 = (-1x + 3)/(3x + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 124 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(2π(-1|x - 2| + 2)) - 1 and g(x) = -1x + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
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