messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-4(-2sin(1πx - 2) + 2) + 5) - 1| - 3 and g(x) = (-2(-5x^2 - 4x - 2) + 1)/(1(-5x^2 - 4x - 2) - 3) 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(3|(4x^2 - 1x - 5) - 3| - 2)^2 - 1 = (-1(-3sin(1πx) - 3) + 3)/(-1(-3sin(1πx) - 3) + 2) have for x 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) = (-2((1(2e^(2x)))/(-1(2e^(2x)) - 2)) + 1)/(-2((1(2e^(2x)))/(-1(2e^(2x)) - 2)) + 3) and g(x) = -2sin(2π(5x^2 - 3x + 5)) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 1481 | MATH |
[
{
"content": "How many solutions does the equation 3((-3((2x - 3)/(-2x - 2)) - 1)/(-3((2x - 3)/(-2x - 2)) + 1)) - 2 = -2sin(1π(5x - 5) - 3) + 3 have for x in the interval [-10, 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) = (-3(5((3x + 2)/(-3x + 2)) + 5) - 3)/(1(5((3x + 2)/(-3x + 2)) + 5) + 3) and g(x) = -1sin(2π(-2x + 3) - 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 45 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(3sin(1π(-3|x - 2| + 1) - 3) + 3)^2 - 3(3sin(1π(-3|x - 2| + 1) - 3) + 3) + 1 and g(x) = (-2((-2x + 3)/(-2x - 2)))/(-1((-2x + 3)/(-2x - 2)) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 55 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(2|(1cos(3πx + 1) - 3) - 3| + 1) + 5 and g(x) = (-2(-3sin(1πx + 2) + 1) - 1)/(-1(-3sin(1πx + 2) + 1) + 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 2(-2(1x^2 - 5x + 2) - 2)^2 - 1 = 3sin(1π((2x - 2)/(1x)) + 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation 2(2|(3x^2 + 2x - 3) - 3|)^2 - 1(2|(3x^2 + 2x - 3) - 3|) - 2 = (-1(5x - 2))/(-3(5x - 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": "Determine the number of intersection points between the functions f(x) = 2|(-2cos(2π(4x + 1) - 1) + 3) - 1| + 2 and g(x) = -2cos(2π(-1x + 1) - 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "How many solutions does the equation -2cos(1π(-2|((-2x - 3)/(2x)) - 3|) - 1) = (-1((1x - 1)/(3x)) + 3)/(3((1x - 1)/(3x))) have for x in the interval [-10, 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) = -3cos(1π(-4(-1|x + 2| + 2) + 5) - 2) - 2 and g(x) = -2e^(2(-2x - 3)) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 47 | MATH |
[
{
"content": "How many solutions does the equation 2(-3|(2sin(2πx - 1) - 2) + 1| + 3) - 4 = (-3(-3cos(3πx + 2) + 2) - 2)/(-2(-3cos(3πx + 2) + 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1sin(2π(1|(1|x - 1| + 3) + 2| + 3)) and g(x) = (1((-1x + 1)/(1x - 3)) + 2)/(-3((-1x + 1)/(1x - 3)) + 4) 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) = -3|(-2cos(3π(-3x - 2) - 3) - 2)| and g(x) = (-1((-3x - 2)/(-1x + 2)) + 2)/(3((-3x - 2)/(-1x + 2))) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 124 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(1cos(1π(1x - 4) + 2) - 2) - 3 and g(x) = 3sin(3π((3x - 1)/(1x + 2)) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 154 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4((1(2|x| - 1) + 1)/(2(2|x| - 1) - 3))^2 - 5((1(2|x| - 1) + 1)/(2(2|x| - 1) - 3)) + 1 and g(x) = 2|(-5x^2 - 4x - 2) - 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 -1|(5(-3x^2 + 2x + 3) + 2) + 2| + 1 = 1(-3|x| - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(3(-1cos(2πx + 1) + 3) + 1) - 3| - 2 and g(x) = 1(-2e^(1x) - 3)^2 + 1(-2e^(1x) - 3) + 1 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) = (1(-4(3x^2 + 4x)^2 + 2(3x^2 + 4x) - 4))/(-3(-4(3x^2 + 4x)^2 + 2(3x^2 + 4x) - 4) + 2) and g(x) = -1cos(3π((-3x)/(3x - 1)) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 35 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(-1(-1x + 2) - 2) + 3| - 2 and g(x) = -3|(4x^2 - 1x - 1) - 3| + 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 -2|(1cos(1π(3|x| + 1) + 2) - 1)| = 2cos(2π(5x^2 + 3) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 658 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(2sin(3π(5x^2 - 2x + 5) - 3) - 3) + 1| - 2 and g(x) = 1e^(-1(-4x^2 + 3x - 2)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(2cos(3π(1|x - 3|) - 3) - 1)^2 + 5 and g(x) = 2(-3x + 5)^2 - 4(-3x + 5) - 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 1cos(3π(-5(-3x + 3) + 4) - 1) + 2 = (3(3cos(3πx + 3)) + 2)/(-3(3cos(3πx + 3)) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 53 | MATH |
[
{
"content": "How many solutions does the equation 1(-3(-5x - 3)^2 + 4(-5x - 3) + 5)^2 - 4(-3(-5x - 3)^2 + 4(-5x - 3) + 5) - 3 = 1sin(1π(1x + 1) - 1) - 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) = 2sin(2π(5(3|x - 1| - 3)^2 - 4(3|x - 1| - 3) - 1) + 1) and g(x) = -4(-2|x - 3| + 1)^2 + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 122 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(2((-1x + 1)/(-3x - 1)) + 2)^2 - 5(2((-1x + 1)/(-3x - 1)) + 2) and g(x) = -3|(2|x - 3| + 3) - 1| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(2(-1|x + 1| + 1)^2 + 3(-1|x + 1| + 1) - 3))/(-3(2(-1|x + 1| + 1)^2 + 3(-1|x + 1| + 1) - 3) + 4) and g(x) = -3|(1|x - 2| - 2) + 2| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(1e^(2(3x^2 - 4)) + 1) - 1| + 1 and g(x) = (-1(-3cos(3πx) - 1) + 2)/(-3(-3cos(3πx) - 1) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many solutions does the equation -1(2(-1x - 1)^2 - 2(-1x - 1) + 4) + 2 = -2sin(3π(3|x + 3|)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(1((1x - 3)/(-3x - 1))^2 + 2((1x - 3)/(-3x - 1)) + 3)^2 + 1(1((1x - 3)/(-3x - 1))^2 + 2((1x - 3)/(-3x - 1)) + 3) + 4 and g(x) = (2(2sin(1πx)) + 1)/(-1(2sin(1πx)) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(-1cos(1π(-4x^2 + 3x - 3) - 2) - 1)^2 + 3 and g(x) = 2|(-1x - 2) - 1| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 108 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(3π(3(-1|x - 3| - 1)^2 - 3) + 1) and g(x) = 3sin(2π(-2|x - 2| - 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 607 | MATH |
[
{
"content": "How many solutions does the equation 1|((3(-2cos(2πx + 1) + 2) - 1)/(1(-2cos(2πx + 1) + 2))) - 3| + 1 = -4((-2x - 1)/(-1x + 2))^2 + 1((-2x - 1)/(-1x + 2)) + 4 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 1|(1(2|x - 2|)) - 3| + 2 = 3(-4x^2 + 4x + 5)^2 + 1(-4x^2 + 4x + 5) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(3π(2(-4x) + 5)) - 2 and g(x) = 4(1e^(-1x) - 3)^2 + 4(1e^(-1x) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "How many solutions does the equation -1((-2(2cos(3πx + 1)) + 3)/(1(2cos(3πx + 1))))^2 - 2((-2(2cos(3πx + 1)) + 3)/(1(2cos(3πx + 1)))) - 4 = -1(2|x| + 2)^2 - 3(2|x| + 2) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 120 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(1π(4(-5x^2 + 5x + 1) - 5) - 3) - 3 and g(x) = 2cos(1π(-2x - 5) + 1) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 981 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(1cos(2π(3|x| - 3)) + 1)^2 - 3(1cos(2π(3|x| - 3)) + 1) + 1 and g(x) = 1cos(1π(-1|x - 2| - 3) + 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(3cos(3π(3|x + 3| - 2) + 2) + 2) - 2)/(-1(3cos(3π(3|x + 3| - 2) + 2) + 2) + 3) and g(x) = -2(-5x - 2)^2 - 4(-5x - 2) + 2 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) = -1cos(2π(2|((3x - 1)/(2x + 1)) + 1| - 1) + 1) and g(x) = -1|(2x^2 - 2) + 1| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(2π((3(4x - 3) + 3)/(2(4x - 3) + 3)) - 2) + 1 and g(x) = 1(2e^(2x) - 1) - 3 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) = -4(-3|(-1cos(2πx - 3)) + 1|) - 4 and g(x) = 3(-1cos(2πx + 3) - 2)^2 + 2(-1cos(2πx + 3) - 2) where -10 ≤ x ≤ 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) = (-2(2cos(1π(-3x^2 + 5x + 5) + 2) + 1))/(1(2cos(1π(-3x^2 + 5x + 5) + 2) + 1)) and g(x) = -1|(2|x + 2| - 2) - 3| + 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) = 4(-1(-2x + 3)^2 - 3(-2x + 3) + 2)^2 + 2(-1(-2x + 3)^2 - 3(-2x + 3) + 2) + 3 and g(x) = 3e^(-1(-1x + 3)) + 3 where -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(1π(2(-1|x - 3| - 3)^2 - 1(-1|x - 3| - 3))) + 3 and g(x) = 2sin(2π(4x^2 + 4) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 760 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1sin(2π((1(1|x + 1| + 1) - 2)/(-1(1|x + 1| + 1) + 2)) + 1) - 1 and g(x) = -3(-3sin(2πx + 3) + 2)^2 - 3(-3sin(2πx + 3) + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1sin(1π(1(3|x + 3| + 1)^2 + 1) - 2) - 1 and g(x) = 5(3sin(3πx - 3) + 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 167 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(1π(-5(4x - 4) - 1) - 3) + 3 and g(x) = 1(5x^2 + 3x + 1)^2 + 1(5x^2 + 3x + 1) - 4 intersect 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 -4((2(-1x + 5) + 2)/(2(-1x + 5) - 1))^2 - 5((2(-1x + 5) + 2)/(2(-1x + 5) - 1)) - 3 = -1(-1sin(3πx + 1) + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many solutions does the equation 3(1|(1x - 4) + 1| - 3) + 5 = -2sin(2π(3|x - 1| - 3) + 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(1((-2x + 3)/(1x)) - 4) - 3)/(1(1((-2x + 3)/(1x)) - 4) - 2) and g(x) = 1(4x^2 + 4x - 5)^2 + 5(4x^2 + 4x - 5) - 3 intersect 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 4((3(-1cos(1πx - 3) - 2) + 2)/(-1(-1cos(1πx - 3) - 2) + 3))^2 - 5((3(-1cos(1πx - 3) - 2) + 2)/(-1(-1cos(1πx - 3) - 2) + 3)) - 2 = 1((-3x + 2)/(-2x - 1))^2 + 3((-3x + 2)/(-2x - 1)) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: ... | 16 | MATH |
[
{
"content": "How many solutions does the equation -4((1(1x + 2) + 2)/(1(1x + 2) - 2)) + 2 = -2sin(1π(-5x^2 - 5x - 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 384 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(3sin(2π(-5x^2 - 1x - 5) + 1) - 1) - 1| - 2 and g(x) = 1sin(3π(2|x + 2| - 1) - 3) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1422 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(1(-2|x + 2| + 1)^2 + 4(-2|x + 2| + 1) - 5)^2 + 4(1(-2|x + 2| + 1)^2 + 4(-2|x + 2| + 1) - 5) - 2 and g(x) = -5(-3x + 2) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 4 | MATH |
[
{
"content": "How many solutions does the equation -3cos(3π(-1((2x - 3)/(-2x - 1)) + 5) + 2) + 2 = -4((-3x - 1)/(-2x - 2)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 28 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-1(-1|x + 2| + 3)^2 + 4(-1|x + 2| + 3) - 5) + 5 and g(x) = 1|(3sin(3πx) + 3) + 2| + 1 in the range -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) = -1cos(3π(4(-2x^2 + 5x - 3) - 1) + 3) - 3 and g(x) = 3(1cos(2πx + 3) - 1) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 437 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(1(-2sin(3πx + 1) + 2)^2 - 5(-2sin(3πx + 1) + 2) + 1)^2 - 5(1(-2sin(3πx + 1) + 2)^2 - 5(-2sin(3πx + 1) + 2) + 1) + 2 and g(x) = -1((-3x - 1)/(3x + 3))^2 - 1((-3x - 1)/(3x + 3)) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer ... | 58 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1sin(3π(-1|(2x) - 2|) - 3) - 2 and g(x) = 3(1x + 2)^2 - 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(-1((2x - 2)/(2x - 1))^2 + 5((2x - 2)/(2x - 1)) + 1)) + 3 = -3cos(1π((2x - 2)/(1x + 3)) - 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 87 | MATH |
[
{
"content": "How many solutions does the equation 3cos(1π(2|(-2|x - 2| - 1) - 3| + 1) + 2) - 1 = 1(-2sin(3πx + 3) - 1)^2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 52 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-3(-2cos(2πx + 3) + 2)^2 - 4(-2cos(2πx + 3) + 2) + 2) and g(x) = (-2(-1sin(1πx + 3) + 3) + 1)/(3(-1sin(1πx + 3) + 3) - 2) 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) = 3(2|(2sin(2πx)) + 3| - 3) - 2 and g(x) = -3|(-1x^2 - 5x + 5) - 2| + 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) = 3(-2sin(3π((-2x - 2)/(2x - 1)) - 3) + 1) - 3 and g(x) = 3cos(3π(1x^2 + 5x - 1) + 1) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 254 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1e^(1(-4((2x - 3)/(2x)) + 3)) + 2 and g(x) = -3sin(3π(-1|x| + 2) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 51 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-3sin(2π((-1x - 1)/(1x + 2)) + 3) - 1) - 3)/(2(-3sin(2π((-1x - 1)/(1x + 2)) + 3) - 1)) and g(x) = (3((2x + 3)/(2x + 2)) - 2)/(-3((2x + 3)/(2x + 2)) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{... | 34 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(1|(2|x - 3| - 1) - 1| + 2)^2 - 4(1|(2|x - 3| - 1) - 1| + 2) + 1 and g(x) = (-2(2cos(3πx + 2)) + 3)/(3(2cos(3πx + 2)) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 8 | MATH |
[
{
"content": "How many solutions does the equation (2((-2(-1cos(2πx + 3) - 2))/(1(-1cos(2πx + 3) - 2) + 2)) + 1)/(-1((-2(-1cos(2πx + 3) - 2))/(1(-1cos(2πx + 3) - 2) + 2)) + 3) = -1(-2x^2 - 2x + 4) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 30 | MATH |
[
{
"content": "How many solutions does the equation -3(-1|(4x^2 - 1x - 5)| + 2) - 5 = 3cos(3π(5x + 4) - 1) have for x in the interval [-10, 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) = -2sin(2π(-1(3x^2 + 3x - 4)^2 - 2(3x^2 + 3x - 4)) + 2) + 3 and g(x) = -2sin(1π((-2x + 1)/(3x - 1)) - 2) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3((1(2x + 2) + 3)/(-1(2x + 2)))^2 - 4((1(2x + 2) + 3)/(-1(2x + 2))) - 3 and g(x) = -1cos(3π(3x - 2) + 2) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many solutions does the equation -1(-2(-2|x - 3|) - 4) = 2sin(1π(3x^2 + 1) - 3) + 2 have for x 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) = 3(-3cos(3π(1|x - 1| + 3) + 2)) + 1 and g(x) = -2(3x + 1) + 2 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(2(1x^2 - 3x - 1) - 2)^2 - 1(2(1x^2 - 3x - 1) - 2) + 4 and g(x) = -5((-2x - 3)/(2x + 2))^2 + 4((-2x - 3)/(2x + 2)) + 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) = (-1(-2cos(2π(1x^2 - 4) + 2) + 1) - 1)/(3(-2cos(2π(1x^2 - 4) + 2) + 1) + 1) and g(x) = -1|(1|x + 1| + 2)| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 73 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(3|(2|x - 1| - 3) - 3| - 2) + 2 and g(x) = (-1(-2x^2 + 5x + 3) - 2)/(-3(-2x^2 + 5x + 3) + 4) 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|(-2(-1e^(-1x) + 1)^2 + 2(-1e^(-1x) + 1) + 1) + 3| - 1 = -1cos(2π(3x^2 - 4x + 3) - 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 526 | MATH |
[
{
"content": "How many solutions does the equation -4(5(-2sin(3πx - 1))^2 - 3(-2sin(3πx - 1)) + 2)^2 + 3(5(-2sin(3πx - 1))^2 - 3(-2sin(3πx - 1)) + 2) + 2 = 4((-3x + 2)/(-2x)) - 4 have for x in the interval [-10, 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) = (1(-2|(-3x^2 + 3x - 1) + 1|) - 1)/(-1(-2|(-3x^2 + 3x - 1) + 1|) - 2) and g(x) = 1(-2|x| + 2)^2 + 1(-2|x| + 2) - 1 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(3|(-1x + 3) - 1| - 3)^2 - 4(3|(-1x + 3) - 1| - 3) - 3 = (-1(-2sin(2πx + 2) - 1) - 1)/(-1(-2sin(2πx + 2) - 1) + 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|(2(4x + 3) + 1) + 2| + 3 and g(x) = 2sin(3π(5x - 1) + 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1cos(1π(-3|((-3x)/(1x - 3)) - 3|) - 3) - 1 and g(x) = -1|(-1sin(1πx - 2) + 2)| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 76 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(1(-3x^2 + 5x + 1)^2 - 2(-3x^2 + 5x + 1) - 1)^2 + 1(1(-3x^2 + 5x + 1)^2 - 2(-3x^2 + 5x + 1) - 1) + 5 and g(x) = -1(-1e^(2x) + 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 4 | MATH |
[
{
"content": "How many solutions does the equation 2sin(3π(1|(-4x^2 + 5x + 2)| - 1)) - 2 = -2|((-2x - 3)/(1x - 3)) + 3| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 319 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|((-2(-3|x + 1| + 2) + 2)/(1(-3|x + 1| + 2) + 3)) + 2| and g(x) = (-3(3cos(2πx + 2) - 2) + 2)/(1(3cos(2πx + 2) - 2) + 2) intersect 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) = 2e^(2((1(-1x - 1) - 1)/(-2(-1x - 1) + 3))) and g(x) = -4(3sin(3πx - 2) - 1)^2 + 5(3sin(3πx - 2) - 1) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 118 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(3cos(3π(1x) - 3) + 3) - 1)/(1(3cos(3π(1x) - 3) + 3) - 3) and g(x) = -3cos(1π(-2x + 3) + 2) - 2 where -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) = -4(2sin(3π((1x + 2)/(-2x + 1))) + 2)^2 + 5(2sin(3π((1x + 2)/(-2x + 1))) + 2) + 4 and g(x) = (-3(5x + 1) - 2)/(3(5x + 1) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 67 | MATH |
[
{
"content": "How many solutions does the equation 4((-3(2x + 1) + 2)/(-3(2x + 1) - 1))^2 - 5((-3(2x + 1) + 2)/(-3(2x + 1) - 1)) + 3 = 3cos(2π(-1|x + 2| - 2) - 3) + 3 have for x 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) = -1|(-3sin(2π((2x + 3)/(2x)) + 1) - 3) - 1| + 1 and g(x) = 2cos(3π((-2x - 3)/(2x + 2)) + 1) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 23 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2((-3(1x^2 + 4x + 3) - 2)/(-2(1x^2 + 4x + 3) + 1)) + 3 and g(x) = -1sin(2π(3x - 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) = 5(3(-1cos(3πx - 1) - 2)^2 - 1(-1cos(3πx - 1) - 2) - 5)^2 - 1(3(-1cos(3πx - 1) - 2)^2 - 1(-1cos(3πx - 1) - 2) - 5) + 1 and g(x) = 1(-2e^(-1x) - 1) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 64 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(1(1x^2 - 2x + 1)^2 - 5(1x^2 - 2x + 1) + 4) - 3| and g(x) = 1|(1|x - 2| + 1) - 1| + 1 in the range -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) = 5(-2cos(3π(-2|x - 1| + 1)) + 1) - 5 and g(x) = 1(1cos(2πx) - 2) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(4(-1sin(2πx + 2) + 1) - 5) + 2| - 2 and g(x) = -3((2x)/(-2x + 3)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 73 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2((-1(-1|x + 2| - 2) - 1)/(1(-1|x + 2| - 2))) + 5 and g(x) = -3cos(3π(-3x^2 - 1x - 4) + 1) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1136 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-1(-1e^(-1x) + 2) + 4) - 3 and g(x) = -5(1cos(2πx - 3) - 2)^2 + 2(1cos(2πx - 3) - 2) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 25 | MATH |
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