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((1(-2|x - 3| + 1) - 1)/(-2(-2|x - 3| + 1))) - 3)/(3((1(-2|x - 3| + 1) - 1)/(-2(-2|x - 3| + 1))) + 1) and g(x) = 1(3cos(3πx - 1))^2 - 1(3cos(3πx - 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 104 | MATH |
[
{
"content": "How many solutions does the equation (-2(1sin(3π(-5x + 5) - 3) + 2) + 3)/(1(1sin(3π(-5x + 5) - 3) + 2)) = -3sin(3π((-3x - 1)/(-1x - 3)) - 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 135 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(2π(2|(-4x^2 + 4x + 5) - 2| - 1) + 2) and g(x) = -3(3cos(3πx - 1) - 2) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 395 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-2e^(-2(2x^2 - 3x + 3)))^2 - 3(-2e^(-2(2x^2 - 3x + 3))) - 2 and g(x) = -2|(3sin(3πx - 2) - 2) + 2| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 120 | MATH |
[
{
"content": "How many solutions does the equation (1(-2cos(3π((2x + 2)/(2x - 1)) + 3)) - 2)/(-3(-2cos(3π((2x + 2)/(2x - 1)) + 3)) - 2) = -1sin(3π(-1|x + 3| + 1) - 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 47 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(1π(2|(2x + 3) - 2| - 3)) + 1 and g(x) = 1e^(1((-3x - 2)/(3x + 2))) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "How many solutions does the equation (3(-3(2|x + 1| - 1)^2 + 3(2|x + 1| - 1) + 4) + 1)/(-2(-3(2|x + 1| - 1)^2 + 3(2|x + 1| - 1) + 4)) = -1e^(-1(1x^2 + 1)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(-1(1x^2 + 2) + 4)| - 2 and g(x) = -1|((-3x - 2)/(-1x - 2))| - 3 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((-1((2x + 1)/(1x - 3)) - 3)/(1((2x + 1)/(1x - 3)) - 2)) - 2)/(3((-1((2x + 1)/(1x - 3)) - 3)/(1((2x + 1)/(1x - 3)) - 2)) - 2) and g(x) = -3cos(2π((2x + 1)/(3x - 2)) + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: ... | 15 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-3sin(3π(2|x - 1| - 3) - 2) + 2) - 2)/(-1(-3sin(3π(2|x - 1| - 3) - 2) + 2) + 2) and g(x) = -3|(-3cos(3πx - 3) - 3) - 2| - 3 intersect in the interval [-10, 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) = 2sin(2π(2(5x^2 - 4x + 1)^2 + 5(5x^2 - 4x + 1) + 3) - 3) and g(x) = (-2((-2x - 1)/(-2x - 2)) - 2)/(2((-2x - 1)/(-2x - 2))) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 187 | MATH |
[
{
"content": "How many solutions does the equation -3|((-2(2|x + 2| + 1) + 2)/(-1(2|x + 2| + 1) + 2)) - 1| = (1(3cos(1πx - 3) - 2) - 3)/(-1(3cos(1πx - 3) - 2)) 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) = -1cos(2π(3(-2|x - 2| + 1)^2 - 3(-2|x - 2| + 1) - 4) + 1) + 2 and g(x) = 2sin(3π((-1x + 2)/(-3x + 1)) - 1) - 1 in the range -10 ≤ x ≤ 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(1sin(2πx - 2))^2 + 3(1sin(2πx - 2)) - 2) - 2 and g(x) = (1((2x + 1)/(-2x + 3)) - 2)/(2((2x + 1)/(-2x + 3)) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 35 | MATH |
[
{
"content": "How many solutions does the equation -1((3(-2x^2 + 1x - 5) - 1)/(2(-2x^2 + 1x - 5) + 2)) + 4 = 1|(4x^2 - 3x - 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": "Determine the number of intersection points between the functions f(x) = -3sin(3π(-1|(-1x + 4) - 3| + 3) - 2) - 1 and g(x) = -1e^(1(-4x^2 - 4x)) - 3 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) = 5((1(1|x + 3| - 2) + 2)/(1(1|x + 3| - 2) + 1))^2 - 5((1(1|x + 3| - 2) + 2)/(1(1|x + 3| - 2) + 1)) - 1 and g(x) = (-1(-5x^2 - 1x - 5) - 2)/(-3(-5x^2 - 1x - 5) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\b... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(4(-2x - 2) + 5) + 3)/(3(4(-2x - 2) + 5) + 1) and g(x) = 2cos(1π(1x + 1) - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2cos(2π(-3(-4x - 5)^2 - 1(-4x - 5) - 3) + 2) - 1 and g(x) = (-1(5x - 5) + 2)/(2(5x - 5)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1510 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(1cos(1π(-3x^2 - 3x - 4) + 3) - 1)^2 + 4(1cos(1π(-3x^2 - 3x - 4) + 3) - 1) - 5 and g(x) = -2|(1x^2 + 2x + 5)| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(2|((-3x - 1)/(-3x + 2)) + 3| - 3))/(2(2|((-3x - 1)/(-3x + 2)) + 3| - 3) + 3) and g(x) = -3cos(1π(-5x + 3)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 98 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-3(-3cos(2πx + 3) - 1) - 3) + 5 and g(x) = (-1(1cos(3πx - 2) + 3) + 1)/(1(1cos(3πx - 2) + 3) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100 | MATH |
[
{
"content": "How many solutions does the equation -3|(1(2|x + 2|)^2 + 1(2|x + 2|) - 1) - 1| + 3 = (-1(2e^(2x) - 3) - 3)/(-3(2e^(2x) - 3) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3cos(1π(4(1x + 3)^2 - 2(1x + 3) + 4) + 1) + 3 and g(x) = 2(-1x^2 + 3x + 3)^2 - 4(-1x^2 + 3x + 3) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(5(-3|x + 3| + 3))^2 - 5(5(-3|x + 3| + 3)) + 4 and g(x) = (1(1cos(3πx + 2) - 2))/(1(1cos(3πx + 2) - 2) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 29 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(2(-3cos(3πx + 3) - 1) - 4) - 2| + 3 and g(x) = 1(1x^2 + 5x + 4) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 26 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2((-2(-1sin(3πx + 3) - 3) + 3)/(-2(-1sin(3πx + 3) - 3) - 1)) + 3 and g(x) = 5(3cos(1πx + 2) - 1) - 3 in the range -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(1π(1x - 1) + 3) + 2) + 3 = -2e^(1(-4x^2 - 2x - 1)) - 1 have for x 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(2|(-2cos(1πx - 3) + 2) - 3| - 2) + 1)/(2(2|(-2cos(1πx - 3) + 2) - 3| - 2) - 3) and g(x) = -3cos(3π(-1|x - 2| + 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "How many solutions does the equation 3|(-1e^(1(-2|x - 3| - 1)))| - 3 = -2cos(2π((1x + 1)/(2x)) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "How many solutions does the equation -3cos(1π(-5(-1|x + 1| - 1)^2 - 5(-1|x + 1| - 1) - 3) + 3) + 2 = (1(1|x| + 1) - 3)/(-2(1|x| + 1) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 865 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(2|(1|x + 1| + 1) - 3|) - 2)/(2(2|(1|x + 1| + 1) - 3|) + 3) and g(x) = 2cos(3π(4x^2 - 2x - 1) - 1) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1576 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(-2sin(3π(-5x - 4) + 2) - 2)^2 - 3(-2sin(3π(-5x - 4) + 2) - 2) - 4 and g(x) = -3cos(1π((-2x - 2)/(-1x)) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 282 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2sin(1π(-3(3x^2 + 5x + 3)^2 - 2) + 3) + 1 and g(x) = 3e^(1((3x + 2)/(-2x + 3))) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 241 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(-2sin(3π(5x + 2)) + 1)| + 2 and g(x) = -3|(2|x + 2| - 3) + 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "How many solutions does the equation 3(-3cos(2π((3x - 3)/(-2x)) - 2) - 1)^2 - 5 = -5(-2|x + 2|) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(3sin(3π(4x - 1) - 1) - 1) and g(x) = 1|(-3x + 4) + 1| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "How many solutions does the equation -2|((-1(-1x + 1))/(-1(-1x + 1) + 2)) + 3| - 1 = 5(-2sin(3πx - 1) - 2)^2 + 5(-2sin(3πx - 1) - 2) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(2sin(3π(-1|x - 2| + 3) - 1))^2 - 4(2sin(3π(-1|x - 2| + 3) - 1)) + 1 and g(x) = 3sin(2π(1|x| - 1)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "How many solutions does the equation -3cos(2π((3(2x^2 - 2) + 2)/(2(2x^2 - 2) - 1)) - 2) + 3 = -3cos(2π(3|x - 3| - 2) + 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 155 | MATH |
[
{
"content": "How many solutions does the equation -2(2cos(3π((-1x + 2)/(3x - 2)) - 3) - 3) - 2 = 2((2x - 3)/(-1x - 3))^2 + 5((2x - 3)/(-1x - 3)) - 2 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) = -5((2(-2cos(3πx + 1) + 1) - 3)/(-2(-2cos(3πx + 1) + 1)))^2 - 1 and g(x) = (-3(-3cos(3πx + 2) + 1) + 1)/(3(-3cos(3πx + 2) + 1) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 93 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1e^(-1(1(5x^2 - 2x))) + 2 and g(x) = 2sin(2π(-2x + 2) + 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-5(-1|x + 2| - 1)^2 + 4(-1|x + 2| - 1)) - 3| + 3 and g(x) = -2(2x^2 + 5x - 5)^2 + 4(2x^2 + 5x - 5) + 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) = 1((3((-2x)/(2x + 3)) + 2)/(-3((-2x)/(2x + 3)) + 4))^2 + 4((3((-2x)/(2x + 3)) + 2)/(-3((-2x)/(2x + 3)) + 4)) - 5 and g(x) = -2(-3cos(1πx - 3) + 3)^2 + 3(-3cos(1πx - 3) + 3) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTe... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((2(3cos(2πx)) - 1)/(-3(3cos(2πx)) + 2))^2 + 3((2(3cos(2πx)) - 1)/(-3(3cos(2πx)) + 2)) + 3 and g(x) = (1(5x^2 - 4x + 1) - 1)/(1(5x^2 - 4x + 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 64 | MATH |
[
{
"content": "How many solutions does the equation 1cos(3π(-1|(2x + 3) - 2| - 3) + 1) + 1 = 3|(-1x + 2) - 3| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(2|(-1|x - 3| + 1) + 1| - 2) - 1)/(3(2|(-1|x - 3| + 1) + 1| - 2) - 2) and g(x) = (3(3|x + 1| + 3) - 3)/(3(3|x + 1| + 3) + 3) 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) = (3(-3sin(3π(1x + 1) + 2) + 2) - 3)/(3(-3sin(3π(1x + 1) + 2) + 2) - 2) and g(x) = (2(-5x + 4) - 3)/(-3(-5x + 4) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 61 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2cos(2π(3(-3|x + 2| - 3) + 1)) - 1 and g(x) = -4(-2x^2 + 3x + 4) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(3(3|x + 1| - 3)^2 + 5(3|x + 1| - 3) + 1)^2 - 1(3(3|x + 1| - 3)^2 + 5(3|x + 1| - 3) + 1) + 2 and g(x) = -2(-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": "Find the number of times the graphs of f(x) = 3sin(2π(-2|(3x) + 1|) + 1) + 3 and g(x) = 3cos(3π(4x + 3) - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 240 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-5(-1cos(3πx + 3) + 1) + 4) - 1)/(2(-5(-1cos(3πx + 3) + 1) + 4) + 2) and g(x) = (3(-1sin(2πx + 3) + 1) + 3)/(-3(-1sin(2πx + 3) + 1)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 49 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(3|(-1x^2 - 3x + 4) + 2| - 1)^2 - 4(3|(-1x^2 - 3x + 4) + 2| - 1) + 4 and g(x) = -5(2|x - 3| - 2)^2 + 4(2|x - 3| - 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) = 4(2|(1x^2 - 4x + 3) + 2| + 1) - 4 and g(x) = (-1(-1cos(1πx - 3)) + 2)/(-3(-1cos(1πx - 3)) + 1) 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) = -3cos(1π(4(2|x| + 2) - 4) + 3) + 3 and g(x) = 1|(-2sin(3πx - 3)) + 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 160 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(2|(4x + 2) + 2| - 1)^2 + 3(2|(4x + 2) + 2| - 1) + 2 and g(x) = (2(-3sin(2πx - 1) - 2) + 3)/(-3(-3sin(2πx - 1) - 2) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 7 | MATH |
[
{
"content": "How many solutions does the equation 2|(5(-1x - 4) + 3) - 1| + 1 = 1|((1x - 2)/(-3x - 3)) - 2| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation -5(-3(-2x + 4) + 1)^2 + 4(-3(-2x + 4) + 1) + 1 = -3((2x + 3)/(-1x))^2 - 5((2x + 3)/(-1x)) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation (2(4(1cos(3πx - 3) - 3) - 3) + 1)/(3(4(1cos(3πx - 3) - 3) - 3) - 1) = -2|(2|x - 2| + 2) - 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 points of intersection exist between the functions f(x) = -5(1cos(2π(1|x + 1|) + 2)) + 2 and g(x) = 2|(-3cos(3πx + 3) + 1) - 1| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "How many solutions does the equation (-2(-2(2cos(3πx - 1) + 2)^2 + 5(2cos(3πx - 1) + 2) - 1) + 3)/(2(-2(2cos(3πx - 1) + 2)^2 + 5(2cos(3πx - 1) + 2) - 1) + 2) = 2|((-1x + 3)/(-1x - 2)) + 3| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 97 | MATH |
[
{
"content": "How many solutions does the equation (-2(-2cos(2π(-1x^2 - 4x) + 1)) - 3)/(2(-2cos(2π(-1x^2 - 4x) + 1)) - 3) = (3(2sin(2πx)) - 2)/(3(2sin(2πx)) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 404 | MATH |
[
{
"content": "How many solutions does the equation -5(3cos(1π(2|x - 3| - 3)) + 2)^2 + 1(3cos(1π(2|x - 3| - 3)) + 2) + 2 = 3((2x + 2)/(1x - 2)) have for x 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 4(3cos(1π(-4x^2 + 4) + 2) + 1)^2 + 1(3cos(1π(-4x^2 + 4) + 2) + 1) + 2 = 1(-1|x + 2| + 3)^2 + 2(-1|x + 2| + 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 561 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(1π(1(4x^2 - 2x + 2)^2 + 3(4x^2 - 2x + 2) - 1) + 1) + 3 and g(x) = (2(-4x^2 - 1x + 1) - 2)/(-2(-4x^2 - 1x + 1) - 3) 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 1(-1sin(2π(1x^2 - 1x + 5) - 1) + 1) + 2 = 1(1|x - 1| - 3) - 5 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) = (2(1e^(-2(1|x - 3| + 3)) + 1) + 2)/(3(1e^(-2(1|x - 3| + 3)) + 1)) and g(x) = (-3(-2sin(2πx + 3) + 2) + 2)/(1(-2sin(2πx + 3) + 2)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 40 | MATH |
[
{
"content": "How many solutions does the equation (-2(-1((-1x + 3)/(2x + 2)) - 5) + 2)/(-3(-1((-1x + 3)/(2x + 2)) - 5) - 1) = -2sin(1π(3x - 2) + 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 57 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5((3(-3cos(3πx) + 3))/(2(-3cos(3πx) + 3) + 3))^2 + 1((3(-3cos(3πx) + 3))/(2(-3cos(3πx) + 3) + 3)) - 5 and g(x) = 3|(-2cos(2πx - 1) + 2) - 2| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 40 | MATH |
[
{
"content": "How many solutions does the equation (3((-2(3sin(1πx - 3) - 2) - 2)/(-1(3sin(1πx - 3) - 2) + 2)))/(2((-2(3sin(1πx - 3) - 2) - 2)/(-1(3sin(1πx - 3) - 2) + 2)) + 3) = -3|(2x^2 - 1x) - 3| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 7 | MATH |
[
{
"content": "How many solutions does the equation -4(2(4x - 1)^2 + 1(4x - 1) - 4) - 4 = (2(1cos(2πx - 1) - 1) + 2)/(-1(1cos(2πx - 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": "Determine the number of intersection points between the functions f(x) = -3(1(1x^2 - 5x + 1)^2 + 4(1x^2 - 5x + 1) + 4)^2 + 5(1(1x^2 - 5x + 1)^2 + 4(1x^2 - 5x + 1) + 4) + 1 and g(x) = -3sin(3π(1x^2 + 3x + 2) + 3) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 38 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-2(-3|x| + 3)) - 2)/(1(-2(-3|x| + 3)) + 1) and g(x) = 1(-3x^2 + 5x - 5) + 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) = 3|(-2((-2x - 2)/(2x + 3)) - 5) + 2| - 2 and g(x) = -3|(-2cos(2πx + 3) + 2) - 3| + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 68 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(3(1cos(3πx - 1) - 1)^2 + 2(1cos(3πx - 1) - 1))^2 - 4(3(1cos(3πx - 1) - 1)^2 + 2(1cos(3πx - 1) - 1)) - 4 and g(x) = 1cos(1π(1x^2 - 1x - 3) - 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 31 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-5(-1|x + 1| + 3) + 2) + 3)/(2(-5(-1|x + 1| + 3) + 2) + 1) and g(x) = 2|((2x - 2)/(-3x - 2)) - 1| - 2 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(3π(1((-3x - 3)/(1x + 2))^2 + 1((-3x - 3)/(1x + 2)) - 2) + 2) - 2 = -2(-2|x - 1|) - 4 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) = (-1((3(3cos(2πx - 2) + 1) - 3)/(2(3cos(2πx - 2) + 1) - 1)) - 3)/(-2((3(3cos(2πx - 2) + 1) - 3)/(2(3cos(2πx - 2) + 1) - 1)) - 2) and g(x) = -3sin(2π(2x - 5) - 3) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: ... | 98 | MATH |
[
{
"content": "How many solutions does the equation (-2(-1(5x^2 - 4x - 5)^2 + 1(5x^2 - 4x - 5) + 1) + 3)/(1(-1(5x^2 - 4x - 5)^2 + 1(5x^2 - 4x - 5) + 1) + 2) = 1|(-2sin(2πx + 2) + 2)| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(1π(1(-3|x| + 1)^2 + 3(-3|x| + 1) - 2) + 1) - 1 and g(x) = (-2(-2x^2 + 3x - 2))/(-3(-2x^2 + 3x - 2) + 4) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 907 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|((-2(-3|x + 1| + 1) - 3)/(3(-3|x + 1| + 1))) - 3| and g(x) = (-1(3|x + 1| - 2) + 1)/(3(3|x + 1| - 2)) 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) = 1|(4(-3x)^2 - 4(-3x) + 3) + 2| + 2 and g(x) = (2(2sin(3πx + 3)))/(-2(2sin(3πx + 3)) + 3) where -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(4(-3x^2 - 1)^2 - 3(-3x^2 - 1) - 5))/(3(4(-3x^2 - 1)^2 - 3(-3x^2 - 1) - 5) + 1) and g(x) = -3sin(3π(-1x^2 + 5x - 5) - 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 638 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(3sin(1π(2x - 3)) - 1) - 1 and g(x) = -5(1x + 4)^2 - 5(1x + 4) + 5 in the range -10 ≤ x ≤ 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(-2sin(2π((-2x - 2)/(-3x)) - 1) - 2) + 3)/(2(-2sin(2π((-2x - 2)/(-3x)) - 1) - 2) - 1) and g(x) = -3sin(1π(1x - 1) - 2) - 1 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 (3(3|(1x^2 + 3)| - 3) + 1)/(-1(3|(1x^2 + 3)| - 3) + 2) = 3cos(3π(-3x + 5) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 162 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(-2(-3sin(2πx + 2) + 2) - 1)^2 - 1(-2(-3sin(2πx + 2) + 2) - 1) - 4 and g(x) = 1(1x) - 4 where -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|(1(-4x^2 + 3x + 5)^2 + 4(-4x^2 + 3x + 5) - 5) - 2| - 3 and g(x) = -5((-1x)/(-1x + 2)) - 5 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) = 1e^(-1((-1(-5x - 3) + 1)/(3(-5x - 3) - 1))) - 2 and g(x) = -2(-1|x + 3| + 2)^2 - 4(-1|x + 3| + 2) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-1((3x)/(-3x - 1)) + 3) - 3)/(3(-1((3x)/(-3x - 1)) + 3) - 2) and g(x) = 3cos(2π(-3x) - 1) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 121 | MATH |
[
{
"content": "How many solutions does the equation -2(2cos(3π((1x + 2)/(2x - 3)) + 1) + 1)^2 + 4(2cos(3π((1x + 2)/(2x - 3)) + 1) + 1) + 2 = 3(-2x + 3)^2 + 2(-2x + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 77 | MATH |
[
{
"content": "How many solutions does the equation -2((3(-2|x + 3| - 2) + 3)/(3(-2|x + 3| - 2) - 1))^2 + 3((3(-2|x + 3| - 2) + 3)/(3(-2|x + 3| - 2) - 1)) - 2 = -2sin(2π(4x^2 - 4x - 1) - 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 489 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(5(-3sin(2πx + 1) + 1)^2 + 1(-3sin(2πx + 1) + 1))| + 3 and g(x) = -1|(3x + 4) + 1| - 3 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) = 1(-2((-1x + 3)/(-3x + 1))^2 + 5((-1x + 3)/(-3x + 1)) + 4)^2 - 5(-2((-1x + 3)/(-3x + 1))^2 + 5((-1x + 3)/(-3x + 1)) + 4) + 1 and g(x) = 2|(-2sin(3πx + 1) + 3) + 3| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-3sin(3π(-1|x + 3| - 1) + 2) - 2)^2 + 5 and g(x) = -3cos(3π(-1x^2 - 1x + 2) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 144 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5((3(-2|x + 1| + 2) + 1)/(-1(-2|x + 1| + 2) - 3)) - 2 and g(x) = 2(3cos(2πx - 1) - 1) - 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 3e^(1(1|((1x - 3)/(2x - 1)) + 3| - 2)) - 1 = 3|(-3sin(1πx - 2)) - 3| have for x 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) = 3(2sin(1π((-1x)/(-1x - 3)) - 3) - 3)^2 + 4(2sin(1π((-1x)/(-1x - 3)) - 3) - 3) + 1 and g(x) = 1((-2x - 2)/(-1x - 2)) - 5 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) = (1((1(-1x^2 + 2x - 1))/(1(-1x^2 + 2x - 1) + 1)) + 1)/(-1((1(-1x^2 + 2x - 1))/(1(-1x^2 + 2x - 1) + 1)) + 2) and g(x) = 3cos(1π(5x^2 - 2x + 1) - 3) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 14 | MATH |
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