messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation 1e^(2((-1(5x^2 + 3x + 2) - 3)/(1(5x^2 + 3x + 2) - 2))) - 2 = 2sin(2π((-1x - 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"
}
] | 112 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((2(2|(1x - 3)| - 2) + 1)/(2(2|(1x - 3)| - 2) + 2))^2 + 5((2(2|(1x - 3)| - 2) + 1)/(2(2|(1x - 3)| - 2) + 2)) - 2 and g(x) = -1(-1cos(3π(1|x + 1| - 3) - 2) + 1)^2 + 4(-1cos(3π(1|x + 1| - 3) - 2) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent... | 56 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-1(1((2x)/(3x + 2)))^2 - 4(1((2x)/(3x + 2))) + 1) + 3| + 1 and g(x) = -4((1(1sin(3πx + 1) - 1) + 1)/(-3(1sin(3πx + 1) - 1)))^2 - 2((1(1sin(3πx + 1) - 1) + 1)/(-3(1sin(3πx + 1) - 1))) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent th... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(2((-3(3sin(2πx - 3)) + 3)/(-2(3sin(2πx - 3)) + 3)) + 2)^2 + 1(2((-3(3sin(2πx - 3)) + 3)/(-2(3sin(2πx - 3)) + 3)) + 2) + 4 and g(x) = -3|(1cos(2π(1x^2 - 3x + 1) - 3))| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex form... | 81 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(1π(-2(-5((-2x - 2)/(-3x - 1))^2 - 1((-2x - 2)/(-3x - 1)) + 2) + 2) + 2) - 1 and g(x) = 1|(5(3x)^2 + 4(3x) - 4) + 3| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 49 | MATH |
[
{
"content": "How many solutions does the equation -1sin(2π(-1|((-2x + 3)/(-2x - 1)) + 3| + 3) + 1) - 1 = (2(-1|x + 2| - 2) - 1)/(-2(-1|x + 2| - 2) + 3) have for x 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) = -3cos(3π(-2|(3x^2) + 2| - 2) + 1) and g(x) = 2|(-2x^2 - 2x) + 3| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 83 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3cos(1π(-2|(-1x + 4) - 2|) + 3) - 2 and g(x) = -2(-3cos(2πx + 3) + 2)^2 - 5(-3cos(2πx + 3) + 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 1e^(-2((-1(2x))/(2(2x) - 3))) - 1 = 2cos(3π((-2x + 2)/(-2x - 1)) - 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": "How many solutions does the equation 2|(2((-2(-1x^2 + 4))/(3(-1x^2 + 4) - 2)) + 4) + 3| + 3 = -5(-1sin(1π(1|x + 3| + 1)) - 2) 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) = 5((2(-2(-2|x - 1| + 2) - 5) - 1)/(-3(-2(-2|x - 1| + 2) - 5) + 4))^2 - 4((2(-2(-2|x - 1| + 2) - 5) - 1)/(-3(-2(-2|x - 1| + 2) - 5) + 4)) - 3 and g(x) = 2(-1|(3x + 1) - 1| + 3)^2 - 3(-1|(3x + 1) - 1| + 3) - 4 intersect in the interval [-10, 10].\n\nPrese... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1cos(3π(-1|(-3x^2 - 1x - 5) + 3| + 1) + 1) + 2 and g(x) = 2cos(1π(2|x - 3| + 3) + 1) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 533 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1sin(1π(1(-4(-1|x + 3| - 2) - 2)^2 - 5(-4(-1|x + 3| - 2) - 2)) + 2) + 2 and g(x) = 2|(-3e^(1((2x - 1)/(1x))) - 2) + 2| - 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(5((-2(-1|x + 2| + 1) - 2)/(-1(-1|x + 2| + 1))) + 1)^2 - 1(5((-2(-1|x + 2| + 1) - 2)/(-1(-1|x + 2| + 1))) + 1) and g(x) = -4(1(-2cos(2πx + 3) - 1) - 1)^2 - 3(1(-2cos(2πx + 3) - 1) - 1) - 4 where -10 ≤ x ≤ 10.\n\nPresent the... | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(3π(4(4(-1|x + 2| + 1)^2 + 1) - 4) - 1) + 3 and g(x) = 2|((-1(-2x^2 - 4x + 1) + 3)/(-2(-2x^2 - 4x + 1) - 3)) + 3| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 129 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-2(3sin(3π(-2|x + 2| - 2) + 2) - 1))^2 + 3(-2(3sin(3π(-2|x + 2| - 2) + 2) - 1)) + 3 and g(x) = -2((-2(2x + 2))/(-1(2x + 2) - 3))^2 + 4((-2(2x + 2))/(-1(2x + 2) - 3)) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex f... | 149 | MATH |
[
{
"content": "How many solutions does the equation 1|((-3(-3sin(2π(-5x - 3) - 3) - 2) - 1)/(-2(-3sin(2π(-5x - 3) - 3) - 2))) + 2| - 1 = 1|(-1cos(1π((-2x + 2)/(-2x + 3)) + 1) + 2) + 1| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 200 | MATH |
[
{
"content": "How many solutions does the equation -1|(1(3(-3cos(3πx + 2) + 3)^2 + 3(-3cos(3πx + 2) + 3) + 5) + 5) - 3| = (2(2(2sin(1πx + 1) - 3) + 4))/(-3(2(2sin(1πx + 1) - 3) + 4) - 3) have for x in the interval [-10, 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) = -2e^(1(-5(3x - 1)^2 + 2(3x - 1) + 5)) and g(x) = -4(2sin(2πx + 1)) + 1 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) = -1(-3sin(1π((3x - 1)/(1x + 3)) - 3) + 3)^2 - 1(-3sin(1π((3x - 1)/(1x + 3)) - 3) + 3) + 1 and g(x) = -1|(1x^2 - 2x + 4) - 1| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 107 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(5(4(3|x - 3|)^2 + 5(3|x - 3|)) + 2) + 1)/(-2(5(4(3|x - 3|)^2 + 5(3|x - 3|)) + 2) + 3) and g(x) = 2|(3sin(3π(3x^2 - 3x) + 2) - 3) - 2| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 29 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(2sin(2π(2|(-5x^2 - 1x + 1) + 1| + 1)) - 1) + 2 and g(x) = -3cos(1π(-2(1x + 1)^2 - 3(1x + 1) - 5) - 2) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 912 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(2sin(2π(2(1x^2 + 3x + 3) - 1) - 1) + 2) + 3| and g(x) = 2|((-2(-5x + 4) - 2)/(-1(-5x + 4) + 2)) - 1| - 1 in the range -10 ≤ x ≤ 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) = -2sin(3π(4(5x^2 - 1x - 3) - 5) + 3) + 2 and g(x) = -3cos(1π(1|x + 2| - 2) + 1) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 530 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-2(2x^2 - 5x) + 1) + 1)/(-3(-2(2x^2 - 5x) + 1) + 2) and g(x) = 2sin(3π(-3x^2 + 1x - 3) + 1) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 975 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(1((1(-3sin(3πx + 1) + 1) - 3)/(2(-3sin(3πx + 1) + 1) - 3))^2 + 5((1(-3sin(3πx + 1) + 1) - 3)/(2(-3sin(3πx + 1) + 1) - 3)) - 4) - 4 and g(x) = -1|(-1sin(1π(-3x^2 + 3x - 5) - 1) - 1) + 2| + 1 intersect in the interval [-10, 10].\n\nPresent the answer i... | 121 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-3|(3cos(1π(2x^2 + 2x - 4) + 3))| - 2) - 5 and g(x) = 1|(2cos(1π(1x - 1) + 2) - 2) + 3| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 797 | MATH |
[
{
"content": "How many solutions does the equation -1|(-3((2(3x^2 + 2x + 4) + 3)/(3(3x^2 + 2x + 4) + 3)) + 5) + 3| + 1 = (-1(2(-2sin(2πx + 2) - 1)^2 - 5(-2sin(2πx + 2) - 1) + 2))/(-3(2(-2sin(2πx + 2) - 1)^2 - 5(-2sin(2πx + 2) - 1) + 2) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex fo... | 13 | MATH |
[
{
"content": "How many solutions does the equation -1(2|(-1((1x - 1)/(-1x - 2)) - 4) + 3| + 2)^2 - 2(2|(-1((1x - 1)/(-1x - 2)) - 4) + 3| + 2) - 2 = (-2(-2(2cos(3πx - 1) - 2)^2 + 1))/(2(-2(2cos(3πx - 1) - 2)^2 + 1) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 17 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(1|(-3((1x)/(3x))^2 - 1((1x)/(3x)) + 1) - 1|) - 5 and g(x) = -3(-2sin(2π((-1x - 1)/(-1x - 2))))^2 - 5(-2sin(2π((-1x - 1)/(-1x - 2)))) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 69 | MATH |
[
{
"content": "How many solutions does the equation -3(-2cos(3π(2x) + 2))^2 + 5(-2cos(3π(2x) + 2)) - 3 = 1(1|x + 2| - 1)^2 + 5(1|x + 2| - 1) + 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 solutions does the equation (-2(2(-3|(2x^2 + 4x - 2) - 1| - 1) - 1) + 3)/(-2(2(-3|(2x^2 + 4x - 2) - 1| - 1) - 1)) = -4(-2(-2sin(2πx + 1) - 3) - 2)^2 - 4(-2(-2sin(2πx + 1) - 3) - 2) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5((3(-1(2|x - 1| - 3)^2 + 4(2|x - 1| - 3) - 4) + 2)/(-3(-1(2|x - 1| - 3)^2 + 4(2|x - 1| - 3) - 4) + 4)) - 3 and g(x) = -1(2(-2sin(1πx + 1) - 3)^2 + 3(-2sin(1πx + 1) - 3) + 4) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTe... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(3(3|x - 1| + 2) - 4) + 1)/(2(3(3|x - 1| + 2) - 4) - 1) and g(x) = 2cos(1π(-5x^2 + 5x - 5) - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 985 | MATH |
[
{
"content": "How many solutions does the equation -3|((-3(-5x - 4) + 3)/(-3(-5x - 4) + 1))| - 2 = -1(3cos(2πx - 1))^2 + 1(3cos(2πx - 1)) + 5 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) = (-1(-1|(2e^(1x) + 1) - 1| - 3) + 2)/(1(-1|(2e^(1x) + 1) - 1| - 3) + 1) and g(x) = -1cos(3π(-1x + 4) - 1) - 3 where -10 ≤ x ≤ 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) = 2|(3e^(-1(1x^2 - 5x + 1)) - 2) - 2| + 3 and g(x) = 1(-2cos(3πx + 1) + 3)^2 + 2(-2cos(3πx + 1) + 3) - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-5(3|(-2cos(1πx + 3)) + 1| - 1) + 3) + 3)/(-1(-5(3|(-2cos(1πx + 3)) + 1| - 1) + 3) + 3) and g(x) = -2sin(3π(3|((-1x - 1)/(2x + 3)) + 3| + 2) - 3) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your... | 48 | MATH |
[
{
"content": "How many solutions does the equation 4((-1(-2(2sin(1πx - 1) - 1)^2 - 4(2sin(1πx - 1) - 1)) + 3)/(-2(-2(2sin(1πx - 1) - 1)^2 - 4(2sin(1πx - 1) - 1)) + 1)) - 3 = -2|(-3sin(2π(1x^2 - 3) - 2)) - 1| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 108 | MATH |
[
{
"content": "How many solutions does the equation -5(3cos(1π((-1x - 3)/(-3x - 3)) + 1))^2 + 5(3cos(1π((-1x - 3)/(-3x - 3)) + 1)) - 1 = -1e^(1((-2x - 3)/(2x - 1))) + 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 points of intersection exist between the functions f(x) = (-3(2|(3cos(3π(2x + 5) - 3) + 2) + 1| - 3) - 1)/(-1(2|(3cos(3π(2x + 5) - 3) + 2) + 1| - 3) + 2) and g(x) = -3|(2e^(-2(2x - 4)) + 1) - 2| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 52 | MATH |
[
{
"content": "How many solutions does the equation (1(1e^(-1(5x^2 - 3x - 5)) - 3) - 3)/(2(1e^(-1(5x^2 - 3x - 5)) - 3) - 1) = -3(-3sin(3πx) - 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 62 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2sin(2π(2|(-5((1x + 1)/(1x - 2))^2 - 1((1x + 1)/(1x - 2)) + 1) + 2| + 1) + 1) - 1 and g(x) = -3((-3(2x + 2) - 3)/(3(2x + 2) - 1))^2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 13 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(4((1(1x + 1) - 3)/(1(1x + 1) + 1))^2 + 1((1(1x + 1) - 3)/(1(1x + 1) + 1)) - 3) - 3| - 2 and g(x) = -4(-5(1sin(3πx + 2) - 1))^2 + 3(-5(1sin(3πx + 2) - 1)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your a... | 56 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-5(-1(-1|x - 3| + 1)^2 - 3(-1|x - 3| + 1) + 3) - 2) - 1)/(3(-5(-1(-1|x - 3| + 1)^2 - 3(-1|x - 3| + 1) + 3) - 2) + 3) and g(x) = 2sin(3π(-1|((-2x - 3)/(1x - 2)) - 2| - 3) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in L... | 168 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3cos(1π((3(-1x) + 2)/(-3(-1x) + 4)) - 1) + 2 and g(x) = 3|((-1x + 1)/(3x - 2))| - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 28 | MATH |
[
{
"content": "How many solutions does the equation -3((1(-3cos(1πx + 2) + 3) + 3)/(-3(-3cos(1πx + 2) + 3)))^2 - 3((1(-3cos(1πx + 2) + 3) + 3)/(-3(-3cos(1πx + 2) + 3))) + 5 = -3|(-3cos(3πx - 1) + 3) - 2| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(1π(-1|(3x + 4) - 1|) - 2) and g(x) = (3(-1e^(1x) + 1) + 2)/(-1(-1e^(1x) + 1) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 33 | MATH |
[
{
"content": "How many solutions does the equation 1(-2|(-2cos(2πx + 2) - 1) + 1| + 2)^2 + 5(-2|(-2cos(2πx + 2) - 1) + 1| + 2) - 3 = -2(2x^2 - 5x - 1) + 5 have for x in the interval [-10, 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) = -2((1(3sin(3πx - 2)) + 1)/(-1(3sin(3πx - 2)) + 3))^2 - 4((1(3sin(3πx - 2)) + 1)/(-1(3sin(3πx - 2)) + 3)) + 3 and g(x) = 2(-2|x - 3| + 2) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 52 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(1π(-5(1x^2 - 4x - 3) + 4) + 1) - 3 and g(x) = -3|(2cos(3πx - 3) - 3) + 1| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 323 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4((-1(2(-1|x + 2| - 1)^2 + 3(-1|x + 2| - 1) + 5) + 2)/(1(2(-1|x + 2| - 1)^2 + 3(-1|x + 2| - 1) + 5) + 3)) + 1 and g(x) = -1(2|(3sin(3πx) - 2) - 2|) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your ... | 54 | MATH |
[
{
"content": "How many solutions does the equation 1sin(3π(5((-1x - 1)/(3x - 1)) - 5) - 1) = (3(1x) - 2)/(-2(1x) + 1) 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 points of intersection exist between the functions f(x) = (-2(1|(-3(3sin(3πx - 2) - 3)^2 - 3(3sin(3πx - 2) - 3) - 1) - 2| - 2) - 3)/(-1(1|(-3(3sin(3πx - 2) - 3)^2 - 3(3sin(3πx - 2) - 3) - 1) - 2| - 2) + 2) and g(x) = 1(-1|((1x - 3)/(1x + 3)) + 2| - 3) - 2 in the range -10 ≤ x ≤ 10?\n\nPre... | 59 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(4(3|((1x - 2)/(3x)) - 3| - 3)^2 + 3(3|((1x - 2)/(3x)) - 3| - 3) - 5) + 4 and g(x) = (2(-1|(2x - 4) - 2| + 2) + 3)/(1(-1|(2x - 4) - 2| + 2)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 6 | MATH |
[
{
"content": "How many solutions does the equation (-1(-3cos(2π(1(4x + 1)^2 + 4(4x + 1) - 2) + 3)) - 3)/(3(-3cos(2π(1(4x + 1)^2 + 4(4x + 1) - 2) + 3))) = (2(1|(1x^2 + 5x + 1) - 2| + 2) + 1)/(1(1|(1x^2 + 5x + 1) - 2| + 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Y... | 344 | MATH |
[
{
"content": "How many solutions does the equation 2|((-1(1x^2 + 2x - 4) + 2)/(-1(1x^2 + 2x - 4) + 3)) - 1| + 3 = 5(3sin(3πx - 1) + 1)^2 - 1(3sin(3πx - 1) + 1) - 4 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) = -1((3(-3|x| - 2) + 2)/(-2(-3|x| - 2) - 2))^2 + 5((3(-3|x| - 2) + 2)/(-2(-3|x| - 2) - 2)) + 3 and g(x) = -5(-2cos(2πx + 2) + 1)^2 + 3(-2cos(2πx + 2) + 1) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{... | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3cos(3π(-4(3|x - 1| - 2) + 4) + 2) + 1 and g(x) = 2cos(2π(2x^2 + 5x + 2)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 755 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(2((1x + 2)/(2x)) - 3)^2 - 1(2((1x + 2)/(2x)) - 3) + 1 and g(x) = -4(3cos(2πx - 3) - 3) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "How many solutions does the equation (3(3|(-1(-3e^(2x) - 1) - 1) + 1| - 1) + 2)/(-3(3|(-1(-3e^(2x) - 1) - 1) + 1| - 1) + 4) = -5(-4(-2cos(3πx + 1) - 2)^2 - 2(-2cos(3πx + 1) - 2) + 1) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(1π((1(-1x^2 - 1x + 4) - 1)/(2(-1x^2 - 1x + 4) - 1)) + 2) + 1 and g(x) = -1(2sin(3πx - 2) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 69 | MATH |
[
{
"content": "How many solutions does the equation 4((-1(-4(2sin(3πx + 1)) + 3) + 2)/(1(-4(2sin(3πx + 1)) + 3) + 1))^2 - 5((-1(-4(2sin(3πx + 1)) + 3) + 2)/(1(-4(2sin(3πx + 1)) + 3) + 1)) + 3 = (3(-3|(3sin(2πx - 3) - 2) + 3| + 2) - 3)/(1(-3|(3sin(2πx - 3) - 2) + 3| + 2) + 3) have for x in the interval [-10, 10]?... | 80 | MATH |
[
{
"content": "How many solutions does the equation (-1(-1sin(3π(4x + 1) - 1) - 3) - 2)/(2(-1sin(3π(4x + 1) - 1) - 3) + 1) = 4(2cos(2πx - 1) + 2) - 5 have for x in the interval [-10, 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) = 2(-2sin(1π(3|x + 3| - 3) - 1)) - 1 and g(x) = 2sin(2π(-3x + 5) + 3) + 2 intersect 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) = 1(-2sin(3π(-2|(-4x + 3) + 2|) + 3) - 2)^2 - 5 and g(x) = -2cos(1π(4(-2x + 4)^2 - 4(-2x + 4) - 2) + 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 478 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(3π((3(2x - 1) + 2)/(1(2x - 1) - 3)) - 2) - 2 and g(x) = -2|(1cos(2πx - 3) + 1) - 1| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 144 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2e^(1(-3|(3((1x - 3)/(-2x + 3))^2 + 2((1x - 3)/(-2x + 3)) + 4) + 1| + 3)) + 2 and g(x) = (2(-2cos(3π(1|x - 1| - 1) - 2) + 2) - 2)/(3(-2cos(3π(1|x - 1| - 1) - 2) + 2) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex fo... | 58 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1cos(3π(1|(-4((1x + 3)/(-3x + 4)) + 4) + 1| - 2) - 2) and g(x) = (1(1|(1x^2) - 1| - 2) - 3)/(-1(1|(1x^2) - 1| - 2)) 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) = -1|(-1(1((-3x + 2)/(2x))^2 + 4((-3x + 2)/(2x)) - 5) - 2) + 1| + 2 and g(x) = 2cos(2π(1((1x - 1)/(1x + 1))^2 - 5((1x - 1)/(1x + 1)) + 3) + 3) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 9 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(2π(-2|((2(5x - 3) - 2)/(1(5x - 3) + 3)) + 3| - 1) - 2) and g(x) = -3|(-2(2sin(3πx - 2) + 1)^2 - 4(2sin(3πx - 2) + 1)) + 2| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 65 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-3(1sin(1πx) + 2)^2 + 1(1sin(1πx) + 2) + 3) - 4 and g(x) = -1|((1x + 1)/(-3x - 1)) + 1| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1cos(1π(1|(4x - 5) + 3| + 1) + 3) + 1 and g(x) = (1(-2|x + 1|) - 2)/(2(-2|x + 1|) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 74 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1cos(3π(-4(-2|x| + 2)^2 + 1(-2|x| + 2)) - 1) + 3 and g(x) = 2e^(-1(-2|x + 3| + 1)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-4(-1cos(3πx + 1)) - 4) + 2)/(-3(-4(-1cos(3πx + 1)) - 4) - 1) and g(x) = 1cos(1π((-1x + 1)/(-1x - 1)) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 53 | MATH |
[
{
"content": "How many solutions does the equation -3(-1|(3cos(3π((-2x - 2)/(-3x + 1)) - 1) - 3) + 1| - 1) - 1 = 1e^(-1(-2(1|x - 3| - 2)^2 - 1(1|x - 3| - 2) + 5)) + 3 have for x in the interval [-10, 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((-1(1sin(3π(-3x - 3) - 2) - 3) - 3)/(1(1sin(3π(-3x - 3) - 2) - 3) + 2))^2 - 4((-1(1sin(3π(-3x - 3) - 2) - 3) - 3)/(1(1sin(3π(-3x - 3) - 2) - 3) + 2)) - 4 and g(x) = 2(2cos(2π(1x^2 + 5x + 3) - 2) + 1) - 3 in the range -10 ≤ x ≤ 10... | 450 | MATH |
[
{
"content": "How many solutions does the equation 1|(3sin(3π((2x)/(-3x + 4)) + 2) + 1)| - 2 = (2(-5x^2 - 5x + 5) + 1)/(-2(-5x^2 - 5x + 5)) have for x in the interval [-10, 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(2(3|(1sin(2πx - 2) + 1) + 1| + 2) + 5)^2 + 2(2(3|(1sin(2πx - 2) + 1) + 1| + 2) + 5) + 2 and g(x) = (1(2sin(3π(2x^2 - 4x - 4) + 2) + 1) + 2)/(-3(2sin(3π(2x^2 - 4x - 4) + 2) + 1) - 3) where -10 ≤ x ≤ 10.\n\nPresent the answe... | 11 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(1cos(1π(-5(5x + 1)^2 + 5(5x + 1) - 3))) - 3)/(2(1cos(1π(-5(5x + 1)^2 + 5(5x + 1) - 3))) + 3) and g(x) = 1((-2(2sin(2πx - 2) + 1) + 3)/(3(2sin(2πx - 2) + 1) - 2))^2 - 1((-2(2sin(2πx - 2) + 1) + 3)/(3(2sin(2πx - 2) + 1) - 2)) - 3 intersect in the inte... | 1131 | MATH |
[
{
"content": "How many solutions does the equation -2|((-1(4x^2 - 2x + 1) - 2)/(1(4x^2 - 2x + 1) - 3)) - 1| + 1 = 1sin(3π((-2x)/(2x - 1))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-1|(3((-2x + 3)/(-1x - 3)) - 2) + 3| + 3)^2 + 2(-1|(3((-2x + 3)/(-1x - 3)) - 2) + 3| + 3) and g(x) = -1((2(-3sin(3πx + 2) + 3) + 2)/(-3(-3sin(3πx + 2) + 3)))^2 - 2((2(-3sin(3πx + 2) + 3) + 2)/(-3(-3sin(3πx + 2) + 3))) + 4 intersect in the interval [-... | 22 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(1(-1(-2|x - 3|)^2 + 1)) - 1)/(1(1(-1(-2|x - 3|)^2 + 1))) and g(x) = -1((-2(1cos(2πx) - 2) + 2)/(-1(1cos(2πx) - 2))) + 4 intersect 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) = -1(-2|((1x + 2)/(2x - 1)) + 1|)^2 + 1(-2|((1x + 2)/(2x - 1)) + 1|) + 4 and g(x) = -1sin(2π(-1x^2 + 3x + 3) - 2) - 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) = -1|(2(-2cos(1π((1x - 3)/(-1x)) - 3) - 3)^2 - 5(-2cos(1π((1x - 3)/(-1x)) - 3) - 3) - 3) + 2| + 3 and g(x) = (-1(-1cos(1π(4x + 3) - 2) + 2) + 2)/(-1(-1cos(1π(4x + 3) - 2) + 2) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in ... | 82 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-1(3(-1|x - 2| + 3) - 4)^2 + 4(3(-1|x - 2| + 3) - 4) - 3) - 3)/(2(-1(3(-1|x - 2| + 3) - 4)^2 + 4(3(-1|x - 2| + 3) - 4) - 3)) and g(x) = 1|((-2(1sin(3πx - 1) + 3) - 2)/(-2(1sin(3πx - 1) + 3) + 1)) - 1| where -10 ≤ x ≤ 10.\n... | 52 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(4((2(3x^2 - 3) - 1)/(-1(3x^2 - 3) - 2)) + 4) - 3| + 1 and g(x) = 5(2cos(2π(-3|x - 3| + 2) + 3) - 2)^2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 116 | MATH |
[
{
"content": "How many solutions does the equation 3(3sin(3π((-2x - 3)/(-1x))) - 1) - 4 = -3|(2cos(3πx + 1) - 3) + 1| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 129 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(3cos(2π((-1x - 3)/(-2x + 3)))) + 5 and g(x) = -2e^(-2((-2x + 1)/(1x - 2))) - 3 in the range -10 ≤ x ≤ 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) = 2sin(1π(-5(-3|x - 2|)^2 + 1(-3|x - 2|) + 2) - 3) - 2 and g(x) = (-3(2sin(1πx + 3) + 3) - 2)/(2(2sin(1πx + 3) + 3) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1460 | MATH |
[
{
"content": "How many solutions does the equation 2|(-3cos(2π((-1(-3x^2 + 3x) - 1)/(-2(-3x^2 + 3x) + 3)) + 3) - 1) - 2| + 1 = (-3(2|(2e^(1x) + 3)| - 1) + 1)/(-2(2|(2e^(1x) + 3)| - 1) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(3(2|x - 3| - 3) + 1) - 3)/(1(3(2|x - 3| - 3) + 1) - 3) and g(x) = 1cos(2π(3x^2 + 4) - 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1168 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3cos(2π(-2(1((-3x - 1)/(2x - 3))^2 - 2) + 4)) - 3 and g(x) = -3(1e^(1((-1x + 2)/(1x))) - 3)^2 in the range -10 ≤ x ≤ 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) = -3|(-2(-2cos(3π((-2x + 2)/(3x - 2)) + 2) - 2)^2 - 5(-2cos(3π((-2x + 2)/(3x - 2)) + 2) - 2) + 1) + 3| + 1 and g(x) = -2(-1sin(3π((2x - 1)/(-1x + 2)) + 3) + 1)^2 + 5(-1sin(3π((2x - 1)/(-1x + 2)) + 3) + 1) - 4 in the range -10 ≤ x ≤ 1... | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(2(2cos(1π((-2x + 1)/(1x)) + 3) + 1) + 2) + 1| - 3 and g(x) = -1e^(2(2|(-4x - 1)| - 2)) - 1 where -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) = 2sin(3π(-1(2x + 5)^2 - 1(2x + 5) - 1) - 1) + 3 and g(x) = -2cos(1π((3x - 1)/(2x + 2)) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 449 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|((-2(5x) + 1)/(3(5x) + 1)) - 3| + 3 and g(x) = 3(1sin(2πx - 3) + 3)^2 - 4(1sin(2πx - 3) + 3) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 42 | MATH |
[
{
"content": "How many solutions does the equation 1sin(2π(-1|((-1x - 2)/(3x)) - 3| - 1) + 1) = -1(-1sin(3πx + 1) - 1) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 73 | MATH |
[
{
"content": "How many solutions does the equation 3|(2(2sin(3πx))^2 + 4(2sin(3πx)) - 3) + 2| = (-2(-2sin(3πx + 3) + 3) - 1)/(1(-2sin(3πx + 3) + 3) - 2) 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) = -4(-2sin(1π(4x - 3)))^2 + 2(-2sin(1π(4x - 3))) + 3 and g(x) = -2|(-1sin(2πx + 1) + 3) - 2| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 160 | MATH |
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