messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation -2cos(1π(2((2x)/(3x - 2)) - 5) + 1) = (2(-1e^(1x) + 2))/(3(-1e^(1x) + 2) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "How many solutions does the equation -2|(1sin(3π((1x - 1)/(-1x + 3)) - 3) + 2) - 1| + 3 = 2cos(1π(1x) + 3) + 1 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(-1|((2x - 2)/(-3x + 4)) + 2| - 1) - 2 = 1(-2cos(3πx + 1) + 1)^2 - 2(-2cos(3πx + 1) + 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 116 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1cos(3π(-2|(4x^2 + 1)| + 1) - 1) - 1 and g(x) = 1sin(1π((-3x - 2)/(3x - 1)) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 198 | MATH |
[
{
"content": "How many solutions does the equation (1(1|(2(3x + 2)^2 - 2(3x + 2) - 3) - 1|) + 3)/(1(1|(2(3x + 2)^2 - 2(3x + 2) - 3) - 1|) - 3) = -4((2(3sin(1πx) - 3) + 1)/(3(3sin(1πx) - 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": "Determine the number of intersection points between the functions f(x) = -3cos(3π((2(2|x - 1| + 3) + 1)/(1(2|x - 1| + 3) + 3)) - 2) and g(x) = -3(-1sin(1πx + 2) + 2) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(-3sin(2π((3x + 3)/(1x - 2))) + 2)^2 + 3(-3sin(2π((3x + 3)/(1x - 2))) + 2) and g(x) = -3e^(-2(-2|x + 1| + 3)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(1sin(1π(-3|(1x^2 - 4x + 4) - 1| + 3) - 3) - 1) - 3)/(-3(1sin(1π(-3|(1x^2 - 4x + 4) - 1| + 3) - 3) - 1) - 1) and g(x) = -4(-2|(3x + 5) + 3| - 2)^2 - 4(-2|(3x + 5) + 3| - 2) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in ... | 9 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|((2(1x - 5) - 1)/(2(1x - 5))) - 1| + 2 and g(x) = 3sin(3π(-4x^2 + 1x - 4) - 2) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1493 | MATH |
[
{
"content": "How many solutions does the equation (-1(-4(3sin(3π(-4x^2 - 4x + 2)) - 3) + 2) + 1)/(1(-4(3sin(3π(-4x^2 - 4x + 2)) - 3) + 2)) = 1((-1(1x^2 + 5x + 2) - 1)/(-2(1x^2 + 5x + 2) - 3)) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 153 | MATH |
[
{
"content": "How many solutions does the equation 2|(-2sin(1π(2((1x)/(-3x - 1)) + 3) - 1) - 3) + 2| = 3cos(1π((3(-1x^2 + 4x + 2) + 2)/(-1(-1x^2 + 4x + 2) - 2)) + 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 72 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-5(-1cos(3πx - 3)) + 4) - 1)/(3(-5(-1cos(3πx - 3)) + 4) + 1) and g(x) = -5(-3|x - 2| + 1)^2 + 5(-3|x - 2| + 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(-5((2(-3|x| + 1) + 2)/(3(-3|x| + 1) + 1)) + 1)^2 - 1(-5((2(-3|x| + 1) + 2)/(3(-3|x| + 1) + 1)) + 1) - 1 = -4(3sin(3π(-2|x + 3| + 3)) + 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 116 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|((-2(1x^2 - 4x) + 2)/(3(1x^2 - 4x) + 1)) - 2| and g(x) = -2cos(3π(4x + 1) - 1) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 217 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-1(-1sin(2πx + 1) - 1) - 3) + 2| and g(x) = (-2(-4x^2 + 1x - 4) + 3)/(-2(-4x^2 + 1x - 4) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-2(1|x - 3| - 2) - 3)^2 - 5(-2(1|x - 3| - 2) - 3) - 2 and g(x) = 1cos(3π(1x^2 - 4x - 3) - 2) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "How many solutions does the equation -1((-3(-1|(-3x) + 1|))/(-3(-1|(-3x) + 1|) + 4))^2 + 3((-3(-1|(-3x) + 1|))/(-3(-1|(-3x) + 1|) + 4)) = (2(3(3cos(1πx - 2) - 3)^2 - 1(3cos(1πx - 2) - 3)) + 1)/(1(3(3cos(1πx - 2) - 3)^2 - 1(3cos(1πx - 2) - 3)) + 2) have for x in the interval [-10, 10]?\n\nPresent t... | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|((2(-2sin(3πx - 2) + 1))/(-2(-2sin(3πx - 2) + 1) - 1)) + 1| + 2 and g(x) = 3e^(-2(-2|x + 3| + 2)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "How many solutions does the equation -5((-1(-2x^2 - 1x - 5) + 2)/(-1(-2x^2 - 1x - 5) + 2)) + 3 = -2|(1sin(3πx + 2) - 2)| + 2 have for x 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) = (2(-1(3sin(2π(-2|x + 3| - 2) - 3) - 1)^2 - 4(3sin(2π(-2|x + 3| - 2) - 3) - 1) + 2) + 1)/(-2(-1(3sin(2π(-2|x + 3| - 2) - 3) - 1)^2 - 4(3sin(2π(-2|x + 3| - 2) - 3) - 1) + 2) + 3) and g(x) = (3(-4(2sin(2πx + 2) + 3) + 5) + 1)/(-1(-4(2... | 108 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(3e^(2(-3|x| - 3)) + 3) + 2)/(2(3e^(2(-3|x| - 3)) + 3) + 3) and g(x) = 1|(3sin(1πx)) + 1| 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) = -2|(1sin(3π(5(-2x + 3)^2 - 5(-2x + 3) - 1) - 1) - 3) + 3| + 3 and g(x) = 4(1sin(3π((-1x + 1)/(1x - 2)) + 3) - 1)^2 - 3(1sin(3π((-1x + 1)/(1x - 2)) + 3) - 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\box... | 96 | MATH |
[
{
"content": "How many solutions does the equation 2e^(-1((1(5(-4x^2 - 3x + 2) - 2) - 1)/(-2(5(-4x^2 - 3x + 2) - 2) - 1))) - 1 = 2|((1(-2sin(2πx + 3) + 2))/(3(-2sin(2πx + 3) + 2) - 2))| + 1 have for x 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(-1|(-4(-3x - 1)^2 + 2(-3x - 1) + 4) - 1| + 2) - 1)/(-2(-1|(-4(-3x - 1)^2 + 2(-3x - 1) + 4) - 1| + 2)) = (-3(1cos(1π(3|x + 1| - 2) - 3) + 2) + 2)/(-1(1cos(1π(3|x + 1| - 2) - 3) + 2) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex for... | 64 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3((2(-3(-2|x - 2| + 3) + 5))/(3(-3(-2|x - 2| + 3) + 5) + 1))^2 - 4((2(-3(-2|x - 2| + 3) + 5))/(3(-3(-2|x - 2| + 3) + 5) + 1)) + 5 and g(x) = 2|(1(5x - 1)^2 - 5(5x - 1) + 2) + 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex fo... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(2|(3(3x^2 + 2x - 4) + 2) - 1| - 2) - 1)/(2(2|(3(3x^2 + 2x - 4) + 2) - 1| - 2) - 1) and g(x) = 1(-1|(3x^2 - 1x + 2)| + 2) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 6 | MATH |
[
{
"content": "How many solutions does the equation -3|((1(3e^(1x) + 2) + 3)/(-3(3e^(1x) + 2) - 1))| - 3 = 2cos(2π((-2x + 2)/(3x)) + 3) - 3 have for x in the interval [-10, 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) = (-1(1|(3sin(3π(5x + 5) - 2) + 3)|) + 2)/(2(1|(3sin(3π(5x + 5) - 2) + 3)|) - 1) and g(x) = (-1(-2e^(1(-1|x - 1| - 2)) + 2) - 2)/(-3(-2e^(1(-1|x - 1| - 2)) + 2) - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\... | 300 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(4(3(3|x - 3| - 1) + 4)^2 - 1(3(3|x - 3| - 1) + 4) - 1) - 2)/(-2(4(3(3|x - 3| - 1) + 4)^2 - 1(3(3|x - 3| - 1) + 4) - 1) - 2) and g(x) = -2cos(3π(1(-2|x - 1| - 2)^2 + 5(-2|x - 1| - 2) - 2) - 2) - 2 intersect in the interval [-10, 10].\n\nPresent the a... | 1511 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|((2(-4(-2x)^2 + 3(-2x) - 1) - 1)/(-1(-4(-2x)^2 + 3(-2x) - 1) + 3))| - 1 and g(x) = (-1(-5(-2sin(3πx + 3) - 2) - 4) - 2)/(-1(-5(-2sin(3πx + 3) - 2) - 4) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|((2(-4x^2 - 5x - 1) + 3)/(-3(-4x^2 - 5x - 1) - 1)) + 3| - 2 and g(x) = -1(1|x + 2|) + 3 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(1|(-2x + 1) + 1|)^2 - 4(1|(-2x + 1) + 1|) + 3 and g(x) = -3cos(3π(4x) + 1) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 51 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3sin(3π(-2(1x^2 + 4x + 5) - 4) + 2) - 1 and g(x) = (1(1x^2 + 4x - 5) - 1)/(-1(1x^2 + 4x - 5) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1150 | MATH |
[
{
"content": "How many solutions does the equation 3cos(1π(-3(-1|x| + 2)^2 + 2(-1|x| + 2) + 2) + 2) - 1 = 3sin(2π(-4x + 2) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 344 | MATH |
[
{
"content": "How many solutions does the equation -2|(4((2(-1e^(1x) - 3) + 1)/(-3(-1e^(1x) - 3) + 1))^2 - 3((2(-1e^(1x) - 3) + 1)/(-3(-1e^(1x) - 3) + 1)) - 1) - 3| + 2 = -4(-2sin(3π((-2x - 3)/(-1x - 2)) - 3) + 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 65 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(1cos(3π(-1(2x + 2)^2 + 5(2x + 2) - 3) + 1) + 3) + 3| + 3 and g(x) = -2|((1(2x - 1) + 3)/(-1(2x - 1) + 3)) + 2| + 3 where -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) = -1|((-2(3(-2x - 1)^2 - 5) + 2)/(1(3(-2x - 1)^2 - 5))) + 2| - 2 and g(x) = -3|(3(2sin(3πx - 2) + 3)^2 - 2(2sin(3πx - 2) + 3) + 1) - 3| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 17 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2e^(1((-2(5(-3|x + 1| - 1) + 3) + 3)/(-1(5(-3|x + 1| - 1) + 3) - 3))) and g(x) = -3(2cos(1π((2x - 3)/(1x + 2))) + 2)^2 - 5(2cos(1π((2x - 3)/(1x + 2))) + 2) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your an... | 86 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2cos(1π(5(1x^2 - 3x - 4) - 2) - 2) - 1 and g(x) = (-2(2e^(-1x) + 1) - 1)/(3(2e^(-1x) + 1) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1009 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-5((2(-2cos(3πx + 1)) - 1)/(-1(-2cos(3πx + 1)) - 2)) - 2)^2 - 2 and g(x) = -4(-1e^(-1((2x + 1)/(1x + 1))) - 1) 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) = 1((-3(4(-3|x - 3| - 1)^2 + 3(-3|x - 3| - 1) + 2) - 2)/(-1(4(-3|x - 3| - 1)^2 + 3(-3|x - 3| - 1) + 2) - 1)) - 5 and g(x) = 3cos(1π((-1(-2x^2 - 2x + 5) - 2)/(1(-2x^2 - 2x + 5) + 1)) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answ... | 23 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(4(3((2x - 1)/(-3x - 2)) + 1)^2 - 5(3((2x - 1)/(-3x - 2)) + 1) - 4) + 3| - 1 and g(x) = (2(5(1cos(3πx - 1) - 1) + 4) - 2)/(-3(5(1cos(3πx - 1) - 1) + 4) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed... | 22 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(3sin(3π(5(-1x + 4)^2 - 2) - 2)) + 1)/(2(3sin(3π(5(-1x + 4)^2 - 2) - 2)) + 2) and g(x) = -2e^(2(-1((-2x + 1)/(1x))^2 - 4((-2x + 1)/(1x)) - 4)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 529 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-1|(3sin(3π(-2x^2 - 5x + 5) - 3) - 3) + 3| + 1) - 1)/(1(-1|(3sin(3π(-2x^2 - 5x + 5) - 3) - 3) + 3| + 1) + 2) and g(x) = (1(2(-2|x + 3| - 1)^2 + 5(-2|x + 3| - 1) - 4) - 1)/(1(2(-2|x + 3| - 1)^2 + 5(-2|x + 3| - 1) - 4) + 3) ... | 21 | MATH |
[
{
"content": "How many solutions does the equation 2(3sin(3π((3(2|x - 3| - 3) + 1)/(-3(2|x - 3| - 3))) - 3)) + 1 = -3((-1(-2x^2 - 5x + 1) + 2)/(1(-2x^2 - 5x + 1) - 2)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 51 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(5((-2(2|x + 1| - 2))/(1(2|x + 1| - 2) + 1)) - 3)^2 - 5(5((-2(2|x + 1| - 2))/(1(2|x + 1| - 2) + 1)) - 3) - 4 and g(x) = -4(-1|((1x)/(-3x + 1)) - 1| - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 6 | MATH |
[
{
"content": "How many solutions does the equation (3(1(-1|(-2sin(3πx + 2) + 1) + 2| + 3)^2 + 3(-1|(-2sin(3πx + 2) + 1) + 2| + 3) - 5) + 2)/(-3(1(-1|(-2sin(3πx + 2) + 1) + 2| + 3)^2 + 3(-1|(-2sin(3πx + 2) + 1) + 2| + 3) - 5) + 2) = 3(-2|(-1cos(3πx) + 1) - 2| + 3) - 3 have for x in the interval [-10, 10]?\n\nPre... | 118 | MATH |
[
{
"content": "How many solutions does the equation -1|(3(3sin(1π((3x - 2)/(-1x - 3)) + 3))^2 + 2(3sin(1π((3x - 2)/(-1x - 3)) + 3)) - 1) - 2| + 3 = -3e^(1(-3((2x + 1)/(3x + 3))^2 + 3((2x + 1)/(3x + 3)) + 2)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 19 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((-1(1sin(1πx - 1)) + 3)/(3(1sin(1πx - 1)) - 2))^2 + 5((-1(1sin(1πx - 1)) + 3)/(3(1sin(1πx - 1)) - 2)) - 2 and g(x) = (-3(1cos(2πx - 3) + 1) + 1)/(2(1cos(2πx - 3) + 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \... | 52 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(1π((-3(2x) + 3)/(1(2x) - 3)) - 3) + 1 and g(x) = 1cos(1π(5x^2 - 2x - 1) - 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 327 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(5(2sin(1πx - 1) - 1) + 4)| - 3 and g(x) = -1|(-2x - 2)| + 1 intersect 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) = (-3(1|(1x^2 + 3x + 2)| - 2) - 3)/(2(1|(1x^2 + 3x + 2)| - 2) + 1) and g(x) = 5(-1cos(3πx - 3) - 1)^2 - 2(-1cos(3πx - 3) - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many solutions does the equation -2(-3cos(3π(1((-1x - 1)/(1x - 2)) + 5) + 3) + 3)^2 - 3(-3cos(3π(1((-1x - 1)/(1x - 2)) + 5) + 3) + 3) + 5 = 4(-5((3x + 3)/(3x))^2 + 1((3x + 3)/(3x)) - 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 98 | MATH |
[
{
"content": "How many solutions does the equation 1(-1(1|((3x - 1)/(-3x - 1))| - 2)^2 + 5(1|((3x - 1)/(-3x - 1))| - 2) - 4) - 3 = (-3(2|(-3cos(2πx - 3) + 2) - 2|) - 1)/(3(2|(-3cos(2πx - 3) + 2) - 2|) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 54 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-2cos(2π(5x^2 + 1) + 1) + 3) + 3 and g(x) = (-3(-1cos(2πx - 1) + 2) + 3)/(2(-1cos(2πx - 1) + 2) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1302 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-2sin(2π(2x^2 - 4x + 1) - 1)) + 3)/(3(-2sin(2π(2x^2 - 4x + 1) - 1)) - 1) and g(x) = -5(1|x + 3| - 1)^2 - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 36 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1cos(3π((3(-2|x + 1| - 2) + 3)/(1(-2|x + 1| - 2) + 3)) + 2) + 2 and g(x) = (-3(1sin(3πx - 2) + 1) + 1)/(1(1sin(3πx - 2) + 1) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 74 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|((-3(4(-5x^2 + 4x + 2) + 2) - 1)/(3(4(-5x^2 + 4x + 2) + 2) - 2))| + 1 and g(x) = -2|((3(1x^2 - 4) + 1)/(2(1x^2 - 4) - 1)) + 2| + 2 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) = (-1(-2|(2cos(2πx - 2) - 2) - 3| + 3))/(-2(-2|(2cos(2πx - 2) - 2) - 3| + 3) + 1) and g(x) = 3cos(2π(3x - 2) - 1) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 120 | MATH |
[
{
"content": "How many solutions does the equation -1cos(1π(-1((2x + 2)/(2x + 3))^2 + 3((2x + 2)/(2x + 3)) + 5) + 3) + 1 = -2cos(2π(2|x - 3| - 1) - 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 89 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(1cos(2π(-2|(-1x^2 + 4x + 2) + 3| - 3) - 3) - 1))/(-2(1cos(2π(-2|(-1x^2 + 4x + 2) + 3| - 3) - 3) - 1) - 3) and g(x) = (-1(-1(-1|x + 2| - 3) - 5) + 2)/(3(-1(-1|x + 2| - 3) - 5) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer... | 76 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(3((2(3|x + 3|) - 2)/(2(3|x + 3|) - 3))^2 + 2) - 3 and g(x) = (-3(2sin(3π(-2|x - 2| - 3) + 2) - 1) - 2)/(2(2sin(3π(-2|x - 2| - 3) + 2) - 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 117 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2((-1(4x^2 + 3x + 5))/(3(4x^2 + 3x + 5) - 2)) + 4 and g(x) = 2sin(2π((-3x)/(-2x + 3)) + 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 70 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(2sin(1π(5(-2x + 2)^2 - 1(-2x + 2) + 4) + 3))| + 3 and g(x) = 3|((-2(2sin(2πx - 3)) - 1)/(3(2sin(2πx - 3)) + 1)) + 1| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 612 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(3sin(3π(5x - 5) + 1) + 1) + 1| - 2 and g(x) = -1|(2x + 3) + 3| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 56 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(1(-1(1|x - 3| + 2)^2 + 2(1|x - 3| + 2) + 4) - 3))/(3(1(-1(1|x - 3| + 2)^2 + 2(1|x - 3| + 2) + 4) - 3) + 1) and g(x) = 4(1sin(3π((3x - 3)/(-2x + 1)) + 3) + 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Yo... | 59 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4((-1(3|x - 2| + 1) - 2)/(-2(3|x - 2| + 1) + 1))^2 - 3((-1(3|x - 2| + 1) - 2)/(-2(3|x - 2| + 1) + 1)) + 3 and g(x) = 3cos(3π(-3x^2 + 2x - 1) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 801 | MATH |
[
{
"content": "How many solutions does the equation 4(2e^(-1(-1x^2 + 5x - 4)) - 2) - 4 = (-1(-2sin(3πx + 3) - 3) - 2)/(2(-2sin(3πx + 3) - 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 points of intersection exist between the functions f(x) = 1e^(1(1(1(-2|x + 2|)^2 + 5(-2|x + 2|) + 4) + 3)) - 2 and g(x) = 3sin(3π(-1(-2|x - 2| + 2)^2 + 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"
... | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-5(2cos(1π(5x - 5) - 3) + 1)^2 + 1(2cos(1π(5x - 5) - 3) + 1) - 4) + 1)/(-1(-5(2cos(1π(5x - 5) - 3) + 1)^2 + 1(2cos(1π(5x - 5) - 3) + 1) - 4) - 3) and g(x) = -2|(-5(-3sin(1πx - 3) - 2)^2 - 5(-3sin(1πx - 3) - 2) + 5)| in the range... | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-1((-3(3e^(-1x) - 3) - 2)/(-1(3e^(-1x) - 3) + 2))^2 - 5((-3(3e^(-1x) - 3) - 2)/(-1(3e^(-1x) - 3) + 2)) - 2) - 1| and g(x) = 5(2((1x - 1)/(2x + 3))^2) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(2π(2((1(-1x^2 + 5x + 4) - 1)/(-1(-1x^2 + 5x + 4) - 1)))) + 1 and g(x) = (1(-5(2cos(2πx - 3) + 1) + 4) - 1)/(-3(-5(2cos(2πx - 3) + 1) + 4)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 88 | MATH |
[
{
"content": "How many solutions does the equation 2sin(3π(-4(-1|((1x)/(-3x + 2))| - 3)^2 + 5) - 3) = 1|((-1(5x + 1) - 3)/(-2(5x + 1) - 3)) - 2| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 130 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(2π(5((-1x + 1)/(-2x + 3))^2 - 5((-1x + 1)/(-2x + 3)) + 5)) - 1 and g(x) = -3|(-3sin(3πx) + 1) - 3| + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 73 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2cos(2π(-2|(1x^2 - 3x - 2) + 3| - 1) - 1) - 3 and g(x) = (-3(1e^(-2x) + 1) + 2)/(2(1e^(-2x) + 1) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 440 | MATH |
[
{
"content": "How many solutions does the equation -1|(-4(-3sin(1π((1x - 2)/(2x - 1)) - 3) + 2) - 1) + 1| = 4(2cos(3π(-1x^2 - 3x - 5) + 1) - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 563 | MATH |
[
{
"content": "How many solutions does the equation (2(-3|(2(-1x + 4)^2 - 5(-1x + 4)) + 1| + 1) - 1)/(2(-3|(2(-1x + 4)^2 - 5(-1x + 4)) + 1| + 1) + 2) = (1(2sin(1π(4x^2 + 4x - 1) - 1) + 1) - 1)/(-2(2sin(1π(4x^2 + 4x - 1) - 1) + 1) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \... | 151 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(-2|(-2x + 5)| + 1) - 1)/(1(-2|(-2x + 5)| + 1) - 3) and g(x) = -5(3cos(1πx + 2) - 2)^2 + 5(3cos(1πx + 2) - 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "How many solutions does the equation -3(2cos(3π((-2x + 1)/(1x - 2))) + 1)^2 - 2(2cos(3π((-2x + 1)/(1x - 2))) + 1) + 1 = (1(1x^2 - 1x + 1) - 3)/(-3(1x^2 - 1x + 1) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 114 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(-1|(3x^2 - 5x - 1) - 2| - 1) + 3)/(3(-1|(3x^2 - 5x - 1) - 2| - 1) + 1) and g(x) = -3sin(1π(-3|x - 2| - 1) + 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(1(2sin(1πx + 2) + 1) - 3)^2 - 5(1(2sin(1πx + 2) + 1) - 3) - 3 and g(x) = 1((-3x - 3)/(-1x - 1))^2 - 4((-3x - 3)/(-1x - 1)) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 20 | MATH |
[
{
"content": "How many solutions does the equation -2sin(2π(2|(2(-2x^2 - 4x - 4) + 5) + 1| - 3) + 3) + 1 = (-3(4(2x^2 - 3x - 1) + 1))/(-2(4(2x^2 - 3x - 1) + 1) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1614 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(2π(4(-4(-1|x + 2| - 2)^2 - 4(-1|x + 2| - 2) + 5) + 2) + 3) - 2 and g(x) = -3((-3(3x - 4) - 1)/(-2(3x - 4)))^2 + 1((-3(3x - 4) - 1)/(-2(3x - 4))) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 212 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(2cos(1π(5(3|x - 3|)^2 + 3(3|x - 3|) - 2) + 3) + 2) - 3 and g(x) = -3|(4((-2x + 2)/(-1x + 3)) + 3) - 3| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 49 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(1π(-3(4x^2 - 5x + 1) - 1) - 1) + 1 and g(x) = (1(-3|x + 3|) - 2)/(-1(-3|x + 3|) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1307 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3cos(2π(3|(3(4x + 4)^2 - 1(4x + 4) + 5) + 2| + 3) + 2) + 1 and g(x) = 1((-2(2e^(-2x) + 1) - 3)/(2(2e^(-2x) + 1) - 1)) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 792 | MATH |
[
{
"content": "How many solutions does the equation (-1(-2|(-3e^(-2(-2x + 1)) + 1) + 3|))/(3(-2|(-3e^(-2(-2x + 1)) + 1) + 3|) + 2) = (2(-2|(-3x^2 + 3x + 3) + 1| - 2) + 3)/(-1(-2|(-3x^2 + 3x + 3) + 1| - 2) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 6 | MATH |
[
{
"content": "How many solutions does the equation (1(5(-3(-2|x - 2| - 1) - 1)^2 + 1(-3(-2|x - 2| - 1) - 1) - 2))/(1(5(-3(-2|x - 2| - 1) - 1)^2 + 1(-3(-2|x - 2| - 1) - 1) - 2)) = -1cos(3π(-2|((2x - 3)/(-3x + 4)) + 3| + 3) + 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\bo... | 15 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|((3(-4x + 3) + 3)/(-2(-4x + 3) - 2)) + 1| - 1 and g(x) = 3sin(1π(2x^2 - 4x + 3) + 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 404 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1((3(1|(-3sin(1πx + 1) + 3) - 2| - 3) + 1)/(-2(1|(-3sin(1πx + 1) + 3) - 2| - 3) - 3)) + 5 and g(x) = -3e^(-1((2(3x - 4) + 1)/(3(3x - 4) - 2))) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 38 | MATH |
[
{
"content": "How many solutions does the equation -1(-1(-2|((-1x + 3)/(-1x - 3)) - 1| - 2)^2 - 3(-2|((-1x + 3)/(-1x - 3)) - 1| - 2) - 2) + 2 = 2|(1cos(2π(5x - 3) + 3) - 2) - 2| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(-1(-2x - 4)^2 + 2(-2x - 4) - 2) + 1)/(1(-1(-2x - 4)^2 + 2(-2x - 4) - 2)) and g(x) = (2(2sin(1πx - 1) + 1))/(-2(2sin(1πx - 1) + 1) + 3) where -10 ≤ x ≤ 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) = -3cos(3π(2(3|(1x^2 - 1x + 4) - 3| + 1) - 3) - 2) - 2 and g(x) = -2(-1e^(-1(1|x| - 2)) + 3)^2 + 4(-1e^(-1(1|x| - 2)) + 3) - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 104 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2sin(1π(-4(3|((1x + 2)/(-2x)) + 2| + 3)^2 + 2(3|((1x + 2)/(-2x)) + 2| + 3) - 3) + 2) - 3 and g(x) = -3cos(2π(1|(1x - 1) + 1| - 2) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 177 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(-5(-2cos(1π((-1x - 1)/(2x)) - 3) - 2)^2 + 5(-2cos(1π((-1x - 1)/(2x)) - 3) - 2) + 1) + 1| + 1 and g(x) = 1(-3|(2sin(3πx + 1) + 3) + 3| - 2) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 29 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(2|(5((-2x + 3)/(-1x + 2)) - 2) + 3| - 3)^2 + 5(2|(5((-2x + 3)/(-1x + 2)) - 2) + 3| - 3) - 4 and g(x) = 3sin(1π(-2(4x + 5)^2 - 5(4x + 5) + 1) - 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-4(-3(-2|x + 2| + 1)^2 - 5(-2|x + 2| + 1) - 4) + 4) + 3)/(1(-4(-3(-2|x + 2| + 1)^2 - 5(-2|x + 2| + 1) - 4) + 4) - 3) and g(x) = 1sin(1π(-1(2|x - 2| + 1) - 1) - 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxe... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((3(-1|(2x^2 - 4x + 5) + 2| - 2) + 3)/(1(-1|(2x^2 - 4x + 5) + 2| - 2))) - 2 and g(x) = 3sin(1π(5(-1|x + 3|) - 5) - 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(1π(-3((1(-2|x - 3|) + 2)/(1(-2|x - 3|)))^2 + 4((1(-2|x - 3|) + 2)/(1(-2|x - 3|))) + 3)) + 1 and g(x) = 2sin(1π(2(2|x + 3| - 1)^2 + 4(2|x + 3| - 1) - 5) - 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Yo... | 1249 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2e^(1(-3(4(2|x + 1| - 3))^2 + 5(4(2|x + 1| - 3)) - 2)) + 1 and g(x) = (1(1sin(1π(-3x - 1) - 2)) + 3)/(-1(1sin(1π(-3x - 1) - 2)) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 60 | MATH |
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