messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation (-3(2sin(1π(-5x - 2) + 2) - 1) + 3)/(2(2sin(1π(-5x - 2) + 2) - 1) - 3) = (-1(-2|x + 3| - 2))/(3(-2|x + 3| - 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|((2(-1(2cos(1πx + 3) - 2)^2 - 4(2cos(1πx + 3) - 2) + 5))/(-2(-1(2cos(1πx + 3) - 2)^2 - 4(2cos(1πx + 3) - 2) + 5) - 1)) - 1| - 2 and g(x) = -2sin(2π((-2(-1x + 1) - 3)/(-3(-1x + 1) - 2)) + 3) + 1 where -10 ≤ x ≤ 10.\n\nPresen... | 38 | MATH |
[
{
"content": "How many solutions does the equation -4(-3sin(2π(-1|(3x - 4) + 1| - 2) + 1) + 3)^2 - 4 = -5((-2(-3sin(2πx - 1) + 2) - 1)/(3(-3sin(2πx - 1) + 2))) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 23 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(1cos(1π(-4x^2 - 2x) + 3) - 3) + 2| - 1 and g(x) = 1cos(3π(2x^2 - 4) - 1) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 800 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(3(-2|(4x^2 - 3x - 5) - 3| + 2) + 3) - 3)/(-1(3(-2|(4x^2 - 3x - 5) - 3| + 2) + 3)) and g(x) = 3(2sin(2π(-5x - 4) - 2))^2 + 4(2sin(2π(-5x - 4) - 2)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 398 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(-1(1|(-5x^2 - 2x + 4) + 1| - 3) - 5) - 3)/(-3(-1(1|(-5x^2 - 2x + 4) + 1| - 3) - 5) - 1) and g(x) = (-1(-5(-3cos(1πx - 3) - 3)^2 + 4(-3cos(1πx - 3) - 3) + 2) + 1)/(-3(-5(-3cos(1πx - 3) - 3)^2 + 4(-3cos(1πx - 3) - 3) + 2)) in the... | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(-1((-3x - 3)/(2x))^2 + 2((-3x - 3)/(2x)) - 2) + 3| - 3 and g(x) = 3(3cos(3πx))^2 + 2(3cos(3πx)) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 106 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(2(-2cos(2π(1x^2 - 3x - 2) + 3)) + 2) + 1| and g(x) = -3(3cos(2π(1|x - 1| + 2) + 2) + 1)^2 - 5(3cos(2π(1|x - 1| + 2) + 2) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 119 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(2cos(1π(4x^2 - 2x + 5) - 2) - 2) + 2| - 1 and g(x) = 1(3|x + 1| - 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 50 | MATH |
[
{
"content": "How many solutions does the equation 1sin(3π(4(-1|((-2x - 3)/(2x + 2))|)^2 - 5(-1|((-2x - 3)/(2x + 2))|)) + 1) + 1 = (2(1|(-3sin(2πx + 1) - 1) + 1| - 1) + 1)/(1(1|(-3sin(2πx + 1) - 1) + 1| - 1) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 26 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(-3((3(-2|x - 1| - 3) - 2)/(2(-2|x - 1| - 3) + 2)) + 2)^2 - 5(-3((3(-2|x - 1| - 3) - 2)/(2(-2|x - 1| - 3) + 2)) + 2) and g(x) = 4(3sin(2π((-2x + 1)/(1x - 3)) - 1) - 3) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex ... | 13 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(1π(-2(-4(2|x - 3| + 1)^2 - 2(2|x - 3| + 1) + 4) + 3) - 1) + 3 and g(x) = 2sin(2π((2(-1x + 5) + 3)/(3(-1x + 5) - 1)) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 763 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(1(2sin(2π(-3|x| + 2) + 1) + 1) - 1) + 3)/(-3(1(2sin(2π(-3|x| + 2) + 1) + 1) - 1) + 4) and g(x) = -5(-1|(3x^2 - 4x + 2) - 3| + 2) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 15 | MATH |
[
{
"content": "How many solutions does the equation 2(1(-2|(2sin(2πx + 3) + 2) - 1| + 2) + 1)^2 + 5(1(-2|(2sin(2πx + 3) + 2) - 1| + 2) + 1) + 4 = 2(1(1|x + 2| - 3)^2 - 3(1|x + 2| - 3) - 5) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 28 | MATH |
[
{
"content": "How many solutions does the equation -5((3(-5(1|x - 3| - 3)^2 - 4(1|x - 3| - 3) + 5) - 3)/(3(-5(1|x - 3| - 3)^2 - 4(1|x - 3| - 3) + 5))) + 5 = 2(1cos(3π(-1x^2 + 4x + 3) - 1) + 1) 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 solutions does the equation (3(-1|(-3x^2 - 1x - 1) - 1| + 1))/(3(-1|(-3x^2 - 1x - 1) - 1| + 1) - 1) = 4(-2cos(3πx + 3) - 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 51 | MATH |
[
{
"content": "How many solutions does the equation -3sin(1π(5(-1|((2x)/(3x + 1)) + 1| - 2)^2 - 1(-1|((2x)/(3x + 1)) + 1| - 2) + 1)) - 3 = -3((1(-1e^(-2x) + 1) - 1)/(-3(-1e^(-2x) + 1)))^2 - 3((1(-1e^(-2x) + 1) - 1)/(-3(-1e^(-2x) + 1))) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex form... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(2cos(1π(-1(-2|x + 1| - 1)^2 + 5(-2|x + 1| - 1) - 5) - 3) + 2))/(3(2cos(1π(-1(-2|x + 1| - 1)^2 + 5(-2|x + 1| - 1) - 5) - 3) + 2) - 2) and g(x) = (1(-2(-1sin(1πx) + 3)) + 2)/(1(-2(-1sin(1πx) + 3)) + 3) intersect in the interval [-10, 10].\n\nPresent t... | 814 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1cos(2π(-2(-2|x + 2| + 3)^2 - 1(-2|x + 2| + 3)) + 2) and g(x) = (-3(-4x - 4) - 3)/(-3(-4x - 4) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 327 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1cos(3π(5(-3((1x - 1)/(2x - 1)) - 5)^2 - 2(-3((1x - 1)/(2x - 1)) - 5) - 2) + 2) - 1 and g(x) = -2|(5(-3sin(3πx) + 1) - 1)| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 31 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1sin(3π(4(5x^2 + 3x + 4) - 1) + 3) and g(x) = -3(2|x + 1| - 3) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 55 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5((2(2|(-1x^2 - 4x + 2)| - 2) + 1)/(3(2|(-1x^2 - 4x + 2)| - 2) - 2)) - 4 and g(x) = -2(2sin(2π(2x + 3) - 2) + 1)^2 + 4(2sin(2π(2x + 3) - 2) + 1) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 150 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3((3(2(-2sin(3πx - 1))^2 + 2(-2sin(3πx - 1)) - 4) - 2)/(2(2(-2sin(3πx - 1))^2 + 2(-2sin(3πx - 1)) - 4) + 1)) + 3 and g(x) = (-1(4(3x - 4)^2 + 2(3x - 4) + 1))/(-3(4(3x - 4)^2 + 2(3x - 4) + 1) - 2) intersect in the interval [-10, 10].\n\nPresent the answ... | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-4(-3sin(2π((2x + 3)/(1x)) + 2) - 2) - 5)^2 + 4(-4(-3sin(2π((2x + 3)/(1x)) + 2) - 2) - 5) - 3 and g(x) = 1(2|((3x + 2)/(3x - 2)) - 2| - 3)^2 - 3(2|((3x + 2)/(3x - 2)) - 2| - 3) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answe... | 46 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3cos(2π(5(1|x - 1| - 2)^2 - 3(1|x - 1| - 2) + 4) - 2) - 2 and g(x) = 1|(5x + 2) - 2| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many solutions does the equation -1e^(-1(5(-5x - 1)^2 - 5(-5x - 1) + 3)) = -2sin(1π(1x - 4) + 2) - 1 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 solutions does the equation (1(2(2sin(3πx - 1) + 1) - 3) + 2)/(-3(2(2sin(3πx - 1) + 1) - 3) + 4) = (-1(1|x + 3| + 2) - 1)/(-1(1|x + 3| + 2) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(2cos(3π(-2|x + 2|) - 3) + 1) + 2 and g(x) = -2(2x + 5)^2 - 2(2x + 5) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1e^(-1((1(5(2|x + 3| + 3) + 3) - 2)/(-2(5(2|x + 3| + 3) + 3) + 3))) and g(x) = (2(3cos(1π(-1|x + 1| + 2) - 2)) - 3)/(1(3cos(1π(-1|x + 1| + 2) - 2))) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 20 | MATH |
[
{
"content": "How many solutions does the equation 1|((-3(3x) - 1)/(1(3x) + 3)) - 1| = -2sin(2π(-5x^2 - 5x - 2) + 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 39 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-5(-3sin(1πx - 1) - 1) - 5) + 3| - 1 and g(x) = 1(-1cos(3πx - 1) - 3)^2 - 2(-1cos(3πx - 1) - 3) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many solutions does the equation -1(3(1|((1x + 3)/(-3x - 2)) + 1|)^2 - 3(1|((1x + 3)/(-3x - 2)) + 1|) + 1) + 4 = -2sin(2π(-1|(1x^2 - 5x + 2)| - 3) + 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(4(1|(-3sin(1πx - 1))| - 1) + 3)^2 + 2(4(1|(-3sin(1πx - 1))| - 1) + 3) + 5 and g(x) = -4(-4(-2|x - 3| + 1)^2 - 4(-2|x - 3| + 1) - 5) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 14 | MATH |
[
{
"content": "How many solutions does the equation -2((3(3cos(2πx - 1) - 2) - 2)/(1(3cos(2πx - 1) - 2) + 2))^2 + 5((3(3cos(2πx - 1) - 2) - 2)/(1(3cos(2πx - 1) - 2) + 2)) - 3 = -3cos(1π(2x^2 - 2) - 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 76 | MATH |
[
{
"content": "How many solutions does the equation (2(-5(-1x - 5)^2 + 5(-1x - 5) + 5) + 2)/(3(-5(-1x - 5)^2 + 5(-1x - 5) + 5) + 1) = -1sin(3π(-4x^2 - 3x - 4)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 13 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(4(-2|x + 2| + 3)^2 - 3(-2|x + 2| + 3) - 3) + 1)/(-3(4(-2|x + 2| + 3)^2 - 3(-2|x + 2| + 3) - 3) - 3) and g(x) = 1sin(2π(-1|x + 1|) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 36 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(4(-3cos(2πx - 3) - 1) - 3) + 1| + 3 and g(x) = -4(3x^2 + 3x + 3) + 2 intersect in the interval [-10, 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((-3(-5(2|x + 1| - 2)^2 + 2(2|x + 1| - 2) + 1) - 3)/(2(-5(2|x + 1| - 2)^2 + 2(2|x + 1| - 2) + 1) + 2)) + 5 and g(x) = (2(-2e^(-1(5x + 5)) - 3) + 2)/(-2(-2e^(-1(5x + 5)) - 3) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex form... | 137 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|((1(4(-1x + 2)^2 - 4(-1x + 2) - 5) + 1)/(-3(4(-1x + 2)^2 - 4(-1x + 2) - 5) - 2)) + 2| and g(x) = -2|(1e^(2((3x - 3)/(-2x - 2)))) - 2| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|((-1(-2(4x - 1)^2 + 5(4x - 1) - 2) - 2)/(3(-2(4x - 1)^2 + 5(4x - 1) - 2) - 2)) + 1| - 1 and g(x) = -2|((-3(-2sin(1πx + 1) + 2))/(-2(-2sin(1πx + 1) + 2) + 3))| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format:... | 22 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|((-1(5(-1x^2 + 2x) - 5) - 3)/(3(5(-1x^2 + 2x) - 5) + 2)) + 1| - 3 and g(x) = -2|(-1((1x + 2)/(3x))^2 - 2((1x + 2)/(3x)) + 4) + 2| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(-2cos(1π(-5x^2 - 1x - 3) - 1) - 3) + 3 and g(x) = -3(1x^2 + 5x + 2) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 70 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3e^(-1(-3((-3x + 1)/(-2x - 1)) + 5)) - 3 and g(x) = -3sin(2π(-3|x| - 3)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 67 | MATH |
[
{
"content": "How many solutions does the equation (1(2cos(2π(-4(2|x - 2| + 2)^2 + 2(2|x - 2| + 2) + 1)) - 3) - 3)/(-2(2cos(2π(-4(2|x - 2| + 2)^2 + 2(2|x - 2| + 2) + 1)) - 3) - 1) = (1(-1(2|x| - 1)^2 + 3(2|x| - 1)) - 1)/(-3(-1(2|x| - 1)^2 + 3(2|x| - 1))) have for x in the interval [-10, 10]?\n\nPresent the answ... | 22 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1((3(4(1cos(1πx - 2) - 2)^2 + 2(1cos(1πx - 2) - 2) - 4) + 3)/(1(4(1cos(1πx - 2) - 2)^2 + 2(1cos(1πx - 2) - 2) - 4) + 1)) + 3 and g(x) = -2(-2cos(2π((-2x - 1)/(2x + 2)) + 3) - 1) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer ... | 34 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(-4(1(1cos(1πx + 3)) - 2)^2 + 3) + 3| + 2 and g(x) = -4(-2|((2x + 2)/(3x + 3)) + 2| - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(1((-2(1x^2 + 5x - 2) + 2)/(2(1x^2 + 5x - 2) + 2)) - 1) + 2| - 2 and g(x) = 5(-1sin(1π(3x + 3) - 2) + 1)^2 - 3(-1sin(1π(3x + 3) - 2) + 1) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 60 | MATH |
[
{
"content": "How many solutions does the equation -3sin(3π(-2|((-2x - 3)/(-2x - 1)) + 1| - 2) - 1) + 2 = -5(-3sin(1πx - 1)) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 28 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(3(1cos(1π(1x + 4) - 2) - 1)^2 + 1(1cos(1π(1x + 4) - 2) - 1) - 4) + 3)/(1(3(1cos(1π(1x + 4) - 2) - 1)^2 + 1(1cos(1π(1x + 4) - 2) - 1) - 4) + 3) and g(x) = -2|((-3(-1sin(1πx - 3) - 3) + 2)/(-3(-1sin(1πx - 3) - 3) + 4)) - 3| - 3 i... | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(-2((1(5x^2 + 4x - 2))/(2(5x^2 + 4x - 2) + 1)) - 5) + 3| + 3 and g(x) = -1(3cos(3π((-2x - 2)/(-2x)) + 1) + 1)^2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1cos(3π((-1(-1|x + 3| + 3) + 1)/(1(-1|x + 3| + 3) - 2)) - 1) - 2 and g(x) = 1cos(3π(2|x - 3|)) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 186 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(3sin(3π(4x - 1) + 3) - 3)^2 - 4(3sin(3π(4x - 1) + 3) - 3) - 4 and g(x) = -3e^(-2(3|x - 1| - 3)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 218 | MATH |
[
{
"content": "How many solutions does the equation 1cos(2π((1(1|(2x + 5)| + 2))/(-2(1|(2x + 5)| + 2) - 2)) + 3) - 3 = 1(1|(-2sin(3πx + 2) + 3)| + 1)^2 - 3(1|(-2sin(3πx + 2) + 3)| + 1) - 3 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) = -1|((-3(4x - 3) - 3)/(1(4x - 3))) - 3| + 1 and g(x) = -2sin(3π(-4x^2 - 1x + 2) + 1) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 298 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|((1(-1(5x^2 - 5x - 1) + 1) + 2)/(2(-1(5x^2 - 5x - 1) + 1) + 3)) + 2| + 2 and g(x) = -1((3(-2x + 3) - 2)/(2(-2x + 3) - 1))^2 + 5((3(-2x + 3) - 2)/(2(-2x + 3) - 1)) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex form... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-1e^(-1(-1x^2 + 1x))) + 2| and g(x) = 3sin(2π((3x + 3)/(1x - 3)) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3cos(2π(-1|(-1((3x)/(2x - 3))^2 - 4((3x)/(2x - 3)) - 5) + 2| - 3) - 1) + 1 and g(x) = -2|(-4((3x - 1)/(1x + 3))^2 + 3((3x - 1)/(1x + 3))) + 3| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 36 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-3((-3(3cos(3πx + 1) - 2) + 2)/(-3(3cos(3πx + 1) - 2) - 1))^2 - 4((-3(3cos(3πx + 1) - 2) + 2)/(-3(3cos(3πx + 1) - 2) - 1)) + 2) + 1| + 3 and g(x) = -3sin(3π(-4(3|x + 2| - 2) - 5) + 2) + 3 where -10 ≤ x ≤ 10.\n\nPresent th... | 35 | MATH |
[
{
"content": "How many solutions does the equation (1(-3(1(2sin(2πx + 1) + 3)^2 - 4(2sin(2πx + 1) + 3) + 5) + 3) - 2)/(3(-3(1(2sin(2πx + 1) + 3)^2 - 4(2sin(2πx + 1) + 3) + 5) + 3) - 2) = -1|((1(3sin(2πx + 2) + 1) - 3)/(1(3sin(2πx + 2) + 1) + 1)) - 2| + 3 have for x in the interval [-10, 10]?\n\nPresent the answ... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|((1(3sin(1π(4x - 1) - 2) - 2) - 2)/(2(3sin(1π(4x - 1) - 2) - 2) - 1)) - 2| + 3 and g(x) = 3(-1(-3|x + 3| + 2)^2 - 1(-3|x + 3| + 2) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(1sin(1π(4x^2 - 2x - 4) + 1) + 1) + 3 and g(x) = 2sin(1π(1x + 1) - 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 799 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-2cos(3π(-1x + 5) - 3) - 1) + 1)/(-2(-2cos(3π(-1x + 5) - 3) - 1) - 1) and g(x) = 2cos(1π((2x - 2)/(2x - 1)) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 47 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(2cos(1π(-1x) + 2) - 3) + 2)/(1(2cos(1π(-1x) + 2) - 3) + 2) and g(x) = -3(-2|x + 2| + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "How many solutions does the equation (2(-2(2|(-1cos(3πx + 2) + 1) - 3| - 3)^2 - 3(2|(-1cos(3πx + 2) + 1) - 3| - 3) + 3) + 2)/(-1(-2(2|(-1cos(3πx + 2) + 1) - 3| - 3)^2 - 3(2|(-1cos(3πx + 2) + 1) - 3| - 3) + 3) + 3) = -3cos(3π(-1(5x)^2 + 1(5x) + 5) - 1) + 3 have for x in the interval [-10, 10]?\n\nP... | 121 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3sin(1π(3((-1x - 3)/(2x - 1))^2 - 4((-1x - 3)/(2x - 1)) + 3) + 1) + 1 and g(x) = 3cos(1π(-3x - 5) - 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 125 | MATH |
[
{
"content": "How many solutions does the equation (-3(-1|(4x^2 + 1x - 5) + 2|) + 2)/(2(-1|(4x^2 + 1x - 5) + 2|) + 1) = 2cos(1π(-1x^2 + 4)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 198 | MATH |
[
{
"content": "How many solutions does the equation -5(1|(-1(1sin(2πx - 3) + 3)^2 + 2(1sin(2πx - 3) + 3) - 3) - 2| - 2) - 3 = -4(-2sin(1π(4x - 3) + 1) - 2)^2 - 2(-2sin(1π(4x - 3) + 1) - 2) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2((-1(-3e^(1(-2|x|)) - 1) + 1)/(-2(-3e^(1(-2|x|)) - 1))) + 2 and g(x) = 1sin(1π(3|((2x + 1)/(1x - 3)) + 1| + 2) - 3) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 46 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(2|(-1((-1x - 1)/(-1x + 2)) - 2) - 3| - 2)^2 - 5(2|(-1((-1x - 1)/(-1x + 2)) - 2) - 3| - 2) + 5 and g(x) = 2e^(2(1(3x - 4)^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) = -3cos(2π(-3(1|((2x - 1)/(1x + 2)) - 1| - 2)^2 - 1(1|((2x - 1)/(1x + 2)) - 1| - 2) - 1) - 3) - 1 and g(x) = 2|((3(1x^2 + 3x + 1))/(3(1x^2 + 3x + 1) - 2)) - 1| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\bo... | 270 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-1(-1e^(-1((-2x + 1)/(1x + 3))) + 1))^2 - 3(-1(-1e^(-1((-2x + 1)/(1x + 3))) + 1)) + 1 and g(x) = (-1(-2sin(3π(1x^2 - 4x + 2) + 1) - 1) - 3)/(3(-2sin(3π(1x^2 - 4x + 2) + 1) - 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex... | 73 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(1cos(1π(-1|((-1x - 1)/(2x + 1)) - 3| + 3) - 3) + 1) - 5 and g(x) = 2((2(-3cos(1πx - 1) + 1) - 2)/(-1(-3cos(1πx - 1) + 1) + 2)) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 20 | MATH |
[
{
"content": "How many solutions does the equation (-2(1(2|x - 2| + 3) + 3) + 1)/(3(1(2|x - 2| + 3) + 3) + 2) = 3cos(1π(4x - 1) - 1) - 2 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 2cos(3π((-1(1x - 5) + 1)/(2(1x - 5) + 2)) - 1) = (1(3x - 2) - 1)/(1(3x - 2) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 61 | MATH |
[
{
"content": "How many solutions does the equation 4((1(-3sin(1πx + 3) + 2) + 1)/(1(-3sin(1πx + 3) + 2) - 3)) - 2 = -5(2|x + 1| + 3) - 4 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(-1cos(3π((-3x - 3)/(-3x)) - 2) + 2)^2 + 2(-1cos(3π((-3x - 3)/(-3x)) - 2) + 2) + 2 and g(x) = -3sin(2π((1x + 3)/(-3x - 2)) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 65 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3((2(-2cos(1π(-2|x + 3| + 3))) - 1)/(2(-2cos(1π(-2|x + 3| + 3))) + 3))^2 - 5 and g(x) = (-3(-3e^(2(-1x^2 + 3x - 2))) - 3)/(1(-3e^(2(-1x^2 + 3x - 2))) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 42 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-5((2x)/(2x - 3)) - 2)^2 + 1(-5((2x)/(2x - 3)) - 2) - 2 and g(x) = (-1(2cos(1πx + 3) - 1) + 2)/(1(2cos(1πx + 3) - 1) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 18 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|((-3(-1(-1x^2 - 3x - 2) - 5))/(2(-1(-1x^2 - 3x - 2) - 5) - 1)) - 1| + 1 and g(x) = -3(-1cos(3π(5x^2 - 2x) - 3) - 3) intersect 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 -3sin(2π(-5(-2|x + 2| - 2)^2 + 2(-2|x + 2| - 2) - 5) + 1) + 3 = -2(-1|x - 1|) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 385 | MATH |
[
{
"content": "How many solutions does the equation (-3(-4(-1x^2 - 4x + 5) + 4) + 2)/(-2(-4(-1x^2 - 4x + 5) + 4) - 2) = 2sin(2π(-1x + 2)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(1cos(3π((-3x - 1)/(-2x + 3))) + 3)^2 + 5(1cos(3π((-3x - 1)/(-2x + 3))) + 3) + 3 and g(x) = 4(-2cos(1πx + 2) - 2)^2 - 2(-2cos(1πx + 2) - 2) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 35 | MATH |
[
{
"content": "How many solutions does the equation 4(1(-3cos(3πx - 1) + 1) - 1)^2 - 5(1(-3cos(3πx - 1) + 1) - 1) + 4 = -4(-1cos(2πx - 1) - 1) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 120 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(-3sin(3π(3((-1x + 2)/(-1x + 3))^2 - 4((-1x + 2)/(-1x + 3)) + 4) + 1) + 3) + 1 and g(x) = 2|(-1((2x)/(-2x + 3)) + 4) - 1| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 150 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|((-1(-2cos(3πx + 3) + 2))/(-2(-2cos(3πx + 3) + 2) + 3)) + 1| and g(x) = (-2(-5x^2 + 2x) - 2)/(-1(-5x^2 + 2x) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 114 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(-2sin(3π(-4x^2 + 1x) + 2) - 1) + 1)/(2(-2sin(3π(-4x^2 + 1x) + 2) - 1) - 1) and g(x) = (2(-2cos(3πx - 3) + 3) + 2)/(-2(-2cos(3πx - 3) + 3) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 167 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(3(-1|(-1cos(2πx + 2) + 2) - 1| + 3) - 5)^2 - 4(3(-1|(-1cos(2πx + 2) + 2) - 1| + 3) - 5) + 1 and g(x) = 1|(2sin(1π((1x - 2)/(-2x + 3)) + 3) - 3) + 2| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 79 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-2cos(1π(-1|((-1x - 2)/(1x)) - 3| - 2)) - 1)^2 and g(x) = -1sin(2π(4(4x^2 - 5x + 1) - 1) + 1) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 221 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(1π(-1|(-2x - 3) - 3| - 3)) - 2 and g(x) = 1sin(1π((-2x)/(2x - 1)) - 2) - 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(-1|(1e^(-1(-1x - 3)) - 1) - 3| + 2)^2 + 2 and g(x) = 2cos(1π(-2(2|x - 2| - 1) + 3) - 2) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 25 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-1(2|(4x + 3) - 3|)^2 - 3(2|(4x + 3) - 3|) - 1) + 1)/(-2(-1(2|(4x + 3) - 3|)^2 - 3(2|(4x + 3) - 3|) - 1) + 3) and g(x) = -2cos(2π((1(-3x^2 - 1x + 4) + 1)/(1(-3x^2 - 1x + 4) - 3)) - 2) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent th... | 47 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(3|((3(-4x^2 + 3x - 1) - 2)/(3(-4x^2 + 3x - 1) - 2)) - 3| + 3) - 3 and g(x) = 2(-3cos(3π((3x)/(3x + 3)) + 3) - 2)^2 - 1(-3cos(3π((3x)/(3x + 3)) + 3) - 2) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxe... | 66 | MATH |
[
{
"content": "How many solutions does the equation -1|(2(1(-3cos(1πx - 2) + 3) - 2)^2 + 4(1(-3cos(1πx - 2) + 3) - 2) + 1)| + 1 = -3(1|(2x^2 + 5x + 1) - 2| - 2) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|((-1(2(-2x + 2)^2 - 5(-2x + 2) + 1) + 3)/(-1(2(-2x + 2)^2 - 5(-2x + 2) + 1)))| + 3 and g(x) = -3sin(2π(-2((1x + 2)/(2x + 1))^2 + 4((1x + 2)/(2x + 1)) - 3) - 3) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: ... | 7 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2sin(2π(1|(1((1x - 3)/(-2x - 3))^2 - 5((1x - 3)/(-2x - 3)) - 5) + 2| + 1)) + 1 and g(x) = (-1(4(-2sin(1πx + 2) - 1)^2 - 2(-2sin(1πx + 2) - 1) - 4) - 2)/(-2(4(-2sin(1πx + 2) - 1)^2 - 2(-2sin(1πx + 2) - 1) - 4) - 1) where -10 ... | 171 | MATH |
[
{
"content": "How many solutions does the equation (2(3|(-1sin(1π(-2x - 3)) - 1) - 2| + 1) - 3)/(3(3|(-1sin(1π(-2x - 3)) - 1) - 2| + 1)) = (-1(2(3cos(1πx + 2) + 1) + 3) - 3)/(-2(2(3cos(1πx + 2) + 1) + 3) - 3) 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) = -4(3|(-3cos(3π(-4x^2 - 1x + 4) - 3) + 3)| - 2) + 4 and g(x) = -2|(-3cos(2π((2x - 3)/(2x - 1)) - 3) - 3)| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1355 | MATH |
[
{
"content": "How many solutions does the equation -1(-1cos(1π(1|((2x - 3)/(3x + 3)) - 1| - 1) - 3) + 1)^2 + 2(-1cos(1π(1|((2x - 3)/(3x + 3)) - 1| - 1) - 3) + 1) + 1 = (1(3(-2|x - 2| - 3)^2 - 2(-2|x - 2| - 3) + 1) + 2)/(1(3(-2|x - 2| - 3)^2 - 2(-2|x - 2| - 3) + 1)) have for x in the interval [-10, 10]?\n\nPrese... | 27 | MATH |
[
{
"content": "How many solutions does the equation 5(2|(2sin(2πx - 1) + 2) - 1|)^2 - 1(2|(2sin(2πx - 1) + 2) - 1|) - 2 = -4((3x - 2)/(1x - 2)) + 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 solutions does the equation (-3(-2|(-2x - 3) + 2| - 2) + 1)/(-1(-2|(-2x - 3) + 2| - 2)) = 1(-3cos(3πx - 2) + 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
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