messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1
value |
|---|---|---|
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(1(-2cos(1πx + 1) + 3) - 3) + 1)/(1(1(-2cos(1πx + 1) + 3) - 3) + 2) and g(x) = (-1(-2sin(2πx - 1) - 2) + 3)/(2(-2sin(2πx - 1) - 2) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 11 | MATH |
[
{
"content": "How many solutions does the equation (3(2(1cos(3π(-2|x + 3| + 2) + 1)) - 1) - 1)/(-3(2(1cos(3π(-2|x + 3| + 2) + 1)) - 1) - 3) = 1e^(-1(-1((1x - 2)/(-2x + 1)) - 3)) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 86 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(3sin(1π((-1x - 2)/(1x - 3)) - 3) - 1) + 2| - 2 and g(x) = -3sin(3π(4x - 4) - 3) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 161 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2sin(1π(2(2|((1x - 3)/(3x + 2)) - 3| + 2) + 2) - 3) and g(x) = (1(1|(5x + 3) + 1| + 3) + 1)/(-1(1|(5x + 3) + 1| + 3)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 77 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(2cos(3π(-3((-3x + 3)/(-2x + 3)) - 1) - 3))^2 + 2(2cos(3π(-3((-3x + 3)/(-2x + 3)) - 1) - 3)) - 2 and g(x) = 3sin(2π((-3(-2x - 4) - 3)/(-2(-2x - 4) + 3)) - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your... | 64 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(2sin(1π(-1x^2 - 4x - 2) - 2) - 1) - 1| - 1 and g(x) = (-3(-1x - 5) + 3)/(3(-1x - 5) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 145 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2cos(2π(-5(3|(3x^2 + 1x - 1) + 2| + 1) + 1) + 2) and g(x) = -1e^(-2(2|(5x + 4) + 2| + 1)) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1135 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-5((1x - 3)/(3x))^2 - 1((1x - 3)/(3x))) + 2| + 3 and g(x) = 2|(2cos(2πx) + 1) + 3| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(1|(4(-4x - 5)^2 + 2(-4x - 5) - 3) - 3| - 1) - 3)/(-2(1|(4(-4x - 5)^2 + 2(-4x - 5) - 3) - 3| - 1) + 1) and g(x) = 2cos(2π(-3(-4x - 2)^2 + 5(-4x - 2) + 3) - 2) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: ... | 1556 | MATH |
[
{
"content": "How many solutions does the equation -3(-3cos(2π(3|x| - 1) - 3) - 3) - 5 = 1(2|x + 3| - 3) + 4 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) = 5(1((2x + 3)/(3x + 1))^2 + 3((2x + 3)/(3x + 1)) - 4) + 4 and g(x) = 1cos(1π(1x^2 + 3x - 1) + 1) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 83 | MATH |
[
{
"content": "How many solutions does the equation 2|((1(-3cos(1π(-1x) - 1) - 1) + 3)/(-3(-3cos(1π(-1x) - 1) - 1) + 4)) + 3| - 1 = 3(3e^(1(1|x + 2|)) - 2)^2 - 2 have for x 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) = 1cos(3π(-2(3|(-4x^2 - 3x + 5) - 1| - 2) + 1)) and g(x) = (1(1cos(3π(-4x^2 - 1x + 2) + 1) + 1) + 3)/(2(1cos(3π(-4x^2 - 1x + 2) + 1) + 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 410 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-1(-3sin(3π((-2x)/(3x - 2)) + 3) - 3))^2 - 2(-1(-3sin(3π((-2x)/(3x - 2)) + 3) - 3)) - 5 and g(x) = 2sin(3π((-1(3x^2 + 5x + 1) - 3)/(1(3x^2 + 5x + 1) - 3)) - 2) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format:... | 35 | MATH |
[
{
"content": "How many solutions does the equation 1((2(1|(-5x - 1) - 3|) + 1)/(1(1|(-5x - 1) - 3|) - 2))^2 + 2((2(1|(-5x - 1) - 3|) + 1)/(1(1|(-5x - 1) - 3|) - 2)) - 3 = (3(3(2sin(3πx - 3) + 1)^2 - 5(2sin(3πx - 3) + 1)) - 2)/(-3(3(2sin(3πx - 3) + 1)^2 - 5(2sin(3πx - 3) + 1)) - 1) have for x in the interval [-1... | 55 | MATH |
[
{
"content": "How many solutions does the equation -1(5(-2|x - 2| + 1))^2 + 1(5(-2|x - 2| + 1)) - 4 = -3((-1x - 3)/(-2x))^2 - 5((-1x - 3)/(-2x)) - 5 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) = 2cos(2π((-2(2(3|x + 2| - 3) + 2) - 1)/(-1(2(3|x + 2| - 3) + 2) + 2)) - 3) - 2 and g(x) = (1(-2(4x^2 - 2x + 1) + 1) - 1)/(-1(-2(4x^2 - 2x + 1) + 1) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your ... | 84 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-1sin(1π(3|x - 2| + 1) + 1) + 3) + 2 and g(x) = -3((2x - 3)/(1x + 2)) - 2 in the range -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) = -1|(1sin(3π((-2x - 3)/(1x + 2)) + 1) + 3) - 2| + 2 and g(x) = (-2(4x) + 2)/(-2(4x) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 63 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-3sin(3π(5(-1x + 2)^2 - 3(-1x + 2)) + 1)) - 2| - 1 and g(x) = -2sin(2π(-5(-1|x - 3| + 2) - 5) + 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 196 | MATH |
[
{
"content": "How many solutions does the equation (2(1(-3|(2cos(2πx - 3) - 2) + 1| + 2) + 3) + 2)/(3(1(-3|(2cos(2πx - 3) - 2) + 1| + 2) + 3) + 2) = 2(2|(-1x^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"
}
] | 10 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(-1(2(-3|x - 3|)^2 + 3(-3|x - 3|) - 4) - 3) - 2)/(-1(-1(2(-3|x - 3|)^2 + 3(-3|x - 3|) - 4) - 3) - 3) and g(x) = -3cos(1π(1(-1|x + 2| - 2)^2 - 1(-1|x + 2| - 2) + 3) - 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \... | 295 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-3((1x + 2)/(-3x)) - 1)^2 + 5(-3((1x + 2)/(-3x)) - 1) + 3 and g(x) = 1cos(2π(-2x - 4) - 1) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-3sin(3π((1(-2x - 2) - 1)/(2(-2x - 2) - 1)) + 1) - 1)^2 - 2(-3sin(3π((1(-2x - 2) - 1)/(2(-2x - 2) - 1)) + 1) - 1) + 3 and g(x) = -1cos(1π(3|((1x - 3)/(2x)) - 1| - 1) - 2) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in L... | 25 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-2|((-2(-5x - 4))/(-3(-5x - 4) - 1)) - 1|)^2 + 4(-2|((-2(-5x - 4))/(-3(-5x - 4) - 1)) - 1|) + 5 and g(x) = 4((-2(-2x - 3) + 1)/(3(-2x - 3) - 2))^2 intersect in the interval [-10, 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(2sin(3πx + 3) + 3))/(-2(2sin(3πx + 3) + 3) + 3)) - 1| + 2 and g(x) = -1(1|x - 2| + 3)^2 + 4(1|x - 2| + 3) + 4 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) = 5(-3sin(1π(2(2|x + 3| - 2) + 5) - 2) + 1)^2 + 3(-3sin(1π(2(2|x + 3| - 2) + 5) - 2) + 1) - 3 and g(x) = -1sin(1π(5((3x - 3)/(3x - 2)) - 2) - 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 160 | MATH |
[
{
"content": "How many solutions does the equation -4(3cos(2π(-4((2x - 3)/(3x - 1)) + 4) - 2) + 2)^2 - 1(3cos(2π(-4((2x - 3)/(3x - 1)) + 4) - 2) + 2) + 2 = -1cos(1π(-3(3x^2 + 1x + 5) - 5)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 126 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(2(3|(5x) - 1|)^2 + 1(3|(5x) - 1|) - 4))/(2(2(3|(5x) - 1|)^2 + 1(3|(5x) - 1|) - 4)) and g(x) = (-1(3sin(1π(4x^2 - 1x - 4) - 2) - 1))/(-1(3sin(1π(4x^2 - 1x - 4) - 2) - 1) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in L... | 748 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(1e^(1(1(-3|x - 2| + 1) + 1)))^2 + 4 and g(x) = -3((-1(3cos(3πx - 2) - 1) + 1)/(-1(3cos(3πx - 2) - 1) - 3)) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-2sin(2π((2(1x^2 + 1x + 4))/(1(1x^2 + 1x + 4) + 2)) + 2) - 1) + 3| - 1 and g(x) = 2(2cos(1π(-2|x + 1|) + 3) - 1) - 3 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) = (3(1sin(1π(4x^2 + 1x - 4) - 1)))/(-1(1sin(1π(4x^2 + 1x - 4) - 1))) and g(x) = 3sin(3π((1x + 3)/(1x)) - 2) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 114 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(2π(-3(2|x| - 3) - 3) + 1) and g(x) = -3e^(-1(-1|x - 2| + 1)) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "How many solutions does the equation 3|((1(4(-1x - 4)^2 + 5(-1x - 4) - 5) - 3)/(2(4(-1x - 4)^2 + 5(-1x - 4) - 5) + 2))| - 2 = 2cos(2π(2(-1|x + 3|)^2 + 1) + 1) - 2 have for x in the interval [-10, 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) = 3e^(2((3(3|(1x - 2) - 3| + 1) + 1)/(2(3|(1x - 2) - 3| + 1) + 2))) - 2 and g(x) = -4(-2|(2sin(2πx + 2) + 3) + 3| - 2) - 2 where -10 ≤ x ≤ 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) = -2cos(1π(2|(4x^2 + 5x + 2) + 3| + 3) + 2) - 3 and g(x) = -2|((-3x)/(1x)) + 3| - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1316 | MATH |
[
{
"content": "How many solutions does the equation 1(-3sin(2π(-1|x - 3| + 1) - 3) + 3) - 1 = 5((3x + 3)/(3x - 1))^2 - 4((3x + 3)/(3x - 1)) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "How many solutions does the equation -3((1(4(3cos(2πx + 1) + 3)^2 + 1(3cos(2πx + 1) + 3) + 1) - 1)/(2(4(3cos(2πx + 1) + 3)^2 + 1(3cos(2πx + 1) + 3) + 1) + 1)) - 2 = -1((1(-2x + 5) - 1)/(2(-2x + 5) - 1))^2 - 5((1(-2x + 5) - 1)/(2(-2x + 5) - 1)) have for x in the interval [-10, 10]?\n\nPresent the a... | 36 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(-1cos(2π((-3x)/(-3x - 3))) - 1) - 1| and g(x) = -4(-1sin(1πx + 1) + 1) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(-4(-2(3|x + 1| - 2)^2 - 1(3|x + 1| - 2) + 3) + 2) + 2)/(-1(-4(-2(3|x + 1| - 2)^2 - 1(3|x + 1| - 2) + 3) + 2) - 1) and g(x) = (3(2(-2|x| + 3)^2 - 3(-2|x| + 3) - 5) - 3)/(2(2(-2|x| + 3)^2 - 3(-2|x| + 3) - 5) - 3) in the range -10 ... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(3sin(3π(1|(-5x - 1) - 1| + 3) + 3) - 1) - 3)/(-1(3sin(3π(1|(-5x - 1) - 1| + 3) + 3) - 1) - 1) and g(x) = -1|(-2cos(3π(-3x^2 - 3) - 3) + 3)| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 606 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(-2|(3sin(2π((2x - 2)/(-1x + 2)) - 3) - 2) - 2| + 1)^2 - 3(-2|(3sin(2π((2x - 2)/(-1x + 2)) - 3) - 2) - 2| + 1) and g(x) = -3((2(1|x + 3| - 3) + 2)/(2(1|x + 3| - 3)))^2 + 5((2(1|x + 3| - 3) + 2)/(2(1|x + 3| - 3))) - 3 intersect in the interval [-10, 1... | 57 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-2|(1x + 3) - 1| + 3) - 2)/(2(-2|(1x + 3) - 1| + 3) - 1) and g(x) = -1sin(3π(3|x + 1|) - 3) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 170 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-2sin(1π(1x - 1) - 3) + 3) - 1)/(3(-2sin(1π(1x - 1) - 3) + 3) + 1) and g(x) = 2cos(2π(-2|x - 1| + 1) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(1cos(2π(2x - 2) - 3) - 1) - 1)/(-1(1cos(2π(2x - 2) - 3) - 1)) and g(x) = -3cos(1π(-3|x| - 2) + 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(3π(1|((3x + 2)/(3x - 2)) - 1| - 3) + 1) - 1 and g(x) = (-1(-5x^2 - 1x - 1) + 2)/(1(-5x^2 - 1x - 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 31 | MATH |
[
{
"content": "How many solutions does the equation -3sin(3π(5(2x^2 + 4x + 4) - 1) + 2) + 1 = 1(1x - 3)^2 - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 330 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2((-2(-4(3|x - 3| + 2)^2 - 3(3|x - 3| + 2) + 5) + 1)/(2(-4(3|x - 3| + 2)^2 - 3(3|x - 3| + 2) + 5) - 1)) - 4 and g(x) = (3(3|(3sin(3πx + 3))| - 2) - 2)/(2(3|(3sin(3πx + 3))| - 2) - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer... | 120 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(2|((-1(3x + 5) - 2)/(3(3x + 5) + 3))|)^2 - 2(2|((-1(3x + 5) - 2)/(3(3x + 5) + 3))|) - 1 and g(x) = -2sin(2π(-1(3x^2 - 4x + 3) - 5)) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 1035 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-2(1x^2 - 3x - 5) - 5) - 3)/(2(-2(1x^2 - 3x - 5) - 5) + 2) and g(x) = -2(3cos(3πx - 1) + 1) + 1 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) = -3cos(1π(-2(-1(2|x - 1| + 2))^2 + 2(-1(2|x - 1| + 2)) - 4) + 2) - 1 and g(x) = 2e^(-1(4(-3|x + 2| + 2))) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 93 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2e^(-2(3((1x - 3)/(2x + 2))^2 - 4((1x - 3)/(2x + 2)) + 1)) - 3 and g(x) = (2(-3cos(2πx - 2) - 2) - 2)/(-1(-3cos(2πx - 2) - 2)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 40 | MATH |
[
{
"content": "How many solutions does the equation 3cos(3π(-2|(3x^2 + 5x - 1)| + 2) - 2) = 3((1x + 3)/(2x - 1)) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 703 | MATH |
[
{
"content": "How many solutions does the equation 2sin(1π(2(-2x - 3)^2 - 5(-2x - 3) + 3) - 2) + 1 = (-1(-1|x + 1| - 3) + 3)/(-1(-1|x + 1| - 3) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1375 | MATH |
[
{
"content": "How many solutions does the equation 2(-3e^(-1(5x + 1)))^2 - 5(-3e^(-1(5x + 1))) + 3 = 5(-3cos(1πx - 3) + 3) 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 2cos(2π(3(2x^2 + 1x + 5) - 1) + 2) - 1 = -1(3cos(3πx) + 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": "Find the number of times the graphs of f(x) = -1((3(1(3|x| + 2)^2 + 3(3|x| + 2) + 5) + 3)/(1(1(3|x| + 2)^2 + 3(3|x| + 2) + 5) - 3)) + 4 and g(x) = -2(2cos(2π((2x + 1)/(2x - 1)) - 3) - 1)^2 + 3(2cos(2π((2x + 1)/(2x - 1)) - 3) - 1) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in La... | 54 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3cos(2π(1(2|x + 1| - 3)^2 - 4(2|x + 1| - 3)) + 2) and g(x) = 3(-1sin(1πx + 2) - 1) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 732 | MATH |
[
{
"content": "How many solutions does the equation (2(3(-3cos(2πx - 2) + 1)^2 + 4(-3cos(2πx - 2) + 1) + 3) + 1)/(-1(3(-3cos(2πx - 2) + 1)^2 + 4(-3cos(2πx - 2) + 1) + 3) + 2) = -2(-1cos(3πx - 1) - 3)^2 - 5(-1cos(3πx - 1) - 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxe... | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(2π(2(-3|x - 1| + 3)^2 - 5(-3|x - 1| + 3) + 4) + 1) + 3 and g(x) = 4((2x - 3)/(2x + 3)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 722 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-1sin(3π((-2(-2x - 3) - 3)/(-3(-2x - 3) - 3)) + 2) - 2)^2 + 1 and g(x) = (1(1|(3cos(3πx - 3) + 3)| - 3) - 1)/(1(1|(3cos(3πx - 3) + 3)| - 3) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 64 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-2sin(1π(-4x^2 + 1x + 2)) + 3)| + 3 and g(x) = -3((-1x + 3)/(1x))^2 + 1((-1x + 3)/(1x)) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 266 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-5((-2(-2x^2 - 5x + 5) - 2)/(-3(-2x^2 - 5x + 5) + 1)) - 3) - 1| - 3 and g(x) = -5(3sin(2π(3|x + 3|) + 2) + 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 120 | MATH |
[
{
"content": "How many solutions does the equation -1(5((-3(-3|x|) + 2)/(-2(-3|x|)))^2 - 2((-3(-3|x|) + 2)/(-2(-3|x|))) + 1) + 2 = -2|(-3(3x^2 - 2x - 3) + 5) + 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) = -1|((3(-4(1sin(1πx - 1))) - 2)/(-3(-4(1sin(1πx - 1))) - 1)) + 3| and g(x) = -2|((3(-3x^2 - 4x - 3) - 3)/(1(-3x^2 - 4x - 3) - 3)) - 1| + 2 in the range -10 ≤ x ≤ 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) = 4((1(1(-3|x + 1| - 1) - 2) + 2)/(-3(1(-3|x + 1| - 1) - 2) + 4))^2 - 5((1(1(-3|x + 1| - 1) - 2) + 2)/(-3(1(-3|x + 1| - 1) - 2) + 4)) and g(x) = -1|(3(-2sin(1πx - 1) + 2) - 2) - 3| + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer ... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(-2|(-2sin(2πx - 1) + 2)| + 3)^2 + 4(-2|(-2sin(2πx - 1) + 2)| + 3) - 1 and g(x) = (-1(-1x^2 + 2x - 5) - 1)/(1(-1x^2 + 2x - 5)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-1((2(-4x) + 3)/(2(-4x) + 1))^2 - 3((2(-4x) + 3)/(2(-4x) + 1)) + 4) - 2| + 1 and g(x) = (-1(-3cos(1π(-2x - 3) - 1) + 2) - 1)/(-2(-3cos(1π(-2x - 3) - 1) + 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your an... | 40 | MATH |
[
{
"content": "How many solutions does the equation -2((1(-3|x - 3| + 2) + 1)/(3(-3|x - 3| + 2) - 2))^2 - 4((1(-3|x - 3| + 2) + 1)/(3(-3|x - 3| + 2) - 2)) + 3 = -2(-2sin(3πx + 3) - 2) - 5 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 -1|(-1(-3sin(3π(-5x + 3) + 1))^2 + 1(-3sin(3π(-5x + 3) + 1)) - 4) + 2| + 2 = -2e^(-2(2((2x + 2)/(1x + 3))^2 + 2((2x + 2)/(1x + 3)) + 1)) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 600 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|((1(3(-1x^2 - 5x) - 5) - 1)/(1(3(-1x^2 - 5x) - 5) - 3)) - 3| + 2 and g(x) = 2|((2(4x + 1) + 1)/(-2(4x + 1) + 3)) - 1| 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) = -5((3(-1|(-4x + 3) - 3| + 2) - 3)/(2(-1|(-4x + 3) - 3| + 2)))^2 - 1((3(-1|(-4x + 3) - 3| + 2) - 3)/(2(-1|(-4x + 3) - 3| + 2))) - 5 and g(x) = -1|(5(-2sin(1πx + 1))^2 - 1(-2sin(1πx + 1)) + 4) - 2| - 3 where -10 ≤ x ≤ 10.\n\nPr... | 36 | MATH |
[
{
"content": "How many solutions does the equation -1(3(1|(-1cos(2πx - 2) - 3) + 1| - 2)^2 - 3(1|(-1cos(2πx - 2) - 3) + 1| - 2) - 2) + 5 = -2((3(-1|x| + 1) + 2)/(3(-1|x| + 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 solutions does the equation -3|((-2(-3cos(2π(3x - 3)) + 1) + 1)/(2(-3cos(2π(3x - 3)) + 1) + 1)) - 1| + 1 = -1((3(1sin(3πx + 1) - 1) - 3)/(3(1sin(3πx + 1) - 1) - 2)) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 120 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(3|(1(-2e^(1x) + 3))| + 1) - 1)/(2(3|(1(-2e^(1x) + 3))| + 1) + 3) and g(x) = 2cos(1π(3|(2x - 4) - 1|) + 3) - 2 where -10 ≤ x ≤ 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) = 1|(1sin(1π(-3x^2 + 1) - 1) - 3) - 3| - 2 and g(x) = -2sin(1π(4x^2 + 4x - 3) + 1) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 723 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2cos(1π(-5(-2|x - 3| - 1)) + 1) + 3 and g(x) = -5(1|x - 2| - 2)^2 + 5(1|x - 2| - 2) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(4(-1(3sin(3πx + 2) - 3)^2 - 4(3sin(3πx + 2) - 3) - 1) - 3) + 3| and g(x) = -2|(-3cos(2π((-2x + 2)/(1x + 2)) - 1) - 2) + 1| + 1 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) = 2|(-3(-2((-3x - 1)/(2x))^2 - 3((-3x - 1)/(2x)) + 4) - 2) + 2| - 1 and g(x) = 3(4((-2x + 3)/(-1x + 2))^2 + 1((-2x + 3)/(-1x + 2)) - 4) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(2π(-3(1|x + 2|) - 1) + 3) - 1 and g(x) = 2(-3x^2 + 1x) 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) = -3|((-2(2(-2x + 1)^2 + 3(-2x + 1) + 4) + 3)/(2(2(-2x + 1)^2 + 3(-2x + 1) + 4) - 1))| + 1 and g(x) = -1|(-2cos(1π(5x) - 3) - 1) - 1| + 1 intersect 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) = -1sin(3π(-1|((-1(-2x - 5) - 2)/(-1(-2x - 5) + 2)) + 2| - 1) - 3) and g(x) = 2((-1(1x^2 + 4x) + 3)/(-2(1x^2 + 4x))) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "How many solutions does the equation -2|((3(-3cos(3π(-1x^2 - 2) + 1)) - 1)/(-3(-3cos(3π(-1x^2 - 2) + 1)) - 2)) + 3| + 2 = (2(1(-2|x - 3| + 3)) + 2)/(2(1(-2|x - 3| + 3)) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 156 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(2π(3|((3(-4x - 1) + 3)/(1(-4x - 1) + 3)) + 3| + 2) - 3) - 1 and g(x) = 3(-2|(-3cos(1πx - 2)) + 3| + 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 27 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4((-2(-1|(1sin(3πx) + 1) - 3| + 1) - 3)/(-1(-1|(1sin(3πx) + 1) - 3| + 1) + 3))^2 - 4((-2(-1|(1sin(3πx) + 1) - 3| + 1) - 3)/(-1(-1|(1sin(3πx) + 1) - 3| + 1) + 3)) + 2 and g(x) = -5((1(-1|x| + 3) - 1)/(-2(-1|x| + 3) + 3)) - 1 in the... | 42 | MATH |
[
{
"content": "How many solutions does the equation 1e^(-2(1|(1((3x + 2)/(1x + 3)) - 4) + 2| + 2)) + 2 = -1sin(1π(2(3|x + 1| - 3)^2 - 3(3|x + 1| - 3) + 4) + 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1508 | MATH |
[
{
"content": "How many solutions does the equation -2|((3(-1x^2 + 5x + 4) + 1)/(3(-1x^2 + 5x + 4) + 3)) - 3| + 2 = (-2(3x^2 - 5x - 5) - 3)/(2(3x^2 - 5x - 5) + 1) have for x 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) = -2|(3cos(1π(3x^2 + 4x - 1) - 2)) + 2| + 2 and g(x) = (-2(1e^(2x) + 2) + 2)/(-1(1e^(2x) + 2) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 354 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-2(5x^2 + 2x - 4)) + 1)/(3(-2(5x^2 + 2x - 4)) - 2) and g(x) = -2sin(3π(2x^2 + 5x + 2) + 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "How many solutions does the equation 2e^(-1(1|(1(1x + 5)^2 + 2(1x + 5) + 5) + 2|)) + 3 = 3|((3(-2cos(1πx + 3) - 1))/(1(-2cos(1πx + 3) - 1) - 3))| + 1 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 solutions does the equation -2((1(-2x^2 + 2x - 2) + 2)/(1(-2x^2 + 2x - 2) - 2)) - 5 = -3cos(2π(-3x - 2) + 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) = -4(3cos(1π((-3x + 2)/(-2x - 3)) + 3) - 2)^2 + 5(3cos(1π((-3x + 2)/(-2x - 3)) + 3) - 2) + 2 and g(x) = -1|(2sin(1πx) + 3) - 2| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 63 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(-2cos(1π(-2x^2 + 2x + 1))) + 3)/(2(-2cos(1π(-2x^2 + 2x + 1))) + 2) and g(x) = 5(-3|x - 3| + 1)^2 - 4(-3|x - 3| + 1) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 17 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(5(1|(-5x + 5) + 2| - 3)^2 - 3(1|(-5x + 5) + 2| - 3)))/(3(5(1|(-5x + 5) + 2| - 3)^2 - 3(1|(-5x + 5) + 2| - 3)) - 1) and g(x) = (-3(3|(-1x^2 - 5x + 2) - 2| - 3) + 1)/(2(3|(-1x^2 - 5x + 2) - 2| - 3) - 1) in the range -10 ≤ x ≤ 10?\... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(3π(-4(-1|x + 2| + 3)^2 + 3(-1|x + 2| + 3) + 2) - 1) - 2 and g(x) = 2sin(1π(1x^2 + 1x - 5) - 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1148 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-3sin(1π(-1|((-3x + 2)/(1x + 3)) + 3| + 1) - 2) + 2) + 4 and g(x) = -3cos(1π(-3(-1x + 2)^2 + 1(-1x + 2) - 3) - 1) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 465 | MATH |
[
{
"content": "How many solutions does the equation 1(-1|(-2((1x + 3)/(2x))^2 - 5((1x + 3)/(2x))) - 2| - 3) - 2 = -2|(3cos(1π((3x - 3)/(3x + 2)))) - 3| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 43 | MATH |
[
{
"content": "How many solutions does the equation 3cos(1π(-4(-3|x - 2| - 1)^2 - 1(-3|x - 2| - 1) + 5) - 1) - 1 = -1(3x)^2 - 4(3x) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 67 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(-5(3sin(1π(2|x - 2|) + 3) + 1) + 2) - 2)/(-2(-5(3sin(1π(2|x - 2|) + 3) + 1) + 2)) and g(x) = 2|(-2((-2x + 3)/(2x - 1))^2 - 3((-2x + 3)/(2x - 1))) - 3| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{You... | 30 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(1π(5((1x + 1)/(2x)))) + 3 and g(x) = 3e^(-2((3x - 3)/(2x + 1))) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
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