messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation -2cos(1π((2(-5x^2 - 2x + 5))/(-2(-5x^2 - 2x + 5) - 3)) + 3) - 3 = 2(-2|x - 2| + 2)^2 + 4(-2|x - 2| + 2) - 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 solutions does the equation (2(-2|(-2cos(3πx) + 1) - 1|) + 1)/(-1(-2|(-2cos(3πx) + 1) - 1|)) = 2(-2e^(-2x) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 61 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-5(-3cos(3π(-4x) - 1) - 3)^2 - 1(-3cos(3π(-4x) - 1) - 3) - 1)| - 3 and g(x) = 2|(2(1cos(1πx + 3)) + 3)| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 240 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-3(1sin(3πx + 3))^2 - 4(1sin(3πx + 3)) + 1) + 1 and g(x) = -1(1x - 3)^2 + 3(1x - 3) + 5 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) = -1|(1((-2(-3e^(-1x) + 1) - 3)/(1(-3e^(-1x) + 1))) + 3)| and g(x) = -3((-1(3cos(3πx + 3) + 2))/(-3(3cos(3πx + 3) + 2) - 1)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 30 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3((2(-3|x| + 1) - 2)/(3(-3|x| + 1) + 3)) + 4 and g(x) = 2|(1sin(1πx) + 3) - 1| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "How many solutions does the equation -3((3(2cos(1πx + 3) - 1) - 2)/(-1(2cos(1πx + 3) - 1) + 2)) + 1 = -1(3cos(1πx - 2)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(1(-1|((-3x - 2)/(-1x - 2)) + 2| + 3)^2 - 1(-1|((-3x - 2)/(-1x - 2)) + 2| + 3) + 1) + 3 and g(x) = (-1(1(-3x^2 - 2x + 5) - 4))/(-3(1(-3x^2 - 2x + 5) - 4) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your an... | 6 | MATH |
[
{
"content": "How many solutions does the equation (1(3sin(1π(-1(-1x^2 - 1x - 5) + 1) - 3)) + 3)/(1(3sin(1π(-1(-1x^2 - 1x - 5) + 1) - 3)) + 2) = -1(-2|((-2x - 3)/(-2x - 2)) - 3| - 3) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 64 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1cos(1π(3|((1(-2x^2 + 5x - 3) + 2)/(-3(-2x^2 + 5x - 3) + 4)) + 3| - 1) + 3) + 3 and g(x) = (-3(-1sin(3π(-2x^2 - 1x + 5) - 2) + 1) - 3)/(3(-1sin(3π(-2x^2 - 1x + 5) - 2) + 1) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in ... | 35 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(1((-3(1x^2 - 5x))/(-2(1x^2 - 5x) + 1))) - 2| - 1 and g(x) = -2|((2(1x^2 + 5x + 3) + 2)/(2(1x^2 + 5x + 3) + 1)) - 3| + 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 3cos(1π(-2((-2(-2x^2 - 2x + 2) + 3)/(-1(-2x^2 - 2x + 2) - 1)) + 1) + 2) = -3(1e^(-2((-3x + 1)/(2x - 3))) - 1)^2 - 3(1e^(-2((-3x + 1)/(2x - 3))) - 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 38 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(1|((1x + 3)/(2x - 1)) + 3| + 3) + 3 and g(x) = 3cos(3π(3x - 2) - 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 172 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(2|((-1x - 3)/(-3x - 2)) + 2| - 3)^2 - 2(2|((-1x - 3)/(-3x - 2)) + 2| - 3) + 4 and g(x) = 3|(-1cos(2πx - 2) - 1) - 3| - 2 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 (1(-2|(-4x^2 - 2x - 1) + 1| + 2) - 3)/(-2(-2|(-4x^2 - 2x - 1) + 1| + 2)) = 2(2cos(1πx - 2)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "How many solutions does the equation (-2(-5(-3sin(3π(-3|x - 2| + 3) - 1) - 3) - 2) + 3)/(1(-5(-3sin(3π(-3|x - 2| + 3) - 1) - 3) - 2) + 1) = -2(2|(-1x + 4) + 1| - 3)^2 + 1(2|(-1x + 4) + 1| - 3) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2sin(3π(-1|(4x - 1)| - 2) - 2) - 2 and g(x) = 3(3sin(3πx + 3) - 3) + 4 where -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) = -4(-2|((-2(4x + 1) - 2)/(-2(4x + 1) - 3)) - 1| + 1)^2 - 3(-2|((-2(4x + 1) - 2)/(-2(4x + 1) - 3)) - 1| + 1) + 2 and g(x) = -3|(3(1x^2 + 1x - 1) - 1) - 2| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(3|(-2cos(3πx - 2) + 3)| + 1)^2 - 5(3|(-2cos(3πx - 2) + 3)| + 1) + 1 and g(x) = 2|(-1x^2 + 4x + 1) - 3| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-3|(-3sin(2π(1x + 3) + 1)) - 1|)^2 - 5(-3|(-3sin(2π(1x + 3) + 1)) - 1|) + 4 and g(x) = 3(-2|(-4x^2 + 4x - 1)|) - 1 in the range -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) = (2(-2|(1(3e^(2x) + 2)^2 - 2(3e^(2x) + 2) - 2) + 2| - 3) - 1)/(3(-2|(1(3e^(2x) + 2)^2 - 2(3e^(2x) + 2) - 2) + 2| - 3) + 2) and g(x) = -5(3|(-3sin(1πx + 3)) - 2| - 3)^2 - 5(3|(-3sin(1πx + 3)) - 2| - 3) + 2 intersect in the interval [-10, 10].\n\nPresent ... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(-1(1|(1sin(1πx - 3) - 3) - 3|) + 2) + 3)/(-2(-1(1|(1sin(1πx - 3) - 3) - 3|) + 2) - 3) and g(x) = 1(-3(1cos(3πx) - 2)^2 - 4(1cos(3πx) - 2)) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(1(-3sin(1π(5x^2 + 4x - 5) - 3) + 3) - 3) - 3)/(-2(1(-3sin(1π(5x^2 + 4x - 5) - 3) + 3) - 3) - 1) and g(x) = 4(-1(-3cos(1πx + 3) + 2) + 4)^2 + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 133 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(2|(1x + 5) - 2| - 3)^2 + 4(2|(1x + 5) - 2| - 3) + 1 and g(x) = 2|((-2x - 3)/(-3x - 3))| + 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) = 2cos(3π(-3((-2x + 3)/(-1x))^2 + 1((-2x + 3)/(-1x)) + 4) - 3) - 2 and g(x) = -1(-1e^(-2x) - 1)^2 - 3(-1e^(-2x) - 1) + 4 where -10 ≤ x ≤ 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(1((-1(3|x| + 2) - 3)/(-1(3|x| + 2) - 1))^2 - 4((-1(3|x| + 2) - 3)/(-1(3|x| + 2) - 1)) - 3) - 4 and g(x) = 3(-3(3sin(3πx + 1) - 2) - 5)^2 - 2(-3(3sin(3πx + 1) - 2) - 5) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex for... | 120 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3cos(3π(-3((-3(1|x + 1|) - 3)/(1(1|x + 1|)))^2 + 3((-3(1|x + 1|) - 3)/(1(1|x + 1|))) + 3) + 2) - 2 and g(x) = -1|((2(2e^(1x) - 1) + 1)/(3(2e^(1x) - 1) - 2)) - 3| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format... | 37 | MATH |
[
{
"content": "How many solutions does the equation 5(2sin(1π((-1x + 2)/(-3x - 1)) - 2) - 3)^2 + 3(2sin(1π((-1x + 2)/(-3x - 1)) - 2) - 3) - 3 = -3e^(2((-1x - 2)/(-3x - 1))) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "How many solutions does the equation 1|(2e^(1((-2(-5x^2 - 5x - 1) - 2)/(-1(-5x^2 - 5x - 1) + 2))) + 1)| - 3 = 3sin(2π(5(-3|x + 1| - 3)) - 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(3π((-2(1|(1x^2 - 2x - 3) + 3|) + 1)/(-1(1|(1x^2 - 2x - 3) + 3|) + 2)) + 3) - 1 and g(x) = -2cos(1π(4(3|x + 2| - 2)^2 + 2(3|x + 2| - 2) - 5) + 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 1070 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-2sin(1π(1(1|x - 3| + 3)^2 - 5(1|x - 3| + 3) + 2)) - 2))/(3(-2sin(1π(1(1|x - 3| + 3)^2 - 5(1|x - 3| + 3) + 2)) - 2) + 2) and g(x) = (-2(-4(-2cos(2πx - 2) + 2)) - 2)/(2(-4(-2cos(2πx - 2) + 2)) - 1) where -10 ≤ x ≤ 10.\n\nPr... | 204 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-1cos(2π(2x - 4) + 1))^2 - 4(-1cos(2π(2x - 4) + 1)) - 5 and g(x) = 2sin(3π(1x - 1)) - 3 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) = -5(2cos(3π(3|((-1x + 1)/(-2x - 2)) + 1| + 3) + 1)) + 3 and g(x) = 1((1(-3|x - 1| - 3))/(-1(-3|x - 1| - 3) - 2)) - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 64 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3cos(1π(3|(2(5x - 5)^2 + 1(5x - 5) - 2) - 1| - 2) + 3) and g(x) = (-2(-4(2x - 2)^2 + 1(2x - 2) + 2) - 3)/(-1(-4(2x - 2)^2 + 1(2x - 2) + 2) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 1284 | MATH |
[
{
"content": "How many solutions does the equation (-1(3cos(3π(-5x^2 - 3x) - 3) + 2) + 2)/(-3(3cos(3π(-5x^2 - 3x) - 3) + 2) - 1) = -2(-1x - 5)^2 - 1(-1x - 5) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 42 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5((-2(1sin(1πx) + 1) + 2)/(-2(1sin(1πx) + 1) - 1))^2 + 5((-2(1sin(1πx) + 1) + 2)/(-2(1sin(1πx) + 1) - 1)) - 5 and g(x) = -5(1e^(1x) + 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1sin(2π(-3(1|((-2x + 1)/(-3x - 3)) - 1| - 2) + 1) + 3) and g(x) = -1|((1(2sin(2πx - 1) - 1) - 2)/(3(2sin(2πx - 1) - 1) + 1)) + 1| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 115 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-2(3|((-1x + 2)/(1x)) + 3| + 3) + 5)^2 + 5(-2(3|((-1x + 2)/(1x)) + 3| + 3) + 5) - 5 and g(x) = 2sin(1π(5(-4x^2 + 1x - 4) + 2) - 1) - 1 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) = (-1(1cos(1π(-3x - 1) - 2) + 2))/(2(1cos(1π(-3x - 1) - 2) + 2) + 1) and g(x) = (-1(1x^2 - 5x + 4) - 3)/(3(1x^2 - 5x + 4) + 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) = -3|(3((3(-1x + 2) - 3)/(-2(-1x + 2) + 1))^2 - 5) + 1| and g(x) = -4(3cos(3π(-1|x - 1| - 2) - 2) + 2) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 56 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(2sin(3π(3x - 1) + 1) + 1) - 1| + 2 and g(x) = 2|(2x) - 1| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(2sin(1π(2((-3x - 3)/(-1x + 2))^2 + 1((-3x - 3)/(-1x + 2)) + 3) + 2) - 2) - 2 and g(x) = -2|(-5((-2x + 3)/(-2x - 1))^2 - 3((-2x + 3)/(-2x - 1)) + 5)| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your a... | 144 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(1(-1e^(-1(2|x| + 2)) + 2)^2 + 5(-1e^(-1(2|x| + 2)) + 2) + 3))/(-3(1(-1e^(-1(2|x| + 2)) + 2)^2 + 5(-1e^(-1(2|x| + 2)) + 2) + 3) - 2) and g(x) = (-2(-2(-2sin(1πx + 1))^2 - 3(-2sin(1πx + 1)) + 1) + 2)/(1(-2(-2sin(1πx + 1))^2 - 3(-2... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(3π(2|(-2(-3x^2 - 5x - 4) - 1)|) - 2) - 3 and g(x) = -3(-1sin(1π(-4x) - 2) + 1)^2 - 5(-1sin(1π(-4x) - 2) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 282 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(2sin(3π((1x + 1)/(1x + 2)) + 2)) - 4 and g(x) = -1(2|x - 3| - 1) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 33 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3e^(-1(2|(-5x - 3)| + 1)) - 2 and g(x) = -2cos(1π(-3|x - 3| - 2) - 2) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "How many solutions does the equation -2((3(-2sin(1πx + 2) + 1) - 1)/(-1(-2sin(1πx + 2) + 1) + 3))^2 + 5((3(-2sin(1πx + 2) + 1) - 1)/(-1(-2sin(1πx + 2) + 1) + 3)) - 1 = (-3(-1x^2 + 3x - 3) + 2)/(-2(-1x^2 + 3x - 3) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\b... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(1((2(-1x + 1) - 3)/(-2(-1x + 1) - 2))^2 + 2((2(-1x + 1) - 3)/(-2(-1x + 1) - 2)) - 5) - 3| + 2 and g(x) = (-2(1cos(3π(1x + 5) + 1) + 1) + 2)/(1(1cos(3π(1x + 5) + 1) + 1) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaT... | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(2|((-3x - 2)/(1x + 1)) + 1| - 3) + 1 and g(x) = 1sin(1π(1|x - 2| + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(2cos(3π(2|x - 2| - 2) + 2) - 2) + 2)/(-1(2cos(3π(2|x - 2| - 2) + 2) - 2) - 1) and g(x) = -2cos(3π(1x - 2) + 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(3π((-3(-4(4x^2 + 1x - 4) - 1) + 1)/(-3(-4(4x^2 + 1x - 4) - 1) - 3)) + 2) - 1 and g(x) = 2(3cos(1π(-3|x + 2| + 2) + 3) + 2) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 63 | MATH |
[
{
"content": "How many solutions does the equation -3|(-2(1sin(1πx - 1) - 2)^2 - 3(1sin(1πx - 1) - 2) - 3) + 2| + 1 = -2sin(3π(4x - 1) + 2) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(2(-3|(-5x^2 + 5x + 1) + 3|) - 1) - 1)/(-2(2(-3|(-5x^2 + 5x + 1) + 3|) - 1) - 1) and g(x) = (2(-2|(1cos(3πx + 3) - 2) + 1| + 2))/(3(-2|(1cos(3πx + 3) - 2) + 1| + 2) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\b... | 60 | MATH |
[
{
"content": "How many solutions does the equation -2(1|(-3cos(1πx - 3) - 2) - 3| - 3) + 3 = 2|((-3x + 2)/(2x + 2)) - 1| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 18 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-5(-3sin(2πx) + 2) - 2) - 2)/(3(-5(-3sin(2πx) + 2) - 2) - 2) and g(x) = -2((-3x + 3)/(3x - 2)) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "How many solutions does the equation 4((-2(3|(-3x + 3)| + 1) + 2)/(-2(3|(-3x + 3)| + 1) + 3))^2 + 5((-2(3|(-3x + 3)| + 1) + 2)/(-2(3|(-3x + 3)| + 1) + 3)) + 1 = 2|((-1(-3sin(1πx + 2) - 2) - 2)/(1(-3sin(1πx + 2) - 2) + 3)) + 1| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTe... | 42 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(2|((-2(-2sin(3πx + 1) - 2) - 1)/(3(-2sin(3πx + 1) - 2) - 2)) + 3| - 1)^2 + 3(2|((-2(-2sin(3πx + 1) - 2) - 1)/(3(-2sin(3πx + 1) - 2) - 2)) + 3| - 1) + 3 and g(x) = (-1(3(1sin(1πx + 1) + 1)) + 2)/(1(3(1sin(1πx + 1) + 1))) in the ra... | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(1cos(3π(-4(-2x - 5)^2 - 2(-2x - 5) + 3) - 2) + 3) + 1)/(2(1cos(3π(-4(-2x - 5)^2 - 2(-2x - 5) + 3) - 2) + 3) + 1) and g(x) = (-1(4(1|x + 2| - 2) - 4) - 1)/(-1(4(1|x + 2| - 2) - 4)) where -10 ≤ x ≤ 10.\n\nPresent the answer ... | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(3sin(1π(-3|x - 1| + 3) + 3)) + 2)/(3(3sin(1π(-3|x - 1| + 3) + 3)) + 2) and g(x) = 1|((1x)/(1x - 3)) + 1| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 57 | MATH |
[
{
"content": "How many solutions does the equation 2sin(1π((-1(5x) + 3)/(3(5x) + 1)) + 2) - 1 = -1e^(1((2x + 1)/(3x - 1))) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "How many solutions does the equation -3sin(2π(4(-3|x + 3| - 2)^2 - 5(-3|x + 3| - 2)) + 3) - 2 = (-3(2|x - 3| - 2) + 2)/(-1(2|x - 3| - 2) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 128 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1sin(3π(1|((-2(1x - 3) - 1)/(-1(1x - 3) + 2)) + 1| + 3) - 1) and g(x) = (-3(3|(4x - 3) - 2|) - 3)/(3(3|(4x - 3) - 2|) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 81 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(-3|(-3sin(3πx + 3) - 3) + 2| + 2) and g(x) = (2(3x - 1) + 2)/(3(3x - 1) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 119 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((2(3cos(1πx + 3) + 1) - 2)/(1(3cos(1πx + 3) + 1)))^2 + 2((2(3cos(1πx + 3) + 1) - 2)/(1(3cos(1πx + 3) + 1))) + 1 and g(x) = -1(2cos(1πx - 2) - 3)^2 + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(3π(-3|(1x^2 - 2x + 5) + 3|) + 3) - 2 and g(x) = 3(-3|x| + 3) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "How many solutions does the equation (2(-3(-1|x - 3| - 3) - 4) - 1)/(2(-3(-1|x - 3| - 3) - 4)) = 2(-3cos(1πx + 3) + 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3e^(-1(-3(-2x^2 + 4x) + 5)) - 2 and g(x) = 1|(1x^2 - 4x - 3) + 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) = -2|(-2sin(1π(2(-3x + 3)^2) + 1) + 3) + 3| + 3 and g(x) = -5(-2((2x + 3)/(-2x + 3)) + 3)^2 + 3(-2((2x + 3)/(-2x + 3)) + 3) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 18 | MATH |
[
{
"content": "How many solutions does the equation (-2(3(3cos(2π(3|x + 2| - 1)) - 2)^2 + 3(3cos(2π(3|x + 2| - 1)) - 2)) - 1)/(-2(3(3cos(2π(3|x + 2| - 1)) - 2)^2 + 3(3cos(2π(3|x + 2| - 1)) - 2))) = (1(2|(-2x + 1) + 2| - 3))/(-2(2|(-2x + 1) + 2| - 3) - 2) have for x in the interval [-10, 10]?\n\nPresent the answe... | 167 | MATH |
[
{
"content": "How many solutions does the equation 3sin(3π(-1|(-3(4x^2 - 3x + 3) - 5) - 2| - 3)) + 1 = -1(1sin(1π(-2x^2 + 2) + 2) - 3) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1101 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3e^(2(-2|(3x + 3) + 1| + 2)) + 1 and g(x) = 1cos(1π(-2|x + 3| - 3) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 19 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3((3(-4(-1|x + 1| + 3)^2 - 4(-1|x + 1| + 3) + 4))/(-2(-4(-1|x + 1| + 3)^2 - 4(-1|x + 1| + 3) + 4) - 3)) and g(x) = -3cos(1π(-3|(3x^2 - 1x - 5) - 1| + 2)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 258 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(1sin(1π(3|((1x + 3)/(3x + 3)) + 1| + 1) + 1) + 2)^2 - 5(1sin(1π(3|((1x + 3)/(3x + 3)) + 1| + 1) + 1) + 2) - 3 and g(x) = -2(-5(1|x - 2| - 3) + 2)^2 - 1(-5(1|x - 2| - 3) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTe... | 18 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-2(3|(-1x) + 1|)^2 - 3))/(2(-2(3|(-1x) + 1|)^2 - 3) - 1) and g(x) = 1(-5(2sin(2πx - 1) + 2)^2 - 5(2sin(2πx - 1) + 2) + 5) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 40 | MATH |
[
{
"content": "How many solutions does the equation (2(-1cos(2π(5(3x^2 + 5x + 5) - 2) + 1)))/(3(-1cos(2π(5(3x^2 + 5x + 5) - 2) + 1)) - 2) = 1|((-1(3cos(1πx - 1) - 3) + 3)/(1(3cos(1πx - 1) - 3) + 1)) - 2| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 287 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-3sin(3π((-1x - 3)/(-2x + 3)) - 1)) and g(x) = 1|((2x - 3)/(2x + 1)) - 1| + 1 where -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|(-1(3sin(2πx + 3) - 1) - 1) - 1| + 1 and g(x) = 4(-3sin(1πx + 2) + 3)^2 + 2(-3sin(1πx + 2) + 3) + 1 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(-1((-2(-3|x + 1| - 2) + 3)/(-2(-3|x + 1| - 2) + 1))^2 - 1) + 3 and g(x) = (3(-1cos(2π(-4x^2 + 2x + 2) + 2) + 1) + 1)/(3(-1cos(2π(-4x^2 + 2x + 2) + 2) + 1) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\... | 251 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-2|(-2((-3x + 3)/(2x - 1)) + 2) - 1| + 2)^2 - 4(-2|(-2((-3x + 3)/(2x - 1)) + 2) - 1| + 2) and g(x) = -1|(1((3x - 1)/(2x))^2 - 2((3x - 1)/(2x)) - 1) + 1| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\box... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-5(-2|x| - 2) + 5) - 2)/(1(-5(-2|x| - 2) + 5) - 3) and g(x) = (1(3cos(2πx + 3) + 1) + 3)/(-2(3cos(2πx + 3) + 1) + 3) in the range -10 ≤ x ≤ 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(2π(-2|x + 2| + 3) + 2) + 2)^2 - 2(1cos(2π(-2|x + 2| + 3) + 2) + 2) + 2 and g(x) = 3|(3sin(3πx + 2) - 3) + 1| - 1 in the range -10 ≤ x ≤ 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) = -2cos(2π(1(-3|((-1x - 3)/(-2x + 3))| - 2) + 4)) - 2 and g(x) = -4(-5(2|x| - 2) + 3)^2 + 4(-5(2|x| - 2) + 3) + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(1((1x + 3)/(-3x + 2)) + 3)^2 and g(x) = -4(1sin(3πx + 1) - 3)^2 + 2(1sin(3πx + 1) - 3) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 52 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(1|(3((2x)/(-1x - 3)) + 2)| - 1)^2 - 2(1|(3((2x)/(-1x - 3)) + 2)| - 1) - 2 and g(x) = 2cos(1π(2|(3x + 5) - 2| - 1) + 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(-1|(2sin(1πx + 1) + 1) - 3| + 1) + 3 and g(x) = 2cos(3π((-3x - 1)/(-1x - 2))) - 1 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) = 3|((2(2sin(3πx + 1) + 1) + 1)/(-2(2sin(3πx + 1) + 1) - 3)) + 1| - 1 and g(x) = 1cos(1π(-1x + 1) - 1) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many solutions does the equation -1|(4(-1cos(2πx + 2) - 1)^2 - 5(-1cos(2πx + 2) - 1)) + 2| - 1 = (-2(2sin(2πx + 1) - 2) + 1)/(3(2sin(2πx + 1) - 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) = (2(-2cos(2π(-1x^2 + 5x)) + 1))/(-1(-2cos(2π(-1x^2 + 5x)) + 1) + 2) and g(x) = -1(3cos(1πx + 1) - 2)^2 - 4(3cos(1πx + 1) - 2) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 279 | MATH |
[
{
"content": "How many solutions does the equation (-2(2(-1|x - 1| + 3) + 5) - 1)/(-3(2(-1|x - 1| + 3) + 5) + 2) = 3sin(2π(2|x|)) - 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) = 1cos(2π(5(2|x - 3| + 1)^2 - 5(2|x - 3| + 1) + 5) + 2) - 2 and g(x) = (-2(-1x^2 - 2x + 5) + 1)/(3(-1x^2 - 2x + 5) - 2) in the range -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|(3e^(2(-1x - 1)) - 3) - 2| + 3 and g(x) = 5(1sin(3πx - 2) + 1)^2 - 5(1sin(3πx - 2) + 1) - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 35 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(3|(1(-3x^2 - 4x + 5) + 1) + 2|) - 1)/(-2(3|(1(-3x^2 - 4x + 5) + 1) + 2|) + 3) and g(x) = 3|(1(3x^2 + 4x - 4) + 5)| + 1 where -10 ≤ x ≤ 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) = -3e^(2((-1(1x + 1))/(2(1x + 1)))) + 1 and g(x) = 3sin(3π(-1|x + 1| - 2) - 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "How many solutions does the equation (-2(-2sin(3π(2|x + 1| - 1)) - 2) + 2)/(-3(-2sin(3π(2|x + 1| - 1)) - 2) - 2) = (-3(2|x - 2| + 2) - 1)/(2(2|x - 2| + 2) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 120 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(2((1x - 1)/(-1x + 2))^2 - 5((1x - 1)/(-1x + 2)) - 5) + 2| + 1 and g(x) = 1sin(1π(-3|x + 3| - 2) - 2) in the range -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(3sin(1π((3(-3x + 5) + 1)/(1(-3x + 5))) + 3) + 2)^2 + 5 and g(x) = 1cos(2π(1(-1|x - 2| - 1) + 4) - 1) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 36 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(-3sin(3π((-2x)/(1x + 3)) + 1))^2 - 2(-3sin(3π((-2x)/(1x + 3)) + 1)) - 1 and g(x) = -5(1|x + 2| - 3) - 5 intersect in the interval [-10, 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) = -2cos(2π(-2|(-1((-1x - 1)/(1x))^2 + 1((-1x - 1)/(1x))) - 3| + 1) - 2) - 2 and g(x) = 1(-3|((-3x + 3)/(1x)) - 3| + 3) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 18 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(3π(1(1x^2 + 5x + 1) + 1)) - 2 and g(x) = 3cos(3π(-4x - 3) - 3) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 474 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((1(-1sin(1π(-1x^2 + 5x + 1)) + 1) - 2)/(2(-1sin(1π(-1x^2 + 5x + 1)) + 1) - 3)) + 5 and g(x) = -2(-3e^(2((2x + 3)/(2x + 1))) + 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 22 | MATH |
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