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) = -3sin(3π(5(2x^2 - 3x + 5)^2 - 5(2x^2 - 3x + 5) - 1) - 1) and g(x) = 2sin(2π(-1x^2 + 1x - 4) - 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 602 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(1|(-1|x| + 3) + 2| - 1) - 1)/(3(1|(-1|x| + 3) + 2| - 1) + 1) and g(x) = 1sin(1π(-3|x + 3| + 1) - 2) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation -2((2(2x^2 + 5x - 2) + 3)/(-1(2x^2 + 5x - 2) + 3))^2 + 3((2(2x^2 + 5x - 2) + 3)/(-1(2x^2 + 5x - 2) + 3)) - 1 = (-3(-3cos(1πx + 3)) + 2)/(-1(-3cos(1πx + 3)) + 2) 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) = (-2(3|((-2x + 1)/(-1x + 3)) + 1| - 1))/(3(3|((-2x + 1)/(-1x + 3)) + 1| - 1) + 3) and g(x) = 5(1x^2 + 3x - 3)^2 + 4(1x^2 + 3x - 3) in the range -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) = 2cos(3π(1|(-1x^2 + 2x + 5) + 3| + 1) + 3) and g(x) = -2|(3cos(2πx - 2) - 1) + 1| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 280 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3sin(2π(3|(-5x^2 + 2x - 3) + 1| - 1) + 2) - 2 and g(x) = (1(-4x^2 + 1x - 5) + 3)/(1(-4x^2 + 1x - 5) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 982 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(3π(5((-3x + 2)/(3x - 2))^2 + 4((-3x + 2)/(3x - 2)) - 1) + 2) + 1 and g(x) = 4(-2|x| + 1)^2 - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(-3(2cos(3πx + 3) + 2) + 2)^2 + 3(-3(2cos(3πx + 3) + 2) + 2) - 2 and g(x) = -2(-3sin(1πx - 2) + 1)^2 + 1(-3sin(1πx - 2) + 1) + 5 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(3cos(1π((3x - 2)/(2x + 3)) - 3) - 2))/(-2(3cos(1π((3x - 2)/(2x + 3)) - 3) - 2) - 1) = (-2(1e^(-1x) + 3) + 2)/(2(1e^(-1x) + 3)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(-1|(1|x - 2| + 2) - 1|) - 3)/(-2(-1|(1|x - 2| + 2) - 1|) - 2) and g(x) = (1(3sin(3πx + 3) + 1) + 2)/(1(3sin(3πx + 3) + 1) - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1((2(-3sin(1πx - 3) - 3))/(2(-3sin(1πx - 3) - 3) - 1)) + 2)/(3((2(-3sin(1πx - 3) - 3))/(2(-3sin(1πx - 3) - 3) - 1))) and g(x) = (1((2x + 1)/(-1x - 3)) - 1)/(1((2x + 1)/(-1x - 3)) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answ... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(2cos(3π(-5x^2 - 1x - 2) - 1) + 3) + 1| - 3 and g(x) = (1(-3x^2 - 4x + 2) + 2)/(-2(-3x^2 - 4x + 2) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "How many solutions does the equation -1(3(-1cos(3πx + 2) + 1) - 3)^2 - 2(3(-1cos(3πx + 2) + 1) - 3) + 2 = (2(-3cos(2πx - 3)) + 1)/(-2(-3cos(2πx - 3))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 117 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(3π(2(1|x - 3| - 1)^2 + 5(1|x - 3| - 1) + 3) - 1) + 3 and g(x) = -2cos(2π(-2|x - 2| - 1) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 528 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5((2(1cos(2πx + 3) - 1) + 3)/(-2(1cos(2πx + 3) - 1) - 1))^2 - 1((2(1cos(2πx + 3) - 1) + 3)/(-2(1cos(2πx + 3) - 1) - 1)) - 4 and g(x) = (1((-2x - 1)/(-1x)) + 1)/(3((-2x - 1)/(-1x)) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answ... | 39 | MATH |
[
{
"content": "How many solutions does the equation 1(2(-1cos(2πx + 2))^2 + 5(-1cos(2πx + 2)) + 2) = 1((2x + 3)/(-1x))^2 - 5((2x + 3)/(-1x)) + 3 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) = -3sin(1π(-5((1x + 3)/(-2x - 2)) - 4) + 3) + 3 and g(x) = -1|((1x - 1)/(1x + 1)) + 1| + 1 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) = -4(-3(2|x + 1| - 1)^2 + 4(2|x + 1| - 1) + 1) and g(x) = 2cos(3π((2x - 1)/(-1x)) - 2) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many solutions does the equation (-1(-5(3sin(1πx + 1) + 3)^2 + 5(3sin(1πx + 1) + 3) + 4) + 3)/(2(-5(3sin(1πx + 1) + 3)^2 + 5(3sin(1πx + 1) + 3) + 4) + 2) = -2sin(1π(3x^2 + 5x + 2) - 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 21 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3e^(1(1(-1|x - 2| + 3) - 3)) and g(x) = 2sin(1π(-4x^2 + 1x) + 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 33 | MATH |
[
{
"content": "How many solutions does the equation -4(3(-1sin(2πx + 3) + 1) - 1) + 5 = -1(-5x - 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(-3sin(3π(2x + 2) + 1) - 2)^2 + 5(-3sin(3π(2x + 2) + 1) - 2) and g(x) = 1(-1sin(2πx) - 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3cos(2π(1(1x - 2)^2 - 1(1x - 2) + 3)) - 3 and g(x) = 4(-2cos(2πx + 2) - 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 140 | MATH |
[
{
"content": "How many solutions does the equation 4((1(-2cos(1πx + 2) + 3) - 3)/(3(-2cos(1πx + 2) + 3) - 2)) + 4 = -3(-3sin(3πx + 1) - 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "How many solutions does the equation -3(2cos(3π(-3x^2 + 2x + 1) - 2) - 1) + 2 = (-3(1e^(1x) - 3) + 1)/(-2(1e^(1x) - 3) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1380 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-4(1|x - 1| - 2)^2 - 5(1|x - 1| - 2) + 4) and g(x) = 1cos(3π(4x^2 - 3x - 3) - 1) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "How many solutions does the equation -1(1|(-2|x + 3|) + 2| - 1)^2 + 1(1|(-2|x + 3|) + 2| - 1) + 1 = (1(1|x + 1| + 2) - 2)/(-2(1|x + 1| + 2) + 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 solutions does the equation (-2(3|(-3cos(3πx - 3)) - 1|) + 1)/(3(3|(-3cos(3πx - 3)) - 1|) + 1) = -1|(4x - 3) + 1| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(1(-3sin(3πx) + 1)^2 - 4(-3sin(3πx) + 1) - 2) + 1| - 1 and g(x) = 1e^(-1(5x^2 - 2x - 4)) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 142 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(1π(-4(-2x + 4)^2 - 3) - 2) - 2 and g(x) = -1|(5x - 4) - 3| - 3 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 -4((-3(1x^2 + 4x - 5) + 2)/(-3(1x^2 + 4x - 5) + 2))^2 + 5((-3(1x^2 + 4x - 5) + 2)/(-3(1x^2 + 4x - 5) + 2)) + 4 = -3(3cos(2πx + 3) - 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(-3(3sin(3πx + 1)) + 4) - 2)/(-1(-3(3sin(3πx + 1)) + 4) - 3) and g(x) = (-1(1|x| + 3) + 3)/(-2(1|x| + 3)) 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 2sin(1π((-1(-3|x - 1| + 1) - 3)/(1(-3|x - 1| + 1) - 2)) - 1) + 2 = 3cos(1π((-3x + 1)/(1x)) - 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 39 | MATH |
[
{
"content": "How many solutions does the equation 1(-3sin(2π(1|x + 3| - 3) + 3) - 1) + 2 = (1(2x^2 + 1x - 4) + 3)/(-1(2x^2 + 1x - 4) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 38 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1cos(1π(-1((1x - 3)/(-3x))^2 + 2((1x - 3)/(-3x)) + 5) - 1) + 2 and g(x) = 1|(-3x + 4) - 1| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 26 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(5(-2x - 5) - 4) - 2| and g(x) = -4(2cos(3πx + 1))^2 - 5(2cos(3πx + 1)) + 1 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) = -2|((2(3sin(1πx - 1) + 1) - 3)/(-2(3sin(1πx - 1) + 1) - 1))| + 1 and g(x) = 3(-1|x + 2| + 3)^2 - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(1(2x - 4) - 5) - 2 and g(x) = 5(-1sin(2πx - 3) + 1)^2 - 1(-1sin(2πx - 3) + 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 9 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(-2sin(2π(-2x^2 + 5x - 4) + 2) + 2) - 1)/(2(-2sin(2π(-2x^2 + 5x - 4) + 2) + 2) + 1) and g(x) = -5((2x + 3)/(-3x)) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 28 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-1((-3x)/(2x + 2))^2 + 4((-3x)/(2x + 2)) - 1) + 3)/(3(-1((-3x)/(2x + 2))^2 + 4((-3x)/(2x + 2)) - 1)) and g(x) = -3(1x^2 - 1x)^2 - 2(1x^2 - 1x) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your a... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1cos(2π(-3(-2|x + 1| + 3)^2 + 3(-2|x + 1| + 3) + 1) - 1) = -1(3sin(2πx + 1) + 2)^2 - 4(3sin(2πx + 1) + 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 points of intersection exist between the functions f(x) = -5(-1cos(3π(5x^2 + 2x - 2) + 3)) + 5 and g(x) = (-3(-3sin(1πx + 2) - 1) + 2)/(-1(-3sin(1πx + 2) - 1) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 768 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(1(-3cos(3πx - 1) - 1)^2 + 5(-3cos(3πx - 1) - 1) + 1) + 3)/(2(1(-3cos(3πx - 1) - 1)^2 + 5(-3cos(3πx - 1) - 1) + 1) + 1) and g(x) = 1(4x - 2) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 22 | MATH |
[
{
"content": "How many solutions does the equation 1(2cos(2π(3x) - 2) + 3)^2 - 5(2cos(2π(3x) - 2) + 3) + 1 = 2sin(1π((-3x)/(-2x - 1)) + 2) + 2 have for x 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) = 3(2sin(3π(5x^2 + 1x + 5) + 1) - 1)^2 + 5(2sin(3π(5x^2 + 1x + 5) + 1) - 1) + 1 and g(x) = -2e^(-2(3x)) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 674 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2e^(1(-5(2|x - 3| + 1) - 2)) and g(x) = 2|(-2x^2 + 1x + 5) - 1| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(3|(-1sin(1πx - 1) + 3) - 1| - 1) + 1)/(1(3|(-1sin(1πx - 1) + 3) - 1| - 1) - 3) and g(x) = 1|(4x^2 + 3x - 2) - 1| - 2 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) = -1cos(3π(2(4x^2 + 2x + 2) + 1) - 3) - 2 and g(x) = 3((1x)/(1x - 2)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 200 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3((2(5x^2 - 1x + 4) + 3)/(-3(5x^2 - 1x + 4) + 4))^2 + 4((2(5x^2 - 1x + 4) + 3)/(-3(5x^2 - 1x + 4) + 4)) - 1 and g(x) = -1(-2|x - 2| + 3)^2 + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(2|(-2|x + 2| + 1) - 2| + 2) + 1 and g(x) = -3|(-3sin(1πx + 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) = (-2(-3|(-3x^2 - 4x - 2) + 1| - 1) + 2)/(1(-3|(-3x^2 - 4x - 2) + 1| - 1) + 2) and g(x) = -3|(-1x - 4) - 1| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2sin(3π((-2((2x)/(2x + 3)) + 2)/(-1((2x)/(2x + 3)) + 3)) + 1) - 1 and g(x) = 4(-3sin(3πx - 2) + 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 63 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(-2(-2x^2 + 5x - 5) + 1))/(2(-2(-2x^2 + 5x - 5) + 1) + 1) and g(x) = -1|(4x^2 + 5x - 3) - 3| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(-1sin(3π(-1x + 1) - 3) - 2))/(-1(-1sin(3π(-1x + 1) - 3) - 2) + 2) and g(x) = 1(-2x^2 + 5x - 1)^2 - 2(-2x^2 + 5x - 1) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(-2sin(3π(2x - 4) + 1) + 2)^2 + 1(-2sin(3π(2x - 4) + 1) + 2) - 3 and g(x) = -3((2x - 1)/(2x + 2))^2 - 2((2x - 1)/(2x + 2)) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 90 | MATH |
[
{
"content": "How many solutions does the equation -3((3(2sin(2πx) + 1))/(3(2sin(2πx) + 1) + 3))^2 + 1((3(2sin(2πx) + 1))/(3(2sin(2πx) + 1) + 3)) + 3 = -2e^(-1(-3|x - 3| + 1)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(3(-1cos(2πx - 3))^2 + 4(-1cos(2πx - 3)) + 1) - 3 and g(x) = -1sin(1π(3|x - 2| + 3) + 1) 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 -3(-2sin(2π(1x + 1)) - 2)^2 + 3(-2sin(2π(1x + 1)) - 2) - 1 = (3(-1|x + 3| - 1) - 3)/(1(-1|x + 3| - 1) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-2e^(-2(-1x^2 + 4x - 3)) + 3)^2 + 3(-2e^(-2(-1x^2 + 4x - 3)) + 3) + 5 and g(x) = 1cos(3π(-5x + 3) - 1) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation -2((-1((-1x + 1)/(-3x + 4)) - 3)/(1((-1x + 1)/(-3x + 4)) - 2)) + 2 = -3|(-4x^2 - 5x + 4)| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-5(4x^2 + 1x - 4) + 5) + 3)/(2(-5(4x^2 + 1x - 4) + 5) - 1) and g(x) = 4(-2cos(2πx + 1) + 3)^2 - 4(-2cos(2πx + 1) + 3) - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(3|(-4x + 1) - 1| + 3) - 1)/(1(3|(-4x + 1) - 1| + 3) + 2) and g(x) = (2(-3cos(1πx - 3) + 2) - 1)/(-3(-3cos(1πx - 3) + 2) + 4) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 22 | MATH |
[
{
"content": "How many solutions does the equation 2(-1|(-2x - 1) - 1| + 2) + 4 = -1cos(2π(5x + 4)) + 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|((-1x + 2)/(3x + 2)) + 3| + 2) + 2| and g(x) = (-1(-3sin(3πx + 1) + 3) - 3)/(3(-3sin(3πx + 1) + 3) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 57 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2((-3(1|x + 1| - 2) + 3)/(-2(1|x + 1| - 2)))^2 - 1((-3(1|x + 1| - 2) + 3)/(-2(1|x + 1| - 2))) - 1 and g(x) = (2(5x^2 - 4x + 4) + 2)/(-2(5x^2 - 4x + 4) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-3sin(3π(-1|x + 3| + 2) + 3) - 3)^2 + 5 and g(x) = -2e^(-1(-1x - 5)) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 23 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(2(1x^2 - 3x - 5) - 1) + 1)/(3(2(1x^2 - 3x - 5) - 1) + 1) and g(x) = -2|(4x^2 - 4x - 3) - 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(2π((1(-1x^2 + 1x))/(2(-1x^2 + 1x) + 3))) - 1 and g(x) = 4(-2x^2 - 5x - 1)^2 + 4(-2x^2 - 5x - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-3(-1cos(2πx + 1) - 2)^2 + 5))/(3(-3(-1cos(2πx + 1) - 2)^2 + 5) - 2) and g(x) = (-2(-3x - 3))/(1(-3x - 3) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 38 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(-5((2x + 2)/(-2x + 3)) - 4) - 2| + 3 and g(x) = -1cos(1π(3x^2 - 2x + 3) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 92 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(1|((2x + 1)/(2x + 2)) - 2| - 2) - 1 and g(x) = -3(-2sin(3πx - 2) + 1)^2 - 3(-2sin(3πx - 2) + 1) - 1 intersect 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 3cos(3π(-1|((-3x - 1)/(1x + 1)) + 3| - 1) + 3) + 3 = (1(-2e^(-2x) + 2) - 3)/(-3(-2e^(-2x) + 2) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3cos(1π(-2|(-3|x - 1|) - 2| - 1) + 2) - 2 and g(x) = 1((3x - 2)/(-1x - 2))^2 + 2((3x - 2)/(-1x - 2)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-3|(1x^2 + 1x - 2) + 3| + 2))/(-2(-3|(1x^2 + 1x - 2) + 3| + 2) + 3) and g(x) = -5(2sin(1πx) - 1) + 2 where -10 ≤ x ≤ 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) = (1((-1((2x)/(2x - 3)) - 1)/(-1((2x)/(2x - 3)) + 2)) + 1)/(2((-1((2x)/(2x - 3)) - 1)/(-1((2x)/(2x - 3)) + 2)) + 3) and g(x) = -2cos(2π((-2x + 2)/(1x + 1)) - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your ... | 37 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|((-3((-1x + 2)/(-1x - 1)))/(2((-1x + 2)/(-1x - 1)) - 1)) + 2| - 3 and g(x) = 3(2sin(3πx - 3) + 1) - 5 where -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) = (-3(3(-1x^2 + 5x)^2 - 4(-1x^2 + 5x) + 5) - 1)/(1(3(-1x^2 + 5x)^2 - 4(-1x^2 + 5x) + 5) - 3) and g(x) = 1(2sin(2πx - 1)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 38 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(3|((-1x + 3)/(1x)) + 1| - 2)^2 - 2(3|((-1x + 3)/(1x)) + 1| - 2) - 1 and g(x) = -2(-1x^2 + 4x + 1)^2 - 2(-1x^2 + 4x + 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(4((-3x)/(-2x + 3))^2 - 3))/(-1(4((-3x)/(-2x + 3))^2 - 3)) and g(x) = 5(-1x^2 - 2x + 2)^2 - 3(-1x^2 - 2x + 2) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation (-3(2cos(2π(-2x + 3) + 2)))/(2(2cos(2π(-2x + 3) + 2)) - 1) = -1(2x^2 - 2x)^2 - 4(2x^2 - 2x) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many solutions does the equation (2(-3|(-1|x + 3| - 1) + 2| - 1) + 3)/(3(-3|(-1|x + 3| - 1) + 2| - 1) + 2) = (-2(3sin(2πx + 1)) - 2)/(-1(3sin(2πx + 1))) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-5((2x + 1)/(-3x + 4))^2 - 5((2x + 1)/(-3x + 4)) - 1)^2 + 4(-5((2x + 1)/(-3x + 4))^2 - 5((2x + 1)/(-3x + 4)) - 1) + 3 and g(x) = -5(-1cos(2πx - 3) - 1)^2 + 5(-1cos(2πx - 3) - 1) + 4 where -10 ≤ x ≤ 10.\n\nPresent the answe... | 34 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3cos(1π((1(2x + 1) + 1)/(2(2x + 1) - 1)) + 3) + 3 and g(x) = (-1((3x + 2)/(2x + 1)))/(-1((3x + 2)/(2x + 1)) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "How many solutions does the equation -1((-3(-1|x - 2| + 2) - 1)/(1(-1|x - 2| + 2) + 3)) - 5 = -1cos(1π(-2|x + 1| - 2) - 1) - 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) = -3cos(2π(1|(-2x - 5) - 2|) + 3) - 2 and g(x) = -3sin(1π(4x - 3) - 1) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "How many solutions does the equation (1(-2sin(2π((-3x + 2)/(-3x + 4)) + 1) + 2) - 3)/(-1(-2sin(2π((-3x + 2)/(-3x + 4)) + 1) + 2) - 2) = 2sin(2π(3|x - 3| - 1) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 124 | MATH |
[
{
"content": "How many solutions does the equation -3|((-1(-2|x - 3|) - 2)/(-3(-2|x - 3|))) + 1| - 3 = -2sin(3π(-5x - 4)) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(-4(-5x)^2 + 4(-5x) + 5) - 1| - 1 and g(x) = -3e^(1(1|x - 2| - 3)) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation 1(2e^(2(-3x - 1)) - 1)^2 + 1(2e^(2(-3x - 1)) - 1) + 2 = 2sin(2π(2x - 4) + 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 41 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2((-3(-3|x + 2| + 2) - 3)/(-3(-3|x + 2| + 2) + 2)))/(-2((-3(-3|x + 2| + 2) - 3)/(-3(-3|x + 2| + 2) + 2)) + 1) and g(x) = -3sin(1π(2x - 3) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 34 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(2(-3x - 4) - 1)^2 - 1(2(-3x - 4) - 1) and g(x) = 3sin(2π(1x^2 - 1x + 4) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation -3|((1(5x^2 + 2x + 3))/(2(5x^2 + 2x + 3))) + 3| + 1 = 5(-3sin(3πx + 2) - 1) - 1 have for x 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(3π(-1(5x + 2) - 1) - 2) and g(x) = 1sin(2π(3x) + 1) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 300 | MATH |
[
{
"content": "How many solutions does the equation -5(-3(2|x - 1| - 1) + 4)^2 + 5 = -1|(4x) - 1| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5((-1(2|x - 1| - 3) + 2)/(1(2|x - 1| - 3) + 1))^2 - 3((-1(2|x - 1| - 3) + 2)/(1(2|x - 1| - 3) + 1)) - 1 and g(x) = 3cos(2π(-4x^2 + 4x) - 3) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 1176 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(2e^(-2(3x + 3))) - 2 and g(x) = 1sin(2π(5x^2 - 4x + 2) + 1) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 724 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(3((1x - 1)/(-1x - 2)) + 5))/(2(3((1x - 1)/(-1x - 2)) + 5) + 3) and g(x) = -3sin(3π((-3x - 2)/(-1x - 3)) + 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 149 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(2π(3|(-3|x - 1|) + 2| + 1) + 2) + 3 and g(x) = 1|((3x + 3)/(3x + 2)) + 1| - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 172 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(2π(1|(-2|x - 2| - 1) + 1|) + 3) - 1 and g(x) = -3|(-5x^2 + 2x + 2) + 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-1|((2x + 1)/(-1x)) + 1| + 1) + 2| and g(x) = 3|(3cos(2πx - 3) - 1) - 3| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
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