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) = -3e^(-2(3|(-2(5x^2 - 2x + 4) + 2) + 3|)) and g(x) = -3sin(3π(-1(-1|x - 3| + 2)^2 + 5(-1|x - 3| + 2) - 1) - 1) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 326 | MATH |
[
{
"content": "How many solutions does the equation 2|(-1(-1cos(1π(-2x - 3) - 3) + 1)^2 - 2(-1cos(1π(-2x - 3) - 3) + 1) + 3) + 3| = 1cos(2π(-4((-2x - 1)/(3x - 1))^2 - 3((-2x - 1)/(3x - 1)) - 1) + 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 90 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(-1sin(3π(3x^2 + 1x - 1) + 3) + 3)| - 2 and g(x) = 3|(1x - 4) + 3| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 212 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-1sin(2π(-4x^2 + 3x - 5) + 2) - 3) and g(x) = 2(-1cos(3πx - 1) + 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 384 | MATH |
[
{
"content": "How many solutions does the equation -3sin(1π(3|(-5(-4x^2 - 4x + 5) - 5) - 3| + 1) + 3) + 3 = -2sin(3π(3((3x + 2)/(-2x - 1))^2 - 3((3x + 2)/(-2x - 1)) + 5) - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 286 | MATH |
[
{
"content": "How many solutions does the equation (2(-2|(2x^2 + 3x)| - 3) - 3)/(-2(-2|(2x^2 + 3x)| - 3) + 1) = -2cos(3π((2x - 1)/(-2x - 2)) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 68 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(-2sin(2π(-4x + 2)) + 2) - 1| + 2 and g(x) = 2sin(3π(1x - 4) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 203 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|((1(-5(1x^2 - 1x - 3) + 5) - 3)/(-2(-5(1x^2 - 1x - 3) + 5) - 1))| + 2 and g(x) = -5(-3sin(2π(1x^2 - 4x - 4) - 1) - 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 416 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5((-1(1sin(2π(2|x + 2| + 2) + 2) - 1) - 1)/(3(1sin(2π(2|x + 2| + 2) + 2) - 1))) - 1 and g(x) = -1(-1((-3x - 1)/(3x)) + 1)^2 - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(-2(1e^(1x))^2 + 4) - 2| and g(x) = -5(2sin(3πx - 3) + 2)^2 - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 35 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(1π(1(-1x^2 - 5x + 3) - 1) - 2) - 2 and g(x) = 1|(-2sin(2πx + 2) + 3) - 3| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 206 | MATH |
[
{
"content": "How many solutions does the equation -2|(2((2x - 3)/(-2x)))| + 1 = -1sin(1π(-3|x + 1|) + 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 26 | MATH |
[
{
"content": "How many solutions does the equation 4((3(1|(2cos(2πx - 1) - 2) + 2|) + 2)/(-3(1|(2cos(2πx - 1) - 2) + 2|) + 2))^2 - 4((3(1|(2cos(2πx - 1) - 2) + 2|) + 2)/(-3(1|(2cos(2πx - 1) - 2) + 2|) + 2)) - 5 = (-2(-5(3x - 3)^2 - 1(3x - 3) - 2) + 1)/(-2(-5(3x - 3)^2 - 1(3x - 3) - 2) - 1) have for x in the int... | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(3((-1x + 3)/(2x)))| - 3 and g(x) = -3|(2sin(3πx + 3) + 3) - 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 79 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(4(-1|(-1sin(3πx + 3) + 3) - 3| + 1)^2 - 4(-1|(-1sin(3πx + 3) + 3) - 3| + 1) + 1) + 1)/(3(4(-1|(-1sin(3πx + 3) + 3) - 3| + 1)^2 - 4(-1|(-1sin(3πx + 3) + 3) - 3| + 1) + 1) - 1) and g(x) = 2((2(2x^2 + 4x + 1))/(-1(2x^2 + 4x + 1) + 2)) - 2 intersect in ... | 231 | MATH |
[
{
"content": "How many solutions does the equation -1|(-3(3((-3x + 3)/(-1x + 2)) - 4)^2 - 2(3((-3x + 3)/(-1x + 2)) - 4) + 2) - 1| - 3 = (1(5(1cos(3πx + 2) - 2)^2 + 5(1cos(3πx + 2) - 2) - 2) - 2)/(-2(5(1cos(3πx + 2) - 2)^2 + 5(1cos(3πx + 2) - 2) - 2) + 1) have for x in the interval [-10, 10]?\n\nPresent the answ... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1sin(1π((-3(-5x^2 - 5x + 4) - 1)/(-3(-5x^2 - 5x + 4) + 2)) + 2) + 1 and g(x) = -1(-2sin(3πx) + 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) = -3sin(1π(3((-2x + 2)/(-2x + 3))^2 - 2((-2x + 2)/(-2x + 3)) - 1) - 3) + 1 and g(x) = 1sin(3π(-3x^2 + 1x - 2) + 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 754 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5((1(1(2|x - 3| - 3)^2 - 4(2|x - 3| - 3) - 5) - 3)/(-2(1(2|x - 3| - 3)^2 - 4(2|x - 3| - 3) - 5) + 3)) + 2 and g(x) = 4((1(-3sin(3πx + 2)) + 2)/(3(-3sin(3πx + 2)) + 1))^2 + 1((1(-3sin(3πx + 2)) + 2)/(3(-3sin(3πx + 2)) + 1)) -... | 120 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-2|((3(5x^2 - 2x - 5) - 2)/(-1(5x^2 - 2x - 5) + 3))|) and g(x) = -1|(-1(-3cos(3πx - 3) - 2)^2 - 1(-3cos(3πx - 3) - 2) + 1) - 3| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 62 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-2(3sin(1πx + 1)) + 2) + 3)/(1(-2(3sin(1πx + 1)) + 2) - 3) and g(x) = -3(2x + 4)^2 - 5(2x + 4) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(2((-2(-1x))/(-1(-1x) + 3))^2 - 3((-2(-1x))/(-1(-1x) + 3))) - 3| + 2 and g(x) = 5((-3(-1|x + 2| + 1) - 2)/(2(-1|x + 2| + 1) - 3))^2 - 4((-3(-1|x + 2| + 1) - 2)/(2(-1|x + 2| + 1) - 3)) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(-2e^(1(-3|x + 1| - 3)) + 2) - 2)/(2(-2e^(1(-3|x + 1| - 3)) + 2) + 3) and g(x) = 1cos(3π(-2|x - 2| + 3) + 2) + 1 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) = 4(-2cos(2π(3|x + 1| + 2) + 2)) and g(x) = -2(3x^2 + 1x - 3) + 5 where -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) = -4((3(-3|(3sin(1πx + 1) - 1) + 3| + 2) - 3)/(-2(-3|(3sin(1πx + 1) - 1) + 3| + 2) - 3))^2 - 3((3(-3|(3sin(1πx + 1) - 1) + 3| + 2) - 3)/(-2(-3|(3sin(1πx + 1) - 1) + 3| + 2) - 3)) - 1 and g(x) = (-3(-2cos(2π(-1x - 1) + 3)) + 3)/(2(-2c... | 56 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(-2|(-1((1x + 2)/(-2x + 3)) + 1) - 3| + 2)^2 - 2(-2|(-1((1x + 2)/(-2x + 3)) + 1) - 3| + 2) + 5 and g(x) = (2(2|(2cos(3πx - 1) - 3) - 1| - 2) + 2)/(1(2|(2cos(3πx - 1) - 3) - 1| - 2) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaT... | 59 | MATH |
[
{
"content": "How many solutions does the equation (2(-2|(1(2x)^2 - 1(2x) - 3) + 1|) + 2)/(-2(-2|(1(2x)^2 - 1(2x) - 3) + 1|) - 3) = -5(-2|((-3x + 2)/(2x - 3)) + 2| + 3)^2 + 1(-2|((-3x + 2)/(2x - 3)) + 2| + 3) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((-1(-2(3|x + 2| + 3)^2 + 5(3|x + 2| + 3) + 5) - 2)/(-2(-2(3|x + 2| + 3)^2 + 5(3|x + 2| + 3) + 5) - 1)) - 5 and g(x) = 2((-1(1cos(1πx + 1) - 1))/(1(1cos(1πx + 1) - 1) + 1)) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format... | 21 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(5(-1|((-2x - 3)/(-3x - 1)) + 3| + 3) + 2)^2 - 4 and g(x) = -2(-1cos(3π(-1x^2 - 1x + 2) + 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 231 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(1sin(1π(-5(1x + 5)^2 + 3)) - 1) + 1| and g(x) = (-3(-5(-1x^2 + 3x - 1) - 3) - 1)/(-1(-5(-1x^2 + 3x - 1) - 3) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(3π((2(2(-3|x + 3| + 1) + 2) - 2)/(-2(2(-3|x + 3| + 1) + 2) - 2))) + 3 and g(x) = -2cos(3π((-1(1x - 1) - 3)/(1(1x - 1) + 1)) - 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 104 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(1|(-5(3x^2 + 1x - 4) - 4) - 3| - 2) + 1)/(-1(1|(-5(3x^2 + 1x - 4) - 4) - 3| - 2) + 2) and g(x) = -1((-3(-1sin(3πx - 2)) + 2)/(1(-1sin(3πx - 2)) - 2)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 52 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1((1(-1sin(1π(2x + 5) + 2) - 1) + 2)/(2(-1sin(1π(2x + 5) + 2) - 1) - 3))^2 - 3((1(-1sin(1π(2x + 5) + 2) - 1) + 2)/(2(-1sin(1π(2x + 5) + 2) - 1) - 3)) - 1 and g(x) = (1(5(2sin(3πx - 1) - 1)^2 + 2(2sin(3πx - 1) - 1)) + 3)/(1(5(... | 40 | MATH |
[
{
"content": "How many solutions does the equation 5(2cos(2π(-3x + 5) - 2) - 1)^2 - 1 = -2(-1x - 5)^2 - 5(-1x - 5) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2((-3(-2(-3|x + 1| + 2)^2 + 4) + 2)/(-2(-2(-3|x + 1| + 2)^2 + 4) - 3)) + 3 and g(x) = 2(1(3cos(1πx - 3) - 1)^2 + 3(3cos(1πx - 3) - 1) - 4) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 26 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-2sin(1π(-3(-2|x + 2| - 1) - 2))) - 3)/(3(-2sin(1π(-3(-2|x + 2| - 1) - 2))) - 2) and g(x) = -3(3(1sin(3πx + 3)) + 5)^2 - 2(3(1sin(3πx + 3)) + 5) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 19 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(4((-1x - 1)/(2x - 1)) + 4) - 1| - 3 and g(x) = -1sin(2π(-2x^2 - 3x - 3) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 344 | MATH |
[
{
"content": "How many solutions does the equation (1(-1|(-4x^2 - 3x - 1) - 3| - 1) + 3)/(-3(-1|(-4x^2 - 3x - 1) - 3| - 1) - 3) = 3sin(1π(1|x + 3| + 2) - 1) - 3 have for x 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) = (1(3|(4x^2 + 4x + 3) - 1| + 1) - 1)/(-3(3|(4x^2 + 4x + 3) - 1| + 1)) and g(x) = -3cos(2π(5x - 1)) 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) = (-2(-1cos(2π(3x^2 + 2x + 5)) + 2) + 1)/(1(-1cos(2π(3x^2 + 2x + 5)) + 2) - 3) and g(x) = -1((-2x + 1)/(2x - 3))^2 - 2((-2x + 1)/(2x - 3)) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 1099 | MATH |
[
{
"content": "How many solutions does the equation 3|((2(-2(-3x - 3)^2 + 2(-3x - 3) + 4) + 3)/(-1(-2(-3x - 3)^2 + 2(-3x - 3) + 4) + 2))| - 1 = -1sin(2π(2(2|x - 1| + 1)^2 - 1(2|x - 1| + 1) + 5) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1cos(3π(-5((-2x - 3)/(-3x - 1))^2 - 2((-2x - 3)/(-3x - 1)) + 3) - 2) + 1 and g(x) = -1cos(2π(-1x^2 + 2x - 3) - 1) + 2 where -10 ≤ x ≤ 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) = -3cos(2π(1((3(-1x^2 + 2x + 5) + 2)/(-2(-1x^2 + 2x + 5) - 2)) - 3) - 2) - 2 and g(x) = -1|(-1((-2x + 1)/(-2x)) - 4) + 1| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 8 | MATH |
[
{
"content": "How many solutions does the equation 2|(1(1x^2 - 3x) + 3) - 2| + 1 = -2cos(2π((1x + 2)/(-1x))) + 2 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 (-1(3|(-1x^2 + 3)| + 1) + 2)/(-2(3|(-1x^2 + 3)| + 1) + 3) = 3cos(2π(3|x + 3| + 3) - 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 122 | MATH |
[
{
"content": "How many solutions does the equation 1(-4((-1(-1|x - 3|) - 2)/(1(-1|x - 3|) + 2)) - 4)^2 + 1(-4((-1(-1|x - 3|) - 2)/(1(-1|x - 3|) + 2)) - 4) + 4 = -3cos(3π(-2(-1x^2 - 4x - 3) + 4) + 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 526 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-2e^(1(-3|x| + 1)) + 3) + 1)/(-3(-2e^(1(-3|x| + 1)) + 3) + 2) and g(x) = -4(-2cos(3πx - 2)) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 62 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(3π(2(-5x - 3)^2 - 5(-5x - 3) + 3) - 3) - 2 and g(x) = (2(2cos(3πx - 2) - 3) - 3)/(-2(2cos(3πx - 2) - 3) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1325 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(-2|((-2x)/(-1x + 2)) - 3| - 1) - 1 and g(x) = -3sin(1π(1x - 4) + 3) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many solutions does the equation -4(-2cos(1π(-2(-2|x - 1| + 3)^2 + 2(-2|x - 1| + 3)) - 3) + 1) - 1 = -2e^(2(-1(-1|x + 3| + 2)^2 + 3(-1|x + 3| + 2) + 3)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 872 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(-2(-2x + 4)^2 + 1(-2x + 4) + 4) - 3| - 3 and g(x) = 5(-2cos(3πx + 3) + 2) 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((2(-2|(1x + 2) + 1| + 1) + 1)/(-3(-2|(1x + 2) + 1| + 1) + 2))^2 - 2((2(-2|(1x + 2) + 1| + 1) + 1)/(-3(-2|(1x + 2) + 1| + 1) + 2)) - 4 and g(x) = -3sin(1π(-2|(-2x^2 + 4x + 5) + 1| + 1)) - 2 where -10 ≤ x ≤ 10.\n\nPresent th... | 756 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3cos(1π(-3|((1(5x^2 - 1x - 2) - 1)/(1(5x^2 - 1x - 2) - 3)) + 3|) + 2) - 1 and g(x) = 3|(-1((1x - 2)/(-2x + 1))^2 + 3((1x - 2)/(-2x + 1)) - 5) + 3| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 22 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(-2sin(2π(4x + 5)) - 3)^2 - 5(-2sin(2π(4x + 5)) - 3) - 2 and g(x) = -2e^(-1(-2|x + 1|)) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many solutions does the equation 3|(2(1x + 5)^2 - 2(1x + 5) - 2)| + 3 = 4(2sin(1πx + 3) - 3)^2 - 5(2sin(1πx + 3) - 3) - 4 have for x 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(2sin(3π((2(-1|x - 3| + 3) + 3)/(-2(-1|x - 3| + 3) + 3))) + 1)^2 - 5 = 5(-1cos(1π(2x^2 - 1x - 1) - 1) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 229 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((-1(-5(1|x - 1| + 3)^2 + 1) - 3)/(-3(-5(1|x - 1| + 3)^2 + 1) - 3)) - 3 and g(x) = 3cos(3π(1((2x + 3)/(3x - 2))^2 - 1((2x + 3)/(3x - 2)) - 4) + 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 135 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(4(1(-2|x + 3| - 3))^2 - 3(1(-2|x + 3| - 3)) + 1) + 3)/(-1(4(1(-2|x + 3| - 3))^2 - 3(1(-2|x + 3| - 3)) + 1) + 2) and g(x) = -1|(-3(3sin(1πx + 2) + 1) + 3) + 1| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Yo... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-5((-3x)/(3x))^2 + 4) - 1 and g(x) = 3sin(3π((2x + 1)/(3x + 2)) - 1) in the range -10 ≤ x ≤ 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) = (3(2|(3cos(3π(1x^2 - 2x + 2) - 2) - 2) - 1| - 2) - 2)/(2(2|(3cos(3π(1x^2 - 2x + 2) - 2) - 2) - 1| - 2) + 1) and g(x) = -5(-3cos(1π(3x^2) - 3) + 3) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 27 | MATH |
[
{
"content": "How many solutions does the equation 4((2(3sin(3πx - 2) + 2) + 3)/(-3(3sin(3πx - 2) + 2) - 1))^2 + 1((2(3sin(3πx - 2) + 2) + 3)/(-3(3sin(3πx - 2) + 2) - 1)) + 3 = 3(-1e^(1x) + 3)^2 - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 73 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(-2(3sin(3π(1|x + 1| - 1) + 1) + 3) + 2)^2 - 2(-2(3sin(3π(1|x + 1| - 1) + 1) + 3) + 2) + 2 and g(x) = (1(1|(-4x^2 - 5x - 4) + 3| + 3) - 3)/(2(1|(-4x^2 - 5x - 4) + 3| + 3)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex ... | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-3cos(3π(-1|((-1x + 3)/(-3x))| + 1) + 2) - 2)^2 - 1(-3cos(3π(-1|((-1x + 3)/(-3x))| + 1) + 2) - 2) and g(x) = (-1(1(-1|x + 3| - 2)^2 + 5(-1|x + 3| - 2) + 3) + 3)/(-1(1(-1|x + 3| - 2)^2 + 5(-1|x + 3| - 2) + 3) - 2) intersect in the interval [-10, 10].\... | 23 | MATH |
[
{
"content": "How many solutions does the equation -1sin(2π(5((-1(1|x - 3| + 3) - 1)/(1(1|x - 3| + 3) - 3))) + 1) - 2 = -1|(5((-1x + 2)/(-1x + 3))^2 + 1((-1x + 2)/(-1x + 3)) - 5) - 1| + 1 have for x 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) = 2((-2(-1cos(3π(2|x - 1| + 3)) + 2) + 3)/(-1(-1cos(3π(2|x - 1| + 3)) + 2) - 2)) + 1 and g(x) = -1(-1sin(1π(-2|x|) + 2) - 2) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(1π(3|(1(-4x^2 + 2x) + 3) - 1| + 2) - 1) + 2 and g(x) = (1(-2cos(2π(-4x + 4))) + 2)/(2(-2cos(2π(-4x + 4))) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 906 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(4(1|(2x^2 - 3x + 4) + 2| + 2) + 3) + 3)/(-1(4(1|(2x^2 - 3x + 4) + 2| + 2) + 3) - 3) and g(x) = (-1(3sin(1π(-1|x + 1| + 1) + 3)) + 3)/(-1(3sin(1π(-1|x + 1| + 1) + 3)) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTe... | 18 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(1π((-1(-2x^2 - 2x + 5) - 3)/(2(-2x^2 - 2x + 5))) + 2) - 1 and g(x) = (3(2|x| - 1) + 1)/(3(2|x| - 1) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 17 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-2(-2|(-3x^2 - 4x - 1) + 3| + 2) + 5))/(-3(-2(-2|(-3x^2 - 4x - 1) + 3| + 2) + 5) + 4) and g(x) = 3cos(2π(4(-1|x + 1| + 2) + 4) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 160 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(1π((-2(2|x - 3| - 2) + 3)/(1(2|x - 3| - 2) - 3)) - 2) + 2 and g(x) = 3(1sin(1πx + 1))^2 - 1(1sin(1πx + 1)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 90 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(3cos(3π(3(-3|x + 2| + 2)^2 - 2(-3|x + 2| + 2) + 2) - 2)) + 2 and g(x) = 1|(-1((2x + 2)/(-2x))^2 - 4((2x + 2)/(-2x)) - 4)| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1373 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-3((-2(3x^2 + 4x + 1))/(3(3x^2 + 4x + 1) + 3)) + 5) + 1| and g(x) = (-1(3(2sin(1πx - 3) - 1)^2 - 2) + 3)/(2(3(2sin(1πx - 3) - 1)^2 - 2) - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(-1(-3sin(3πx + 2) + 3)^2 + 3(-3sin(3πx + 2) + 3)) and g(x) = -1(-2e^(2x) - 3) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(2π(2|(2(5x + 1)^2 + 4(5x + 1) - 3) - 1| + 3) - 3) - 1 and g(x) = 1cos(3π(2(-1x^2 - 3x - 4) - 2) + 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1343 | MATH |
[
{
"content": "How many solutions does the equation -4(-1|(-1(-1sin(3πx - 3) + 1) - 5) + 1| - 1)^2 - 2(-1|(-1(-1sin(3πx - 3) + 1) - 5) + 1| - 1) = (2(3|(1sin(1πx - 2) + 2) - 2| - 2) + 3)/(2(3|(1sin(1πx - 2) + 2) - 2| - 2) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Y... | 29 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(3cos(2π((3x + 1)/(1x - 3)) - 2) + 3) - 1| + 2 and g(x) = (-1(3e^(-2x) - 2) - 1)/(-2(3e^(-2x) - 2) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 157 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2cos(3π(-3(2x - 1)^2 + 5(2x - 1) + 1) + 1) - 2 and g(x) = -3sin(2π((-2x + 2)/(3x + 1)) + 2) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 143 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(2(-4x + 4)^2 + 5(-4x + 4) - 1) - 1)/(2(2(-4x + 4)^2 + 5(-4x + 4) - 1) + 2) and g(x) = -3sin(1π(-4x^2 + 5x - 4)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 787 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(3sin(3π(2(-3x + 5)^2 + 1(-3x + 5) + 1)) + 1) + 3)/(-2(3sin(3π(2(-3x + 5)^2 + 1(-3x + 5) + 1)) + 1) - 2) and g(x) = -3e^(-1(1(-2|x + 2| - 2)^2 - 5(-2|x + 2| - 2) + 2)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\... | 258 | MATH |
[
{
"content": "How many solutions does the equation -5(1((1(-2|x - 3| + 1) + 1)/(2(-2|x - 3| + 1) - 3))^2 - 5((1(-2|x - 3| + 1) + 1)/(2(-2|x - 3| + 1) - 3)) + 2) + 4 = 5((3(3cos(2πx + 1) + 2))/(2(3cos(2πx + 1) + 2) + 3)) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{You... | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(3((-1(2x - 1) - 2)/(-1(2x - 1) + 2))^2 + 3) - 2| and g(x) = 3(-3cos(3π((3x - 2)/(2x + 3)) - 1) - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 104 | MATH |
[
{
"content": "How many solutions does the equation (3(3|(-2(-3x^2 + 2x + 1) + 1) + 1| + 3) + 2)/(-1(3|(-2(-3x^2 + 2x + 1) + 1) + 1| + 3) + 2) = (-2(3cos(3π(1|x - 2| + 2) - 3) + 2))/(-1(3cos(3π(1|x - 2| + 2) - 3) + 2) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your ... | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(2cos(2π(-4x^2 - 1x + 5) + 2) - 2) - 5 and g(x) = 1(-5x^2 - 1x) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|((2(3cos(2π(-4x^2 + 2x + 4) - 3)) - 2)/(-2(3cos(2π(-4x^2 + 2x + 4) - 3)) + 3)) - 3| + 3 and g(x) = 1|(1(-1x^2 + 2x + 4) - 3) + 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 14 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-2|(1x^2 + 4x - 2) - 3| + 3) + 2)/(2(-2|(1x^2 + 4x - 2) - 3| + 3) - 1) and g(x) = -3(3|x + 1| - 3)^2 + 2(3|x + 1| - 3) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(3cos(2π((1(-2|x + 1| - 3))/(-2(-2|x + 1| - 3) + 3)) - 2) + 2)^2 + 3(3cos(2π((1(-2|x + 1| - 3))/(-2(-2|x + 1| - 3) + 3)) - 2) + 2) - 4 and g(x) = (-3(-3cos(2π(1x + 4)) + 2) - 2)/(2(-3cos(2π(1x + 4)) + 2) - 1) in the range -10 ≤ x ... | 38 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(2(3((2x + 1)/(-1x + 2))^2 - 4((2x + 1)/(-1x + 2)) - 2) + 3) - 2| + 2 and g(x) = -2|(1((2x + 1)/(-3x - 2)) - 3) - 1| - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5((-2(-2sin(3π(3|x - 2| - 2) + 3) + 1) - 1)/(-2(-2sin(3π(3|x - 2| - 2) + 3) + 1) - 3))^2 + 3((-2(-2sin(3π(3|x - 2| - 2) + 3) + 1) - 1)/(-2(-2sin(3π(3|x - 2| - 2) + 3) + 1) - 3)) + 4 and g(x) = 3((2(3|x + 3| + 3) - 1)/(-1(3|x + 3| + 3) - 2))^2 - 2 inter... | 193 | MATH |
[
{
"content": "How many solutions does the equation (1(1(2sin(3πx - 3) + 3)^2 - 4(2sin(3πx - 3) + 3) - 5) - 3)/(3(1(2sin(3πx - 3) + 3)^2 - 4(2sin(3πx - 3) + 3) - 5) + 3) = (2(-4x + 1))/(-2(-4x + 1) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 29 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-1|((3(4x^2 - 2x - 2) - 2)/(-3(4x^2 - 2x - 2) - 3)) - 2| + 1) + 1 and g(x) = -1((3(-1|x + 3|))/(2(-1|x + 3|) + 2)) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 6 | MATH |
[
{
"content": "How many solutions does the equation -1((-3(-1|(-3x + 3)| - 3))/(3(-1|(-3x + 3)| - 3) - 1))^2 + 2((-3(-1|(-3x + 3)| - 3))/(3(-1|(-3x + 3)| - 3) - 1)) - 1 = -3(1sin(3π((-2x - 3)/(-1x - 2)) - 2) + 1)^2 - 1(1sin(3π((-2x - 3)/(-1x - 2)) - 2) + 1) + 4 have for x in the interval [-10, 10]?\n\nPresent th... | 62 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(3π(3|((2x)/(-2x - 3)) - 1| + 1) + 2) and g(x) = 4(1|x| - 2)^2 - 4(1|x| - 2) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 32 | MATH |
[
{
"content": "How many solutions does the equation -1e^(1((-2(-4x^2 - 4x + 4) + 1)/(-3(-4x^2 - 4x + 4)))) + 2 = -3cos(3π(-3x^2 + 5x + 1) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1442 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(2sin(3π((3x - 3)/(3x + 3)) + 2) + 2) - 1 and g(x) = 2cos(3π(-1x - 3) - 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 44 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(2π(1(-4x^2 + 4) + 4) + 3) - 1 and g(x) = -1(-2cos(1πx - 2) - 1)^2 + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 355 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2((-3(-1|(-3x^2 + 3x + 4) + 1| + 1) - 2)/(-2(-1|(-3x^2 + 3x + 4) + 1| + 1) + 3)) - 3 and g(x) = -3cos(1π(1((-1x - 1)/(-2x + 3)) + 5) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-5(-2sin(3π(1|x - 2| + 3) + 2) - 1)^2 + 4(-2sin(3π(1|x - 2| + 3) + 2) - 1) + 4) - 5 and g(x) = -5(-1|((-1x - 3)/(1x - 2)) - 3| - 2)^2 + 2(-1|((-1x - 3)/(1x - 2)) - 3| - 2) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in L... | 34 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(1π(-1(1x^2 + 4x + 1) + 3) + 1) + 2 and g(x) = (1(1|x + 1| + 2) + 1)/(3(1|x + 1| + 2) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 208 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(2((3(1cos(2πx + 2) - 2) + 2)/(1(1cos(2πx + 2) - 2) + 3))^2 + 1((3(1cos(2πx + 2) - 2) + 2)/(1(1cos(2πx + 2) - 2) + 3)) - 5) + 3| + 3 and g(x) = 3((-1(-1|x + 3| + 2) - 1)/(3(-1|x + 3| + 2) - 2))^2 - 3((-1(-1|x + 3| + 2) - 1)/(3(-1|x + 3| + 2) - 2)) + ... | 12 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(3π(-2|(5(-2x^2 - 4x - 3) - 2)|)) - 1 and g(x) = (1(1(5x^2 - 4x - 5) + 5))/(2(1(5x^2 - 4x - 5) + 5) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1145 | MATH |
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