messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation (2(-1(1(-2sin(2πx - 2) - 1)^2 - 5(-2sin(2πx - 2) - 1) - 3) - 2) - 2)/(-3(-1(1(-2sin(2πx - 2) - 1)^2 - 5(-2sin(2πx - 2) - 1) - 3) - 2) - 3) = 5(1cos(3π((1x - 2)/(1x + 3)) + 1) - 2)^2 - 4(1cos(3π((1x - 2)/(1x + 3)) + 1) - 2) - 3 have for x in the interval [-10, 1... | 13 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(-3sin(1π(1(3x^2 - 5x + 3) + 5) - 3) + 1) - 3| + 3 and g(x) = -1((-1(4x^2 - 1x + 4) + 1)/(-2(4x^2 - 1x + 4) + 1)) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}... | 390 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-1sin(1π(-1|(-1x + 3) + 2| + 3) - 2) + 1)^2 + 2(-1sin(1π(-1|(-1x + 3) + 2| + 3) - 2) + 1) - 2 and g(x) = 1(5((1x + 3)/(-2x + 1))^2 + 5((1x + 3)/(-2x + 1)) + 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Yo... | 16 | MATH |
[
{
"content": "How many solutions does the equation 3((-2(2(-1sin(2πx + 1) - 3)^2 + 5(-1sin(2πx + 1) - 3) - 5) - 3)/(2(2(-1sin(2πx + 1) - 3)^2 + 5(-1sin(2πx + 1) - 3) - 5))) - 3 = (2(3(3cos(3πx - 2) + 2) + 5) - 3)/(-1(3(3cos(3πx - 2) + 2) + 5) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in L... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(2|(1((3x)/(1x - 3)) + 2) + 3| - 1)^2 and g(x) = 3(-1cos(1π((2x + 3)/(-1x + 2)) + 2) - 3) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 7 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(3π(-4((2(1|x|) - 2)/(-1(1|x|) - 1)) + 1) - 3) - 3 and g(x) = (-3(3|(1cos(2πx - 1) + 2)| - 2) - 2)/(1(3|(1cos(2πx - 1) + 2)| - 2) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 88 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(-3(1cos(1πx + 1) + 1)^2 - 1(1cos(1πx + 1) + 1) - 2) - 5 and g(x) = 3sin(2π(-3x - 3) + 3) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many solutions does the equation 4(2(3e^(-1((2x - 2)/(-2x - 3))))^2 + 1(3e^(-1((2x - 2)/(-2x - 3))))) + 3 = 2((2(-1sin(1πx - 3)) - 3)/(1(-1sin(1πx - 3)))) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(5((1(4x - 3) - 1)/(-2(4x - 3) - 2))^2 + 1((1(4x - 3) - 1)/(-2(4x - 3) - 2)) - 4) - 3| and g(x) = -2cos(3π(-3(-1x^2 - 2x - 4) + 5) + 1) - 1 in the range -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) = -2((1(2(3|x + 2| - 3) + 2) - 3)/(3(2(3|x + 2| - 3) + 2) - 2))^2 - 5((1(2(3|x + 2| - 3) + 2) - 3)/(3(2(3|x + 2| - 3) + 2) - 2)) + 3 and g(x) = -4(-2|(-1cos(1πx - 3)) - 1| + 3) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in ... | 24 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(1|((-1(-1sin(1πx + 3) - 2) + 3)/(3(-1sin(1πx + 3) - 2) + 2)) + 1|)^2 - 3(1|((-1(-1sin(1πx + 3) - 2) + 3)/(3(-1sin(1πx + 3) - 2) + 2)) + 1|) - 3 and g(x) = 2|((1(-2cos(2πx - 2) + 3) - 2)/(-3(-2cos(2πx - 2) + 3) - 3)) - 2| + 2 in t... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5((1(-3x^2 + 2x + 2) + 3)/(2(-3x^2 + 2x + 2))) - 1 and g(x) = -2cos(1π(1x - 3) + 2) intersect in the interval [-10, 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) = (3(-5(1(1e^(-2x) - 3) + 4)^2 + 5(1(1e^(-2x) - 3) + 4) + 5))/(-3(-5(1(1e^(-2x) - 3) + 4)^2 + 5(1(1e^(-2x) - 3) + 4) + 5) + 4) and g(x) = -3cos(3π((-3(-5x) + 1)/(2(-5x) + 2))) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format... | 35 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(-3cos(1π(-2|(3x + 1) + 3| - 2)) + 1)^2 + 5(-3cos(1π(-2|(3x + 1) + 3| - 2)) + 1) + 3 and g(x) = 2(1|((3x - 1)/(-2x - 2)) - 3| - 1)^2 + 4(1|((3x - 1)/(-2x - 2)) - 3| - 1) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex... | 118 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(1|(-1(5x + 4)^2 + 1(5x + 4) - 2) - 3|) + 1)/(2(1|(-1(5x + 4)^2 + 1(5x + 4) - 2) - 3|)) and g(x) = -5((3(-3sin(3πx - 2) - 1) + 2)/(-2(-3sin(3πx - 2) - 1) - 3))^2 + 5((3(-3sin(3πx - 2) - 1) + 2)/(-2(-3sin(3πx - 2) - 1) - 3)) + 1 intersect in the inter... | 110 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-1|((-3(1x^2 - 2x - 4) + 2)/(-1(1x^2 - 2x - 4))) + 1| + 2) - 2 and g(x) = 2sin(1π(-2|(2x)| + 3) - 2) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 72 | MATH |
[
{
"content": "How many solutions does the equation (-3(-1(2|(-1cos(1πx - 1) + 2) + 3|) - 2) - 3)/(-2(-1(2|(-1cos(1πx - 1) + 2) + 3|) - 2) - 1) = 3(-3sin(3π(-3|x + 1| - 2) + 1) - 1)^2 - 5(-3sin(3π(-3|x + 1| - 2) + 1) - 1) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Yo... | 180 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-5(3cos(2π(2|x - 1| - 3) + 3)) + 4) - 2)/(-1(-5(3cos(2π(2|x - 1| - 3) + 3)) + 4) + 3) and g(x) = 3|(2(1sin(3πx) - 2)^2 - 4)| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 143 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(5((1(1x^2 + 3x - 3) + 2)/(3(1x^2 + 3x - 3) - 2)) + 5) - 2| + 3 and g(x) = -1|(5(1cos(1πx + 3) - 1) + 4)| + 2 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 -2sin(3π(4(2(-3|x + 2| + 3)^2 - 4(-3|x + 2| + 3) + 2) - 1) - 3) - 3 = -2sin(3π((-1(4x + 4) + 1)/(-3(4x + 4) + 1)) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 49 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-1|(1sin(3π(-5x^2 - 4x + 2) - 1) + 1) - 3|) + 1)/(-1(-1|(1sin(3π(-5x^2 - 4x + 2) - 1) + 1) - 3|) - 2) and g(x) = 5((-1(1|x + 2| + 2) + 2)/(3(1|x + 2| + 2) - 1))^2 - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxe... | 629 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(-2(-2sin(2π(-1|x + 2| + 1) - 3) - 1)^2 - 2(-2sin(2π(-1|x + 2| + 1) - 3) - 1) - 5))/(3(-2(-2sin(2π(-1|x + 2| + 1) - 3) - 1)^2 - 2(-2sin(2π(-1|x + 2| + 1) - 3) - 1) - 5) - 1) and g(x) = 1(-5(-2cos(1πx - 2) + 2) + 4)^2 + 2(-5(-2cos(1πx - 2) + 2) + 4) +... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-1(4x - 2)^2 - 3(4x - 2) - 2))/(-1(-1(4x - 2)^2 - 3(4x - 2) - 2) - 3) and g(x) = (-1(3sin(2πx - 3) + 3) + 2)/(2(3sin(2πx - 3) + 3) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-1(-4x - 1)^2 - 4(-4x - 1)) + 3)/(-2(-1(-4x - 1)^2 - 4(-4x - 1)) + 3) and g(x) = -2cos(2π(4x^2 + 4x + 5) - 3) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1080 | MATH |
[
{
"content": "How many solutions does the equation 3|((1(-4(-1cos(3πx + 2) - 1) - 1) + 3)/(3(-4(-1cos(3πx + 2) - 1) - 1) - 2)) + 2| - 3 = 2|(5(2x^2 - 2x + 5) - 5) + 1| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(2|(-1x - 1) + 1| - 1) - 1)/(-1(2|(-1x - 1) + 1| - 1) + 2) and g(x) = -2(-3sin(1πx - 2) - 2)^2 - 5(-3sin(1πx - 2) - 2) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(-1|(-4x^2 + 3x) + 3| + 3) + 2 and g(x) = -1|(2cos(3πx - 2) + 1) - 1| + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(1|(-2((-1x + 1)/(-3x + 1))^2 - 4((-1x + 1)/(-3x + 1)) - 5) + 1| - 3) + 1 and g(x) = (3(3sin(3π(-2|x| + 1) + 2) + 2) - 1)/(-3(3sin(3π(-2|x| + 1) + 2) + 2) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\box... | 79 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-3cos(3π(3|x + 3|) + 3) - 2) - 2)/(-1(-3cos(3π(3|x + 3|) + 3) - 2) + 2) and g(x) = 3(2sin(1πx + 2)) - 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) = -3cos(1π(-1|(-5(5x - 4)^2 + 5(5x - 4) + 5) - 1| + 3) + 3) - 1 and g(x) = -5(3|(-2x + 5) - 2| - 2)^2 + 4(3|(-2x + 5) - 2| - 2) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 24 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(-1(2x^2 - 1x - 5) + 2) + 3)/(-2(-1(2x^2 - 1x - 5) + 2) - 2) and g(x) = 2cos(2π((-2x - 3)/(1x + 2)) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-1(3(-1|x + 3| - 2) + 3)^2 - 3))/(-2(-1(3(-1|x + 3| - 2) + 3)^2 - 3) + 1) and g(x) = 1(-3(2sin(2πx + 3) - 1)^2 + 4(2sin(2πx + 3) - 1) + 3) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(-1((3(1cos(3πx + 3) + 2) + 2)/(-3(1cos(3πx + 3) + 2) + 2)) - 1)| + 1 and g(x) = -2|(-4(-2e^(-1x) + 3) + 5) - 3| + 2 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 (-3(-1(-3sin(2π(1|x| - 3) + 3) - 1) - 1))/(2(-1(-3sin(2π(1|x| - 3) + 3) - 1) - 1) - 1) = 4(-2sin(1π(2x^2 + 1x - 3) + 3) - 2) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 404 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(3|(3sin(1π(-5x^2 - 4x + 2) - 2) - 1)| + 2) and g(x) = -5(1(-2sin(2πx - 1) + 3)^2 + 1(-2sin(2πx - 1) + 3) + 5) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 429 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(2π(-2(1x + 1)^2 - 3(1x + 1) + 2)) - 1 and g(x) = -5((2x)/(3x - 1))^2 + 2((2x)/(3x - 1)) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 236 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3cos(1π(1(1|(4x - 5)| + 3)^2 + 5(1|(4x - 5)| + 3) + 1) - 2) - 3 and g(x) = 3cos(2π(4(3x + 5)^2 + 3) + 3) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 533 | MATH |
[
{
"content": "How many solutions does the equation (-2(3cos(1π(-3|x + 1| + 2) - 3) + 1) - 3)/(-2(3cos(1π(-3|x + 1| + 2) - 3) + 1) + 3) = -2(2|x - 3| - 1)^2 + 2(2|x - 3| - 1) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 23 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(-2cos(3π(-2|(3x^2 + 3x - 4) - 1| + 3) - 1) - 2) and g(x) = 3((-3(2|x + 1| + 3) - 1)/(-2(2|x + 1| + 3)))^2 + 4((-3(2|x + 1| + 3) - 1)/(-2(2|x + 1| + 3))) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxe... | 1578 | MATH |
[
{
"content": "How many solutions does the equation -1(5(-2|(1cos(2πx) + 1) + 1| + 2) + 5)^2 - 4(5(-2|(1cos(2πx) + 1) + 1| + 2) + 5) = -5((3(1|x - 2| + 2) + 3)/(2(1|x - 2| + 2) - 1))^2 - 3((3(1|x - 2| + 2) + 3)/(2(1|x - 2| + 2) - 1)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format... | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(2sin(2π(-1x^2 - 1x + 2) - 1) - 2) + 2| + 2 and g(x) = (3(-5x + 5) + 3)/(1(-5x + 5) - 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) = 1e^(2((1(4x - 5))/(1(4x - 5)))) - 3 and g(x) = 2cos(3π((-2x - 3)/(-2x + 3)) - 1) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 98 | MATH |
[
{
"content": "How many solutions does the equation 5(-1|(-1((-2x - 3)/(-2x + 1))^2 + 2((-2x - 3)/(-2x + 1)) - 4) + 2| + 1) + 5 = (2(2|(-2sin(3πx) + 1) - 1|) + 2)/(3(2|(-2sin(3πx) + 1) - 1|) - 2) have for x 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((1(-1cos(3π(-4x^2 - 4x + 1) + 3) + 1) - 3)/(-1(-1cos(3π(-4x^2 - 4x + 1) + 3) + 1) + 3)) - 2 = 1(-1|(1sin(3πx + 1)) - 1| - 1)^2 + 1(-1|(1sin(3πx + 1)) - 1| - 1) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 30 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(4(-1cos(2πx) + 2)^2 + 2(-1cos(2πx) + 2) + 1) - 3| + 3 and g(x) = (-3(3cos(2πx - 1)) + 1)/(3(3cos(2πx - 1))) 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 3|(1(-2sin(1πx)) - 4) + 1| - 1 = 2|(1x - 3) + 1| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(-5((3(-1cos(3πx + 3) + 1) + 1)/(1(-1cos(3πx + 3) + 1) - 3))^2 + 4) - 3| + 2 and g(x) = -4((-2(-3x^2 + 4) + 1)/(1(-3x^2 + 4) + 2)) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 14 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(4(4((-3x - 2)/(2x + 1)) - 3)^2 - 1(4((-3x - 2)/(2x + 1)) - 3) + 1) - 3| - 3 and g(x) = (-2(-5(1cos(2πx - 1))^2 + 3(1cos(2πx - 1)) + 5) + 2)/(1(-5(1cos(2πx - 1))^2 + 3(1cos(2πx - 1)) + 5) - 3) intersect in the interval [-10, 10].\n\nPresent the answ... | 44 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(3(-2cos(3π((3x + 2)/(3x - 2)) + 1) + 2) - 4) - 2| - 3 and g(x) = (-3(-2cos(2π(-2|x - 2| + 1) - 3)) + 3)/(-2(-2cos(2π(-2|x - 2| + 1) - 3)) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 45 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2e^(-1(5((2x + 3)/(1x - 3))^2 - 2((2x + 3)/(1x - 3)) - 2)) - 1 and g(x) = -3cos(3π(1x^2) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 528 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1sin(3π(1(-2|(2x^2 - 4x + 1) - 3|) + 4) + 3) and g(x) = (1(-4(2cos(3πx + 2) - 3)^2 - 3(2cos(3πx + 2) - 3) - 4) + 1)/(2(-4(2cos(3πx + 2) - 3)^2 - 3(2cos(3πx + 2) - 3) - 4) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: ... | 1493 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-3|(2sin(2πx + 2) - 1)| + 2))/(1(-3|(2sin(2πx + 2) - 1)| + 2) + 1) and g(x) = -1sin(1π((3x + 2)/(-3x + 4))) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(-3sin(3π((-1(5x + 5) - 2)/(-2(5x + 5))) - 2) + 3)^2 - 5(-3sin(3π((-1(5x + 5) - 2)/(-2(5x + 5))) - 2) + 3) + 2 and g(x) = -4(1|(2sin(2πx - 1) - 1) + 3| + 3) - 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\b... | 36 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-5(2sin(1πx + 2) + 1) + 5) + 1)/(-3(-5(2sin(1πx + 2) + 1) + 5) - 3) and g(x) = (2(-3|x - 1| + 1) - 1)/(1(-3|x - 1| + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1sin(2π(2(2|(5x^2 - 1x - 4) - 2| - 3) - 3) + 3) + 1 and g(x) = -4(1sin(1π((-2x + 3)/(-1x - 1)) - 2) - 2)^2 - 2(1sin(1π((-2x + 3)/(-1x - 1)) - 2) - 2) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{You... | 229 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-2cos(3π(3|(1x^2 - 1x - 5)| + 3) - 3)) + 2)/(-2(-2cos(3π(3|(1x^2 - 1x - 5)| + 3) - 3))) and g(x) = 2(1e^(2(-3|x + 1| + 3)) - 3) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 204 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(2sin(2π(-5(-1|x - 1| - 1) + 3) + 3) + 1)^2 - 5(2sin(2π(-5(-1|x - 1| - 1) + 3) + 3) + 1) + 2 and g(x) = 3|((1(-1x + 3) + 2)/(-1(-1x + 3) - 1)) - 1| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 216 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(1(-2cos(1πx) + 3) - 3)^2 - 4 and g(x) = -2(-2e^(1x) + 3) + 3 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(2x + 5)^2 + 2(2x + 5) - 5) - 3| - 2 = 1sin(2π(4x^2 - 2)) - 1 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(1sin(3π(3|x + 1| - 3) - 1) + 2) - 1 and g(x) = -5(3x^2 - 3x - 1) - 2 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 (-2(-3|(-2x + 3) + 2|) + 1)/(3(-3|(-2x + 3) + 2|) - 2) = -3cos(3π((-1x - 1)/(1x))) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 46 | MATH |
[
{
"content": "How many solutions does the equation 2(-1(-1|((1x - 3)/(3x - 2))| + 3)^2 - 4(-1|((1x - 3)/(3x - 2))| + 3) + 5) - 4 = 3sin(2π(-3|((-3x + 2)/(1x + 1)) - 1| - 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": "Find the number of times the graphs of f(x) = (2(5(2cos(2π(-1|x + 1| + 3) + 2) - 1) - 5))/(-2(5(2cos(2π(-1|x + 1| + 3) + 2) - 1) - 5) + 1) and g(x) = -2sin(2π((-1(-4x - 4) - 2)/(1(-4x - 4) - 2)) + 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 55 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(-3((-3(3x + 3) - 1)/(1(3x + 3) - 3))^2 + 3((-3(3x + 3) - 1)/(1(3x + 3) - 3))) - 3| + 3 and g(x) = (1(3cos(2π(-1x^2 - 5x + 1) + 3) + 1) - 2)/(-2(3cos(2π(-1x^2 - 5x + 1) + 3) + 1) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the an... | 21 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-1|(1e^(2(5x^2 - 5x + 5)) + 2) - 3| - 2))/(-3(-1|(1e^(2(5x^2 - 5x + 5)) + 2) - 3| - 2) + 2) and g(x) = (-2(3(3cos(3πx - 1)) - 2) + 3)/(-2(3(3cos(3πx - 1)) - 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-4(5x + 1)^2 + 2(5x + 1) + 5) + 1)/(3(-4(5x + 1)^2 + 2(5x + 1) + 5) - 2) and g(x) = 4(-3cos(1πx + 1) + 1) - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 21 | MATH |
[
{
"content": "How many solutions does the equation 3|(1(-3sin(3π(1x^2 - 2x + 3) + 3) - 3) - 2) + 1| - 2 = -1sin(2π(3|((-1x - 1)/(-2x - 2)) + 1|) + 3) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 546 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((-2(-4x^2 - 1x - 4) + 3)/(-1(-4x^2 - 1x - 4) + 2)) - 4 and g(x) = 3cos(1π(-5x - 4) + 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100 | MATH |
[
{
"content": "How many solutions does the equation -2|(-1e^(-1(-2x^2 + 1x + 4)) - 1) + 1| + 2 = 1sin(2π((1x - 1)/(-2x - 3)) + 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many solutions does the equation 2cos(3π(3((-1x + 2)/(2x - 3))^2 - 3((-1x + 2)/(2x - 3)) + 4) + 2) = (2(3x + 4) - 1)/(-1(3x + 4)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 37 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(1π(-3|((3x)/(-1x + 2)) - 1|)) + 2 and g(x) = 3(2x^2 - 1x - 3) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 27 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(2sin(3π(-5((2x - 1)/(-3x + 4))^2 + 2((2x - 1)/(-3x + 4)) - 5) - 3) + 1) - 5 and g(x) = 2e^(1(5((-1x + 3)/(-2x + 3)) + 5)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 50 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(-1cos(3π(2|((-1x - 3)/(-1x))|)) + 3)^2 + 1(-1cos(3π(2|((-1x - 3)/(-1x))|)) + 3) - 4 and g(x) = (3(2cos(3π(-3|x + 1| - 1) + 1) + 1) + 3)/(-1(2cos(3π(-3|x + 1| - 1) + 1) + 1)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaT... | 109 | MATH |
[
{
"content": "How many solutions does the equation 5((1(3cos(3π(1|x + 3| - 2) - 1) + 1) + 3)/(-3(3cos(3π(1|x + 3| - 2) - 1) + 1) + 1))^2 - 5((1(3cos(3π(1|x + 3| - 2) - 1) + 1) + 3)/(-3(3cos(3π(1|x + 3| - 2) - 1) + 1) + 1)) - 4 = (2(-2|(-2x^2 + 1x) - 1| + 3) + 3)/(1(-2|(-2x^2 + 1x) - 1| + 3)) have for x in the i... | 118 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(3|(-1sin(3π(-1x + 1) - 1)) - 2| - 1)^2 + 1(3|(-1sin(3π(-1x + 1) - 1)) - 2| - 1) and g(x) = 3(-2sin(2π(3x^2 + 4x - 3)) + 2) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 373 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2cos(3π(1|(3((2x + 2)/(2x)) + 5) - 1| + 2)) - 2 and g(x) = 5((-1(4x^2 + 2x + 4) + 3)/(2(4x^2 + 2x + 4) - 1)) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34 | MATH |
[
{
"content": "How many solutions does the equation 1(2|(-2((3x - 1)/(-3x + 2))^2 - 3((3x - 1)/(-3x + 2)) + 3) - 2| + 3) - 3 = 4(3cos(1π(1|x - 2| - 1) - 1) + 2)^2 + 4(3cos(1π(1|x - 2| - 1) - 1) + 2) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 30 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2sin(2π(4((1x - 1)/(-2x + 3))^2 - 4((1x - 1)/(-2x + 3)) - 2) - 3) + 2 and g(x) = -3e^(-2(1x^2 + 2x + 3)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 35 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|((2(-4(3sin(1πx) + 3)^2 + 3(3sin(1πx) + 3) + 1) + 3)/(-1(-4(3sin(1πx) + 3)^2 + 3(3sin(1πx) + 3) + 1) + 3)) - 2| + 1 and g(x) = (-3(3(-1x^2 - 1) + 1) - 1)/(-1(3(-1x^2 - 1) + 1) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer... | 60 | MATH |
[
{
"content": "How many solutions does the equation 2sin(2π(-3((-1x + 2)/(-2x - 1)) - 3) + 1) + 2 = -1sin(1π((-3x - 2)/(-2x)) - 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 99 | MATH |
[
{
"content": "How many solutions does the equation 3(-3cos(1π(-4(1|x + 1| - 1) + 3) + 1) - 3)^2 - 3(-3cos(1π(-4(1|x + 1| - 1) + 3) + 1) - 3) + 1 = 3sin(3π(-1|(-1x - 4) - 1|) + 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 36 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1cos(2π(3|(2(4x + 4)^2 - 2(4x + 4) - 3) + 1| + 1) + 3) + 1 and g(x) = 1(-4(1|x + 1| + 1)^2 + 5) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(2sin(3π(2x) - 1)) - 2)/(2(2sin(3π(2x) - 1)) - 1) and g(x) = 2|(3x^2 + 3x + 5) + 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4((-1(-4(-1|x + 3| + 1) - 4) - 2)/(2(-4(-1|x + 3| + 1) - 4) + 2))^2 - 1((-1(-4(-1|x + 3| + 1) - 4) - 2)/(2(-4(-1|x + 3| + 1) - 4) + 2)) - 4 and g(x) = 3(1(2|x - 1| - 1) - 3)^2 - 4(1(2|x - 1| - 1) - 3) - 2 in the range -10 ≤ x ≤ 10?... | 6 | MATH |
[
{
"content": "How many solutions does the equation 1((1(3cos(1πx + 3) + 2) - 1)/(-2(3cos(1πx + 3) + 2) + 3))^2 - 2((1(3cos(1πx + 3) + 2) - 1)/(-2(3cos(1πx + 3) + 2) + 3)) - 5 = -2|(3cos(1πx) - 1) - 2| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(-3(3sin(3πx - 3) - 3)^2 - 1(3sin(3πx - 3) - 3) - 3) + 1| - 1 and g(x) = -1|((-3x + 1)/(1x - 2)) - 1| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(2π((-3(1|x + 1| - 1) + 1)/(-1(1|x + 1| - 1) + 2)) + 3) and g(x) = 3((-1x + 2)/(-2x - 2)) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 135 | MATH |
[
{
"content": "How many solutions does the equation 1(-4((-1(-1|x - 1| - 3) + 3)/(1(-1|x - 1| - 3) - 2))^2 + 3((-1(-1|x - 1| - 3) + 3)/(1(-1|x - 1| - 3) - 2)) + 5) - 1 = -3(1cos(1π(-2|x + 1| - 2) - 3) - 1)^2 + 4(1cos(1π(-2|x + 1| - 2) - 3) - 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in L... | 40 | MATH |
[
{
"content": "How many solutions does the equation 5(-3cos(2π(1|x + 2| - 1) - 1) + 2) - 3 = -3((-2x - 3)/(3x - 1))^2 - 1((-2x - 3)/(3x - 1)) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 36 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3((-1(-3|x - 2|) + 3)/(2(-3|x - 2|) - 3))^2 + 5((-1(-3|x - 2|) + 3)/(2(-3|x - 2|) - 3)) + 4 and g(x) = -1sin(3π(1x^2 + 1x + 4) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 580 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2((1(3|(-1x - 5) + 2| - 2) + 3)/(-1(3|(-1x - 5) + 2| - 2)))^2 - 5((1(3|(-1x - 5) + 2| - 2) + 3)/(-1(3|(-1x - 5) + 2| - 2))) - 4 and g(x) = -2cos(1π(1(4x^2 + 5x - 5) - 4) - 1) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in... | 730 | MATH |
[
{
"content": "How many solutions does the equation -3(2sin(2π((-2(-1|x + 2| + 3) - 1)/(1(-1|x + 2| + 3) + 3)) + 2) + 2)^2 + 1 = -1cos(2π(2(3x^2 + 3x - 4)) + 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 252 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-1|(3cos(3πx + 2) + 3) - 3| + 2) + 2)/(-3(-1|(3cos(3πx + 2) + 3) - 3| + 2) - 3) and g(x) = -3(2|x| - 1)^2 + 2(2|x| - 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 115 | MATH |
[
{
"content": "How many solutions does the equation (2(1|(5(-2sin(3πx)) - 4) + 3| - 3) - 1)/(-1(1|(5(-2sin(3πx)) - 4) + 3| - 3)) = -1(5(3e^(-1x) + 1)^2 + 5(3e^(-1x) + 1) + 2) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-1|(2((-2x - 3)/(-3x + 2))^2 + 5((-2x - 3)/(-3x + 2)) - 4) + 3| + 3) + 3 and g(x) = (1(-3(-2|x + 1| + 1)^2 - 3(-2|x + 1| + 1) + 4) + 1)/(3(-3(-2|x + 1| + 1)^2 - 3(-2|x + 1| + 1) + 4)) in the range -10 ≤ x ≤ 10?\n\nPresent the an... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(-5(-4(2|x - 3| + 2)^2 + 4)) + 2)/(-2(-5(-4(2|x - 3| + 2)^2 + 4)) + 3) and g(x) = -5(-3|(1cos(2πx + 3) - 1) + 2| + 2)^2 - 4(-3|(1cos(2πx + 3) - 1) + 2| + 2) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your ... | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1sin(1π(-5((-3(3x) + 3)/(-1(3x)))^2 + 5((-3(3x) + 3)/(-1(3x))) + 2) - 2) + 2 and g(x) = (2(-4(-2|x + 3| - 3) - 1) + 3)/(1(-4(-2|x + 3| - 3) - 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 161 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(-1sin(1π(-5x - 4) - 1) + 1)^2 - 4(-1sin(1π(-5x - 4) - 1) + 1) - 1 and g(x) = 2sin(3π(-1|x - 3| + 1) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "How many solutions does the equation 5(5((-1(3sin(1πx + 3)) - 3)/(-3(3sin(1πx + 3)) + 1)))^2 + 2(5((-1(3sin(1πx + 3)) - 3)/(-3(3sin(1πx + 3)) + 1))) - 3 = -2|(-3((-3x + 3)/(-2x + 1)) + 5) + 1| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 18 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(3sin(3π(2(-3x^2 - 5x - 4) - 5)) + 2)| - 3 and g(x) = 5((3(-1|x| + 2) - 2)/(-1(-1|x| + 2) + 2)) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
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