messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation -1sin(2π(-5(1|x| - 3) + 2) + 1) + 1 = 1(-1|x - 1| + 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(2cos(3π(-4(-1|x| + 2) + 3) + 3))^2 - 4(2cos(3π(-4(-1|x| + 2) + 3) + 3)) - 5 and g(x) = 3(-2((-3x + 1)/(-1x + 3)) - 4)^2 + 5(-2((-3x + 1)/(-1x + 3)) - 4) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed... | 12 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-3sin(3π((-3x + 3)/(2x + 2))) + 1) + 4 and g(x) = 2(-1x - 1)^2 + 5(-1x - 1) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 48 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(-4(2|(2x^2 - 1x - 2) - 1| - 3)) + 3)/(1(-4(2|(2x^2 - 1x - 2) - 1| - 3)) + 2) and g(x) = -4(3cos(1π((-1x - 1)/(-1x - 2)) + 3) + 3)^2 + 1(3cos(1π((-1x - 1)/(-1x - 2)) + 3) + 3) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer ... | 26 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(5(-1|x - 2| - 2) + 2) - 1)/(1(5(-1|x - 2| - 2) + 2) + 3) and g(x) = -2|(1sin(2πx - 3) - 1) + 2| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(-1sin(1π(-2|(4x^2 - 1x + 5) - 2| + 3) - 3) - 2) - 5 and g(x) = -4((-2(-4x^2 - 4x + 4) + 1)/(-3(-4x^2 - 4x + 4) + 2)) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1222 | MATH |
[
{
"content": "How many solutions does the equation (3(-4(-5x^2 + 1x - 4) - 2) + 2)/(-3(-4(-5x^2 + 1x - 4) - 2) + 4) = 2sin(1π(-3x^2 - 4x + 3) + 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 604 | MATH |
[
{
"content": "How many solutions does the equation -4((3(3sin(3πx + 1)) + 1)/(-1(3sin(3πx + 1)) + 2)) - 5 = -1(3|x| - 1)^2 - 3(3|x| - 1) + 2 have for x 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 2(-2|(-1cos(1π((-1x - 1)/(2x - 3)) + 1)) - 2| + 2)^2 + 3(-2|(-1cos(1π((-1x - 1)/(2x - 3)) + 1)) - 2| + 2) + 5 = 2|(-1(1x^2 - 2) - 1)| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 23 | MATH |
[
{
"content": "How many solutions does the equation 2sin(3π((3(-5x) - 3)/(2(-5x))) + 2) - 2 = 1(-2sin(3πx + 3) + 1)^2 + 5(-2sin(3πx + 3) + 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 62 | MATH |
[
{
"content": "How many solutions does the equation -1cos(1π(5(-3|x - 1| - 2)^2 + 5) - 1) = 2sin(1π(2x^2 + 1x - 5) - 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 701 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(2π((-2(3x^2 - 4x + 4) + 2)/(-1(3x^2 - 4x + 4) - 1)) - 1) + 3 and g(x) = -3sin(3π((-2x + 3)/(1x + 1)) + 1) - 1 in the range -10 ≤ x ≤ 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) = -3sin(2π(-1((-1x + 3)/(-2x - 1))) + 2) - 1 and g(x) = (-1(2e^(1x) - 2) + 2)/(-2(2e^(1x) - 2) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 66 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(2e^(-1(2x - 5)) - 3) - 1| - 3 and g(x) = 3sin(1π(-2x - 1) + 3) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3((-2(-1e^(1x) + 1) - 2)/(2(-1e^(1x) + 1) - 1)) + 3 and g(x) = 1|(-2cos(3πx + 2) + 1) + 1| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 27 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(1sin(2π(-1x^2 + 5x - 1) + 3) - 2)| - 1 and g(x) = -4(2e^(1x) + 1)^2 + 1(2e^(1x) + 1) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 270 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2((-1(-1sin(1πx + 3) - 3) - 2)/(-1(-1sin(1πx + 3) - 3)))^2 + 1((-1(-1sin(1πx + 3) - 3) - 2)/(-1(-1sin(1πx + 3) - 3))) and g(x) = -2sin(3π(5x) - 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 258 | MATH |
[
{
"content": "How many solutions does the equation 2((1(2|(-1cos(2πx - 3) + 3) + 3| - 2) + 3)/(2(2|(-1cos(2πx - 3) + 3) + 3| - 2) + 1))^2 + 2((1(2|(-1cos(2πx - 3) + 3) + 3| - 2) + 3)/(2(2|(-1cos(2πx - 3) + 3) + 3| - 2) + 1)) - 4 = -5(3sin(3π(5x + 5) + 3) + 2)^2 + 2(3sin(3π(5x + 5) + 3) + 2) + 2 have for x in th... | 600 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-1|((2x - 2)/(-2x - 1)) + 3| - 2) - 2 and g(x) = (-2(2cos(1πx - 3) - 1) - 2)/(2(2cos(1πx - 3) - 1) - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "How many solutions does the equation 3(-3|((-1(-1x^2 - 5x - 3) - 1)/(2(-1x^2 - 5x - 3)))| - 2) + 5 = 1(1sin(3π(1x^2 + 1x - 2) + 2)) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 44 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(2sin(1π(-4x - 3) - 3) + 1) - 3)/(2(2sin(1π(-4x - 3) - 3) + 1) - 1) and g(x) = 1|(1x^2 + 4x - 1)| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 49 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(-3|((2x + 3)/(-3x))| + 1) - 3 and g(x) = 3|(1cos(3πx - 2) + 2) - 3| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 24 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(2sin(2π(-3|(-1x^2 - 2x + 4) + 1| - 1) - 3) + 3) and g(x) = -1((1(2cos(2πx + 3) + 1) - 3)/(-2(2cos(2πx + 3) + 1) - 1))^2 - 3((1(2cos(2πx + 3) + 1) - 3)/(-2(2cos(2πx + 3) + 1) - 1)) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the an... | 130 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(2((-2(3sin(2πx - 2) + 2) - 1)/(3(3sin(2πx - 2) + 2))) - 1)^2 + 5(2((-2(3sin(2πx - 2) + 2) - 1)/(3(3sin(2πx - 2) + 2))) - 1) - 4 and g(x) = -1|(-5(-1sin(3πx + 3) + 3)^2 - 1)| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex for... | 100 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3cos(3π(4(2x^2 + 3x + 4)) - 1) - 1 and g(x) = 4(-2cos(2πx + 3))^2 + 2(-2cos(2πx + 3)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 210 | MATH |
[
{
"content": "How many solutions does the equation -2cos(3π(3|(2(1x^2 + 2x + 2) + 3) - 1| + 2) - 1) + 3 = -2|(3(1x^2 - 4x + 4)) - 2| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 25 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(3((1x - 2)/(-1x - 1))^2 + 4((1x - 2)/(-1x - 1)) + 2) - 2| + 3 and g(x) = -2sin(2π(4x - 5) - 1) - 3 where -10 ≤ x ≤ 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) = 2((-2(2cos(2πx + 2) - 2) - 2)/(-3(2cos(2πx + 2) - 2) - 1)) + 4 and g(x) = -3|(-1e^(-1x)) - 3| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 25 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(-3cos(2π(4x^2 - 5x) - 1) - 1) - 4 and g(x) = (1(1|x + 3| + 1) + 3)/(-3(1|x + 3| + 1)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 1131 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3cos(2π(1((2(2x + 3) - 3)/(-1(2x + 3) - 1))^2 + 1) + 3) and g(x) = (-2(-3|(-4x + 2) + 1| + 2) + 2)/(-3(-3|(-4x + 2) + 1| + 2) - 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 165 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1((-2(-1x - 4) - 3)/(2(-1x - 4) + 2))^2 + 5((-2(-1x - 4) - 3)/(2(-1x - 4) + 2)) + 5 and g(x) = -2(3sin(2πx - 2) + 2)^2 + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 26 | MATH |
[
{
"content": "How many solutions does the equation (-2(3cos(3π(-1x^2 - 1x + 2)) + 3) + 3)/(-1(3cos(3π(-1x^2 - 1x + 2)) + 3)) = 3sin(1π(4x - 4) - 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 495 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(-1sin(1π(-2(-4x + 4)^2 - 1(-4x + 4) + 1) + 2) - 3) + 2| - 3 and g(x) = 5(-5((1x - 3)/(1x + 3)) - 3)^2 - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 14 | MATH |
[
{
"content": "How many solutions does the equation 3cos(1π(5(-3|x + 1| + 1)^2) - 1) + 3 = -1|((2x)/(-2x - 2))| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 399 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(3π(-5(2|(-4x^2 - 4x - 2)| + 1) + 1) + 1) and g(x) = 1(-4(-2|x - 1| + 1)^2 - 2(-2|x - 1| + 1) + 5) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 35 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1e^(1(-1(-3|(-5x^2 - 3x + 1) + 1| + 3) - 1)) - 1 and g(x) = 2cos(2π(1(2|x - 2| + 3) - 2)) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-1(4(2|x + 1| - 3) + 4)^2 + 4(4(2|x + 1| - 3) + 4) + 5) - 1)/(-1(-1(4(2|x + 1| - 3) + 4)^2 + 4(4(2|x + 1| - 3) + 4) + 5) + 2) and g(x) = 1(-2sin(1π(-3x) + 2))^2 + 2(-2sin(1π(-3x) + 2)) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent t... | 62 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-3sin(1π(2x^2 - 2) + 1) + 2) - 3)/(-1(-3sin(1π(2x^2 - 2) + 1) + 2) - 1) and g(x) = (1(2x^2 - 3x + 4) - 2)/(-3(2x^2 - 3x + 4) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 294 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(3π(-5((1(4x + 4))/(-1(4x + 4) - 3))^2 - 5((1(4x + 4))/(-1(4x + 4) - 3)) + 3) - 3) - 1 and g(x) = (1(2(-1|x - 3| + 1)) - 2)/(2(2(-1|x - 3| + 1)) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Yo... | 122 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-3sin(3π((3(-1|x - 3| - 1) + 1)/(2(-1|x - 3| - 1) - 3)) - 2) - 1) - 4 and g(x) = -2(-2cos(3π(3|x + 1| + 1) + 2) + 1)^2 - 3(-2cos(3π(3|x + 1| + 1) + 2) + 1) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\... | 115 | MATH |
[
{
"content": "How many solutions does the equation -3(1|(3cos(1πx + 3) - 1) - 2| - 3)^2 + 2(1|(3cos(1πx + 3) - 1) - 2| - 3) + 3 = 3e^(-1(-3x + 5)) + 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) = -5((-1(2cos(3πx + 1) + 3) + 3)/(-2(2cos(3πx + 1) + 3)))^2 - 3((-1(2cos(3πx + 1) + 3) + 3)/(-2(2cos(3πx + 1) + 3))) - 2 and g(x) = 2cos(3π(-5x + 1) - 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 300 | MATH |
[
{
"content": "How many solutions does the equation -4((-1(1|(-2x + 2) + 1| - 1))/(1(1|(-2x + 2) + 1| - 1)))^2 - 2((-1(1|(-2x + 2) + 1| - 1))/(1(1|(-2x + 2) + 1| - 1))) + 4 = -3cos(1π(3((2x - 1)/(2x))^2 - 2((2x - 1)/(2x))) - 3) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\bo... | 67 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3((-1(4(-1|x - 1| + 1) - 3) - 2)/(2(4(-1|x - 1| + 1) - 3)))^2 + 1((-1(4(-1|x - 1| + 1) - 3) - 2)/(2(4(-1|x - 1| + 1) - 3))) + 3 and g(x) = 2|(-4((1x)/(2x + 1)) - 2) - 1| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxe... | 6 | MATH |
[
{
"content": "How many solutions does the equation 5((1(-1e^(-1x) + 3) + 3)/(-2(-1e^(-1x) + 3) + 3))^2 - 5((1(-1e^(-1x) + 3) + 3)/(-2(-1e^(-1x) + 3) + 3)) = 3sin(1π(1x - 2) + 1) + 1 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 points of intersection exist between the functions f(x) = 2(1|(2cos(2πx + 3) + 2)|)^2 + 5(1|(2cos(2πx + 3) + 2)|) - 2 and g(x) = (-2(2cos(3πx - 1) + 2))/(-1(2cos(3πx - 1) + 2) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(1|(-1(3e^(-1x) - 3) + 1) - 3|)^2 + 1 and g(x) = -1(1sin(2π(-3|x + 1| - 3)) - 2)^2 + 1(1sin(2π(-3|x + 1| - 3)) - 2) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(4(-3sin(3π(-5x^2 - 3x + 2) + 2) - 3) - 1) - 1| - 2 and g(x) = (1(1cos(1π(1|x - 1|)) - 3))/(-1(1cos(1π(1|x - 1|)) - 3) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 583 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(-1(2cos(2πx - 2) - 1)^2 + 3(2cos(2πx - 2) - 1) + 1) - 5 and g(x) = -1e^(1((-2x - 3)/(2x - 1))) + 1 intersect 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 -5((-1(-1|x + 2|) - 3)/(1(-1|x + 2|) + 2))^2 + 4((-1(-1|x + 2|) - 3)/(1(-1|x + 2|) + 2)) + 5 = -1cos(2π(3x^2 + 3x + 5) + 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 423 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(2π(4((1(-1x + 3))/(-3(-1x + 3) + 2))^2 + 3((1(-1x + 3))/(-3(-1x + 3) + 2)) - 2) + 3) - 2 and g(x) = (2(-3|(-3x^2 + 1x + 3) - 1|))/(-1(-3|(-3x^2 + 1x + 3) - 1|) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxe... | 54 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(-1|((-3(3cos(1πx - 1) - 3))/(2(3cos(1πx - 1) - 3) - 3)) + 1| - 2) and g(x) = 4(1sin(2π(-1|x|) - 3) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-1(2e^(-1(-1|x| + 2)) + 2)^2 + 3(2e^(-1(-1|x| + 2)) + 2) - 5) - 4 and g(x) = -5(3sin(2π(-1x - 1) + 2) + 3)^2 - 1(3sin(2π(-1x - 1) + 2) + 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(-2|((3(-1x^2 + 5x + 2) + 2)/(1(-1x^2 + 5x + 2) + 3)) + 2| + 2) + 2 and g(x) = (-3(2e^(-1(3x^2 - 5x - 4)) - 2))/(2(2e^(-1(3x^2 - 5x - 4)) - 2) - 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 6 | MATH |
[
{
"content": "How many solutions does the equation 4(-2sin(2π(1|x + 1|) - 2)) + 4 = (-3(-2x^2 - 5x - 5) - 2)/(-1(-2x^2 - 5x - 5) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(-3cos(2π(-1|x - 3| - 3) + 2) + 1) - 2)/(-2(-3cos(2π(-1|x - 3| - 3) + 2) + 1) + 3) and g(x) = 2(-1cos(1πx - 1) + 2)^2 - 3(-1cos(1πx - 1) + 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 46 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(-2((3(2sin(1πx - 3) + 2) - 1)/(-3(2sin(1πx - 3) + 2) + 1))^2 - 3((3(2sin(1πx - 3) + 2) - 1)/(-3(2sin(1πx - 3) + 2) + 1)) - 1) - 2 and g(x) = -4(-3cos(2π(-5x - 5) - 3) - 3)^2 + 2(-3cos(2π(-5x - 5) - 3) - 3) + 5 intersect in the interval [-10, 10].\n\n... | 200 | MATH |
[
{
"content": "How many solutions does the equation -5((2(-3cos(1π(2x - 3) + 1)) + 3)/(3(-3cos(1π(2x - 3) + 1)) + 2))^2 - 2((2(-3cos(1π(2x - 3) + 1)) + 3)/(3(-3cos(1π(2x - 3) + 1)) + 2)) + 5 = -3(2|(1x^2 + 1x) - 1| + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your... | 77 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(5(-3x^2 + 2x + 3) + 4) - 1)/(-2(5(-3x^2 + 2x + 3) + 4) + 3) and g(x) = -2(3sin(2πx + 2) + 2) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 42 | MATH |
[
{
"content": "How many solutions does the equation -1|((2(3sin(2πx + 1) + 2) + 3)/(-2(3sin(2πx + 1) + 2) + 1)) - 3| + 3 = 4(-3sin(1πx - 2) - 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-3sin(3π(-1|x - 1| + 2) - 3) - 2) + 1)/(2(-3sin(3π(-1|x - 1| + 2) - 3) - 2) - 3) and g(x) = (-3(3sin(3πx + 2) + 1) - 3)/(2(3sin(3πx + 2) + 1) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 60 | MATH |
[
{
"content": "How many solutions does the equation (-2(-5(2(-1e^(-2x) + 2) + 5)^2 - 5) + 2)/(-3(-5(2(-1e^(-2x) + 2) + 5)^2 - 5) + 4) = -2sin(1π(3(3|x| - 1) + 1) + 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 180 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(1((-2(-2sin(1πx + 2) + 3) - 3)/(-3(-2sin(1πx + 2) + 3))) + 2)^2 - 4(1((-2(-2sin(1πx + 2) + 3) - 3)/(-3(-2sin(1πx + 2) + 3))) + 2) - 2 and g(x) = 3(-2cos(2π((1x + 3)/(-2x + 3)) + 1) - 2)^2 + 4(-2cos(2π((1x + 3)/(-2x + 3)) + ... | 78 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-4(-1cos(3π((1x)/(-2x - 2)) + 3) - 1)^2 - 1(-1cos(3π((1x)/(-2x - 2)) + 3) - 1) + 2) - 2 and g(x) = (2(-3(5x^2 + 1x - 2) + 4) - 3)/(2(-3(5x^2 + 1x - 2) + 4) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\b... | 11 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(-3((3x)/(-2x - 3))^2 - 5((3x)/(-2x - 3)) - 1)| and g(x) = 1|(1cos(3πx - 2) + 3) + 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 25 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-2cos(1π(-3|x| + 1) + 2) + 3)^2 + 4(-2cos(1π(-3|x| + 1) + 2) + 3) and g(x) = 1(2cos(3πx + 2) - 2)^2 + 3(2cos(3πx + 2) - 2) - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 58 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((-3(2|(-3sin(3πx - 1)) - 1| - 1) + 2)/(-2(2|(-3sin(3πx - 1)) - 1| - 1) - 3))^2 - 2((-3(2|(-3sin(3πx - 1)) - 1| - 1) + 2)/(-2(2|(-3sin(3πx - 1)) - 1| - 1) - 3)) - 2 and g(x) = (-2(-2|(-1x^2 + 2x) + 2|) - 3)/(3(-2|(-1x^2 + 2x) + 2|) + 2) intersect in t... | 124 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-3sin(2π(2(1x^2 - 5x - 5) + 1)) - 2) - 2)/(-3(-3sin(2π(2(1x^2 - 5x - 5) + 1)) - 2) + 1) and g(x) = (-2(3(-1sin(1πx + 1) + 1) + 4))/(-1(3(-1sin(1πx + 1) + 1) + 4) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\box... | 349 | MATH |
[
{
"content": "How many solutions does the equation 1((3(-5(3|x + 2| - 2) + 4) - 1)/(1(-5(3|x + 2| - 2) + 4)))^2 - 4((3(-5(3|x + 2| - 2) + 4) - 1)/(1(-5(3|x + 2| - 2) + 4))) - 4 = (2(-3sin(2π(-2|x + 2|) + 2) - 1) - 3)/(-2(-3sin(2π(-2|x + 2|) + 2) - 1) - 2) have for x in the interval [-10, 10]?\n\nPresent the ans... | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-1(-3|(-5x^2 - 1) - 2| - 1) + 4) - 3)/(-2(-1(-3|(-5x^2 - 1) - 2| - 1) + 4)) and g(x) = -1(-2|(-1cos(1πx) + 2) - 2| + 1)^2 - 4(-2|(-1cos(1πx) + 2) - 2| + 1) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3((3(2x^2 - 2x + 3) - 1)/(3(2x^2 - 2x + 3) - 2)) + 3 and g(x) = 2cos(1π(-5x - 1) - 3) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((-2(4x + 2) + 2)/(1(4x + 2) + 1))^2 + 1((-2(4x + 2) + 2)/(1(4x + 2) + 1)) + 3 and g(x) = (-1(1cos(2πx + 3) - 2) + 1)/(1(1cos(2πx + 3) - 2) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(3(-1sin(2π(2|x - 2| + 2) - 2)) + 3) + 1)/(3(3(-1sin(2π(2|x - 2| + 2) - 2)) + 3) - 1) and g(x) = 5((-1(-3|x + 1| - 2) - 2)/(-3(-3|x + 1| - 2)))^2 + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your a... | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(1sin(1π(-3|((-1x)/(2x - 1))| - 3) - 1) - 1) - 3 and g(x) = (2(-1(-3|x + 3| + 2)^2 + 5(-3|x + 3| + 2) + 4) + 2)/(2(-1(-3|x + 3| + 2)^2 + 5(-3|x + 3| + 2) + 4)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\bo... | 32 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1cos(3π((-3(-5(5x^2 + 2x + 5) + 2) + 1)/(1(-5(5x^2 + 2x + 5) + 2) + 3)) + 2) - 2 and g(x) = 2sin(1π(3|(1x - 2) + 2| + 1) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 48 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2((2(3sin(3π(-3|x + 3| + 1) - 1) + 1) - 2)/(-2(3sin(3π(-3|x + 3| + 1) - 1) + 1) - 3))^2 + 2((2(3sin(3π(-3|x + 3| + 1) - 1) + 1) - 2)/(-2(3sin(3π(-3|x + 3| + 1) - 1) + 1) - 3)) + 1 and g(x) = -2(2|(4x^2 - 1x)|) - 5 where -10 ... | 254 | MATH |
[
{
"content": "How many solutions does the equation 5(-2|(1x^2 - 5x - 5) + 2| + 1) + 1 = (1(-3cos(2πx - 1) + 2) - 2)/(-2(-3cos(2πx - 1) + 2) + 3) have for x in the interval [-10, 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) = 2|(-2sin(2π(-2x^2 - 2x + 1) - 2) + 3) - 1| + 2 and g(x) = -4(2x^2 + 3x - 5) + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many solutions does the equation (-3(-3cos(2π(-1|x - 1| + 2) + 3) + 1) - 2)/(1(-3cos(2π(-1|x - 1| + 2) + 3) + 1)) = (-2(-3|x - 2| + 3) - 2)/(-3(-3|x - 2| + 3) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 42 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(-1(1cos(3πx) - 2)^2 - 2(1cos(3πx) - 2) - 5) + 3| + 2 and g(x) = 3(3cos(2πx - 2) + 1)^2 + 5(3cos(2πx - 2) + 1) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 40 | MATH |
[
{
"content": "How many solutions does the equation -1|(-3cos(2π(-4(1x)^2 - 3(1x) + 2)) - 1) - 1| + 1 = 4(-2((1x - 1)/(-2x - 1))^2 + 5((1x - 1)/(-2x - 1)) + 2) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 456 | MATH |
[
{
"content": "How many solutions does the equation 3cos(1π(-3(-3|(-5x^2 + 4x - 5) + 2| - 3) - 2)) - 2 = -3|(-1(-3x - 4)^2 + 5(-3x - 4) - 3) + 3| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 45 | MATH |
[
{
"content": "How many solutions does the equation 1|((2(-5x^2 - 1x + 5) - 1)/(3(-5x^2 - 1x + 5) - 2))| - 1 = 3cos(1π(1|x - 3| - 2) - 2) have for x 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) = 2(3sin(1π((-1x - 1)/(-2x + 1))) + 2) - 4 and g(x) = (2(-2x^2 + 2x + 3) - 3)/(1(-2x^2 + 2x + 3) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 34 | MATH |
[
{
"content": "How many solutions does the equation -2cos(1π(1(-5x^2 - 4x) - 4) - 3) - 1 = -1cos(3π(-2|x + 2| + 2) + 1) - 1 have for x 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) = -1(2cos(1π(3|((1x)/(3x - 2)) + 2| + 2) - 3) - 1)^2 + 5(2cos(1π(3|((1x)/(3x - 2)) + 2| + 2) - 3) - 1) - 1 and g(x) = -1|(-5(-2cos(2πx + 1) - 3) + 5)| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 19 | MATH |
[
{
"content": "How many solutions does the equation 1(1|(1((-1x - 2)/(1x + 3)))| + 3)^2 - 1 = 2|(3sin(3π((3x)/(-1x + 2)) - 1) + 3) + 3| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 155 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1sin(1π(1|((-1(1x + 5) - 1)/(-3(1x + 5) + 4)) - 3| + 3) + 3) - 1 and g(x) = 1(3|(-2x^2 - 5x + 5) + 1| - 2) - 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) = (-3(-1sin(1π(3|(-1x^2 - 2x + 1) - 1| - 3) + 2) - 1) - 2)/(1(-1sin(1π(3|(-1x^2 - 2x + 1) - 1| - 3) + 2) - 1)) and g(x) = (2(-1(2x - 5)^2 + 4(2x - 5) + 1) + 2)/(-3(-1(2x - 5)^2 + 4(2x - 5) + 1) + 2) in the range -10 ≤ x ≤ 10?\n\nPres... | 600 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(1e^(-2(3|(-1x^2 + 3x - 1) + 3| - 3))) - 3)/(1(1e^(-2(3|(-1x^2 + 3x - 1) + 3| - 3))) + 2) and g(x) = -4(2cos(3π((-1x + 3)/(3x + 2)) + 1) - 1)^2 + 4(2cos(3π((-1x + 3)/(3x + 2)) + 1) - 1) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent ... | 66 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(3|(1x - 3) + 2| + 1) + 3)/(1(3|(1x - 3) + 2| + 1) - 3) and g(x) = (-1(1sin(1πx) - 2) - 2)/(2(1sin(1πx) - 2) + 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) = 1cos(2π(-5(-1|x - 3| - 2)^2 - 2(-1|x - 3| - 2) + 1) + 1) + 3 and g(x) = -5((2x - 2)/(-3x - 2)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 524 | MATH |
[
{
"content": "How many solutions does the equation (3(2cos(3π(2(-1|x| - 1)^2 + 5(-1|x| - 1) - 3) + 1) - 3) + 1)/(-1(2cos(3π(2(-1|x| - 1)^2 + 5(-1|x| - 1) - 3) + 1) - 3) + 2) = (-2(2sin(1π(5x^2 + 3x + 3) - 1) - 3) + 3)/(3(2sin(1π(5x^2 + 3x + 3) - 1) - 3) + 1) have for x in the interval [-10, 10]?\n\nPresent the ... | 1146 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(3(2sin(3π((-3x - 2)/(-1x - 1))) + 1)^2 - 2(2sin(3π((-3x - 2)/(-1x - 1))) + 1) - 1) + 1| and g(x) = -3(2|((-2x - 2)/(3x - 2)) - 2|)^2 - 5(2|((-2x - 2)/(3x - 2)) - 2|) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-1cos(1π(1(-2x - 5)^2 + 4(-2x - 5) + 2) - 2))| - 2 and g(x) = 3(-3e^(1((3x + 3)/(-2x - 3))))^2 - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 11 | MATH |
[
{
"content": "How many solutions does the equation 3cos(2π(-2|((-2(-2x^2 - 5x - 3) - 3)/(-1(-2x^2 - 5x - 3) + 3)) - 2| - 2) - 2) - 3 = 2|(-1((1x + 1)/(-1x + 3))^2 + 1((1x + 1)/(-1x + 3))) + 1| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2e^(1((-3(4(2|x + 2| + 1)^2 + 5(2|x + 2| + 1) - 4) - 2)/(-3(4(2|x + 2| + 1)^2 + 5(2|x + 2| + 1) - 4) + 2))) - 2 and g(x) = 1sin(1π(-1|(4x + 5) + 3| - 3) + 3) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\bo... | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-1sin(2π(-2|(1x) + 2| - 1) + 2) - 1) - 2)/(1(-1sin(2π(-2|(1x) + 2| - 1) + 2) - 1) + 3) and g(x) = -2(-2cos(3π(-1|x - 1| + 3))) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(3|(3((-3x)/(3x + 2)) - 1) + 3| - 2)^2 + 4(3|(3((-3x)/(3x + 2)) - 1) + 3| - 2) + 2 and g(x) = (-2(-3sin(1π(2x^2 + 1x + 5) + 2) + 3) + 2)/(3(-3sin(1π(2x^2 + 1x + 5) + 2) + 3) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex form... | 41 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-2e^(-1(-3(2|x| + 2)^2 - 4(2|x| + 2) + 1)) + 1) + 2)/(-1(-2e^(-1(-3(2|x| + 2)^2 - 4(2|x| + 2) + 1)) + 1) + 2) and g(x) = -3cos(1π(-3|((-3x - 1)/(-3x + 1)) - 1|) + 3) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex ... | 51 | MATH |
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