messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(1π(2|(2((-1((-2x + 2)/(-1x + 3)) - 3)/(-1((-2x + 2)/(-1x + 3)) - 2)) - 5) - 2| + 2) + 3) - 1 and g(x) = (1((2(1(3e^(1x) - 1) + 3))/(1(1(3e^(1x) - 1) + 3))) + 3)/(-3((2(1(3e^(1x) - 1) + 3))/(1(1(3e^(1x) - 1) + 3))) - 1) intersect in the interval [... | 35 | MATH |
[
{
"content": "How many solutions does the equation 3(2((1(-1(-4x^2 - 2x + 2)^2 - 2(-4x^2 - 2x + 2) + 2) - 1)/(-2(-1(-4x^2 - 2x + 2)^2 - 2(-4x^2 - 2x + 2) + 2) - 3)) - 2) - 3 = (-1(2|(2sin(1π((-2x)/(-1x + 2)) + 3) - 2) + 2| - 1) - 3)/(-3(2|(2sin(1π((-2x)/(-1x + 2)) + 3) - 2) + 2| - 1) - 1) have for x in the inte... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(5(2|(1(1cos(1πx + 1) + 2) - 4)| - 1)^2 + 1(2|(1(1cos(1πx + 1) + 2) - 4)| - 1) + 3) - 2 and g(x) = 1(-2|(-1(4x + 1)^2 + 5(4x + 1) + 5) + 2| - 3)^2 + 4(-2|(-1(4x + 1)^2 + 5(4x + 1) + 5) + 2| - 3) + 4 intersect in the interval [-10, 10].\n\nPresent the ... | 4 | MATH |
[
{
"content": "How many solutions does the equation (-3(2sin(2π(-1|(-3(2x + 2) - 1) + 2| + 2) - 1)) + 1)/(2(2sin(2π(-1|(-3(2x + 2) - 1) + 2| + 2) - 1)) + 2) = 4(-2|((1(-1|x - 2| + 2) + 2)/(3(-1|x - 2| + 2) - 2))| - 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your a... | 116 | MATH |
[
{
"content": "How many solutions does the equation -1|(-2cos(1π(-2|(-2|(3x - 3) - 2| + 2) - 3|) + 1) - 2)| + 3 = (-2(2(1(2|x - 1| + 1))^2 - 5(1(2|x - 1| + 1)) + 2) + 2)/(3(2(1(2|x - 1| + 1))^2 - 5(1(2|x - 1| + 1)) + 2) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{You... | 226 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-5(4((2(-1x) + 1)/(1(-1x) + 3)) - 3)^2 + 2(4((2(-1x) + 1)/(1(-1x) + 3)) - 3) - 4) + 3)/(1(-5(4((2(-1x) + 1)/(1(-1x) + 3)) - 3)^2 + 2(4((2(-1x) + 1)/(1(-1x) + 3)) - 3) - 4) + 1) and g(x) = -1(-3sin(2π(1|(-4x + 3) - 2|) - 3) - 2) ... | 160 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(-3(-1cos(2π(-1(-1x - 4)^2 + 1(-1x - 4)) - 1) - 3) - 2) - 2)/(-1(-3(-1cos(2π(-1(-1x - 4)^2 + 1(-1x - 4)) - 1) - 3) - 2) - 1) and g(x) = 1(-2(-3cos(3π((-1x + 2)/(2x - 1)) - 2))^2 - 2(-3cos(3π((-1x + 2)/(2x - 1)) - 2)) + 3) -... | 79 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-2((1(-2(2|x - 3| - 1)^2 - 5) - 3)/(2(-2(2|x - 3| - 1)^2 - 5) - 1))^2 - 4) - 1| - 1 and g(x) = (-2((-3(2cos(3π(1|x - 2| - 1)) - 1) + 3)/(-2(2cos(3π(1|x - 2| - 1)) - 1) - 2)) - 3)/(-3((-3(2cos(3π(1|x - 2| - 1)) - 1) + 3)/(-2(2cos... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(2π(-2|(4(-5(5x - 4) + 5)^2 - 1(-5(5x - 4) + 5) - 3) - 2| - 1)) + 1 and g(x) = -3sin(3π(-4((1(-1|x - 1| + 3) - 3)/(-2(-1|x - 1| + 3) - 1)) - 5) - 2) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Y... | 318 | MATH |
[
{
"content": "How many solutions does the equation -5((-3(-2|((3(2x + 5) + 2)/(-2(2x + 5) - 3)) - 1| + 3) + 2)/(2(-2|((3(2x + 5) + 2)/(-2(2x + 5) - 3)) - 1| + 3) + 2))^2 - 2((-3(-2|((3(2x + 5) + 2)/(-2(2x + 5) - 3)) - 1| + 3) + 2)/(2(-2|((3(2x + 5) + 2)/(-2(2x + 5) - 3)) - 1| + 3) + 2)) = 5(-2sin(2π(5(4x^2 - 3x... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-3(1(-2cos(1π(1|x - 1| + 2) - 2) - 3)^2 + 2(-2cos(1π(1|x - 1| + 2) - 2) - 3) + 4) + 1)^2 - 1(-3(1(-2cos(1π(1|x - 1| + 2) - 2) - 3)^2 + 2(-2cos(1π(1|x - 1| + 2) - 2) - 3) + 4) + 1) - 5 and g(x) = 1e^(-2((-3(4((-3x - 1)/(1x +... | 11 | MATH |
[
{
"content": "How many solutions does the equation -2(1|((-2(-5(2|x + 1| - 3)^2 + 4) + 3)/(3(-5(2|x + 1| - 3)^2 + 4) - 2))| - 1) + 2 = 1(3(1|(3x - 2) + 2| - 3)) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(3(-1(-2|(2x - 4) + 1| - 1)) - 1) + 4 and g(x) = 2|((3(4(-4x - 1)^2 + 4(-4x - 1) - 5) - 3)/(3(4(-4x - 1)^2 + 4(-4x - 1) - 5) + 2)) + 1| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|((3(-1(-1(-2|x - 2| + 2)^2 - 4(-2|x - 2| + 2) + 1) + 3) + 1)/(-1(-1(-1(-2|x - 2| + 2)^2 - 4(-2|x - 2| + 2) + 1) + 3) + 3)) - 3| + 2 and g(x) = 3|(-5((-1(1cos(3πx - 2) + 3) + 2)/(-1(1cos(3πx - 2) + 3) - 2))^2 - 2((-1(1cos(3π... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-1|(-2sin(2π(-4(-1|x + 1| + 1) - 2) - 3) - 2) - 2| + 3) - 2)/(3(-1|(-2sin(2π(-4(-1|x + 1| + 1) - 2) - 3) - 2) - 2| + 3)) and g(x) = 5(-3sin(1π(-3(2|x - 3| + 3)^2 + 5(2|x - 3| + 3) + 5) - 2) - 1) - 2 in the range -10 ≤ x ≤ 10?\n... | 1596 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-3|((2(1(1cos(2πx + 3) - 3)^2 + 1(1cos(2πx + 3) - 3) - 2) + 1)/(-2(1(1cos(2πx + 3) - 3)^2 + 1(1cos(2πx + 3) - 3) - 2) + 3)) + 2| - 2) + 3| and g(x) = -2cos(2π(5(1(-4x) - 1) - 4) - 3) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent th... | 499 | MATH |
[
{
"content": "How many solutions does the equation 1((2(2e^(1((1(2x^2 + 4x - 4) - 1)/(-1(2x^2 + 4x - 4) + 2)))) + 1)/(3(2e^(1((1(2x^2 + 4x - 4) - 1)/(-1(2x^2 + 4x - 4) + 2)))) + 2))^2 - 3((2(2e^(1((1(2x^2 + 4x - 4) - 1)/(-1(2x^2 + 4x - 4) + 2)))) + 1)/(3(2e^(1((1(2x^2 + 4x - 4) - 1)/(-1(2x^2 + 4x - 4) + 2)))) +... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4((-1((-2(1|(1x - 2) + 1| + 1) + 1)/(1(1|(1x - 2) + 1| + 1) - 3)))/(-2((-2(1|(1x - 2) + 1| + 1) + 1)/(1(1|(1x - 2) + 1| + 1) - 3)) + 3)) - 3 and g(x) = -5((3((1(-1|x| - 1) + 2)/(2(-1|x| - 1) + 3)) - 3)/(-3((1(-1|x| - 1) + 2)... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(2π(-3(-2(-4(-2x + 4) - 1) + 3)^2 + 2(-2(-4(-2x + 4) - 1) + 3) + 1) + 2) and g(x) = (-1(4(2|(-1sin(2πx - 1) + 2) + 2| - 1)^2 - 4(2|(-1sin(2πx - 1) + 2) + 2| - 1) - 2) + 3)/(-3(4(2|(-1sin(2πx - 1) + 2) + 2| - 1)^2 - 4(2|(-... | 1474 | MATH |
[
{
"content": "How many solutions does the equation (-3(2sin(1π((-3((1(-5x^2 + 3x + 4) + 1)/(2(-5x^2 + 3x + 4) - 1)) + 1)/(2((1(-5x^2 + 3x + 4) + 1)/(2(-5x^2 + 3x + 4) - 1)) - 1)) - 2) + 2) - 2)/(2(2sin(1π((-3((1(-5x^2 + 3x + 4) + 1)/(2(-5x^2 + 3x + 4) - 1)) + 1)/(2((1(-5x^2 + 3x + 4) + 1)/(2(-5x^2 + 3x + 4) - 1... | 299 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(2cos(1π(2(3|(3|x + 1| - 2) + 1| + 1)^2 + 3(3|(3|x + 1| - 2) + 1| + 1) - 1) - 1) - 3)^2 + 2(2cos(1π(2(3|(3|x + 1| - 2) + 1| + 1)^2 + 3(3|(3|x + 1| - 2) + 1| + 1) - 1) - 1) - 3) - 4 and g(x) = (-2(4((-3(1|x - 3| + 3) - 1)/(2(1|x - 3| + 3)))^2 + 4((-3(... | 19 | MATH |
[
{
"content": "How many solutions does the equation (-2(2(-5(-1(1x^2 + 3x + 3) + 5) - 1)^2 + 1(-5(-1(1x^2 + 3x + 3) + 5) - 1) - 2) - 3)/(3(2(-5(-1(1x^2 + 3x + 3) + 5) - 1)^2 + 1(-5(-1(1x^2 + 3x + 3) + 5) - 1) - 2) + 1) = (2(-1|(4((-1x - 3)/(-1x + 2))^2 - 3((-1x - 3)/(-1x + 2)) + 3) + 2|) + 3)/(3(-1|(4((-1x - 3)/... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(1(5(1(2cos(2πx + 1) - 1) - 2)^2 + 2(1(2cos(2πx + 1) - 1) - 2) - 4) - 1) + 1 and g(x) = -3e^(1((-2(4(3|x - 1| - 1)) - 1)/(-3(4(3|x - 1| - 1))))) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your an... | 42 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(3π((1((3(2|(3x^2 + 3x + 5) + 3| - 1) + 1)/(1(2|(3x^2 + 3x + 5) + 3| - 1) + 2)) - 1)/(1((3(2|(3x^2 + 3x + 5) + 3| - 1) + 1)/(1(2|(3x^2 + 3x + 5) + 3| - 1) + 2)) + 2)) - 2) - 2 and g(x) = (-3((-3(4(2sin(1πx + 1) + 1) + 2) - 3)/(1(4(2sin(1πx + 1) + ... | 20 | MATH |
[
{
"content": "How many solutions does the equation 2((-1(2|(-4(2sin(3πx + 3) - 2)^2 - 1(2sin(3πx + 3) - 2)) + 1| + 3) - 1)/(-1(2|(-4(2sin(3πx + 3) - 2)^2 - 1(2sin(3πx + 3) - 2)) + 1| + 3) - 2))^2 - 5((-1(2|(-4(2sin(3πx + 3) - 2)^2 - 1(2sin(3πx + 3) - 2)) + 1| + 3) - 1)/(-1(2|(-4(2sin(3πx + 3) - 2)^2 - 1(2sin(3π... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2((3(2(2sin(2π(-1|x| + 1) + 2) + 3) + 1) - 3)/(1(2(2sin(2π(-1|x| + 1) + 2) + 3) + 1) - 2)) - 2)/(2((3(2(2sin(2π(-1|x| + 1) + 2) + 3) + 1) - 3)/(1(2(2sin(2π(-1|x| + 1) + 2) + 3) + 1) - 2)) + 2) and g(x) = -3(-2cos(1π(4(3x - ... | 1356 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(2(3cos(1π(3|(-4x + 2) - 2| + 2) - 3) - 3)) and g(x) = (1(-3|(-2(-1x + 2)) + 3|) + 3)/(-1(-3|(-2(-1x + 2)) + 3|) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 232 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(3|(-3(-2(3x^2 + 3x - 5) - 5)^2 - 1(-2(3x^2 + 3x - 5) - 5) + 4) + 2| - 2)^2 + 5(3|(-3(-2(3x^2 + 3x - 5) - 5)^2 - 1(-2(3x^2 + 3x - 5) - 5) + 4) + 2| - 2) - 5 and g(x) = -2|(-1e^(-2((-2(-4x + 3) + 3)/(-1(-4x + 3) + 2))) + 3) - 1| + 2 intersect in the in... | 10 | MATH |
[
{
"content": "How many solutions does the equation -1|(-4(-2|(3e^(2(5x + 4))) - 3| + 1)^2 + 5(-2|(3e^(2(5x + 4))) - 3| + 1) - 5) + 3| - 1 = -3|(3|((3(-2x - 5) - 2)/(1(-2x - 5) - 3))| + 1) - 1| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(5(3sin(1π((-2(2|x - 3| - 1) - 3)/(2(2|x - 3| - 1) - 1))) + 3) - 3)^2 - 4(5(3sin(1π((-2(2|x - 3| - 1) - 3)/(2(2|x - 3| - 1) - 1))) + 3) - 3) - 4 and g(x) = 2(1(-2(1x + 1)^2 + 3(1x + 1) - 3) + 2) where -10 ≤ x ≤ 10.\n\nPrese... | 59 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(3cos(1π(2|(2(3x + 1)^2 + 4(3x + 1) + 4) + 1| - 2) + 2) + 3)^2 - 4(3cos(1π(2|(2(3x + 1)^2 + 4(3x + 1) + 4) + 1| - 2) + 2) + 3) - 2 and g(x) = 3(4((1((-2x)/(-2x + 3)) - 2)/(-1((-2x)/(-2x + 3)) + 2))^2 - 2((1((-2x)/(-2x + 3)) ... | 961 | MATH |
[
{
"content": "How many solutions does the equation 3|(2(2cos(2π(-2|((-2x - 3)/(2x + 3)) - 1| + 1) - 3) + 2)^2 - 4(2cos(2π(-2|((-2x - 3)/(2x + 3)) - 1| + 1) - 3) + 2) - 2) - 2| + 2 = 2(-1(3|(-4x^2 - 3x - 4) + 3| - 3)^2 + 1(3|(-4x^2 - 3x - 4) + 3| - 3) + 3)^2 + 4(-1(3|(-4x^2 - 3x - 4) + 3| - 3)^2 + 1(3|(-4x^2 - 3... | 6 | MATH |
[
{
"content": "How many solutions does the equation 1|(-1((2((-3(1sin(1πx - 2) + 1) + 3)/(1(1sin(1πx - 2) + 1))) - 1)/(-3((-3(1sin(1πx - 2) + 1) + 3)/(1(1sin(1πx - 2) + 1))) + 1))^2 - 5((2((-3(1sin(1πx - 2) + 1) + 3)/(1(1sin(1πx - 2) + 1))) - 1)/(-3((-3(1sin(1πx - 2) + 1) + 3)/(1(1sin(1πx - 2) + 1))) + 1)) - 2)|... | 30 | MATH |
[
{
"content": "How many solutions does the equation 3|(-2sin(1π(4(1(3|x + 2| - 2))^2 - 3(1(3|x + 2| - 2)) - 4) + 3) - 3) + 2| + 3 = 2|(-2|(-4((2x - 1)/(3x - 2)) + 2) - 3| - 3) + 3| + 2 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) = -1cos(3π(3(2(3|(1x^2 - 5x + 2)| - 1)^2 - 4(3|(1x^2 - 5x + 2)| - 1) + 5) + 5) + 1) - 1 and g(x) = 3cos(1π(5((3(-1x^2 - 2) + 1)/(-1(-1x^2 - 2) + 2))^2 - 5((3(-1x^2 - 2) + 1)/(-1(-1x^2 - 2) + 2)) + 2) + 3) - 3 where -10 ≤ x ≤ 10... | 277 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-1|(-4(1|(1cos(1πx + 1) + 1) - 3| + 2)^2 - 2(1|(1cos(1πx + 1) + 1) - 3| + 2) + 5) + 1| + 3) - 2 and g(x) = -2(3|(-1|(-2e^(1x) + 3) - 1| - 1) + 2| + 1)^2 - 4(3|(-1|(-2e^(1x) + 3) - 1| - 1) + 2| + 1) + 3 where -10 ≤ x ≤ 10.\n... | 7 | MATH |
[
{
"content": "How many solutions does the equation -1|(-3(2cos(2π(3|(4x^2 + 5x - 2)| - 3) - 2)) - 2) + 2| = -3(1((3((-3x - 2)/(3x + 1)) + 1)/(-2((-3x - 2)/(3x + 1)) + 3))^2 - 4((3((-3x - 2)/(3x + 1)) + 1)/(-2((-3x - 2)/(3x + 1)) + 3)) + 1)^2 + 4(1((3((-3x - 2)/(3x + 1)) + 1)/(-2((-3x - 2)/(3x + 1)) + 3))^2 - 4(... | 169 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(-3sin(2π(4(-2(-4x - 5)^2 + 2(-4x - 5) + 4)^2 + 3(-2(-4x - 5)^2 + 2(-4x - 5) + 4) - 3) + 2) + 1))/(2(-3sin(2π(4(-2(-4x - 5)^2 + 2(-4x - 5) + 4)^2 + 3(-2(-4x - 5)^2 + 2(-4x - 5) + 4) - 3) + 2) + 1) - 3) and g(x) = -1(2sin(2π(-2(2x + 4) + 2) + 2) + 2)... | 10 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-2(-2(-1(-3x^2 + 3x - 1) - 4)^2 - 2(-1(-3x^2 + 3x - 1) - 4) + 5)^2 + 3) - 5 and g(x) = -1e^(-2(-1((-2(2|x + 3| - 3) - 2)/(3(2|x + 3| - 3) - 2)) - 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 9 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(-2sin(1π(3((-1((2x - 1)/(-3x)) - 1)/(1((2x - 1)/(-3x)))) + 1) + 3)) - 1 and g(x) = -2|(1(3|((-1x - 2)/(-1x - 1)) - 2|)^2 - 5(3|((-1x - 2)/(-1x - 1)) - 2|) + 3) + 2| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\bo... | 53 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(2π(-3((3((2(4x) + 1)/(1(4x) + 3)) - 3)/(-1((2(4x) + 1)/(1(4x) + 3)) + 3))^2 + 2((3((2(4x) + 1)/(1(4x) + 3)) - 3)/(-1((2(4x) + 1)/(1(4x) + 3)) + 3)) - 4) - 2) - 1 and g(x) = (-2(3((-2(1x^2 + 3x + 1) + 3)/(2(1x^2 + 3x + 1) + 2)) + 2) - 3)/(-2(3((-2... | 206 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-4(1e^(-2(-2(1x^2 + 1)^2 - 1(1x^2 + 1) + 5)) - 1)^2 + 5(1e^(-2(-2(1x^2 + 1)^2 - 1(1x^2 + 1) + 5)) - 1)) + 2)/(-3(-4(1e^(-2(-2(1x^2 + 1)^2 - 1(1x^2 + 1) + 5)) - 1)^2 + 5(1e^(-2(-2(1x^2 + 1)^2 - 1(1x^2 + 1) + 5)) - 1)) + 1) and g(... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(3π(-5((-1(-5((-3x)/(-3x + 2)) + 4) + 1)/(-2(-5((-3x)/(-3x + 2)) + 4) + 3))^2 - 1((-1(-5((-3x)/(-3x + 2)) + 4) + 1)/(-2(-5((-3x)/(-3x + 2)) + 4) + 3)) - 3)) - 2 and g(x) = 1sin(1π(-1|((2(1x) - 1)/(1(1x))) - 2| - 1)) where... | 50 | MATH |
[
{
"content": "How many solutions does the equation 1((2(5((-2(-2|x - 3|) + 1)/(1(-2|x - 3|) + 1)) - 3) - 3)/(-3(5((-2(-2|x - 3|) + 1)/(1(-2|x - 3|) + 1)) - 3) - 2)) - 4 = 1(5(2cos(3π((-2x + 3)/(-2x - 3)) + 2)) + 2) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 109 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(-2|(-3(1((2x - 2)/(1x))^2 - 2((2x - 2)/(1x)))^2 + 2(1((2x - 2)/(1x))^2 - 2((2x - 2)/(1x))) - 2) + 3| + 3)^2 and g(x) = -3cos(2π(4(-2((-3x + 2)/(-2x + 1)) + 3)^2 - 5(-2((-3x + 2)/(-2x + 1)) + 3) - 4) - 1) - 2 intersect in the interval [-10, 10].\n\nPr... | 4 | MATH |
[
{
"content": "How many solutions does the equation (-2(1((1(-4(2|x| - 1)^2 + 2(2|x| - 1)))/(2(-4(2|x| - 1)^2 + 2(2|x| - 1)) - 1))) - 1)/(-2(1((1(-4(2|x| - 1)^2 + 2(2|x| - 1)))/(2(-4(2|x| - 1)^2 + 2(2|x| - 1)) - 1))) + 1) = -1|(-2sin(2π(-3|((-3x + 1)/(1x + 2)) + 2| + 3)) + 3) - 1| + 1 have for x in the interval ... | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2sin(1π(2|((1(1(-1x + 2) - 4) - 3)/(-2(1(-1x + 2) - 4) - 1)) - 2| - 2) - 3) + 3 and g(x) = 1(4((-2((-3x + 3)/(-1x)) - 1)/(-3((-3x + 3)/(-1x)))) + 2)^2 - 4(4((-2((-3x + 3)/(-1x)) - 1)/(-3((-3x + 3)/(-1x)))) + 2) - 2 intersect in the interval [-10, 10].\... | 69 | MATH |
[
{
"content": "How many solutions does the equation -5(4((3(-1(3|x - 3| + 3)^2 + 3(3|x - 3| + 3) + 5) + 1)/(-2(-1(3|x - 3| + 3)^2 + 3(3|x - 3| + 3) + 5) - 2))^2 + 3((3(-1(3|x - 3| + 3)^2 + 3(3|x - 3| + 3) + 5) + 1)/(-2(-1(3|x - 3| + 3)^2 + 3(3|x - 3| + 3) + 5) - 2)) - 2)^2 + 4(4((3(-1(3|x - 3| + 3)^2 + 3(3|x - 3... | 228 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(4(-4(-1(1x^2 + 4x) - 4) - 5))^2 - 4(4(-4(-1(1x^2 + 4x) - 4) - 5)) + 1 and g(x) = -3((-2((1(2cos(3πx + 2) - 1))/(3(2cos(3πx + 2) - 1) - 2)) - 1)/(-3((1(2cos(3πx + 2) - 1))/(3(2cos(3πx + 2) - 1) - 2)) - 3))^2 + 4((-2((1(2cos(3πx +... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-3(-2cos(2π(-1((-1x)/(-2x - 1))^2 - 5((-1x)/(-2x - 1)) - 1) - 3) - 1)^2 + 5(-2cos(2π(-1((-1x)/(-2x - 1))^2 - 5((-1x)/(-2x - 1)) - 1) - 3) - 1) + 1) - 3 and g(x) = 2((2(-1|(-1x + 1) + 2| - 3) - 3)/(-1(-1|(-1x + 1) + 2| - 3) ... | 57 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(3cos(1π(1(1|(3|x - 1| - 2)|) + 4)))| + 3 and g(x) = 5((3(3cos(3π(2x^2 + 3x - 1) - 3) - 2) + 1)/(2(3cos(3π(2x^2 + 3x - 1) - 3) - 2)))^2 + 1((3(3cos(3π(2x^2 + 3x - 1) - 3) - 2) + 1)/(2(3cos(3π(2x^2 + 3x - 1) - 3) - 2))) - 4 intersect in the interval [... | 505 | MATH |
[
{
"content": "How many solutions does the equation -3cos(1π(-5(1|(3|(1|x - 2| + 3) - 1| - 3) + 1| + 2)^2 + 1(1|(3|(1|x - 2| + 3) - 1| - 3) + 1| + 2) + 3) + 3) = 1((-2(2((-1x - 3)/(1x)) - 2) - 1)/(3(2((-1x - 3)/(1x)) - 2) + 1)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\bo... | 1160 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-3|(3(4(-3sin(2πx - 1))^2 - 2(-3sin(2πx - 1)) + 5)^2 + 1(4(-3sin(2πx - 1))^2 - 2(-3sin(2πx - 1)) + 5)) + 2| - 3) + 1| - 3 and g(x) = -4(3|(1(2x + 3)^2 - 1(2x + 3) + 2)| - 3) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer i... | 63 | MATH |
[
{
"content": "How many solutions does the equation 3((-3(3sin(1π(-5(-3x + 2)^2 + 1(-3x + 2)) + 2) + 1) - 2)/(3(3sin(1π(-5(-3x + 2)^2 + 1(-3x + 2)) + 2) + 1))) + 4 = -2|(5(-2(-2x^2 - 1)) - 4) - 1| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(1(-2|(-1|(2|x - 3| - 3) + 2| + 1) + 2|) + 5) + 3)/(-1(1(-2|(-1|(2|x - 3| - 3) + 2| + 1) + 2|) + 5) - 1) and g(x) = 1((2((2(-2cos(1πx - 2) - 1) - 2)/(2(-2cos(1πx - 2) - 1) + 1)) + 3)/(-1((2(-2cos(1πx - 2) - 1) - 2)/(2(-2cos(1πx -... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(2(-1|(1|(1x^2 - 5x + 4) - 2| - 3)| + 1)^2 + 1(-1|(1|(1x^2 - 5x + 4) - 2| - 3)| + 1) + 1) - 3| + 3 and g(x) = -1|(-2|((-1(3cos(1πx + 3) + 3) + 2)/(2(3cos(1πx + 3) + 3))) + 3|) - 1| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the a... | 14 | MATH |
[
{
"content": "How many solutions does the equation -3(1(4(-3cos(3π(2x^2 - 2x - 5) + 1))^2 + 4(-3cos(3π(2x^2 - 2x - 5) + 1)) - 1))^2 - 4(1(4(-3cos(3π(2x^2 - 2x - 5) + 1))^2 + 4(-3cos(3π(2x^2 - 2x - 5) + 1)) - 1)) + 2 = (1(-1|(3((-1x + 1)/(1x + 2)) + 3) + 2| + 3))/(1(-1|(3((-1x + 1)/(1x + 2)) + 3) + 2| + 3)) have... | 519 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(2e^(1(4(-1(1|x + 1| - 3)^2 + 2)^2)) - 2) - 3| - 3 and g(x) = 1(-4(5(1|x - 1|)^2 - 3(1|x - 1|) + 3))^2 - 4(-4(5(1|x - 1|)^2 - 3(1|x - 1|) + 3)) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your an... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((-1(-2|(2(-3e^(-2x) + 3)^2 + 1) - 3| + 3) + 3)/(-1(-2|(2(-3e^(-2x) + 3)^2 + 1) - 3| + 3) - 3))^2 - 5((-1(-2|(2(-3e^(-2x) + 3)^2 + 1) - 3| + 3) + 3)/(-1(-2|(2(-3e^(-2x) + 3)^2 + 1) - 3| + 3) - 3)) - 3 and g(x) = -2sin(1π(2((2(1x + 4))/(-2(1x + 4) + 3)... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(1((-3((3((1x - 2)/(-2x - 2)) - 2)/(3((1x - 2)/(-2x - 2)) + 2)) - 3)/(3((3((1x - 2)/(-2x - 2)) - 2)/(3((1x - 2)/(-2x - 2)) + 2))))^2 + 5((-3((3((1x - 2)/(-2x - 2)) - 2)/(3((1x - 2)/(-2x - 2)) + 2)) - 3)/(3((3((1x - 2)/(-2x ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3((-2((3(-2cos(1π((3x - 2)/(2x + 2)) + 3) - 2) + 3)/(-3(-2cos(1π((3x - 2)/(2x + 2)) + 3) - 2) + 1)) + 2)/(-3((3(-2cos(1π((3x - 2)/(2x + 2)) + 3) - 2) + 3)/(-3(-2cos(1π((3x - 2)/(2x + 2)) + 3) - 2) + 1)) - 2)) - 2)/(3((-2((3(-2cos... | 43 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2((3(-1|(2|(-1|x - 3| - 1) - 1| - 3) + 2| - 2) - 3)/(3(-1|(2|(-1|x - 3| - 1) - 1| - 3) + 2| - 2) - 2)) - 3 and g(x) = -2|(2|(-2|(-1x + 3) + 3| + 1) + 3| + 1) - 3| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3((-2(-2sin(2π(2|((-2x - 2)/(2x)) + 3|) + 2)) + 3)/(-2(-2sin(2π(2|((-2x - 2)/(2x)) + 3|) + 2)) - 2)) - 1 and g(x) = (3((3(-2sin(1π(-4x^2 - 5x + 3) + 1) - 2) - 2)/(-3(-2sin(1π(-4x^2 - 5x + 3) + 1) - 2) + 4)) + 1)/(-3((3(-2sin(1π(-4x^2 - 5x + 3) + 1) - ... | 63 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4((-1(5(3e^(-1(-1x + 3)) - 1) + 2) - 1)/(-3(5(3e^(-1(-1x + 3)) - 1) + 2) - 2))^2 + 3((-1(5(3e^(-1(-1x + 3)) - 1) + 2) - 1)/(-3(5(3e^(-1(-1x + 3)) - 1) + 2) - 2)) + 3 and g(x) = 3|(-1|(-3(1x - 2)^2 - 3(1x - 2) + 3) + 1|) + 2| + 3 intersect in the inter... | 8 | MATH |
[
{
"content": "How many solutions does the equation -1(-5(1|(3(-2x - 5)^2 + 3(-2x - 5) - 1) - 3|) + 1) + 5 = 2|(1(-1cos(1π(-1x^2 + 4x + 5) + 3)) + 4) + 3| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation (-2(4((-3(2(-1|x + 1| + 2) + 5) - 3)/(-1(2(-1|x + 1| + 2) + 5) - 2))^2 - 5) + 2)/(-2(4((-3(2(-1|x + 1| + 2) + 5) - 3)/(-1(2(-1|x + 1| + 2) + 5) - 2))^2 - 5)) = -3|(4(-3cos(2π(-3x^2 - 4x - 4) - 1) - 1) + 3) - 2| + 2 have for x in the interval [-10, 10]?\n\nPrese... | 89 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(3π(3(5(1|((2x - 3)/(2x - 3)) + 3| - 3)^2 - 5(1|((2x - 3)/(2x - 3)) + 3| - 3) - 2)^2 + 2(5(1|((2x - 3)/(2x - 3)) + 3| - 3)^2 - 5(1|((2x - 3)/(2x - 3)) + 3| - 3) - 2) + 2) + 1) - 3 and g(x) = (1(1(3cos(2π(4x^2 + 1x + 3) - 3) + ... | 134 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(-5(1|(2cos(2π(-4x^2 - 2) + 1) - 1)| - 2) - 5) - 3 and g(x) = -4(3sin(2π(5((2x + 1)/(-2x)) + 3) - 3) + 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 695 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(5((2(2|((-1x + 3)/(1x + 3)) + 2| - 3) + 1)/(-1(2|((-1x + 3)/(1x + 3)) + 2| - 3) - 3)) - 1)^2 + 3(5((2(2|((-1x + 3)/(1x + 3)) + 2| - 3) + 1)/(-1(2|((-1x + 3)/(1x + 3)) + 2| - 3) - 3)) - 1) + 4 and g(x) = 3(4(2sin(2π(1x - 4) + 1) -... | 15 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2((-2(-2(-1(-3cos(2πx - 1) + 2)^2 + 4(-3cos(2πx - 1) + 2) + 2) - 2) - 3)/(1(-2(-1(-3cos(2πx - 1) + 2)^2 + 4(-3cos(2πx - 1) + 2) + 2) - 2) + 1)) - 5 and g(x) = 2(-2|((1(3e^(-1x) + 1) - 1)/(2(3e^(-1x) + 1) - 1)) + 1| + 1)^2 + 5(-2|((1(3e^(-1x) + 1) - 1)/... | 40 | MATH |
[
{
"content": "How many solutions does the equation 2sin(1π(1(1|(3(-5x + 5) + 2) - 2| - 3)^2 - 1(1|(3(-5x + 5) + 2) - 2| - 3) + 4)) + 3 = (3(1(3((2x - 3)/(2x + 1)) + 1) + 2) + 3)/(1(1(3((2x - 3)/(2x + 1)) + 1) + 2) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 1576 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(1|((3(2|(4x^2 - 2) + 2| - 3) - 3)/(2(2|(4x^2 - 2) + 2| - 3) + 1)) - 3| - 1) - 3 and g(x) = -1|(-3|((1(-5x^2 + 1x + 1) - 1)/(1(-5x^2 + 1x + 1) + 2)) + 1| + 1) + 3| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex form... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-1|(-3sin(3π(-1((-3x - 2)/(2x + 2)) - 5) - 2) + 2) - 3| + 3)^2 - 1(-1|(-3sin(3π(-1((-3x - 2)/(2x + 2)) - 5) - 2) + 2) - 3| + 3) - 3 and g(x) = (-2(-5(-4(-4x + 3) + 3)^2 - 2(-4(-4x + 3) + 3) - 1) - 3)/(-2(-5(-4(-4x + 3) + 3)^2 - 2(-4(-4x + 3) + 3) - 1... | 48 | MATH |
[
{
"content": "How many solutions does the equation 2(-1(3cos(3π((-1(2x + 5) + 2)/(1(2x + 5) + 1))) + 1) + 4) - 1 = -1|(3(-2|((-1x + 1)/(-2x - 2)) + 2| + 3) + 1) + 2| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-5(3(-2cos(2π(2x - 1) - 1) + 3)^2 + 3(-2cos(2π(2x - 1) - 1) + 3) - 1) + 1) + 2| - 1 and g(x) = -3|(-1(-4(1x^2 + 5x - 5)^2 - 3(1x^2 + 5x - 5) + 5)^2 - 4(-4(1x^2 + 5x - 5)^2 - 3(1x^2 + 5x - 5) + 5) + 3) + 3| + 2 where -10 ≤... | 6 | MATH |
[
{
"content": "How many solutions does the equation -5(3|(1((1(-1x - 4) + 2)/(-2(-1x - 4)))^2) + 2|) - 4 = (1(-2sin(3π(5(-2|x - 1| - 3) - 2) - 1) - 2) - 2)/(-2(-2sin(3π(5(-2|x - 1| - 3) - 2) - 1) - 2) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-1(3cos(1π(3|((-1x - 3)/(-1x + 3)) - 1| - 3) - 3) + 3) + 1) + 2 and g(x) = 3cos(2π(-2(-4((-2x)/(-3x - 2))) + 3) - 1) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 168 | MATH |
[
{
"content": "How many solutions does the equation 1cos(1π((-1((1(-3|((2x - 3)/(2x - 1)) + 1| + 2) - 1)/(1(-3|((2x - 3)/(2x - 1)) + 1| + 2) + 1)) - 1)/(1((1(-3|((2x - 3)/(2x - 1)) + 1| + 2) - 1)/(1(-3|((2x - 3)/(2x - 1)) + 1| + 2) + 1)) - 2))) - 3 = -1(-2|(-1cos(1π(-2x^2 + 5x - 2) - 2) + 2)| + 1)^2 - 4(-2|(-1co... | 196 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-1(-2e^(-1((-1(-5x) + 1)/(1(-5x)))) + 3)^2 + 4(-2e^(-1((-1(-5x) + 1)/(1(-5x)))) + 3) + 5) - 1| + 3 and g(x) = 1(1((1(4x - 3))/(-2(4x - 3) - 2))^2 + 3)^2 - 1(1((1(4x - 3))/(-2(4x - 3) - 2))^2 + 3) - 3 intersect in the interval [-10, 10].\n\nPresent t... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(-1e^(1((-1(1(-3x^2 - 4x)^2 + 5(-3x^2 - 4x) + 2) + 3)/(1(1(-3x^2 - 4x)^2 + 5(-3x^2 - 4x) + 2) + 2))) - 1)^2 - 5(-1e^(1((-1(1(-3x^2 - 4x)^2 + 5(-3x^2 - 4x) + 2) + 3)/(1(1(-3x^2 - 4x)^2 + 5(-3x^2 - 4x) + 2) + 2))) - 1) - 2 and... | 350 | MATH |
[
{
"content": "How many solutions does the equation (-2(1(3(3(-2|x - 3| - 3) + 5) + 1) - 3) + 1)/(1(1(3(3(-2|x - 3| - 3) + 5) + 1) - 3) + 2) = -3|(-5(1|(2x - 3)| - 3)^2 - 2(1|(2x - 3)| - 3) - 1) + 3| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(1sin(3π(-1|(-4((-1x - 3)/(-1x + 2)) - 3) - 3| + 3) + 2) - 2)^2 + 2(1sin(3π(-1|(-4((-1x - 3)/(-1x + 2)) - 3) - 3| + 3) + 2) - 2) - 2 and g(x) = -1(1(-1|(-2e^(1x) + 1) + 1| + 3)^2 + 3(-1|(-2e^(1x) + 1) + 1| + 3) + 5)^2 + 2(1(-1|(-2e^(1x) + 1) + 1| + 3... | 32 | MATH |
[
{
"content": "How many solutions does the equation 3(-5(-2cos(3π(-1(1x - 4) + 1) + 3) - 3)^2 - 5(-2cos(3π(-1(1x - 4) + 1) + 3) - 3) + 3) + 3 = 3(-3(2(-1x^2 - 2x + 2) - 3) - 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "How many solutions does the equation -1(-5(1sin(2π(5(5x - 4) - 1) - 3) + 2) + 2)^2 + 1(-5(1sin(2π(5(5x - 4) - 1) - 3) + 2) + 2) + 2 = 5(2(-3sin(3π((-3x + 1)/(1x - 2)) + 3) - 3) + 1) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 575 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(-1cos(3π(-2(4((-2x + 2)/(2x + 2)) + 5) - 5) + 3) - 3)| + 2 and g(x) = (2(-3|((-3(-1x^2 + 1x + 1) - 3)/(3(-1x^2 + 1x + 1) - 2))| + 2) - 3)/(1(-3|((-3(-1x^2 + 1x + 1) - 3)/(3(-1x^2 + 1x + 1) - 2))| + 2)) in the range -10 ≤ x ≤ 10?... | 17 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(-1((2(1(1|x - 2| + 1)^2 - 2) + 3)/(-3(1(1|x - 2| + 1)^2 - 2) - 1))^2 + 5((2(1(1|x - 2| + 1)^2 - 2) + 3)/(-3(1(1|x - 2| + 1)^2 - 2) - 1)) - 4) + 3| + 1 and g(x) = (-3(-4(-2(2x - 1)^2 + 1(2x - 1) + 5)^2 - 3(-2(2x - 1)^2 + 1(... | 4 | MATH |
[
{
"content": "How many solutions does the equation -2|(-1((1((-1(1x + 5) + 2)/(-3(1x + 5) - 3)) + 1)/(3((-1(1x + 5) + 2)/(-3(1x + 5) - 3)) - 1))^2 - 5((1((-1(1x + 5) + 2)/(-3(1x + 5) - 3)) + 1)/(3((-1(1x + 5) + 2)/(-3(1x + 5) - 3)) - 1)) + 2) - 3| - 1 = -1|((-3(-4(1|x - 1| - 1)^2 - 2(1|x - 1| - 1) - 2) + 1)/(1(... | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-1cos(3π(-2((-1(3x^2 + 4x + 5) + 2)/(3(3x^2 + 4x + 5) - 2)) + 1) + 2) + 3) + 3| and g(x) = 1(5((-1((3x - 1)/(-2x - 2)) - 3)/(1((3x - 1)/(-2x - 2)) + 2))^2 + 2((-1((3x - 1)/(-2x - 2)) - 3)/(1((3x - 1)/(-2x - 2)) + 2)) - 5)^2 - 5 ... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1(2sin(1π(-3|(1(-2|x + 2| - 2)^2 + 2(-2|x + 2| - 2) + 3) - 2| - 2) - 1) - 1) - 3 = -4((-1(-1|(-2cos(2πx + 2) - 3) - 3| + 2) + 1)/(1(-1|(-2cos(2πx + 2) - 3) - 3| + 2)))^2 - 2((-1(-1|(-2cos(2πx + 2) - 3) - 3| + 2) + 1)/(1(-1|(-2cos(2πx + 2) - 3) - 3| + 2))) + 1 ... | 1157 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-1(3sin(1π(3|(-2x^2 + 5x + 2) + 3| - 1) + 3) + 2)^2 - 2(3sin(1π(3|(-2x^2 + 5x + 2) + 3| - 1) + 3) + 2) + 5)^2 + 5(-1(3sin(1π(3|(-2x^2 + 5x + 2) + 3| - 1) + 3) + 2)^2 - 2(3sin(1π(3|(-2x^2 + 5x + 2) + 3| - 1) + 3) + 2) + 5) +... | 544 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(3cos(2π(4((1(3x^2 - 1x - 5) + 2)/(1(3x^2 - 1x - 5) - 2))^2 - 2((1(3x^2 - 1x - 5) + 2)/(1(3x^2 - 1x - 5) - 2)) + 3) - 1) - 1)| + 2 and g(x) = 1sin(1π(2|(3|(5x - 3) + 1| - 1) - 3| + 2) - 2) - 3 where -10 ≤ x ≤ 10.\n\nPresen... | 453 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3((1(-3|(-2|(-3sin(2πx - 3) + 1)| - 1) - 3| + 3) - 3)/(-1(-3|(-2|(-3sin(2πx - 3) + 1)| - 1) - 3| + 3) + 3)) - 3)/(1((1(-3|(-2|(-3sin(2πx - 3) + 1)| - 1) - 3| + 3) - 3)/(-1(-3|(-2|(-3sin(2πx - 3) + 1)| - 1) - 3| + 3) + 3)) - 2) an... | 437 | MATH |
[
{
"content": "How many solutions does the equation 2|((1(-3sin(3π(-1|(-1|x - 2| - 1) - 2| + 2) - 3) + 1) - 2)/(2(-3sin(3π(-1|(-1|x - 2| - 1) - 2| + 2) - 3) + 1) - 1))| - 2 = -3|(2|(-3(4x + 2)^2 + 5) - 2| - 2) - 2| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(-2((-1((-2(4x^2 - 4x + 1) - 2)/(1(4x^2 - 4x + 1) + 3)) + 2)/(-3((-2(4x^2 - 4x + 1) - 2)/(1(4x^2 - 4x + 1) + 3)) - 3))^2 - 5((-1((-2(4x^2 - 4x + 1) - 2)/(1(4x^2 - 4x + 1) + 3)) + 2)/(-3((-2(4x^2 - 4x + 1) - 2)/(1(4x^2 - 4x +... | 6 | MATH |
[
{
"content": "How many solutions does the equation -1(-5(2|(-2(1sin(3πx + 2) - 2)^2 - 4(1sin(3πx + 2) - 2) - 5) + 2| + 3) + 3) - 5 = 1(-3|(-2(1|x - 1| - 3)^2 - 1(1|x - 1| - 3) - 1) + 1| + 1)^2 + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "us... | 10 | MATH |
[
{
"content": "How many solutions does the equation -1|((3(-2(-3(-3|x + 1|) + 3)^2 + 2(-3(-3|x + 1|) + 3) + 4) - 1)/(2(-2(-3(-3|x + 1|) + 3)^2 + 2(-3(-3|x + 1|) + 3) + 4) + 2)) + 3| = -1(3|(1sin(2π(-4x^2 - 5x + 2))) - 2| - 3) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxe... | 1371 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(1((2(-4((-3x + 1)/(2x - 1))^2 + 2((-3x + 1)/(2x - 1)) - 2) + 2)/(-1(-4((-3x + 1)/(2x - 1))^2 + 2((-3x + 1)/(2x - 1)) - 2))) + 2)^2 + 4(1((2(-4((-3x + 1)/(2x - 1))^2 + 2((-3x + 1)/(2x - 1)) - 2) + 2)/(-1(-4((-3x + 1)/(2x - ... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2((-3(-5(-2|(1sin(1πx + 1) - 2) - 1| + 2) + 2) - 2)/(2(-5(-2|(1sin(1πx + 1) - 2) - 1| + 2) + 2) - 1)) - 3)/(-1((-3(-5(-2|(1sin(1πx + 1) - 2) - 1| + 2) + 2) - 2)/(2(-5(-2|(1sin(1πx + 1) - 2) - 1| + 2) + 2) - 1))) and g(x) = -1|(-3|(5(-2x - 2) + 5) - 1|... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(4(-1|(3(-3x^2 - 1x + 4) + 4)| + 2)^2 - 4(-1|(3(-3x^2 - 1x + 4) + 4)| + 2) - 4) - 2 and g(x) = 3|((-1(5(-2e^(2x) + 2) - 5) - 3)/(3(5(-2e^(2x) + 2) - 5) - 1)) + 1| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex forma... | 8 | MATH |
[
{
"content": "How many solutions does the equation 2sin(2π(-1(-2(1(2x + 3) - 2) + 5) + 1) + 2) = 2(-2e^(2(2(2x - 2) - 5)))^2 - 1(-2e^(2(2(2x - 2) - 5))) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 97 | MATH |
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