messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(4(-3(-3(-4x^2 - 5x - 4) + 5)^2 + 5(-3(-4x^2 - 5x - 4) + 5) + 1)^2 - 4(-3(-3(-4x^2 - 5x - 4) + 5)^2 + 5(-3(-4x^2 - 5x - 4) + 5) + 1) + 1) + 1)/(-1(4(-3(-3(-4x^2 - 5x - 4) + 5)^2 + 5(-3(-4x^2 - 5x - 4) + 5) + 1)^2 - 4(-3(-3(-4x^2 ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(1(2(-5(-3x^2 + 3x + 3)^2 + 4(-3x^2 + 3x + 3))^2 - 5(-5(-3x^2 + 3x + 3)^2 + 4(-3x^2 + 3x + 3)) - 4) - 2) - 2)/(1(1(2(-5(-3x^2 + 3x + 3)^2 + 4(-3x^2 + 3x + 3))^2 - 5(-5(-3x^2 + 3x + 3)^2 + 4(-3x^2 + 3x + 3)) - 4) - 2) - 3) and g(x) = -2|(3e^(2(-1|(-3... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(1(3|(-2sin(2π((3x)/(-3x - 2)) - 1) + 3)| - 1) + 3) + 2)/(-1(1(3|(-2sin(2π((3x)/(-3x - 2)) - 1) + 3)| - 1) + 3) - 2) and g(x) = -3(3(-5(2|x + 1| - 3)^2 + 4(2|x + 1| - 3) + 2) + 3) - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the ans... | 4 | MATH |
[
{
"content": "How many solutions does the equation 2|(5(-1(3|(1x^2 - 3x + 2) - 1| + 2) + 4)^2 + 3(-1(3|(1x^2 - 3x + 2) - 1| + 2) + 4) - 5) - 3| + 1 = -1((-1(3(-4x - 4) - 3) - 1)/(1(3(-4x - 4) - 3) - 2)) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-2|(3sin(2π(1|(4x^2 + 4x + 5) + 3|) - 2)) + 1| + 3) and g(x) = 1|(1(-3|(3x - 2) + 2| + 2) + 2) - 3| - 2 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 (1(1(5(-2(-2|x| + 2) - 1) + 4) + 3) - 1)/(1(1(5(-2(-2|x| + 2) - 1) + 4) + 3)) = (-2(3(-1|((-2x - 3)/(2x - 1)) + 3| + 2) - 3) - 2)/(2(3(-1|((-2x - 3)/(2x - 1)) + 3| + 2) - 3)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Y... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(-4((-2(2|(1|x + 3| - 2) - 2| - 1) + 2)/(-3(2|(1|x + 3| - 2) - 2| - 1)))^2 + 5((-2(2|(1|x + 3| - 2) - 2| - 1) + 2)/(-3(2|(1|x + 3| - 2) - 2| - 1))) - 4) - 3 and g(x) = -3((-2(1((3x - 3)/(-1x - 1)) - 3) + 1)/(-1(1((3x - 3)/(-... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(3(3cos(2π(-1(2|x + 1|)^2 + 5(2|x + 1|) - 5) - 1) + 2)^2 - 3(3cos(2π(-1(2|x + 1|)^2 + 5(2|x + 1|) - 5) - 1) + 2) - 4) and g(x) = (-1(1|((2(2x^2 - 1x - 2) - 3)/(-2(2x^2 - 1x - 2))) + 1| + 1) + 2)/(-1(1|((2(2x^2 - 1x - 2) - 3)... | 1242 | MATH |
[
{
"content": "How many solutions does the equation -3|(3|(1|(1cos(3π(5x) - 1) - 3) + 2| - 3) - 3|) - 2| - 2 = (-2(-2sin(1π(3((3x + 1)/(-2x - 3))^2 - 2((3x + 1)/(-2x - 3)) + 4) - 1) + 3) + 3)/(1(-2sin(1π(3((3x + 1)/(-2x - 3))^2 - 2((3x + 1)/(-2x - 3)) + 4) - 1) + 3) - 3) have for x in the interval [-10, 10]?\n\n... | 29 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(-1|(1|(-2|(-3|x - 2| + 1)|) + 2| - 1) - 2| - 1)^2 + 4(-1|(1|(-2|(-3|x - 2| + 1)|) + 2| - 1) - 2| - 1) - 4 and g(x) = (-3((1(2sin(2π(2|x + 3| - 1) + 1)) - 2)/(3(2sin(2π(2|x + 3| - 1) + 1)) + 1)) - 1)/(-2((1(2sin(2π(2|x + 3| ... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-1(-3(-1|(-5x - 2) + 2| + 3) - 1))^2 - 5(-1(-3(-1|(-5x - 2) + 2| + 3) - 1)) + 2 and g(x) = (-3((3(3(1|x - 2| - 2) + 3) + 2)/(3(3(1|x - 2| - 2) + 3) - 1)))/(3((3(3(1|x - 2| - 2) + 3) + 2)/(3(3(1|x - 2| - 2) + 3) - 1)) + 2) in the... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-3cos(3π(2|((3(1x^2 - 5x) - 3)/(-1(1x^2 - 5x) - 2)) - 2| + 2) + 2) + 3) - 2)/(-3(-3cos(3π(2|((3(1x^2 - 5x) - 3)/(-1(1x^2 - 5x) - 2)) - 2| + 2) + 2) + 3) - 1) and g(x) = -1(2((-2(2|x| + 2) + 3)/(-2(2|x| + 2) - 3)) - 5) - 2 in th... | 72 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(1(1|(1((1x - 2)/(2x))^2 + 2((1x - 2)/(2x)) - 4) + 2| + 1)) + 1| - 2 and g(x) = (2(-2(-1|(1sin(1πx - 1) + 3) - 2|)^2 + 4(-1|(1sin(1πx - 1) + 3) - 2|) - 1))/(-2(-2(-1|(1sin(1πx - 1) + 3) - 2|)^2 + 4(-1|(1sin(1πx - 1) + 3) - 2|) - 1) - 3) intersect in ... | 19 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(5(-1cos(2π(3(5x^2 - 4) + 3) + 3) - 2) - 3)^2 - 4(5(-1cos(2π(3(5x^2 - 4) + 3) + 3) - 2) - 3) - 2 and g(x) = -4(1sin(1π(-1|(3x + 4)| + 3) + 1) - 3)^2 - 1(1sin(1π(-1|(3x + 4)| + 3) + 1) - 3) + 1 where -10 ≤ x ≤ 10.\n\nPresent... | 450 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(2|(-3(4((1x + 2)/(-2x + 1)) - 3)^2 + 5(4((1x + 2)/(-2x + 1)) - 3) + 4) - 1| - 1) - 5 and g(x) = -1sin(2π(-1((-2(5x^2 - 2x + 4))/(2(5x^2 - 2x + 4) + 1)) + 5) - 2) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1(-1|(-1|((-2((-2x)/(2x)) - 1)/(1((-2x)/(2x)) - 2)) - 3| + 2) - 2| + 3)^2 - 3(-1|(-1|((-2((-2x)/(2x)) - 1)/(1((-2x)/(2x)) - 2)) - 3| + 2) - 2| + 3) = 1(-1(-2cos(1π(-1x + 2) - 3) - 3)^2 + 1(-2cos(1π(-1x + 2) - 3) - 3) + 4)^2 - 2(-1(-2cos(1π(-1x + 2) - 3) - 3)^2 ... | 20 | MATH |
[
{
"content": "How many solutions does the equation 1(1(1cos(1π((-1((1x)/(3x + 1)) + 3)/(1((1x)/(3x + 1)) + 2)) - 2) - 3))^2 - 3(1(1cos(1π((-1((1x)/(3x + 1)) + 3)/(1((1x)/(3x + 1)) + 2)) - 2) - 3)) - 2 = 1|(-3(3(-3x^2 + 2) + 2) + 4) + 2| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex fo... | 15 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1((2(5(-2(4x^2 + 4x - 2) - 5)^2 + 3(-2(4x^2 + 4x - 2) - 5) - 2) - 2)/(-3(5(-2(4x^2 + 4x - 2) - 5)^2 + 3(-2(4x^2 + 4x - 2) - 5) - 2) - 3)) + 2)/(-2((2(5(-2(4x^2 + 4x - 2) - 5)^2 + 3(-2(4x^2 + 4x - 2) - 5) - 2) - 2)/(-3(5(-2(4x^2 + ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-2|(2(-3(-3sin(2πx + 2) - 3) - 3)) + 3| - 1)^2 - 2(-2|(2(-3(-3sin(2πx + 2) - 3) - 3)) + 3| - 1) + 2 and g(x) = 5(1|((-2((-1x + 2)/(-1x + 2)) - 3)/(2((-1x + 2)/(-1x + 2)) - 3)) + 2|)^2 + 3(1|((-2((-1x + 2)/(-1x + 2)) - 3)/(2((-1x ... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(2|(3(-4(3e^(1x))^2 + 5(3e^(1x)) + 5) - 3) - 3| - 1) + 2| + 1 and g(x) = -2(1|((2(-5x^2 - 5x - 1) + 1)/(-1(-5x^2 - 5x - 1) - 2)) + 2| - 1) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(1|(-2|(2(2cos(3πx + 2))^2 - 3(2cos(3πx + 2))) + 2| + 2)| - 2)^2 + 5(1|(-2|(2(2cos(3πx + 2))^2 - 3(2cos(3πx + 2))) + 2| + 2)| - 2) - 5 and g(x) = 1sin(2π(-3(-2(-2|x - 3| + 2) - 4)^2 - 5(-2(-2|x - 3| + 2) - 4) - 2) - 1) - 3 w... | 186 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(1(5(-2(-3sin(2πx - 2) + 1)^2 + 4(-3sin(2πx - 2) + 1) + 1) + 3) - 3) + 2)/(2(1(5(-2(-3sin(2πx - 2) + 1)^2 + 4(-3sin(2πx - 2) + 1) + 1) + 3) - 3) - 3) and g(x) = (2(1((1(-2x^2 - 4x) + 3)/(1(-2x^2 - 4x) + 1))^2 - 5) - 1)/(3(1... | 82 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-1(1e^(-1(3(4x)^2 - 1(4x) + 4)) - 1) - 1) + 1| + 3 and g(x) = 5(3sin(1π((-2(5x^2 - 3x - 3) - 2)/(2(5x^2 - 3x - 3) - 1)) + 3) + 3)^2 + 1(3sin(1π((-2(5x^2 - 3x - 3) - 2)/(2(5x^2 - 3x - 3) - 1)) + 3) + 3) + 2 intersect in the interval [-10, 10].\n\nPre... | 28 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(-5((-2(-3(2x - 4) + 4) + 1)/(2(-3(2x - 4) + 4)))^2 - 5((-2(-3(2x - 4) + 4) + 1)/(2(-3(2x - 4) + 4)))) - 3)/(3(-5((-2(-3(2x - 4) + 4) + 1)/(2(-3(2x - 4) + 4)))^2 - 5((-2(-3(2x - 4) + 4) + 1)/(2(-3(2x - 4) + 4))))) and g(x) = 3cos(1π(1(-4(-4x - 4))^2 ... | 164 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1sin(3π(5(1(3|(2|x - 2| + 1)|) - 4) + 3) - 2) - 1 and g(x) = -3cos(2π(5((-1(-3x + 3))/(1(-3x + 3) + 2))^2 + 5((-1(-3x + 3))/(1(-3x + 3) + 2)) + 5) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 460 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(1π(-5((2((-3(4x + 4) - 1)/(2(4x + 4) + 2)) + 2)/(-2((-3(4x + 4) - 1)/(2(4x + 4) + 2)) - 3)) - 3) - 2) and g(x) = (-1(-5(-1((-2x + 2)/(3x + 3)) - 4)^2 + 1) + 1)/(-3(-5(-1((-2x + 2)/(3x + 3)) - 4)^2 + 1) - 1) in the range -10 ≤ ... | 202 | MATH |
[
{
"content": "How many solutions does the equation 2|(2|((3(-3(4x^2 - 3x) - 1) - 3)/(1(-3(4x^2 - 3x) - 1) + 2)) + 3| - 2) - 1| - 3 = 1(3sin(3π(-3(-1|x| - 2)^2 - 5(-1|x| - 2) + 3) + 2) + 2) - 2 have for x in the interval [-10, 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) = -2|(-1(2(-4(3|x + 3| - 3)^2 - 2(3|x + 3| - 3) + 5)^2 - 3(-4(3|x + 3| - 3)^2 - 2(3|x + 3| - 3) + 5) - 4) - 5) + 2| + 1 and g(x) = -1|(-2|(-1|(1x + 5) + 1| - 2) + 1| - 1) + 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format... | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1((-3((1(-2cos(3π(-3x - 4) - 2) - 1) + 3)/(3(-2cos(3π(-3x - 4) - 2) - 1) + 2)) + 3)/(-2((1(-2cos(3π(-3x - 4) - 2) - 1) + 3)/(3(-2cos(3π(-3x - 4) - 2) - 1) + 2)) + 1)) and g(x) = (3(-1(4(3sin(3πx + 2) - 1) - 5)^2 + 4(4(3sin(3π... | 180 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(1π(3|(-1|(2(-5x + 3) - 5) - 1| - 2) + 2| - 1) + 3) + 3 and g(x) = 2|(-3|((3(4x^2 - 3x - 4) + 1)/(3(4x^2 - 3x - 4) - 1)) + 1| + 2)| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(2e^(1(-3|((-1(-1x - 2) - 1)/(-2(-1x - 2) + 3)) - 3| + 2)) - 3) - 2)/(-2(2e^(1(-3|((-1(-1x - 2) - 1)/(-2(-1x - 2) + 3)) - 3| + 2)) - 3) + 3) and g(x) = 1(2cos(2π(2|(-3|x|)| - 2) - 1) + 1)^2 - 4(2cos(2π(2|(-3|x|)| - 2) - 1) + 1) + 2 intersect in the ... | 239 | MATH |
[
{
"content": "How many solutions does the equation 5(-3|((-3(3|((-3x)/(1x + 1))| - 2) - 1)/(-3(3|((-3x)/(1x + 1))| - 2))) + 1| + 3) + 5 = 2(-4(-1|(-2|x + 2| + 2) + 3|)^2 - 1(-1|(-2|x + 2| + 2) + 3|) + 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 8 | MATH |
[
{
"content": "How many solutions does the equation 3|(-1(-5(-1(-5x + 2)))^2 - 1(-5(-1(-5x + 2))) - 1)| + 3 = 1((-3(1(3sin(2πx - 1) + 2)^2 + 4(3sin(2πx - 1) + 2) + 5) - 2)/(-1(1(3sin(2πx - 1) + 2)^2 + 4(3sin(2πx - 1) + 2) + 5) + 3))^2 + 2((-3(1(3sin(2πx - 1) + 2)^2 + 4(3sin(2πx - 1) + 2) + 5) - 2)/(-1(1(3sin(2πx... | 42 | MATH |
[
{
"content": "How many solutions does the equation (-3(3e^(-2(-2(4(-3x + 3)^2 - 5(-3x + 3) - 3) - 2)) - 2))/(-3(3e^(-2(-2(4(-3x + 3)^2 - 5(-3x + 3) - 3) - 2)) - 2) - 1) = 2(4(-2|(2cos(1πx + 2) - 3) + 1|)^2 + 2(-2|(2cos(1πx + 2) - 3) + 1|) - 3) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in L... | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(-4(-1(-2|(2|x - 2| - 2) - 2| - 1) - 3)^2 + 1(-1(-2|(2|x - 2| - 2) - 2| - 1) - 3) + 3) + 1| - 2 and g(x) = -2((-2(1(2sin(3πx + 1) + 1) + 5) - 1)/(1(1(2sin(3πx + 1) + 1) + 5) + 3)) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answ... | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(-1(2sin(1π(-3|(4x^2 + 3x - 3) + 1| + 2)) + 2)^2 + 3(2sin(1π(-3|(4x^2 + 3x - 3) + 1| + 2)) + 2) + 4) + 4 and g(x) = -2e^(-2((-1((2(-3|x - 2| - 3))/(-3(-3|x - 2| - 3) + 1)))/(2((2(-3|x - 2| - 3))/(-3(-3|x - 2| - 3) + 1)) - 1... | 1023 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(3cos(3π(3|(-1|((-3x - 3)/(-1x + 2)) + 3| + 2) - 2| + 1) + 1) + 2) - 2 and g(x) = (1(3|(2|(1|x - 2| + 2)| - 2) - 3| + 1) - 3)/(-3(3|(2|(1|x - 2| + 2)| - 2) - 3| + 1) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\bo... | 295 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(-3|(-5(-1(2cos(3πx + 3) + 1) + 1) + 3) + 3|) + 3| + 2 and g(x) = 3|(3sin(3π(-5((2x - 1)/(-2x + 3))^2 - 1((2x - 1)/(-2x + 3)) - 2) - 3) - 1)| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 74 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(-3(2|(-5(1|x - 3| - 1)^2 + 2) + 2| - 3)^2 + 4)^2 + 5(-3(2|(-5(1|x - 3| - 1)^2 + 2) + 2| - 3)^2 + 4) + 4 and g(x) = -2|(-4(-2|(-4x^2 + 3) + 2| + 1) + 3)| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxe... | 18 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(3|(-1|(5(3e^(2x) - 1) - 3) - 2| + 3) - 2| + 2) + 3 and g(x) = 2sin(3π((2((-1(1|x - 1| - 3) - 2)/(2(1|x - 1| - 3) + 2)) - 1)/(-2((-1(1|x - 1| - 3) - 2)/(2(1|x - 1| - 3) + 2)))) - 2) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in La... | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(2(1(-1|(3x - 1) - 1|) + 2) + 4) - 1| - 2 and g(x) = -3|((-1(-2|((1x + 2)/(-1x - 2)) + 2| - 2) + 1)/(2(-2|((1x + 2)/(-1x - 2)) + 2| - 2) + 3)) - 2| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{You... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(5(-3(-2sin(3π(-2x^2 + 1x + 4) + 3) + 3) + 4)^2 + 3(-3(-2sin(3π(-2x^2 + 1x + 4) + 3) + 3) + 4) - 1)^2 + 5(5(-3(-2sin(3π(-2x^2 + 1x + 4) + 3) + 3) + 4)^2 + 3(-3(-2sin(3π(-2x^2 + 1x + 4) + 3) + 3) + 4) - 1) and g(x) = -3(2|(-2(-1|x... | 581 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(2cos(2π(-3|((-2(-4x^2 + 2x + 5) + 1)/(1(-4x^2 + 2x + 5) + 3)) - 3| + 2) + 1)) + 3)/(-3(2cos(2π(-3|((-2(-4x^2 + 2x + 5) + 1)/(1(-4x^2 + 2x + 5) + 3)) - 3| + 2) + 1)) + 2) and g(x) = -1|(2(-3(-1x + 1) - 1)^2 - 2(-3(-1x + 1)... | 25 | MATH |
[
{
"content": "How many solutions does the equation -2((-2(2cos(1π(2(-3x^2 + 5x - 2)^2 - 2(-3x^2 + 5x - 2) - 2) + 3) - 1) - 2)/(-2(2cos(1π(2(-3x^2 + 5x - 2)^2 - 2(-3x^2 + 5x - 2) - 2) + 3) - 1) + 3))^2 + 4((-2(2cos(1π(2(-3x^2 + 5x - 2)^2 - 2(-3x^2 + 5x - 2) - 2) + 3) - 1) - 2)/(-2(2cos(1π(2(-3x^2 + 5x - 2)^2 - 2... | 720 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(-5((1(1sin(2π(5x + 2) - 1) + 2))/(-2(1sin(2π(5x + 2) - 1) + 2) - 1)) - 3) - 1| - 3 and g(x) = 1(2(-2sin(1π((2x)/(2x - 3)) - 2) - 3) - 2) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 67 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4((-2(2cos(2π(-2(-5x + 5)^2 - 4(-5x + 5) - 2) - 1) - 3) + 3)/(-1(2cos(2π(-2(-5x + 5)^2 - 4(-5x + 5) - 2) - 1) - 3) + 2))^2 + 1((-2(2cos(2π(-2(-5x + 5)^2 - 4(-5x + 5) - 2) - 1) - 3) + 3)/(-1(2cos(2π(-2(-5x + 5)^2 - 4(-5x + 5) - 2) - 1) - 3) + 2)) - 2 a... | 463 | MATH |
[
{
"content": "How many solutions does the equation 3(-5(2|(-5(3sin(2πx + 2) - 2)^2 + 1(3sin(2πx + 2) - 2)) + 2| - 1)^2 + 2(2|(-5(3sin(2πx + 2) - 2)^2 + 1(3sin(2πx + 2) - 2)) + 2| - 1) + 2) - 1 = 1|((3(2(3x^2)^2 - 1(3x^2) + 4) + 1)/(-2(2(3x^2)^2 - 1(3x^2) + 4) - 3)) + 2| have for x in the interval [-10, 10]?\n\n... | 278 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-1(2sin(3π((2((1x + 2)/(3x)))/(1((1x + 2)/(3x)) + 1)) - 1) - 3)^2 + 4(2sin(3π((2((1x + 2)/(3x)))/(1((1x + 2)/(3x)) + 1)) - 1) - 3) - 4) - 2)/(2(-1(2sin(3π((2((1x + 2)/(3x)))/(1((1x + 2)/(3x)) + 1)) - 1) - 3)^2 + 4(2sin(3π((2((1x + 2)/(3x)))/(1((1x ... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(1cos(3π(2|((-2((-1x + 1)/(2x - 1)) - 3)/(2((-1x + 1)/(2x - 1)) + 2)) - 3| + 1) - 3) - 2)^2 - 1(1cos(3π(2|((-2((-1x + 1)/(2x - 1)) - 3)/(2((-1x + 1)/(2x - 1)) + 2)) - 3| + 1) - 3) - 2) + 1 and g(x) = 3|(-3sin(2π(3(4x - 5)^2 + 1(4x - 5) - 1) + 1)) - 2|... | 1308 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((-2(2|(3(5x^2 + 4x - 2) + 4) - 1| - 2) - 3)/(-1(2|(3(5x^2 + 4x - 2) + 4) - 1| - 2) + 3)) and g(x) = 4(-1(4(-1sin(1πx - 1) - 1)^2 + 5(-1sin(1πx - 1) - 1) - 5)^2 - 3(4(-1sin(1πx - 1) - 1)^2 + 5(-1sin(1πx - 1) - 1) - 5)) + 2 intersect in the interval [-... | 41 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(3(2|((-1(-1|x + 2| + 2) + 1)/(2(-1|x + 2| + 2) + 2)) - 1| + 1)^2 - 3(2|((-1(-1|x + 2| + 2) + 1)/(2(-1|x + 2| + 2) + 2)) - 1| + 1) + 2) + 1)/(-1(3(2|((-1(-1|x + 2| + 2) + 1)/(2(-1|x + 2| + 2) + 2)) - 1| + 1)^2 - 3(2|((-1(-1... | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(1cos(2π(-1(-2|(-4x + 2) + 1|) - 2) - 2) - 3) - 2 and g(x) = 3cos(1π(3|(3(-2|x + 1|) - 1) + 1| - 2) + 3) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 360 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1((1(2e^(1((2(-5x - 2) + 2)/(-1(-5x - 2) + 2)))) - 2)/(2(2e^(1((2(-5x - 2) + 2)/(-1(-5x - 2) + 2)))) - 1)) - 2)/(1((1(2e^(1((2(-5x - 2) + 2)/(-1(-5x - 2) + 2)))) - 2)/(2(2e^(1((2(-5x - 2) + 2)/(-1(-5x - 2) + 2)))) - 1))) and g(x) = 2(-3|(-3|(5x^2 - 3... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(-1cos(1π(2|(3|(-3x^2 + 4x - 3)| + 3) + 2| + 2) + 1) + 2) + 1| - 3 and g(x) = 2|((2(2(2x + 1)^2 - 1(2x + 1) + 2) + 3)/(1(2(2x + 1)^2 - 1(2x + 1) + 2) - 2))| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your ... | 419 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5((2((-3((-3(5x^2 - 5x - 4))/(-3(5x^2 - 5x - 4) + 1)) + 2)/(1((-3(5x^2 - 5x - 4))/(-3(5x^2 - 5x - 4) + 1)) - 2)) - 1)/(-1((-3((-3(5x^2 - 5x - 4))/(-3(5x^2 - 5x - 4) + 1)) + 2)/(1((-3(5x^2 - 5x - 4))/(-3(5x^2 - 5x - 4) + 1)) - 2)) ... | 474 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(2|(-4(1cos(2π((2x)/(-1x + 3)) - 2) + 3)^2 - 1(1cos(2π((2x)/(-1x + 3)) - 2) + 3) - 5) + 1| - 3) + 3)/(-1(2|(-4(1cos(2π((2x)/(-1x + 3)) - 2) + 3)^2 - 1(1cos(2π((2x)/(-1x + 3)) - 2) + 3) - 5) + 1| - 3) - 2) and g(x) = -2(3((2(4x^2... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1((-3((1(-4(3|x - 1| - 3)^2 - 4(3|x - 1| - 3) + 2) + 3)/(1(-4(3|x - 1| - 3)^2 - 4(3|x - 1| - 3) + 2) - 2)) + 3)/(-2((1(-4(3|x - 1| - 3)^2 - 4(3|x - 1| - 3) + 2) + 3)/(1(-4(3|x - 1| - 3)^2 - 4(3|x - 1| - 3) + 2) - 2))))^2 - 4 and g(... | 31 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(-2(1|(-5((1x - 2)/(-2x + 3)) + 2) + 3| - 3) + 2)^2 + 4(-2(1|(-5((1x - 2)/(-2x + 3)) + 2) + 3| - 3) + 2) and g(x) = 3|(-2(-2(-2x^2 - 4x) - 4) + 1) + 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(4(-1(-3(1x - 5))^2 + 2(-3(1x - 5)) - 3)^2 - 3(-1(-3(1x - 5))^2 + 2(-3(1x - 5)) - 3) + 3) - 2)/(-2(4(-1(-3(1x - 5))^2 + 2(-3(1x - 5)) - 3)^2 - 3(-1(-3(1x - 5))^2 + 2(-3(1x - 5)) - 3) + 3) + 3) and g(x) = 2|(-2sin(3π(-1|(-1x + 2) + 1| - 2) + 2) - 3) ... | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(3sin(3π(-5((-2(1|x + 2| + 1) - 2)/(2(1|x + 2| + 1) - 1))^2 - 4((-2(1|x + 2| + 1) - 2)/(2(1|x + 2| + 1) - 1)) - 3) - 1) + 1) - 1)/(-2(3sin(3π(-5((-2(1|x + 2| + 1) - 2)/(2(1|x + 2| + 1) - 1))^2 - 4((-2(1|x + 2| + 1) - 2)/(2(1|x + 2| + 1) - 1)) - 3) -... | 114 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2e^(1(-1((-2((-2(2x + 5) - 2)/(2(2x + 5) - 1)) + 3)/(3((-2(2x + 5) - 2)/(2(2x + 5) - 1)) + 3)) - 1)) and g(x) = (-1((3(-2cos(2π(-1x + 5) - 2)) - 2)/(-3(-2cos(2π(-1x + 5) - 2)) - 1)) - 2)/(1((3(-2cos(2π(-1x + 5) - 2)) - 2)/(-3(-2cos(2π(-1x + 5) - 2)) - ... | 18 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(1π(1(-1(1((2x - 3)/(1x + 1))^2 - 4((2x - 3)/(1x + 1)) + 1)^2 - 3(1((2x - 3)/(1x + 1))^2 - 4((2x - 3)/(1x + 1)) + 1) - 2) - 4) - 2) and g(x) = -2|(-1((3(-3|x + 1|))/(2(-3|x + 1|) + 3))^2 + 5((3(-3|x + 1|))/(2(-3|x + 1|) + 3)) ... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-2(-2(-3(-1x - 4)^2 - 3(-1x - 4) - 5)^2 - 4(-3(-1x - 4)^2 - 3(-1x - 4) - 5) - 5) - 5)| + 2 and g(x) = -3(-3|(-1e^(1(-3|x + 1| + 3))) + 3| - 2)^2 + 2(-3|(-1e^(1(-3|x + 1| + 3))) + 3| - 2) + 2 where -10 ≤ x ≤ 10.\n\nPresent... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|((2(-3(-1|(-1|x + 3| + 3) - 1| + 1)^2 + 1(-1|(-1|x + 3| + 3) - 1| + 1) + 2) - 2)/(1(-3(-1|(-1|x + 3| + 3) - 1| + 1)^2 + 1(-1|(-1|x + 3| + 3) - 1| + 1) + 2) + 1)) - 1| + 2 and g(x) = 3((2((-3((3x - 2)/(-1x - 3)) + 1)/(-1((3x - 2)/(-1x - 3)))) + 2)/(2(... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(2sin(3π(-2|(-2(-2x^2 - 2x + 5)^2 - 4(-2x^2 - 2x + 5)) - 2|)) + 1)^2 + 4(2sin(3π(-2|(-2(-2x^2 - 2x + 5)^2 - 4(-2x^2 - 2x + 5)) - 2|)) + 1) - 1 and g(x) = 4((-2(-5(-2|x| - 2) + 5) + 2)/(-2(-5(-2|x| - 2) + 5)))^2 - 2((-2(-5(-2... | 756 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((-2((-3((-3(-5x^2 - 5) - 1)/(-3(-5x^2 - 5) + 1)) - 2)/(-1((-3(-5x^2 - 5) - 1)/(-3(-5x^2 - 5) + 1)))) + 2)/(1((-3((-3(-5x^2 - 5) - 1)/(-3(-5x^2 - 5) + 1)) - 2)/(-1((-3(-5x^2 - 5) - 1)/(-3(-5x^2 - 5) + 1)))) + 3))^2 - 5((-2((-3((-3(-5x^2 - 5) - 1)/(-3... | 913 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(2|(-1sin(1π(-4(-5x^2 - 3x - 2) + 1) - 2) - 1) - 1| - 3) - 1)/(1(2|(-1sin(1π(-4(-5x^2 - 3x - 2) + 1) - 2) - 1) - 1| - 3) - 3) and g(x) = -1(-3(-2cos(2π((-3x - 2)/(1x)) + 3) - 2) - 2)^2 + 1(-3(-2cos(2π((-3x - 2)/(1x)) + 3) ... | 333 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|((3(-3(-1e^(-1(3x^2 + 5x + 2)) - 3)^2 + 2(-1e^(-1(3x^2 + 5x + 2)) - 3) - 4))/(1(-3(-1e^(-1(3x^2 + 5x + 2)) - 3)^2 + 2(-1e^(-1(3x^2 + 5x + 2)) - 3) - 4) + 3)) - 1| - 1 and g(x) = -3(-1((1(-4x^2 - 1x + 5) + 2)/(3(-4x^2 - 1x + 5) + 2))^2 + 1((1(-4x^2 -... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(1|(-2|(-2e^(-1(2|x - 1| - 3)) + 3) + 1| + 2) + 1| + 2)^2 - 2(1|(-2|(-2e^(-1(2|x - 1| - 3)) + 3) + 1| + 2) + 1| + 2) + 2 and g(x) = -2|(-3(1((3x - 3)/(1x + 3)) + 2) - 1) - 1| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in... | 6 | MATH |
[
{
"content": "How many solutions does the equation -2|(-1cos(2π(1|((2(5x - 3) - 2)/(-3(5x - 3))) + 3| + 3) + 1) + 2) - 1| - 2 = -2|((1(-1(-2x^2 - 4x) - 5) - 3)/(-1(-1(-2x^2 - 4x) - 5))) + 3| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 4 | MATH |
[
{
"content": "How many solutions does the equation (-1(4(-2|(1|(1cos(3πx + 2)) + 1| + 2) + 3| + 2)^2 - 3) + 2)/(-2(4(-2|(1|(1cos(3πx + 2)) + 1| + 2) + 3| + 2)^2 - 3) + 3) = 1((1((-3(2x^2 - 5x - 4) + 3)/(2(2x^2 - 5x - 4))))/(2((-3(2x^2 - 5x - 4) + 3)/(2(2x^2 - 5x - 4))) - 1))^2 - 3((1((-3(2x^2 - 5x - 4) + 3)/(2(... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(5(-3((-1((3x - 1)/(-2x + 3)) + 2)/(-2((3x - 1)/(-2x + 3)) + 3)) + 3)^2 + 2(-3((-1((3x - 1)/(-2x + 3)) + 2)/(-2((3x - 1)/(-2x + 3)) + 3)) + 3) - 4)^2 + 4(5(-3((-1((3x - 1)/(-2x + 3)) + 2)/(-2((3x - 1)/(-2x + 3)) + 3)) + 3)^2 + 2(-3((-1((3x - 1)/(-2x ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2((-1(-5(3|(2|x + 3| + 2) + 3| - 2)^2 - 2(3|(2|x + 3| + 2) + 3| - 2) - 3) + 3)/(3(-5(3|(2|x + 3| + 2) + 3| - 2)^2 - 2(3|(2|x + 3| + 2) + 3| - 2) - 3) - 2)) + 1)/(1((-1(-5(3|(2|x + 3| + 2) + 3| - 2)^2 - 2(3|(2|x + 3| + 2) + 3| - 2) - 3) + 3)/(3(-5(3|(... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(1((3(1sin(3π(3|x + 3| - 2) - 1) + 1) + 3)/(-2(1sin(3π(3|x + 3| - 2) - 1) + 1) + 1))^2 + 3((3(1sin(3π(3|x + 3| - 2) - 1) + 1) + 3)/(-2(1sin(3π(3|x + 3| - 2) - 1) + 1) + 1)) - 1) - 3)/(-3(1((3(1sin(3π(3|x + 3| - 2) - 1) + 1) + 3)/(-2(1sin(3π(3|x + 3| ... | 180 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(1(-1cos(1π(-1(1x - 5)^2 - 4(1x - 5) - 2) - 2))) - 2)/(2(1(-1cos(1π(-1(1x - 5)^2 - 4(1x - 5) - 2) - 2))) - 1) and g(x) = 2|(1((3(2x^2 + 3x + 2) + 2)/(2(2x^2 + 3x + 2) - 1)) + 3) - 2| + 3 where -10 ≤ x ≤ 10.\n\nPresent the ... | 192 | MATH |
[
{
"content": "How many solutions does the equation 2sin(1π(-1|(-3|(3|(-1|x - 1| - 3) - 2| - 2) - 2| + 3)| + 1)) = (-1((2(4(3|x| - 2) - 3) + 1)/(-2(4(3|x| - 2) - 3) - 2)) + 3)/(-3((2(4(3|x| - 2) - 3) + 1)/(-2(4(3|x| - 2) - 3) - 2)) + 4) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format:... | 184 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5((2(-4(4(-2cos(3πx - 1) - 2)^2 + 1(-2cos(3πx - 1) - 2) - 4) - 3) + 2)/(1(-4(4(-2cos(3πx - 1) - 2)^2 + 1(-2cos(3πx - 1) - 2) - 4) - 3) - 3))^2 + 4((2(-4(4(-2cos(3πx - 1) - 2)^2 + 1(-2cos(3πx - 1) - 2) - 4) - 3) + 2)/(1(-4(4(... | 108 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|((1((1(-3(-2x^2 - 3x - 5)^2 + 4(-2x^2 - 3x - 5) - 2) + 2)/(-3(-3(-2x^2 - 3x - 5)^2 + 4(-2x^2 - 3x - 5) - 2) + 4)))/(1((1(-3(-2x^2 - 3x - 5)^2 + 4(-2x^2 - 3x - 5) - 2) + 2)/(-3(-3(-2x^2 - 3x - 5)^2 + 4(-2x^2 - 3x - 5) - 2) + 4)) + 2)) - 1| - 1 and g(x... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-3|(-5(2|(-3cos(3πx + 1) - 2) - 3| + 3) + 4)|) - 3)/(3(-3|(-5(2|(-3cos(3πx + 1) - 2) - 3| + 3) + 4)|) - 2) and g(x) = -4(1sin(3π(-5(-2|x - 2| + 1)^2 - 5(-2|x - 2| + 1) - 4) + 3) - 1) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in... | 1149 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-3cos(3π(-3|(4(-1x^2 - 4)) + 1| - 3) - 2) - 2)^2 + 3(-3cos(3π(-3|(4(-1x^2 - 4)) + 1| - 3) - 2) - 2) and g(x) = (-3(1e^(-1(-1|(1|x - 3| - 1) + 3| - 1)) + 3) + 3)/(-2(1e^(-1(-1|(1|x - 3| - 1) + 3| - 1)) + 3) + 3) intersect in the interval [-10, 10].\n\... | 1580 | MATH |
[
{
"content": "How many solutions does the equation -3(3e^(-1(-4(-4((-2x + 2)/(1x - 3))^2 + 5((-2x + 2)/(1x - 3)) + 4)^2 + 2(-4((-2x + 2)/(1x - 3))^2 + 5((-2x + 2)/(1x - 3)) + 4) + 4)) - 3) = -1(-1sin(3π(-3((1x + 3)/(3x)) - 3) + 1) + 2)^2 + 2(-1sin(3π(-3((1x + 3)/(3x)) - 3) + 1) + 2) + 1 have for x in the interv... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3((3(-2(-4(-3x^2 + 5x + 2)^2 + 5(-3x^2 + 5x + 2) - 4) - 2) + 2)/(2(-2(-4(-3x^2 + 5x + 2)^2 + 5(-3x^2 + 5x + 2) - 4) - 2) - 1))^2 + 3((3(-2(-4(-3x^2 + 5x + 2)^2 + 5(-3x^2 + 5x + 2) - 4) - 2) + 2)/(2(-2(-4(-3x^2 + 5x + 2)^2 + 5(-3x^2 + 5x + 2) - 4) - 2)... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2((1((2(-1(3x^2 + 4x) + 5) - 1)/(2(-1(3x^2 + 4x) + 5) - 3)) + 2)/(-3((2(-1(3x^2 + 4x) + 5) - 1)/(2(-1(3x^2 + 4x) + 5) - 3)) + 2)) + 3 and g(x) = -3sin(2π(3|(-4(-3x^2 + 4x + 1)^2 - 2(-3x^2 + 4x + 1) + 1)| + 2)) where -10 ≤ x ... | 17 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(-2sin(3π(4(-3|(-2|x| - 1) + 1| - 1) + 2) + 2)) + 1| and g(x) = -1|((3(1e^(-2(5x^2 - 2x)) + 1))/(-3(1e^(-2(5x^2 - 2x)) + 1) + 4)) - 3| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 719 | MATH |
[
{
"content": "How many solutions does the equation 1(3|(3|(-4(-4x^2 - 2x + 2))|) - 2| + 2)^2 - 5(3|(3|(-4(-4x^2 - 2x + 2))|) - 2| + 2) = (-3(-5(-2|(-1e^(1x) + 2) + 3|) - 5) - 2)/(3(-5(-2|(-1e^(1x) + 2) + 3|) - 5) - 1) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-3sin(3π(4(3(-5x + 4) - 4) + 1) + 2) - 2) + 2| - 2 and g(x) = -1(5(3cos(2π(2|x - 3| + 3) + 3) - 1) - 5) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 111 | MATH |
[
{
"content": "How many solutions does the equation -3(-3|(-5(-1sin(1π(-2|x - 1| + 2) - 2) + 1) + 3) - 1| + 1) - 2 = -4(-1|((2(-1x^2 - 2x - 4) + 2)/(-2(-1x^2 - 2x - 4) - 1))| - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-2cos(2π(1(4(1|x| - 1)^2 + 5(1|x| - 1) + 3)^2 - 5(4(1|x| - 1)^2 + 5(1|x| - 1) + 3) - 4)) + 3) + 1| - 2 and g(x) = -1|((-2(2sin(1π(-1x^2 - 5x) + 2) - 1))/(1(2sin(1π(-1x^2 - 5x) + 2) - 1) + 3)) + 2| + 1 in the range -10 ≤ x ≤ 10?... | 105 | MATH |
[
{
"content": "How many solutions does the equation 3(2|(-1(2((2x)/(-1x - 1))^2 - 2((2x)/(-1x - 1)) - 4) - 1) + 3| - 1) - 2 = 1e^(2((-2((2(-3|x + 2| - 3) - 2)/(2(-3|x + 2| - 3) - 1)) - 1)/(-2((2(-3|x + 2| - 3) - 2)/(2(-3|x + 2| - 3) - 1)) + 3))) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-1(-3|(2|((-1x)/(-2x - 2)) - 1| - 1)| - 1)^2 - 5(-3|(2|((-1x)/(-2x - 2)) - 1| - 1)| - 1) + 5) + 1 and g(x) = -4(5(2cos(1π((1x - 1)/(1x + 1)) + 2) - 3) - 2)^2 - 2(5(2cos(1π((1x - 1)/(1x + 1)) + 2) - 3) - 2) - 4 in the range -10 ≤ ... | 17 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1cos(2π((3(-4((1((-1x - 3)/(2x)) - 3)/(-2((-1x - 3)/(2x)) - 1))^2 - 3((1((-1x - 3)/(2x)) - 3)/(-2((-1x - 3)/(2x)) - 1)) + 5) + 3)/(-3(-4((1((-1x - 3)/(2x)) - 3)/(-2((-1x - 3)/(2x)) - 1))^2 - 3((1((-1x - 3)/(2x)) - 3)/(-2((-1x - 3)... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(2cos(1π(5(-1(-5x + 2) + 5) + 1) - 1) - 1) + 1| - 2 and g(x) = 4(-1|((-1(2|x - 3| - 2) - 2)/(-1(2|x - 3| - 2) - 1)) + 3| - 1) + 5 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(-3cos(2π((2((2((2x - 1)/(1x + 2)) - 1)/(1((2x - 1)/(1x + 2)))))/(-2((2((2x - 1)/(1x + 2)) - 1)/(1((2x - 1)/(1x + 2)))) + 1)) - 3) + 3) + 4 and g(x) = (-2(1((1(-3x + 4) + 1)/(-1(-3x + 4) - 3)) - 3))/(3(1((1(-3x + 4) + 1)/(-... | 38 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2sin(3π(-4(2(-1((-1x + 1)/(-3x + 4)) + 4)^2 - 5) + 4) + 2) + 2 and g(x) = (2(5(-2e^(-1(3x^2 + 2x - 3)) - 1) - 1) - 3)/(3(5(-2e^(-1(3x^2 + 2x - 3)) - 1) - 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 150 | MATH |
[
{
"content": "How many solutions does the equation -2(-1(3|((-1(1x^2 + 4x + 2) - 3)/(-2(1x^2 + 4x + 2) - 2)) - 1| - 2) + 3)^2 - 4(-1(3|((-1(1x^2 + 4x + 2) - 3)/(-2(1x^2 + 4x + 2) - 2)) - 1| - 2) + 3) = (-1(-1(2(2sin(2πx - 3) - 3)^2 - 5(2sin(2πx - 3) - 3) - 1)^2 + 4(2(2sin(2πx - 3) - 3)^2 - 5(2sin(2πx - 3) - 3) ... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(5(-5(-3sin(2π(2x + 2) - 3))^2 + 2)^2 + 3(-5(-3sin(2π(2x + 2) - 3))^2 + 2) - 2) + 1)/(-3(5(-5(-3sin(2π(2x + 2) - 3))^2 + 2)^2 + 3(-5(-3sin(2π(2x + 2) - 3))^2 + 2) - 2) - 1) and g(x) = -3(-3(-1((-1x + 3)/(3x - 1)) + 2)^2 - 2(-1((... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(1|(2(1(1|x - 3| - 1)^2 + 5(1|x - 3| - 1)) - 1) + 3| - 2)^2 - 1(1|(2(1(1|x - 3| - 1)^2 + 5(1|x - 3| - 1)) - 1) + 3| - 2) - 3 and g(x) = (2(-2|(-3|(-2sin(2πx + 3) + 1) - 2| + 2)| + 3) - 3)/(-3(-2|(-3|(-2sin(2πx + 3) + 1) - 2... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1e^(-2(-1(-2|((1((-1x - 1)/(-2x + 3)) - 1)/(-2((-1x - 1)/(-2x + 3)) - 1)) - 3| + 3) + 2)) + 2 and g(x) = (-2(-2(-1cos(3π((-1x + 1)/(1x - 3))) + 1)^2 + 2(-1cos(3π((-1x + 1)/(1x - 3))) + 1) + 1) + 1)/(3(-2(-1cos(3π((-1x + 1)/(1x - 3))) + 1)^2 + 2(-1cos(... | 28 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-2|(1(-4(3cos(2πx - 1) - 1)^2 - 4(3cos(2πx - 1) - 1) + 1) + 3) - 1|) + 3)/(-2(-2|(1(-4(3cos(2πx - 1) - 1)^2 - 4(3cos(2πx - 1) - 1) + 1) + 3) - 1|) - 2) and g(x) = (-3(-5(2(1|x - 3| + 1)^2 - 4(1|x - 3| + 1) - 3) + 1) + 2)/(-2(-5(2(1|x - 3| + 1)^2 - 4... | 45 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(1|((-1(2|(3x - 2) + 1| - 3) + 3)/(-1(2|(3x - 2) + 1| - 3) + 2)) - 1| - 3) - 4 and g(x) = 4((2(-1(2|x - 3|) + 2) - 2)/(2(-1(2|x - 3|) + 2) - 1))^2 + 3((2(-1(2|x - 3|) + 2) - 2)/(2(-1(2|x - 3|) + 2) - 1)) - 4 intersect in the interval [-10, 10].\n\nPr... | 8 | MATH |
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