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) = (-1(-1((3((2((-2x - 1)/(1x)) + 1)/(3((-2x - 1)/(1x)) + 2)))/(-3((2((-2x - 1)/(1x)) + 1)/(3((-2x - 1)/(1x)) + 2)) + 1)) - 4) + 2)/(2(-1((3((2((-2x - 1)/(1x)) + 1)/(3((-2x - 1)/(1x)) + 2)))/(-3((2((-2x - 1)/(1x)) + 1)/(3((-2x - 1)/(1... | 55 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|((-1((3(4(-1x^2 - 1x + 3) - 5) - 2)/(-3(4(-1x^2 - 1x + 3) - 5) + 1)) - 2)/(3((3(4(-1x^2 - 1x + 3) - 5) - 2)/(-3(4(-1x^2 - 1x + 3) - 5) + 1)))) + 1| + 1 and g(x) = 1(2e^(-1((1(2x + 5))/(3(2x + 5) - 1))) + 1)^2 - 4(2e^(-1((1(2x + ... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3((2(-1(-1|(-3|x - 1| - 3) + 1|) - 4) + 2)/(3(-1(-1|(-3|x - 1| - 3) + 1|) - 4) - 2))^2 + 1 and g(x) = (2(1cos(2π(-3(-1|x - 3| + 1)^2 - 5(-1|x - 3| + 1) + 1) - 1)) + 2)/(2(1cos(2π(-3(-1|x - 3| + 1)^2 - 5(-1|x - 3| + 1) + 1) - 1)) - 3) intersect in the ... | 875 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((-3(-3|(-3((2x + 1)/(1x)) + 3) + 3| + 1) - 2)/(-3(-3|(-3((2x + 1)/(1x)) + 3) + 3| + 1) + 1))^2 + 3((-3(-3|(-3((2x + 1)/(1x)) + 3) + 3| + 1) - 2)/(-3(-3|(-3((2x + 1)/(1x)) + 3) + 3| + 1) + 1)) + 3 and g(x) = 1|(1(-3(-2x^2 + 3) + 5) - 2)| + 1 intersect... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(-2|(-2(-1(1cos(2πx - 2)) + 2) + 2) + 2|) + 2)/(1(-2|(-2(-1(1cos(2πx - 2)) + 2) + 2) + 2|) + 3) and g(x) = -4(4((-2((-1x - 1)/(-1x + 2)) - 2)/(-2((-1x - 1)/(-1x + 2))))) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: ... | 78 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(4(-5(1(4x^2 - 4x - 3)^2 - 2(4x^2 - 4x - 3) - 1) - 3)^2 + 2(-5(1(4x^2 - 4x - 3)^2 - 2(4x^2 - 4x - 3) - 1) - 3) - 5)^2 + 4(4(-5(1(4x^2 - 4x - 3)^2 - 2(4x^2 - 4x - 3) - 1) - 3)^2 + 2(-5(1(4x^2 - 4x - 3)^2 - 2(4x^2 - 4x - 3) - ... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(2(3|(2cos(2π(1x^2 - 4x - 5) - 3) - 3) - 1| - 3) - 4) - 2| - 1 and g(x) = 4(3(-3cos(3π(2x^2 + 2x - 1) + 1) + 1)^2 - 1(-3cos(3π(2x^2 + 2x - 1) + 1) + 1) + 2)^2 + 2(3(-3cos(3π(2x^2 + 2x - 1) + 1) + 1)^2 - 1(-3cos(3π(2x^2 + 2x - 1) + 1) + 1) + 2) inters... | 87 | MATH |
[
{
"content": "How many solutions does the equation 5(-3|(-2e^(-2(5(-1|x - 2| - 2)^2 + 1(-1|x - 2| - 2))) - 3) + 3|) - 1 = 2(4(1|(-5x^2 - 5x + 4) + 3| - 3) + 5) - 4 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 5((-2(-1|(4(-1x^2 + 2x + 4) - 5) + 2| + 3) + 2)/(-2(-1|(4(-1x^2 + 2x + 4) - 5) + 2| + 3) + 3))^2 - 1((-2(-1|(4(-1x^2 + 2x + 4) - 5) + 2| + 3) + 2)/(-2(-1|(4(-1x^2 + 2x + 4) - 5) + 2| + 3) + 3)) + 2 = 2|((1((-1(2x^2 + 3x - 2) + 2)/(-1(2x^2 + 3x - 2) + 2)) + 3)/(... | 8 | MATH |
[
{
"content": "How many solutions does the equation 1((-2(2|(3|(-2sin(2πx - 3))| - 1)| - 2) - 3)/(3(2|(3|(-2sin(2πx - 3))| - 1)| - 2) + 3)) - 4 = -3(-2cos(3π((-2(-5x + 2) + 2)/(2(-5x + 2) - 1))) + 1)^2 + 3(-2cos(3π((-2(-5x + 2) + 2)/(2(-5x + 2) - 1))) + 1) - 4 have for x in the interval [-10, 10]?\n\nPresent the... | 57 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(3(-3|(4(-3x^2 + 1x - 2)) - 1| + 2) - 4) + 3 and g(x) = 2(4(1|(-3sin(2πx - 1) - 2) - 2| + 3)^2 - 3(1|(-3sin(2πx - 1) - 2) - 2| + 3) + 2)^2 + 2(4(1|(-3sin(2πx - 1) - 2) - 2| + 3)^2 - 3(1|(-3sin(2πx - 1) - 2) - 2| + 3) + 2) +... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3sin(2π(2(-3(3|(-4x) + 2|) - 5)^2 + 5(-3(3|(-4x) + 2|) - 5) + 5) + 2) and g(x) = 4((-3(2|((-3x + 2)/(3x + 1)) + 2| - 1) + 3)/(-3(2|((-3x + 2)/(3x + 1)) + 2| - 1))) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{... | 248 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(-3(3sin(2π(-4(-1x - 4)) - 3) + 1)^2 + 1(3sin(2π(-4(-1x - 4)) - 3) + 1)) - 3 and g(x) = 1((2(-1(1x + 3) + 1) - 1)/(3(-1(1x + 3) + 1)))^2 - 5((2(-1(1x + 3) + 1) - 1)/(3(-1(1x + 3) + 1))) - 1 intersect in the interval [-10, 10].\n\nPresent the answer i... | 26 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2((-3(-2cos(2π(1|(1x^2 + 4x + 1) + 1|) + 2)) - 1)/(-3(-2cos(2π(1|(1x^2 + 4x + 1) + 1|) + 2)) + 2)) + 1)/(-3((-3(-2cos(2π(1|(1x^2 + 4x + 1) + 1|) + 2)) - 1)/(-3(-2cos(2π(1|(1x^2 + 4x + 1) + 1|) + 2)) + 2))) and g(x) = 1|(-1(... | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(-3(-3cos(1π((1(-1x^2 + 1x - 1) + 1)/(-1(-1x^2 + 1x - 1) + 2))) - 1) - 1) + 2)/(-3(-3(-3cos(1π((1(-1x^2 + 1x - 1) + 1)/(-1(-1x^2 + 1x - 1) + 2))) - 1) - 1) - 3) and g(x) = (-1(2(-2(-1|x - 1| + 2))^2 + 3(-2(-1|x - 1| + 2)) ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(-1|(-2|(2|(-1sin(2πx - 3)) + 1| + 1) + 1| + 2) - 2| - 3)^2 - 2(-1|(-2|(2|(-1sin(2πx - 3)) + 1| + 1) + 1| + 2) - 2| - 3) - 5 and g(x) = 3e^(-1(-1|(5(-1|x - 2| + 3) - 3) + 3| + 3)) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answe... | 4 | MATH |
[
{
"content": "How many solutions does the equation 2((3(1sin(2π((-1(-3x^2 + 1x - 1))/(2(-3x^2 + 1x - 1) + 2))) - 1))/(-3(1sin(2π((-1(-3x^2 + 1x - 1))/(2(-3x^2 + 1x - 1) + 2))) - 1) - 1)) - 4 = 2|((-2(-2(2|x - 1| - 2) - 2) - 1)/(-1(-2(2|x - 1| - 2) - 2) + 3)) - 2| + 3 have for x in the interval [-10, 10]?\n\nPre... | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(-1((2((-3(3|x - 1| - 1) + 3)/(3(3|x - 1| - 1) - 2)) + 2)/(-2((-3(3|x - 1| - 1) + 3)/(3(3|x - 1| - 1) - 2)) + 3))^2 - 4((2((-3(3|x - 1| - 1) + 3)/(3(3|x - 1| - 1) - 2)) + 2)/(-2((-3(3|x - 1| - 1) + 3)/(3(3|x - 1| - 1) - 2))... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(1|(1(-5(-5x + 2)^2 + 4(-5x + 2) + 4) - 1) - 1|)^2 + 4(1|(1(-5(-5x + 2)^2 + 4(-5x + 2) + 4) - 1) - 1|) + 4 and g(x) = 4((-3((3((1x - 2)/(1x + 1)) - 1)/(1((1x - 2)/(1x + 1)) + 3)) - 3)/(-3((3((1x - 2)/(1x + 1)) - 1)/(1((1x - 2)/(1x + 1)) + 3)) - 2))^2... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(1π(-1|((-1(1(-1x^2 - 3x + 4) + 4) + 3)/(1(1(-1x^2 - 3x + 4) + 4) + 2)) - 3| + 2) - 3) + 2 and g(x) = -3sin(1π((2(-3(3|x + 2| - 2)^2 - 2(3|x + 2| - 2) + 4))/(3(-3(3|x + 2| - 2)^2 - 2(3|x + 2| - 2) + 4) + 1)) + 2) + 1 wher... | 69 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(-3((3(-1(-3|x - 2|) + 5) - 1)/(1(-1(-3|x - 2|) + 5) + 1)) - 2) + 1 and g(x) = -2(-1|((-3(1x^2 + 4x - 5) + 1)/(-3(1x^2 + 4x - 5) - 3))| - 3)^2 - 3(-1|((-3(1x^2 + 4x - 5) + 1)/(-3(1x^2 + 4x - 5) - 3))| - 3) - 3 where -10 ≤ x ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-5(-1(-4(-4x^2 - 3x + 5)^2 - 5(-4x^2 - 3x + 5) - 4)^2 - 5(-4(-4x^2 - 3x + 5)^2 - 5(-4x^2 - 3x + 5) - 4) + 1)^2 + 4(-1(-4(-4x^2 - 3x + 5)^2 - 5(-4x^2 - 3x + 5) - 4)^2 - 5(-4(-4x^2 - 3x + 5)^2 - 5(-4x^2 - 3x + 5) - 4) + 1) + 2)^2 - 1(-5(-1(-4(-4x^2 - 3... | 6 | MATH |
[
{
"content": "How many solutions does the equation -3sin(2π(-1(-2|(-1(-4x^2 + 4x + 5) - 3) + 2|)^2 - 3(-2|(-1(-4x^2 + 4x + 5) - 3) + 2|) + 2) + 3) = 5(-1e^(-2(-2(1x^2 - 3x - 2)^2 - 3(1x^2 - 3x - 2) + 2)) + 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",... | 12 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(1π(-3(3(1(-3|x + 2| + 1) + 3) - 3) + 5) + 2) + 3 and g(x) = 1(-3(-3((-1x + 3)/(-2x + 3))^2 - 2((-1x + 3)/(-2x + 3)) - 1) - 3)^2 - 1(-3(-3((-1x + 3)/(-2x + 3))^2 - 2((-1x + 3)/(-2x + 3)) - 1) - 3) - 4 in the range -10 ≤ x ≤ 10... | 55 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(-3|(-1|((-3(5x + 3) - 2)/(-1(5x + 3) + 2)) + 2| + 3) + 1|) + 1| - 2 and g(x) = 1(3|(-1|(5x - 5) - 3| + 3)| - 2) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(1cos(3π(-5((1(-1x - 2))/(2(-1x - 2) + 3))^2 - 5((1(-1x - 2))/(2(-1x - 2) + 3)) - 5) - 1) - 1) + 1 and g(x) = (3(-2(3cos(3π(2|x + 2| + 1) - 3) - 2)) - 1)/(2(-2(3cos(3π(2|x + 2| + 1) - 3) - 2)) - 1) intersect in the interval [-10, 10].\n\nPresent the ... | 212 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1((-1((-1((-3(3sin(2πx + 3) + 1))/(-1(3sin(2πx + 3) + 1) + 2)))/(-3((-3(3sin(2πx + 3) + 1))/(-1(3sin(2πx + 3) + 1) + 2)) + 4)) - 2)/(2((-1((-3(3sin(2πx + 3) + 1))/(-1(3sin(2πx + 3) + 1) + 2)))/(-3((-3(3sin(2πx + 3) + 1))/(-1(3sin(2... | 1062 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-2|((2(5(3sin(3πx) - 1)^2 + 2(3sin(3πx) - 1) - 5) - 1)/(2(5(3sin(3πx) - 1)^2 + 2(3sin(3πx) - 1) - 5) + 1)) + 3|))/(-3(-2|((2(5(3sin(3πx) - 1)^2 + 2(3sin(3πx) - 1) - 5) - 1)/(2(5(3sin(3πx) - 1)^2 + 2(3sin(3πx) - 1) - 5) + 1)) + ... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(3π((1(1(-1|(3|x - 1| + 2)| - 1)^2 + 4(-1|(3|x - 1| + 2)| - 1) + 2) - 1)/(-3(1(-1|(3|x - 1| + 2)| - 1)^2 + 4(-1|(3|x - 1| + 2)| - 1) + 2) - 2)) - 1) + 3 and g(x) = 1|(-3(-5(-2e^(-1x) + 3)^2 + 5(-2e^(-1x) + 3) + 5)^2 + 2(-5(-2e... | 4 | MATH |
[
{
"content": "How many solutions does the equation 2(-4(3e^(-1((-3(-3x^2 + 4x - 2) - 3)/(-1(-3x^2 + 4x - 2) - 2))) - 3)^2 - 1) + 4 = -2(-3|(-3|(-4x - 3)| + 2) + 1| + 3)^2 + 4(-3|(-3|(-4x - 3)| + 2) + 1| + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-3(-4((-2(-2|x - 3| + 1) + 3)/(-1(-2|x - 3| + 1) - 3))^2 - 4((-2(-2|x - 3| + 1) + 3)/(-1(-2|x - 3| + 1) - 3)) + 1)^2 + 3(-4((-2(-2|x - 3| + 1) + 3)/(-1(-2|x - 3| + 1) - 3))^2 - 4((-2(-2|x - 3| + 1) + 3)/(-1(-2|x - 3| + 1) - 3)) ... | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(1sin(2π(3(5(5x^2 - 1x)^2 + 4(5x^2 - 1x) - 2) + 2) + 1)) + 3 and g(x) = 3|(1(-4(-1x^2 + 1x + 3) + 4)^2 + 1(-4(-1x^2 + 1x + 3) + 4) - 3) + 2| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 14 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(1|((1(1(3x + 5) + 1) + 3)/(3(1(3x + 5) + 1) - 1)) - 3| - 2) - 3 and g(x) = -2|((-3(-3(3x - 2) - 1) + 3)/(3(-3(3x - 2) - 1) - 2)) + 3| + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 4 | MATH |
[
{
"content": "How many solutions does the equation (2(-1(3sin(3π(-4((-3x + 2)/(-1x - 3))^2 - 5((-3x + 2)/(-1x - 3)) - 3) - 2) + 1)^2 - 4(3sin(3π(-4((-3x + 2)/(-1x - 3))^2 - 5((-3x + 2)/(-1x - 3)) - 3) - 2) + 1) - 1) - 2)/(2(-1(3sin(3π(-4((-3x + 2)/(-1x - 3))^2 - 5((-3x + 2)/(-1x - 3)) - 3) - 2) + 1)^2 - 4(3sin(... | 198 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(1|(1e^(-1(-4(3|x - 1| - 2) + 5))) - 3| - 3) - 3| - 3 and g(x) = 3|(2|(1((-2x + 1)/(-1x - 1))^2 + 1((-2x + 1)/(-1x - 1)) - 2) + 1| + 1) - 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 8 | MATH |
[
{
"content": "How many solutions does the equation -3sin(3π(-2((1((1((2x + 1)/(-2x - 1)))/(1((2x + 1)/(-2x - 1)))) - 1)/(-2((1((2x + 1)/(-2x - 1)))/(1((2x + 1)/(-2x - 1)))) - 3))^2 - 3((1((1((2x + 1)/(-2x - 1)))/(1((2x + 1)/(-2x - 1)))) - 1)/(-2((1((2x + 1)/(-2x - 1)))/(1((2x + 1)/(-2x - 1)))) - 3)) + 3) - 1) =... | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|((-2(1|(4((-2x + 1)/(2x + 3)) + 2) + 3|))/(2(1|(4((-2x + 1)/(2x + 3)) + 2) + 3|) - 3)) + 1| - 3 and g(x) = (3(-3(-5(4x^2 + 2x - 5)^2 + 2(4x^2 + 2x - 5) - 2) - 4) - 2)/(-2(-3(-5(4x^2 + 2x - 5)^2 + 2(4x^2 + 2x - 5) - 2) - 4) + 3) intersect in the inter... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(2π(2((-2(-3(1|x - 3|) + 2) - 3)/(3(-3(1|x - 3|) + 2))) - 2) + 1) + 1 and g(x) = (-3(1e^(-1((1((2x + 2)/(-3x + 4)) - 3)/(-1((2x + 2)/(-3x + 4)) - 1))) - 2))/(-3(1e^(-1((1((2x + 2)/(-3x + 4)) - 3)/(-1((2x + 2)/(-3x + 4)) -... | 85 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3((-3(2(5(-1|x + 1| + 1) + 3) + 2) + 2)/(2(2(5(-1|x + 1| + 1) + 3) + 2) - 1)) + 5 and g(x) = -3cos(3π(5(-2((2x - 1)/(1x + 1))^2 - 2((2x - 1)/(1x + 1)) + 4) + 4)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Yo... | 544 | MATH |
[
{
"content": "How many solutions does the equation 2((-2((-3(-2(3|x|) - 5) - 1)/(-2(-2(3|x|) - 5) + 3)) - 1)/(2((-3(-2(3|x|) - 5) - 1)/(-2(-2(3|x|) - 5) + 3)) - 1))^2 + 5((-2((-3(-2(3|x|) - 5) - 1)/(-2(-2(3|x|) - 5) + 3)) - 1)/(2((-3(-2(3|x|) - 5) - 1)/(-2(-2(3|x|) - 5) + 3)) - 1)) + 4 = -1(2(-1|((-2x + 2)/(2x ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(-2(1((2(-5x^2 + 2x) - 1)/(-1(-5x^2 + 2x) + 2)) + 5)^2 + 4(1((2(-5x^2 + 2x) - 1)/(-1(-5x^2 + 2x) + 2)) + 5) - 3) + 1| + 1 and g(x) = -4((-1(3e^(-1(-1x^2 - 5x - 1))) - 1)/(2(3e^(-1(-1x^2 - 5x - 1))) - 1))^2 + 4((-1(3e^(-1(-... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(-5(-2sin(2π(3(-1x - 1)^2 + 3(-1x - 1) + 1)) + 1)^2 - 3(-2sin(2π(3(-1x - 1)^2 + 3(-1x - 1) + 1)) + 1) + 4) - 1)/(-1(-5(-2sin(2π(3(-1x - 1)^2 + 3(-1x - 1) + 1)) + 1)^2 - 3(-2sin(2π(3(-1x - 1)^2 + 3(-1x - 1) + 1)) + 1) + 4) + 2) and g(x) = (1(-1|(2(3|... | 195 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-4((-1((-3(-1|x - 2| - 3) + 3)/(-1(-1|x - 2| - 3) - 2)))/(-1((-3(-1|x - 2| - 3) + 3)/(-1(-1|x - 2| - 3) - 2)) + 3))^2 + 5((-1((-3(-1|x - 2| - 3) + 3)/(-1(-1|x - 2| - 3) - 2)))/(-1((-3(-1|x - 2| - 3) + 3)/(-1(-1|x - 2| - 3) - 2)) + 3)) + 1) + 3| and... | 4 | MATH |
[
{
"content": "How many solutions does the equation -3(5(-3|(1|(1x - 2) + 2| - 3) - 1| + 3)^2 - 3(-3|(1|(1x - 2) + 2| - 3) - 1| + 3) + 5) - 1 = -4(2(2sin(1π(1x - 2) - 3) + 2)^2 - 1(2sin(1π(1x - 2) - 3) + 2) - 3) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(-3|(-2(-3((-3x + 1)/(1x - 2))^2 + 1)^2 + 3(-3((-3x + 1)/(1x - 2))^2 + 1))| - 3)^2 - 4(-3|(-2(-3((-3x + 1)/(1x - 2))^2 + 1)^2 + 3(-3((-3x + 1)/(1x - 2))^2 + 1))| - 3) - 3 and g(x) = -3e^(-1(-1|(1|(-2|x - 2| - 2) + 3| - 1) -... | 5 | MATH |
[
{
"content": "How many solutions does the equation (-3((2(-5((3(-3sin(2πx + 3) - 2) + 3)/(1(-3sin(2πx + 3) - 2) - 3))^2 + 3((3(-3sin(2πx + 3) - 2) + 3)/(1(-3sin(2πx + 3) - 2) - 3)) + 1) - 1)/(1(-5((3(-3sin(2πx + 3) - 2) + 3)/(1(-3sin(2πx + 3) - 2) - 3))^2 + 3((3(-3sin(2πx + 3) - 2) + 3)/(1(-3sin(2πx + 3) - 2) -... | 59 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(3|(3|((2(2x + 1) - 3)/(2(2x + 1) - 1))| + 2) + 2|) - 3| and g(x) = 5(4((-3(-2x^2 + 1x + 2) + 2)/(1(-2x^2 + 1x + 2) + 1)) - 1)^2 + 1(4((-3(-2x^2 + 1x + 2) + 2)/(1(-2x^2 + 1x + 2) + 1)) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent t... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(-1(1(-1|(1e^(1x)) + 2| + 2) - 5) + 3) - 1| + 1 and g(x) = (-3(1sin(2π(3|(1|x + 3|) + 3| + 2) + 2)) + 1)/(3(1sin(2π(3|(1|x + 3|) + 3| + 2) + 2)) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your ... | 43 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(1π(1(-3((3(4x^2 + 3x - 2) - 2)/(2(4x^2 + 3x - 2) - 1)) - 2)^2 + 2(-3((3(4x^2 + 3x - 2) - 2)/(2(4x^2 + 3x - 2) - 1)) - 2) + 2) + 1) + 1 and g(x) = (1(2|(-3|(-1x^2 + 2x - 5) + 2| + 3) + 1| + 1) + 3)/(-3(2|(-3|(-1x^2 + 2x - 5) + 2| + 3) + 1| + 1) + 2... | 69 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(-1(-2(2|(-1|x - 1| + 1)| - 1)^2 - 3(2|(-1|x - 1| + 1)| - 1) - 3)^2 + 1(-2(2|(-1|x - 1| + 1)| - 1)^2 - 3(2|(-1|x - 1| + 1)| - 1) - 3) + 4) - 3)/(3(-1(-2(2|(-1|x - 1| + 1)| - 1)^2 - 3(2|(-1|x - 1| + 1)| - 1) - 3)^2 + 1(-2(2|(-1|x - 1| + 1)| - 1)^2 - ... | 1378 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(2|((1(-3cos(2π((-3x + 3)/(-1x - 2)) + 3) - 2) - 3)/(-3(-3cos(2π((-3x + 3)/(-1x - 2)) + 3) - 2) - 3)) - 1| + 3)| - 2 and g(x) = 1(1|(1|(-3e^(-1x) - 1) - 3| + 1) - 1| + 2)^2 + 2(1|(1|(-3e^(-1x) - 1) - 3| + 1) - 1| + 2) - 2 in the ... | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(2(-1|(-3cos(3π(5x^2 + 5x + 2) + 2) + 3) + 1|)^2 + 5(-1|(-3cos(3π(5x^2 + 5x + 2) + 2) + 3) + 1|) - 3) - 5 and g(x) = (-3(-2((-1(1|x - 1|) - 2)/(-1(1|x - 1|) + 2))^2 + 3((-1(1|x - 1|) - 2)/(-1(1|x - 1|) + 2)) - 3) - 3)/(1(-2... | 1527 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(-4(2e^(-1((1(2|x - 1| + 1) - 3)/(-2(2|x - 1| + 1) + 3)))))^2 - 3(-4(2e^(-1((1(2|x - 1| + 1) - 3)/(-2(2|x - 1| + 1) + 3))))) + 2 and g(x) = -4(-2|(3cos(3π(1x - 2)) - 3)| + 1)^2 - 2(-2|(3cos(3π(1x - 2)) - 3)| + 1) + 3 intersect in the interval [-10, 1... | 62 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1((-3(-4(-3(-2cos(1πx - 1) - 2)^2 + 2(-2cos(1πx - 1) - 2) + 4) - 4) + 2)/(1(-4(-3(-2cos(1πx - 1) - 2)^2 + 2(-2cos(1πx - 1) - 2) + 4) - 4)))^2 - 5((-3(-4(-3(-2cos(1πx - 1) - 2)^2 + 2(-2cos(1πx - 1) - 2) + 4) - 4) + 2)/(1(-4(-3(-2cos... | 17 | MATH |
[
{
"content": "How many solutions does the equation -3|((3(-2|((2(-5x - 5) - 1)/(-2(-5x - 5) - 3)) + 3| + 3) + 1)/(-1(-2|((2(-5x - 5) - 1)/(-2(-5x - 5) - 3)) + 3| + 3) - 3)) - 3| + 3 = -2(5((1(-1x^2 - 3x - 2))/(-2(-1x^2 - 3x - 2) + 3))^2 + 2((1(-1x^2 - 3x - 2))/(-2(-1x^2 - 3x - 2) + 3)) + 2) + 1 have for x in th... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1(-4(-2(5(-2sin(1πx + 1) + 2)^2 - 5(-2sin(1πx + 1) + 2) - 1)^2 + 1(5(-2sin(1πx + 1) + 2)^2 - 5(-2sin(1πx + 1) + 2) - 1) + 4) + 2) - 3 = 1e^(-2(3(4(4x - 4) - 4)^2 - 5(4(4x - 4) - 4) + 4)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex form... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|((-2(-2(-2cos(1π(2x^2 + 4x - 3) + 1) - 1)^2 - 2(-2cos(1π(2x^2 + 4x - 3) + 1) - 1) + 2))/(-2(-2(-2cos(1π(2x^2 + 4x - 3) + 1) - 1)^2 - 2(-2cos(1π(2x^2 + 4x - 3) + 1) - 1) + 2) - 2)) - 2| - 1 and g(x) = -1(-1|((-1(2x^2 - 3x +... | 154 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(1(2(5(-3cos(1πx + 1) - 2) + 2)^2 - 3(5(-3cos(1πx + 1) - 2) + 2) + 3)^2 + 2(2(5(-3cos(1πx + 1) - 2) + 2)^2 - 3(5(-3cos(1πx + 1) - 2) + 2) + 3)) + 3)/(2(1(2(5(-3cos(1πx + 1) - 2) + 2)^2 - 3(5(-3cos(1πx + 1) - 2) + 2) + 3)^2 ... | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-5((-1(3sin(3π(2|x| - 2) + 3) - 3) - 3)/(-2(3sin(3π(2|x| - 2) + 3) - 3) - 1))^2 - 2((-1(3sin(3π(2|x| - 2) + 3) - 3) - 3)/(-2(3sin(3π(2|x| - 2) + 3) - 3) - 1)) + 4) + 3| - 1 and g(x) = 1sin(2π(-1|(1(3|x - 1| - 3) + 2) - 3|) - 2) + 2 intersect in the ... | 136 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(-2|(-1|(1|(-1x + 1) + 3| + 2)| + 1) + 2| - 2) - 4 and g(x) = -2|((-2(-2|(2e^(2x) - 1) - 1| + 1) - 2)/(-1(-2|(2e^(2x) - 1) - 1| + 1))) - 2| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(2(5(2e^(1(1x - 3)) - 2)^2 + 3(2e^(1(1x - 3)) - 2) - 3)^2 + 1(5(2e^(1(1x - 3)) - 2)^2 + 3(2e^(1(1x - 3)) - 2) - 3) - 5) + 3| and g(x) = 1cos(3π((-2(2|(3|x - 2|) - 2| + 1) + 3)/(-2(2|(3|x - 2|) - 2| + 1) - 3)) + 1) + 2 in the rang... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(1π((2((-1(3(-2|x - 1| - 3) + 4) - 1)/(2(3(-2|x - 1| - 3) + 4) - 1)) + 2)/(1((-1(3(-2|x - 1| - 3) + 4) - 1)/(2(3(-2|x - 1| - 3) + 4) - 1)))) + 2) + 3 and g(x) = 2(-5(2(3x + 4)^2 - 3(3x + 4)) + 1)^2 - 4(-5(2(3x + 4)^2 - 3(3x + 4)) + 1) - 2 intersec... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(-2(-2|(3((1x + 3)/(3x + 2))^2 - 5((1x + 3)/(3x + 2))) - 2| - 3) + 1) + 3| + 3 and g(x) = (-3(4(-3cos(1π((3x + 2)/(-1x - 3)) - 3) + 1)) - 3)/(1(4(-3cos(1π((3x + 2)/(-1x - 3)) - 3) + 1)) + 3) where -10 ≤ x ≤ 10.\n\nPresent ... | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((-2(5(1|(-1|x + 3| + 1) + 1|)^2 + 1) + 2)/(-3(5(1|(-1|x + 3| + 1) + 1|)^2 + 1) + 4))^2 - 5((-2(5(1|(-1|x + 3| + 1) + 1|)^2 + 1) + 2)/(-3(5(1|(-1|x + 3| + 1) + 1|)^2 + 1) + 4)) - 2 and g(x) = -2(2(-3|(1e^(2x) - 1) - 2| + 3)^2 - 3(-3|(1e^(2x) - 1) - 2|... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1sin(2π((3(-3|(-1|(2x - 5) - 3| + 1) + 1|) - 3)/(-3(-3|(-1|(2x - 5) - 3| + 1) + 1|) - 3)) - 3) = (-1(-5(1(-1x^2 - 5x + 5) - 1) - 5) + 1)/(3(-5(1(-1x^2 - 5x + 5) - 1) - 5) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed... | 104 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-2|(-2(1(2x + 5)^2 - 3(2x + 5) - 4) + 5)| - 2) - 1)/(3(-2|(-2(1(2x + 5)^2 - 3(2x + 5) - 4) + 5)| - 2) + 2) and g(x) = 3((2((-1(3|x| - 2) - 1)/(-1(3|x| - 2) + 2)) + 1)/(1((-1(3|x| - 2) - 1)/(-1(3|x| - 2) + 2)) + 1))^2 + 2((2((-1(... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(3(2((1(2|x + 2| - 2) - 3)/(3(2|x + 2| - 2) + 1))^2 + 1((1(2|x + 2| - 2) - 3)/(3(2|x + 2| - 2) + 1)) - 3) - 3) - 4 and g(x) = -1((1(1e^(1(-3|x + 3| + 2)) + 3))/(-2(1e^(1(-3|x + 3| + 2)) + 3) - 1)) - 5 where -10 ≤ x ≤ 10.\n\n... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|((1(-3|(2cos(2π(-1x - 5) - 1) + 3) - 2| - 1) - 3)/(3(-3|(2cos(2π(-1x - 5) - 1) + 3) - 2| - 1) + 2)) - 2| - 1 and g(x) = 2|(2(-1(-3sin(2πx + 1) + 2)^2 + 3(-3sin(2πx + 1) + 2) + 3)^2 - 2) + 1| - 3 where -10 ≤ x ≤ 10.\n\nPrese... | 120 | MATH |
[
{
"content": "How many solutions does the equation -3(-1(-2|(-1|((-2x + 2)/(-3x - 3)) - 3| - 1) + 2| - 3)^2 + 4(-2|(-1|((-2x + 2)/(-3x - 3)) - 3| - 1) + 2| - 3) - 4) + 5 = 5(2|(4(-2e^(-2x) + 2) - 1) + 3| + 3)^2 + 5(2|(4(-2e^(-2x) + 2) - 1) + 3| + 3) + 4 have for x in the interval [-10, 10]?\n\nPresent the answe... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(2sin(2π(-5(2(2x - 3)^2 - 2(2x - 3) + 3)^2 - 4(2(2x - 3)^2 - 2(2x - 3) + 3) + 1) + 1) - 1) - 1)/(-1(2sin(2π(-5(2(2x - 3)^2 - 2(2x - 3) + 3)^2 - 4(2(2x - 3)^2 - 2(2x - 3) + 3) + 1) + 1) - 1) - 2) and g(x) = -4(1(-2|(2|x + 1| - 2)|... | 24 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(4((3(-2|(3x - 5)| + 3))/(-3(-2|(3x - 5)| + 3) + 4)) - 1) + 1)/(2(4((3(-2|(3x - 5)| + 3))/(-3(-2|(3x - 5)| + 3) + 4)) - 1) + 2) and g(x) = -1|(1(3(3|x + 1| - 3)^2 - 4(3|x + 1| - 3) - 1) - 5) - 2| + 2 in the range -10 ≤ x ≤ 10?\n... | 9 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5((1(1|((-3(-5x^2 + 1x + 1) - 3)/(3(-5x^2 + 1x + 1) - 2)) + 1| + 1) - 3)/(-3(1|((-3(-5x^2 + 1x + 1) - 3)/(3(-5x^2 + 1x + 1) - 2)) + 1| + 1) - 1))^2 - 2((1(1|((-3(-5x^2 + 1x + 1) - 3)/(3(-5x^2 + 1x + 1) - 2)) + 1| + 1) - 3)/(-3(1|((-3(-5x^2 + 1x + 1) -... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2((2(-2|(-3(-5x^2 - 1x + 2)^2 - 4(-5x^2 - 1x + 2)) + 1|) - 3)/(-3(-2|(-3(-5x^2 - 1x + 2)^2 - 4(-5x^2 - 1x + 2)) + 1|) + 1)) - 1)/(1((2(-2|(-3(-5x^2 - 1x + 2)^2 - 4(-5x^2 - 1x + 2)) + 1|) - 3)/(-3(-2|(-3(-5x^2 - 1x + 2)^2 - 4(-5x^2 - 1x + 2)) + 1|) + ... | 907 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(1(2|(-5(-1|x - 3| - 2)^2 - 2(-1|x - 3| - 2) - 1) - 2| - 2) - 5)^2 + 2(1(2|(-5(-1|x - 3| - 2)^2 - 2(-1|x - 3| - 2) - 1) - 2| - 2) - 5) - 2 and g(x) = -5(-3(1sin(2π(2x^2 - 3x + 5) - 3) - 2)^2 + 4(1sin(2π(2x^2 - 3x + 5) - 3) ... | 18 | MATH |
[
{
"content": "How many solutions does the equation (2((2(-2cos(2π((3(4x^2 + 2x + 1) - 2)/(3(4x^2 + 2x + 1) + 2)) - 2)) - 2)/(1(-2cos(2π((3(4x^2 + 2x + 1) - 2)/(3(4x^2 + 2x + 1) + 2)) - 2)))) + 3)/(-1((2(-2cos(2π((3(4x^2 + 2x + 1) - 2)/(3(4x^2 + 2x + 1) + 2)) - 2)) - 2)/(1(-2cos(2π((3(4x^2 + 2x + 1) - 2)/(3(4x^2... | 52 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(2sin(1π(1|((-1(-1x - 4) + 1)/(-1(-1x - 4) + 3)) - 3| + 3) + 2) + 1) - 1)/(1(2sin(1π(1|((-1(-1x - 4) + 1)/(-1(-1x - 4) + 3)) - 3| + 3) + 2) + 1)) and g(x) = (1(4(2e^(-2(-2|x| - 3)) - 1) + 2) - 2)/(-2(4(2e^(-2(-2|x| - 3)) - 1) + ... | 30 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4((-2((-1(-5(-5x^2 - 3x + 2)^2 + 5(-5x^2 - 3x + 2) + 4) - 2)/(-3(-5(-5x^2 - 3x + 2)^2 + 5(-5x^2 - 3x + 2) + 4) - 1)) - 3)/(-3((-1(-5(-5x^2 - 3x + 2)^2 + 5(-5x^2 - 3x + 2) + 4) - 2)/(-3(-5(-5x^2 - 3x + 2)^2 + 5(-5x^2 - 3x + 2)... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1|((-1(-2cos(2π(-3(-2|x + 3| - 3) + 5) + 2) + 3) - 1)/(-1(-2cos(2π(-3(-2|x + 3| - 3) + 5) + 2) + 3))) - 3| - 2 = (1(2(2cos(1π(-5x^2 - 2x - 5) - 2) + 3) + 4) - 3)/(-2(2(2cos(1π(-5x^2 - 2x - 5) - 2) + 3) + 4) + 1) have for x in the interval [-10, 10]?\n\nPresent ... | 272 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-1(1|(-1(-1x^2 + 5x - 5)^2 - 2(-1x^2 + 5x - 5) + 3) + 3| - 3)^2 - 4(1|(-1(-1x^2 + 5x - 5)^2 - 2(-1x^2 + 5x - 5) + 3) + 3| - 3)) - 3)/(3(-1(1|(-1(-1x^2 + 5x - 5)^2 - 2(-1x^2 + 5x - 5) + 3) + 3| - 3)^2 - 4(1|(-1(-1x^2 + 5x - 5)^2 - 2(-1x^2 + 5x - 5) ... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3((1(-3|(5(-3x^2 + 2)^2 + 4(-3x^2 + 2) + 3) - 2|) - 1)/(2(-3|(5(-3x^2 + 2)^2 + 4(-3x^2 + 2) + 3) - 2|) + 2)) - 2)/(2((1(-3|(5(-3x^2 + 2)^2 + 4(-3x^2 + 2) + 3) - 2|) - 1)/(2(-3|(5(-3x^2 + 2)^2 + 4(-3x^2 + 2) + 3) - 2|) + 2)) + 3) a... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(1|(-3|((3((1x + 3)/(-1x + 2)) + 1)/(1((1x + 3)/(-1x + 2)) + 1)) - 2| + 2) - 2| - 1) + 2)/(-2(1|(-3|((3((1x + 3)/(-1x + 2)) + 1)/(1((1x + 3)/(-1x + 2)) + 1)) - 2| + 2) - 2| - 1) - 1) and g(x) = -3((1(2|(-2x - 1) + 2| - 1) ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(5(1|(-1(-5x^2 - 4x + 1)^2 + 5(-5x^2 - 4x + 1) + 2) - 2| - 1)^2 - 5(1|(-1(-5x^2 - 4x + 1)^2 + 5(-5x^2 - 4x + 1) + 2) - 2| - 1) + 1) + 4 and g(x) = 1(-3sin(3π((-2(1x^2 - 4x - 5) + 2)/(2(1x^2 - 4x - 5) + 2)) - 1)) + 3 in the range ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(2sin(3π(3|((3((-1x - 2)/(2x - 1)) - 3)/(-3((-1x - 2)/(2x - 1)) + 1)) + 3|) - 1) - 3) - 1)/(-2(2sin(3π(3|((3((-1x - 2)/(2x - 1)) - 3)/(-3((-1x - 2)/(2x - 1)) + 1)) + 3|) - 1) - 3) - 3) and g(x) = -1|(-3|(2(-5x^2 - 4x + 2)^2 + 4)|... | 23 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(1e^(-1(1(1(-1|x - 3|) - 4) + 4)) - 2)^2 + 4(1e^(-1(1(1(-1|x - 3|) - 4) + 4)) - 2) + 4 and g(x) = 2e^(2((-3((1(1|x + 3| + 3) - 3)/(2(1|x + 3| + 3) + 2)) - 3)/(1((1(1|x + 3| + 3) - 3)/(2(1|x + 3| + 3) + 2)) + 3))) - 2 where ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|((-2(1(3((2x + 1)/(-1x + 2)) + 2)^2 + 1(3((2x + 1)/(-1x + 2)) + 2) - 4) - 3)/(2(1(3((2x + 1)/(-1x + 2)) + 2)^2 + 1(3((2x + 1)/(-1x + 2)) + 2) - 4))) - 2| - 1 and g(x) = -1(1|((3(1x^2 - 1x - 2) + 3)/(1(1x^2 - 1x - 2) + 3)) - 1| - ... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(2sin(1π(1(2(3|x + 2| - 2)^2 + 1(3|x + 2| - 2) - 4)^2 - 3(2(3|x + 2| - 2)^2 + 1(3|x + 2| - 2) - 4)) - 3) + 1) + 3)/(1(2sin(1π(1(2(3|x + 2| - 2)^2 + 1(3|x + 2| - 2) - 4)^2 - 3(2(3|x + 2| - 2)^2 + 1(3|x + 2| - 2) - 4)) - 3) +... | 357 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(3e^(-2(2|(-3(-1x^2 - 2x + 1)^2 + 5(-1x^2 - 2x + 1) + 2) + 1| + 1)) + 1) - 2 and g(x) = -4(3cos(2π(-1|((1x - 2)/(3x + 1)) - 2| + 3) + 1) - 1) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 41 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3((-2(2(4(4x - 5)^2 - 3(4x - 5) + 1) - 4) - 1)/(1(2(4(4x - 5)^2 - 3(4x - 5) + 1) - 4) + 3)) - 1)/(-3((-2(2(4(4x - 5)^2 - 3(4x - 5) + 1) - 4) - 1)/(1(2(4(4x - 5)^2 - 3(4x - 5) + 1) - 4) + 3)) - 1) and g(x) = -3|(-1(2(3|x - 2|... | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(4(-2sin(2π((-2((-2x + 1)/(-2x - 2)) + 3)/(-3((-2x + 1)/(-2x - 2)))) - 2) + 1) + 1) + 2)/(1(4(-2sin(2π((-2((-2x + 1)/(-2x - 2)) + 3)/(-3((-2x + 1)/(-2x - 2)))) - 2) + 1) + 1) - 3) and g(x) = (2((-2(-3(-1|x + 1| + 3)^2 + 5(-1|x + ... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-5(-1(-3|(2sin(3πx - 2)) - 1| - 1)^2 - 2(-3|(2sin(3πx - 2)) - 1| - 1) + 2)^2 - 2(-1(-3|(2sin(3πx - 2)) - 1| - 1)^2 - 2(-3|(2sin(3πx - 2)) - 1| - 1) + 2)) - 3| + 2 and g(x) = 1(-3|(5(3x^2 + 2x + 3) - 1)| + 1)^2 + 2(-3|(5(3x^2 + 2x + 3) - 1)| + 1) + 4... | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-2(-1(2|(-2x^2 + 5) - 3| - 2) + 4) + 2) + 3)/(-3(-2(-1(2|(-2x^2 + 5) - 3| - 2) + 4) + 2) - 3) and g(x) = 5((-3(-1|(-3x + 3) - 3| + 2) + 3)/(-3(-1|(-3x + 3) - 3| + 2) - 2))^2 - 5((-3(-1|(-3x + 3) - 3| + 2) + 3)/(-3(-1|(-3x ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(-2|(4(3cos(1π((2x + 1)/(2x - 1)) - 2) - 1) + 1) + 1| + 2)^2 - 3(-2|(4(3cos(1π((2x + 1)/(2x - 1)) - 2) - 1) + 1) + 1| + 2) + 1 and g(x) = 2(-1(1e^(-2(4x - 2)))^2 + 2(1e^(-2(4x - 2))) + 5) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent ... | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(4(-1((1(4x + 2) + 3)/(-1(4x + 2) + 3))^2 + 3((1(4x + 2) + 3)/(-1(4x + 2) + 3))) + 1)^2 + 2 and g(x) = -4(4(-3sin(2π(-1x^2 + 5x + 1) + 2))^2 - 1(-3sin(2π(-1x^2 + 5x + 1) + 2)) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answer ... | 123 | MATH |
[
{
"content": "How many solutions does the equation -3|(3sin(1π(1|(1|(-3x^2 - 2x - 1) + 2| + 2) - 2| + 3) + 2)) - 2| + 1 = (2((-2(1|(-3x + 2) + 1|) - 3)/(-2(1|(-3x + 2) + 1|) + 3)) - 2)/(2((-2(1|(-3x + 2) + 1|) - 3)/(-2(1|(-3x + 2) + 1|) + 3)) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in L... | 446 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(1|((1(-2(1x^2 + 2x - 2) + 5) + 2)/(2(-2(1x^2 + 2x - 2) + 5) - 3)) + 2| - 3)^2 + 2(1|((1(-2(1x^2 + 2x - 2) + 5) + 2)/(2(-2(1x^2 + 2x - 2) + 5) - 3)) + 2| - 3) - 5 and g(x) = -1|(1e^(1(-4((-1x - 3)/(3x - 2)) + 1)) + 3) + 2| - 1 in... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(2|((-1((-2((3x)/(-2x + 1)) - 1)/(-1((3x)/(-2x + 1)) + 2)) - 2)/(-2((-2((3x)/(-2x + 1)) - 1)/(-1((3x)/(-2x + 1)) + 2)) + 1))| - 2)^2 + 3(2|((-1((-2((3x)/(-2x + 1)) - 1)/(-1((3x)/(-2x + 1)) + 2)) - 2)/(-2((-2((3x)/(-2x + 1)) - 1)/... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(3((-1(-2|(1x) - 2| + 3) + 1)/(2(-2|(1x) - 2| + 3) + 2))) - 2| - 3 and g(x) = -2(-2|(-3(-3|x - 2| + 1) + 3) - 2| - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(-2((-2(-3|((3x - 3)/(2x - 3)) - 1| + 2))/(-3(-3|((3x - 3)/(2x - 3)) - 1| + 2))) + 2) - 3| + 2 and g(x) = (-2(-1(-1(1|x + 3| - 1)^2 + 5(1|x + 3| - 1) - 4) - 3) + 2)/(1(-1(-1(1|x + 3| - 1)^2 + 5(1|x + 3| - 1) - 4) - 3)) in the ra... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(-3|(-3(5(-4x^2 - 2x + 5)^2 + 3(-4x^2 - 2x + 5) - 3)^2 + 3) + 1| + 2) - 4 and g(x) = -3sin(2π(-3((-1(-1x + 5) + 2)/(-1(-1x + 5) + 2))^2 + 5((-1(-1x + 5) + 2)/(-1(-1x + 5) + 2)) + 3) - 1) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent ... | 12 | MATH |
[
{
"content": "How many solutions does the equation -1|(5(2(3(2x^2 - 4x - 2)^2 + 4(2x^2 - 4x - 2) - 1)^2 + 5(3(2x^2 - 4x - 2)^2 + 4(2x^2 - 4x - 2) - 1) + 2) + 3) + 1| + 2 = -3(1|(-3cos(1π(4x^2 - 2x - 1) - 2) - 2) - 3|) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your ... | 8 | MATH |
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