messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1cos(1π(-1|(-1(-5(2x^2 + 1x - 3)^2 + 2(2x^2 + 1x - 3) - 2)^2 + 2(-5(2x^2 + 1x - 3)^2 + 2(2x^2 + 1x - 3) - 2) + 3) - 2| - 1) - 1) + 3 and g(x) = (1(1|((2(2x^2 - 2x - 4) - 3)/(3(2x^2 - 2x - 4))) - 1| - 1) - 2)/(2(1|((2(2x^2 - ... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(2(2sin(2π((-2(-5x^2 + 5x - 2) + 1)/(-1(-5x^2 + 5x - 2) - 3)) + 3) + 1) + 1) - 2)/(1(2(2sin(2π((-2(-5x^2 + 5x - 2) + 1)/(-1(-5x^2 + 5x - 2) - 3)) + 3) + 1) + 1)) and g(x) = -5((-3(-1|(1x + 5) - 1| + 1) - 3)/(-1(-1|(1x + 5) - 1| ... | 32 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1cos(1π(-3((1((1(-1x + 3) - 2)/(-2(-1x + 3) + 3)))/(-3((1(-1x + 3) - 2)/(-2(-1x + 3) + 3)) - 1))^2 - 1((1((1(-1x + 3) - 2)/(-2(-1x + 3) + 3)))/(-3((1(-1x + 3) - 2)/(-2(-1x + 3) + 3)) - 1)) - 1) - 2) - 1 and g(x) = 3(-1|((3(-3x^2 -... | 4 | MATH |
[
{
"content": "How many solutions does the equation (3((2(1|((1(5x - 1) + 2)/(1(5x - 1)))|) - 3)/(1(1|((1(5x - 1) + 2)/(1(5x - 1)))|) + 2)) + 3)/(2((2(1|((1(5x - 1) + 2)/(1(5x - 1)))|) - 3)/(1(1|((1(5x - 1) + 2)/(1(5x - 1)))|) + 2)) + 3) = 2(-1|(2e^(-2(-1x^2 - 4x - 1)) - 2) + 1| + 1) - 1 have for x in the interv... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-2sin(1π(2|(-3(2x^2 + 4x - 5) - 3) - 2| + 1) + 1) + 2) - 1)/(-1(-2sin(1π(2|(-3(2x^2 + 4x - 5) - 3) - 2| + 1) + 1) + 2) - 2) and g(x) = -1|(1e^(-2((3(-1x^2))/(1(-1x^2) + 1))) - 3) + 1| where -10 ≤ x ≤ 10.\n\nPresent the ans... | 18 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(1|((3(-2(-4x + 1) - 2) + 1)/(-1(-2(-4x + 1) - 2) + 3)) + 3| - 2) + 2| - 2 and g(x) = -1(1(4(-1x - 5)^2 - 1(-1x - 5) + 1)) + 4 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(-5(2|(-1|(-1x + 5)| + 1) - 1| - 3)^2 - 4(2|(-1|(-1x + 5)| + 1) - 1| - 3) + 4)^2 - 2(-5(2|(-1|(-1x + 5)| + 1) - 1| - 3)^2 - 4(2|(-1|(-1x + 5)| + 1) - 1| - 3) + 4) - 2 and g(x) = -3(2|(2cos(2π(-1|x + 3| - 1) - 1) + 1) + 1| +... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1((2(3((-3(-1x^2 + 5x - 5) + 3)/(1(-1x^2 + 5x - 5) + 3))^2 + 4((-3(-1x^2 + 5x - 5) + 3)/(1(-1x^2 + 5x - 5) + 3)) - 3) + 1)/(1(3((-3(-1x^2 + 5x - 5) + 3)/(1(-1x^2 + 5x - 5) + 3))^2 + 4((-3(-1x^2 + 5x - 5) + 3)/(1(-1x^2 + 5x - 5) + ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(5(-1((3(2x) - 3)/(-2(2x) + 3))^2 - 5((3(2x) - 3)/(-2(2x) + 3)) + 2)^2 - 1(-1((3(2x) - 3)/(-2(2x) + 3))^2 - 5((3(2x) - 3)/(-2(2x) + 3)) + 2) + 4) - 1| + 3 and g(x) = -4(1(3|(2|x + 1| - 3) + 1| + 2)^2 + 1(3|(2|x + 1| - 3) + 1| + ... | 6 | MATH |
[
{
"content": "How many solutions does the equation 4((3(-2(-3sin(2π(-3|x - 1| - 1) + 1)) - 5) - 1)/(2(-2(-3sin(2π(-3|x - 1| - 1) + 1)) - 5) - 1)) - 5 = 3|(1(-1sin(1π(-3x^2 + 4x - 3) - 1) + 2)^2 + 1) + 2| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 35 | MATH |
[
{
"content": "How many solutions does the equation -1|(-1|(2(-5(3|x + 3| - 3)^2 - 3(3|x + 3| - 3) + 1)) + 1|) + 2| + 3 = -2(-1|(-1(-1|x - 1| - 2) - 4) - 3| + 2)^2 - 5(-1|(-1(-1|x - 1| - 2) - 4) - 3| + 2) - 2 have for x 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|(5(-1(-3sin(3π(2|x + 2| + 3) + 3) - 3)^2 + 3(-3sin(3π(2|x + 2| + 3) + 3) - 3) + 5) - 5) - 2| + 1 and g(x) = 1(-5((-1((3x - 1)/(1x - 3)) - 1)/(2((3x - 1)/(1x - 3)) - 1)) + 1) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex form... | 222 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(3(-1((-2(-4x^2 - 1x + 2) - 1)/(3(-4x^2 - 1x + 2) + 1)) + 1) + 4)^2 + 5(3(-1((-2(-4x^2 - 1x + 2) - 1)/(3(-4x^2 - 1x + 2) + 1)) + 1) + 4) + 2 and g(x) = (2(4((2(1x + 2) + 2)/(-1(1x + 2) + 2))^2 - 4) - 3)/(-2(4((2(1x + 2) + 2)/(-1(1x + 2) + 2))^2 - 4) ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(5(-3|(3((1x - 2)/(2x - 1)) + 2) - 1| + 2) - 3)^2 - 4 and g(x) = (-1(1((-1(1|x + 1| - 1) + 1)/(3(1|x + 1| - 1) - 2))^2 - 3((-1(1|x + 1| - 1) + 1)/(3(1|x + 1| - 1) - 2)) + 2) - 2)/(-3(1((-1(1|x + 1| - 1) + 1)/(3(1|x + 1| - 1) - 2))^2 - 3((-1(1|x + 1| -... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1e^(1(-4(4((-3((-3x - 2)/(2x + 2)) - 1)/(1((-3x - 2)/(2x + 2)) - 2))^2 - 5((-3((-3x - 2)/(2x + 2)) - 1)/(1((-3x - 2)/(2x + 2)) - 2)) + 4) + 5)) + 1 and g(x) = (-2(1|((3(2|x + 2| - 3) + 3)/(3(2|x + 2| - 3) + 1)) - 2| - 2) + 1)... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(-2sin(3π(2|(3((-2x)/(1x - 2))^2 + 1((-2x)/(1x - 2)) + 3)| + 2) + 2) + 3) - 3 and g(x) = 3sin(3π(-2|((3((-3x + 3)/(1x - 3)))/(1((-3x + 3)/(1x - 3)) + 2)) + 2| - 2) + 1) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex form... | 111 | MATH |
[
{
"content": "How many solutions does the equation 3((-1(1(-2((-3x + 1)/(-2x + 3)) - 2)^2 + 3(-2((-3x + 1)/(-2x + 3)) - 2) + 3))/(3(1(-2((-3x + 1)/(-2x + 3)) - 2)^2 + 3(-2((-3x + 1)/(-2x + 3)) - 2) + 3) - 2)) + 5 = 4(-4(3cos(3π(5x + 3) - 3)) + 5)^2 - 5(-4(3cos(3π(5x + 3) - 3)) + 5) have for x in the interval [-... | 396 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-4(-2sin(1π((2(-3x^2 - 1x + 4) + 2)/(-1(-3x^2 - 1x + 4) + 3)) - 2) - 1)^2 + 4(-2sin(1π((2(-3x^2 - 1x + 4) + 2)/(-1(-3x^2 - 1x + 4) + 3)) - 2) - 1) - 5)^2 + 5(-4(-2sin(1π((2(-3x^2 - 1x + 4) + 2)/(-1(-3x^2 - 1x + 4) + 3)) - 2) - 1)^2 + 4(-2sin(1π((2(-3... | 114 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1((1(-1|(3((-2x - 2)/(-3x)) - 2) + 3| + 3) + 2)/(2(-1|(3((-2x - 2)/(-3x)) - 2) + 3| + 3) - 1))^2 - 4((1(-1|(3((-2x - 2)/(-3x)) - 2) + 3| + 3) + 2)/(2(-1|(3((-2x - 2)/(-3x)) - 2) + 3| + 3) - 1)) - 3 and g(x) = (-1(-3(2e^(2(-2x^2 - 3... | 4 | MATH |
[
{
"content": "How many solutions does the equation -3(2(1(1cos(1π(-1x - 1) + 1) + 3)^2 - 2(1cos(1π(-1x - 1) + 1) + 3) - 3) - 5)^2 - 2(2(1(1cos(1π(-1x - 1) + 1) + 3)^2 - 2(1cos(1π(-1x - 1) + 1) + 3) - 3) - 5) - 4 = -2(5(-3|((-2x + 1)/(2x - 3))| - 3)^2 + 2(-3|((-2x + 1)/(2x - 3))| - 3) + 1) - 1 have for x in the ... | 12 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(-3|(-2|(-3(3e^(2x))^2 - 3(3e^(2x)) - 2) + 1| - 2) + 1| - 2) - 3)/(-1(-3|(-2|(-3(3e^(2x))^2 - 3(3e^(2x)) - 2) + 1| - 2) + 1| - 2) + 2) and g(x) = 3e^(-1(-2(3|(3|x - 1|) - 3| - 3) + 1)) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent t... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3((-2(-1(2|(3cos(2πx - 2) + 1) - 2| - 2)^2 - 2(2|(3cos(2πx - 2) + 1) - 2| - 2) + 3) - 3)/(-1(-1(2|(3cos(2πx - 2) + 1) - 2| - 2)^2 - 2(2|(3cos(2πx - 2) + 1) - 2| - 2) + 3))) + 4 and g(x) = (2(-2sin(3π(2|(-2x + 3) - 3| - 3) - 1) - 2... | 179 | MATH |
[
{
"content": "How many solutions does the equation -3(-1(-1(5(4x^2 - 1x - 5)^2 - 4(4x^2 - 1x - 5) - 1)^2 - 4(5(4x^2 - 1x - 5)^2 - 4(4x^2 - 1x - 5) - 1) - 3)) - 3 = (-2(-2|(5(2cos(1πx + 1) + 2) + 4) - 1| + 2) + 3)/(2(-2|(5(2cos(1πx + 1) + 2) + 4) - 1| + 2) + 2) have for x in the interval [-10, 10]?\n\nPresent th... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(-1sin(1π((3((3(1x^2 - 3x + 2) - 3)/(-1(1x^2 - 3x + 2) - 2)) - 1)/(1((3(1x^2 - 3x + 2) - 3)/(-1(1x^2 - 3x + 2) - 2)) - 2)) - 3))^2 + 5(-1sin(1π((3((3(1x^2 - 3x + 2) - 3)/(-1(1x^2 - 3x + 2) - 2)) - 1)/(1((3(1x^2 - 3x + 2) - 3)/(-1(1x^2 - 3x + 2) - 2))... | 87 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3((1(-1|(-1cos(3π(1|x - 2| - 1) + 2) - 2) + 3| + 1))/(2(-1|(-1cos(3π(1|x - 2| - 1) + 2) - 2) + 3| + 1) + 3)) + 2)/(1((1(-1|(-1cos(3π(1|x - 2| - 1) + 2) - 2) + 3| + 1))/(2(-1|(-1cos(3π(1|x - 2| - 1) + 2) - 2) + 3| + 1) + 3))) and ... | 134 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(-3(-3(2(-3x + 4)^2 - 4(-3x + 4) + 4)^2 + 5(2(-3x + 4)^2 - 4(-3x + 4) + 4) + 5)^2 + 3(-3(2(-3x + 4)^2 - 4(-3x + 4) + 4)^2 + 5(2(-3x + 4)^2 - 4(-3x + 4) + 4) + 5) + 3)^2 - 1(-3(-3(2(-3x + 4)^2 - 4(-3x + 4) + 4)^2 + 5(2(-3x + 4)^2 - 4(-3x + 4) + 4) + 5)... | 4 | MATH |
[
{
"content": "How many solutions does the equation 5((-3(1e^(-2(1((2x + 2)/(2x - 3)) - 5)) + 2) - 1)/(3(1e^(-2(1((2x + 2)/(2x - 3)) - 5)) + 2) - 2))^2 + 4((-3(1e^(-2(1((2x + 2)/(2x - 3)) - 5)) + 2) - 1)/(3(1e^(-2(1((2x + 2)/(2x - 3)) - 5)) + 2) - 2)) - 5 = -1|(1(3sin(3π(-1x + 4) + 3) - 3)^2 + 1(3sin(3π(-1x + 4)... | 115 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(2|(3|((3(-1sin(1πx + 2) + 2) - 3)/(-1(-1sin(1πx + 2) + 2) - 2))| - 1) - 3| - 2) + 3)/(-3(2|(3|((3(-1sin(1πx + 2) + 2) - 3)/(-1(-1sin(1πx + 2) + 2) - 2))| - 1) - 3| - 2) - 1) and g(x) = 4((3(-1(-3x + 4)^2 + 3(-3x + 4) - 2) ... | 22 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(2((-1((3((2x - 1)/(3x - 2)) + 3)/(-1((2x - 1)/(3x - 2)) - 3)) - 2)/(-1((3((2x - 1)/(3x - 2)) + 3)/(-1((2x - 1)/(3x - 2)) - 3)) + 2))^2 - 1((-1((3((2x - 1)/(3x - 2)) + 3)/(-1((2x - 1)/(3x - 2)) - 3)) - 2)/(-1((3((2x - 1)/(3x - 2))... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5((-1(-4((2(-3x^2 - 3x + 2) + 3)/(-1(-3x^2 - 3x + 2))) - 5) - 1)/(3(-4((2(-3x^2 - 3x + 2) + 3)/(-1(-3x^2 - 3x + 2))) - 5) + 2))^2 - 3((-1(-4((2(-3x^2 - 3x + 2) + 3)/(-1(-3x^2 - 3x + 2))) - 5) - 1)/(3(-4((2(-3x^2 - 3x + 2) + 3)/(-1(-3x^2 - 3x + 2))) - ... | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-5((2(-2(4x^2 + 3x + 1)^2 + 3(4x^2 + 3x + 1) - 2) - 1)/(2(-2(4x^2 + 3x + 1)^2 + 3(4x^2 + 3x + 1) - 2) + 1))^2 + 3((2(-2(4x^2 + 3x + 1)^2 + 3(4x^2 + 3x + 1) - 2) - 1)/(2(-2(4x^2 + 3x + 1)^2 + 3(4x^2 + 3x + 1) - 2) + 1)) - 4) - 1 ... | 4 | MATH |
[
{
"content": "How many solutions does the equation 3|((3((2(2(3x - 3)^2 + 5(3x - 3)))/(-1(2(3x - 3)^2 + 5(3x - 3)) - 3)))/(-3((2(2(3x - 3)^2 + 5(3x - 3)))/(-1(2(3x - 3)^2 + 5(3x - 3)) - 3)) - 1))| - 3 = (1(-2cos(2π(-5(-1|x + 3| + 2)^2 - 5(-1|x + 3| + 2) + 4)) + 2) + 3)/(-2(-2cos(2π(-5(-1|x + 3| + 2)^2 - 5(-1|x ... | 10 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-3(1(2|(2|x - 3| + 1)|) - 5)^2 - 2(1(2|(2|x - 3| + 1)|) - 5) - 1) - 1)/(-1(-3(1(2|(2|x - 3| + 1)|) - 5)^2 - 2(1(2|(2|x - 3| + 1)|) - 5) - 1) - 3) and g(x) = (3((-1(3cos(1π(-3x^2 + 3x + 5)) + 3) + 3)/(3(3cos(1π(-3x^2 + 3x + 5)) + 3) + 2)))/(-1((-1(3... | 8 | MATH |
[
{
"content": "How many solutions does the equation (1(-1(-5((-1(1|x - 3| - 1) + 2)/(3(1|x - 3| - 1) - 1)) - 2)^2 + 4(-5((-1(1|x - 3| - 1) + 2)/(3(1|x - 3| - 1) - 1)) - 2) + 5) + 3)/(1(-1(-5((-1(1|x - 3| - 1) + 2)/(3(1|x - 3| - 1) - 1)) - 2)^2 + 4(-5((-1(1|x - 3| - 1) + 2)/(3(1|x - 3| - 1) - 1)) - 2) + 5) - 2) =... | 6 | MATH |
[
{
"content": "How many solutions does the equation -3(5(2(-2|(-2cos(3πx - 3) + 1) + 2| - 1)^2 + 4(-2|(-2cos(3πx - 3) + 1) + 2| - 1) + 5) - 3) - 1 = -4(1(2|(2x - 5)| - 3) + 2)^2 + 5(1(2|(2x - 5)| - 3) + 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 32 | MATH |
[
{
"content": "How many solutions does the equation 3|(-5(-3(-2(3x^2 - 3))^2 + 4(-2(3x^2 - 3)) + 3) - 5) - 1| - 2 = -4(-1(-1|(-2x - 1) + 1| - 2)^2 + 2(-1|(-2x - 1) + 1| - 2) - 4) 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) = 1(2e^(-1((-2(4(4x^2 + 5x - 5) - 4))/(-1(4(4x^2 + 5x - 5) - 4) - 3))) - 3) + 3 and g(x) = -1cos(3π(-3|(3|(-4x + 1) - 1| - 3) - 3| - 3) - 2) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 7 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(2sin(3π(-4(-2|(2|x + 1| + 3) - 2|)^2 - 3(-2|(2|x + 1| + 3) - 2|) - 4) + 1) - 1) + 3| + 2 and g(x) = -1sin(3π(4(-3|(-2x^2 - 2x - 5)|)^2 + 1(-3|(-2x^2 - 2x - 5)|))) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex form... | 175 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-5((-2(-1(-3sin(2πx - 2) - 1) + 1) - 2)/(3(-1(-3sin(2πx - 2) - 1) + 1) + 1)) + 2) + 5 and g(x) = 1|(-1(-3cos(3π((-1x + 2)/(1x)) + 3) + 2) - 3) - 2| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{You... | 45 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2cos(2π(-5(-2((-3(-4x - 4) - 1)/(2(-4x - 4) - 1)) + 4) - 2) + 2) - 1 and g(x) = 1|((-1(-1|(-1x^2 - 3x - 2) - 2| + 3) - 1)/(3(-1|(-1x^2 - 3x - 2) - 2| + 3) + 1)) - 2| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex for... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(2π(-3|(2((-3(-1x^2 + 3x + 5) - 1)/(2(-1x^2 + 3x + 5) + 2)) + 5) + 3| + 3) + 1) - 1 and g(x) = -1|(1|((3(-3x + 2) - 1)/(3(-3x + 2) + 1)) + 1| - 2) + 1| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxe... | 95 | MATH |
[
{
"content": "How many solutions does the equation 4(-3(-4(-3|(2x - 2) - 3| - 1)^2 - 3(-3|(2x - 2) - 3| - 1) + 2))^2 - 3(-3(-4(-3|(2x - 2) - 3| - 1)^2 - 3(-3|(2x - 2) - 3| - 1) + 2)) - 4 = 2|(-1(-3|(2e^(-1x) - 1) + 1| - 1)^2 - 4) + 1| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex form... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(1(-3|(1(1|x + 2| + 1) - 5)| + 3) + 5) + 5 and g(x) = -3((-1(-2(-4x^2 - 4x - 4)^2 - 5(-4x^2 - 4x - 4) + 5) - 2)/(2(-2(-4x^2 - 4x - 4)^2 - 5(-4x^2 - 4x - 4) + 5) + 1)) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\bo... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2e^(1(1((-2((-2(2x - 5) + 1)/(2(2x - 5))) - 1)/(-3((-2(2x - 5) + 1)/(2(2x - 5))) - 2)) - 3)) + 3 and g(x) = -4(3(-5(3cos(2πx + 3)) - 1)) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 39 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(3((2(2((-2x)/(1x + 2))^2 - 3((-2x)/(1x + 2)) - 3) - 1)/(3(2((-2x)/(1x + 2))^2 - 3((-2x)/(1x + 2)) - 3) - 1))^2 + 3((2(2((-2x)/(1x + 2))^2 - 3((-2x)/(1x + 2)) - 3) - 1)/(3(2((-2x)/(1x + 2))^2 - 3((-2x)/(1x + 2)) - 3) - 1)) ... | 241 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(2cos(1π(-2(2((-1x)/(3x - 2)) + 4)^2 + 4(2((-1x)/(3x - 2)) + 4) - 3) + 2)) + 4 and g(x) = -5((-1(4(2|x - 2| - 2)^2 - 2(2|x - 2| - 2) - 2) - 2)/(3(4(2|x - 2| - 2)^2 - 2(2|x - 2| - 2) - 2) - 1)) + 5 where -10 ≤ x ≤ 10.\n\nPres... | 54 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2((2((-2((3(-3x^2 - 1x + 4) - 1)/(-3(-3x^2 - 1x + 4) - 2)) + 1)/(2((3(-3x^2 - 1x + 4) - 1)/(-3(-3x^2 - 1x + 4) - 2)) + 3)) - 3)/(2((-2((3(-3x^2 - 1x + 4) - 1)/(-3(-3x^2 - 1x + 4) - 2)) + 1)/(2((3(-3x^2 - 1x + 4) - 1)/(-3(-3x^2 - 1x... | 4 | MATH |
[
{
"content": "How many solutions does the equation -4(-3(3(2|(-2x^2 + 3x) + 1| - 2) + 3) - 2) - 2 = (2((1(-2e^(1(3|x - 3| - 1)) + 1) - 3)/(3(-2e^(1(3|x - 3| - 1)) + 1) + 1)))/(-2((1(-2e^(1(3|x - 3| - 1)) + 1) - 3)/(3(-2e^(1(3|x - 3| - 1)) + 1) + 1)) + 1) have for x in the interval [-10, 10]?\n\nPresent the answ... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3e^(-1(1|(5(-5((-3x + 2)/(-1x - 3)) - 2)^2 + 2(-5((-3x + 2)/(-1x - 3)) - 2) - 3) - 3| - 3)) + 3 and g(x) = (-3(-2((-1(2x) - 3)/(-2(2x))) + 2) + 2)/(-3(-2((-1(2x) - 3)/(-2(2x))) + 2)) where -10 ≤ x ≤ 10.\n\nPresent the answer... | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(2e^(-2(4(2|(2|x + 3| - 1) - 3| - 1) + 1)) + 1)^2 + 2(2e^(-2(4(2|(2|x + 3| - 1) - 3| - 1) + 1)) + 1) + 5 and g(x) = 4((-1((-1(4x + 2) + 3)/(1(4x + 2) - 2)) + 2)/(-1((-1(4x + 2) + 3)/(1(4x + 2) - 2)) - 1))^2 - 1((-1((-1(4x + 2) + 3... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-1(-1(5(5x^2 + 1x - 4)^2 + 1(5x^2 + 1x - 4) + 4)^2 - 4(5(5x^2 + 1x - 4)^2 + 1(5x^2 + 1x - 4) + 4) - 2)^2 - 3(-1(5(5x^2 + 1x - 4)^2 + 1(5x^2 + 1x - 4) + 4)^2 - 4(5(5x^2 + 1x - 4)^2 + 1(5x^2 + 1x - 4) + 4) - 2) + 3) + 3)/(3... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(3|(-1(2(4x)^2 - 4(4x) - 5)^2 - 3(2(4x)^2 - 4(4x) - 5) + 5) + 1| + 1) + 3| - 3 and g(x) = 5(3e^(-1((-3(-1x^2 - 4x + 2) + 2)/(-3(-1x^2 - 4x + 2) + 1))) + 3) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your a... | 9 | MATH |
[
{
"content": "How many solutions does the equation -2(-2(-1(1(-5x^2 - 2x - 5)^2 + 4(-5x^2 - 2x - 5) - 3) + 2) - 2)^2 + 3 = (1(4(2sin(2π((3x - 2)/(-3x - 3)) + 3) - 2) + 2))/(1(4(2sin(2π((3x - 2)/(-3x - 3)) + 3) - 2) + 2)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your a... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1|(3(4(-1(1sin(3πx + 3) + 1)^2 + 4(1sin(3πx + 3) + 1) - 3) - 4) - 1) - 1| + 1 = 1|((2(-1((2x + 1)/(-1x - 1))^2 + 2((2x + 1)/(-1x - 1)) - 3) + 3)/(-3(-1((2x + 1)/(-1x - 1))^2 + 2((2x + 1)/(-1x - 1)) - 3) - 1)) - 3| + 3 have for x in the interval [-10, 10]?\n\nPr... | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-1(3cos(3π(-5((-3x - 2)/(-1x + 3)) - 2) + 2) - 2)^2 - 4(3cos(3π(-5((-3x - 2)/(-1x + 3)) - 2) + 2) - 2) - 3)| - 2 and g(x) = -3|(-2sin(2π(1|(2x^2 - 1x - 4)| - 1) - 3) + 2) - 3| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex form... | 852 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(5(3(-3sin(1π(-2x^2 - 3) + 1) - 1) - 2) + 2) + 1| + 2 and g(x) = (1(-3cos(3π(3|((1x + 1)/(-1x + 3)) - 3| - 1) - 3) + 1) - 1)/(3(-3cos(3π(3|((1x + 1)/(-1x + 3)) - 3| - 1) - 3) + 1) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the a... | 148 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(-1(-4(1|(4x^2 + 2x + 2) - 1| - 1)^2 + 1(1|(4x^2 + 2x + 2) - 1| - 1) + 1)^2 - 4(-4(1|(4x^2 + 2x + 2) - 1| - 1)^2 + 1(1|(4x^2 + 2x + 2) - 1| - 1) + 1) + 3) - 3)/(-1(-1(-4(1|(4x^2 + 2x + 2) - 1| - 1)^2 + 1(1|(4x^2 + 2x + 2) - 1| -... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-5(5(-3(3|x + 3| - 3) + 4) + 3)^2 - 2(5(-3(3|x + 3| - 3) + 4) + 3) - 4) - 2)/(-1(-5(5(-3(3|x + 3| - 3) + 4) + 3)^2 - 2(5(-3(3|x + 3| - 3) + 4) + 3) - 4) - 2) and g(x) = -3(5(1|(2|x - 2| - 1) - 2| - 2)^2 - 3) + 4 intersect in the interval [-10, 10].... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(1(-1(-2|((2x - 2)/(3x - 2)) + 1| - 1) - 2)^2 - 2) - 2| + 3 and g(x) = 1((2(1(-2|x + 1|)^2 + 2(-2|x + 1|) + 1) + 3)/(-2(1(-2|x + 1|)^2 + 2(-2|x + 1|) + 1) - 2)) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(5(2|((-2(3sin(1πx + 1)) + 2)/(1(3sin(1πx + 1)) + 3)) + 1| - 3)^2 - 5(2|((-2(3sin(1πx + 1)) + 2)/(1(3sin(1πx + 1)) + 3)) + 1| - 3) - 2) + 2 and g(x) = (-3((-3(3(-2|x - 2| + 2)^2 + 4(-2|x - 2| + 2) - 1) - 3)/(-3(3(-2|x - 2| + 2)^2 + 4(-2|x - 2| + 2) - ... | 41 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-5((3(-3cos(1π(3x^2 - 5x + 1) + 1) - 1) - 2)/(3(-3cos(1π(3x^2 - 5x + 1) + 1) - 1))) - 5) - 3| - 3 and g(x) = -5(3sin(1π(2|(2|x + 3| - 2)| + 1) + 3) + 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{... | 104 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(-1|(1((-1(-3cos(1πx - 1) - 2) + 2)/(3(-3cos(1πx - 1) - 2) - 2))^2 - 2((-1(-3cos(1πx - 1) - 2) + 2)/(3(-3cos(1πx - 1) - 2) - 2)) - 2) + 2| - 2) - 2| and g(x) = -2|(-5(5(2x - 4)^2 - 2(2x - 4) - 3)^2 - 3(5(2x - 4)^2 - 2(2x - 4) - 3) - 5) - 2| - 2 inte... | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(-3sin(2π(-4(5((-2x + 3)/(2x)) + 1) + 4) + 2)) - 5 and g(x) = -1cos(1π(-1((3(-1x^2 + 5x + 1) - 2)/(-2(-1x^2 + 5x + 1) - 3)) + 4) + 1) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 260 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-1|((-2(-4(-1|x + 3| + 1) + 1) + 1)/(1(-4(-1|x + 3| + 1) + 1) - 3))| + 3) + 1)/(3(-1|((-2(-4(-1|x + 3| + 1) + 1) + 1)/(1(-4(-1|x + 3| + 1) + 1) - 3))| + 3) + 3) and g(x) = (-1(-3sin(3π(3|(2|x - 1| + 1) + 3| - 3) + 1) + 3) - 1)/(... | 92 | MATH |
[
{
"content": "How many solutions does the equation (3(1(1|(2|(3|x + 2|) + 1| - 2) + 2| - 3) + 5) - 1)/(2(1(1|(2|(3|x + 2|) + 1| - 2) + 2| - 3) + 5)) = (2(-1(2cos(1π((3x + 2)/(1x + 3))))^2 - 3(2cos(1π((3x + 2)/(1x + 3)))) + 3) + 1)/(3(-1(2cos(1π((3x + 2)/(1x + 3))))^2 - 3(2cos(1π((3x + 2)/(1x + 3)))) + 3) - 1) h... | 27 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(1|(-4(2cos(1π(-1x + 4) - 3) - 1)^2 + 1) + 1| - 1) - 1| and g(x) = (3(-1(4(-2x^2 + 5) - 2)^2 - 5(4(-2x^2 + 5) - 2)) + 3)/(-1(-1(4(-2x^2 + 5) - 2)^2 - 5(4(-2x^2 + 5) - 2)) + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex forma... | 102 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(4(1((2((1x + 1)/(-3x + 2)) - 2)/(2((1x + 1)/(-3x + 2)) + 1))^2 + 3((2((1x + 1)/(-3x + 2)) - 2)/(2((1x + 1)/(-3x + 2)) + 1)))^2 - 1(1((2((1x + 1)/(-3x + 2)) - 2)/(2((1x + 1)/(-3x + 2)) + 1))^2 + 3((2((1x + 1)/(-3x + 2)) - ... | 4 | MATH |
[
{
"content": "How many solutions does the equation 3|(-2sin(1π(-2(2((-1x + 3)/(2x - 3)) + 2) - 4) + 3) + 1) - 3| = (1(-3(2|(-1x + 2)| + 3)^2) - 2)/(2(-3(2|(-1x + 2)| + 3)^2) + 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 45 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(2(2(1((1x + 1)/(1x))^2 + 4((1x + 1)/(1x))) + 4)^2 - 4(2(1((1x + 1)/(1x))^2 + 4((1x + 1)/(1x))) + 4) - 4) + 4 and g(x) = -5((1(-1cos(1π((2x + 3)/(1x - 3)) + 2) + 2) + 3)/(-1(-1cos(1π((2x + 3)/(1x - 3)) + 2) + 2) - 3))^2 + 3((1(-1cos(1π((2x + 3)/(1x -... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(-3(-5(1|(1cos(3πx + 1) + 1) + 1| - 2)^2 - 5(1|(1cos(3πx + 1) + 1) + 1| - 2) - 2)^2 + 1(-5(1|(1cos(3πx + 1) + 1) + 1| - 2)^2 - 5(1|(1cos(3πx + 1) + 1) + 1| - 2) - 2)) - 2| - 2 and g(x) = -4(2|(3cos(1π(-4x - 1) - 3) + 2) + 1| + 3) - 2 intersect in th... | 100 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(3|((-2(5((2x - 3)/(-2x + 3)) + 4) - 3)/(1(5((2x - 3)/(-2x + 3)) + 4))) - 2| + 2)^2 - 3(3|((-2(5((2x - 3)/(-2x + 3)) + 4) - 3)/(1(5((2x - 3)/(-2x + 3)) + 4))) - 2| + 2) + 4 and g(x) = 2(4((1(-2|x - 1|))/(-2(-2|x - 1|) - 2))^2 + 2((1(-2|x - 1|))/(-2(-2... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(-3(2|((-2(-3sin(3πx)) + 2)/(-2(-3sin(3πx)))) - 2|) + 3) - 5 and g(x) = (-3(2(-1|(2x^2 + 3x + 1) - 2| + 1)^2 - 4(-1|(2x^2 + 3x + 1) - 2| + 1) + 3) + 1)/(-3(2(-1|(2x^2 + 3x + 1) - 2| + 1)^2 - 4(-1|(2x^2 + 3x + 1) - 2| + 1) +... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|((1((-1(3sin(2π((3x + 2)/(-1x + 2)) + 2) - 3) - 3)/(3(3sin(2π((3x + 2)/(-1x + 2)) + 2) - 3) - 1)) - 3)/(-1((-1(3sin(2π((3x + 2)/(-1x + 2)) + 2) - 3) - 3)/(3(3sin(2π((3x + 2)/(-1x + 2)) + 2) - 3) - 1)) + 2))| + 2 and g(x) = ... | 58 | MATH |
[
{
"content": "How many solutions does the equation 3cos(1π(3(-5(2|(3|x - 1|) - 1| + 2)^2 - 4(2|(3|x - 1|) - 1| + 2) + 1)^2 - 4(-5(2|(3|x - 1|) - 1| + 2)^2 - 4(2|(3|x - 1|) - 1| + 2) + 1) - 3)) - 1 = 3sin(2π((2(4((-1x - 1)/(1x))^2 - 2((-1x - 1)/(1x)) + 4) - 2)/(3(4((-1x - 1)/(1x))^2 - 2((-1x - 1)/(1x)) + 4) - 2)... | 207 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(1|((-1(3(-3x^2 - 4x - 1)^2 + 4(-3x^2 - 4x - 1) - 1) + 3)/(-3(3(-3x^2 - 4x - 1)^2 + 4(-3x^2 - 4x - 1) - 1) - 2)) - 3| + 3) + 1)/(-1(1|((-1(3(-3x^2 - 4x - 1)^2 + 4(-3x^2 - 4x - 1) - 1) + 3)/(-3(3(-3x^2 - 4x - 1)^2 + 4(-3x^2 ... | 73 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((1(-2e^(1((-2((1x + 3)/(1x)) + 1)/(3((1x + 3)/(1x)) - 2))) + 1) + 1)/(-2(-2e^(1((-2((1x + 3)/(1x)) + 1)/(3((1x + 3)/(1x)) - 2))) + 1) - 1))^2 + 1((1(-2e^(1((-2((1x + 3)/(1x)) + 1)/(3((1x + 3)/(1x)) - 2))) + 1) + 1)/(-2(-2e^(1((-2((1x + 3)/(1x)) + 1)/... | 39 | MATH |
[
{
"content": "How many solutions does the equation -4(4(-2(1sin(1π((-2x)/(-1x + 2)) + 3) - 2) - 5)^2 + 3(-2(1sin(1π((-2x)/(-1x + 2)) + 3) - 2) - 5) - 2)^2 - 1(4(-2(1sin(1π((-2x)/(-1x + 2)) + 3) - 2) - 5)^2 + 3(-2(1sin(1π((-2x)/(-1x + 2)) + 3) - 2) - 5) - 2) = -3((1(-1sin(1π(-2x - 1) + 2) - 2))/(-1(-1sin(1π(-2x ... | 13 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(1|(-3|(3(2|x - 2| - 1)^2 - 5(2|x - 2| - 1) - 5) - 2| - 1) - 1| + 2) - 2| + 3 and g(x) = -2e^(2(5(-1(-1x^2 - 4x) - 2)^2 + 3(-1(-1x^2 - 4x) - 2))) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(2|(1(5(-2|x - 3| + 3) + 5) - 2)|) - 1| + 1 and g(x) = -2|(-4(-3(-1|x + 1| + 3)^2 - 2(-1|x + 1| + 3) + 3) + 1) + 2| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(-2(1e^(1(-4(4x) - 1)) + 1) + 3)^2 - 2(-2(1e^(1(-4(4x) - 1)) + 1) + 3) + 1 and g(x) = (1(1|(-1(-2cos(3πx - 3) + 1) - 2) - 1| - 3) + 1)/(-2(1|(-1(-2cos(3πx - 3) + 1) - 2) - 1| - 3) + 3) where -10 ≤ x ≤ 10.\n\nPresent the answ... | 30 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(2π(5(-2((-1(-1x + 1) - 1)/(-1(-1x + 1) - 3)) - 5) - 5) - 1) + 2 and g(x) = 2e^(-1(-5(-2(3|x - 1|)^2 - 2(3|x - 1|) - 5) - 3)) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 233 | MATH |
[
{
"content": "How many solutions does the equation -1sin(2π((-2(1|(3|(-1x^2 - 3x - 4) - 3|) - 1| - 2) - 2)/(-2(1|(3|(-1x^2 - 3x - 4) - 3|) - 1| - 2) + 1)) - 3) - 1 = 5(-2sin(2π(-1(-1|x + 3| - 1)) - 3) - 1)^2 - 5(-2sin(2π(-1(-1|x + 3| - 1)) - 3) - 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answe... | 40 | MATH |
[
{
"content": "How many solutions does the equation 3|(3sin(3π(-1(1(-2|x + 2| - 1)^2 + 1(-2|x + 2| - 1) - 4) - 3) - 3) + 1)| - 3 = (1(-4(2sin(1π(4x + 1) - 1) + 3)^2 + 5(2sin(1π(4x + 1) - 1) + 3)) - 3)/(-3(-4(2sin(1π(4x + 1) - 1) + 3)^2 + 5(2sin(1π(4x + 1) - 1) + 3))) have for x in the interval [-10, 10]?\n\nPres... | 1067 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(2π(3(4(-2|(2|x - 3| - 1) + 2|)^2 + 2(-2|(2|x - 3| - 1) + 2|) - 5) + 1) + 2) - 2 and g(x) = -2((2((-2((-2x + 3)/(-1x - 2)))/(-2((-2x + 3)/(-1x - 2)) - 3)) - 3)/(-1((-2((-2x + 3)/(-1x - 2)))/(-2((-2x + 3)/(-1x - 2)) - 3)) - 2))... | 26 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(4(-4((-3(-1sin(2πx + 3) - 1) - 2)/(-2(-1sin(2πx + 3) - 1) - 3))^2 - 4)^2 + 1(-4((-3(-1sin(2πx + 3) - 1) - 2)/(-2(-1sin(2πx + 3) - 1) - 3))^2 - 4) + 3) + 4 and g(x) = -2(-4(1(1x + 5) - 2)^2 - 1)^2 - 5(-4(1(1x + 5) - 2)^2 - 1) + 4... | 34 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(1|((2(-5(-3e^(-1x) + 3) + 4) - 3)/(2(-5(-3e^(-1x) + 3) + 4) - 1)) + 2| + 1) - 2)/(2(1|((2(-5(-3e^(-1x) + 3) + 4) - 3)/(2(-5(-3e^(-1x) + 3) + 4) - 1)) + 2| + 1) - 1) and g(x) = (2(-2(-3|((3x - 3)/(-2x - 2)) - 3| + 2)^2 - 1(-3|((... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(-1|(-2(4(-2x^2 + 5x + 4)^2 + 2(-2x^2 + 5x + 4) + 2)^2 + 3(4(-2x^2 + 5x + 4)^2 + 2(-2x^2 + 5x + 4) + 2) + 5)| + 1)^2 + 1(-1|(-2(4(-2x^2 + 5x + 4)^2 + 2(-2x^2 + 5x + 4) + 2)^2 + 3(4(-2x^2 + 5x + 4)^2 + 2(-2x^2 + 5x + 4) + 2) ... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5((-1(3((-2(-1|x - 3| - 3) - 1)/(2(-1|x - 3| - 3) + 2)) + 5) + 3)/(1(3((-2(-1|x - 3| - 3) - 1)/(2(-1|x - 3| - 3) + 2)) + 5) + 1)) - 4 and g(x) = -1(3|(-3|(4x^2 + 3x - 4) + 2| - 3) + 3| + 1)^2 - 2(3|(-3|(4x^2 + 3x - 4) + 2| - 3) + 3| + 1) + 5 intersect ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(2((3(1|(3|x - 1| - 2) + 1| - 3))/(-2(1|(3|x - 1| - 2) + 1| - 3) - 1)) + 3)^2 - 4(2((3(1|(3|x - 1| - 2) + 1| - 3))/(-2(1|(3|x - 1| - 2) + 1| - 3) - 1)) + 3) + 1 and g(x) = 4((3(2(1x^2 + 3x - 1) - 4) - 2)/(-2(2(1x^2 + 3x - 1) - 4)... | 6 | MATH |
[
{
"content": "How many solutions does the equation 1|((-2(-3cos(2π(-3(2|x| - 3) + 2) + 1) + 1))/(1(-3cos(2π(-3(2|x| - 3) + 2) + 1) + 1) + 2)) - 1| - 1 = (-1((2(-2(4x^2 - 2x + 3)^2 - 1(4x^2 - 2x + 3) + 2))/(2(-2(4x^2 - 2x + 3)^2 - 1(4x^2 - 2x + 3) + 2) - 3)) - 3)/(-3((2(-2(4x^2 - 2x + 3)^2 - 1(4x^2 - 2x + 3) + 2... | 240 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-2(1(-4(4x^2 - 5x - 5)^2 + 4(4x^2 - 5x - 5) + 2)^2 + 3(-4(4x^2 - 5x - 5)^2 + 4(4x^2 - 5x - 5) + 2) + 4) - 2) and g(x) = 2|(-2sin(1π(-2(-4x^2 + 5x - 4)^2 + 5(-4x^2 + 5x - 4) - 5)) + 3) + 3| in the range -10 ≤ x ≤ 10?\n\nPresent t... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(5(4((-3(2x - 3) + 3)/(-2(2x - 3) + 3))^2 + 1((-3(2x - 3) + 3)/(-2(2x - 3) + 3)) - 2) + 1) - 5 and g(x) = -3cos(1π(-2(-4(1|x - 2| - 1)^2 + 2(1|x - 2| - 1) - 3)^2 - 4(-4(1|x - 2| - 1)^2 + 2(1|x - 2| - 1) - 3) + 4)) - 3 where... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(2|(-2|(-4((-2x - 1)/(-2x - 3))^2 - 3((-2x - 1)/(-2x - 3)) - 5) + 3| + 1) - 2| - 3) + 1| + 2 and g(x) = 2|((2(2cos(2π(1x^2 - 3x) - 2) - 1) - 3)/(-1(2cos(2π(1x^2 - 3x) - 2) - 1) - 3)) - 2| + 3 intersect in the interval [-10, 10].\n\nPresent the answer... | 381 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1cos(2π(-1((2(-1((3x + 3)/(1x + 1))^2 + 5((3x + 3)/(1x + 1)) + 4) + 3)/(-3(-1((3x + 3)/(1x + 1))^2 + 5((3x + 3)/(1x + 1)) + 4) - 3)) - 5) - 1) + 2 and g(x) = 4(4(4(-3sin(3πx + 2) + 3) - 5))^2 - 5(4(4(-3sin(3πx + 2) + 3) - 5)... | 100 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-2|(1|((-1(1x^2 + 4x - 4) + 3)/(-1(1x^2 + 4x - 4) + 2)) - 2| + 1) - 2| + 2) + 3| + 2 and g(x) = -3|(3cos(2π(-5((3x + 3)/(-2x - 1)) + 3) - 1) - 1) + 1| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 119 | MATH |
[
{
"content": "How many solutions does the equation 3|(-1sin(1π(3|(-1(4x)^2 - 5(4x) - 5) + 1| - 1) - 1)) - 3| + 3 = (-3(3|(1(2x^2 - 3x + 2) - 3) - 3| - 2) - 2)/(-3(3|(1(2x^2 - 3x + 2) - 3) - 3| - 2)) have for x in the interval [-10, 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) = (-1(-2sin(2π(-3(-1(2x + 3)^2 + 1(2x + 3) - 3)^2 - 4(-1(2x + 3)^2 + 1(2x + 3) - 3) + 1) + 3)) - 3)/(1(-2sin(2π(-3(-1(2x + 3)^2 + 1(2x + 3) - 3)^2 - 4(-1(2x + 3)^2 + 1(2x + 3) - 3) + 1) + 3)) + 2) and g(x) = 5(-2((-3(-2|x + 3| ... | 242 | MATH |
[
{
"content": "How many solutions does the equation (2((-1((-3(-3e^(-2((2x - 1)/(2x + 3)))) + 3)/(3(-3e^(-2((2x - 1)/(2x + 3)))) + 2)) + 1)/(-2((-3(-3e^(-2((2x - 1)/(2x + 3)))) + 3)/(3(-3e^(-2((2x - 1)/(2x + 3)))) + 2)) + 1)) - 3)/(-3((-1((-3(-3e^(-2((2x - 1)/(2x + 3)))) + 3)/(3(-3e^(-2((2x - 1)/(2x + 3)))) + 2)... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1|(-2((-2((-2((-3x)/(1x - 2)) - 3)/(3((-3x)/(1x - 2)) - 2)) + 3)/(-2((-2((-3x)/(1x - 2)) - 3)/(3((-3x)/(1x - 2)) - 2)) - 3))^2 + 5((-2((-2((-3x)/(1x - 2)) - 3)/(3((-3x)/(1x - 2)) - 2)) + 3)/(-2((-2((-3x)/(1x - 2)) - 3)/(3((-3x)/(1x - 2)) - 2)) - 3)) - 4) + 3| -... | 41 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-5(-3(3(-1|x + 3| - 2)^2 + 5(-1|x + 3| - 2) - 1) + 5) + 4) + 2 and g(x) = -4(-5((2(4x^2 + 3x - 4) - 3)/(-2(4x^2 + 3x - 4) - 2)) - 3)^2 + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 6 | MATH |
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