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(-3(-5(-2sin(1π(-2(-3x^2 - 1x + 2) - 4) + 2) + 1)^2 - 4(-2sin(1π(-2(-3x^2 - 1x + 2) - 4) + 2) + 1) - 2)^2 + 1(-5(-2sin(1π(-2(-3x^2 - 1x + 2) - 4) + 2) + 1)^2 - 4(-2sin(1π(-2(-3x^2 - 1x + 2) - 4) + 2) + 1) - 2) + 2) - 2 and g(x) = ... | 868 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(3(1cos(3π(-1|(-1|(3x^2 - 1x - 3) + 3| - 2) + 1| - 2) + 3) + 2) - 5)^2 + 3(3(1cos(3π(-1|(-1|(3x^2 - 1x - 3) + 3| - 2) + 1| - 2) + 3) + 2) - 5) - 1 and g(x) = (3(-1sin(3π(2|(-2(-1|x + 1| + 1)^2 - 2) - 2| + 3)) - 1) - 3)/(-3(-... | 261 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(-2|(1|(-3(2cos(3π(1x) + 3))^2 + 2(2cos(3π(1x) + 3)) - 3) - 1| - 3) + 3| + 2) - 3| and g(x) = (-3(2(1(-1sin(2π(2x) - 1) + 3)^2 - 1(-1sin(2π(2x) - 1) + 3) - 4)^2 - 5(1(-1sin(2π(2x) - 1) + 3)^2 - 1(-1sin(2π(2x) - 1) + 3) - 4) + 3)... | 29 | MATH |
[
{
"content": "How many solutions does the equation -3(3sin(1π((1(-3|(-3((-3x + 3)/(-1x - 2))^2 + 4((-3x + 3)/(-1x - 2)) - 2) - 3| + 3) - 2)/(1(-3|(-3((-3x + 3)/(-1x - 2))^2 + 4((-3x + 3)/(-1x - 2)) - 2) - 3| + 3) - 2)) - 1) + 3) + 4 = -1((3(-5(-5(3cos(1πx - 3) + 2) - 3)) + 3)/(-2(-5(-5(3cos(1πx - 3) + 2) - 3)) ... | 9 | MATH |
[
{
"content": "How many solutions does the equation (-1(3|(1sin(1π(3(1|((2x + 3)/(-1x + 2)) + 3|) - 2) + 2) + 3) - 2| + 1))/(2(3|(1sin(1π(3(1|((2x + 3)/(-1x + 2)) + 3|) - 2) + 2) + 3) - 2| + 1) - 3) = 4((-2(-3(-3cos(1π(2x + 2) - 1) + 1) - 4) - 1)/(1(-3(-3cos(1π(2x + 2) - 1) + 1) - 4) + 2)) + 3 have for x in the ... | 78 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5((2(3(5(5(-1x^2 + 3x + 1)^2 - 3) + 5)^2 - 4(5(5(-1x^2 + 3x + 1)^2 - 3) + 5) + 5))/(1(3(5(5(-1x^2 + 3x + 1)^2 - 3) + 5)^2 - 4(5(5(-1x^2 + 3x + 1)^2 - 3) + 5) + 5))) and g(x) = -3|(3|(-4((-2(-2x^2 - 2x - 3) - 3)/(-1(-2x^2 - 2x - 3) + 3))^2 + 3((-2(-2x^... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1sin(3π((-3(2(1((-1(-4x^2 - 1x - 2) - 3)/(-3(-4x^2 - 1x - 2) - 2))^2 + 5((-1(-4x^2 - 1x - 2) - 3)/(-3(-4x^2 - 1x - 2) - 2)) + 4) - 4) + 3)/(-2(2(1((-1(-4x^2 - 1x - 2) - 3)/(-3(-4x^2 - 1x - 2) - 2))^2 + 5((-1(-4x^2 - 1x - 2) - 3)/(-... | 116 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2((-1((-3(4(4((3x + 1)/(-1x + 3)) + 5)^2 - 4(4((3x + 1)/(-1x + 3)) + 5) - 3))/(-1(4(4((3x + 1)/(-1x + 3)) + 5)^2 - 4(4((3x + 1)/(-1x + 3)) + 5) - 3) + 3)) + 1)/(-1((-3(4(4((3x + 1)/(-1x + 3)) + 5)^2 - 4(4((3x + 1)/(-1x + 3)... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2((-3((-2(3|(2(-1|x - 1|)^2 - 4(-1|x - 1|) + 2) - 3| - 1) - 1)/(-1(3|(2(-1|x - 1|)^2 - 4(-1|x - 1|) + 2) - 3| - 1))) + 1)/(3((-2(3|(2(-1|x - 1|)^2 - 4(-1|x - 1|) + 2) - 3| - 1) - 1)/(-1(3|(2(-1|x - 1|)^2 - 4(-1|x - 1|) + 2) ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(4(2|(2|((3(4x^2 + 4x))/(-3(4x^2 + 4x) - 1)) - 3| - 3) - 1| - 2) + 3) + 5 and g(x) = -2|(-4((1(-2|(-1x + 2) + 3| - 2))/(-1(-2|(-1x + 2) + 3| - 2)))^2 - 1) - 2| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: ... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(2cos(3π(-2|(2|((-2(-2|x - 2| - 1) - 1)/(2(-2|x - 2| - 1) - 1))| + 3)| + 1) + 1) + 3)^2 + 3(2cos(3π(-2|(2|((-2(-2|x - 2| - 1) - 1)/(2(-2|x - 2| - 1) - 1))| + 3)| + 1) + 1) + 3) - 5 and g(x) = (3((2(-5(-1(5x + 5)^2 + 5(5x + 5) + 4)... | 17 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(1(-4(-1(3(2sin(3πx + 1) + 2) - 3) + 2) - 1) - 4) - 3 and g(x) = -3|((-3((2(1(3x + 2) - 4) - 1)/(-3(1(3x + 2) - 4) - 1)) + 3)/(2((2(1(3x + 2) - 4) - 1)/(-3(1(3x + 2) - 4) - 1)) - 3)) - 3| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent... | 61 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(2|((3((-1((-1(1|x + 2| - 2) + 3)/(-2(1|x + 2| - 2) + 3)) - 3)/(-1((-1(1|x + 2| - 2) + 3)/(-2(1|x + 2| - 2) + 3)) - 2)))/(1((-1((-1(1|x + 2| - 2) + 3)/(-2(1|x + 2| - 2) + 3)) - 3)/(-1((-1(1|x + 2| - 2) + 3)/(-2(1|x + 2| - 2) + 3))... | 4 | MATH |
[
{
"content": "How many solutions does the equation (3(4(4(-1|(-1|(3|x + 2|) + 1| + 3) + 1| + 1) - 3) + 2) - 1)/(2(4(4(-1|(-1|(3|x + 2|) + 1| + 3) + 1| + 1) - 3) + 2) + 3) = 2sin(2π(-3(-2|((-2(-3x + 2))/(-2(-3x + 2) + 3)) - 2| - 1)^2 - 1(-2|((-2(-3x + 2))/(-2(-3x + 2) + 3)) - 2| - 1) - 4) - 1) - 3 have for x in ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3((-3(2|(4(-2sin(2π(2|x + 3| - 3) - 3) - 2)^2 - 2(-2sin(2π(2|x + 3| - 3) - 3) - 2) + 3)| - 3) + 2)/(-1(2|(4(-2sin(2π(2|x + 3| - 3) - 3) - 2)^2 - 2(-2sin(2π(2|x + 3| - 3) - 3) - 2) + 3)| - 3) + 3)) + 3)/(-1((-3(2|(4(-2sin(2π(2|x +... | 1260 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-2(-1|((1(-2|(3cos(1πx - 3) - 3) - 3| - 3) + 2)/(1(-2|(3cos(1πx - 3) - 3) - 3| - 3))) - 3| - 2)^2 - 1(-1|((1(-2|(3cos(1πx - 3) - 3) - 3| - 3) + 2)/(1(-2|(3cos(1πx - 3) - 3) - 3| - 3))) - 3| - 2) + 5)^2 - 2(-2(-1|((1(-2|(3cos(1πx... | 4 | MATH |
[
{
"content": "How many solutions does the equation 3(-3(-3cos(2π(-5(-3|((-2x + 2)/(1x - 3)) - 3| + 3) - 2) - 1))^2 - 3(-3cos(2π(-5(-3|((-2x + 2)/(1x - 3)) - 3| + 3) - 2) - 1)) - 5) + 5 = -3|(1|((-3(-1(1x^2 - 4x + 4)^2 + 3(1x^2 - 4x + 4) + 5) - 2)/(-2(-1(1x^2 - 4x + 4)^2 + 3(1x^2 - 4x + 4) + 5) + 3)) + 3|)| + 3 ... | 332 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3((3(-1|(4(3|(-2x^2 - 1x + 4) + 1| + 2)^2 - 2(3|(-2x^2 - 1x + 4) + 1| + 2) + 2) - 1|) + 3)/(-2(-1|(4(3|(-2x^2 - 1x + 4) + 1| + 2)^2 - 2(3|(-2x^2 - 1x + 4) + 1| + 2) + 2) - 1|) + 3)) - 2 and g(x) = (-2(-2sin(3π(-1((-3(5x^2 - 5x) + 3... | 23 | MATH |
[
{
"content": "How many solutions does the equation -1(-5(3|(1|((1(-1cos(3πx + 1) - 1) - 2)/(-3(-1cos(3πx + 1) - 1) - 1)) + 2| - 3) - 3| - 1)^2 + 2(3|(1|((1(-1cos(3πx + 1) - 1) - 2)/(-3(-1cos(3πx + 1) - 1) - 1)) + 2| - 3) - 3| - 1) + 3)^2 - 3(-5(3|(1|((1(-1cos(3πx + 1) - 1) - 2)/(-3(-1cos(3πx + 1) - 1) - 1)) + 2... | 124 | MATH |
[
{
"content": "How many solutions does the equation -1|(3((-2(-5(1e^(-1(3|x - 3|)))^2 - 4(1e^(-1(3|x - 3|))) - 5) + 1)/(-2(-5(1e^(-1(3|x - 3|)))^2 - 4(1e^(-1(3|x - 3|))) - 5) - 3))^2 + 2((-2(-5(1e^(-1(3|x - 3|)))^2 - 4(1e^(-1(3|x - 3|))) - 5) + 1)/(-2(-5(1e^(-1(3|x - 3|)))^2 - 4(1e^(-1(3|x - 3|))) - 5) - 3)) - 5... | 11 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(2|((-1(-5(-2cos(3π(1x + 5)) - 2) - 4) - 2)/(-2(-5(-2cos(3π(1x + 5)) - 2) - 4))) - 1| + 2)^2 - 1(2|((-1(-5(-2cos(3π(1x + 5)) - 2) - 4) - 2)/(-2(-5(-2cos(3π(1x + 5)) - 2) - 4))) - 1| + 2) + 5 and g(x) = 5(-4(-1(2(1x^2 + 4x + ... | 13 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(2|(2|(3|(2|(-2x^2 - 3x - 1)| + 3) + 1| - 1)| + 1) + 3| - 2) - 3| and g(x) = -5((-2(3(-1|((-3x - 1)/(2x + 1)) - 3| + 3)^2 + 4(-1|((-3x - 1)/(2x + 1)) - 3| + 3) - 5) - 2)/(3(3(-1|((-3x - 1)/(2x + 1)) - 3| + 3)^2 + 4(-1|((-3... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(1|(2((1(5((1x + 3)/(-1x + 2))^2 - 1((1x + 3)/(-1x + 2)) + 1))/(-1(5((1x + 3)/(-1x + 2))^2 - 1((1x + 3)/(-1x + 2)) + 1) - 2))^2 - 1((1(5((1x + 3)/(-1x + 2))^2 - 1((1x + 3)/(-1x + 2)) + 1))/(-1(5((1x + 3)/(-1x + 2))^2 - 1((1x + 3)/(-1x + 2)) + 1) - 2... | 4 | MATH |
[
{
"content": "How many solutions does the equation 2|(-1(1|(-2|(-1e^(1(2|x| - 1))) - 1| + 2) - 3| - 2)^2 + 5(1|(-2|(-1e^(1(2|x| - 1))) - 1| + 2) - 3| - 2) + 3) - 3| = (-1(3(2|(2(-1x - 4) - 3) + 1| + 2)^2 - 3(2|(2(-1x - 4) - 3) + 1| + 2) + 5) + 2)/(-3(3(2|(2(-1x - 4) - 3) + 1| + 2)^2 - 3(2|(2(-1x - 4) - 3) + 1| ... | 4 | MATH |
[
{
"content": "How many solutions does the equation -3|(-2((2(2sin(2π(-2|(2x - 4) - 1|))) - 3)/(-3(2sin(2π(-2|(2x - 4) - 1|))) + 1))) + 1| + 3 = -1sin(2π(-5(-1|(1(-1|x + 2| - 3)^2 + 3(-1|x + 2| - 3) + 4) + 2| - 3) - 1) + 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{... | 324 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(3(-2(-2(1(-2x - 3) - 5)^2 - 2(1(-2x - 3) - 5) - 3)^2 - 5(-2(1(-2x - 3) - 5)^2 - 2(1(-2x - 3) - 5) - 3) + 4)^2 - 3(-2(-2(1(-2x - 3) - 5)^2 - 2(1(-2x - 3) - 5) - 3)^2 - 5(-2(1(-2x - 3) - 5)^2 - 2(1(-2x - 3) - 5) - 3) + 4) - 4) + 1 and g(x) = -1(1(-4((... | 4 | MATH |
[
{
"content": "How many solutions does the equation (-3(2|(-2(-5(-1sin(1π(-4x + 2) - 1) - 1) + 3)^2 + 3(-5(-1sin(1π(-4x + 2) - 1) - 1) + 3) + 5) - 3| - 2))/(2(2|(-2(-5(-1sin(1π(-4x + 2) - 1) - 1) + 3)^2 + 3(-5(-1sin(1π(-4x + 2) - 1) - 1) + 3) + 5) - 3| - 2) - 3) = -3|((1(1(-4(2|x - 2| - 3) + 1) - 5) + 1)/(-3(1(-... | 4 | MATH |
[
{
"content": "How many solutions does the equation -3sin(3π(3(2|((2(-2|((2x - 1)/(-3x + 4)) - 2| + 3) + 1)/(-3(-2|((2x - 1)/(-3x + 4)) - 2| + 3) + 4)) - 2| + 1)^2 - 3(2|((2(-2|((2x - 1)/(-3x + 4)) - 2| + 3) + 1)/(-3(-2|((2x - 1)/(-3x + 4)) - 2| + 3) + 4)) - 2| + 1) + 2) + 2) - 3 = 1(-1|(2|(-4(4x^2 - 3x - 3) + 1... | 15 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(-3cos(2π(-1((-2(5(-3x - 1) - 1) + 3)/(-3(5(-3x - 1) - 1) + 2))^2 - 2((-2(5(-3x - 1) - 1) + 3)/(-3(5(-3x - 1) - 1) + 2)) + 3) + 3)) + 1)/(3(-3cos(2π(-1((-2(5(-3x - 1) - 1) + 3)/(-3(5(-3x - 1) - 1) + 2))^2 - 2((-2(5(-3x - 1... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(-4(3(-2cos(3π(1(-2|x + 2| + 2)^2 - 1) + 1))^2 + 5(-2cos(3π(1(-2|x + 2| + 2)^2 - 1) + 1)) - 5)^2 + 5(3(-2cos(3π(1(-2|x + 2| + 2)^2 - 1) + 1))^2 + 5(-2cos(3π(1(-2|x + 2| + 2)^2 - 1) + 1)) - 5))^2 - 2(-4(3(-2cos(3π(1(-2|x + 2| + 2)^2 - 1) + 1))^2 + 5(-2... | 70 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4((3(-4(-1|((3(3|x + 1| + 3) - 2)/(1(3|x + 1| + 3) - 2)) - 2| + 2)^2 - 4(-1|((3(3|x + 1| + 3) - 2)/(1(3|x + 1| + 3) - 2)) - 2| + 2) - 5) + 1)/(2(-4(-1|((3(3|x + 1| + 3) - 2)/(1(3|x + 1| + 3) - 2)) - 2| + 2)^2 - 4(-1|((3(3|x + 1| + 3) - 2)/(1(3|x + 1| +... | 5 | MATH |
[
{
"content": "How many solutions does the equation 2|(-2(-1(-2e^(-2(-1(2|x - 2|)^2 - 2(2|x - 2|) + 2)) - 1)^2 - 5(-2e^(-2(-1(2|x - 2|)^2 - 2(2|x - 2|) + 2)) - 1) + 5)^2 + 3(-1(-2e^(-2(-1(2|x - 2|)^2 - 2(2|x - 2|) + 2)) - 1)^2 - 5(-2e^(-2(-1(2|x - 2|)^2 - 2(2|x - 2|) + 2)) - 1) + 5))| - 3 = 3|(-1e^(2(5((1((3x + ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(2|((-2(-3(4(2x + 2)^2 - 5(2x + 2) + 3) + 2) - 2)/(-3(-3(4(2x + 2)^2 - 5(2x + 2) + 3) + 2) + 1)) + 1| - 2) + 2| - 2 and g(x) = -3|(1(4(2|(1x^2 - 4)| - 2)^2 - 3(2|(1x^2 - 4)| - 2) + 5)^2 - 5(4(2|(1x^2 - 4)| - 2)^2 - 3(2|(1x^2 - 4)| - 2) + 5) - 2) + 2... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3((2((1(-2((2((3x + 3)/(2x - 3)) - 2)/(-1((3x + 3)/(2x - 3)) - 3)) + 3) + 3)/(1(-2((2((3x + 3)/(2x - 3)) - 2)/(-1((3x + 3)/(2x - 3)) - 3)) + 3) + 1)) + 2)/(2((1(-2((2((3x + 3)/(2x - 3)) - 2)/(-1((3x + 3)/(2x - 3)) - 3)) + 3)... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(2(3cos(3π(-1|(-4(1|x + 1| - 1) + 3) - 1| + 3) - 1) + 2)^2 + 1(3cos(3π(-1|(-4(1|x + 1| - 1) + 3) - 1| + 3) - 1) + 2) + 4) - 2 and g(x) = 3(1((2((-1(2x - 4) - 3)/(-1(2x - 4) + 2)) - 2)/(3((-1(2x - 4) - 3)/(-1(2x - 4) + 2)) - 1))^2 + 3)^2 - 5(1((2((-1(2... | 210 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3((3(5(1|(-1((1x - 1)/(3x))^2 + 2((1x - 1)/(3x)) - 3) + 2| + 1) - 2) - 3)/(2(5(1|(-1((1x - 1)/(3x))^2 + 2((1x - 1)/(3x)) - 3) + 2| + 1) - 2) - 1)) - 2)/(1((3(5(1|(-1((1x - 1)/(3x))^2 + 2((1x - 1)/(3x)) - 3) + 2| + 1) - 2) - 3)/(2(... | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(-1|(-1(3((-1((-2x)/(2x - 1)) - 1)/(-1((-2x)/(2x - 1)) + 2)) - 1)^2 + 1(3((-1((-2x)/(2x - 1)) - 1)/(-1((-2x)/(2x - 1)) + 2)) - 1) - 4) - 3| + 2)^2 - 2(-1|(-1(3((-1((-2x)/(2x - 1)) - 1)/(-1((-2x)/(2x - 1)) + 2)) - 1)^2 + 1(3(... | 97 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(1(1|(-2((-3(3|x| - 1) + 3)/(1(3|x| - 1) + 1)) - 5) + 1| - 2)^2 + 3(1|(-2((-3(3|x| - 1) + 3)/(1(3|x| - 1) + 1)) - 5) + 1| - 2) + 1)^2 - 2(1(1|(-2((-3(3|x| - 1) + 3)/(1(3|x| - 1) + 1)) - 5) + 1| - 2)^2 + 3(1|(-2((-3(3|x| - 1) + 3)/(1(3|x| - 1) + 1)) - ... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(2cos(3π(-3(-1(2(-5x + 4)^2 + 4(-5x + 4) - 3) - 2)^2 + 2(-1(2(-5x + 4)^2 + 4(-5x + 4) - 3) - 2) - 4) - 1) + 3)^2 - 5(2cos(3π(-3(-1(2(-5x + 4)^2 + 4(-5x + 4) - 3) - 2)^2 + 2(-1(2(-5x + 4)^2 + 4(-5x + 4) - 3) - 2) - 4) - 1) + 3) - 5 and g(x) = 2|(2|(-1|... | 125 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-2((-2(-1(1cos(1π(4x - 1) + 2) - 3) - 4) + 1)/(-3(-1(1cos(1π(4x - 1) + 2) - 3) - 4) - 3))^2 + 3((-2(-1(1cos(1π(4x - 1) + 2) - 3) - 4) + 1)/(-3(-1(1cos(1π(4x - 1) + 2) - 3) - 4) - 3)) + 1) + 2| + 1 and g(x) = 1|(-3(2|(-1e^... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(2π(-4(1|((-1((-2(-1x^2 + 4) - 2)/(-1(-1x^2 + 4) + 3)) - 3)/(1((-2(-1x^2 + 4) - 2)/(-1(-1x^2 + 4) + 3)))) + 2| + 2)) - 2) and g(x) = 5(2e^(2((-2(1((-3x + 3)/(-3x + 4)) + 1) + 3)/(3(1((-3x + 3)/(-3x + 4)) + 1) - 2))) - 3) ... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-2cos(2π(3(2((-3(-1x - 3) + 2)/(2(-1x - 3)))^2 + 4((-3(-1x - 3) + 2)/(2(-1x - 3))) + 4) - 4) - 1) + 3)^2 - 3(-2cos(2π(3(2((-3(-1x - 3) + 2)/(2(-1x - 3)))^2 + 4((-3(-1x - 3) + 2)/(2(-1x - 3))) + 4) - 4) - 1) + 3) - 3 and g(x) = (2(2(1(-4(1|x| - 1)^2 +... | 153 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1((3(5(1(2sin(2π(-4x + 5) - 1) - 2) + 4) - 5) - 1)/(-2(5(1(2sin(2π(-4x + 5) - 1) - 2) + 4) - 5) - 1)) + 3)/(3((3(5(1(2sin(2π(-4x + 5) - 1) - 2) + 4) - 5) - 1)/(-2(5(1(2sin(2π(-4x + 5) - 1) - 2) + 4) - 5) - 1))) and g(x) = -1cos(1π... | 61 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(4(1(-5(1sin(3π(-1x - 1) - 1) + 3)^2 - 4(1sin(3π(-1x - 1) - 1) + 3) + 5)^2 + 1(-5(1sin(3π(-1x - 1) - 1) + 3)^2 - 4(1sin(3π(-1x - 1) - 1) + 3) + 5) + 4) - 5) + 2| - 2 and g(x) = 4(3(1(2((2x - 2)/(-1x + 2)) + 3)^2 + 1(2((2x - 2)/(-... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(4(1|(1e^(-1(-1|(-1x - 5)| + 2)) + 1)| - 2)^2 - 4(1|(1e^(-1(-1|(-1x - 5)| + 2)) + 1)| - 2) - 1) - 3| - 1 and g(x) = (-3(-1cos(1π(-1(-2((-1x)/(3x - 1)) + 3)^2 - 3(-2((-1x)/(3x - 1)) + 3) - 4) + 3) - 2))/(1(-1cos(1π(-1(-2((-1... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(-1(3(-4((-3(1cos(3πx - 2) + 1))/(3(1cos(3πx - 2) + 1) + 2)))^2 + 5(-4((-3(1cos(3πx - 2) + 1))/(3(1cos(3πx - 2) + 1) + 2))) - 4) + 3) + 1 and g(x) = -2((3(1(3|((2x - 3)/(-1x - 2)) - 2| + 1)^2 + 1(3|((2x - 3)/(-1x - 2)) - 2| + 1) +... | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(-3cos(1π(-1(-2(2|(-2|x - 1| - 3)| - 1) - 1) - 2) + 1) - 3) + 1)/(2(-3cos(1π(-1(-2(2|(-2|x - 1| - 3)| - 1) - 1) - 2) + 1) - 3) + 3) and g(x) = 2e^(1(1|(5(-1|((-1x - 3)/(2x - 1)) + 1|)^2 - 4(-1|((-1x - 3)/(2x - 1)) + 1|) + 5) - 3... | 24 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-3(2(2sin(3π(-2|(-2|x + 3| - 3) + 3| + 2) - 1))^2 - 3(2sin(3π(-2|(-2|x + 3| - 3) + 3| + 2) - 1)) - 2)^2 + 5(2(2sin(3π(-2|(-2|x + 3| - 3) + 3| + 2) - 1))^2 - 3(2sin(3π(-2|(-2|x + 3| - 3) + 3| + 2) - 1)) - 2) - 1))/(-1(-3(2(2sin(3π(-2|(-2|x + 3| - 3) ... | 122 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(1(-1|(2e^(-1(3|(3|x - 1|) + 2| + 1)) + 2) + 2| - 1) + 4) + 1)/(-2(1(-1|(2e^(-1(3|(3|x - 1|) + 2| + 1)) + 2) + 2| - 1) + 4) + 1) and g(x) = -3cos(3π((-2(2(1(1x + 3)^2 + 2(1x + 3) - 4) + 2) - 3)/(1(2(1(1x + 3)^2 + 2(1x + 3) - 4) + 2) + 3)) + 2) inter... | 86 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(1(-1cos(3π(4((2(5x^2 - 1x - 3))/(3(5x^2 - 1x - 3) + 1))^2 - 5((2(5x^2 - 1x - 3))/(3(5x^2 - 1x - 3) + 1)) + 2)) - 2) + 5) + 5 and g(x) = (1(1cos(2π(5(3(-1x + 4)^2 + 4(-1x + 4) - 2) - 4) + 2) - 3) + 2)/(-1(1cos(2π(5(3(-1x + 4)^2 +... | 164 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-3((-1(-2sin(3π((-2((-2x - 1)/(-3x + 2)))/(-2((-2x - 1)/(-3x + 2)) + 3)) + 2) - 3))/(2(-2sin(3π((-2((-2x - 1)/(-3x + 2)))/(-2((-2x - 1)/(-3x + 2)) + 3)) + 2) - 3) + 1)) + 2) + 2 and g(x) = 2|(2cos(3π((-1((1((3x + 1)/(2x - 1)) + 2)/(2((3x + 1)/(2x - 1... | 9 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-3(-3(4(-1|(-1sin(3πx + 2) + 2) + 3| + 1)) - 5)^2 + 3(-3(4(-1|(-1sin(3πx + 2) + 2) + 3| + 1)) - 5) - 1) + 3)/(2(-3(-3(4(-1|(-1sin(3πx + 2) + 2) + 3| + 1)) - 5)^2 + 3(-3(4(-1|(-1sin(3πx + 2) + 2) + 3| + 1)) - 5) - 1) - 1) and g(... | 887 | MATH |
[
{
"content": "How many solutions does the equation 1(1cos(1π(-4(-3|(-2(1x^2 - 2x + 3) + 1) - 3| - 1)^2 + 2(-3|(-2(1x^2 - 2x + 3) + 1) - 3| - 1) - 4)) + 3) + 3 = 3|(1(-2sin(3π(-1((-1x)/(1x + 2))^2 + 5((-1x)/(1x + 2)) + 2)) - 1)^2 - 2(-2sin(3π(-1((-1x)/(1x + 2))^2 + 5((-1x)/(1x + 2)) + 2)) - 1) - 4) - 3| have for... | 97 | MATH |
[
{
"content": "How many solutions does the equation (2(1(4(-1cos(2π(4(1|x - 2| + 3) + 3) - 3) - 2) - 4) + 4))/(3(1(4(-1cos(2π(4(1|x - 2| + 3) + 3) - 3) - 2) - 4) + 4) + 2) = (-1(3(1cos(3π(4((-2x - 1)/(2x - 1)) - 5) - 3) - 1) + 1) - 1)/(3(3(1cos(3π(4((-2x - 1)/(2x - 1)) - 5) - 3) - 1) + 1) - 2) have for x in the ... | 74 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(3(-1|(-1|(-4(-1x - 2) + 1) + 2|) + 3| - 3) + 5) + 2)/(-2(3(-1|(-1|(-4(-1x - 2) + 1) + 2|) + 3| - 3) + 5) - 3) and g(x) = (-3(-1|(-1(1(-3|x| + 1) + 4)^2 - 3(1(-3|x| + 1) + 4) - 4) + 2| + 1) - 3)/(2(-1|(-1(1(-3|x| + 1) + 4)^... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1|(2|(-2((2(-1|((-2x + 3)/(3x + 3)) + 1| + 2) - 2)/(-1(-1|((-2x + 3)/(3x + 3)) + 1| + 2) + 2)) - 1) - 2| - 2)| + 3 = 1(5(-1sin(2π((2(-2|x + 1| + 1) + 2)/(-2(-2|x + 1| + 1) - 3)) + 2)) - 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTe... | 52 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1sin(3π(-2|(-4((-2(-1((1x + 1)/(-2x)) - 3) + 3)/(3(-1((1x + 1)/(-2x)) - 3)))^2 - 5((-2(-1((1x + 1)/(-2x)) - 3) + 3)/(3(-1((1x + 1)/(-2x)) - 3))) - 2)| + 3) + 3) + 2 and g(x) = (1(2((-1(2|(2sin(3πx - 2) - 1) + 3| + 3) + 1)/(-2(2|(2s... | 46 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(2(2cos(3π((2((2((-3x + 2)/(3x + 2)) + 3)/(1((-3x + 2)/(3x + 2)) - 3)))/(1((2((-3x + 2)/(3x + 2)) + 3)/(1((-3x + 2)/(3x + 2)) - 3)) + 1)) + 2) + 3) - 2) - 3)/(1(2(2cos(3π((2((2((-3x + 2)/(3x + 2)) + 3)/(1((-3x + 2)/(3x + 2)... | 46 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-2sin(2π((-1(-5(4(2|x + 2| - 3)^2 - 3) - 5))/(-3(-5(4(2|x + 2| - 3)^2 - 3) - 5) - 1)) - 1) - 1) - 3)/(-2(-2sin(2π((-1(-5(4(2|x + 2| - 3)^2 - 3) - 5))/(-3(-5(4(2|x + 2| - 3)^2 - 3) - 5) - 1)) - 1) - 1) - 2) and g(x) = -1|(2|(-3c... | 183 | MATH |
[
{
"content": "How many solutions does the equation (-2((-1((3(-2|(4((3x)/(3x + 2)) - 4) - 1|) + 2)/(3(-2|(4((3x)/(3x + 2)) - 4) - 1|) + 1)))/(1((3(-2|(4((3x)/(3x + 2)) - 4) - 1|) + 2)/(3(-2|(4((3x)/(3x + 2)) - 4) - 1|) + 1)))) - 1)/(2((-1((3(-2|(4((3x)/(3x + 2)) - 4) - 1|) + 2)/(3(-2|(4((3x)/(3x + 2)) - 4) - 1|... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-2sin(3π(-1(3|((-1(5x) - 1)/(2(5x) + 3)) - 2| + 3)^2 - 2(3|((-1(5x) - 1)/(2(5x) + 3)) - 2| + 3) + 4) + 2) + 1) + 3| + 3 and g(x) = (-2(1(3(2(4x^2 + 5x - 5) + 5) - 3) - 2) + 2)/(2(1(3(2(4x^2 + 5x - 5) + 5) - 3) - 2) - 1) w... | 116 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2sin(1π(1(4(5(-1|(1|x + 3| + 3) + 1| + 1) - 2)^2 - 2(5(-1|(1|x + 3| + 3) + 1| + 1) - 2) + 3)^2 + 1(4(5(-1|(1|x + 3| + 3) + 1| + 1) - 2)^2 - 2(5(-1|(1|x + 3| + 3) + 1| + 1) - 2) + 3) - 4) - 3) - 1 and g(x) = -2((1((2((-2(-5x ... | 52 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-2(2sin(2π(3(5(-1x^2 - 1x) - 5) + 1) + 3) - 3)^2) - 2)/(1(-2(2sin(2π(3(5(-1x^2 - 1x) - 5) + 1) + 3) - 3)^2) - 3) and g(x) = -3|(-3((-3(-5(-2e^(1x) + 2)^2 - 4(-2e^(1x) + 2) + 2) - 2)/(-3(-5(-2e^(1x) + 2)^2 - 4(-2e^(1x) + 2) + 2) + 2))^2 - 2((-3(-5(-... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(2|(-2|(-2|((3(-3|x - 2| + 1) + 3)/(-1(-3|x - 2| + 1) + 3)) + 1| + 2) - 2|) + 1| - 3) + 2 and g(x) = -3sin(2π((-2(-1|(-3(5x^2 - 1x + 1) - 3) + 2| + 2) - 2)/(-1(-1|(-3(5x^2 - 1x + 1) - 3) + 2| + 2) + 2)) + 1) + 3 intersect in the interval [-10, 10].\n... | 6 | MATH |
[
{
"content": "How many solutions does the equation (1((3(-4(5(2|(5x - 4) + 2| + 1) + 2)) - 2)/(1(-4(5(2|(5x - 4) + 2| + 1) + 2)) + 3)) - 3)/(2((3(-4(5(2|(5x - 4) + 2| + 1) + 2)) - 2)/(1(-4(5(2|(5x - 4) + 2| + 1) + 2)) + 3)) - 1) = 1((-1(-1(-3|(-1x^2 - 4x - 1) + 3| - 3)^2 - 4(-3|(-1x^2 - 4x - 1) + 3| - 3) + 4) -... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(1|(5(1e^(-1(-1(1x + 3)^2 - 1(1x + 3) + 1)) - 1)^2 - 4(1e^(-1(-1(1x + 3)^2 - 1(1x + 3) + 1)) - 1) - 1) - 3| - 3) - 3)/(-3(1|(5(1e^(-1(-1(1x + 3)^2 - 1(1x + 3) + 1)) - 1)^2 - 4(1e^(-1(-1(1x + 3)^2 - 1(1x + 3) + 1)) - 1) - 1)... | 1434 | MATH |
[
{
"content": "How many solutions does the equation -4(-1|((-1(1e^(1(-5(2|x - 1| + 3) + 2)) - 3) - 3)/(1(1e^(1(-5(2|x - 1| + 3) + 2)) - 3))) + 1| - 1) - 5 = -5(1(1|(-2(-4x^2 + 1x + 2) - 4) - 1| - 2)^2 - 4(1|(-2(-4x^2 + 1x + 2) - 4) - 1| - 2) + 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(1(-3(2|(-4(-2cos(3πx + 3) - 1) + 2) - 3|)^2 - 3(2|(-4(-2cos(3πx + 3) - 1) + 2) - 3|))^2 - 5(-3(2|(-4(-2cos(3πx + 3) - 1) + 2) - 3|)^2 - 3(2|(-4(-2cos(3πx + 3) - 1) + 2) - 3|))) - 2| - 2 and g(x) = (-1(2sin(3π(1|(-3(1x^2 - ... | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|((1(5(5(3(-3x) - 5)^2 - 2(3(-3x) - 5) - 5)^2 + 1) + 2)/(-3(5(5(3(-3x) - 5)^2 - 2(3(-3x) - 5) - 5)^2 + 1))) + 1| and g(x) = (3(-2|(5(-3cos(2π(-2|x + 1| + 3) + 2) - 2) + 1)| + 2) + 3)/(-3(-2|(5(-3cos(2π(-2|x + 1| + 3) + 2) - 2) + 1)| + 2) + 2) intersec... | 108 | MATH |
[
{
"content": "How many solutions does the equation -3(-5(1(-1|(-1|(-2x^2 + 1x - 3) + 3| + 1) - 2| - 2) + 4)^2 - 2(1(-1|(-1|(-2x^2 + 1x - 3) + 3| + 1) - 2| - 2) + 4) + 1) + 3 = 1|(-3|(-1|((-3(3e^(1x) - 2) + 3)/(-1(3e^(1x) - 2) + 2)) + 1| - 3) + 3| - 1) - 3| + 3 have for x in the interval [-10, 10]?\n\nPresent th... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(4(-2sin(2π(-2((3(4x - 5) - 1)/(-2(4x - 5) + 1))^2 + 5((3(4x - 5) - 1)/(-2(4x - 5) + 1)) - 5) - 3) + 1)^2 - 5(-2sin(2π(-2((3(4x - 5) - 1)/(-2(4x - 5) + 1))^2 + 5((3(4x - 5) - 1)/(-2(4x - 5) + 1)) - 5) - 3) + 1) + 4)^2 - 5(4(-2sin(2π(-2((3(4x - 5) - 1)... | 22 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(4((-2(-3(-5(-3|x + 1|) + 3) + 2) + 2)/(-2(-3(-5(-3|x + 1|) + 3) + 2) - 1)) - 5) + 1 and g(x) = (-3(-3|(2cos(1π((3(-2x - 5) + 2)/(1(-2x - 5))) - 3) - 2) + 1|) - 1)/(3(-3|(2cos(1π((3(-2x - 5) + 2)/(1(-2x - 5))) - 3) - 2) + 1|... | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-1|(1(-2(3|(-4x^2 + 1x) + 1| - 2)) + 2) - 2|) + 2)/(1(-1|(1(-2(3|(-4x^2 + 1x) + 1| - 2)) + 2) - 2|)) and g(x) = (2((2(-1e^(-2(-1|(1x^2 - 5x - 3)| + 3)) + 3))/(-1(-1e^(-2(-1|(1x^2 - 5x - 3)| + 3)) + 3) + 2)) - 3)/(-1((2(-1... | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-1(-3|(5(-2sin(2π(4x + 1) - 2) - 2) + 4) + 1| + 3) - 1)^2 + 3(-1(-3|(5(-2sin(2π(4x + 1) - 2) - 2) + 4) + 1| + 3) - 1) - 4 and g(x) = -3(-1(2(-4(1x^2 + 3x) - 3) + 1)^2 + 3(2(-4(1x^2 + 3x) - 3) + 1) + 4)^2 + 3(-1(2(-4(1x^2 +... | 16 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(2cos(1π((-3(4(4(-3|x - 2| + 3)^2 - 3(-3|x - 2| + 3) - 5) + 3) + 1)/(-2(4(4(-3|x - 2| + 3)^2 - 3(-3|x - 2| + 3) - 5) + 3) + 3))) + 2)^2 - 3 and g(x) = 1|(5(-3e^(-1(1((2x + 3)/(3x + 3))^2 - 1((2x + 3)/(3x + 3)) + 5)) - 2)^2 - 5(-3e... | 18 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(-2sin(2π(-3|(-1|(-3|(-2|x - 3| - 1) + 2| - 3) + 2| - 2) - 2| + 2) - 1))^2 - 4(-2sin(2π(-3|(-1|(-3|(-2|x - 3| - 1) + 2| - 3) + 2| - 2) - 2| + 2) - 1)) and g(x) = -3(3|(-2|(5((-2x - 1)/(-2x - 2)) - 1) - 1| + 3) - 1| - 1) - 2... | 63 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(-2|(2(-4((-1((-3x - 1)/(-1x + 2)) + 3)/(1((-3x - 1)/(-1x + 2)) + 2)) - 3)^2 - 4(-4((-1((-3x - 1)/(-1x + 2)) + 3)/(1((-3x - 1)/(-1x + 2)) + 2)) - 3) - 4) + 3|) - 1 and g(x) = (2(-1(-1((-1(-3cos(3πx - 3) - 1) - 2)/(2(-3cos(3πx - 3) - 1) + 1))^2 + 2((-... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(3π(3(-1|(-2|((-1(2|x + 1| + 1) + 2)/(1(2|x + 1| + 1))) - 3| - 1) - 2| - 1) + 5)) - 3 and g(x) = -2(4(-5(-3(-2x + 5)^2 + 2) - 2)^2 + 1(-5(-3(-2x + 5)^2 + 2) - 2))^2 - 4(4(-5(-3(-2x + 5)^2 + 2) - 2)^2 + 1(-5(-3(-2x + 5)^2 ... | 4 | MATH |
[
{
"content": "How many solutions does the equation 3(-1|(-3(-1(1|((3x + 1)/(-3x - 3)) + 3|)^2 - 5(1|((3x + 1)/(-3x - 3)) + 3|) - 1)) - 3| + 3)^2 + 5(-1|(-3(-1(1|((3x + 1)/(-3x - 3)) + 3|)^2 - 5(1|((3x + 1)/(-3x - 3)) + 3|) - 1)) - 3| + 3) + 3 = 1(5((1(-3sin(2π(1x + 4) - 2) - 3) - 2)/(3(-3sin(2π(1x + 4) - 2) - 3... | 41 | MATH |
[
{
"content": "How many solutions does the equation 3|(2sin(2π(1|(2|((-2(-4x + 4) + 3)/(-1(-4x + 4) + 2)) - 3| + 2) - 3| + 1) - 2)) - 1| - 3 = -2cos(3π((1((1((3(3x - 4) + 1)/(3(3x - 4) - 2)))/(2((3(3x - 4) + 1)/(3(3x - 4) - 2)) + 1)))/(-1((1((3(3x - 4) + 1)/(3(3x - 4) - 2)))/(2((3(3x - 4) + 1)/(3(3x - 4) - 2)) +... | 47 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(2|((1(-3|(4((2x - 2)/(2x - 1)) + 3) + 2| - 3) + 1)/(-1(-3|(4((2x - 2)/(2x - 1)) + 3) + 2| - 3) + 2))|) - 3 and g(x) = -2|(1((1(3(-1x^2 + 4x + 3) + 4) + 3)/(1(3(-1x^2 + 4x + 3) + 4)))^2 - 3) + 3| + 3 intersect in the interval [-10, 10].\n\nPresent the... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1(3(-2|(5(1((-2x + 1)/(1x - 2)))^2 + 5(1((-2x + 1)/(1x - 2))) - 1) + 1| + 2) + 3)^2 - 4(3(-2|(5(1((-2x + 1)/(1x - 2)))^2 + 5(1((-2x + 1)/(1x - 2))) - 1) + 1| + 2) + 3) + 3 = 2|(-1|(2(2cos(1π(-4x^2 + 3x + 5)) - 2)^2 + 4(2cos(1π(-4x^2 + 3x + 5)) - 2)) - 1|) - 1|... | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(-1cos(1π(-5(2|(-1|(1x - 3) + 1|) - 3|) - 1)) + 2) - 4 and g(x) = 1(-5((2(-1((2x + 3)/(-2x - 2))^2 + 4((2x + 3)/(-2x - 2)) + 3) - 3)/(2(-1((2x + 3)/(-2x - 2))^2 + 4((2x + 3)/(-2x - 2)) + 3))) - 1) - 1 in the range -10 ≤ x ≤ 10?\n\... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1(-2((-2(1|(-3(1x + 5)^2 - 1(1x + 5) - 1) + 3| - 2) - 3)/(2(1|(-3(1x + 5)^2 - 1(1x + 5) - 1) + 3| - 2) - 1))^2 - 4((-2(1|(-3(1x + 5)^2 - 1(1x + 5) - 1) + 3| - 2) - 3)/(2(1|(-3(1x + 5)^2 - 1(1x + 5) - 1) + 3| - 2) - 1)) + 4)^2 + 4(-2((-2(1|(-3(1x + 5)^2 - 1(1x ... | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(4((2(1(-2|(1|x + 1| - 3) - 2| + 1)^2 - 1(-2|(1|x + 1| - 3) - 2| + 1)) - 3)/(-2(1(-2|(1|x + 1| - 3) - 2| + 1)^2 - 1(-2|(1|x + 1| - 3) - 2| + 1)) + 1)) - 3) + 1| + 1 and g(x) = -3(-1((-1((1(-2x + 1) - 2)/(-2(-2x + 1) - 3)) -... | 5 | MATH |
[
{
"content": "How many solutions does the equation 2(-2(1|(5(-3|(2|x + 3| - 1) - 2| + 1) + 1) - 1| + 3) - 3) = -2|(-2|((-2(-2sin(3π(-1|x + 3| - 3))) + 2)/(1(-2sin(3π(-1|x + 3| - 3))))) - 3| - 2) - 3| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 54 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-3|(2|(5(-1((-1x - 2)/(-2x - 2))^2 + 4((-1x - 2)/(-2x - 2)) - 4)^2 - 2(-1((-1x - 2)/(-2x - 2))^2 + 4((-1x - 2)/(-2x - 2)) - 4)) - 1| + 2) - 3| + 2) - 2 and g(x) = 2cos(1π((-1(-4(3((-1x + 1)/(-3x - 1))^2 + 3((-1x + 1)/(-3x - 1)) + 3) - 5) - 2)/(-1(-4(... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(1π(-3(2|(1|(-2|(-1|x - 3| + 1) - 2| - 1) + 2| + 1) + 2| - 1)^2 - 2(2|(1|(-2|(-1|x - 3| + 1) - 2| - 1) + 2| + 1) + 2| - 1) + 1) + 3) - 2 and g(x) = -1|(1((-1(-1(3cos(3πx - 1) - 2)^2 + 2(3cos(3πx - 1) - 2) - 1) - 2)/(-1(-1... | 1581 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3sin(2π(1|(-3(1(-2|(1|x + 3|) - 2| - 3) - 3)^2 + 4(1(-2|(1|x + 3|) - 2| - 3) - 3) - 1) + 2| + 1) - 2) - 1 and g(x) = 2(-3(1(-4(-2|x - 3| + 3)) - 2)) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your... | 16 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-1|(-2sin(2π(-1((-1(3|x + 2| + 1) - 3)/(-1(3|x + 2| + 1) + 3))^2 + 5((-1(3|x + 2| + 1) - 3)/(-1(3|x + 2| + 1) + 3)) - 2) - 2)) + 2| - 2) + 1| - 3 and g(x) = -2cos(3π((-2(2((-2((3x + 2)/(3x + 2)))/(-3((3x + 2)/(3x + 2))))^2 + 1((... | 202 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(3|(1sin(1π(1(1((3x + 2)/(2x - 1)) - 3)^2 - 3(1((3x + 2)/(2x - 1)) - 3) - 3) + 2) + 3) - 2| - 2) - 3)/(3(3|(1sin(1π(1(1((3x + 2)/(2x - 1)) - 3)^2 - 3(1((3x + 2)/(2x - 1)) - 3) - 3) + 2) + 3) - 2| - 2) - 2) and g(x) = 1|(-3(-3|((-3(3cos(2πx + 1) - 1)... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1sin(3π((-1(4((3(2(-3|x + 3|) + 1) + 1)/(-1(2(-3|x + 3|) + 1) + 3))^2 + 5((3(2(-3|x + 3|) + 1) + 1)/(-1(2(-3|x + 3|) + 1) + 3)) - 5))/(1(4((3(2(-3|x + 3|) + 1) + 1)/(-1(2(-3|x + 3|) + 1) + 3))^2 + 5((3(2(-3|x + 3|) + 1) + 1)/(-1(2(-3|x + 3|) + 1) + 3)) - 5) - ... | 30 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(-4(-3(3|(5((-2x + 1)/(-1x + 2))^2 + 3((-2x + 1)/(-1x + 2)) - 1) - 1| + 1)^2 + 1(3|(5((-2x + 1)/(-1x + 2))^2 + 3((-2x + 1)/(-1x + 2)) - 1) - 1| + 1) + 4) - 2) - 4 and g(x) = 2((1(-4(1(3x)^2 - 3(3x) + 1)^2 - 1(1(3x)^2 - 3(3x) + 1) + 4) - 3)/(-1(-4(1(3... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-2(5(-1|(-2(1x^2 - 3x + 1)^2 + 5(1x^2 - 3x + 1) + 2) - 1| + 2)^2 - 1(-1|(-2(1x^2 - 3x + 1)^2 + 5(1x^2 - 3x + 1) + 2) - 1| + 2))^2 + 4(5(-1|(-2(1x^2 - 3x + 1)^2 + 5(1x^2 - 3x + 1) + 2) - 1| + 2)^2 - 1(-1|(-2(1x^2 - 3x + 1)^2 + 5(1x^2 - 3x + 1) + 2) - ... | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(1((3(-3(-2sin(2π(-4x - 2) + 3) - 2)^2 + 1(-2sin(2π(-4x - 2) + 3) - 2) + 5) + 1)/(-1(-3(-2sin(2π(-4x - 2) + 3) - 2)^2 + 1(-2sin(2π(-4x - 2) + 3) - 2) + 5) + 2)) + 3)^2 - 1(1((3(-3(-2sin(2π(-4x - 2) + 3) - 2)^2 + 1(-2sin(2π(-... | 160 | MATH |
[
{
"content": "How many solutions does the equation 3|(3(-3(1((-2(-2x^2 - 3x))/(2(-2x^2 - 3x)))^2 + 1((-2(-2x^2 - 3x))/(2(-2x^2 - 3x))) + 4)^2 + 2(1((-2(-2x^2 - 3x))/(2(-2x^2 - 3x)))^2 + 1((-2(-2x^2 - 3x))/(2(-2x^2 - 3x))) + 4) + 2)^2 + 4(-3(1((-2(-2x^2 - 3x))/(2(-2x^2 - 3x)))^2 + 1((-2(-2x^2 - 3x))/(2(-2x^2 - 3... | 6 | MATH |
[
{
"content": "How many solutions does the equation (3(-3cos(3π(-1|(2|((-1(-1|x + 2| - 1) - 3)/(2(-1|x + 2| - 1) + 2)) - 3| - 3) + 3| - 3) - 2)) + 1)/(2(-3cos(3π(-1|(2|((-1(-1|x + 2| - 1) - 3)/(2(-1|x + 2| - 1) + 2)) - 3| - 3) + 3| - 3) - 2)) + 1) = (1(-3((-3((-2(-4x - 3) - 3)/(2(-4x - 3) - 3)))/(2((-2(-4x - 3) ... | 32 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(1(1|((-1(-4(-3|x + 1|) + 1) + 2)/(-2(-4(-3|x + 1|) + 1) + 3)) + 1| - 1)) - 1)/(2(1(1|((-1(-4(-3|x + 1|) + 1) + 2)/(-2(-4(-3|x + 1|) + 1) + 3)) + 1| - 1)) + 2) and g(x) = -2|(2e^(1(4((3(-1x^2 - 1x + 2) + 3)/(-2(-1x^2 - 1x + 2)))... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3((-3((-1(-1(-3(3x + 2) + 2)^2 - 3(-3(3x + 2) + 2) - 5) + 1)/(-1(-1(-3(3x + 2) + 2)^2 - 3(-3(3x + 2) + 2) - 5))) + 2)/(-2((-1(-1(-3(3x + 2) + 2)^2 - 3(-3(3x + 2) + 2) - 5) + 1)/(-1(-1(-3(3x + 2) + 2)^2 - 3(-3(3x + 2) + 2) - 5))) ... | 345 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(5(-1((3(5(1|x| + 2) - 2) + 1)/(3(5(1|x| + 2) - 2) - 2))^2 - 2((3(5(1|x| + 2) - 2) + 1)/(3(5(1|x| + 2) - 2) - 2)) + 5)^2 + 3) - 1)/(-2(5(-1((3(5(1|x| + 2) - 2) + 1)/(3(5(1|x| + 2) - 2) - 2))^2 - 2((3(5(1|x| + 2) - 2) + 1)/(... | 4 | MATH |
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