messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation 4((-1(-1|(-3sin(2π(2(3x^2 + 4x + 3) - 5) - 2) + 2) - 2| - 1) - 1)/(-3(-1|(-3sin(2π(2(3x^2 + 4x + 3) - 5) - 2) + 2) - 2| - 1))) + 1 = -3(-3sin(2π(1|((2(-3|x - 3| + 1) - 1)/(1(-3|x - 3| + 1))) + 1| - 2) + 2) - 3)^2 + 5(-3sin(2π(1|((2(-3|x - 3| + 1) - 1)/(1(-3|x -... | 30 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-5(-3e^(1(-1(1(2|x - 2|)^2 + 4) + 3)) - 2)^2 + 4(-3e^(1(-1(1(2|x - 2|)^2 + 4) + 3)) - 2) + 5)^2 + 2(-5(-3e^(1(-1(1(2|x - 2|)^2 + 4) + 3)) - 2)^2 + 4(-3e^(1(-1(1(2|x - 2|)^2 + 4) + 3)) - 2) + 5) + 4 and g(x) = -3|(-5((2(4((-2x - 1)/(-3x + 4))^2 - 2((... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5((-2(-2|((3(1sin(2π((1x - 3)/(-2x - 1)) + 1) - 1) + 3)/(1(1sin(2π((1x - 3)/(-2x - 1)) + 1) - 1) + 3)) - 3| + 3) - 2)/(2(-2|((3(1sin(2π((1x - 3)/(-2x - 1)) + 1) - 1) + 3)/(1(1sin(2π((1x - 3)/(-2x - 1)) + 1) - 1) + 3)) - 3| +... | 1993 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-1(1|(-1((1(-1x^2 + 4x + 3))/(-1(-1x^2 + 4x + 3) + 3))^2 + 3((1(-1x^2 + 4x + 3))/(-1(-1x^2 + 4x + 3) + 3)) + 1) - 2| - 1)^2 - 3(1|(-1((1(-1x^2 + 4x + 3))/(-1(-1x^2 + 4x + 3) + 3))^2 + 3((1(-1x^2 + 4x + 3))/(-1(-1x^2 + 4x +... | 8 | MATH |
[
{
"content": "How many solutions does the equation 2|(3|(5((3(3|(3|x - 1|) - 2| - 3) + 2)/(-1(3|(3|x - 1|) - 2| - 3)))) + 3| + 3) + 3| - 1 = 3((-2(3|(-2(1cos(3πx + 1) + 1)^2 - 2(1cos(3πx + 1) + 1) - 2) - 3| - 1))/(1(3|(-2(1cos(3πx + 1) + 1)^2 - 2(1cos(3πx + 1) + 1) - 2) - 3| - 1) - 2))^2 - 1((-2(3|(-2(1cos(3πx ... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(4(1(-1|(-4(-3|x| - 1)^2 - 4(-3|x| - 1) - 2) - 1| + 3)^2 + 2) + 1) + 1)/(-3(4(1(-1|(-4(-3|x| - 1)^2 - 4(-3|x| - 1) - 2) - 1| + 3)^2 + 2) + 1) - 3) and g(x) = 3sin(3π(4(3(-2|(-3|x + 3|) - 2| - 2)^2 - 4(-2|(-3|x + 3|) - 2| - 2))^2 ... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(3((-3((1(-3cos(2π(-5x^2 + 4x - 1) - 2)) - 3)/(3(-3cos(2π(-5x^2 + 4x - 1) - 2)) - 1)) - 3)/(1((1(-3cos(2π(-5x^2 + 4x - 1) - 2)) - 3)/(3(-3cos(2π(-5x^2 + 4x - 1) - 2)) - 1))))^2 + 2((-3((1(-3cos(2π(-5x^2 + 4x - 1) - 2)) - 3)/(3(-3cos(2π(-5x^2 + 4x - ... | 187 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(1(2((-1(3|(-4x^2 + 1x - 1) + 3| + 2) - 3)/(-2(3|(-4x^2 + 1x - 1) + 3| + 2) + 1)) + 5) - 3) - 3)/(1(1(2((-1(3|(-4x^2 + 1x - 1) + 3| + 2) - 3)/(-2(3|(-4x^2 + 1x - 1) + 3| + 2) + 1)) + 5) - 3) + 1) and g(x) = -4(-3cos(3π(2|(-3|(3x... | 495 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5((1(-1|((-1(-1|(-3|x - 2| - 1) + 1|) - 3)/(-2(-1|(-3|x - 2| - 1) + 1|) - 1))| - 2) + 1)/(3(-1|((-1(-1|(-3|x - 2| - 1) + 1|) - 3)/(-2(-1|(-3|x - 2| - 1) + 1|) - 1))| - 2) - 2)) - 3 and g(x) = -3(-2|((2(3(3e^(-2x))^2 + 4(3e^(-2x)) ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|((2(-1(5((-2(2x^2 - 2x - 5) + 2)/(2(2x^2 - 2x - 5) - 1)) + 4)^2 + 3(5((-2(2x^2 - 2x - 5) + 2)/(2(2x^2 - 2x - 5) - 1)) + 4) - 1) - 3)/(-1(-1(5((-2(2x^2 - 2x - 5) + 2)/(2(2x^2 - 2x - 5) - 1)) + 4)^2 + 3(5((-2(2x^2 - 2x - 5) ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-3(-5(4(1|(-2x + 3) - 2| + 1)^2 + 1(1|(-2x + 3) - 2| + 1) + 3) - 4)^2 + 1(-5(4(1|(-2x + 3) - 2| + 1)^2 + 1(1|(-2x + 3) - 2| + 1) + 3) - 4) + 1) + 2)/(1(-3(-5(4(1|(-2x + 3) - 2| + 1)^2 + 1(1|(-2x + 3) - 2| + 1) + 3) - 4)^2 + 1(-5(4(1|(-2x + 3) - 2| +... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-3(-2(3|(2((-2x - 2)/(-2x - 2)) + 1) + 3| + 3) + 4)^2 - 4(-2(3|(2((-2x - 2)/(-2x - 2)) + 1) + 3| + 3) + 4) + 4))/(3(-3(-2(3|(2((-2x - 2)/(-2x - 2)) + 1) + 3| + 3) + 4)^2 - 4(-2(3|(2((-2x - 2)/(-2x - 2)) + 1) + 3| + 3) + 4) + 4) - 1) and g(x) = -5(-... | 40 | MATH |
[
{
"content": "How many solutions does the equation -5(2(1((1((-1((-3x)/(1x)) + 2)/(3((-3x)/(1x)) - 2)))/(-1((-1((-3x)/(1x)) + 2)/(3((-3x)/(1x)) - 2)) - 3))^2 - 5((1((-1((-3x)/(1x)) + 2)/(3((-3x)/(1x)) - 2)))/(-1((-1((-3x)/(1x)) + 2)/(3((-3x)/(1x)) - 2)) - 3)) - 2))^2 - 4(2(1((1((-1((-3x)/(1x)) + 2)/(3((-3x)/(1x... | 6 | MATH |
[
{
"content": "How many solutions does the equation -3|((2(-1(-5(4((3x + 3)/(1x + 3))^2 + 5((3x + 3)/(1x + 3)) + 4)^2 + 4(4((3x + 3)/(1x + 3))^2 + 5((3x + 3)/(1x + 3)) + 4))^2 + 2(-5(4((3x + 3)/(1x + 3))^2 + 5((3x + 3)/(1x + 3)) + 4)^2 + 4(4((3x + 3)/(1x + 3))^2 + 5((3x + 3)/(1x + 3)) + 4)) - 4) + 3)/(-1(-1(-5(4... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-2cos(3π(2((3((-1(-5x) - 2)/(-1(-5x) + 2)) + 3)/(-3((-1(-5x) - 2)/(-1(-5x) + 2)) - 2))^2 - 4((3((-1(-5x) - 2)/(-1(-5x) + 2)) + 3)/(-3((-1(-5x) - 2)/(-1(-5x) + 2)) - 2)) + 3) - 1) + 3) + 5 and g(x) = 1|(5(1|(-5((-3x + 2)/(1x + 3))... | 7 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(4(-2(-5((-3(-4x) - 2)/(3(-4x) + 3))^2 - 5((-3(-4x) - 2)/(3(-4x) + 3)) + 1)^2 - 1(-5((-3(-4x) - 2)/(3(-4x) + 3))^2 - 5((-3(-4x) - 2)/(3(-4x) + 3)) + 1) - 5) + 1) - 3)/(-2(4(-2(-5((-3(-4x) - 2)/(3(-4x) + 3))^2 - 5((-3(-4x) - 2)/(3... | 865 | MATH |
[
{
"content": "How many solutions does the equation -1|(-1(-2((1((1((2x + 3)/(-1x + 2)) + 1)/(2((2x + 3)/(-1x + 2)) - 3)) + 2)/(3((1((2x + 3)/(-1x + 2)) + 1)/(2((2x + 3)/(-1x + 2)) - 3))))^2 - 3((1((1((2x + 3)/(-1x + 2)) + 1)/(2((2x + 3)/(-1x + 2)) - 3)) + 2)/(3((1((2x + 3)/(-1x + 2)) + 1)/(2((2x + 3)/(-1x + 2))... | 5 | MATH |
[
{
"content": "How many solutions does the equation -1(4(-3sin(2π(-5(2|(2|x + 3| - 3) - 3|) - 2) + 3) + 2) + 3)^2 - 2(4(-3sin(2π(-5(2|(2|x + 3| - 3) - 3|) - 2) + 3) + 2) + 3) + 1 = 1((-3(3cos(1π(1|((2x)/(-1x + 2))| + 1) - 1) + 3) + 2)/(-1(3cos(1π(1|((2x)/(-1x + 2))| + 1) - 1) + 3) + 2))^2 + 4((-3(3cos(1π(1|((2x)... | 142 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(5(1((2((-3(2x + 5))/(2(2x + 5) + 3)) + 1)/(2((-3(2x + 5))/(2(2x + 5) + 3)) - 1))^2 - 4((2((-3(2x + 5))/(2(2x + 5) + 3)) + 1)/(2((-3(2x + 5))/(2(2x + 5) + 3)) - 1)) + 2)^2 + 1)^2 - 3(5(1((2((-3(2x + 5))/(2(2x + 5) + 3)) + 1)/(2((-... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3sin(3π(1|(-1|(-3(-4(5x^2 + 2x)^2 + 2(5x^2 + 2x) + 5) + 3) + 2| - 1) + 1| + 3)) + 3 and g(x) = 2cos(1π((-3(3((-1(2|x - 2| + 2) + 2)/(-1(2|x - 2| + 2) + 2))^2 - 3((-1(2|x - 2| + 2) + 2)/(-1(2|x - 2| + 2) + 2)) + 3) - 1)/(3(3((-1(2|x - 2| + 2) + 2)/(-1(2... | 341 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(-2(2|(-2cos(1π(1((3x + 1)/(2x - 1)) + 4) + 1)) + 1| + 2)^2 - 1(2|(-2cos(1π(1((3x + 1)/(2x - 1)) + 4) + 1)) + 1| + 2) - 3) - 3)/(3(-2(2|(-2cos(1π(1((3x + 1)/(2x - 1)) + 4) + 1)) + 1| + 2)^2 - 1(2|(-2cos(1π(1((3x + 1)/(2x - ... | 17 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-3((2(2cos(1π(-3|(-1x^2 - 1) - 1| + 1) + 2) - 2) + 1)/(-2(2cos(1π(-3|(-1x^2 - 1) - 1| + 1) + 2) - 2) - 3))^2 - 4((2(2cos(1π(-3|(-1x^2 - 1) - 1| + 1) + 2) - 2) + 1)/(-2(2cos(1π(-3|(-1x^2 - 1) - 1| + 1) + 2) - 2) - 3)) - 2) - 2)/... | 30 | MATH |
[
{
"content": "How many solutions does the equation 3|(-1((2((2((-3(1x - 2) - 2)/(3(1x - 2) + 1)) + 3)/(2((-3(1x - 2) - 2)/(3(1x - 2) + 1)) + 3)) + 2)/(-1((2((-3(1x - 2) - 2)/(3(1x - 2) + 1)) + 3)/(2((-3(1x - 2) - 2)/(3(1x - 2) + 1)) + 3)) - 2))^2 + 3((2((2((-3(1x - 2) - 2)/(3(1x - 2) + 1)) + 3)/(2((-3(1x - 2) -... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-2(1((-3(-3(-2|x + 3| - 1)^2 + 1(-2|x + 3| - 1) + 5))/(-1(-3(-2|x + 3| - 1)^2 + 1(-2|x + 3| - 1) + 5) + 2))^2 + 4((-3(-3(-2|x + 3| - 1)^2 + 1(-2|x + 3| - 1) + 5))/(-1(-3(-2|x + 3| - 1)^2 + 1(-2|x + 3| - 1) + 5) + 2)) + 4)) + 1)/(-2(-2(1((-3(-3(-2|x ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(1(-3sin(2π(3(-2|(2x - 1) - 2| - 1) + 4) + 3) + 1)^2 - 1) + 2 and g(x) = (-3(-3cos(3π(-3(-2(1x^2 + 3x + 4)^2 + 4(1x^2 + 3x + 4) - 1) - 1) + 3) - 1) - 1)/(1(-3cos(3π(-3(-2(1x^2 + 3x + 4)^2 + 4(1x^2 + 3x + 4) - 1) - 1) + 3) - 1) + ... | 200 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2cos(2π(-5(-2|((-2(-1|(1x - 3) + 3| + 1) + 1)/(-2(-1|(1x - 3) + 3| + 1) + 3)) - 2| - 3)^2 + 5(-2|((-2(-1|(1x - 3) + 3| + 1) + 1)/(-2(-1|(1x - 3) + 3| + 1) + 3)) - 2| - 3) + 4) - 2) - 2 and g(x) = 2(1|(2(1((-3x + 3)/(-2x + 3)... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1((1(-2|(2(3(1|x - 1|)^2 + 4(1|x - 1|) + 5)^2 - 1(3(1|x - 1|)^2 + 4(1|x - 1|) + 5) + 3) - 2| + 3))/(-1(-2|(2(3(1|x - 1|)^2 + 4(1|x - 1|) + 5)^2 - 1(3(1|x - 1|)^2 + 4(1|x - 1|) + 5) + 3) - 2| + 3) + 2)) + 2)/(-2((1(-2|(2(3(1|x - 1|... | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(3((3(2|(-1((-3x + 1)/(1x + 3))^2 + 4((-3x + 1)/(1x + 3)) - 5) + 3| - 1) - 3)/(1(2|(-1((-3x + 1)/(1x + 3))^2 + 4((-3x + 1)/(1x + 3)) - 5) + 3| - 1) + 2))^2 + 4((3(2|(-1((-3x + 1)/(1x + 3))^2 + 4((-3x + 1)/(1x + 3)) - 5) + 3| - 1) - 3)/(1(2|(-1((-3x + ... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3((2(2(-2sin(3π(2|((-3x + 1)/(-2x + 3)) - 2| - 3) + 1) + 3)^2 + 3(-2sin(3π(2|((-3x + 1)/(-2x + 3)) - 2| - 3) + 1) + 3) - 5) - 3)/(1(2(-2sin(3π(2|((-3x + 1)/(-2x + 3)) - 2| - 3) + 1) + 3)^2 + 3(-2sin(3π(2|((-3x + 1)/(-2x + 3)) - 2|... | 11 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(3|(-3sin(1π(1((3(2|x - 3|))/(1(2|x - 3|) + 1))^2 + 4((3(2|x - 3|))/(1(2|x - 3|) + 1)))) + 2) + 3| + 3)| - 3 and g(x) = 3(-3cos(1π(-2((-3((1x + 3)/(-2x - 3)) - 3)/(2((1x + 3)/(-2x - 3)) - 3)) + 4) - 3) - 3) - 5 in the range -10 ... | 53 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-3e^(1(-1|(-1|(-4(3x^2 - 4x)^2 - 1(3x^2 - 4x) - 4)| - 1) + 3| + 3)) - 2) - 1)/(3(-3e^(1(-1|(-1|(-4(3x^2 - 4x)^2 - 1(3x^2 - 4x) - 4)| - 1) + 3| + 3)) - 2) - 2) and g(x) = -2|(-1(-3|(2(-5x + 4) + 4) + 1| + 2) - 3) - 2| + 2 in the ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-3((2((2(2|(-1|x + 1| + 2) + 3| - 1) - 1)/(-1(2|(-1|x + 1| + 2) + 3| - 1) + 3)))/(-2((2(2|(-1|x + 1| + 2) + 3| - 1) - 1)/(-1(2|(-1|x + 1| + 2) + 3| - 1) + 3)) + 3)) - 4) - 3 and g(x) = (-3(-1((1(1(1x + 3)^2 - 4(1x + 3) - 5)... | 8 | MATH |
[
{
"content": "How many solutions does the equation 1(-3|((-2(-3cos(3π(-2(2|x| - 3) - 4) - 1) + 2) + 2)/(-2(-3cos(3π(-2(2|x| - 3) - 4) - 1) + 2) - 3)) + 2|)^2 + 1(-3|((-2(-3cos(3π(-2(2|x| - 3) - 4) - 1) + 2) + 2)/(-2(-3cos(3π(-2(2|x| - 3) - 4) - 1) + 2) - 3)) + 2|) - 5 = 4(1(-1(-4(1|x - 1| + 1)^2 + 4) + 5)^2 - 5... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|((2(4(1|((1(3|x - 2|) - 3)/(-2(3|x - 2|))) - 1| + 1)^2 - 5(1|((1(3|x - 2|) - 3)/(-2(3|x - 2|))) - 1| + 1) - 4) - 2)/(-2(4(1|((1(3|x - 2|) - 3)/(-2(3|x - 2|))) - 1| + 1)^2 - 5(1|((1(3|x - 2|) - 3)/(-2(3|x - 2|))) - 1| + 1) -... | 7 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1((-3(-2|(-2cos(3π(-3(-1|x| - 1) - 3) - 2) + 2)| - 1) + 2)/(-3(-2|(-2cos(3π(-3(-1|x| - 1) - 3) - 2) + 2)| - 1) + 1))^2 + 2((-3(-2|(-2cos(3π(-3(-1|x| - 1) - 3) - 2) + 2)| - 1) + 2)/(-3(-2|(-2cos(3π(-3(-1|x| - 1) - 3) - 2) + 2)... | 65 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(1|(5(-4(1sin(1π(-3|x + 3| + 3) + 1) + 1)^2 + 1(1sin(1π(-3|x + 3| + 3) + 1) + 1) + 1)^2 - 5(-4(1sin(1π(-3|x + 3| + 3) + 1) + 1)^2 + 1(1sin(1π(-3|x + 3| + 3) + 1) + 1) + 1) + 3) - 1| + 2) - 4 and g(x) = 2|(3sin(3π(-4((1(-4x^2 - 1x - 2) + 2)/(-1(-4x^2 -... | 110 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(1π(-2(1|(3|(-1|(2|x| + 3)| - 3) - 1| - 1) - 2| - 1)^2 + 1(1|(3|(-1|(2|x| + 3)| - 3) - 1| - 1) - 2| - 1) - 4)) - 2 and g(x) = (-1((3((1(3|((-1x - 2)/(-1x + 2)) + 2| - 2) + 3)/(-3(3|((-1x - 2)/(-1x + 2)) + 2| - 2) + 4)) + ... | 1238 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1cos(2π(-1(2(5(3|(1x^2 + 3x - 4) - 1| - 1)^2 + 5(3|(1x^2 + 3x - 4) - 1| - 1))^2 - 1) - 2) + 3) + 1 and g(x) = (3(2(-3|(-1(1x - 3) - 4) + 3| + 1) + 5) - 3)/(-1(2(-3|(-1(1x - 3) - 4) + 3| + 1) + 5) - 1) in the range -10 ≤ x ≤ 10?\n\... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4((3(3(5((-2(-3|x + 2| + 1))/(-1(-3|x + 2| + 1) + 3)) + 5) - 4) - 2)/(2(3(5((-2(-3|x + 2| + 1))/(-1(-3|x + 2| + 1) + 3)) + 5) - 4) - 3))^2 - 3((3(3(5((-2(-3|x + 2| + 1))/(-1(-3|x + 2| + 1) + 3)) + 5) - 4) - 2)/(2(3(5((-2(-3|... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(2|(-3|(4(-3e^(1(-3|x|)))^2 + 1(-3e^(1(-3|x|))) + 2) + 2| - 1) - 1| + 1) + 3)/(-2(2|(-3|(4(-3e^(1(-3|x|)))^2 + 1(-3e^(1(-3|x|))) + 2) + 2| - 1) - 1| + 1)) and g(x) = -1(-1(3cos(1π(5(1|x - 2| - 1)^2 + 2(1|x - 2| - 1) + 3) -... | 804 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(2(1(-2|(2|(-2sin(2πx - 3) - 1) + 1| + 1) - 2|) - 5) + 1) - 1 and g(x) = -1|(-1|(-1|((-1(-1|x - 2| + 2) - 1)/(1(-1|x - 2| + 2))) - 1|) + 3| + 2) - 3| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your a... | 4 | MATH |
[
{
"content": "How many solutions does the equation (1(-1((3((1(2cos(2π(1x - 2) - 2) - 2) + 1)/(-2(2cos(2π(1x - 2) - 2) - 2) + 3)) - 2)/(-1((1(2cos(2π(1x - 2) - 2) - 2) + 1)/(-2(2cos(2π(1x - 2) - 2) - 2) + 3)) - 3)) + 3) + 3)/(-2(-1((3((1(2cos(2π(1x - 2) - 2) - 2) + 1)/(-2(2cos(2π(1x - 2) - 2) - 2) + 3)) - 2)/(-... | 20 | MATH |
[
{
"content": "How many solutions does the equation (2(5(-2e^(1(-3|(2((2x - 2)/(2x - 1)) + 1) + 3|)))^2 + 3(-2e^(1(-3|(2((2x - 2)/(2x - 1)) + 1) + 3|))) + 2))/(2(5(-2e^(1(-3|(2((2x - 2)/(2x - 1)) + 1) + 3|)))^2 + 3(-2e^(1(-3|(2((2x - 2)/(2x - 1)) + 1) + 3|))) + 2) - 1) = -1|(-3|((-3((3((-3x - 1)/(2x)) + 3)/(-3((... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(3|((-1(-1(2cos(2π((-2x + 2)/(-2x - 1)) + 2) + 2)^2 + 1(2cos(2π((-2x + 2)/(-2x - 1)) + 2) + 2) + 2))/(1(-1(2cos(2π((-2x + 2)/(-2x - 1)) + 2) + 2)^2 + 1(2cos(2π((-2x + 2)/(-2x - 1)) + 2) + 2) + 2) + 2)) + 2| - 1) - 2)/(-2(3|((-1(... | 96 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-4(-1e^(-1((1(-4(5x - 5) - 4) + 1)/(1(-4(5x - 5) - 4))))) + 2) + 1)/(3(-4(-1e^(-1((1(-4(5x - 5) - 4) + 1)/(1(-4(5x - 5) - 4))))) + 2) + 2) and g(x) = -3cos(1π(-5(1|(-1(-4x - 3) + 2)|) - 1)) - 3 intersect in the interval [-10, 10].\n\nPresent the an... | 400 | MATH |
[
{
"content": "How many solutions does the equation -3(2cos(1π(-1|(-3(5(2|x| + 2) - 2) + 1) - 2| + 2) + 1) + 1) + 4 = 2((2(3(5(-1|x + 1| + 1) - 1)^2 - 4) - 2)/(-1(3(5(-1|x + 1| + 1) - 1)^2 - 4) - 1))^2 - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 237 | MATH |
[
{
"content": "How many solutions does the equation 3((-1((2(-3(-2((3x - 3)/(-1x + 3))^2 - 4((3x - 3)/(-1x + 3)) - 1) - 4) + 2)/(3(-3(-2((3x - 3)/(-1x + 3))^2 - 4((3x - 3)/(-1x + 3)) - 1) - 4) - 1)) + 2)/(-2((2(-3(-2((3x - 3)/(-1x + 3))^2 - 4((3x - 3)/(-1x + 3)) - 1) - 4) + 2)/(3(-3(-2((3x - 3)/(-1x + 3))^2 - 4(... | 4 | MATH |
[
{
"content": "How many solutions does the equation (2(-2((-2(-1(3sin(1π(-3x - 4) + 1) + 1)^2 - 5(3sin(1π(-3x - 4) + 1) + 1) + 5) + 3)/(1(-1(3sin(1π(-3x - 4) + 1) + 1)^2 - 5(3sin(1π(-3x - 4) + 1) + 1) + 5))) + 3) - 2)/(3(-2((-2(-1(3sin(1π(-3x - 4) + 1) + 1)^2 - 5(3sin(1π(-3x - 4) + 1) + 1) + 5) + 3)/(1(-1(3sin(1... | 49 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(1((1(-2(1(1|x - 3| + 1)^2 - 5(1|x - 3| + 1) + 5)^2 + 2(1(1|x - 3| + 1)^2 - 5(1|x - 3| + 1) + 5) + 2) + 3)/(-3(-2(1(1|x - 3| + 1)^2 - 5(1|x - 3| + 1) + 5)^2 + 2(1(1|x - 3| + 1)^2 - 5(1|x - 3| + 1) + 5) + 2) + 4))^2 - 3((1(-... | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-3cos(3π(2|((1((1((3x + 1)/(-1x - 2)) - 3)/(-3((3x + 1)/(-1x - 2)) + 2)) + 1)/(-3((1((3x + 1)/(-1x - 2)) - 3)/(-3((3x + 1)/(-1x - 2)) + 2)) + 1)) - 1|) + 1) + 1) - 3)/(-3(-3cos(3π(2|((1((1((3x + 1)/(-1x - 2)) - 3)/(-3((3x... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(-1sin(1π(-5((-3(-3(-1x^2 - 3x + 1)^2 + 3(-1x^2 - 3x + 1) - 4) + 3)/(-1(-3(-1x^2 - 3x + 1)^2 + 3(-1x^2 - 3x + 1) - 4) + 3))^2 + 2((-3(-3(-1x^2 - 3x + 1)^2 + 3(-1x^2 - 3x + 1) - 4) + 3)/(-1(-3(-1x^2 - 3x + 1)^2 + 3(-1x^2 - 3x + 1) - 4) + 3)) - 4) - 2)... | 75 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-2|((-3((-3(2|(-1x - 4) - 2| - 1) + 2)/(-1(2|(-1x - 4) - 2| - 1) - 1)))/(2((-3(2|(-1x - 4) - 2| - 1) + 2)/(-1(2|(-1x - 4) - 2| - 1) - 1)) + 3)) - 2| + 1) - 3| - 2 and g(x) = 1cos(3π((1(2(1(3|x + 1| + 1) - 4) - 1) + 1)/(-1(2(1(3|x + 1| + 1) - 4) - 1)... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1((-1(3cos(1π(-3|(5((-1x + 1)/(2x - 1)) + 2) + 1| - 2) - 2)) + 2)/(-3(3cos(1π(-3|(5((-1x + 1)/(2x - 1)) + 2) + 1| - 2) - 2)))) + 3 and g(x) = (2(-1|(2(-3e^(2(-4x - 3)) - 2) - 3) - 1| + 3) - 2)/(-1(-1|(2(-3e^(2(-4x - 3)) - 2)... | 18 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2((1(4(1|(-1(2sin(2πx - 1)) - 2) - 1| + 1)^2 + 2(1|(-1(2sin(2πx - 1)) - 2) - 1| + 1) + 5) - 2)/(-3(4(1|(-1(2sin(2πx - 1)) - 2) - 1| + 1)^2 + 2(1|(-1(2sin(2πx - 1)) - 2) - 1| + 1) + 5) + 2)) + 3)/(-3((1(4(1|(-1(2sin(2πx - 1)... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(-4(3(-2|((2((2x + 3)/(3x)) - 3)/(2((2x + 3)/(3x)) + 1)) + 1| - 1) + 4) + 5) - 3 and g(x) = -1(2sin(3π((1((1(-1|x + 1| + 3) - 3)/(2(-1|x + 1| + 3) - 1)) - 2)/(-2((1(-1|x + 1| + 3) - 3)/(2(-1|x + 1| + 3) - 1)) + 1))) - 3)^2 + 4(2sin(3π((1((1(-1|x + 1|... | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(-2((1((2((2(3x - 1) + 2)/(-2(3x - 1) + 3)) + 1)/(-3((2(3x - 1) + 2)/(-2(3x - 1) + 3)) - 2)) + 1)/(-2((2((2(3x - 1) + 2)/(-2(3x - 1) + 3)) + 1)/(-3((2(3x - 1) + 2)/(-2(3x - 1) + 3)) - 2)) + 3)) + 4) - 1| + 3 and g(x) = -1(-... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2e^(-1(4(-1(4(-1(1|x + 2| - 3)^2 - 4(1|x + 2| - 3) - 3) - 3) + 1) + 3)) - 2 and g(x) = 2|(2(-1|(1(3cos(3πx + 1) + 2)^2 - 2(3cos(3πx + 1) + 2) - 4)| + 1)^2 - 5(-1|(1(3cos(3πx + 1) + 2)^2 - 2(3cos(3πx + 1) + 2) - 4)| + 1) - 1) - 3| -... | 115 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2((3(1|((-2(-1(1|x - 3| - 3) - 1) + 2)/(2(-1(1|x - 3| - 3) - 1) + 1)) + 1| - 1) + 1)/(-2(1|((-2(-1(1|x - 3| - 3) - 1) + 2)/(2(-1(1|x - 3| - 3) - 1) + 1)) + 1| - 1) - 3)) + 3)/(-2((3(1|((-2(-1(1|x - 3| - 3) - 1) + 2)/(2(-1(1|x - 3| - 3) - 1) + 1)) + 1|... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(4(-1sin(2π(-4(-2|(-4x^2 + 4x - 4) - 2| - 1)^2 + 1(-2|(-4x^2 + 4x - 4) - 2| - 1) + 5) + 2))^2 + 1(-1sin(2π(-4(-2|(-4x^2 + 4x - 4) - 2| - 1)^2 + 1(-2|(-4x^2 + 4x - 4) - 2| - 1) + 5) + 2)) + 3) + 2| - 1 and g(x) = (-3(3sin(1π((1(5(... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(1e^(-1(-2|(4(2|((1x - 1)/(1x - 2)) + 2| - 2) - 2) + 3| + 2)) - 2) - 1| + 1 and g(x) = -2sin(3π((-3(-1|(-1(1x + 4)^2 - 1(1x + 4) - 5)| - 2) + 2)/(-2(-1|(-1(1x + 4)^2 - 1(1x + 4) - 5)| - 2))) - 1) + 1 intersect in the interval [-10, 10].\n\nPresent t... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(2(2(-1|(3(-1|x + 1| + 1))| - 1) + 2)^2 + 5(2(-1|(3(-1|x + 1| + 1))| - 1) + 2) + 2) - 2 and g(x) = (-1((-2((2(-2sin(2π(-2x + 2) - 1)) + 1)/(-2(-2sin(2π(-2x + 2) - 1)))) + 3)/(-3((2(-2sin(2π(-2x + 2) - 1)) + 1)/(-2(-2sin(2π(-... | 14 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-2((1(-5(-1sin(2π((3x + 1)/(2x + 3)) + 1) - 2)^2 + 4(-1sin(2π((3x + 1)/(2x + 3)) + 1) - 2) + 4) + 2)/(-2(-5(-1sin(2π((3x + 1)/(2x + 3)) + 1) - 2)^2 + 4(-1sin(2π((3x + 1)/(2x + 3)) + 1) - 2) + 4) + 3))^2 - 2((1(-5(-1sin(2π((3x + ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(-2|((3(-1|(3|(-4x^2 - 2x + 5) - 3| + 1)| + 2) - 3)/(-1(-1|(3|(-4x^2 - 2x + 5) - 3| + 1)| + 2) - 3)) - 3| + 3)^2 + 2(-2|((3(-1|(3|(-4x^2 - 2x + 5) - 3| + 1)| + 2) - 3)/(-1(-1|(3|(-4x^2 - 2x + 5) - 3| + 1)| + 2) - 3)) - 3| +... | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3((2(4(3(-1(-2e^(-1x) - 2) - 4)^2 - 4(-1(-2e^(-1x) - 2) - 4) - 3) - 4) + 3)/(2(4(3(-1(-2e^(-1x) - 2) - 4)^2 - 4(-1(-2e^(-1x) - 2) - 4) - 3) - 4)))^2 + 1((2(4(3(-1(-2e^(-1x) - 2) - 4)^2 - 4(-1(-2e^(-1x) - 2) - 4) - 3) - 4) + 3... | 4 | MATH |
[
{
"content": "How many solutions does the equation -3cos(1π(2|(3|((2(-4((1x + 1)/(2x + 3))^2 - 4((1x + 1)/(2x + 3)) + 4))/(2(-4((1x + 1)/(2x + 3))^2 - 4((1x + 1)/(2x + 3)) + 4) + 3)) - 3| + 1)| - 2) + 3) - 2 = (-2((1(3((2(-3x - 5) + 1)/(-1(-3x - 5) - 2))^2 - 1((2(-3x - 5) + 1)/(-1(-3x - 5) - 2)) - 3) + 3)/(-3(3... | 49 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(-1sin(2π(3|(-3|(-1((-2x - 3)/(-3x - 3))) + 3| + 3)| - 2) + 3) - 1)^2 - 5(-1sin(2π(3|(-3|(-1((-2x - 3)/(-3x - 3))) + 3| + 3)| - 2) + 3) - 1) + 4 and g(x) = -5((2(-1sin(1π(-1|(2x - 3) + 2| + 3) - 3) - 1) + 2)/(2(-1sin(1π(-1|(2x - ... | 86 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(2π(1(1(-2(-1(5x + 3) + 1) + 4))^2 - 1(1(-2(-1(5x + 3) + 1) + 4)) - 3) - 3) + 3 and g(x) = 4(-1((1(3cos(1π(-4x + 5) + 1) - 3) - 3)/(3(3cos(1π(-4x + 5) + 1) - 3) + 3)) + 2)^2 - 1(-1((1(3cos(1π(-4x + 5) + 1) - 3) - 3)/(3(3cos(1π... | 94 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(-2sin(1π(-3|(-4((-2(-1|x + 3| - 3) + 3)/(2(-1|x + 3| - 3) + 1)) + 1)| - 3)) - 3) - 1 and g(x) = -3(-3((-1(2|(4x) - 1| - 2) - 1)/(-2(2|(4x) - 1| - 2)))^2 - 1((-1(2|(4x) - 1| - 2) - 1)/(-2(2|(4x) - 1| - 2))) + 1) where -10 ≤... | 4 | MATH |
[
{
"content": "How many solutions does the equation -2|(-2(5(2(-1((-2x - 1)/(3x - 2))^2 + 4((-2x - 1)/(3x - 2)) - 1)^2 + 2(-1((-2x - 1)/(3x - 2))^2 + 4((-2x - 1)/(3x - 2)) - 1) - 3)^2 - 2(2(-1((-2x - 1)/(3x - 2))^2 + 4((-2x - 1)/(3x - 2)) - 1)^2 + 2(-1((-2x - 1)/(3x - 2))^2 + 4((-2x - 1)/(3x - 2)) - 1) - 3) - 4)... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(1(-1(-5(1((-2x - 1)/(1x + 3)) - 3)^2 + 1(1((-2x - 1)/(1x + 3)) - 3) + 1)^2 + 2(-5(1((-2x - 1)/(1x + 3)) - 3)^2 + 1(1((-2x - 1)/(1x + 3)) - 3) + 1) - 2)) - 2)/(1(1(-1(-5(1((-2x - 1)/(1x + 3)) - 3)^2 + 1(1((-2x - 1)/(1x + 3)) - 3) + 1)^2 + 2(-5(1((-2... | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(1π((-2((3(2(-4(3x^2 + 5x - 4) - 5) - 2) - 1)/(-1(2(-4(3x^2 + 5x - 4) - 5) - 2) + 3)) + 3)/(1((3(2(-4(3x^2 + 5x - 4) - 5) - 2) - 1)/(-1(2(-4(3x^2 + 5x - 4) - 5) - 2) + 3)))) - 2) and g(x) = 1(-1((1(1sin(2π(-1x^2 + 4x + 5)... | 265 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|((1(2|(1|(2|(1|x + 1| + 1) - 3| - 2) + 2| - 1) + 3| + 2) + 3)/(-1(2|(1|(2|(1|x + 1| + 1) - 3| - 2) + 2| - 1) + 3| + 2) - 3)) + 2| - 3 and g(x) = -4((-2(-3cos(2π(-4(-3x + 3)^2 + 4) + 3) - 2) + 2)/(-2(-3cos(2π(-4(-3x + 3)^2 + 4) + ... | 1130 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|((1(3|(1|(3(5x - 3)^2 + 3) + 3| - 2) + 3| + 1) + 1)/(3(3|(1|(3(5x - 3)^2 + 3) + 3| - 2) + 3| + 1) - 2))| + 1 and g(x) = 2|(-1(2sin(3π((-1(3x - 2) - 2)/(-3(3x - 2) + 4)))) - 4) + 3| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaT... | 39 | MATH |
[
{
"content": "How many solutions does the equation 2(-5(3sin(2π(3|(-1(3x - 4) - 5) - 2| + 3) + 1) + 3)^2 + 4(3sin(2π(3|(-1(3x - 4) - 5) - 2| + 3) + 1) + 3) + 3) - 2 = (1(-2sin(3π(4(3(5x^2 + 1) - 5) + 1) + 3) - 1) + 1)/(-2(-2sin(3π(4(3(5x^2 + 1) - 5) + 1) + 3) - 1)) have for x in the interval [-10, 10]?\n\nPrese... | 215 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(2(4(3(2(2|x - 3| + 2)^2 - 3(2|x - 3| + 2) + 2)^2 - 2) + 3) - 1) + 3)/(-1(2(4(3(2(2|x - 3| + 2)^2 - 3(2|x - 3| + 2) + 2)^2 - 2) + 3) - 1) + 2) and g(x) = -3|((-3(-1(3(-1x^2 + 4x - 3) + 1)^2 + 5(3(-1x^2 + 4x - 3) + 1) - 5) + 1)/(3(-1(3(-1x^2 + 4x - 3)... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(1|(-2((-2(1|(1e^(1x))| + 2) + 2)/(-3(1|(1e^(1x))| + 2) - 2))^2 + 4) - 1| + 2) + 3| - 3 and g(x) = -2((-1(-5(-1sin(1π(-1|x + 2| + 2) + 2) - 1)^2 + 2(-1sin(1π(-1|x + 2| + 2) + 2) - 1) + 1) - 2)/(2(-5(-1sin(1π(-1|x + 2| + 2)... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1((2((1((2(2|(-1|x + 2| - 3) - 2| - 2) - 2)/(-1(2|(-1|x + 2| - 3) - 2| - 2) + 2)) + 3)/(-3((2(2|(-1|x + 2| - 3) - 2| - 2) - 2)/(-1(2|(-1|x + 2| - 3) - 2| - 2) + 2)) + 4)) - 3)/(2((1((2(2|(-1|x + 2| - 3) - 2| - 2) - 2)/(-1(2|(-1|x ... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-3cos(1π(-2(-1((3((-2x + 1)/(-1x + 2)) - 2)/(-2((-2x + 1)/(-1x + 2)) - 1))^2 + 1((3((-2x + 1)/(-1x + 2)) - 2)/(-2((-2x + 1)/(-1x + 2)) - 1)) - 2) - 3) - 3) + 1) + 1| and g(x) = -3sin(3π(-5(4(5((1x)/(-2x + 3)) - 4) + 5)^2 + 5(4(5... | 996 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(3|((-3(5((2(1|x + 1| - 1) - 3)/(-2(1|x + 1| - 1) - 2)) + 5) - 3)/(2(5((2(1|x + 1| - 1) - 3)/(-2(1|x + 1| - 1) - 2)) + 5) - 1)) + 2| - 3) - 4 and g(x) = 1(1(-2sin(1π(1((2x)/(-1x + 2)) + 3) + 1) + 2) - 1) - 1 in the range -10 ≤ x ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4((3(-3cos(2π(-3(-3((1x + 2)/(-2x + 3)) + 3)^2 - 2(-3((1x + 2)/(-2x + 3)) + 3) - 2))) - 2)/(1(-3cos(2π(-3(-3((1x + 2)/(-2x + 3)) + 3)^2 - 2(-3((1x + 2)/(-2x + 3)) + 3) - 2))) + 3))^2 + 5((3(-3cos(2π(-3(-3((1x + 2)/(-2x + 3)) ... | 706 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5((-2(2((-1(2sin(3π((1x)/(-2x - 2)) - 1) - 3) + 2)/(2(2sin(3π((1x)/(-2x - 2)) - 1) - 3) - 1))^2 - 5((-1(2sin(3π((1x)/(-2x - 2)) - 1) - 3) + 2)/(2(2sin(3π((1x)/(-2x - 2)) - 1) - 3) - 1)) + 5) - 2)/(1(2((-1(2sin(3π((1x)/(-2x - 2)) - 1) - 3) + 2)/(2(2sin(... | 15 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1((2(1(-1(1((-3x + 2)/(-2x + 3))^2 + 4((-3x + 2)/(-2x + 3)) - 1) + 1)^2 - 4(-1(1((-3x + 2)/(-2x + 3))^2 + 4((-3x + 2)/(-2x + 3)) - 1) + 1) + 2) - 2)/(-1(1(-1(1((-3x + 2)/(-2x + 3))^2 + 4((-3x + 2)/(-2x + 3)) - 1) + 1)^2 - 4... | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3((2(-5(-3((1(-2cos(2πx + 3)) + 1)/(3(-2cos(2πx + 3)) + 2)) - 1) + 4) + 2)/(-3(-5(-3((1(-2cos(2πx + 3)) + 1)/(3(-2cos(2πx + 3)) + 2)) - 1) + 4) + 1)) - 1)/(2((2(-5(-3((1(-2cos(2πx + 3)) + 1)/(3(-2cos(2πx + 3)) + 2)) - 1) + 4) + 2)/(-3(-5(-3((1(-2cos(2... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-2(3|(1|((-3(-1|x| - 3))/(3(-1|x| - 3) + 2)) - 2| + 2) + 2| + 3)) + 1| - 1 and g(x) = 2|(-2|((-1(-5(2sin(3πx - 1)) - 3) + 2)/(2(-5(2sin(3πx - 1)) - 3) + 1)) + 3| - 1) + 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex forma... | 37 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(1(1|((-1((-2(1x^2 + 4x) + 2)/(-2(1x^2 + 4x) - 2)) + 2)/(2((-2(1x^2 + 4x) + 2)/(-2(1x^2 + 4x) - 2)) + 3)) - 1| + 2)^2 + 2(1|((-1((-2(1x^2 + 4x) + 2)/(-2(1x^2 + 4x) - 2)) + 2)/(2((-2(1x^2 + 4x) + 2)/(-2(1x^2 + 4x) - 2)) + 3)... | 330 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(3|((-1(-1(1(-3x - 4)^2 + 3(-3x - 4) - 1)^2 + 1) - 1)/(-2(-1(1(-3x - 4)^2 + 3(-3x - 4) - 1)^2 + 1) - 3)) + 3| - 2)^2 - 3(3|((-1(-1(1(-3x - 4)^2 + 3(-3x - 4) - 1)^2 + 1) - 1)/(-2(-1(1(-3x - 4)^2 + 3(-3x - 4) - 1)^2 + 1) - 3)... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(3((-2(-5(5(1x - 1) + 1) + 3) + 3)/(2(-5(5(1x - 1) + 1) + 3) + 2)) - 3) - 3)/(2(3((-2(-5(5(1x - 1) + 1) + 3) + 3)/(2(-5(5(1x - 1) + 1) + 3) + 2)) - 3) + 3) and g(x) = (-2(2(-3(-2|(-1cos(2πx - 2) + 1) - 2| + 2)^2 + 2(-2|(-1cos(2πx - 2) + 1) - 2| + 2) ... | 81 | MATH |
[
{
"content": "How many solutions does the equation (3(-1(-4(2sin(1π((1(-4x^2 + 1x - 5) + 2)/(1(-4x^2 + 1x - 5))) + 3) + 3) + 1)^2 + 4(-4(2sin(1π((1(-4x^2 + 1x - 5) + 2)/(1(-4x^2 + 1x - 5))) + 3) + 3) + 1) - 1) - 2)/(-3(-1(-4(2sin(1π((1(-4x^2 + 1x - 5) + 2)/(1(-4x^2 + 1x - 5))) + 3) + 3) + 1)^2 + 4(-4(2sin(1π((1... | 6 | MATH |
[
{
"content": "How many solutions does the equation -1(4(2cos(1π(-5(-3|((-2x + 1)/(-1x + 3))| + 2) + 3) - 3) + 1) + 3)^2 - 3 = -1(2|((-1((3(-3x - 1) + 3)/(-2(-3x - 1))) - 2)/(-1((3(-3x - 1) + 3)/(-2(-3x - 1))) - 3)) - 1| + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxe... | 254 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4((2((-1(-3|(-4(2x^2 + 5x + 2)^2 + 3(2x^2 + 5x + 2) + 3) + 1| + 1) + 1)/(2(-3|(-4(2x^2 + 5x + 2)^2 + 3(2x^2 + 5x + 2) + 3) + 1| + 1) + 2)) - 2)/(-2((-1(-3|(-4(2x^2 + 5x + 2)^2 + 3(2x^2 + 5x + 2) + 3) + 1| + 1) + 1)/(2(-3|(-4(2x^2 +... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-1sin(2π((-1((1(-4(1|x - 3| - 3)^2 - 2(1|x - 3| - 3) + 5) + 2)/(2(-4(1|x - 3| - 3)^2 - 2(1|x - 3| - 3) + 5) - 3)) + 2)/(-2((1(-4(1|x - 3| - 3)^2 - 2(1|x - 3| - 3) + 5) + 2)/(2(-4(1|x - 3| - 3)^2 - 2(1|x - 3| - 3) + 5) - 3)) - 3))) - 2) - 1)/(-2(-1si... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(1|(2((-3((3((2x)/(-1x - 1)) + 3)/(2((2x)/(-1x - 1)) - 3)) + 1)/(-1((3((2x)/(-1x - 1)) + 3)/(2((2x)/(-1x - 1)) - 3)) + 2))^2 + 1((-3((3((2x)/(-1x - 1)) + 3)/(2((2x)/(-1x - 1)) - 3)) + 1)/(-1((3((2x)/(-1x - 1)) + 3)/(2((2x)/... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(3sin(1π(3((-2((2(-4x - 1) + 2)/(3(-4x - 1) - 2)) + 3)/(-3((2(-4x - 1) + 2)/(3(-4x - 1) - 2)) - 2))^2 + 1((-2((2(-4x - 1) + 2)/(3(-4x - 1) - 2)) + 3)/(-3((2(-4x - 1) + 2)/(3(-4x - 1) - 2)) - 2)) + 4) + 2) - 2)^2 - 4(3sin(1π... | 22 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(3e^(1(-4(5(2((2x + 1)/(-3x - 1)) + 4)^2 + 4(2((2x + 1)/(-3x - 1)) + 4) - 1) - 3)) - 1) + 3 and g(x) = 2|(1(4(2|(2|x + 2| - 2)|)^2 - 5(2|(2|x + 2| - 2)|) - 4)^2 + 3(4(2|(2|x + 2| - 2)|)^2 - 5(2|(2|x + 2| - 2)|) - 4) + 1) + 3| + 1... | 8 | MATH |
[
{
"content": "How many solutions does the equation -3|(1cos(1π(3(-5(-3|(1|x - 3| + 3) - 2| - 2) + 1) - 4)) + 1) + 2| - 1 = -2|((3(-5(-3(3x^2 + 3x + 2)^2 - 1(3x^2 + 3x + 2) + 5) - 5) - 2)/(-1(-5(-3(3x^2 + 3x + 2)^2 - 1(3x^2 + 3x + 2) + 5) - 5) - 2)) + 1| - 3 have for x in the interval [-10, 10]?\n\nPresent the a... | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(-4(-3|((1((-1(-1x^2 + 2x + 1) - 3)/(-3(-1x^2 + 2x + 1) + 2)) - 3)/(1((-1(-1x^2 + 2x + 1) - 3)/(-3(-1x^2 + 2x + 1) + 2)))) - 3| + 1)) and g(x) = (2(2|(3|(4(-1sin(1πx + 3) + 2)^2 - 1(-1sin(1πx + 3) + 2)) + 2| + 3) - 2| - 2))/(-2(2|... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(1π(1|(-3(4((-2((-2x + 3)/(-2x + 1)) - 2)/(-3((-2x + 3)/(-2x + 1)) + 2)) + 2)^2 + 3(4((-2((-2x + 3)/(-2x + 1)) - 2)/(-3((-2x + 3)/(-2x + 1)) + 2)) + 2)) + 3| - 2) + 1) - 2 and g(x) = 2|(1|(1|(1e^(-1(3|x + 2| + 1))) + 1|)| - 3) ... | 552 | MATH |
[
{
"content": "How many solutions does the equation -2|(2(-2|(1|(-1((2x + 1)/(2x))^2 - 1((2x + 1)/(2x)) - 1) - 2| + 2) - 2| - 3) + 2) + 3| = -2|(4(3(-5(2sin(1πx + 1) + 1)^2 + 4(2sin(1πx + 1) + 1) + 4)^2 + 3(-5(2sin(1πx + 1) + 1)^2 + 4(2sin(1πx + 1) + 1) + 4) + 5))| have for x in the interval [-10, 10]?\n\nPresen... | 75 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(1((-3(4(-2(-2x^2 - 2x - 3) - 4)) - 2)/(-3(4(-2(-2x^2 - 2x - 3) - 4)) + 2)) + 5) + 2 and g(x) = 5(-2((2(4(2x^2 - 1x - 5) - 5))/(-3(4(2x^2 - 1x - 5) - 5) - 1))^2 + 1)^2 + 5(-2((2(4(2x^2 - 1x - 5) - 5))/(-3(4(2x^2 - 1x - 5) - 5) - 1... | 4 | MATH |
[
{
"content": "How many solutions does the equation -3(-4((3((-2(2|(-3x + 4) + 2| - 3) + 3)/(3(2|(-3x + 4) + 2| - 3) + 1)) + 1)/(2((-2(2|(-3x + 4) + 2| - 3) + 3)/(3(2|(-3x + 4) + 2| - 3) + 1)) + 1)) + 4)^2 - 5(-4((3((-2(2|(-3x + 4) + 2| - 3) + 3)/(3(2|(-3x + 4) + 2| - 3) + 1)) + 1)/(2((-2(2|(-3x + 4) + 2| - 3) +... | 4 | MATH |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.