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) = -2cos(1π((-2((-2(1|(-4(-5x - 4)^2 - 1(-5x - 4) + 3) + 1| + 3) - 1)/(-2(1|(-4(-5x - 4)^2 - 1(-5x - 4) + 3) + 1| + 3) - 3)) - 3)/(-2((-2(1|(-4(-5x - 4)^2 - 1(-5x - 4) + 3) + 1| + 3) - 1)/(-2(1|(-4(-5x - 4)^2 - 1(-5x - 4) + 3) +... | 42 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(2(-2|((-3(1(-3x + 4)^2 + 4(-3x + 4) - 3))/(1(1(-3x + 4)^2 + 4(-3x + 4) - 3) + 3)) - 3| + 3))^2 - 1(2(-2|((-3(1(-3x + 4)^2 + 4(-3x + 4) - 3))/(1(1(-3x + 4)^2 + 4(-3x + 4) - 3) + 3)) - 3| + 3)) + 3 and g(x) = -4(2sin(3π(1(-1((3x - 1)/(2x - 1))^2 + 5((... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1((3(-5(-1((2(1cos(1πx - 1) + 2) + 1)/(2(1cos(1πx - 1) + 2) - 3)) + 5)^2 - 1(-1((2(1cos(1πx - 1) + 2) + 1)/(2(1cos(1πx - 1) + 2) - 3)) + 5) + 4) + 2)/(-2(-5(-1((2(1cos(1πx - 1) + 2) + 1)/(2(1cos(1πx - 1) + 2) - 3)) + 5)^2 - 1... | 36 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|((1(3cos(2π(5(1|((3x - 1)/(3x - 2)) - 2| - 1)^2 - 5(1|((3x - 1)/(3x - 2)) - 2| - 1) + 4) + 2) + 1) - 1)/(-1(3cos(2π(5(1|((3x - 1)/(3x - 2)) - 2| - 1)^2 - 5(1|((3x - 1)/(3x - 2)) - 2| - 1) + 4) + 2) + 1))) + 2| + 2 and g(x) ... | 35 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(-1|(-2|(4((2(-1sin(3πx - 3) + 1) - 2)/(2(-1sin(3πx - 3) + 1) - 1))^2 + 1((2(-1sin(3πx - 3) + 1) - 2)/(2(-1sin(3πx - 3) + 1) - 1)) - 3) - 1| - 1) + 3|) + 3 and g(x) = -1|((2(-2cos(1π(-2|(-3x + 2) - 2| - 1) - 1)) + 2)/(-2(-2cos(1π(... | 159 | MATH |
[
{
"content": "How many solutions does the equation (1(-1sin(2π(-4(-1((-2(3x - 5) + 1)/(-1(3x - 5) - 2)) + 2) + 2) - 1) - 1) - 3)/(3(-1sin(2π(-4(-1((-2(3x - 5) + 1)/(-1(3x - 5) - 2)) + 2) + 2) - 1) - 1) - 2) = 1((1(-3|(-3|(-4x^2 + 3x + 3) + 1|)| + 2) + 3)/(-2(-3|(-3|(-4x^2 + 3x + 3) + 1|)| + 2) - 1)) - 2 have fo... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-1|(2(-3(-2(-5x - 1)^2 + 2(-5x - 1) + 3)^2 - 3(-2(-5x - 1)^2 + 2(-5x - 1) + 3) - 4)^2 + 5(-3(-2(-5x - 1)^2 + 2(-5x - 1) + 3)^2 - 3(-2(-5x - 1)^2 + 2(-5x - 1) + 3) - 4) - 4) + 3| + 1) - 3)/(-1(-1|(2(-3(-2(-5x - 1)^2 + 2(-5x - 1)... | 80 | MATH |
[
{
"content": "How many solutions does the equation (2((-1((-2(-5(-1(-3cos(1πx + 1) + 3) - 5) - 5))/(3(-5(-1(-3cos(1πx + 1) + 3) - 5) - 5) - 1)) - 3)/(3((-2(-5(-1(-3cos(1πx + 1) + 3) - 5) - 5))/(3(-5(-1(-3cos(1πx + 1) + 3) - 5) - 5) - 1)) + 3)) + 1)/(-1((-1((-2(-5(-1(-3cos(1πx + 1) + 3) - 5) - 5))/(3(-5(-1(-3cos... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(2sin(1π(-2((1(-5((-1x + 3)/(-1x - 1))^2 - 3((-1x + 3)/(-1x - 1)) + 4) + 1)/(-2(-5((-1x + 3)/(-1x - 1))^2 - 3((-1x + 3)/(-1x - 1)) + 4) + 1)) + 3) + 3) - 2)^2 - 5(2sin(1π(-2((1(-5((-1x + 3)/(-1x - 1))^2 - 3((-1x + 3)/(-1x -... | 47 | MATH |
[
{
"content": "How many solutions does the equation -3(2(3((1(-2((1x + 3)/(-2x - 3))^2 - 5((1x + 3)/(-2x - 3)) + 5) + 1)/(1(-2((1x + 3)/(-2x - 3))^2 - 5((1x + 3)/(-2x - 3)) + 5)))^2 + 1((1(-2((1x + 3)/(-2x - 3))^2 - 5((1x + 3)/(-2x - 3)) + 5) + 1)/(1(-2((1x + 3)/(-2x - 3))^2 - 5((1x + 3)/(-2x - 3)) + 5))) - 3)^2... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1e^(1((-1((2(1(2|(-4x^2 - 5x + 4) + 3| + 3)^2 - 1(2|(-4x^2 - 5x + 4) + 3| + 3) - 1) + 3)/(1(1(2|(-4x^2 - 5x + 4) + 3| + 3)^2 - 1(2|(-4x^2 - 5x + 4) + 3| + 3) - 1) - 3)) - 3)/(-2((2(1(2|(-4x^2 - 5x + 4) + 3| + 3)^2 - 1(2|(-4x... | 4 | MATH |
[
{
"content": "How many solutions does the equation (1(-1|(-2|((1(-3(-2|x - 1| - 3)^2 - 4(-2|x - 1| - 3)) + 3)/(2(-3(-2|x - 1| - 3)^2 - 4(-2|x - 1| - 3)) + 3)) - 1| - 3) + 1| + 3) + 2)/(1(-1|(-2|((1(-3(-2|x - 1| - 3)^2 - 4(-2|x - 1| - 3)) + 3)/(2(-3(-2|x - 1| - 3)^2 - 4(-2|x - 1| - 3)) + 3)) - 1| - 3) + 1| + 3) ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(1(-1(-2(3sin(2π(-3x^2 - 1x - 2) - 2) + 2) + 5) + 3)^2 - 3(-1(-2(3sin(2π(-3x^2 - 1x - 2) - 2) + 2) + 5) + 3) + 5)^2 + 2(1(-1(-2(3sin(2π(-3x^2 - 1x - 2) - 2) + 2) + 5) + 3)^2 - 3(-1(-2(3sin(2π(-3x^2 - 1x - 2) - 2) + 2) + 5) + 3) + ... | 42 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(5(-3sin(1π(-3(1(-2x^2 - 2x + 4)^2 + 5(-2x^2 - 2x + 4) + 3)) - 3) - 2) - 2) - 1)/(1(5(-3sin(1π(-3(1(-2x^2 - 2x + 4)^2 + 5(-2x^2 - 2x + 4) + 3)) - 3) - 2) - 2) - 3) and g(x) = 5((1(2(2sin(3π(-5x - 4) - 3) - 2) - 5) - 3)/(-2(2(2si... | 167 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2((3(-1(-2cos(1π(2(-2|x| - 3)^2 + 5(-2|x| - 3) + 3)) - 1) - 2) - 3)/(-1(-1(-2cos(1π(2(-2|x| - 3)^2 + 5(-2|x| - 3) + 3)) - 1) - 2) - 1))^2 - 3((3(-1(-2cos(1π(2(-2|x| - 3)^2 + 5(-2|x| - 3) + 3)) - 1) - 2) - 3)/(-1(-1(-2cos(1π(2(-2|x... | 198 | MATH |
[
{
"content": "How many solutions does the equation (1(-3cos(3π(1((-1(-1|(2x + 1)| - 3) - 2)/(-2(-1|(2x + 1)| - 3) - 2))^2 + 3((-1(-1|(2x + 1)| - 3) - 2)/(-2(-1|(2x + 1)| - 3) - 2))) - 1) - 3) + 3)/(1(-3cos(3π(1((-1(-1|(2x + 1)| - 3) - 2)/(-2(-1|(2x + 1)| - 3) - 2))^2 + 3((-1(-1|(2x + 1)| - 3) - 2)/(-2(-1|(2x + ... | 6 | MATH |
[
{
"content": "How many solutions does the equation (3(3|(4(2(1cos(1π(2x^2 + 1x - 1) + 1) + 3)^2 - 3(1cos(1π(2x^2 + 1x - 1) + 1) + 3) - 2)^2 - 5(2(1cos(1π(2x^2 + 1x - 1) + 1) + 3)^2 - 3(1cos(1π(2x^2 + 1x - 1) + 1) + 3) - 2) - 3)| - 3) + 1)/(2(3|(4(2(1cos(1π(2x^2 + 1x - 1) + 1) + 3)^2 - 3(1cos(1π(2x^2 + 1x - 1) +... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(-2|((3(1(2|(-3|x - 2| + 1)| - 1)^2 + 5(2|(-3|x - 2| + 1)| - 1) - 3) + 3)/(1(1(2|(-3|x - 2| + 1)| - 1)^2 + 5(2|(-3|x - 2| + 1)| - 1) - 3))) - 3| + 1) - 3)/(3(-2|((3(1(2|(-3|x - 2| + 1)| - 1)^2 + 5(2|(-3|x - 2| + 1)| - 1) -... | 10 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(2((1(2|(1(-3|x + 3| - 2) + 3) - 1| - 1) - 2)/(3(2|(1(-3|x + 3| - 2) + 3) - 1| - 1) - 2))^2 - 4((1(2|(1(-3|x + 3| - 2) + 3) - 1| - 1) - 2)/(3(2|(1(-3|x + 3| - 2) + 3) - 1| - 1) - 2)) + 3) - 3| - 3 and g(x) = -2(-2((1(-1|(-3|x - 3| + 2) + 3| - 1) - 1... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1((2((-3(-3cos(3π((-2((1x + 3)/(-2x - 2)) + 2)/(2((1x + 3)/(-2x - 2)))) + 1) + 1))/(-1(-3cos(3π((-2((1x + 3)/(-2x - 2)) + 2)/(2((1x + 3)/(-2x - 2)))) + 1) + 1) + 2)) + 3)/(2((-3(-3cos(3π((-2((1x + 3)/(-2x - 2)) + 2)/(2((1x + 3)/(-2x - 2)))) + 1) + 1))/(-1(-3cos... | 158 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(2(1sin(1π(-5((-3(-3x - 3))/(3(-3x - 3) + 1)) + 2) - 1) + 3)^2 - 2(1sin(1π(-5((-3(-3x - 3))/(3(-3x - 3) + 1)) + 2) - 1) + 3) + 1) - 3| - 1 and g(x) = -3((-3((-2(5(1x^2 + 4x - 2) - 3) - 2)/(3(5(1x^2 + 4x - 2) - 3) + 1)))/(-1((-2(5(1x^2 + 4x - 2) - 3) ... | 25 | MATH |
[
{
"content": "How many solutions does the equation (1((-1(3(-1((-2(-2|x + 2| + 3) - 1)/(2(-2|x + 2| + 3) + 2))^2 + 3((-2(-2|x + 2| + 3) - 1)/(2(-2|x + 2| + 3) + 2)) + 1)^2 - 1(-1((-2(-2|x + 2| + 3) - 1)/(2(-2|x + 2| + 3) + 2))^2 + 3((-2(-2|x + 2| + 3) - 1)/(2(-2|x + 2| + 3) + 2)) + 1)) + 1)/(1(3(-1((-2(-2|x + 2... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2((2(3sin(1π(-2|(2(2x - 2)^2 - 4(2x - 2) + 5) - 2| + 3) + 1) - 3) - 2)/(-3(3sin(1π(-2|(2(2x - 2)^2 - 4(2x - 2) + 5) - 2| + 3) + 1) - 3) - 1)) - 4 and g(x) = 2(2sin(1π(-3|((1(-1|x| + 2) - 2)/(1(-1|x| + 2) + 3)) + 1| + 2)))^2 - 4(2sin(1π(-3|((1(-1|x| + ... | 303 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(-2|(-3cos(1π(1(2|((2x + 3)/(-1x - 1)) - 1| + 3)^2 - 3(2|((2x + 3)/(-1x - 1)) - 1| + 3) + 2) + 2) - 2)| - 3)^2 - 2(-2|(-3cos(1π(1(2|((2x + 3)/(-1x - 1)) - 1| + 3)^2 - 3(2|((2x + 3)/(-1x - 1)) - 1| + 3) + 2) + 2) - 2)| - 3) + 1 and g(x) = -2(1(-1(1(-4... | 139 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-2cos(3π(-4(-1|(-2(1|x - 2| - 2) + 3) - 1| + 1)^2 + 1(-1|(-2(1|x - 2| - 2) + 3) - 1| + 1) + 2) + 3) + 1) + 4 and g(x) = 3(-1|(-2(-4((1x - 2)/(-1x - 3)) - 4) - 4) + 1| + 3)^2 - 5(-1|(-2(-4((1x - 2)/(-1x - 3)) - 4) - 4) + 1| + 3) - 1 intersect in the i... | 325 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(3|(-3(-1sin(1π(3(3|x + 1| - 1)^2 + 3) - 1) + 3) + 3) + 3| - 1) and g(x) = 1(1|((-3(-1|(-2|x - 3| + 1) + 3| - 2) + 1)/(1(-1|(-2|x - 3| + 1) + 3| - 2) - 2)) - 2| - 3)^2 + 2(1|((-3(-1|(-2|x - 3| + 1) + 3| - 2) + 1)/(1(-1|(-2|x... | 1494 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|((-1((-3((-1(-1(2e^(1x) - 3) + 5) - 2)/(-3(-1(2e^(1x) - 3) + 5))) + 3)/(1((-1(-1(2e^(1x) - 3) + 5) - 2)/(-3(-1(2e^(1x) - 3) + 5))) + 2)) - 3)/(1((-3((-1(-1(2e^(1x) - 3) + 5) - 2)/(-3(-1(2e^(1x) - 3) + 5))) + 3)/(1((-1(-1(2e^(1x)... | 10 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5((3(2sin(3π(-2|(-1(3|x + 2| - 3)^2 + 5(3|x + 2| - 3) + 5) + 1| + 1) - 3) - 1) + 2)/(-3(2sin(3π(-2|(-1(3|x + 2| - 3)^2 + 5(3|x + 2| - 3) + 5) + 1| + 1) - 3) - 1) + 4))^2 - 4((3(2sin(3π(-2|(-1(3|x + 2| - 3)^2 + 5(3|x + 2| - 3) + 5) + 1| + 1) - 3) - 1) +... | 152 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(3π(1|(1|(-1(-2(-1x - 1)^2 - 5(-1x - 1) - 5)^2 - 5(-2(-1x - 1)^2 - 5(-1x - 1) - 5) - 3) + 2| - 2) - 3| - 1) - 3) - 3 and g(x) = -1|(-1((3(-2((-2x - 2)/(2x - 3)) + 3) - 2)/(2(-2((-2x - 2)/(2x - 3)) + 3) - 3)) + 5) - 2| - 1 intersect in the interval ... | 1188 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1e^(1((2(1|(-1|(1(-1x^2 + 5x + 2)^2 + 2(-1x^2 + 5x + 2) + 1) - 2| + 3)| - 2) + 1)/(-1(1|(-1|(1(-1x^2 + 5x + 2)^2 + 2(-1x^2 + 5x + 2) + 1) - 2| + 3)| - 2) + 2))) - 3 and g(x) = -1(-3|(1|((-1(5x^2 + 2x - 2) + 3)/(-2(5x^2 + 2x - 2) + 1)) + 3| + 3) + 1| -... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1((1(-1(-1(2((-2x - 2)/(-3x - 2))^2 - 3((-2x - 2)/(-3x - 2)) + 5)^2 + 3(2((-2x - 2)/(-3x - 2))^2 - 3((-2x - 2)/(-3x - 2)) + 5) + 3)^2 - 4(-1(2((-2x - 2)/(-3x - 2))^2 - 3((-2x - 2)/(-3x - 2)) + 5)^2 + 3(2((-2x - 2)/(-3x - 2))^2 - 3((-2x - 2)/(-3x - 2)) + 5) + 3)... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-3|(-3sin(1π(-1(3(-3x^2 + 2)^2 + 3(-3x^2 + 2) - 4) - 3) + 2) + 3) - 2| + 3) - 3)/(3(-3|(-3sin(1π(-1(3(-3x^2 + 2)^2 + 3(-3x^2 + 2) - 4) - 3) + 2) + 3) - 2| + 3) + 2) and g(x) = -1|(-1(1(-1(3cos(2πx - 2) + 2)^2 - 1(3cos(2πx - 2) ... | 25 | MATH |
[
{
"content": "How many solutions does the equation 2(-2|(2|(2|((3(-2|x + 3| + 3) - 2)/(-1(-2|x + 3| + 3) + 3)) - 3| + 3) - 3|) - 2| + 2) + 4 = -4(-3(-1(-1|(3cos(2πx - 3) - 3) + 3| + 2) + 1)^2 - 5(-1(-1|(3cos(2πx - 3) - 3) + 3| + 2) + 1) + 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaT... | 4 | MATH |
[
{
"content": "How many solutions does the equation 2cos(2π((-1(-3|(5(-1(-1x - 3)^2 + 2(-1x - 3) - 2)^2 - 5(-1(-1x - 3)^2 + 2(-1x - 3) - 2) - 1) - 2|) + 1)/(2(-3|(5(-1(-1x - 3)^2 + 2(-1x - 3) - 2)^2 - 5(-1(-1x - 3)^2 + 2(-1x - 3) - 2) - 1) - 2|) - 1))) = -1(3cos(3π(3|(1(3x + 2) + 1) + 2| + 2) + 2) - 3)^2 - 4(3co... | 542 | MATH |
[
{
"content": "How many solutions does the equation 5(2e^(-2(-1|(2|(1((-1x + 1)/(-1x))^2 + 3((-1x + 1)/(-1x)) + 2) - 3| - 1) - 2| - 1)) - 3)^2 + 4(2e^(-2(-1|(2|(1((-1x + 1)/(-1x))^2 + 3((-1x + 1)/(-1x)) + 2) - 3| - 1) - 2| - 1)) - 3) - 3 = 5(2|(-2(-4(-1x^2 - 3x - 1) + 1)^2 + 5(-4(-1x^2 - 3x - 1) + 1) + 5) - 2| -... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1|(-2|(-5(2|(1e^(1((-2x - 1)/(-3x + 4))) - 2) + 3| - 2)^2 + 1(2|(1e^(1((-2x - 1)/(-3x + 4))) - 2) + 3| - 2) + 5) + 3| + 3) + 2| + 3 = 1|(1cos(1π(-3|((-1(4x^2 - 3x + 3) + 1)/(-2(4x^2 - 3x + 3) + 3)) - 3|) + 3) - 2) - 3| - 3 have for x in the interval [-10, 10]?... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(4(3sin(1π((2(1|(5x^2 - 2x - 4) + 2| - 1) - 1)/(-3(1|(5x^2 - 2x - 4) + 2| - 1) - 3)) - 2) - 1) + 4)^2 - 4 and g(x) = 3(1e^(2(4(3(-3|x| - 3) + 2) + 4)))^2 - 5(1e^(2(4(3(-3|x| - 3) + 2) + 4))) - 2 intersect in the interval [-10, 10].\n\nPresent the answ... | 15 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(-1(1e^(-2(4(-2|(-4x - 4)|)^2 - 3(-2|(-4x - 4)|) - 2)) - 2) + 4)^2 - 5(-1(1e^(-2(4(-2|(-4x - 4)|)^2 - 3(-2|(-4x - 4)|) - 2)) - 2) + 4) and g(x) = 1(3(3((-1(3x^2 - 2x - 3) - 2)/(1(3x^2 - 2x - 3) - 3)))) - 4 where -10 ≤ x ≤ 10... | 6 | MATH |
[
{
"content": "How many solutions does the equation (-2(-2(1(-1|(3cos(2π(-1x^2 - 3x - 5) - 2) - 2) + 3| + 1) + 3) + 4) + 2)/(1(-2(1(-1|(3cos(2π(-1x^2 - 3x - 5) - 2) - 2) + 3| + 1) + 3) + 4) - 2) = 3cos(2π(-3|(-2(-2|(-1x^2 + 5x - 5) - 3| + 2) - 2) - 2| - 2) - 2) + 2 have for x in the interval [-10, 10]?\n\nPresen... | 190 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-1|(2|((2(-3|(-2x^2 + 3x + 5) + 2| - 3) + 1)/(2(-3|(-2x^2 + 3x + 5) + 2| - 3) + 2)) - 1| - 1) + 1|)^2 + 1(-1|(2|((2(-3|(-2x^2 + 3x + 5) + 2| - 3) + 1)/(2(-3|(-2x^2 + 3x + 5) + 2| - 3) + 2)) - 1| - 1) + 1|) + 4 and g(x) = 1... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2((1((1((3(-2(3cos(2πx - 1) + 1)^2 - 4(3cos(2πx - 1) + 1) + 1) - 2)/(-2(-2(3cos(2πx - 1) + 1)^2 - 4(3cos(2πx - 1) + 1) + 1) + 3)) + 1)/(-1((3(-2(3cos(2πx - 1) + 1)^2 - 4(3cos(2πx - 1) + 1) + 1) - 2)/(-2(-2(3cos(2πx - 1) + 1)^2 - 4... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1((2(3(-3(-2(1sin(2πx + 2) + 3)^2 + 2(1sin(2πx + 2) + 3) + 3)^2 - 2(-2(1sin(2πx + 2) + 3)^2 + 2(1sin(2πx + 2) + 3) + 3) + 2) - 5) + 2)/(-3(3(-3(-2(1sin(2πx + 2) + 3)^2 + 2(1sin(2πx + 2) + 3) + 3)^2 - 2(-2(1sin(2πx + 2) + 3)^2... | 1125 | MATH |
[
{
"content": "How many solutions does the equation -2|((-1(-5(3(1sin(1π(-4x^2 - 1x - 2) - 3)) - 2)) + 1)/(2(-5(3(1sin(1π(-4x^2 - 1x - 2) - 3)) - 2)) + 1)) + 3| - 2 = -1|(-1|(-2(1e^(1(-4x + 4)) - 2)^2 - 3(1e^(1(-4x + 4)) - 2) - 2) + 3| - 3)| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTe... | 100 | MATH |
[
{
"content": "How many solutions does the equation (-1((-3(-1|((-2((-1((-2x + 3)/(2x - 1)) + 2)/(1((-2x + 3)/(2x - 1)) + 3)) - 1)/(3((-1((-2x + 3)/(2x - 1)) + 2)/(1((-2x + 3)/(2x - 1)) + 3)) - 1)) + 2| - 3) + 1)/(-1(-1|((-2((-1((-2x + 3)/(2x - 1)) + 2)/(1((-2x + 3)/(2x - 1)) + 3)) - 1)/(3((-1((-2x + 3)/(2x - 1)... | 148 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|((-2((-1(-3sin(2π(5((2x - 3)/(-3x - 1))^2 - 2((2x - 3)/(-3x - 1)) - 1))) + 2)/(-1(-3sin(2π(5((2x - 3)/(-3x - 1))^2 - 2((2x - 3)/(-3x - 1)) - 1))) + 3)))/(-2((-1(-3sin(2π(5((2x - 3)/(-3x - 1))^2 - 2((2x - 3)/(-3x - 1)) - 1)... | 130 | MATH |
[
{
"content": "How many solutions does the equation (-2(-1|(4(-1sin(1π(-3|(-1x^2 - 3x + 3) - 1| + 2) - 3))^2 + 2(-1sin(1π(-3|(-1x^2 - 3x + 3) - 1| + 2) - 3)) - 3) - 1| + 2) - 2)/(-2(-1|(4(-1sin(1π(-3|(-1x^2 - 3x + 3) - 1| + 2) - 3))^2 + 2(-1sin(1π(-3|(-1x^2 - 3x + 3) - 1| + 2) - 3)) - 3) - 1| + 2) + 1) = (-3(-1c... | 646 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(3π((3(1(-2(-3(2x^2 + 2x - 3) - 2)^2 - 4(-3(2x^2 + 2x - 3) - 2) - 2) - 2) - 2)/(3(1(-2(-3(2x^2 + 2x - 3) - 2)^2 - 4(-3(2x^2 + 2x - 3) - 2) - 2) - 2) - 1)) - 1) and g(x) = (-3(-5(-1(-4(5x^2 - 5x - 1)^2 + 5(5x^2 - 5x - 1) + 5)^2 ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(1|(-1(-2((2(3x^2 - 3) - 1)/(-3(3x^2 - 3) - 1))^2 + 4((2(3x^2 - 3) - 1)/(-3(3x^2 - 3) - 1)) - 4) + 3) - 2| - 1) + 2 and g(x) = -3(1cos(2π(-2(1((-2x - 3)/(1x + 2)))^2 + 4(1((-2x - 3)/(1x + 2))) - 2) - 2) - 1) where -10 ≤ x ≤ ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(1π((1((2((2(-1|((1x)/(1x)) + 1| - 1) + 2)/(-3(-1|((1x)/(1x)) + 1| - 1) + 4)))/(2((2(-1|((1x)/(1x)) + 1| - 1) + 2)/(-3(-1|((1x)/(1x)) + 1| - 1) + 4)) - 1)) + 1)/(-2((2((2(-1|((1x)/(1x)) + 1| - 1) + 2)/(-3(-1|((1x)/(1x)) +... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(2e^(-1(1(-5(2|((-1x - 1)/(-2x)) + 1| - 3)^2 + 5(2|((-1x - 1)/(-2x)) + 1| - 3) + 2) + 5)) - 1)^2 - 2(2e^(-1(1(-5(2|((-1x - 1)/(-2x)) + 1| - 3)^2 + 5(2|((-1x - 1)/(-2x)) + 1| - 3) + 2) + 5)) - 1) + 5 and g(x) = (-2(-4(4(2((-2x - 3... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(5(4(1(3((3x)/(-1x + 3))^2 + 3((3x)/(-1x + 3)) - 4)^2 - 2(3((3x)/(-1x + 3))^2 + 3((3x)/(-1x + 3)) - 4) - 5)^2 - 5(1(3((3x)/(-1x + 3))^2 + 3((3x)/(-1x + 3)) - 4)^2 - 2(3((3x)/(-1x + 3))^2 + 3((3x)/(-1x + 3)) - 4) - 5) - 3) + ... | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(4(1((3(2(-2|x + 2| + 2) + 4) - 2)/(-1(2(-2|x + 2| + 2) + 4) + 2)) - 1) - 1) + 2| - 3 and g(x) = 1(-2e^(1(-4(-4(2|x - 2| + 1))^2 + 4(-4(2|x - 2| + 1)) - 5)) + 2) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(1(1(-3(-4(1x^2 + 3x - 1) + 1) - 5)^2 - 2(-3(-4(1x^2 + 3x - 1) + 1) - 5)) - 4) - 1| and g(x) = -1(2sin(3π(-3|(1|(-3x^2 - 4x - 5) - 3| + 3) + 1| + 3)) - 1)^2 - 5(2sin(3π(-3|(1|(-3x^2 - 4x - 5) - 3| + 3) + 1| + 3)) - 1) - 3 in the ... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(2|(-1(-3|(-3|(-3sin(2πx) + 1) - 3| + 3)| - 1) - 5)| - 3) - 1)/(-1(2|(-1(-3|(-3|(-3sin(2πx) + 1) - 3| + 3)| - 1) - 5)| - 3) + 2) and g(x) = -3sin(2π((-2(-4((-3((-2x - 2)/(2x + 3)))/(-1((-2x - 2)/(2x + 3)) + 3))^2 + 2((-3((-2x - 2)/(2x + 3)))/(-1((-2... | 186 | MATH |
[
{
"content": "How many solutions does the equation -1cos(2π((-2((1(-2|(2(2|x - 3| + 1) - 1) - 2| + 3) - 3)/(1(-2|(2(2|x - 3| + 1) - 1) - 2| + 3) - 2)))/(-1((1(-2|(2(2|x - 3| + 1) - 1) - 2| + 3) - 3)/(1(-2|(2(2|x - 3| + 1) - 1) - 2| + 3) - 2)) - 1)) + 2) + 1 = -1(1cos(3π((2(-3|((-3x - 2)/(1x + 1)) + 1| + 3) + 1)... | 16 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(-1((1(-5(-2(2|x + 2| - 1)^2 + 5(2|x + 2| - 1) - 1)^2 - 1(-2(2|x + 2| - 1)^2 + 5(2|x + 2| - 1) - 1)) + 3)/(-2(-5(-2(2|x + 2| - 1)^2 + 5(2|x + 2| - 1) - 1)^2 - 1(-2(2|x + 2| - 1)^2 + 5(2|x + 2| - 1) - 1)) - 1))^2 + 1((1(-5(-2(2|x... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(3(4(-5(2cos(1π(5x + 4) - 2) - 1))^2 + 2(-5(2cos(1π(5x + 4) - 2) - 1)) - 2) - 1) - 2| - 3 and g(x) = -3(2e^(-1(4(2(-4x^2 + 4))^2 - 5(2(-4x^2 + 4)) - 2)) - 3) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your... | 196 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(2π(3|(-1((-1((-3(-2x^2 - 2) - 2)/(-1(-2x^2 - 2) - 1)) + 2)/(1((-3(-2x^2 - 2) - 2)/(-1(-2x^2 - 2) - 1)) + 1))^2 + 5((-1((-3(-2x^2 - 2) - 2)/(-1(-2x^2 - 2) - 1)) + 2)/(1((-3(-2x^2 - 2) - 2)/(-1(-2x^2 - 2) - 1)) + 1)) - 3) + 1| ... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(1(2(3(1((-3x - 2)/(-2x - 3)) - 5) - 3))^2 + 3(2(3(1((-3x - 2)/(-2x - 3)) - 5) - 3)) - 4) - 2)/(-1(1(2(3(1((-3x - 2)/(-2x - 3)) - 5) - 3))^2 + 3(2(3(1((-3x - 2)/(-2x - 3)) - 5) - 3)) - 4) - 3) and g(x) = 3(2(-3cos(1π(-5(2x^... | 274 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-2(3|(5(-2|((-3x - 1)/(2x - 3)) + 3| - 1)^2 - 5(-2|((-3x - 1)/(2x - 3)) + 3| - 1) - 3) - 1| + 2)))/(-3(-2(3|(5(-2|((-3x - 1)/(2x - 3)) + 3| - 1)^2 - 5(-2|((-3x - 1)/(2x - 3)) + 3| - 1) - 3) - 1| + 2)) - 3) and g(x) = -3|(-... | 1200 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(1(2(5(2(2|x - 2| - 3) + 2)^2 - 5(2(2|x - 2| - 3) + 2) + 3) - 4) + 4) + 1)/(-1(1(2(5(2(2|x - 2| - 3) + 2)^2 - 5(2(2|x - 2| - 3) + 2) + 3) - 4) + 4) + 3) and g(x) = -2(-3((-1(-3e^(-2(-3|x - 1|)) + 2) - 3)/(1(-3e^(-2(-3|x - 1|)) + ... | 4 | MATH |
[
{
"content": "How many solutions does the equation 2cos(3π(2|(-5(4((1(-1x + 3) + 1)/(2(-1x + 3) + 1)) + 4) - 1)| + 3) - 2) + 2 = (-1((-2((-2(-3(2cos(2πx)) + 2) - 1)/(-3(-3(2cos(2πx)) + 2) - 3)) + 2)/(1((-2(-3(2cos(2πx)) + 2) - 1)/(-3(-3(2cos(2πx)) + 2) - 3)) + 3)) + 3)/(-3((-2((-2(-3(2cos(2πx)) + 2) - 1)/(-3(-3... | 205 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-3|(1|((-1(-3|(-5x - 1) - 3| + 3) - 3)/(2(-3|(-5x - 1) - 3| + 3) - 3)) - 2| - 3) - 3| + 2)^2 - 3(-3|(1|((-1(-3|(-5x - 1) - 3| + 3) - 3)/(2(-3|(-5x - 1) - 3| + 3) - 3)) - 2| - 3) - 3| + 2) + 2 and g(x) = 5(-3((-2(4((1x - 2)/(3x - 2))^2 + 2((1x - 2)/(3... | 7 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3((-3(-2((-2(-2|(-1|x + 2| + 1)| - 2) + 3)/(-1(-2|(-1|x + 2| + 1)| - 2)))^2 - 4((-2(-2|(-1|x + 2| + 1)| - 2) + 3)/(-1(-2|(-1|x + 2| + 1)| - 2))) - 5) - 2)/(2(-2((-2(-2|(-1|x + 2| + 1)| - 2) + 3)/(-1(-2|(-1|x + 2| + 1)| - 2)))^2 - 4... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(3(-2|(-2(4(-3cos(3πx - 1) - 3) + 2)^2 + 3(4(-3cos(3πx - 1) - 3) + 2) - 3) + 1| + 1) - 1) + 2)/(-1(3(-2|(-2(4(-3cos(3πx - 1) - 3) + 2)^2 + 3(4(-3cos(3πx - 1) - 3) + 2) - 3) + 1| + 1) - 1) - 2) and g(x) = -1(-3|(4(-4(-1x - 5)^2 -... | 6 | MATH |
[
{
"content": "How many solutions does the equation -2|(-3cos(1π(4(-4(-5(4x + 1) - 2)^2 - 5(-5(4x + 1) - 2) + 3) + 3) - 3)) - 3| + 2 = (2((2((-2(3(2|x + 2| - 1) - 3) - 2)/(-1(3(2|x + 2| - 1) - 3) + 3)) - 1)/(-1((-2(3(2|x + 2| - 1) - 3) - 2)/(-1(3(2|x + 2| - 1) - 3) + 3)))) + 2)/(2((2((-2(3(2|x + 2| - 1) - 3) - 2... | 438 | MATH |
[
{
"content": "How many solutions does the equation 5(-3(1|(2((-2(-1|x| + 1) - 3)/(2(-1|x| + 1) - 1))^2 - 5((-2(-1|x| + 1) - 3)/(2(-1|x| + 1) - 1)) - 2) + 2| - 3) - 1)^2 + 3(-3(1|(2((-2(-1|x| + 1) - 3)/(2(-1|x| + 1) - 1))^2 - 5((-2(-1|x| + 1) - 3)/(2(-1|x| + 1) - 1)) - 2) + 2| - 3) - 1) - 3 = 3(1sin(2π(-3|(-1|((... | 28 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(1|(-2((2(3e^(2(3x^2 + 1x - 4)) - 2))/(-1(3e^(2(3x^2 + 1x - 4)) - 2) - 2))^2 + 5((2(3e^(2(3x^2 + 1x - 4)) - 2))/(-1(3e^(2(3x^2 + 1x - 4)) - 2) - 2)) + 2) - 1| - 1) - 1| + 1 and g(x) = 5((2((2(1sin(1π((-1x + 2)/(-3x - 3)) + 1)) + 1)/(-1(1sin(1π((-1x ... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3((3(-1|(-2|(3cos(3π((-3x)/(1x - 2)) - 3)) + 2| + 1)|) - 3)/(1(-1|(-2|(3cos(3π((-3x)/(1x - 2)) - 3)) + 2| + 1)|))) + 1 and g(x) = 1(-2|(-3sin(2π(3(3x^2 - 2x + 2) + 4) + 2))| - 3) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex ... | 825 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(3π(-1|((1(-5((1(1|x - 1| + 2) + 1)/(3(1|x - 1| + 2)))^2 + 1((1(1|x - 1| + 2) + 1)/(3(1|x - 1| + 2))) - 3) + 3)/(3(-5((1(1|x - 1| + 2) + 1)/(3(1|x - 1| + 2)))^2 + 1((1(1|x - 1| + 2) + 1)/(3(1|x - 1| + 2))) - 3) - 2)) - 1|) - 1) + 3 and g(x) = 2|(1|... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((-1(3|(1e^(1(3|(-2|x + 2| + 3) - 2| + 1)) + 3)|) + 1)/(1(3|(1e^(1(3|(-2|x + 2| + 3) - 2| + 1)) + 3)|))) - 5 and g(x) = -3|((3(3sin(2π(-1(-2|x| - 2)^2 - 3(-2|x| - 2) - 4) - 1) + 1) + 2)/(3(3sin(2π(-1(-2|x| - 2)^2 - 3(-2|x| - 2) - 4) - 1) + 1))) + 3| +... | 177 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(-1(4(2((2(1x + 4) + 3)/(-3(1x + 4) + 2))^2 + 4((2(1x + 4) + 3)/(-3(1x + 4) + 2)) - 5) + 4)^2 - 5(4(2((2(1x + 4) + 3)/(-3(1x + 4) + 2))^2 + 4((2(1x + 4) + 3)/(-3(1x + 4) + 2)) - 5) + 4) - 1)| + 2 and g(x) = -1(-2cos(3π(5(2((-2x ... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1((1(4(-3(-5(2x^2)^2 - 1(2x^2) + 5)^2 - 2(-5(2x^2)^2 - 1(2x^2) + 5) + 2) + 1) + 2)/(1(4(-3(-5(2x^2)^2 - 1(2x^2) + 5)^2 - 2(-5(2x^2)^2 - 1(2x^2) + 5) + 2) + 1))) + 1)/(1((1(4(-3(-5(2x^2)^2 - 1(2x^2) + 5)^2 - 2(-5(2x^2)^2 - 1(2x^2) + 5) + 2) + 1) + 2)/(... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|((-2(-3|(3(2(-2|x + 2| + 2))^2 - 3(2(-2|x + 2| + 2))) + 1| + 2) - 1)/(3(-3|(3(2(-2|x + 2| + 2))^2 - 3(2(-2|x + 2| + 2))) + 1| + 2) + 2))| - 2 and g(x) = 3cos(2π((-2(-1((3(1x^2 - 5x) + 3)/(2(1x^2 - 5x) - 3))^2 + 2) + 2)/(2(... | 300 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-1((1(-4(2(-3|x - 3| + 3)^2 + 5(-3|x - 3| + 3) - 4) - 4) - 1)/(2(-4(2(-3|x - 3| + 3)^2 + 5(-3|x - 3| + 3) - 4) - 4) + 2))^2 - 1((1(-4(2(-3|x - 3| + 3)^2 + 5(-3|x - 3| + 3) - 4) - 4) - 1)/(2(-4(2(-3|x - 3| + 3)^2 + 5(-3|x - ... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(-3((-3((-2(3(-1|x - 2| + 1)^2 + 5(-1|x - 2| + 1) - 3) + 2)/(2(3(-1|x - 2| + 1)^2 + 5(-1|x - 2| + 1) - 3) - 1)) - 1)/(1((-2(3(-1|x - 2| + 1)^2 + 5(-1|x - 2| + 1) - 3) + 2)/(2(3(-1|x - 2| + 1)^2 + 5(-1|x - 2| + 1) - 3) - 1))... | 8 | MATH |
[
{
"content": "How many solutions does the equation -4(-2(2|(2((3(2|x + 2| - 3) + 3)/(-2(2|x + 2| - 3) - 1))^2 + 4((3(2|x + 2| - 3) + 3)/(-2(2|x + 2| - 3) - 1))) + 1| - 3) + 5) - 3 = 3sin(2π(2|(-1(5(-5x^2 + 2x - 1) + 5)^2 - 5(5(-5x^2 + 2x - 1) + 5) + 2) + 2| - 3) + 3) have for x in the interval [-10, 10]?\n\nPre... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2sin(2π(-1(1(1|(5((2x)/(-2x - 1))^2 + 2((2x)/(-2x - 1)) - 3) + 2| + 2)^2 + 5(1|(5((2x)/(-2x - 1))^2 + 2((2x)/(-2x - 1)) - 3) + 2| + 2) + 4)^2 + 4(1(1|(5((2x)/(-2x - 1))^2 + 2((2x)/(-2x - 1)) - 3) + 2| + 2)^2 + 5(1|(5((2x)/(-2x - 1))^2 + 2((2x)/(-2x - 1... | 1627 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-4((1(-1cos(1π(1|(-3x - 3) - 3|) - 1) + 1) - 3)/(-2(-1cos(1π(1|(-3x - 3) - 3|) - 1) + 1) + 3))) - 1| and g(x) = -1|(1(-1|(2e^(1(-2|x + 2|)) - 1) - 2| - 1) - 4) - 1| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex f... | 120 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(2(-1sin(3π(-5((-2(1x^2 - 1) + 2)/(-2(1x^2 - 1) + 3))) + 3))^2 - 5(-1sin(3π(-5((-2(1x^2 - 1) + 2)/(-2(1x^2 - 1) + 3))) + 3)) - 2) + 2 and g(x) = (-2(2|(-5((3(2cos(2πx + 3)) - 1)/(2(2cos(2πx + 3)) - 1))^2 - 2((3(2cos(2πx + 3)... | 100 | MATH |
[
{
"content": "How many solutions does the equation 2(-2|(2sin(1π(2(2|(-2x^2 + 4x - 3) - 1| + 1) + 4) - 2) + 3) + 3| + 1)^2 - 1(-2|(2sin(1π(2(2|(-2x^2 + 4x - 3) - 1| + 1) + 4) - 2) + 3) + 3| + 1) - 4 = 3|(-1(-3|(4(-5x + 1) + 4) - 2| + 2)) - 1| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex ... | 100 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(1|((1(-1|((-2(-2cos(3πx + 1) + 1) + 1)/(-3(-2cos(3πx + 1) + 1) + 2)) + 2| - 3) + 3)/(-2(-1|((-2(-2cos(3πx + 1) + 1) + 1)/(-3(-2cos(3πx + 1) + 1) + 2)) + 2| - 3) - 2))| - 1) - 4 and g(x) = (3(-2|(2(1|(5x - 2) + 1| - 3)^2 + 1) - 3| + 3) + 2)/(-3(-2|(2... | 23 | MATH |
[
{
"content": "How many solutions does the equation 3|(4(5(-3|(3(2x^2 + 5x - 3)) + 2| + 1) + 5) + 5) - 2| - 1 = -3sin(1π(-1|(-2(-1(-3|x + 1| + 3) + 5)^2 - 1(-1(-3|x + 1| + 3) + 5) - 3) - 3|) + 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(3(-2(3cos(1π((2(-3x + 2) + 2)/(-1(-3x + 2) + 3)) - 2) + 2)^2 + 4(3cos(1π((2(-3x + 2) + 2)/(-1(-3x + 2) + 3)) - 2) + 2) + 2)^2 - 2(-2(3cos(1π((2(-3x + 2) + 2)/(-1(-3x + 2) + 3)) - 2) + 2)^2 + 4(3cos(1π((2(-3x + 2) + 2)/(-1(-3x + 2... | 54 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(1((-3(3sin(2π(-1|(1x + 3) + 2| + 3) - 2) + 3))/(2(3sin(2π(-1|(1x + 3) + 2| + 3) - 2) + 3) + 1))^2 - 5((-3(3sin(2π(-1|(1x + 3) + 2| + 3) - 2) + 3))/(2(3sin(2π(-1|(1x + 3) + 2| + 3) - 2) + 3) + 1)) - 4)^2 - 4(1((-3(3sin(2π(-1|(1x + 3) + 2| + 3) - 2) +... | 179 | MATH |
[
{
"content": "How many solutions does the equation (-2((2(2((2(-5(2|x + 2| - 3) - 5) - 3)/(1(-5(2|x + 2| - 3) - 5) + 2)) - 3) - 3)/(-3(2((2(-5(2|x + 2| - 3) - 5) - 3)/(1(-5(2|x + 2| - 3) - 5) + 2)) - 3) + 4)) + 1)/(1((2(2((2(-5(2|x + 2| - 3) - 5) - 3)/(1(-5(2|x + 2| - 3) - 5) + 2)) - 3) - 3)/(-3(2((2(-5(2|x + 2... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2((3(2(-3((-3(4x^2 + 3x + 4) + 1)/(-1(4x^2 + 3x + 4))))^2 - 1(-3((-3(4x^2 + 3x + 4) + 1)/(-1(4x^2 + 3x + 4)))) + 3) - 2)/(-2(2(-3((-3(4x^2 + 3x + 4) + 1)/(-1(4x^2 + 3x + 4))))^2 - 1(-3((-3(4x^2 + 3x + 4) + 1)/(-1(4x^2 + 3x + 4)))... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3((-2((2(2(-4((-2x + 2)/(1x - 2))^2 + 4((-2x + 2)/(1x - 2)) - 1)^2 + 2) + 3)/(1(2(-4((-2x + 2)/(1x - 2))^2 + 4((-2x + 2)/(1x - 2)) - 1)^2 + 2) - 3)) - 3)/(-3((2(2(-4((-2x + 2)/(1x - 2))^2 + 4((-2x + 2)/(1x - 2)) - 1)^2 + 2) + 3)/(1(2(-4((-2x + 2)/(1x ... | 6 | MATH |
[
{
"content": "How many solutions does the equation 3sin(1π(-2|(-2(2|(-3(5x - 1) + 2) - 2| - 2)^2 + 2(2|(-3(5x - 1) + 2) - 2| - 2) + 4) + 2| - 2) - 1) - 3 = -3|((-3(-3(4(-2e^(-2x) + 1)^2 - 3(-2e^(-2x) + 1) + 2)^2 - 4(4(-2e^(-2x) + 1)^2 - 3(-2e^(-2x) + 1) + 2) + 5))/(-2(-3(4(-2e^(-2x) + 1)^2 - 3(-2e^(-2x) + 1) + ... | 1248 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(1(3(1((2(3|x - 1| - 1) - 2)/(-1(3|x - 1| - 1) - 3)) + 1) - 4)^2 - 1(3(1((2(3|x - 1| - 1) - 2)/(-1(3|x - 1| - 1) - 3)) + 1) - 4) + 3) + 3 and g(x) = 5(-1cos(3π(2(-3(3|x + 2| - 1)^2 - 5(3|x + 2| - 1) + 3) + 5) + 1) - 2)^2 + 4(-1co... | 72 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3cos(1π(1(5(-4(3|((2x + 2)/(-2x - 1)) - 1| - 3) - 1) + 5) - 5) - 2) and g(x) = -2cos(2π((2((-1((2((2x + 2)/(1x + 3)) + 2)/(-3((2x + 2)/(1x + 3)) + 4)) - 3)/(2((2((2x + 2)/(1x + 3)) + 2)/(-3((2x + 2)/(1x + 3)) + 4)))) - 2)/(-3((-1(... | 101 | MATH |
[
{
"content": "How many solutions does the equation 5((-1((-2(3((-2(-2cos(3πx + 2) + 1))/(-1(-2cos(3πx + 2) + 1) + 3)) + 5) + 2)/(-3(3((-2(-2cos(3πx + 2) + 1))/(-1(-2cos(3πx + 2) + 1) + 3)) + 5) - 2)) - 3)/(-3((-2(3((-2(-2cos(3πx + 2) + 1))/(-1(-2cos(3πx + 2) + 1) + 3)) + 5) + 2)/(-3(3((-2(-2cos(3πx + 2) + 1))/(... | 81 | MATH |
[
{
"content": "How many solutions does the equation -3(5(-3cos(3π((2(-2((1x - 1)/(-3x - 2))^2 - 1((1x - 1)/(-3x - 2)) + 1) + 1)/(-1(-2((1x - 1)/(-3x - 2))^2 - 1((1x - 1)/(-3x - 2)) + 1) + 2)) - 2) + 2))^2 + 1(5(-3cos(3π((2(-2((1x - 1)/(-3x - 2))^2 - 1((1x - 1)/(-3x - 2)) + 1) + 1)/(-1(-2((1x - 1)/(-3x - 2))^2 - ... | 30 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-2((-1(-1(3|((-3x - 2)/(3x + 3))| - 2)^2 - 2) + 2)/(2(-1(3|((-3x - 2)/(3x + 3))| - 2)^2 - 2) - 1)) + 3)^2 and g(x) = -1|((3(-3(2cos(3π(-1|x + 3| + 2) - 1) - 1) - 1) - 1)/(-1(-3(2cos(3π(-1|x + 3| + 2) - 1) - 1) - 1) + 3)) - 1| - ... | 120 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|((-3(-4((1(5(1x - 1)^2 + 2(1x - 1)) - 2)/(-2(5(1x - 1)^2 + 2(1x - 1)) + 3))^2 + 4((1(5(1x - 1)^2 + 2(1x - 1)) - 2)/(-2(5(1x - 1)^2 + 2(1x - 1)) + 3)) + 1) - 3)/(2(-4((1(5(1x - 1)^2 + 2(1x - 1)) - 2)/(-2(5(1x - 1)^2 + 2(1x ... | 5 | MATH |
[
{
"content": "How many solutions does the equation 1|(1(-2|((1(-2cos(3π(4x + 2) - 3) + 1) - 1)/(1(-2cos(3π(4x + 2) - 3) + 1) - 3)) + 1| + 1) - 4) + 1| - 1 = 2|(5(3|(1e^(-1(-4x + 3)) + 2) - 3| - 2) - 2)| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"... | 145 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(1(-3(3|(-1|(1|x| - 1) - 2|) + 1| - 2)^2 + 2(3|(-1|(1|x| - 1) - 2|) + 1| - 2) - 3) + 4)^2 + 4(1(-3(3|(-1|(1|x| - 1) - 2|) + 1| - 2)^2 + 2(3|(-1|(1|x| - 1) - 2|) + 1| - 2) - 3) + 4) - 3 and g(x) = (-1(3(1cos(2π(3|(2x^2 + 5x +... | 67 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2((-1(-3(2(4(-3cos(2πx - 1) - 3) + 2)^2 - 1(4(-3cos(2πx - 1) - 3) + 2) - 2) + 1) + 2)/(1(-3(2(4(-3cos(2πx - 1) - 3) + 2)^2 - 1(4(-3cos(2πx - 1) - 3) + 2) - 2) + 1) + 3)) + 2 and g(x) = (-2((-2(2(2((-1x - 1)/(3x)) - 3)^2 - 1(2((-1x... | 75 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-1|(2(-1(-2sin(2π(3|x - 1|) + 3) + 2))^2 - 5(-1(-2sin(2π(3|x - 1|) + 3) + 2)) - 4) - 1| + 1) + 5 and g(x) = (-2(2(-2cos(1π(-4((-2x)/(-1x)) - 1) - 1)) - 1) - 2)/(-3(2(-2cos(1π(-4((-2x)/(-1x)) - 1) - 1)) - 1) - 3) in the range -10 ... | 240 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(2(2|((-1(1(1|x + 2| - 3)^2 - 3(1|x + 2| - 3) + 5) - 1)/(1(1(1|x + 2| - 3)^2 - 3(1|x + 2| - 3) + 5) - 3)) - 3| + 1))^2 - 2(2(2|((-1(1(1|x + 2| - 3)^2 - 3(1|x + 2| - 3) + 5) - 1)/(1(1(1|x + 2| - 3)^2 - 3(1|x + 2| - 3) + 5) - ... | 344 | MATH |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.