messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Find the number of times the graphs of f(x) = 1|(1(1|(-4(-2(-2sin(1πx) - 1)^2 - 3(-2sin(1πx) - 1) + 1) - 2) + 3| + 2) + 1)| - 3 and g(x) = (2(2e^(-2((-1((-2(-3x + 4) - 3)/(-2(-3x + 4))) - 2)/(1((-2(-3x + 4) - 3)/(-2(-3x + 4))))))) + 3)/(1(2e^(-2((-1((-2(-3x + 4) - 3)/(-2(-3x + 4))) - 2)/(1((-2(-3x... | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|((3((-1(2|((1(1|x - 2| + 3) - 1)/(-1(1|x - 2| + 3))) - 3| - 3) + 2)/(-1(2|((1(1|x - 2| + 3) - 1)/(-1(1|x - 2| + 3))) - 3| - 3) + 2)) + 3)/(-2((-1(2|((1(1|x - 2| + 3) - 1)/(-1(1|x - 2| + 3))) - 3| - 3) + 2)/(-1(2|((1(1|x - 2| + 3... | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5((-1(3((-2(5(3sin(2πx - 3) - 2) - 1) + 2)/(-1(5(3sin(2πx - 3) - 2) - 1) + 2)) + 5) + 3)/(2(3((-2(5(3sin(2πx - 3) - 2) - 1) + 2)/(-1(5(3sin(2πx - 3) - 2) - 1) + 2)) + 5)))^2 + 2((-1(3((-2(5(3sin(2πx - 3) - 2) - 1) + 2)/(-1(5(3sin(... | 19 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3((-3(-1cos(3π(3((1(-2|x - 3| - 2) - 3)/(-2(-2|x - 3| - 2) - 2)) + 4) + 1) + 2) - 1)/(-3(-1cos(3π(3((1(-2|x - 3| - 2) - 3)/(-2(-2|x - 3| - 2) - 2)) + 4) + 1) + 2) - 2)) + 1)/(-3((-3(-1cos(3π(3((1(-2|x - 3| - 2) - 3)/(-2(-2|x - 3|... | 719 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(5(2((-3(2(-1|x - 1| - 1) + 3) - 1)/(2(2(-1|x - 1| - 1) + 3) - 1)) - 3)^2 - 2(2((-3(2(-1|x - 1| - 1) + 3) - 1)/(2(2(-1|x - 1| - 1) + 3) - 1)) - 3) + 4) - 3 and g(x) = 3|(-3(1|(4(1x + 2)^2 - 5(1x + 2) - 4) - 1| + 2)^2 + 5(1|(... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(3sin(1π((1(4(-2(-1x^2 - 3x + 3) + 3)^2 - 2(-2(-1x^2 - 3x + 3) + 3) + 2) - 2)/(3(4(-2(-1x^2 - 3x + 3) + 3)^2 - 2(-2(-1x^2 - 3x + 3) + 3) + 2) - 1)) + 1) - 1)^2 - 4(3sin(1π((1(4(-2(-1x^2 - 3x + 3) + 3)^2 - 2(-2(-1x^2 - 3x + 3) + 3) + 2) - 2)/(3(4(-2(-1... | 247 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(3|(4(-3|(-4(-1x^2 + 3x - 2)^2 - 5(-1x^2 + 3x - 2) - 2) + 2| + 3)^2 + 5(-3|(-4(-1x^2 + 3x - 2)^2 - 5(-1x^2 + 3x - 2) - 2) + 2| + 3)) + 2| - 3) + 4 and g(x) = (1(-3|(-1(-2(1|x| + 3) + 3) + 1) + 1| + 3))/(-3(-3|(-1(-2(1|x| + ... | 16 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(3π(2((1((-2(-2|(3x) - 1| + 1) - 3)/(2(-2|(3x) - 1| + 1) + 2)) - 3)/(1((-2(-2|(3x) - 1| + 1) - 3)/(2(-2|(3x) - 1| + 1) + 2)) - 2))^2 + 3) + 1) - 1 and g(x) = (2(-1|((1(2((-3x + 3)/(-2x + 3))^2 - 3((-3x + 3)/(-2x + 3))) - 3)/(-... | 13 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1((2(5(2|(-2e^(-1(-1|x - 3| - 2)) - 1) + 1| + 1) + 4) + 3)/(-1(5(2|(-2e^(-1(-1|x - 3| - 2)) - 1) + 1| + 1) + 4) - 1)) + 2 and g(x) = 1|(-4(1|((-1(-3x^2 + 3x + 2) - 2)/(-2(-3x^2 + 3x + 2) + 3)) - 3| - 3)^2 + 4(1|((-1(-3x^2 + 3x + 2... | 10 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(-2(-1sin(3π(-2|(-4(-5x^2 + 5x - 5)) + 3| + 3) + 1) + 2) - 1)^2 - 5(-2(-1sin(3π(-2|(-4(-5x^2 + 5x - 5)) + 3| + 3) + 1) + 2) - 1) - 2 and g(x) = 3(-3(1(2e^(1(1x^2 + 4x - 2)) + 2)^2 + 1(2e^(1(1x^2 + 4x - 2)) + 2) - 5)^2 - 1(1(2e^(1(1x^2 + 4x - 2)) + 2)... | 190 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2((2(-3|(-2(-2(2x - 5)^2 + 5(2x - 5))^2 + 4(-2(2x - 5)^2 + 5(2x - 5)) + 3) - 2| - 1) - 3)/(2(-3|(-2(-2(2x - 5)^2 + 5(2x - 5))^2 + 4(-2(2x - 5)^2 + 5(2x - 5)) + 3) - 2| - 1) + 3)) + 3)/(2((2(-3|(-2(-2(2x - 5)^2 + 5(2x - 5))^2... | 401 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5((-2((-1(3((-2(2|x - 2| - 2) - 2)/(1(2|x - 2| - 2) + 3))^2 - 4((-2(2|x - 2| - 2) - 2)/(1(2|x - 2| - 2) + 3))) - 2)/(-2(3((-2(2|x - 2| - 2) - 2)/(1(2|x - 2| - 2) + 3))^2 - 4((-2(2|x - 2| - 2) - 2)/(1(2|x - 2| - 2) + 3))))) + ... | 10 | MATH |
[
{
"content": "How many solutions does the equation (1(-2cos(1π(1|(-5((1((2x - 2)/(-2x - 3)) + 2)/(2((2x - 2)/(-2x - 3)) + 2)) - 4) - 2| + 2) + 2) - 2) - 3)/(3(-2cos(1π(1|(-5((1((2x - 2)/(-2x - 3)) + 2)/(2((2x - 2)/(-2x - 3)) + 2)) - 4) - 2| + 2) + 2) - 2) + 2) = (-2(2|(3(-3|(2cos(3πx - 3) - 2) + 1| - 3)^2 - 2(-... | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-5(1cos(1π(-3(-4(-1x^2 - 5)^2 - 3(-1x^2 - 5) + 2) - 5) + 2)) - 4) + 1)/(-2(-5(1cos(1π(-3(-4(-1x^2 - 5)^2 - 3(-1x^2 - 5) + 2) - 5) + 2)) - 4) + 3) and g(x) = 1sin(2π(-1|((-3(-1|((-3x)/(1x - 2)) + 3| - 2) + 3)/(1(-1|((-3x)/(... | 116 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(2|(-2|(4(-2|(2|x - 1| + 2) + 2| + 3) - 4) + 3| - 3) - 3| - 3) - 1)/(-2(2|(-2|(4(-2|(2|x - 1| + 2) + 2| + 3) - 4) + 3| - 3) - 3| - 3) + 3) and g(x) = (2(5((-3(-2|(1x^2 - 3x + 4) - 3| - 1) - 1)/(3(-2|(1x^2 - 3x + 4) - 3| - 1) - 1)... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1((1(-3(2sin(3π(-1(1x + 1) + 4) - 1) + 1) - 1) + 1)/(2(-3(2sin(3π(-1(1x + 1) + 4) - 1) + 1) - 1)))^2 + 1 = 1sin(3π(1(4(-5(-1|x + 1|)^2 - 5(-1|x + 1|) + 2) - 5)^2 + 1(4(-5(-1|x + 1|)^2 - 5(-1|x + 1|) + 2) - 5) + 4) + 2) have for x in the interval [-10, 10]?\n\n... | 364 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-3|(-3|(1((1(3|x| + 1) - 2)/(3(3|x| + 1) - 2)) + 3) - 3| + 1) - 2| - 1) + 3)/(1(-3|(-3|(1((1(3|x| + 1) - 2)/(3(3|x| + 1) - 2)) + 3) - 3| + 1) - 2| - 1) - 3) and g(x) = 1(-1(-1|(3sin(3π(-3x - 3)) - 2)|)^2 - 2(-1|(3sin(3π(-3x - 3)... | 259 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(3sin(1π(-1|(-1|(-2|(-1|x + 1|) - 2| - 1) - 2| + 2) + 2| - 3) + 2)) - 4 and g(x) = (-3(-1|(-5(1((3x)/(2x + 3)) + 5)^2 + 5(1((3x)/(2x + 3)) + 5) + 4) - 2| + 1) + 3)/(-3(-1|(-5(1((3x)/(2x + 3)) + 5)^2 + 5(1((3x)/(2x + 3)) + 5)... | 42 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(-1(-1(-1|(-3cos(3π(-2x - 3) - 2)) - 2| - 2) + 4)^2 + 3(-1(-1|(-3cos(3π(-2x - 3) - 2)) - 2| - 2) + 4) - 3) + 1| and g(x) = 3|(1(-2(-1(1|x|)^2 - 4(1|x|) + 3)^2 - 3)^2 - 2(-2(-1(1|x|)^2 - 4(1|x|) + 3)^2 - 3) + 4) + 1| + 1 whe... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(5(2|((3(1(5x^2 + 3x + 5)^2 + 1(5x^2 + 3x + 5) - 4) - 2)/(-3(1(5x^2 + 3x + 5)^2 + 1(5x^2 + 3x + 5) - 4) - 2)) - 3| + 2) - 5)| and g(x) = 2((1(-3|(4(-1x + 4) - 1) - 2| + 1))/(1(-3|(4(-1x + 4) - 1) - 2| + 1) + 3))^2 + 2((1(-3... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(2(-5(2(-2(3sin(3πx + 2) + 3)^2 - 1(3sin(3πx + 2) + 3) - 4)^2 - 1(-2(3sin(3πx + 2) + 3)^2 - 1(3sin(3πx + 2) + 3) - 4) + 1) + 4) + 1))/(-2(2(-5(2(-2(3sin(3πx + 2) + 3)^2 - 1(3sin(3πx + 2) + 3) - 4)^2 - 1(-2(3sin(3πx + 2) + 3)^2 - ... | 263 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(3(-2sin(2π(2|((2(1|x - 2| + 2) - 2)/(-2(1|x - 2| + 2) - 1))| - 3) + 3))^2 - 3(-2sin(2π(2|((2(1|x - 2| + 2) - 2)/(-2(1|x - 2| + 2) - 1))| - 3) + 3))) - 1| - 3 and g(x) = -1|((-1(5(1((2x)/(-2x + 3))^2 - 1((2x)/(-2x + 3)) - 2) + 2... | 10 | MATH |
[
{
"content": "How many solutions does the equation -3(3|((-2(3sin(2π(-2(4x + 2)^2 - 4(4x + 2) - 1)) + 3))/(-3(3sin(2π(-2(4x + 2)^2 - 4(4x + 2) - 1)) + 3) + 1))| - 1) + 2 = -1(-2|(-1e^(-1((1((1x + 3)/(-3x)) - 2)/(3((1x + 3)/(-3x)))))) - 1| - 2) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in L... | 99 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3((2(-3(4((-2((-3x - 3)/(-2x - 1)) + 2)/(-2((-3x - 3)/(-2x - 1))))^2 - 3((-2((-3x - 3)/(-2x - 1)) + 2)/(-2((-3x - 3)/(-2x - 1)))) + 1)^2 - 5(4((-2((-3x - 3)/(-2x - 1)) + 2)/(-2((-3x - 3)/(-2x - 1))))^2 - 3((-2((-3x - 3)/(-2x - 1)) + 2)/(-2((-3x - 3)/(... | 524 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(1(-3|((-2(1|((-3x - 3)/(1x + 1)) - 1| + 3) + 2)/(3(1|((-3x - 3)/(1x + 1)) - 1| + 3) + 2)) + 2| + 2)^2 + 5(-3|((-2(1|((-3x - 3)/(1x + 1)) - 1| + 3) + 2)/(3(1|((-3x - 3)/(1x + 1)) - 1| + 3) + 2)) + 2| + 2))| - 1 and g(x) = 1(-5(-5... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3cos(2π(-3(-3|((-1(1(-1x)^2 + 2(-1x) + 4) - 3)/(3(1(-1x)^2 + 2(-1x) + 4) + 2)) - 2|)^2 - 2(-3|((-1(1(-1x)^2 + 2(-1x) + 4) - 3)/(3(1(-1x)^2 + 2(-1x) + 4) + 2)) - 2|) + 4) - 3) - 2 and g(x) = -4((-3(-1(3(-4x + 5)^2 - 4(-4x + 5) + 1)^2 - 3(3(-4x + 5)^2 -... | 103 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(-3|((-1(-5(1((1x - 2)/(2x - 1)) - 4) + 5) - 2)/(-1(-5(1((1x - 2)/(2x - 1)) - 4) + 5) - 1)) - 2| + 1) + 3)/(2(-3|((-1(-5(1((1x - 2)/(2x - 1)) - 4) + 5) - 2)/(-1(-5(1((1x - 2)/(2x - 1)) - 4) + 5) - 1)) - 2| + 1) + 2) and g(x) = -1... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1sin(3π(2|(3(-1|((-1(1|x - 3|) - 2)/(-1(1|x - 3|) - 2)) + 2| - 2)^2 - 2(-1|((-1(1|x - 3|) - 2)/(-1(1|x - 3|) - 2)) + 2| - 2) - 5) + 3| + 3) - 1) + 1 = 3|((-1(1(-1(-3|x + 2| + 2)^2 + 3(-3|x + 2| + 2) + 4)^2 + 4(-1(-3|x + 2| + 2)^2 + 3(-3|x + 2| + 2) + 4) - 4) + ... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(3((3(-2((-2((1x + 3)/(-2x - 1)) + 2)/(-3((1x + 3)/(-2x - 1)) - 2)) + 5) - 2)/(1(-2((-2((1x + 3)/(-2x - 1)) + 2)/(-3((1x + 3)/(-2x - 1)) - 2)) + 5) - 3)) + 2) + 1)/(2(3((3(-2((-2((1x + 3)/(-2x - 1)) + 2)/(-3((1x + 3)/(-2x -... | 316 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-1|(3|(-1e^(2(-5(2|x + 3| - 3)^2 + 4(2|x + 3| - 3) + 1)) + 2) - 3| - 3) + 2| + 1) - 2 and g(x) = -5((-2(-3|((2(3x^2 + 2x - 4) - 1)/(-1(3x^2 + 2x - 4) - 1)) - 3| + 3) - 2)/(-1(-3|((2(3x^2 + 2x - 4) - 1)/(-1(3x^2 + 2x - 4) - 1)) - ... | 9 | MATH |
[
{
"content": "How many solutions does the equation 2(4(2(-2(-2((-2x + 2)/(1x))^2 + 2((-2x + 2)/(1x)) + 4) - 3)^2 + 5(-2(-2((-2x + 2)/(1x))^2 + 2((-2x + 2)/(1x)) + 4) - 3) - 4) + 1) - 3 = -1((1(1(-1((-1x + 3)/(-3x + 4))^2 + 1((-1x + 3)/(-3x + 4)) + 1)^2 + 4(-1((-1x + 3)/(-3x + 4))^2 + 1((-1x + 3)/(-3x + 4)) + 1)... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(2(-4((-2(-1((-2x - 1)/(-1x - 3)) - 5) - 2)/(2(-1((-2x - 1)/(-1x - 3)) - 5) - 1))^2 + 1((-2(-1((-2x - 1)/(-1x - 3)) - 5) - 2)/(2(-1((-2x - 1)/(-1x - 3)) - 5) - 1)) + 1)^2 - 2(-4((-2(-1((-2x - 1)/(-1x - 3)) - 5) - 2)/(2(-1((-2x - 1)/(-1x - 3)) - 5) -... | 17 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-3cos(1π(1(-5(-2(-1x - 5) - 3) + 4)^2 + 1(-5(-2(-1x - 5) - 3) + 4) + 1) + 1) - 3))/(-3(-3cos(1π(1(-5(-2(-1x - 5) - 3) + 4)^2 + 1(-5(-2(-1x - 5) - 3) + 4) + 1) + 1) - 3) - 3) and g(x) = 1(2cos(1π(2((-1(1x^2 - 3x + 3) - 3)/(... | 293 | MATH |
[
{
"content": "How many solutions does the equation (-1((-1(-2|(-3(-2(-1x) + 1)^2 + 3(-2(-1x) + 1) - 4) - 2| - 2) - 3)/(-3(-2|(-3(-2(-1x) + 1)^2 + 3(-2(-1x) + 1) - 4) - 2| - 2) - 1)) + 2)/(1((-1(-2|(-3(-2(-1x) + 1)^2 + 3(-2(-1x) + 1) - 4) - 2| - 2) - 3)/(-3(-2|(-3(-2(-1x) + 1)^2 + 3(-2(-1x) + 1) - 4) - 2| - 2) -... | 34 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(-3|(2sin(1π(4(-1(-1|x + 2| - 3)^2 - 4(-1|x + 2| - 3) - 2) - 1) + 1) + 1) - 2| - 2) - 2| + 2 and g(x) = -3((-3(-3|(-2(2|x + 3| - 2) - 5) + 3| + 1) + 3)/(-2(-3|(-2(2|x + 3| - 2) - 5) + 3| + 1) + 1)) + 2 in the range -10 ≤ x ≤ 10?\... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(1sin(1π(-2((1(5((1x + 1)/(-2x)) - 2) - 1)/(-1(5((1x + 1)/(-2x)) - 2) + 2))^2 + 3((1(5((1x + 1)/(-2x)) - 2) - 1)/(-1(5((1x + 1)/(-2x)) - 2) + 2)) - 4) - 3)) + 1| - 2 and g(x) = 1(-1(-1(-1((3x - 1)/(2x))^2 - 3((3x - 1)/(2x)))^2 + 3(-1((3x - 1)/(2x))^2... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|((-3((-3(-3(2(2x^2 + 4x - 5) - 4) + 5) - 2)/(-3(-3(2(2x^2 + 4x - 5) - 4) + 5) - 1)) + 1)/(-2((-3(-3(2(2x^2 + 4x - 5) - 4) + 5) - 2)/(-3(-3(2(2x^2 + 4x - 5) - 4) + 5) - 1)) - 3)) - 1| - 3 and g(x) = -2|(-5(2(-2|(-3sin(2πx + 1) + ... | 160 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(1sin(1π(-1(-5(-2((-2x - 1)/(3x - 2))^2 - 2((-2x - 1)/(3x - 2)) + 3)) + 1) + 2)) - 2| + 1 and g(x) = 2(1((-2(1((-2x + 2)/(1x)) - 1) + 1)/(3(1((-2x + 2)/(1x)) - 1) - 2))^2 - 1((-2(1((-2x + 2)/(1x)) - 1) + 1)/(3(1((-2x + 2)/(... | 57 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(1|(-2(4(1cos(1π(-1x + 4)))^2 + 2(1cos(1π(-1x + 4))) + 5)^2 - 3(4(1cos(1π(-1x + 4)))^2 + 2(1cos(1π(-1x + 4))) + 5) - 5) - 1| - 3)^2 + 2(1|(-2(4(1cos(1π(-1x + 4)))^2 + 2(1cos(1π(-1x + 4))) + 5)^2 - 3(4(1cos(1π(-1x + 4)))^2 +... | 18 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5((-2(-5((-3(3|(-3|x + 3| + 3) + 3|) + 2)/(-1(3|(-3|x + 3| + 3) + 3|) - 2))^2 + 2) + 3)/(-1(-5((-3(3|(-3|x + 3| + 3) + 3|) + 2)/(-1(3|(-3|x + 3| + 3) + 3|) - 2))^2 + 2) + 3)) - 2 and g(x) = 2(-2|(-2e^(-1((1(-3|x| - 1) + 2)/(3(-3|x| - 1) - 1))) + 2) + ... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1|(-2e^(2((-2((-3(3(-2x^2 - 5x + 3) - 5) - 1)/(-1(3(-2x^2 - 5x + 3) - 5) - 2)) + 1)/(-2((-3(3(-2x^2 - 5x + 3) - 5) - 1)/(-1(3(-2x^2 - 5x + 3) - 5) - 2)) + 1))) + 3) + 1| - 2 = 1|((3((2((-3(2x^2 - 2x - 1) - 1)/(-2(2x^2 - 2x - 1) + 3)) + 2)/(-3((-3(2x^2 - 2x - 1)... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-1(1|((2((3(-1|x + 3| + 3) + 3)/(-3(-1|x + 3| + 3))) - 3)/(2((3(-1|x + 3| + 3) + 3)/(-3(-1|x + 3| + 3))) - 1)) + 3|) + 3) - 3 and g(x) = -2|(-2((-1(2(2|x + 1| + 1) + 4) - 2)/(-3(2(2|x + 1| + 1) + 4) - 2)) + 5) - 3| + 3 whe... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-3cos(2π((2(-4(3(-3x^2 - 4x - 2) + 4) + 1) + 1)/(1(-4(3(-3x^2 - 4x - 2) + 4) + 1))) - 1) + 1)^2 + 1(-3cos(2π((2(-4(3(-3x^2 - 4x - 2) + 4) + 1) + 1)/(1(-4(3(-3x^2 - 4x - 2) + 4) + 1))) - 1) + 1) + 2 and g(x) = -4(-1e^(1(-3|... | 21 | MATH |
[
{
"content": "How many solutions does the equation 1|(1(5(2(-4(2sin(2πx) - 2) - 1)^2 + 5(-4(2sin(2πx) - 2) - 1) + 2)^2 - 5(2(-4(2sin(2πx) - 2) - 1)^2 + 5(-4(2sin(2πx) - 2) - 1) + 2) + 2)^2 - 5(5(2(-4(2sin(2πx) - 2) - 1)^2 + 5(-4(2sin(2πx) - 2) - 1) + 2)^2 - 5(2(-4(2sin(2πx) - 2) - 1)^2 + 5(-4(2sin(2πx) - 2) - 1... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(-1(-1|(3|(3(-2sin(3πx - 2) - 2)^2 - 1(-2sin(3πx - 2) - 2) - 4)| - 2) + 1| - 1) + 2) + 2| - 3 and g(x) = -3e^(1(3((1((-1(-3|x - 1| - 1))/(-1(-3|x - 1| - 1) + 2)) + 1)/(2((-1(-3|x - 1| - 1))/(-1(-3|x - 1| - 1) + 2)) + 3)) + 1)) + 1 intersect in the i... | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-1|((1(1|(2|(-2x^2 - 4x) - 1| - 3) + 3| + 3) + 3)/(1(1|(2|(-2x^2 - 4x) - 1| - 3) + 3| + 3) + 1)) + 3| + 2)^2 + 4(-1|((1(1|(2|(-2x^2 - 4x) - 1| - 3) + 3| + 3) + 3)/(1(1|(2|(-2x^2 - 4x) - 1| - 3) + 3| + 3) + 1)) + 3| + 2) - 2 and g(x) = (1(2|((2((1(-4x... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(3(-1(3|((-3(3x + 1) - 3)/(-1(3x + 1) + 2)) - 2| - 3)^2 - 3(3|((-3(3x + 1) - 3)/(-1(3x + 1) + 2)) - 2| - 3) + 5) - 3)^2 - 5(3(-1(3|((-3(3x + 1) - 3)/(-1(3x + 1) + 2)) - 2| - 3)^2 - 3(3|((-3(3x + 1) - 3)/(-1(3x + 1) + 2)) - 2| - 3)... | 17 | MATH |
[
{
"content": "How many solutions does the equation (3(3cos(1π((-1(-5((2(-5x^2 + 2x + 1) + 3)/(-3(-5x^2 + 2x + 1) + 4))^2 + 4((2(-5x^2 + 2x + 1) + 3)/(-3(-5x^2 + 2x + 1) + 4)) + 3) - 2)/(1(-5((2(-5x^2 + 2x + 1) + 3)/(-3(-5x^2 + 2x + 1) + 4))^2 + 4((2(-5x^2 + 2x + 1) + 3)/(-3(-5x^2 + 2x + 1) + 4)) + 3) - 2)) + 1)... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-3(-3(-1(-2sin(1π(-2x^2 - 4) - 2)))^2 - 1(-1(-2sin(1π(-2x^2 - 4) - 2))) - 4)^2 - 5(-3(-1(-2sin(1π(-2x^2 - 4) - 2)))^2 - 1(-1(-2sin(1π(-2x^2 - 4) - 2))) - 4) - 5) + 2| + 3 and g(x) = 3|(4(-3(2(1x^2) - 3) - 3)^2 + 4(-3(2(1x^2) - 3... | 22 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3((-3(1(2cos(3π(2|(-1|x - 3|) + 3|) + 3) - 3)^2 - 3(2cos(3π(2|(-1|x - 3|) + 3|) + 3) - 3) + 4) + 2)/(-3(1(2cos(3π(2|(-1|x - 3|) + 3|) + 3) - 3)^2 - 3(2cos(3π(2|(-1|x - 3|) + 3|) + 3) - 3) + 4) + 4)) + 3)/(1((-3(1(2cos(3π(2|(... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(1sin(1π(-1|(5(5(-4x - 1)^2 + 1(-4x - 1) + 2) + 5) - 3|) - 1) + 1)^2 - 2(1sin(1π(-1|(5(5(-4x - 1)^2 + 1(-4x - 1) + 2) + 5) - 3|) - 1) + 1) + 1 and g(x) = 2cos(3π((-2((-3((-2(3|x|) - 2)/(-2(3|x|) + 1)) + 1)/(2((-2(3|x|) - 2)/(-2(3|... | 48 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-3|(1e^(-2(-2|(1(-2x^2 - 3)^2 - 3(-2x^2 - 3) - 5) - 3| - 2)) - 1) - 3| - 3))/(-3(-3|(1e^(-2(-2|(1(-2x^2 - 3)^2 - 3(-2x^2 - 3) - 5) - 3| - 2)) - 1) - 3| - 3) + 2) and g(x) = 2(2cos(2π(4(-1(-4x^2 - 5)^2 - 1(-4x^2 - 5) + 2) + 1) - 1) + 2) - 5 intersec... | 623 | MATH |
[
{
"content": "How many solutions does the equation (2(-2|(-1(-1((-2(3|x + 1| - 3) + 1)/(-1(3|x + 1| - 3) - 3)) + 1)^2 - 3(-1((-2(3|x + 1| - 3) + 1)/(-1(3|x + 1| - 3) - 3)) + 1) + 5) + 3| + 1) - 1)/(2(-2|(-1(-1((-2(3|x + 1| - 3) + 1)/(-1(3|x + 1| - 3) - 3)) + 1)^2 - 3(-1((-2(3|x + 1| - 3) + 1)/(-1(3|x + 1| - 3) ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(1|(4((-1(-2((2x + 3)/(-2x + 1))^2 - 2((2x + 3)/(-2x + 1)) + 4))/(-3(-2((2x + 3)/(-2x + 1))^2 - 2((2x + 3)/(-2x + 1)) + 4) - 1)) - 2) - 3| + 3) - 2| - 2 and g(x) = -1(-2|(5(3|(3sin(2πx + 2)) + 3| - 3)^2 - 4) - 2| - 3)^2 + ... | 24 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(3π(1(1(-5((-2(-3|x + 3| + 2) + 2)/(1(-3|x + 3| + 2)))^2 - 5((-2(-3|x + 3| + 2) + 2)/(1(-3|x + 3| + 2))) + 2) - 2)^2 - 5(1(-5((-2(-3|x + 3| + 2) + 2)/(1(-3|x + 3| + 2)))^2 - 5((-2(-3|x + 3| + 2) + 2)/(1(-3|x + 3| + 2))) + 2) - ... | 779 | MATH |
[
{
"content": "How many solutions does the equation (1(-3|((-3((3(3|(-2|x - 3| - 3) - 2| + 1) - 3)/(-3(3|(-2|x - 3| - 3) - 2| + 1) + 2)) - 3)/(1((3(3|(-2|x - 3| - 3) - 2| + 1) - 3)/(-3(3|(-2|x - 3| - 3) - 2| + 1) + 2)) - 3)) - 2| + 3) - 2)/(1(-3|((-3((3(3|(-2|x - 3| - 3) - 2| + 1) - 3)/(-3(3|(-2|x - 3| - 3) - 2|... | 4 | MATH |
[
{
"content": "How many solutions does the equation 4(-3(-1e^(1(1|(1(-5x^2 - 4x + 1)^2 + 5(-5x^2 - 4x + 1) - 2) - 2| - 1)) + 1)^2 - 3(-1e^(1(1|(1(-5x^2 - 4x + 1)^2 + 5(-5x^2 - 4x + 1) - 2) - 2| - 1)) + 1) - 5) + 1 = -3(2(-1(-1|(-2|x - 2| - 2) - 3| - 2)^2 - 2(-1|(-2|x - 2| - 2) - 3| - 2) - 2) - 2)^2 - 5(2(-1(-1|(... | 6 | MATH |
[
{
"content": "How many solutions does the equation 1sin(3π(-1(-2|(3(-2(5x^2 - 2x - 5)^2 - 2(5x^2 - 2x - 5) - 3) - 2) - 3| + 3)^2 + 5(-2|(3(-2(5x^2 - 2x - 5)^2 - 2(5x^2 - 2x - 5) - 3) - 2) - 3| + 3) + 5) + 2) - 3 = -3((-1(3(-1(4x - 5) - 2) - 3) - 2)/(-2(3(-1(4x - 5) - 2) - 3) - 1)) - 2 have for x in the interval... | 113 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3((2(-5(-1(1sin(2π(3x + 4) + 2) - 1)^2 - 2(1sin(2π(3x + 4) + 2) - 1))^2 + 1(-1(1sin(2π(3x + 4) + 2) - 1)^2 - 2(1sin(2π(3x + 4) + 2) - 1)) + 3) - 2)/(2(-5(-1(1sin(2π(3x + 4) + 2) - 1)^2 - 2(1sin(2π(3x + 4) + 2) - 1))^2 + 1(-1(1sin(... | 19 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-1|(3(2|(2((-2x - 3)/(2x - 1))^2 - 5((-2x - 3)/(2x - 1)) - 4) + 3| - 1)^2 - 5(2|(2((-2x - 3)/(2x - 1))^2 - 5((-2x - 3)/(2x - 1)) - 4) + 3| - 1) - 1) - 1|)^2 + 2(-1|(3(2|(2((-2x - 3)/(2x - 1))^2 - 5((-2x - 3)/(2x - 1)) - 4) ... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1((1(-1|(-3(3(-2|x + 2| + 1)) - 3) - 1| + 3) - 1)/(-1(-1|(-3(3(-2|x + 2| + 1)) - 3) - 1| + 3) + 2)) - 2)/(1((1(-1|(-3(3(-2|x + 2| + 1)) - 3) - 1| + 3) - 1)/(-1(-1|(-3(3(-2|x + 2| + 1)) - 3) - 1| + 3) + 2))) and g(x) = 5((-1(... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3((2(5(-2|((3(-4x^2 + 4x - 3))/(2(-4x^2 + 4x - 3) + 3)) + 2| + 1) - 4) - 3)/(-1(5(-2|((3(-4x^2 + 4x - 3))/(2(-4x^2 + 4x - 3) + 3)) + 2| + 1) - 4) + 2)) + 3)/(-1((2(5(-2|((3(-4x^2 + 4x - 3))/(2(-4x^2 + 4x - 3) + 3)) + 2| + 1) - 4) ... | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(2|(5(2(-4(-3x - 3)^2 - 4(-3x - 3) + 1) + 3) + 3) + 1| - 3)| + 3 and g(x) = (-2(2|(-3|(-2(2cos(1πx - 2) - 1)^2 - 1(2cos(1πx - 2) - 1)) + 3| - 3) - 1| - 1) + 3)/(-2(2|(-3|(-2(2cos(1πx - 2) - 1)^2 - 1(2cos(1πx - 2) - 1)) + 3| - 3)... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(3(-2(-5(2(2cos(3πx) + 2) - 3) + 5)^2 + 1(-5(2(2cos(3πx) + 2) - 3) + 5) + 2) + 3) + 2)/(-2(3(-2(-5(2(2cos(3πx) + 2) - 3) + 5)^2 + 1(-5(2(2cos(3πx) + 2) - 3) + 5) + 2) + 3) - 3) and g(x) = (1(-2cos(2π((-1(2|(3x^2 - 1x - 5) - 3| - 1) + 3)/(-2(2|(3x^2 -... | 39 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-1((1(-1((2(1|x + 3| - 2) - 2)/(-1(1|x + 3| - 2) - 1))^2 + 1((2(1|x + 3| - 2) - 2)/(-1(1|x + 3| - 2) - 1)) + 1))/(-1(-1((2(1|x + 3| - 2) - 2)/(-1(1|x + 3| - 2) - 1))^2 + 1((2(1|x + 3| - 2) - 2)/(-1(1|x + 3| - 2) - 1)) + 1) - 2)) + 3)^2 and g(x) = 3|(... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(4(-1|(3|((-2(-4x + 1) - 3)/(-3(-4x + 1) - 1)) - 2|)| + 2)^2 + 1(-1|(3|((-2(-4x + 1) - 3)/(-3(-4x + 1) - 1)) - 2|)| + 2))^2 + 3(4(-1|(3|((-2(-4x + 1) - 3)/(-3(-4x + 1) - 1)) - 2|)| + 2)^2 + 1(-1|(3|((-2(-4x + 1) - 3)/(-3(-4x + 1)... | 115 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5((2(-3sin(2π(2(-2|((-3x + 2)/(-2x - 3)) + 1| + 2)^2 + 5(-2|((-3x + 2)/(-2x - 3)) + 1| + 2) - 3) - 1) - 3) + 2)/(-3(-3sin(2π(2(-2|((-3x + 2)/(-2x - 3)) + 1| + 2)^2 + 5(-2|((-3x + 2)/(-2x - 3)) + 1| + 2) - 3) - 1) - 3) - 1))^2 + 1(... | 300 | MATH |
[
{
"content": "How many solutions does the equation (1(-1(1(-3cos(1π(-1(-1x) + 3) + 3) + 3)^2 + 3(-3cos(1π(-1(-1x) + 3) + 3) + 3) - 3) + 1) + 2)/(1(-1(1(-3cos(1π(-1(-1x) + 3) + 3) + 3)^2 + 3(-3cos(1π(-1(-1x) + 3) + 3) + 3) - 3) + 1) - 3) = (2(-2cos(3π((-2(-3|(1|x - 1| + 1) - 3|) - 3)/(-2(-3|(1|x - 1| + 1) - 3|) ... | 20 | MATH |
[
{
"content": "How many solutions does the equation 1(1((2(-3|(4(-5x - 3) - 5) + 1| + 1) - 2)/(1(-3|(4(-5x - 3) - 5) + 1| + 1) + 3))^2 - 5((2(-3|(4(-5x - 3) - 5) + 1| + 1) - 2)/(1(-3|(4(-5x - 3) - 5) + 1| + 1) + 3)) + 1)^2 + 1(1((2(-3|(4(-5x - 3) - 5) + 1| + 1) - 2)/(1(-3|(4(-5x - 3) - 5) + 1| + 1) + 3))^2 - 5((... | 122 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(-1(1|(-1|(2sin(3π(1|x + 1| - 3) - 3) - 1) + 1| - 2) - 2| - 1) + 3) and g(x) = 3cos(2π((-2(2|(-4(-1x^2 + 1x) + 2) - 3| - 1) - 1)/(-1(2|(-4(-1x^2 + 1x) + 2) - 3| - 1) + 2)) + 3) - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer i... | 128 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(-3((2((-3(-2(4x - 3)^2 + 4(4x - 3) - 1) - 2)/(1(-2(4x - 3)^2 + 4(4x - 3) - 1) + 3)) - 2)/(1((-3(-2(4x - 3)^2 + 4(4x - 3) - 1) - 2)/(1(-2(4x - 3)^2 + 4(4x - 3) - 1) + 3)))) + 5) + 3)/(1(-3((2((-3(-2(4x - 3)^2 + 4(4x - 3) - ... | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(1sin(1π(3(1(-2|(2|x - 2| - 3) + 2| - 3) - 1) - 3) - 3) + 2)^2 + 5(1sin(1π(3(1(-2|(2|x - 2| - 3) + 2| - 3) - 1) - 3) - 3) + 2) and g(x) = -3|(2|(1((-2(5x^2 + 2x - 4) - 1)/(1(5x^2 + 2x - 4) - 2)))| + 2) - 1| - 3 in the range -10 ≤... | 232 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(-4(-4(-2(2e^(1(-4x - 5)) - 2) - 5) + 1)^2 - 4(-4(-2(2e^(1(-4x - 5)) - 2) - 5) + 1) + 4) - 1 and g(x) = -3|(3(-4(-3(1x^2 - 4x + 2) - 3)^2 + 4(-3(1x^2 - 4x + 2) - 3) + 1)^2 - 4(-4(-3(1x^2 - 4x + 2) - 3)^2 + 4(-3(1x^2 - 4x + 2) - 3) + 1) + 5) - 2| - 3 i... | 5 | MATH |
[
{
"content": "How many solutions does the equation 2(2|(-1e^(2((2(1(-3x + 2)^2 - 2(-3x + 2) - 5) - 2)/(1(1(-3x + 2)^2 - 2(-3x + 2) - 5) + 1))) - 2) + 1| - 1) + 4 = 2(5(3(-3|(3x^2 - 5x - 5) - 3|)^2 + 2(-3|(3x^2 - 5x - 5) - 3|) + 4) + 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex for... | 6 | MATH |
[
{
"content": "How many solutions does the equation 5(1cos(2π(-4(2(-1(-1x - 4)^2 + 5(-1x - 4) - 4)^2 + 4(-1(-1x - 4)^2 + 5(-1x - 4) - 4) - 5) + 5) - 3) - 1) + 3 = -3e^(1(-1(2((3((3x)/(-2x - 2)) - 3)/(-3((3x)/(-2x - 2)) - 3)) - 1) + 5)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: ... | 112 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(-4(-3sin(2π((-3(-3|((3x + 3)/(-1x - 3)) + 3| - 1) - 2)/(-3(-3|((3x + 3)/(-1x - 3)) + 3| - 1) + 2)) - 1))^2 - 2(-3sin(2π((-3(-3|((3x + 3)/(-1x - 3)) + 3| - 1) - 2)/(-3(-3|((3x + 3)/(-1x - 3)) + 3| - 1) + 2)) - 1)) - 1) - 2| and ... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1|((2((-2(3|(-2(1x + 2) - 5) + 2| + 2) - 2)/(2(3|(-2(1x + 2) - 5) + 2| + 2))) - 1)/(3((-2(3|(-2(1x + 2) - 5) + 2| + 2) - 2)/(2(3|(-2(1x + 2) - 5) + 2| + 2))) - 2)) - 3| + 1 = -2|(4(-5(3(1|x + 2| - 1)^2 - 5(1|x + 2| - 1) + 1)^2 - 1(3(1|x + 2| - 1)^2 - 5(1|x + 2... | 12 | MATH |
[
{
"content": "How many solutions does the equation -2|(2(-5(4((3((-3x + 1)/(-1x + 2)) + 3)/(-2((-3x + 1)/(-1x + 2)) + 1)) - 1)^2 + 4(4((3((-3x + 1)/(-1x + 2)) + 3)/(-2((-3x + 1)/(-1x + 2)) + 1)) - 1) + 1) + 4) + 1| + 3 = -3(1|((-1(2((1x + 1)/(-1x + 2))^2 - 5) + 3)/(3(2((1x + 1)/(-1x + 2))^2 - 5) + 3))| - 1) hav... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-1((-3(-2(4((2x + 3)/(1x))) - 5) - 2)/(-2(-2(4((2x + 3)/(1x))) - 5) + 3))^2 + 1((-3(-2(4((2x + 3)/(1x))) - 5) - 2)/(-2(-2(4((2x + 3)/(1x))) - 5) + 3)) + 1) + 5 and g(x) = -3(-2((2(1|((-2x - 1)/(-1x + 2)) + 1| - 1) + 1)/(-3(1|((-2... | 4 | MATH |
[
{
"content": "How many solutions does the equation 5((-3(1(-2|(5(5x)^2 + 2(5x) - 5) - 3| + 3) + 2) + 1)/(-2(1(-2|(5(5x)^2 + 2(5x) - 5) - 3| + 3) + 2) - 1)) - 3 = -1(-4(3(2cos(1π(2|x + 1| + 2) + 1) + 1) + 1)^2 - 5(3(2cos(1π(2|x + 1| + 2) + 1) + 1) + 1) + 5) - 3 have for x in the interval [-10, 10]?\n\nPresent th... | 42 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(3|(2|(2|((-3(2|x - 3| - 1) + 2)/(-1(2|x - 3| - 1) - 2)) + 1|) - 3|) - 2| - 2) - 2)/(2(3|(2|(2|((-3(2|x - 3| - 1) + 2)/(-1(2|x - 3| - 1) - 2)) + 1|) - 3|) - 2| - 2)) and g(x) = 5(2cos(2π(3(-2|(3x - 1) - 2| - 2)^2 + 1(-2|(3x - 1)... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((-1(1(-3e^(-1(-3(-4x^2 - 4x + 2)^2 + 3(-4x^2 - 4x + 2) + 1)) + 3) + 2) + 3)/(1(1(-3e^(-1(-3(-4x^2 - 4x + 2)^2 + 3(-4x^2 - 4x + 2) + 1)) + 3) + 2) - 3)) - 4 and g(x) = -3|(-4(-1(-1(1x^2 + 1x - 3) - 2)^2 + 5(-1(1x^2 + 1x - 3) - 2) - 1) - 3) + 2| + 3 i... | 4 | MATH |
[
{
"content": "How many solutions does the equation 2|(-2|(-3((2(-4(-1x^2 + 5x - 5)^2 - 3(-1x^2 + 5x - 5) + 4) + 3)/(-1(-4(-1x^2 + 5x - 5)^2 - 3(-1x^2 + 5x - 5) + 4) + 2)) - 5) + 2| + 1) + 3| = (2(-2|(-5(2((-1x)/(1x + 2))^2 - 1((-1x)/(1x + 2)) + 3)^2 + 2(2((-1x)/(1x + 2))^2 - 1((-1x)/(1x + 2)) + 3) - 3) + 3|) + ... | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(-1(2cos(3π(-5((1(2|x + 2| - 3) + 3)/(1(2|x + 2| - 3) - 3))^2 + 5((1(2|x + 2| - 3) + 3)/(1(2|x + 2| - 3) - 3)) - 2) - 1) - 1) - 2)^2 - 2(-1(2cos(3π(-5((1(2|x + 2| - 3) + 3)/(1(2|x + 2| - 3) - 3))^2 + 5((1(2|x + 2| - 3) + 3)/(1(2|x + 2| - 3) - 3)) - 2)... | 591 | MATH |
[
{
"content": "How many solutions does the equation -2((1(-1|(-1(-2sin(2π(-1x^2 + 4) + 3) - 3)^2 - 1(-2sin(2π(-1x^2 + 4) + 3) - 3) + 4) - 2| + 3) - 1)/(-2(-1|(-1(-2sin(2π(-1x^2 + 4) + 3) - 3)^2 - 1(-2sin(2π(-1x^2 + 4) + 3) - 3) + 4) - 2| + 3)))^2 + 4 = -2((1((1(2(2x^2 + 5x + 2)^2 + 5(2x^2 + 5x + 2)) - 1)/(-3(2(2... | 124 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|((-1(3(-1(1(-1x + 2) + 1) - 4)^2 - 5(-1(1(-1x + 2) + 1) - 4) + 3))/(-2(3(-1(1(-1x + 2) + 1) - 4)^2 - 5(-1(1(-1x + 2) + 1) - 4) + 3) + 1))| - 3 and g(x) = -1|(-1((1((1(-3|x + 2| + 3) - 1)/(-3(-3|x + 2| + 3) + 1)) - 3)/(1((1(... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(4(-3cos(3π(3(2|(-3x^2 + 5x - 3) - 1| + 1)^2 + 3(2|(-3x^2 + 5x - 3) - 1| + 1) - 5) - 1) + 2)^2 - 5(-3cos(3π(3(2|(-3x^2 + 5x - 3) - 1| + 1)^2 + 3(2|(-3x^2 + 5x - 3) - 1| + 1) - 5) - 1) + 2) + 4) + 2| + 3 and g(x) = 2(-1(2sin... | 317 | MATH |
[
{
"content": "How many solutions does the equation 3e^(2(-1(2(-5((-1(3x^2 - 3) - 3)/(-1(3x^2 - 3) + 2)))^2 - 1(-5((-1(3x^2 - 3) - 3)/(-1(3x^2 - 3) + 2))) + 5) - 1)) + 1 = -2(-2e^(1(-1|(-4(-3x^2 - 5x) + 2) - 1| + 3)) + 1)^2 - 5(-2e^(1(-1|(-4(-3x^2 - 5x) + 2) - 1| + 3)) + 1) + 5 have for x in the interval [-10, 1... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(-4(2|(-5(-1((-1x + 3)/(2x + 2)) - 3)^2 - 2(-1((-1x + 3)/(2x + 2)) - 3)) + 3| - 2) + 4) - 4 and g(x) = -5(4(-3|(2cos(2π(2x - 5) + 3) + 2) + 3| + 3) - 2) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 24 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-4(2|(-4(-5(5x - 1)^2 + 1(5x - 1) - 3)^2 + 1(-5(5x - 1)^2 + 1(5x - 1) - 3) - 4) + 2| + 1) - 4) - 1)/(-3(-4(2|(-4(-5(5x - 1)^2 + 1(5x - 1) - 3)^2 + 1(-5(5x - 1)^2 + 1(5x - 1) - 3) - 4) + 2| + 1) - 4) + 1) and g(x) = 1sin(1π(-1(-3... | 1360 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2e^(-1(2(-3((2(2|((-2x + 3)/(2x - 1)) + 3| - 2))/(-1(2|((-2x + 3)/(2x - 1)) + 3| - 2))) - 1)^2 + 3(-3((2(2|((-2x + 3)/(2x - 1)) + 3| - 2))/(-1(2|((-2x + 3)/(2x - 1)) + 3| - 2))) - 1) + 5)) + 1 and g(x) = 3(1|(2(2(-2|x| - 1) + 2) + 2) + 2| + 1) - 5 inte... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|((-2(-2|((-3(-1|(1|x - 2| + 1)|) + 1)/(-1(-1|(1|x - 2| + 1)|) + 3)) - 2|) - 3)/(3(-2|((-3(-1|(1|x - 2| + 1)|) + 1)/(-1(-1|(1|x - 2| + 1)|) + 3)) - 2|) + 1)) + 3| + 2 and g(x) = -2|(1sin(2π((2(-1|(3x^2 + 3) - 3| - 3) - 2)/(-2(-1|(3x^2 + 3) - 3| - 3) ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(4(-3((-2(-3|(-3|x + 2|) + 3| + 3) - 1)/(-2(-3|(-3|x + 2|) + 3| + 3) + 3)) - 2) - 3) - 3| + 3 and g(x) = (3(-1|(-1|(-4(1x - 1) + 2) + 2| + 3) + 2| - 3) - 3)/(3(-1|(-1|(-4(1x - 1) + 2) + 2| + 3) + 2| - 3) - 2) in the range -10 ≤ ... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(-2(-2cos(3π(4(-3((-3x - 1)/(-1x)) + 2)^2 - 5(-3((-3x - 1)/(-1x)) + 2) - 2) + 1) - 1) - 1) and g(x) = 2sin(2π(3(4((2(5x^2 - 3x) + 1)/(1(5x^2 - 3x))) + 5)^2 - 5(4((2(5x^2 - 3x) + 1)/(1(5x^2 - 3x))) + 5) + 1) + 1) where -10 ≤... | 731 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1((-3(1(-2cos(2π(1(1x - 3)^2 - 4(1x - 3) + 4) + 2) - 2) - 4) + 2)/(3(1(-2cos(2π(1(1x - 3)^2 - 4(1x - 3) + 4) + 2) - 2) - 4)))^2 - 2((-3(1(-2cos(2π(1(1x - 3)^2 - 4(1x - 3) + 4) + 2) - 2) - 4) + 2)/(3(1(-2cos(2π(1(1x - 3)^2 - 4(1x - ... | 4 | MATH |
[
{
"content": "How many solutions does the equation (-2(-1(1((3(4(1x^2 + 3x - 5) + 5) + 2)/(1(4(1x^2 + 3x - 5) + 5) - 3)) - 4)^2 + 4(1((3(4(1x^2 + 3x - 5) + 5) + 2)/(1(4(1x^2 + 3x - 5) + 5) - 3)) - 4) - 3) - 3)/(-1(-1(1((3(4(1x^2 + 3x - 5) + 5) + 2)/(1(4(1x^2 + 3x - 5) + 5) - 3)) - 4)^2 + 4(1((3(4(1x^2 + 3x - 5)... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(4(-1sin(1π((3(2|(1x^2 - 5x - 2) + 2| - 1) + 1)/(3(2|(1x^2 - 5x - 2) + 2| - 1) + 2)) + 2) - 1)^2 + 4(-1sin(1π((3(2|(1x^2 - 5x - 2) + 2| - 1) + 1)/(3(2|(1x^2 - 5x - 2) + 2| - 1) + 2)) + 2) - 1) - 5)^2 - 4(4(-1sin(1π((3(2|(1x^2 - 5x - 2) + 2| - 1) + 1)/... | 7 | MATH |
[
{
"content": "How many solutions does the equation 1|(1|(5(1(1(3e^(2x) + 3) - 3)^2 + 5(1(3e^(2x) + 3) - 3) + 4) - 5) - 1|) + 3| + 3 = 2|(-3|((1(-1(-2|x| + 2)^2 + 3) - 2)/(3(-1(-2|x| + 2)^2 + 3) + 2)) - 3| + 1) + 1| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-4(1|((-3(-3(3x + 5) + 1) + 2)/(2(-3(3x + 5) + 1) + 1)) - 3| + 3)^2 - 1(1|((-3(-3(3x + 5) + 1) + 2)/(2(-3(3x + 5) + 1) + 1)) - 3| + 3) - 3) - 1 and g(x) = -5(5(-1((1((3x)/(-1x - 3)) - 2)/(-2((3x)/(-1x - 3)) + 3)) - 2)^2 - 2... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(-5((-2((-2(-1(3x^2 + 5x - 3)^2 - 4(3x^2 + 5x - 3) - 1) - 2)/(1(-1(3x^2 + 5x - 3)^2 - 4(3x^2 + 5x - 3) - 1) - 2)) - 2)/(-2((-2(-1(3x^2 + 5x - 3)^2 - 4(3x^2 + 5x - 3) - 1) - 2)/(1(-1(3x^2 + 5x - 3)^2 - 4(3x^2 + 5x - 3) - 1) - 2)) + 3)) + 4) + 2| and g... | 8 | MATH |
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