messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Find the number of times the graphs of f(x) = 5(-2(3sin(2π(-2((2(5x - 3))/(3(5x - 3) - 2))^2 + 3((2(5x - 3))/(3(5x - 3) - 2)) + 1) - 2) + 1)^2 + 5(3sin(2π(-2((2(5x - 3))/(3(5x - 3) - 2))^2 + 3((2(5x - 3))/(3(5x - 3) - 2)) + 1) - 2) + 1) + 3) - 2 and g(x) = -2(-1(1|((3((1x + 2)/(-3x + 4)) - 2)/(1((... | 24 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(3(5(-1(1(2e^(-1x))^2 - 4(2e^(-1x)) + 3) - 5)^2 + 2(-1(1(2e^(-1x))^2 - 4(2e^(-1x)) + 3) - 5) - 5) + 2) - 2)/(-2(3(5(-1(1(2e^(-1x))^2 - 4(2e^(-1x)) + 3) - 5)^2 + 2(-1(1(2e^(-1x))^2 - 4(2e^(-1x)) + 3) - 5) - 5) + 2) - 3) and g(x) = -5(2(-3|((2(-2|x + 1... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-1(-3((-1(-1(-3x^2 + 3x - 2)^2 + 2(-3x^2 + 3x - 2)) + 1)/(-3(-1(-3x^2 + 3x - 2)^2 + 2(-3x^2 + 3x - 2)) - 3))^2 + 4((-1(-1(-3x^2 + 3x - 2)^2 + 2(-3x^2 + 3x - 2)) + 1)/(-3(-1(-3x^2 + 3x - 2)^2 + 2(-3x^2 + 3x - 2)) - 3)) - 4) + 1)... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(3|(3(-3|(1cos(3π((-1x - 1)/(3x - 2))) + 2) - 2| + 2)^2 + 4(-3|(1cos(3π((-1x - 1)/(3x - 2))) + 2) - 2| + 2) - 2) + 3| + 1) - 1| - 2 and g(x) = -2|(2|(2|(1e^(2(-5x^2 - 2x)) - 1) + 3| + 3) + 3| - 2)| - 3 where -10 ≤ x ≤ 10.\... | 57 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-2(3((-3(1|(4x^2 - 4) + 1| + 2) + 3)/(1(1|(4x^2 - 4) + 1| + 2))) - 4) - 3) + 1 and g(x) = 1(-3(5(-2|(-4x^2 + 1x + 3)| + 1) + 4) + 1)^2 + 4(-3(5(-2|(-4x^2 + 1x + 3)| + 1) + 4) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex fo... | 8 | MATH |
[
{
"content": "How many solutions does the equation (-3(-2(-2|(-3|((-2(-2|x + 3| + 2) - 3)/(-2(-2|x + 3| + 2) - 2)) + 3| - 2)| - 2) + 1) + 1)/(-3(-2(-2|(-3|((-2(-2|x + 3| + 2) - 3)/(-2(-2|x + 3| + 2) - 2)) + 3| - 2)| - 2) + 1) + 4) = -4(-2cos(2π(3(-2((2x)/(3x - 2))^2 + 4((2x)/(3x - 2)) + 3) - 2) - 1) - 3)^2 - 4(... | 67 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(1(2|(1|(-1|(1|x + 2| - 3) - 2|) + 3| - 1)| - 2)^2 + 5(2|(1|(-1|(1|x + 2| - 3) - 2|) + 3| - 1)| - 2) + 2) + 4 and g(x) = -3|((3(3cos(1π((1(4x^2 + 2x + 1) + 1)/(-1(4x^2 + 2x + 1) + 2)) + 1) + 2) - 3)/(-2(3cos(1π((1(4x^2 + 2x + 1) +... | 42 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(-1|(2|((-2(1e^(2(-1x^2 + 3)) - 2) + 3)/(-2(1e^(2(-1x^2 + 3)) - 2) + 3))| - 2) + 2| + 3)^2 - 3(-1|(2|((-2(1e^(2(-1x^2 + 3)) - 2) + 3)/(-2(1e^(2(-1x^2 + 3)) - 2) + 3))| - 2) + 2| + 3) + 2 and g(x) = -1|(-1sin(2π(2(2((2x + 1)... | 22 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|((1(2((-2((1(-1x - 3) + 1)/(1(-1x - 3) + 1)) - 2)/(2((1(-1x - 3) + 1)/(1(-1x - 3) + 1)) - 1)) - 5) + 1)/(2(2((-2((1(-1x - 3) + 1)/(1(-1x - 3) + 1)) - 2)/(2((1(-1x - 3) + 1)/(1(-1x - 3) + 1)) - 1)) - 5) + 3)) + 1| + 3 and g(x) = ... | 181 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-3((3((-1(3|(2sin(3πx + 3) - 3) + 3| - 3) + 1)/(1(3|(2sin(3πx + 3) - 3) + 3| - 3) + 3)) - 2)/(-3((-1(3|(2sin(3πx + 3) - 3) + 3| - 3) + 1)/(1(3|(2sin(3πx + 3) - 3) + 3| - 3) + 3)) - 2))) - 3)/(1(-3((3((-1(3|(2sin(3πx + 3) -... | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((-2((1((2(2sin(2π(-4x - 1) - 3)) - 1)/(-1(2sin(2π(-4x - 1) - 3)) + 3)) - 1)/(-2((2(2sin(2π(-4x - 1) - 3)) - 1)/(-1(2sin(2π(-4x - 1) - 3)) + 3)) + 1)) + 1)/(2((1((2(2sin(2π(-4x - 1) - 3)) - 1)/(-1(2sin(2π(-4x - 1) - 3)) + 3)) - 1)/(-2((2(2sin(2π(-4x ... | 225 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(2π(-3(1((3(-3|(-2x + 5) - 1| + 1) + 3)/(2(-3|(-2x + 5) - 1| + 1) + 3))^2 + 1((3(-3|(-2x + 5) - 1| + 1) + 3)/(2(-3|(-2x + 5) - 1| + 1) + 3)) - 3) + 4) - 2) + 2 and g(x) = (3(-3((2(-3(1x^2 - 2x) + 2) + 3)/(2(-3(1x^2 - 2x) + 2) ... | 17 | MATH |
[
{
"content": "How many solutions does the equation (1(3(5(-3(3(-3x^2 - 2x - 1)^2 + 4(-3x^2 - 2x - 1) - 5) + 1)^2 - 3(-3(3(-3x^2 - 2x - 1)^2 + 4(-3x^2 - 2x - 1) - 5) + 1) - 4) - 4) + 3)/(1(3(5(-3(3(-3x^2 - 2x - 1)^2 + 4(-3x^2 - 2x - 1) - 5) + 1)^2 - 3(-3(3(-3x^2 - 2x - 1)^2 + 4(-3x^2 - 2x - 1) - 5) + 1) - 4) - 4... | 9 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(2((-2(1e^(-2(1(4x^2 - 3x - 2) + 2)) + 1) - 3)/(-3(1e^(-2(1(4x^2 - 3x - 2) + 2)) + 1) - 2)) - 3)^2 - 2(2((-2(1e^(-2(1(4x^2 - 3x - 2) + 2)) + 1) - 3)/(-3(1e^(-2(1(4x^2 - 3x - 2) + 2)) + 1) - 2)) - 3) + 4 and g(x) = 1(-2|(5(3(2|x + ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-3(-2|(5((1((-3x + 2)/(1x)) - 1)/(-2((-3x + 2)/(1x)) + 3))^2) + 3| - 1) + 2) - 1)/(-3(-3(-2|(5((1((-3x + 2)/(1x)) - 1)/(-2((-3x + 2)/(1x)) + 3))^2) + 3| - 1) + 2) - 2) and g(x) = -3(-2(-1((2(-3|x + 3| + 1))/(-2(-3|x + 3| + 1) + ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(-3|(-1(3|(3(-1cos(2πx + 2) - 1)^2 + 2(-1cos(2πx + 2) - 1) + 4) + 2| + 2)^2 - 4(3|(3(-1cos(2πx + 2) - 1)^2 + 2(-1cos(2πx + 2) - 1) + 4) + 2| + 2) - 1) - 2| + 3) + 2| + 3 and g(x) = -4(3(2|(2(3x + 1) - 3) + 2|) - 4)^2 - 1(3(2|(2(... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1cos(3π(3(1(1|(-5(-2x) + 5)| + 1)^2 + 3(1|(-5(-2x) + 5)| + 1) + 2) + 1) - 1) - 2 and g(x) = -5(1(-5((-2((2x - 1)/(1x + 3)) - 2)/(1((2x - 1)/(1x + 3)) - 2))^2 + 4((-2((2x - 1)/(1x + 3)) - 2)/(1((2x - 1)/(1x + 3)) - 2)) - 2)^2 + 2(-5((-2((2x - 1)/(1x + 3... | 12 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(1|((-3(-3|(3|(1x^2) + 3|) - 1| - 3))/(2(-3|(3|(1x^2) + 3|) - 1| - 3) + 3)) - 3| - 1)^2 - 2(1|((-3(-3|(3|(1x^2) + 3|) - 1| - 3))/(2(-3|(3|(1x^2) + 3|) - 1| - 3) + 3)) - 3| - 1) + 3 and g(x) = 4(2(1sin(3π(-1|(4x^2)| - 3) - 1) + 2)^... | 1384 | MATH |
[
{
"content": "How many solutions does the equation -1cos(1π((2((-3(-1|(3(1|x - 2| + 2) - 5) - 2| + 2) - 1)/(3(-1|(3(1|x - 2| + 2) - 5) - 2| + 2))) - 2)/(1((-3(-1|(3(1|x - 2| + 2) - 5) - 2| + 2) - 1)/(3(-1|(3(1|x - 2| + 2) - 5) - 2| + 2))))) - 1) + 2 = 3|(-1(-2cos(2π((3(-1x - 5) - 3)/(-1(-1x - 5) - 3)) - 2) + 1)... | 69 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(2|((-2(3|(-1sin(3π(-2x - 5) - 2) - 3) + 3| - 2))/(3(3|(-1sin(3π(-2x - 5) - 2) - 3) + 3| - 2) + 1)) - 1| + 1) - 1| + 1 and g(x) = -1|((-3(-2(1(1e^(-2x) + 2) - 5) - 3) + 1)/(1(-2(1(1e^(-2x) + 2) - 5) - 3) - 3)) - 3| - 2 in the ra... | 172 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(3|(1sin(1π(3|((-2(-1x^2 - 5x) - 2)/(-1(-1x^2 - 5x))) + 2| - 1) + 2) - 2) + 3| + 3) - 3)/(2(3|(1sin(1π(3|((-2(-1x^2 - 5x) - 2)/(-1(-1x^2 - 5x))) + 2| - 1) + 2) - 2) + 3| + 3) + 3) and g(x) = -2|((3(1e^(-1(3(3|x + 1| - 1) -... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(3|(-1|(-1|(3cos(2π((-3x - 2)/(-2x + 3))) - 3) + 1|) - 1|) + 3| + 1) - 2| + 1 and g(x) = -4(-1|(3sin(1π(1((-2x + 1)/(3x - 2)) + 2) - 1)) - 1| + 1) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your... | 27 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-1(4(1|((-2(3|x + 2| - 1) + 1)/(-3(3|x + 2| - 1) + 1))|)^2 - 4(1|((-2(3|x + 2| - 1) + 1)/(-3(3|x + 2| - 1) + 1))|) + 2) + 4)| - 1 and g(x) = -3|(2|(-2cos(3π(3|(-1x + 5) - 2| - 2) - 3) - 2) - 1| - 2) + 3| - 3 where -10 ≤ x... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|((-1(1(-1(3(-5x^2 - 5x + 5) - 1) - 4)^2 + 3(-1(3(-5x^2 - 5x + 5) - 1) - 4) + 1) - 1)/(3(1(-1(3(-5x^2 - 5x + 5) - 1) - 4)^2 + 3(-1(3(-5x^2 - 5x + 5) - 1) - 4) + 1))) - 2| and g(x) = (3(4(-1|(-1(3|x + 1| - 1) + 3) - 2| + 1) ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5((1((3(3(-1(3e^(1x) + 2) + 3) - 3) - 1)/(-1(3(-1(3e^(1x) + 2) + 3) - 3) - 3)) + 2)/(-2((3(3(-1(3e^(1x) + 2) + 3) - 3) - 1)/(-1(3(-1(3e^(1x) + 2) + 3) - 3) - 3)) - 3))^2 + 3((1((3(3(-1(3e^(1x) + 2) + 3) - 3) - 1)/(-1(3(-1(3e^(1x) + 2) + 3) - 3) - 3)) +... | 240 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(1(1(-3|(4(3x^2 + 2x - 1) + 4)| + 3)^2 - 5(-3|(4(3x^2 + 2x - 1) + 4)| + 3) + 1) - 2) + 3)/(-1(1(1(-3|(4(3x^2 + 2x - 1) + 4)| + 3)^2 - 5(-3|(4(3x^2 + 2x - 1) + 4)| + 3) + 1) - 2) - 1) and g(x) = 1sin(1π(-3|(-2|(-2|(3x^2 + 4x + 4) + 3| - 1) + 3| + 1) ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(-3(-4((3(-2|((1x - 2)/(1x)) - 3| + 2) + 2)/(-1(-2|((1x - 2)/(1x)) - 3| + 2) + 3)) + 1) + 2)| - 2 and g(x) = 5(2cos(1π(4(3(3|x| - 3) + 5)^2 + 3(3(3|x| - 3) + 5) - 2) - 3))^2 - 1(2cos(1π(4(3(3|x| - 3) + 5)^2 + 3(3(3|x| - 3) + 5) -... | 451 | MATH |
[
{
"content": "How many solutions does the equation (-2((-1(-3sin(3π(-2(-4(-3x^2 + 5x)^2 - 5(-3x^2 + 5x) - 4) - 3) + 2) - 1))/(3(-3sin(3π(-2(-4(-3x^2 + 5x)^2 - 5(-3x^2 + 5x) - 4) - 3) + 2) - 1) + 2)) - 2)/(3((-1(-3sin(3π(-2(-4(-3x^2 + 5x)^2 - 5(-3x^2 + 5x) - 4) - 3) + 2) - 1))/(3(-3sin(3π(-2(-4(-3x^2 + 5x)^2 - 5... | 45 | MATH |
[
{
"content": "How many solutions does the equation (-2(-2e^(-1(3(-1(4(-4x)) + 4)^2 + 4))) - 3)/(2(-2e^(-1(3(-1(4(-4x)) + 4)^2 + 4))) + 2) = -2(-3|(1(1(5x^2 + 4x - 1) - 5) - 1) - 1| + 2)^2 - 2(-3|(1(1(5x^2 + 4x - 1) - 5) - 1) - 1| + 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex form... | 8 | MATH |
[
{
"content": "How many solutions does the equation 1((2((-2(2sin(1π((2(-1x + 1) + 1)/(2(-1x + 1) + 2)) + 3) + 3) - 2)/(2(2sin(1π((2(-1x + 1) + 1)/(2(-1x + 1) + 2)) + 3) + 3) - 1)))/(-3((-2(2sin(1π((2(-1x + 1) + 1)/(2(-1x + 1) + 2)) + 3) + 3) - 2)/(2(2sin(1π((2(-1x + 1) + 1)/(2(-1x + 1) + 2)) + 3) + 3) - 1)) + 2... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2((3((2(-4(-2cos(3π(-5x^2 - 4) - 3) - 3)) - 1)/(-3(-4(-2cos(3π(-5x^2 - 4) - 3) - 3)))) - 2)/(-1((2(-4(-2cos(3π(-5x^2 - 4) - 3) - 3)) - 1)/(-3(-4(-2cos(3π(-5x^2 - 4) - 3) - 3)))) - 3)) - 1)/(-1((3((2(-4(-2cos(3π(-5x^2 - 4) - 3) - 3... | 1576 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-1|(-2sin(3π(-1(-3|((3x + 2)/(2x - 1)) + 2| - 3)^2 - 3(-3|((3x + 2)/(2x - 1)) + 2| - 3) + 1) - 1) + 3)| - 1)^2 + 3(-1|(-2sin(3π(-1(-3|((3x + 2)/(2x - 1)) + 2| - 3)^2 - 3(-3|((3x + 2)/(2x - 1)) + 2| - 3) + 1) - 1) + 3)| - 1)... | 8 | MATH |
[
{
"content": "How many solutions does the equation -3(-2(1e^(-1(-2|(3(-3x^2 + 4x + 5) - 4) - 2| + 2)) - 1)^2 + 2) - 3 = 5(1(3(2|((2x - 1)/(3x + 3)) - 3| + 2)^2 + 5(2|((2x - 1)/(3x + 3)) - 3| + 2) - 5)^2 - 3(3(2|((2x - 1)/(3x + 3)) - 3| + 2)^2 + 5(2|((2x - 1)/(3x + 3)) - 3| + 2) - 5) - 5) + 5 have for x in the i... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(1|(-3cos(2π(2|((-2((1x - 2)/(-3x - 1)) + 3)/(2((1x - 2)/(-3x - 1)) - 1)) + 2|) - 1) - 1) + 3| - 3) - 3| + 1 and g(x) = (-1(-4(2|((-3(1sin(3πx + 1)) - 1)/(3(1sin(3πx + 1))))| + 1) + 5) + 3)/(-2(-4(2|((-3(1sin(3πx + 1)) - 1)/(3(1sin(3πx + 1))))| + 1)... | 65 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(1(-5(2(-3(1x + 3)) + 5) + 1)^2 - 2(-5(2(-3(1x + 3)) + 5) + 1) - 3) - 3)/(-2(1(-5(2(-3(1x + 3)) + 5) + 1)^2 - 2(-5(2(-3(1x + 3)) + 5) + 1) - 3)) and g(x) = -2(3|((1(1|(-1x + 2) - 3| - 3) - 3)/(-1(1|(-1x + 2) - 3| - 3) + 2)) - 2| + 1) + 4 intersect i... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(2(2(1sin(1π(-2(1|x - 1| + 3) - 1) + 1) + 1) - 1) - 1))/(-1(2(2(1sin(1π(-2(1|x - 1| + 3) - 1) + 1) + 1) - 1) - 1) + 3) and g(x) = 2|(-3cos(2π((2(1|(-3x^2 + 4x - 5) - 1| + 3) + 1)/(2(1|(-3x^2 + 4x - 5) - 1| + 3) - 1))) - 3) - 1| + 3 intersect in the i... | 40 | MATH |
[
{
"content": "How many solutions does the equation -2|(-1cos(1π((2(-3|(-5((-3x - 1)/(-1x + 2)) - 3) - 1| - 3) - 1)/(-1(-3|(-5((-3x - 1)/(-1x + 2)) - 3) - 1| - 3) - 2)) - 3) - 3) + 3| - 1 = -5(-2sin(1π(1((-3(-3x + 3) - 3)/(1(-3x + 3) - 3))^2 - 3((-3(-3x + 3) - 3)/(1(-3x + 3) - 3)) - 5)) - 2)^2 + 1(-2sin(1π(1((-3... | 111 | MATH |
[
{
"content": "How many solutions does the equation 1|(-3sin(1π(-1(-5(1(1x - 5) + 5)^2 + 2(1(1x - 5) + 5) + 1)^2 + 5(-5(1(1x - 5) + 5)^2 + 2(1(1x - 5) + 5) + 1) + 4) + 2)) - 3| - 3 = 5((-2((3(2|(-5x^2 - 1x) + 1| + 1) + 3)/(-2(2|(-5x^2 - 1x) + 1| + 1) - 3)) - 1)/(2((3(2|(-5x^2 - 1x) + 1| + 1) + 3)/(-2(2|(-5x^2 - ... | 1034 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-5((-3(1(-4(1cos(3πx - 3) + 1) - 3) + 5) + 1)/(3(1(-4(1cos(3πx - 3) + 1) - 3) + 5) + 2)) + 2) + 3| + 3 and g(x) = 3|(2(-4(2(-3cos(3πx - 3) - 3)^2 + 5(-3cos(3πx - 3) - 3)) + 1) + 1) - 2| + 1 intersect in the interval [-10, 10].\n\nPresent the answer ... | 260 | MATH |
[
{
"content": "How many solutions does the equation -1|(3cos(2π((-3(-3((3(-5x - 4) + 1)/(-1(-5x - 4) - 2))^2 + 3((3(-5x - 4) + 1)/(-1(-5x - 4) - 2))) + 2)/(2(-3((3(-5x - 4) + 1)/(-1(-5x - 4) - 2))^2 + 3((3(-5x - 4) + 1)/(-1(-5x - 4) - 2))) - 1))) - 3) + 2| = -3|((-1(3((-3((-3x + 1)/(1x + 1)) - 1)/(-1((-3x + 1)/(... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-1|(-4(-5((1(-1|x| + 1) + 3)/(-3(-1|x| + 1) + 2)) + 2) - 5) - 2|)^2 + 1 and g(x) = (-2(3e^(1(-2(4(-3|x + 3|) - 1)^2 + 2(4(-3|x + 3|) - 1) + 4)) - 2) - 1)/(1(3e^(1(-2(4(-3|x + 3|) - 1)^2 + 2(4(-3|x + 3|) - 1) + 4)) - 2) + 1... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(3cos(2π(-3(3|(-1|(1x + 4) - 1|) + 3| - 3)^2 + 1) - 2) + 2)| - 2 and g(x) = -2|(-3|(2|(-2|((3x)/(1x + 1)) - 1| + 2)| - 2) - 3| + 1) + 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 251 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(1(-2e^(-1(-4(-2|(4x^2 - 5x - 2) + 1| - 1)^2 - 2(-2|(4x^2 - 5x - 2) + 1| - 1) - 2)) - 1) + 2) - 2)/(-2(1(-2e^(-1(-4(-2|(4x^2 - 5x - 2) + 1| - 1)^2 - 2(-2|(4x^2 - 5x - 2) + 1| - 1) - 2)) - 1) + 2)) and g(x) = -2cos(1π((-2((2((1(4x... | 10 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(3|(-5(-1((-1(-1x^2 - 5x - 4) - 3)/(1(-1x^2 - 5x - 4)))^2 - 5((-1(-1x^2 - 5x - 4) - 3)/(1(-1x^2 - 5x - 4))) - 2) - 3) + 3| - 1) - 1| - 1 and g(x) = 2|(-3sin(1π(-4(-3(1|x + 2| - 2)^2 - 5(1|x + 2| - 2) - 2) + 5) + 2)) + 1| - 2 intersect in the interva... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2sin(1π(2(-4(-4(-1|(-1|x - 2| - 2) + 1| + 2)^2 - 1(-1|(-1|x - 2| - 2) + 1| + 2) + 4)^2 + 2(-4(-1|(-1|x - 2| - 2) + 1| + 2)^2 - 1(-1|(-1|x - 2| - 2) + 1| + 2) + 4) - 4)^2 + 4(-4(-4(-1|(-1|x - 2| - 2) + 1| + 2)^2 - 1(-1|(-1|x - 2| - ... | 364 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-5(-5(-3(5((2x - 2)/(1x))^2 - 5((2x - 2)/(1x)) - 1) + 5)^2 - 4(-3(5((2x - 2)/(1x))^2 - 5((2x - 2)/(1x)) - 1) + 5) + 4) - 5) + 4 and g(x) = (1(3((1(1(-3|x| - 3) - 1) - 3)/(3(1(-3|x| - 3) - 1) + 1)) + 4) + 2)/(1(3((1(1(-3|x| - 3) -... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1e^(1(-3|(1(1|(4(2x^2 + 1x) - 3) - 3|) + 2) + 1| + 2)) - 2 and g(x) = 2(-2(-1(-1(-1|x - 3| + 1)^2 - 4(-1|x - 3| + 1) - 3)^2 + 5(-1(-1|x - 3| + 1)^2 - 4(-1|x - 3| + 1) - 3) + 2) - 5) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the ans... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(2|(-3(-5(4(-2x^2 - 2x - 5)^2 + 3(-2x^2 - 2x - 5) + 5) - 5) + 1) - 3| - 1) - 3)/(1(2|(-3(-5(4(-2x^2 - 2x - 5)^2 + 3(-2x^2 - 2x - 5) + 5) - 5) + 1) - 3| - 1) - 3) and g(x) = 3((-1(5(-1(-2cos(3πx - 1) + 3)^2 + 3(-2cos(3πx - ... | 76 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-3((2(4(-2|(-1|x + 2| + 3) - 2| + 2) - 3) + 2)/(1(4(-2|(-1|x + 2| + 3) - 2| + 2) - 3) + 3)) + 3) - 1| - 2 and g(x) = 1|(4(3|((-3(1|x|) + 3)/(-2(1|x|) - 2)) - 3| + 1)^2 - 4(3|((-3(1|x|) + 3)/(-2(1|x|) - 2)) - 3| + 1) - 1) - 1| - 3 intersect in the in... | 5 | MATH |
[
{
"content": "How many solutions does the equation 2(1(-2(-3(3(1|x - 2| - 1)^2 - 4(1|x - 2| - 1) - 2) - 1) - 2))^2 - 4(1(-2(-3(3(1|x - 2| - 1)^2 - 4(1|x - 2| - 1) - 2) - 1) - 2)) + 1 = (3((3((1(-2|(-3x + 2) + 2| - 3) + 2)/(2(-2|(-3x + 2) + 2| - 3) - 3)) + 3)/(2((1(-2|(-3x + 2) + 2| - 3) + 2)/(2(-2|(-3x + 2) + 2... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(1((2(4(-1(-2|x - 3| + 2) + 5) - 3) + 1)/(2(4(-1(-2|x - 3| + 2) + 5) - 3) - 1)) - 4) - 1)/(-3(1((2(4(-1(-2|x - 3| + 2) + 5) - 3) + 1)/(2(4(-1(-2|x - 3| + 2) + 5) - 3) - 1)) - 4) - 2) and g(x) = -1(-1cos(2π(-1|((-1((-2x + 2... | 21 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3((-2(-5(3e^(1(-4(-3x + 2)^2 + 4(-3x + 2) + 2)) + 2) + 4) - 2)/(2(-5(3e^(1(-4(-3x + 2)^2 + 4(-3x + 2) + 2)) + 2) + 4) - 1)) + 3 and g(x) = 1((1(-4(5(-2|x - 3| + 1) + 2) - 3) - 1)/(-3(-4(5(-2|x - 3| + 1) + 2) - 3) - 1))^2 - 1((1(-4(5(-2|x - 3| + 1) + 2... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2cos(2π(-3(-1|(-1(5(4x^2 + 1x - 5)^2 + 1(4x^2 + 1x - 5) - 4)^2 - 1(5(4x^2 + 1x - 5)^2 + 1(4x^2 + 1x - 5) - 4) + 4) - 2| + 1)^2 - 3(-1|(-1(5(4x^2 + 1x - 5)^2 + 1(4x^2 + 1x - 5) - 4)^2 - 1(5(4x^2 + 1x - 5)^2 + 1(4x^2 + 1x - 5) - 4) + 4) - 2| + 1) - 5)) ... | 40 | MATH |
[
{
"content": "How many solutions does the equation 4(4(3sin(1π(-3|((-3((-2x)/(-2x + 3)) - 3)/(-2((-2x)/(-2x + 3)) - 3)) + 3| + 3) - 2) + 2) + 5)^2 + 2(4(3sin(1π(-3|((-3((-2x)/(-2x + 3)) - 3)/(-2((-2x)/(-2x + 3)) - 3)) + 3| + 3) - 2) + 2) + 5) - 5 = -2cos(1π(-2(3|(1((1x + 3)/(-1x - 2))^2 - 1((1x + 3)/(-1x - 2)) ... | 15 | MATH |
[
{
"content": "How many solutions does the equation (3((3(2sin(3π(2(-3|(-5x^2 - 3x + 3) + 3|)) + 2) - 3) - 2)/(-2(2sin(3π(2(-3|(-5x^2 - 3x + 3) + 3|)) + 2) - 3) - 3)) + 1)/(2((3(2sin(3π(2(-3|(-5x^2 - 3x + 3) + 3|)) + 2) - 3) - 2)/(-2(2sin(3π(2(-3|(-5x^2 - 3x + 3) + 3|)) + 2) - 3) - 3)) + 2) = -4(-2cos(3π(2(-2|(-... | 1315 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(5(1e^(-2(-5(-3|(-3x - 5) - 1| - 2))) - 3)^2 + 5(1e^(-2(-5(-3|(-3x - 5) - 1| - 2))) - 3) + 2) + 4 and g(x) = 2e^(-1(-4(1(4(5x - 3)^2 - 4(5x - 3) + 4)^2 - 4(4(5x - 3)^2 - 4(5x - 3) + 4) + 1)^2 + 3(1(4(5x - 3)^2 - 4(5x - 3) + ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(1((1((3(-2(-2x) - 1) + 2)/(-1(-2(-2x) - 1) + 3)) + 3)/(-3((3(-2(-2x) - 1) + 2)/(-1(-2(-2x) - 1) + 3)) - 3)) + 1) + 1| + 3 and g(x) = 2|((-3((-1(-1|((1x - 3)/(-3x)) + 2| + 1) + 1)/(3(-1|((1x - 3)/(-3x)) + 2| + 1) - 1)) - 2)/(2((-1(-1|((1x - 3)/(-3x))... | 6 | MATH |
[
{
"content": "How many solutions does the equation 1(-3cos(3π(1(-2|((-1(4x^2 + 5x - 1))/(-1(4x^2 + 5x - 1) + 2)) - 3| + 3) + 2) + 2) - 3)^2 + 4(-3cos(3π(1(-2|((-1(4x^2 + 5x - 1))/(-1(4x^2 + 5x - 1) + 2)) - 3| + 3) + 2) + 2) - 3) - 2 = (-2(-3(3(1|(-3x + 1) + 2| - 3) + 5) - 3) + 2)/(1(-3(3(1|(-3x + 1) + 2| - 3) +... | 100 | MATH |
[
{
"content": "How many solutions does the equation -1(5(-1(-2((3(-4x + 2))/(-3(-4x + 2) + 2))^2 - 5((3(-4x + 2))/(-3(-4x + 2) + 2)))^2 - 1(-2((3(-4x + 2))/(-3(-4x + 2) + 2))^2 - 5((3(-4x + 2))/(-3(-4x + 2) + 2)))) + 4)^2 - 5(5(-1(-2((3(-4x + 2))/(-3(-4x + 2) + 2))^2 - 5((3(-4x + 2))/(-3(-4x + 2) + 2)))^2 - 1(-2... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(3π(-4(1|(3(3|((-1x - 2)/(-1x - 1)) - 2|) + 2) - 2| + 3) + 5) - 3) - 1 and g(x) = (3(1|(-2(4(5x - 2)^2 - 3(5x - 2) + 3)^2 + 4(4(5x - 2)^2 - 3(5x - 2) + 3) - 3) + 1| + 3) + 2)/(-2(1|(-2(4(5x - 2)^2 - 3(5x - 2) + 3)^2 + 4(4... | 364 | MATH |
[
{
"content": "How many solutions does the equation -1|(1(-3sin(1π((2((-2(2x^2 + 5x - 3) + 3)/(1(2x^2 + 5x - 3) + 1)) - 3)/(-3((-2(2x^2 + 5x - 3) + 3)/(1(2x^2 + 5x - 3) + 1)) + 1)) + 3))^2 - 3(-3sin(1π((2((-2(2x^2 + 5x - 3) + 3)/(1(2x^2 + 5x - 3) + 1)) - 3)/(-3((-2(2x^2 + 5x - 3) + 3)/(1(2x^2 + 5x - 3) + 1)) + 1... | 30 | MATH |
[
{
"content": "How many solutions does the equation 2(4(-1(1|((-1(-1cos(2πx + 3) + 2) + 2)/(-2(-1cos(2πx + 3) + 2) + 3)) - 1| - 1) + 4) + 5)^2 + 3(4(-1(1|((-1(-1cos(2πx + 3) + 2) + 2)/(-2(-1cos(2πx + 3) + 2) + 3)) - 1| - 1) + 4) + 5) - 5 = (-1((1(-1((-1(1x))/(-2(1x)))^2 - 4((-1(1x))/(-2(1x))) + 4) + 2)/(2(-1((-1... | 18 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(2cos(2π(-2(-1|(-2(-3x + 5)^2 - 2(-3x + 5) - 4) - 1| + 2) + 1) + 3) + 3) + 5 and g(x) = (3(2(-2(3|(5x - 5)| - 2) + 2) - 2) - 3)/(1(2(-2(3|(5x - 5)| - 2) + 2) - 2) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxe... | 12 | MATH |
[
{
"content": "How many solutions does the equation (-2(-3|(5((-1((-3(-2|x|) - 3)/(1(-2|x|) + 1)) + 1)/(3((-3(-2|x|) - 3)/(1(-2|x|) + 1)) - 2)) - 2) + 1| + 2) + 1)/(1(-3|(5((-1((-3(-2|x|) - 3)/(1(-2|x|) + 1)) + 1)/(3((-3(-2|x|) - 3)/(1(-2|x|) + 1)) - 2)) - 2) + 1| + 2) - 3) = -2cos(2π(-1|(4(-4((3x + 2)/(-2x + 3)... | 663 | MATH |
[
{
"content": "How many solutions does the equation 4(-2sin(1π(4((-3(-5(2|x - 3| + 2)) + 3)/(2(-5(2|x - 3| + 2)) - 1)) + 4) + 2) + 1) + 5 = -1(1sin(2π(-2((2(3x - 1) - 3)/(3(3x - 1) - 1))^2 - 4((2(3x - 1) - 3)/(3(3x - 1) - 1)) + 5) - 3) + 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex... | 63 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(4(1(-4(-3(-2x^2 + 1x - 4)^2 + 5(-2x^2 + 1x - 4) + 1) - 3) - 2) + 3) - 2)/(-1(4(1(-4(-3(-2x^2 + 1x - 4)^2 + 5(-2x^2 + 1x - 4) + 1) - 3) - 2) + 3) - 3) and g(x) = 1(1(2(1|((2x)/(-2x + 3)) - 1| - 1) - 3) - 3)^2 - 5(1(2(1|((2... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((1((1(-3|(-4((3x + 3)/(2x - 1))^2 + 4((3x + 3)/(2x - 1)) + 4)| - 2) - 2)/(-3(-3|(-4((3x + 3)/(2x - 1))^2 + 4((3x + 3)/(2x - 1)) + 4)| - 2))))/(2((1(-3|(-4((3x + 3)/(2x - 1))^2 + 4((3x + 3)/(2x - 1)) + 4)| - 2) - 2)/(-3(-3|(-4((3x + 3)/(2x - 1))^2 + ... | 58 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1sin(3π((3(3(2|(-1|(-1x - 1) + 1| - 3) + 3| + 1)^2 + 1(2|(-1|(-1x - 1) + 1| - 3) + 3| + 1) - 4) + 3)/(1(3(2|(-1|(-1x - 1) + 1| - 3) + 3| + 1)^2 + 1(2|(-1|(-1x - 1) + 1| - 3) + 3| + 1) - 4) - 3)) - 3) + 3 and g(x) = 4(-3cos(1π(-1((-3((2x + 2)/(3x + 1)) ... | 77 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(4(-5(2cos(2π(2(1x^2 - 1x - 5)^2 + 1(1x^2 - 1x - 5)) + 3) + 1) - 2)^2 + 3(-5(2cos(2π(2(1x^2 - 1x - 5)^2 + 1(1x^2 - 1x - 5)) + 3) + 1) - 2)) + 1)/(2(4(-5(2cos(2π(2(1x^2 - 1x - 5)^2 + 1(1x^2 - 1x - 5)) + 3) + 1) - 2)^2 + 3(-5... | 64 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-4((2(-1cos(3π(-3((3x - 2)/(-2x - 2)) + 4) - 2) - 1) - 3)/(2(-1cos(3π(-3((3x - 2)/(-2x - 2)) + 4) - 2) - 1) + 1)) - 5) + 1 and g(x) = -1|(1(-2|(3sin(1π(1x) + 1) - 1) + 3| - 3) - 5) - 1| intersect in the interval [-10, 10].\n\nPresent the answer in L... | 109 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2((-3((-2(-2|(-1(-1sin(2πx - 2) + 2) - 4) + 3| - 2) + 2)/(1(-2|(-1(-1sin(2πx - 2) + 2) - 4) + 3| - 2) + 3)) - 1)/(3((-2(-2|(-1(-1sin(2πx - 2) + 2) - 4) + 3| - 2) + 2)/(1(-2|(-1(-1sin(2πx - 2) + 2) - 4) + 3| - 2) + 3)) - 2)) + 2 and g(x) = 3(1sin(1π(-1... | 730 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(3(2|(-1(-2(-2sin(3πx - 3) - 3)^2 + 1(-2sin(3πx - 3) - 3) + 3)^2 + 1(-2(-2sin(3πx - 3) - 3)^2 + 1(-2sin(3πx - 3) - 3) + 3)) - 2| + 2)^2 - 1(2|(-1(-2(-2sin(3πx - 3) - 3)^2 + 1(-2sin(3πx - 3) - 3) + 3)^2 + 1(-2(-2sin(3πx - 3) - 3)^2 + 1(-2sin(3πx - 3) ... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1cos(1π(-3((2(1((2((-3x + 3)/(-1x)) - 3)/(3((-3x + 3)/(-1x)) + 1))^2 + 4((2((-3x + 3)/(-1x)) - 3)/(3((-3x + 3)/(-1x)) + 1)) + 2) + 3)/(1(1((2((-3x + 3)/(-1x)) - 3)/(3((-3x + 3)/(-1x)) + 1))^2 + 4((2((-3x + 3)/(-1x)) - 3)/(3(... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(1(-1|((-3((2(4x^2 + 5))/(-2(4x^2 + 5) - 3)) - 3)/(-3((2(4x^2 + 5))/(-2(4x^2 + 5) - 3)) + 1)) + 3| + 2)^2 - 5) - 1| + 1 and g(x) = 3(-5(-2(-5(4x^2 + 1x - 1)^2 - 1)^2 - 4(-5(4x^2 + 1x - 1)^2 - 1)) + 1)^2 + 4(-5(-2(-5(4x^2 + ... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(2|(-1|((1(-3(1|x + 2| - 3) - 5) + 2)/(-2(-3(1|x + 2| - 3) - 5))) + 1| + 1) + 3| - 3)^2 - 5(2|(-1|((1(-3(1|x + 2| - 3) - 5) + 2)/(-2(-3(1|x + 2| - 3) - 5))) + 1| + 1) + 3| - 3) - 1 and g(x) = -2|(2(-1((3((-2x - 3)/(1x + 2)) - 1)/(1((-2x - 3)/(1x + 2)... | 17 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3((-3(2(1(-2cos(2π((-3x - 2)/(-1x + 3)) + 3) - 3)^2 - 5(-2cos(2π((-3x - 2)/(-1x + 3)) + 3) - 3) + 2) + 5) - 2)/(-3(2(1(-2cos(2π((-3x - 2)/(-1x + 3)) + 3) - 3)^2 - 5(-2cos(2π((-3x - 2)/(-1x + 3)) + 3) - 3) + 2) + 5) - 2)) - 1)/(-2... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(-5(2|(-3|(-3|(-5x - 2) + 1| + 3) + 3| - 3) + 3| - 3) - 3) + 2| - 2 and g(x) = -3|(-2((3(-2(3|x + 2| + 2)^2 - 4(3|x + 2| + 2) + 2))/(3(-2(3|x + 2| + 2)^2 - 4(3|x + 2| + 2) + 2)))^2 - 2((3(-2(3|x + 2| + 2)^2 - 4(3|x + 2| + 2) + 2... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(-1(-2|(2(2((-1x - 3)/(1x)) + 2)^2 + 2(2((-1x - 3)/(1x)) + 2) - 1) - 2| + 2) - 3) + 2 and g(x) = (-1(3cos(2π(-2(-3|(-1|x - 3| - 2) - 1| - 3) + 1) + 3) + 1))/(3(3cos(2π(-2(-3|(-1|x - 3| - 2) - 1| - 3) + 1) + 3) + 1) + 2) whe... | 34 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-1|(-1(2e^(-1(4(-2x + 5)^2 - 3(-2x + 5) - 1)) - 1)^2 + 5(2e^(-1(4(-2x + 5)^2 - 3(-2x + 5) - 1)) - 1) + 1) - 3| + 1) + 1)/(-1(-1|(-1(2e^(-1(4(-2x + 5)^2 - 3(-2x + 5) - 1)) - 1)^2 + 5(2e^(-1(4(-2x + 5)^2 - 3(-2x + 5) - 1)) -... | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2((1(1|(1|(-5((-3x + 3)/(-3x + 4))^2 + 4((-3x + 3)/(-3x + 4)) + 5) + 2| + 3) + 1| + 1) + 1)/(3(1|(1|(-5((-3x + 3)/(-3x + 4))^2 + 4((-3x + 3)/(-3x + 4)) + 5) + 2| + 3) + 1| + 1) - 1)) + 2)/(1((1(1|(1|(-5((-3x + 3)/(-3x + 4))^... | 57 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(-2(-1(1(-1|(4x + 5) - 1| - 2)^2 + 3(-1|(4x + 5) - 1| - 2) - 3) + 5) - 1) + 2| - 3 and g(x) = -1sin(1π(2|(-2(3|(1x + 1) - 2|)^2 - 4(3|(1x + 1) - 2|) + 2) - 3| - 1) + 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex... | 7 | MATH |
[
{
"content": "How many solutions does the equation -4(-3((2(1(3sin(1π(5x + 5) + 3)) + 1) - 3)/(-3(1(3sin(1π(5x + 5) + 3)) + 1) + 1)) + 5) + 5 = -1|(2|((3(-1e^(-1(-1x^2 + 3x - 3)) - 1))/(3(-1e^(-1(-1x^2 + 3x - 3)) - 1) - 2))| + 1) + 3| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex form... | 100 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(1((2(-1((-1((2x + 1)/(2x + 1)) + 1)/(-1((2x + 1)/(2x + 1)) + 2))^2 - 1((-1((2x + 1)/(2x + 1)) + 1)/(-1((2x + 1)/(2x + 1)) + 2)) + 1) - 2)/(2(-1((-1((2x + 1)/(2x + 1)) + 1)/(-1((2x + 1)/(2x + 1)) + 2))^2 - 1((-1((2x + 1)/(... | 1446 | MATH |
[
{
"content": "How many solutions does the equation (1((-1((2(-2((2(-3e^(-2x) + 1) + 3)/(-3(-3e^(-2x) + 1) + 1)) + 5))/(2(-2((2(-3e^(-2x) + 1) + 3)/(-3(-3e^(-2x) + 1) + 1)) + 5) - 1)) + 3)/(-1((2(-2((2(-3e^(-2x) + 1) + 3)/(-3(-3e^(-2x) + 1) + 1)) + 5))/(2(-2((2(-3e^(-2x) + 1) + 3)/(-3(-3e^(-2x) + 1) + 1)) + 5) -... | 13 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4((1(-4(2sin(3π(-1(2x + 1)^2 - 4(2x + 1) - 3) + 2) + 2)^2 + 5) + 1)/(-2(-4(2sin(3π(-1(2x + 1)^2 - 4(2x + 1) - 3) + 2) + 2)^2 + 5) - 3)) + 3 and g(x) = (-2(-4(-3|(-3((1x - 3)/(-2x - 2))^2 + 4((1x - 3)/(-2x - 2)) + 1) + 2| - 1) - 5) ... | 65 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2((-3((2((-2((1(-3|x - 3| + 3) - 3)/(1(-3|x - 3| + 3) + 1)) - 1)/(3((1(-3|x - 3| + 3) - 3)/(1(-3|x - 3| + 3) + 1)) - 2)))/(1((-2((1(-3|x - 3| + 3) - 3)/(1(-3|x - 3| + 3) + 1)) - 1)/(3((1(-3|x - 3| + 3) - 3)/(1(-3|x - 3| + 3) ... | 18 | MATH |
[
{
"content": "How many solutions does the equation (2(3sin(2π(2((3(3(3|x + 1| + 1) - 2) + 1)/(-1(3(3|x + 1| + 1) - 2) - 2)) + 2)) - 2))/(-1(3sin(2π(2((3(3(3|x + 1| + 1) - 2) + 1)/(-1(3(3|x + 1| + 1) - 2) - 2)) + 2)) - 2) - 1) = 1((-2(-1|(1(-2cos(2πx - 1) - 1)^2 - 4)| + 1) + 1)/(1(-1|(1(-2cos(2πx - 1) - 1)^2 - 4... | 8 | MATH |
[
{
"content": "How many solutions does the equation 5(1((2(4(-2sin(1π(-3x^2 + 2) + 1) + 2)^2 - 2(-2sin(1π(-3x^2 + 2) + 1) + 2) + 2) - 3)/(1(4(-2sin(1π(-3x^2 + 2) + 1) + 2)^2 - 2(-2sin(1π(-3x^2 + 2) + 1) + 2) + 2) + 2)) - 1)^2 - 5(1((2(4(-2sin(1π(-3x^2 + 2) + 1) + 2)^2 - 2(-2sin(1π(-3x^2 + 2) + 1) + 2) + 2) - 3)/... | 592 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(2e^(2((2(4(5(4x - 2)^2 + 3(4x - 2) - 2) - 1) - 1)/(-2(4(5(4x - 2)^2 + 3(4x - 2) - 2) - 1) - 2))) + 3) + 1)/(2(2e^(2((2(4(5(4x - 2)^2 + 3(4x - 2) - 2) - 1) - 1)/(-2(4(5(4x - 2)^2 + 3(4x - 2) - 2) - 1) - 2))) + 3) - 3) and g(x) = ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1((-2((-3(3(3sin(2π(-2|x + 3| + 3) - 3)) + 3) - 3)/(-1(3(3sin(2π(-2|x + 3| + 3) - 3)) + 3) + 2)) - 1)/(1((-3(3(3sin(2π(-2|x + 3| + 3) - 3)) + 3) - 3)/(-1(3(3sin(2π(-2|x + 3| + 3) - 3)) + 3) + 2)) - 3))^2 - 2((-2((-3(3(3sin(2π(-2|x... | 160 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-3e^(-1(1(2|((1((1x + 3)/(-2x + 1)) + 3)/(-1((1x + 3)/(-2x + 1)) - 3)) + 2| + 2) - 4)) - 2) - 3)/(-2(-3e^(-1(1(2|((1((1x + 3)/(-2x + 1)) + 3)/(-1((1x + 3)/(-2x + 1)) - 3)) + 2| + 2) - 4)) - 2) - 1) and g(x) = 1|(-2|(2sin(2π(1|(3x^2 + 4x - 4) + 1|) -... | 162 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(2(1((1(1((1x - 3)/(-2x - 1))))/(-1(1((1x - 3)/(-2x - 1))) + 3))^2 + 2((1(1((1x - 3)/(-2x - 1))))/(-1(1((1x - 3)/(-2x - 1))) + 3)) + 4) + 3) + 1)/(-2(2(1((1(1((1x - 3)/(-2x - 1))))/(-1(1((1x - 3)/(-2x - 1))) + 3))^2 + 2((1(1((1x ... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(-1|((-1(-5((3(4x + 3) - 2)/(3(4x + 3) + 1)) - 3))/(-2(-5((3(4x + 3) - 2)/(3(4x + 3) + 1)) - 3) + 3)) + 2| + 2)^2 + 1(-1|((-1(-5((3(4x + 3) - 2)/(3(4x + 3) + 1)) - 3))/(-2(-5((3(4x + 3) - 2)/(3(4x + 3) + 1)) - 3) + 3)) + 2| + 2) - 3 and g(x) = -1(1((-... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(2π(3(-4(2|(4(-4x^2 - 3x + 3)^2 - 4) - 1| + 2) + 1) + 1) - 2) - 3 and g(x) = -3(-2|(2(-1cos(3π(-2x - 4)) + 1)^2 - 1(-1cos(3π(-2x - 4)) + 1) - 3) - 3| + 3)^2 + 3(-2|(2(-1cos(3π(-2x - 4)) + 1)^2 - 1(-1cos(3π(-2x - 4)) + 1) - 3) - 3| + 3) - 4 interse... | 27 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((2((-1(4((2(3e^(1x) - 2) - 1)/(2(3e^(1x) - 2) + 2)) + 4) + 3)/(1(4((2(3e^(1x) - 2) - 1)/(2(3e^(1x) - 2) + 2)) + 4) - 2)) + 1)/(1((-1(4((2(3e^(1x) - 2) - 1)/(2(3e^(1x) - 2) + 2)) + 4) + 3)/(1(4((2(3e^(1x) - 2) - 1)/(2(3e^(1x) - 2) + 2)) + 4) - 2)))) ... | 41 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1e^(1((3(2(4(-2|(-1x - 1) + 2| + 2)^2 + 4(-2|(-1x - 1) + 2| + 2) - 4)^2 - 3(4(-2|(-1x - 1) + 2| + 2)^2 + 4(-2|(-1x - 1) + 2| + 2) - 4) - 1) - 1)/(-3(2(4(-2|(-1x - 1) + 2| + 2)^2 + 4(-2|(-1x - 1) + 2| + 2) - 4)^2 - 3(4(-2|(-1... | 119 | MATH |
[
{
"content": "How many solutions does the equation -3(1|(5((-1(-1(4x + 5)^2 - 3(4x + 5)) - 3)/(-2(-1(4x + 5)^2 - 3(4x + 5)) + 1))^2 - 1((-1(-1(4x + 5)^2 - 3(4x + 5)) - 3)/(-2(-1(4x + 5)^2 - 3(4x + 5)) + 1)) - 5) + 1| - 2) - 2 = (-1(-3|((-2((-2(-4x^2 + 4x) - 3)/(1(-4x^2 + 4x) + 2)) + 3)/(-3((-2(-4x^2 + 4x) - 3)/... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-1(5((3(3((-1x + 1)/(2x - 1))^2 - 1((-1x + 1)/(2x - 1)) + 2) - 3)/(-3(3((-1x + 1)/(2x - 1))^2 - 1((-1x + 1)/(2x - 1)) + 2) - 3)))^2 + 3(5((3(3((-1x + 1)/(2x - 1))^2 - 1((-1x + 1)/(2x - 1)) + 2) - 3)/(-3(3((-1x + 1)/(2x - 1))^2 - 1((-1x + 1)/(2x - 1... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5((-2(2(-3(2e^(-1(3x^2 + 3x + 1)) - 1) + 1)) + 3)/(3(2(-3(2e^(-1(3x^2 + 3x + 1)) - 1) + 1)) - 1))^2 + 5((-2(2(-3(2e^(-1(3x^2 + 3x + 1)) - 1) + 1)) + 3)/(3(2(-3(2e^(-1(3x^2 + 3x + 1)) - 1) + 1)) - 1)) + 4 and g(x) = 2(1|(-3e^(-2(3|(... | 4 | MATH |
[
{
"content": "How many solutions does the equation -4((1(-2((-1(-2((-1x - 3)/(1x))^2 + 1((-1x - 3)/(1x)) - 1) - 1)/(-3(-2((-1x - 3)/(1x))^2 + 1((-1x - 3)/(1x)) - 1))) - 3) + 3)/(2(-2((-1(-2((-1x - 3)/(1x))^2 + 1((-1x - 3)/(1x)) - 1) - 1)/(-3(-2((-1x - 3)/(1x))^2 + 1((-1x - 3)/(1x)) - 1))) - 3) + 1)) + 3 = -5(5(... | 4 | MATH |
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