messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation -3cos(1π(-2(-3(-1(1(5x^2 + 2x + 1) - 2)^2 - 4(1(5x^2 + 2x + 1) - 2))^2 + 1(-1(1(5x^2 + 2x + 1) - 2)^2 - 4(1(5x^2 + 2x + 1) - 2)) + 2)^2 + 5(-3(-1(1(5x^2 + 2x + 1) - 2)^2 - 4(1(5x^2 + 2x + 1) - 2))^2 + 1(-1(1(5x^2 + 2x + 1) - 2)^2 - 4(1(5x^2 + 2x + 1) - 2)) + 2)... | 109 | MATH |
[
{
"content": "How many solutions does the equation 2(-2e^(2(-2((3(-1|(2|x - 1| - 2) + 2| - 1) + 1)/(-2(-1|(2|x - 1| - 2) + 2| - 1)))))) + 4 = -3|(2|((-3(1(-3cos(3πx)) - 3) + 3)/(1(1(-3cos(3πx)) - 3))) + 1| - 2) - 1| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|((1((-1((-2((-1(4x^2 + 2x - 1))/(1(4x^2 + 2x - 1) + 2)) + 3)/(3((-1(4x^2 + 2x - 1))/(1(4x^2 + 2x - 1) + 2)) - 2)))/(3((-2((-1(4x^2 + 2x - 1))/(1(4x^2 + 2x - 1) + 2)) + 3)/(3((-1(4x^2 + 2x - 1))/(1(4x^2 + 2x - 1) + 2)) - 2)) + 1))... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(4(5(3((1((2x)/(-2x + 1)) + 3)/(2((2x)/(-2x + 1)) + 3)) + 2)^2 + 5(3((1((2x)/(-2x + 1)) + 3)/(2((2x)/(-2x + 1)) + 3)) + 2) - 1)^2)^2 - 4(4(5(3((1((2x)/(-2x + 1)) + 3)/(2((2x)/(-2x + 1)) + 3)) + 2)^2 + 5(3((1((2x)/(-2x + 1)) + 3)/(... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(5(-3|(2|(1((2x + 2)/(1x)) + 4) + 3| - 3) + 1| + 2)^2 + 3(-3|(2|(1((2x + 2)/(1x)) + 4) + 3| - 3) + 1| + 2) - 4) - 2 and g(x) = (1((-3(2(5(-1|x - 1| + 2) - 1)^2 + 5(5(-1|x - 1| + 2) - 1) - 5) - 2)/(-1(2(5(-1|x - 1| + 2) - 1)... | 7 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-3sin(3π(-5(3(2|(2x - 1) + 3| - 2) - 4) + 1) + 3)) - 3| - 1 and g(x) = -3(2|((2(1((-3x - 2)/(2x + 2))^2 - 2((-3x - 2)/(2x + 2)) - 2) - 1)/(1(1((-3x - 2)/(2x + 2))^2 - 2((-3x - 2)/(2x + 2)) - 2) - 3))| + 2) + 4 where -10 ≤... | 65 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(1|(2sin(1π(-4((2(-1x^2 + 5x + 5) - 2)/(-1(-1x^2 + 5x + 5) + 3))^2 + 2((2(-1x^2 + 5x + 5) - 2)/(-1(-1x^2 + 5x + 5) + 3)) - 3) + 2)) + 1| - 3)^2 + 5(1|(2sin(1π(-4((2(-1x^2 + 5x + 5) - 2)/(-1(-1x^2 + 5x + 5) + 3))^2 + 2((2(-1x^2 + 5x + 5) - 2)/(-1(-1x^... | 23 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(1((-1(2(-1|(3sin(3πx - 2) - 2) + 3| + 1)^2 - 5(-1|(3sin(3πx - 2) - 2) + 3| + 1)) - 2)/(-1(2(-1|(3sin(3πx - 2) - 2) + 3| + 1)^2 - 5(-1|(3sin(3πx - 2) - 2) + 3| + 1)) - 1))^2 - 2((-1(2(-1|(3sin(3πx - 2) - 2) + 3| + 1)^2 - 5(... | 129 | MATH |
[
{
"content": "How many solutions does the equation (2((-2((1(5((-3((-2x)/(-3x - 1)) + 1)/(-2((-2x)/(-3x - 1)) - 3))^2 + 2((-3((-2x)/(-3x - 1)) + 1)/(-2((-2x)/(-3x - 1)) - 3)) - 5) - 3)/(-1(5((-3((-2x)/(-3x - 1)) + 1)/(-2((-2x)/(-3x - 1)) - 3))^2 + 2((-3((-2x)/(-3x - 1)) + 1)/(-2((-2x)/(-3x - 1)) - 3)) - 5) - 2)... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(1e^(-1(3|((-2((-2(4x^2 - 3x - 3) - 2)/(1(4x^2 - 3x - 3) + 3)) - 2)/(1((-2(4x^2 - 3x - 3) - 2)/(1(4x^2 - 3x - 3) + 3)) + 2)) + 3| - 2)) + 2)^2 - 2(1e^(-1(3|((-2((-2(4x^2 - 3x - 3) - 2)/(1(4x^2 - 3x - 3) + 3)) - 2)/(1((-2(4x^2 - 3x - 3) - 2)/(1(4x^2 -... | 160 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(-1(-4((1(2sin(3π(4x^2 - 3x + 5) - 2) + 1))/(3(2sin(3π(4x^2 - 3x + 5) - 2) + 1) - 1))^2 + 4) + 4) and g(x) = 1((2((-2((1((3x - 3)/(-2x)) + 2)/(-2((3x - 3)/(-2x)))) + 1)/(2((1((3x - 3)/(-2x)) + 2)/(-2((3x - 3)/(-2x)))))) + 1)/(3((... | 294 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(2sin(2π((2(1((3(2|x| - 2) - 1)/(-2(2|x| - 2) - 3)) + 2) + 1)/(1(1((3(2|x| - 2) - 1)/(-2(2|x| - 2) - 3)) + 2) - 2)))) - 4 and g(x) = 2cos(1π(3|(-3|(5(3x^2 + 4x - 2)) - 2| + 2) + 2| - 2) - 1) intersect in the interval [-10, 10].\n\nPresent the answer i... | 623 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(-1(-3|(-1(-1cos(1π(1|x - 1| + 2) + 3) - 2)^2 - 2(-1cos(1π(1|x - 1| + 2) + 3) - 2) + 1)| - 3)^2 - 3(-3|(-1(-1cos(1π(1|x - 1| + 2) + 3) - 2)^2 - 2(-1cos(1π(1|x - 1| + 2) + 3) - 2) + 1)| - 3) + 4) + 4 and g(x) = -2(1cos(2π(-2(... | 42 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(-5(-3|(-5(-3cos(3π(4x^2 - 3x + 2) - 3) - 2)^2 + 2(-3cos(3π(4x^2 - 3x + 2) - 3) - 2) + 3) - 3| + 3) - 2) + 3)/(-3(-5(-3|(-5(-3cos(3π(4x^2 - 3x + 2) - 3) - 2)^2 + 2(-3cos(3π(4x^2 - 3x + 2) - 3) - 2) + 3) - 3| + 3) - 2) - 3)... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(1(5(-2(1|(-1|x + 1| - 1) + 3| - 1)^2 + 4(1|(-1|x + 1| - 1) + 3| - 1) - 1)^2 + 1(-2(1|(-1|x + 1| - 1) + 3| - 1)^2 + 4(1|(-1|x + 1| - 1) + 3| - 1) - 1) - 4)^2 + 5(5(-2(1|(-1|x + 1| - 1) + 3| - 1)^2 + 4(1|(-1|x + 1| - 1) + 3|... | 24 | MATH |
[
{
"content": "How many solutions does the equation -2e^(-2(-4(3(-2|(1(-5x^2 - 3x + 4)) + 2| + 2) - 2)^2 + 5)) = -2(2cos(3π((-3(3(-1x^2 + 3x + 2)^2 + 1(-1x^2 + 3x + 2) - 5) - 1)/(-3(3(-1x^2 + 3x + 2)^2 + 1(-1x^2 + 3x + 2) - 5) + 1)) + 3) - 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaT... | 8 | MATH |
[
{
"content": "How many solutions does the equation -1((2(3(2(1|(-1sin(2πx + 3) - 3) + 2| - 2) + 3)^2 - 4(2(1|(-1sin(2πx + 3) - 3) + 2| - 2) + 3) - 1))/(3(3(2(1|(-1sin(2πx + 3) - 3) + 2| - 2) + 3)^2 - 4(2(1|(-1sin(2πx + 3) - 3) + 2| - 2) + 3) - 1) + 2))^2 - 3((2(3(2(1|(-1sin(2πx + 3) - 3) + 2| - 2) + 3)^2 - 4(2(... | 152 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(-2(4(4(-2cos(3π((1x - 3)/(3x + 2))) + 3)^2 + 5(-2cos(3π((1x - 3)/(3x + 2))) + 3) - 5) - 4) + 1) + 3| and g(x) = -2|(-5(-5(-2|((1x - 1)/(1x + 2)) - 3|) + 4) - 4) + 3| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex... | 67 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-3|((2(-4(3|((3x - 2)/(2x)) + 3| - 2)^2 + 1(3|((3x - 2)/(2x)) + 3| - 2) + 1) + 1)/(-1(-4(3|((3x - 2)/(2x)) + 3| - 2)^2 + 1(3|((3x - 2)/(2x)) + 3| - 2) + 1) + 2)) - 2| + 3)^2 - 5(-3|((2(-4(3|((3x - 2)/(2x)) + 3| - 2)^2 + 1(3|((3x - 2)/(2x)) + 3| - 2) ... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1((2(2(3(-2|(-1|x + 1|) + 2| + 3) - 5) - 1) + 1)/(2(2(3(-2|(-1|x + 1|) + 2| + 3) - 5) - 1) - 3))^2 + 3((2(2(3(-2|(-1|x + 1|) + 2| + 3) - 5) - 1) + 1)/(2(2(3(-2|(-1|x + 1|) + 2| + 3) - 5) - 1) - 3)) + 5 and g(x) = -4(2(-1cos(... | 23 | MATH |
[
{
"content": "How many solutions does the equation -4(3sin(2π(5(-2|((2(-1x + 5) - 2)/(2(-1x + 5) - 1)) + 3| - 1)) + 1) + 3) - 2 = -3|((-3(-1cos(1π(-2(-2x - 5)^2 - 5(-2x - 5) - 3) - 2) - 3) - 3)/(-3(-1cos(1π(-2(-2x - 5)^2 - 5(-2x - 5) - 3) - 2) - 3) - 2)) + 1| have for x in the interval [-10, 10]?\n\nPresent the... | 106 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-1|(-3(3((2(-5x - 3))/(3(-5x - 3) - 1))^2 + 4((2(-5x - 3))/(3(-5x - 3) - 1)) - 1) - 1) + 2|) + 3)/(-2(-1|(-3(3((2(-5x - 3))/(3(-5x - 3) - 1))^2 + 4((2(-5x - 3))/(3(-5x - 3) - 1)) - 1) - 1) + 2|)) and g(x) = -2|(1((-3(2(1|x... | 8 | MATH |
[
{
"content": "How many solutions does the equation -4((-1(-4(-4(-3|(-1x^2 - 5x - 5)| - 2) - 1) - 3) - 1)/(1(-4(-4(-3|(-1x^2 - 5x - 5)| - 2) - 1) - 3) - 2))^2 + 5((-1(-4(-4(-3|(-1x^2 - 5x - 5)| - 2) - 1) - 3) - 1)/(1(-4(-4(-3|(-1x^2 - 5x - 5)| - 2) - 1) - 3) - 2)) - 3 = -1(-1(5(-1|(-1x^2 - 5x) + 1| + 3)^2 + 5(-1... | 16 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-3|(1(3(-1(-1|x + 2| + 1) - 4) + 2) + 1)|)^2 - 3(-3|(1(3(-1(-1|x + 2| + 1) - 4) + 2) + 1)|) - 3 and g(x) = 3(-2(-5((-2((-3x)/(1x + 1)) - 2)/(2((-3x)/(1x + 1)) - 1))^2 + 3((-2((-3x)/(1x + 1)) - 2)/(2((-3x)/(1x + 1)) - 1)) - 2) + 2... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(4(3(-4(5((1x + 1)/(-3x + 2)) + 1) + 5) - 5) + 3) + 3)/(-3(4(3(-4(5((1x + 1)/(-3x + 2)) + 1) + 5) - 5) + 3) + 1) and g(x) = 2cos(1π(1(-3(3|((2x - 2)/(-1x - 1)) + 1| - 1)^2 - 3(3|((2x - 2)/(-1x - 1)) + 1| - 1) + 2) - 3) + 3)... | 467 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(-1(-1((2((-2(2sin(1πx + 3) - 1))/(2(2sin(1πx + 3) - 1) - 3)) - 1)/(-2((-2(2sin(1πx + 3) - 1))/(2(2sin(1πx + 3) - 1) - 3)) + 3)) - 3)^2 + 4(-1((2((-2(2sin(1πx + 3) - 1))/(2(2sin(1πx + 3) - 1) - 3)) - 1)/(-2((-2(2sin(1πx + 3... | 140 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(5((2(1(2sin(3π(1x^2 - 2x - 2) + 1) - 2) - 4) + 1)/(1(1(2sin(3π(1x^2 - 2x - 2) + 1) - 2) - 4) - 2)) - 3)^2 + 1(5((2(1(2sin(3π(1x^2 - 2x - 2) + 1) - 2) - 4) + 1)/(1(1(2sin(3π(1x^2 - 2x - 2) + 1) - 2) - 4) - 2)) - 3) + 3 and g(x) =... | 10 | MATH |
[
{
"content": "How many solutions does the equation -2(4((-2(-3|(5((1x - 1)/(-1x + 2)) - 1)| + 1) + 2)/(3(-3|(5((1x - 1)/(-1x + 2)) - 1)| + 1) + 3)) - 1) - 4 = (-1(-3(2|(-5((-3x - 1)/(-1x - 1))^2 + 4((-3x - 1)/(-1x - 1)) + 1) + 2| - 1) + 3))/(-1(-3(2|(-5((-3x - 1)/(-1x - 1))^2 + 4((-3x - 1)/(-1x - 1)) + 1) + 2| ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(5(2(2cos(3π(-2((-1x - 1)/(2x)) - 5) + 3) + 1)^2 - 4)^2 + 5(2(2cos(3π(-2((-1x - 1)/(2x)) - 5) + 3) + 1)^2 - 4) - 5) - 5 and g(x) = -4(-1|(-4(-2(-5x^2 + 2x - 3)^2 - 1(-5x^2 + 2x - 3) + 1)^2 - 5) + 3| + 2) + 2 in the range -10 ≤ x ≤... | 13 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-5(-1(3(5((-2x + 3)/(-2x + 3)) + 3)^2 - 1(5((-2x + 3)/(-2x + 3)) + 3) + 1) + 4)^2 - 2(-1(3(5((-2x + 3)/(-2x + 3)) + 3)^2 - 1(5((-2x + 3)/(-2x + 3)) + 3) + 1) + 4) - 1) - 2)/(3(-5(-1(3(5((-2x + 3)/(-2x + 3)) + 3)^2 - 1(5((-2x + 3)/(-2x + 3)) + 3) + 1... | 131 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|((2(-4((3((-1(2x + 4) - 2)/(2(2x + 4) - 3)) - 2)/(-2((-1(2x + 4) - 2)/(2(2x + 4) - 3)) - 1))^2 + 2((3((-1(2x + 4) - 2)/(2(2x + 4) - 3)) - 2)/(-2((-1(2x + 4) - 2)/(2(2x + 4) - 3)) - 1)) + 1) + 3)/(1(-4((3((-1(2x + 4) - 2)/(2(2x + 4) - 3)) - 2)/(-2((-1... | 22 | MATH |
[
{
"content": "How many solutions does the equation 4(-5((3(-1e^(-2(-3(-4x + 1) + 2)) + 3) + 2)/(2(-1e^(-2(-3(-4x + 1) + 2)) + 3) - 1)) + 5) - 3 = -2(3cos(1π(-1((1(3x^2 - 5x - 3) + 3)/(2(3x^2 - 5x - 3) + 2))^2 - 1((1(3x^2 - 5x - 3) + 3)/(2(3x^2 - 5x - 3) + 2)) + 1)))^2 - 4 have for x in the interval [-10, 10]?\n... | 26 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(4(-4((-1((2((-3x + 1)/(-3x)) - 1)/(3((-3x + 1)/(-3x)) + 1)) + 3)/(-3((2((-3x + 1)/(-3x)) - 1)/(3((-3x + 1)/(-3x)) + 1)) - 3)) - 1) + 1) - 2 and g(x) = (1(2|(1|(-1(5x - 5)^2 + 5(5x - 5)) - 3|) - 3| - 3) + 2)/(1(2|(1|(-1(5x - 5)^2... | 7 | MATH |
[
{
"content": "How many solutions does the equation 5(1|(1cos(3π((1(-1(3|x - 3| + 3) - 5) + 1)/(-1(-1(3|x - 3| + 3) - 5) + 3)) + 2) + 2) - 1| + 2) - 4 = 2|(-2|(-3sin(2π(-1(2x^2 + 3x - 2)^2 - 3(2x^2 + 3x - 2) + 5) - 1) - 2) - 2| + 1) - 2| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex fo... | 1189 | MATH |
[
{
"content": "How many solutions does the equation 3|(-2|(-3((-1(1|((1x + 3)/(-3x - 3)) + 3| - 3) - 1)/(-1(1|((1x + 3)/(-3x - 3)) + 3| - 3) - 3))^2 + 2((-1(1|((1x + 3)/(-3x - 3)) + 3| - 3) - 1)/(-1(1|((1x + 3)/(-3x - 3)) + 3| - 3) - 3)) + 2) + 3| + 2) + 2| - 2 = (3(1|(1(-2|(-3cos(1πx - 3) - 3)| + 3)^2 + 3(-2|(-... | 7 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2((-1(-2|(-2|(3e^(1(1x^2 + 1x))) - 3| - 3) + 1| - 1) + 1)/(3(-2|(-2|(3e^(1(1x^2 + 1x))) - 3| - 3) + 1| - 1) - 2))^2 + 1((-1(-2|(-2|(3e^(1(1x^2 + 1x))) - 3| - 3) + 1| - 1) + 1)/(3(-2|(-2|(3e^(1(1x^2 + 1x))) - 3| - 3) + 1| - 1) - 2)... | 890 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(5(-1((-1((3(4x + 4) + 3)/(2(4x + 4))) - 2)/(-1((3(4x + 4) + 3)/(2(4x + 4))) + 2)) - 3) + 4) + 3| + 2 and g(x) = 2((3((-1(1(3|x| - 1) + 3) + 1)/(1(1(3|x| - 1) + 3) - 3)) + 2)/(2((-1(1(3|x| - 1) + 3) + 1)/(1(1(3|x| - 1) + 3) - 3)) + 3))^2 - 3((3((-1(1... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|((-2(1|(3|(1(3|x - 3| - 2) - 1) - 3| + 1) + 1| - 3) + 2)/(-1(1|(3|(1(3|x - 3| - 2) - 1) - 3| + 1) + 1| - 3) + 2)) - 2| - 3 and g(x) = -2|(2cos(1π(-1|(-1|(-1|x - 1| + 3) + 3| - 2) + 2| - 1)) - 2) + 1| - 2 intersect in the interval [-10, 10].\n\nPrese... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(3(-3(1(-2(-2x^2 - 1x)^2 - 2(-2x^2 - 1x))^2 + 1(-2(-2x^2 - 1x)^2 - 2(-2x^2 - 1x)) + 3) - 3) + 4) - 3| and g(x) = 5(-2|(3(3|((2x - 3)/(3x - 2)) - 2| + 1)^2 - 5(3|((2x - 3)/(3x - 2)) - 2| + 1) - 4) + 2| + 3)^2 + 2(-2|(3(3|((2... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-2|(1(2|(5(-2sin(3πx + 2) + 3)^2 - 5(-2sin(3πx + 2) + 3))| - 2) + 4) - 3| + 3) - 2)/(1(-2|(1(2|(5(-2sin(3πx + 2) + 3)^2 - 5(-2sin(3πx + 2) + 3))| - 2) + 4) - 3| + 3)) and g(x) = -1e^(1(-2|(-1|(-1(2|x + 1| + 2) + 4) - 3| + 2) - 3| + 2)) + 2 intersec... | 32 | MATH |
[
{
"content": "How many solutions does the equation -4(3(-2cos(2π(-5(5(-4x + 3)^2 + 5(-4x + 3) - 3) + 3) + 2) - 2) + 3)^2 + 4(3(-2cos(2π(-5(5(-4x + 3)^2 + 5(-4x + 3) - 3) + 3) + 2) - 2) + 3) - 2 = -3cos(3π((-3(2(-3|(1|x - 3| + 3) - 1| + 1)^2 + 4(-3|(1|x - 3| + 3) - 1| + 1) - 3))/(-1(2(-3|(1|x - 3| + 3) - 1| + 1)... | 179 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(1|(3((1(-5(-2|x + 1|)^2 + 2(-2|x + 1|) - 1))/(3(-5(-2|x + 1|)^2 + 2(-2|x + 1|) - 1) + 2))^2 + 2((1(-5(-2|x + 1|)^2 + 2(-2|x + 1|) - 1))/(3(-5(-2|x + 1|)^2 + 2(-2|x + 1|) - 1) + 2))) - 2| + 2) + 1 and g(x) = -2|(-2|(3((3(-3... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-2((-2(-3|((3(2|x + 1|) - 2)/(2(2|x + 1|) + 3))| + 3) + 3)/(-2(-3|((3(2|x + 1|) - 2)/(2(2|x + 1|) + 3))| + 3) + 1)) - 2)^2 + 2(-2((-2(-3|((3(2|x + 1|) - 2)/(2(2|x + 1|) + 3))| + 3) + 3)/(-2(-3|((3(2|x + 1|) - 2)/(2(2|x + 1|) + 3))| + 3) + 1)) - 2) + ... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1|(4(-4(-2|(3(-3x^2 - 2x + 2)^2 + 2) - 2|)^2 - 3(-2|(3(-3x^2 - 2x + 2)^2 + 2) - 2|) - 4) + 4) + 2| + 3 = -3(-1(1cos(2π(-3((2x - 3)/(-3x - 2))^2 + 5((2x - 3)/(-3x - 2)) + 2) + 3) - 2)^2 - 1(1cos(2π(-3((2x - 3)/(-3x - 2))^2 + 5((2x - 3)/(-3x - 2)) + 2) + 3) - 2))... | 16 | MATH |
[
{
"content": "How many solutions does the equation -2|(3(1|(-2|((-1(2x + 1) - 2)/(-3(2x + 1) + 1)) + 1| + 2) + 2| - 3)^2 - 4(1|(-2|((-1(2x + 1) - 2)/(-3(2x + 1) + 1)) + 1| + 2) + 2| - 3) - 2) + 2| - 2 = 4((-1(-2|((-2(-4x - 3))/(1(-4x - 3) + 3)) - 2| + 3) + 2)/(3(-2|((-2(-4x - 3))/(1(-4x - 3) + 3)) - 2| + 3) - 2... | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-1|(1|(2(2((-3x)/(-2x - 1))^2 - 3((-3x)/(-2x - 1)) - 1)^2 + 1(2((-3x)/(-2x - 1))^2 - 3((-3x)/(-2x - 1)) - 1) - 2) - 1| - 1) - 3|) + 3 and g(x) = 1|((1(-1(-2((1x + 2)/(2x - 1))^2 + 3((1x + 2)/(2x - 1)) + 5) - 5) + 1)/(3(-1(-2((1x + 2)/(2x - 1))^2 + 3(... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-1(-2cos(1π(4(1|(3x) + 2| + 3) + 5) + 3) - 1))^2 - 3 and g(x) = 4(2sin(1π(1|((-2(-1|x - 3| - 3) + 1)/(3(-1|x - 3| - 3) + 3)) - 1| - 1))) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 46 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(1(-1sin(3π(1(-5(1|x + 1| + 1)^2 + 2(1|x + 1| + 1) - 5) + 4) - 3) - 3) - 2) - 2| + 3 and g(x) = -2(1|(-5(-3sin(2π(3x^2 + 5x - 4) + 3) - 2)^2 + 5(-3sin(2π(3x^2 + 5x - 4) + 3) - 2) + 1) - 3| + 1)^2 + 3(1|(-5(-3sin(2π(3x^2 + 5x - 4... | 855 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(2|((3(-5(-3|(3|x - 3| - 1) + 1| + 1) + 3) - 1)/(2(-5(-3|(3|x - 3| - 1) + 1| + 1) + 3) - 1))|))/(1(2|((3(-5(-3|(3|x - 3| - 1) + 1| + 1) + 3) - 1)/(2(-5(-3|(3|x - 3| - 1) + 1| + 1) + 3) - 1))|) + 1) and g(x) = -3((2(4(1cos(... | 80 | MATH |
[
{
"content": "How many solutions does the equation 3e^(2(-4(-2(-1|(-4(-1x + 2)^2 - 1(-1x + 2) - 3) - 2| + 2) - 3)^2 - 4(-2(-1|(-4(-1x + 2)^2 - 1(-1x + 2) - 3) - 2| + 2) - 3) - 4)) - 3 = -4(3(3cos(1π(-4(-1|x - 2| + 1) - 2) - 3) + 1) - 2) - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex fo... | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(1|(2((2(3|(5x^2 - 2) - 3| + 2))/(-2(3|(5x^2 - 2) - 3| + 2))) + 1) - 1| + 1)^2 - 3(1|(2((2(3|(5x^2 - 2) - 3| + 2))/(-2(3|(5x^2 - 2) - 3| + 2))) + 1) - 1| + 1) + 5 and g(x) = (1(1|(1cos(3π(-3(3|x| - 2) + 1) + 3) - 1) - 2| - ... | 116 | MATH |
[
{
"content": "How many solutions does the equation (2(2sin(1π(3(1|((-1(1|x - 2| + 3) - 2)/(-1(1|x - 2| + 3) + 2)) + 1| + 2) - 1) - 3) - 2) + 2)/(3(2sin(1π(3(1|((-1(1|x - 2| + 3) - 2)/(-1(1|x - 2| + 3) + 2)) + 1| + 2) - 1) - 3) - 2) - 1) = (2(2|((1(3|((-1x)/(-1x - 3))| - 2))/(-2(3|((-1x)/(-1x - 3))| - 2) + 1))| ... | 10 | MATH |
[
{
"content": "How many solutions does the equation 1(5(-1(1(-3|(3|x + 2| - 3) - 3| - 1)^2 + 4(-3|(3|x + 2| - 3) - 3| - 1) - 4) + 1) - 4)^2 + 5(5(-1(1(-3|(3|x + 2| - 3) - 3| - 1)^2 + 4(-3|(3|x + 2| - 3) - 3| - 1) - 4) + 1) - 4) - 1 = 2((-2(-2cos(2π(-5(1x^2 + 3x - 1)^2 + 4(1x^2 + 3x - 1) - 4) - 3) + 2) - 3)/(1(-2... | 11 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(-2sin(2π(5(-5(-1|(-1x + 1) + 1| - 1)^2 - 2(-1|(-1x + 1) + 1| - 1) - 5) - 5) - 1) + 2)^2 + 5(-2sin(2π(5(-5(-1|(-1x + 1) + 1| - 1)^2 - 2(-1|(-1x + 1) + 1| - 1) - 5) - 5) - 1) + 2) + 4 and g(x) = -2e^(-1(2((-1((-2(-5x - 5))/(... | 1566 | MATH |
[
{
"content": "How many solutions does the equation 4(2|(-2(4(-1|(1|x + 1| - 2) - 3|) + 4) + 3) - 1| - 1)^2 - 1(2|(-2(4(-1|(1|x + 1| - 2) - 3|) + 4) + 3) - 1| - 1) = -4(-1sin(2π(4(-1(-1x + 2)^2 + 4(-1x + 2) - 1)^2 + 2(-1(-1x + 2)^2 + 4(-1x + 2) - 1) + 2) - 1) - 1) - 3 have for x in the interval [-10, 10]?\n\nPre... | 90 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3e^(2((-1(-1(4(2|(2|x + 1| - 3) - 1| + 3)^2 + 4(2|(2|x + 1| - 3) - 1| + 3) - 4)^2 - 1(4(2|(2|x + 1| - 3) - 1| + 3)^2 + 4(2|(2|x + 1| - 3) - 1| + 3) - 4) - 4) - 3)/(-1(-1(4(2|(2|x + 1| - 3) - 1| + 3)^2 + 4(2|(2|x + 1| - 3) - ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(1|(-5((3(3sin(3π(-1x - 3) - 3) + 1))/(-3(3sin(3π(-1x - 3) - 3) + 1) - 2))^2 + 4((3(3sin(3π(-1x - 3) - 3) + 1))/(-3(3sin(3π(-1x - 3) - 3) + 1) - 2)) + 2) - 2| - 3) + 1 and g(x) = -2cos(1π(5((-3(2|(-3|x| - 1) + 2| - 2))/(-3(2... | 120 | MATH |
[
{
"content": "How many solutions does the equation -2(3|(-2|((-1(1(-3x - 3) + 5) - 3)/(-1(1(-3x - 3) + 5) - 1)) + 3| + 1) - 1| - 3) + 5 = -2(2(2|(-2|(1x^2 + 5x) - 3| + 3)|) - 4) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(1|(-1|(-1|(-4(-3x^2 - 2x + 5)) + 3| - 2) + 1|) + 2| - 3) and g(x) = -1(-5(-3sin(2π((-2(5x - 5) + 2)/(-1(5x - 5) + 2)) - 3) + 3) + 5)^2 + 1(-5(-3sin(2π((-2(5x - 5) + 2)/(-1(5x - 5) + 2)) - 3) + 3) + 5) + 4 where -10 ≤ x ≤ 10... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4((-2((2(-4(-2|(-3x + 3) + 3| - 2) - 4) - 1)/(3(-4(-2|(-3x + 3) + 3| - 2) - 4) + 1)) - 1)/(2((2(-4(-2|(-3x + 3) + 3| - 2) - 4) - 1)/(3(-4(-2|(-3x + 3) + 3| - 2) - 4) + 1)) + 3)) - 1 and g(x) = (-1(1|(-2|(4(1sin(2πx + 1) + 2) - 1) ... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-2|((2(2|(2e^(-2(1x - 3))) + 2|) - 3)/(2(2|(2e^(-2(1x - 3))) + 2|) - 1)) - 1|) + 4 and g(x) = -1|(-1(-2|(3(-1|x - 3| - 1)^2 + 5(-1|x - 3| - 1) + 4) - 2| + 2)^2 - 4(-2|(3(-1|x - 3| - 1)^2 + 5(-1|x - 3| - 1) + 4) - 2| + 2) - 3) + 1| + 3 intersect in th... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1((3((3((2(3cos(2π((2x - 1)/(1x)) + 1) - 1) + 2)/(1(3cos(2π((2x - 1)/(1x)) + 1) - 1))) - 1)/(-1((2(3cos(2π((2x - 1)/(1x)) + 1) - 1) + 2)/(1(3cos(2π((2x - 1)/(1x)) + 1) - 1))) - 3)) + 2)/(1((3((2(3cos(2π((2x - 1)/(1x)) + 1) - 1) + ... | 102 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(-3((-3(-2|(3|(1|x|) + 2| + 1) + 3| - 2) + 1)/(1(-2|(3|(1|x|) + 2| + 1) + 3| - 2)))^2 + 2((-3(-2|(3|(1|x|) + 2| + 1) + 3| - 2) + 1)/(1(-2|(3|(1|x|) + 2| + 1) + 3| - 2))) + 2) - 1| and g(x) = 1(2((2(2|(2cos(1πx - 3) - 1) - 2| - 3) + 3)/(3(2|(2cos(1πx ... | 40 | MATH |
[
{
"content": "How many solutions does the equation 5(-2((-3((-1(3cos(2π(3x + 1) + 3) - 3) - 2)/(-3(3cos(2π(3x + 1) + 3) - 3) - 2)) - 1)/(-3((-1(3cos(2π(3x + 1) + 3) - 3) - 2)/(-3(3cos(2π(3x + 1) + 3) - 3) - 2)) + 2))^2 + 3((-3((-1(3cos(2π(3x + 1) + 3) - 3) - 2)/(-3(3cos(2π(3x + 1) + 3) - 3) - 2)) - 1)/(-3((-1(3... | 14 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|((3(2|(1sin(3π(-3(-1x - 3) - 5) - 2) + 1) + 1|) - 3)/(1(2|(1sin(3π(-3(-1x - 3) - 5) - 2) + 1) + 1|) + 1)) + 3| - 1 and g(x) = 1|(2(1|(-1|(-5x - 4)| + 3) + 2| - 1)) - 1| + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTe... | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(-1(-2e^(-1(-1|(-4(2|x - 2| - 1) + 3) + 2| + 1)))) - 5 and g(x) = 2(2|(-2cos(2π((-3(5x^2 - 5x - 4))/(-1(5x^2 - 5x - 4) - 2)) - 1) + 3) - 3| + 1)^2 - 4(2|(-2cos(2π((-3(5x^2 - 5x - 4))/(-1(5x^2 - 5x - 4) - 2)) - 1) + 3) - 3| +... | 24 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(-1|(1(-2|(1|(3x^2 - 2x - 5)| - 1) + 2| + 2)^2 - 2(-2|(1|(3x^2 - 2x - 5)| - 1) + 2| + 2) - 4) - 1| + 3) - 1| + 2 and g(x) = -1(-1(-1|((-1(-4x^2 + 5x + 2) + 2)/(2(-4x^2 + 5x + 2) - 1)) + 1| + 2)^2 + 4(-1|((-1(-4x^2 + 5x + 2) + 2)/(2(-4x^2 + 5x + 2) -... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(3π(-3|(1(-1|(-2(-5x - 4)^2 - 3(-5x - 4) - 4) - 3| + 3)^2 + 1(-1|(-2(-5x - 4)^2 - 3(-5x - 4) - 4) - 3| + 3) + 2) + 3| + 2) - 3) + 1 and g(x) = 1|(-4(2|((1(-1x^2 + 4) + 3)/(1(-1x^2 + 4) - 2)) + 3|)^2 + 3(2|((1(-1x^2 + 4) + 3)/(1(-1x^2 + 4) - 2)) + 3... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(-3sin(2π(2(5(1(-3|x + 2| - 1) - 4) - 4)^2 - 2(5(1(-3|x + 2| - 1) - 4) - 4) - 1) + 2) - 1) - 1)/(1(-3sin(2π(2(5(1(-3|x + 2| - 1) - 4) - 4)^2 - 2(5(1(-3|x + 2| - 1) - 4) - 4) - 1) + 2) - 1)) and g(x) = 1(5(4((1((3x - 2)/(-1x - 1)... | 23 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|((-2(-5(-1(3|(-1cos(3πx - 3) + 2) + 1| - 2) + 4)^2 + 5(-1(3|(-1cos(3πx - 3) + 2) + 1| - 2) + 4) - 1) - 1)/(2(-5(-1(3|(-1cos(3πx - 3) + 2) + 1| - 2) + 4)^2 + 5(-1(3|(-1cos(3πx - 3) + 2) + 1| - 2) + 4) - 1) + 3)) - 3| + 1 and g(x)... | 21 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-3((-1((-1(1cos(1π(4x^2 + 3x - 4) + 2) - 3) - 3)/(1(1cos(1π(4x^2 + 3x - 4) + 2) - 3) - 3)) - 2)/(1((-1(1cos(1π(4x^2 + 3x - 4) + 2) - 3) - 3)/(1(1cos(1π(4x^2 + 3x - 4) + 2) - 3) - 3)) + 2)) + 5) + 3)/(3(-3((-1((-1(1cos(1π(4x^2 +... | 1227 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2((1(-1((2(-1sin(3π((2x + 3)/(1x + 3)) - 1) - 1) + 3)/(2(-1sin(3π((2x + 3)/(1x + 3)) - 1) - 1) - 3)) - 5) - 1)/(-3(-1((2(-1sin(3π((2x + 3)/(1x + 3)) - 1) - 1) + 3)/(2(-1sin(3π((2x + 3)/(1x + 3)) - 1) - 1) - 3)) - 5) - 1)) + 5 and g(x) = -1(-3(-4(1|(3x ... | 4 | MATH |
[
{
"content": "How many solutions does the equation (2(-2(5(5(5(-3cos(1πx + 1)) - 1) + 5)^2 + 5(5(5(-3cos(1πx + 1)) - 1) + 5) + 2) - 3) - 2)/(-2(-2(5(5(5(-3cos(1πx + 1)) - 1) + 5)^2 + 5(5(5(-3cos(1πx + 1)) - 1) + 5) + 2) - 3) - 2) = (3(-3(-1cos(1π(-2|(3x - 2) + 2| - 1) - 1) - 1)^2 + 3(-1cos(1π(-2|(3x - 2) + 2| -... | 119 | MATH |
[
{
"content": "How many solutions does the equation 2|(3(-1(5((3(-5x^2 + 4) + 1)/(1(-5x^2 + 4) - 3)) + 1) - 5)^2 - 4(-1(5((3(-5x^2 + 4) + 1)/(1(-5x^2 + 4) - 3)) + 1) - 5) - 5) + 3| - 3 = (1(-1|(-3|(1cos(1π(-5x) + 3) + 3) - 1|) + 2| + 2) + 1)/(-1(-1|(-3|(1cos(1π(-5x) + 3) + 3) - 1|) + 2| + 2) - 1) have for x in t... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3sin(1π((-3(5((2(2|(2x^2 - 5x + 4) - 1| - 2) - 3)/(2(2|(2x^2 - 5x + 4) - 1| - 2)))^2 - 1((2(2|(2x^2 - 5x + 4) - 1| - 2) - 3)/(2(2|(2x^2 - 5x + 4) - 1| - 2))) - 3) + 2)/(2(5((2(2|(2x^2 - 5x + 4) - 1| - 2) - 3)/(2(2|(2x^2 - 5x ... | 11 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-2|(-2sin(3π(3(5(-1|x + 3| + 1) + 2) - 5) + 1) + 2) + 1|) + 1 and g(x) = -3|(2|(-2sin(3π((2((3x + 1)/(-3x + 4)) + 3)/(2((3x + 1)/(-3x + 4)) + 1)) - 2) + 1) - 1| + 3) - 1| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format:... | 807 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-3sin(3π(1(-5((1(5x^2 - 4x + 1) + 1)/(-1(5x^2 - 4x + 1) + 2))^2 - 1((1(5x^2 - 4x + 1) + 1)/(-1(5x^2 - 4x + 1) + 2)) + 2)^2 - 2(-5((1(5x^2 - 4x + 1) + 1)/(-1(5x^2 - 4x + 1) + 2))^2 - 1((1(5x^2 - 4x + 1) + 1)/(-1(5x^2 - 4x +... | 134 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(2(5(-4(2cos(2π(5x^2 + 1x + 3) + 1) - 1) + 5) - 1)^2 + 4(5(-4(2cos(2π(5x^2 + 1x + 3) + 1) - 1) + 5) - 1) - 5) - 1)/(1(2(5(-4(2cos(2π(5x^2 + 1x + 3) + 1) - 1) + 5) - 1)^2 + 4(5(-4(2cos(2π(5x^2 + 1x + 3) + 1) - 1) + 5) - 1) - 5) - 3) and g(x) = -2((3(... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2e^(-2(-1|(2|(3((2(1|x + 3|))/(2(1|x + 3|) + 3)) - 3) + 1|) - 2| - 1)) + 1 and g(x) = -3(3|(-2sin(2π(4((1x + 1)/(-3x)) - 5) + 3) + 1) + 2| + 3)^2 - 3(3|(-2sin(2π(4((1x + 1)/(-3x)) - 5) + 3) + 1) + 2| + 3) - 2 intersect in the interval [-10, 10].\n\nPr... | 56 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(2(-1((-1(-3(1|x - 2| + 2)^2 - 4(1|x - 2| + 2) - 1) + 2)/(-1(-3(1|x - 2| + 2)^2 - 4(1|x - 2| + 2) - 1) + 3)) - 1) - 3)| - 3 and g(x) = (3((-2(2(4(-2cos(1πx + 2) - 1) + 5)^2 - 2(4(-2cos(1πx + 2) - 1) + 5) - 5))/(-1(2(4(-2cos(1πx ... | 40 | MATH |
[
{
"content": "How many solutions does the equation 1((2(2|(-2(-3cos(3π(-5x^2 + 1x + 5) + 3) + 1)^2 - 5(-3cos(3π(-5x^2 + 1x + 5) + 3) + 1) + 5) + 2|) + 2)/(2(2|(-2(-3cos(3π(-5x^2 + 1x + 5) + 3) + 1)^2 - 5(-3cos(3π(-5x^2 + 1x + 5) + 3) + 1) + 5) + 2|) + 3)) + 4 = 1|((1(-2|(-2((-1x - 2)/(-2x - 1)) + 3)| + 3) - 3)/... | 9 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3((-2((-1(-1(-4(-4x - 5))^2 + 5(-4(-4x - 5)) - 3) - 1)/(-1(-1(-4(-4x - 5))^2 + 5(-4(-4x - 5)) - 3))) - 3)/(2((-1(-1(-4(-4x - 5))^2 + 5(-4(-4x - 5)) - 3) - 1)/(-1(-1(-4(-4x - 5))^2 + 5(-4(-4x - 5)) - 3))) - 3))^2 - 5((-2((-1(-1(-4(-4x - 5))^2 + 5(-4(-4x... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3sin(3π(2(1(1(-2(2|x|)^2 + 3(2|x|) - 3)^2 + 2(-2(2|x|)^2 + 3(2|x|) - 3) + 2)^2 - 3(1(-2(2|x|)^2 + 3(2|x|) - 3)^2 + 2(-2(2|x|)^2 + 3(2|x|) - 3) + 2) + 2) + 2) + 1) + 1 and g(x) = -2((1((-1((2(2|x + 1|) - 1)/(2(2|x + 1|))) + 3... | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(2(5(-1e^(-2(4(-1x^2 - 1x + 5))) - 2) - 5) + 4) - 1| - 2 and g(x) = 3(-4(-5((2(-3x^2 - 2x + 4) + 2)/(-1(-3x^2 - 2x + 4) + 2))^2 + 2((2(-3x^2 - 2x + 4) + 2)/(-1(-3x^2 - 2x + 4) + 2)))^2 + 1(-5((2(-3x^2 - 2x + 4) + 2)/(-1(-3... | 6 | MATH |
[
{
"content": "How many solutions does the equation (-1(2|(-1(4((2(1x - 5) - 2)/(-2(1x - 5) + 3)) - 2) - 4)| + 3) + 1)/(2(2|(-1(4((2(1x - 5) - 2)/(-2(1x - 5) + 3)) - 2) - 4)| + 3)) = -4((-2(3|(-3cos(2π((-1x + 1)/(-1x)) - 3) + 2) - 3| - 1) - 1)/(2(3|(-3cos(2π((-1x + 1)/(-1x)) - 3) + 2) - 3| - 1) + 2)) + 2 have fo... | 17 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3((1(-1|(-2sin(3π((3(3x + 5) - 1)/(-1(3x + 5))) - 3) + 3)| + 2))/(-3(-1|(-2sin(3π((3(3x + 5) - 1)/(-1(3x + 5))) - 3) + 3)| + 2) + 4)) + 1 and g(x) = -1((2((3(2|(3x^2 - 2) - 1| + 1) - 3)/(2(2|(3x^2 - 2) - 1| + 1) + 1)) - 3)/(... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(-1(-1(1|(5(-4x + 2) + 2) - 3| - 1)^2 + 2) + 2) + 2 and g(x) = 2cos(1π((3(-1(5(3x + 3) - 2)^2 + 3(5(3x + 3) - 2)))/(-1(-1(5(3x + 3) - 2)^2 + 3(5(3x + 3) - 2)) + 3)) - 2) + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex... | 4 | MATH |
[
{
"content": "How many solutions does the equation 3|(1(-4(-3|(1|((2x + 3)/(-2x + 1)) - 2|) - 2|)^2 - 2(-3|(1|((2x + 3)/(-2x + 1)) - 2|) - 2|) + 5) - 5) + 1| - 2 = -1sin(1π(1|(3(-3|((2x + 1)/(1x - 2)) + 3| - 3) - 1)| - 1) - 1) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\bo... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1((-1(-5(-1(2(-3|x + 1|)^2 - 5(-3|x + 1|) - 5)^2 + 3(2(-3|x + 1|)^2 - 5(-3|x + 1|) - 5) - 2)^2 + 5(-1(2(-3|x + 1|)^2 - 5(-3|x + 1|) - 5)^2 + 3(2(-3|x + 1|)^2 - 5(-3|x + 1|) - 5) - 2) + 2))/(-2(-5(-1(2(-3|x + 1|)^2 - 5(-3|x + ... | 120 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|((3(3(-1cos(2π((2(-1|x + 1| - 2) - 1)/(1(-1|x + 1| - 2) + 3)) + 1))^2 + 4(-1cos(2π((2(-1|x + 1| - 2) - 1)/(1(-1|x + 1| - 2) + 3)) + 1)) + 3) + 2)/(2(3(-1cos(2π((2(-1|x + 1| - 2) - 1)/(1(-1|x + 1| - 2) + 3)) + 1))^2 + 4(-1c... | 22 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(1((1(-1|(1|(4x - 5) - 3| - 2) + 3| - 2) - 2)/(-1(-1|(1|(4x - 5) - 3| - 2) + 3| - 2) - 1)) - 3)^2 - 2(1((1(-1|(1|(4x - 5) - 3| - 2) + 3| - 2) - 2)/(-1(-1|(1|(4x - 5) - 3| - 2) + 3| - 2) - 1)) - 3) - 2 and g(x) = -5(-1(3|(-3(... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|((1(1((2((2((3x + 3)/(1x + 3)) - 1)/(-2((3x + 3)/(1x + 3)) - 3)) + 1)/(-1((2((3x + 3)/(1x + 3)) - 1)/(-2((3x + 3)/(1x + 3)) - 3)) + 2))^2 + 2((2((2((3x + 3)/(1x + 3)) - 1)/(-2((3x + 3)/(1x + 3)) - 3)) + 1)/(-1((2((3x + 3)/(... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(3(-1|((1((-2(-2|x - 2| + 3) + 3)/(1(-2|x - 2| + 3) + 3)) - 2)/(-3((-2(-2|x - 2| + 3) + 3)/(1(-2|x - 2| + 3) + 3)))) + 2| + 3)^2 + 1(-1|((1((-2(-2|x - 2| + 3) + 3)/(1(-2|x - 2| + 3) + 3)) - 2)/(-3((-2(-2|x - 2| + 3) + 3)/(1... | 10 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-1|(-2|(-3|(3(-1|x + 3| + 2) + 5) - 2| + 3) - 3|) + 2| - 3)^2 - 1(-1|(-2|(-3|(3(-1|x + 3| + 2) + 5) - 2| + 3) - 3|) + 2| - 3) + 4 and g(x) = 4(5(1e^(-1((-1(2|x + 2| + 2) - 1)/(2(2|x + 2| + 2) + 2))))^2 + 4(1e^(-1((-1(2|x + 2| + 2) - 1)/(2(2|x + 2| + ... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1e^(-1(-2((-2(3((1(2|x + 1| - 3) - 1)/(1(2|x + 1| - 3))) + 1) - 3)/(3(3((1(2|x + 1| - 3) - 1)/(1(2|x + 1| - 3))) + 1)))^2 - 1((-2(3((1(2|x + 1| - 3) - 1)/(1(2|x + 1| - 3))) + 1) - 3)/(3(3((1(2|x + 1| - 3) - 1)/(1(2|x + 1| - 3))) +... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(-4(-2(1|(-1(1x + 3)^2 - 1(1x + 3)) + 3| - 2) - 2) - 5) + 3| + 1 and g(x) = -1(-1|(3(5(-1x - 2) + 3) - 3)| + 2) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(2(3|(-1|(3(-3|x| - 3) - 2) + 3| - 2) - 1| + 1)^2 - 1(3|(-1|(3(-3|x| - 3) - 2) + 3| - 2) - 1| + 1) - 5) - 2)/(3(2(3|(-1|(3(-3|x| - 3) - 2) + 3| - 2) - 1| + 1)^2 - 1(3|(-1|(3(-3|x| - 3) - 2) + 3| - 2) - 1| + 1) - 5) + 3) and... | 85 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4((-1(-1((2(2|(-3|x + 1| + 3) - 3| + 3) + 1)/(-1(2|(-3|x + 1| + 3) - 3| + 3) - 1))^2 + 1((2(2|(-3|x + 1| + 3) - 3| + 3) + 1)/(-1(2|(-3|x + 1| + 3) - 3| + 3) - 1)) + 3) + 3)/(-3(-1((2(2|(-3|x + 1| + 3) - 3| + 3) + 1)/(-1(2|(-3|x + 1| + 3) - 3| + 3) - 1)... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2((-3(3((2((-1(-3|x - 1| - 2))/(2(-3|x - 1| - 2) - 3)) - 1)/(-3((-1(-3|x - 1| - 2))/(2(-3|x - 1| - 2) - 3)) - 2)) + 5) - 2)/(1(3((2((-1(-3|x - 1| - 2))/(2(-3|x - 1| - 2) - 3)) - 1)/(-3((-1(-3|x - 1| - 2))/(2(-3|x - 1| - 2) - 3)) - 2)) + 5) - 3)) - 2)... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1|(-2|((-3((-1(-2((-2x - 3)/(2x + 2)) - 4) + 2)/(-2(-2((-2x - 3)/(2x + 2)) - 4) + 3)))/(-3((-1(-2((-2x - 3)/(2x + 2)) - 4) + 2)/(-2(-2((-2x - 3)/(2x + 2)) - 4) + 3)) - 1)) - 3| - 2) - 1| - 1 = -4(-3(-3|(2|(1x^2 + 4x + 4) - 2| + 1) - 1| + 1)^2 - 2(-3|(2|(1x^2 + ... | 6 | MATH |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.