messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation (1((1(-1(-4(1(1x + 3) - 5) - 2)^2 + 2(-4(1(1x + 3) - 5) - 2) + 2) + 1)/(3(-1(-4(1(1x + 3) - 5) - 2)^2 + 2(-4(1(1x + 3) - 5) - 2) + 2) - 1)) + 1)/(-1((1(-1(-4(1(1x + 3) - 5) - 2)^2 + 2(-4(1(1x + 3) - 5) - 2) + 2) + 1)/(3(-1(-4(1(1x + 3) - 5) - 2)^2 + 2(-4(1(1x +... | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(-2(-1|(-1(3|(-3x^2 - 2x + 4) + 1| + 1)^2 + 2(3|(-3x^2 - 2x + 4) + 1| + 1) - 2) - 2| - 3) - 3) - 1| + 3 and g(x) = 3(2(1sin(3π(5(2x^2 + 3x + 2)^2 + 1(2x^2 + 3x + 2) + 3)) + 1) + 5)^2 + 4 where -10 ≤ x ≤ 10.\n\nPresent the a... | 39 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-1|((1((-1(-4(-3x^2 - 5x - 2) + 4) + 3)/(-1(-4(-3x^2 - 5x - 2) + 4) - 3)))/(2((-1(-4(-3x^2 - 5x - 2) + 4) + 3)/(-1(-4(-3x^2 - 5x - 2) + 4) - 3)) - 1)) + 3| - 1) - 3| - 3 and g(x) = -2|((2(1|(-2cos(1π(5x^2 - 1x + 1) - 3) - 2)|) - 1)/(-2(1|(-2cos(1π(... | 481 | MATH |
[
{
"content": "How many solutions does the equation 5(-1e^(1((-3(4(3|(3x^2 - 5x + 5) - 1| + 2) + 3) - 2)/(3(4(3|(3x^2 - 5x + 5) - 1| + 2) + 3)))) + 2) - 1 = (-1(-2sin(2π(-1|(-5(-1x^2 + 1x - 2) - 5) + 3| - 2) + 1) - 2))/(1(-2sin(2π(-1|(-5(-1x^2 + 1x - 2) - 5) + 3| - 2) + 1) - 2) + 1) have for x in the interval [-... | 157 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-4(-3(-3cos(1π(-3|((-2x + 2)/(-2x - 3)) + 3| + 1) - 3)) - 3) - 1) + 4 and g(x) = (-2((-2((1(2e^(2(5x + 5)) + 2) - 1)/(2(2e^(2(5x + 5)) + 2) - 1)) - 1)/(-2((1(2e^(2(5x + 5)) + 2) - 1)/(2(2e^(2(5x + 5)) + 2) - 1)))) - 1)/(-3((-2((1... | 99 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(1(3((-3(-5(2x^2 - 2x + 1)^2 + 1(2x^2 - 2x + 1) - 5) + 3)/(3(-5(2x^2 - 2x + 1)^2 + 1(2x^2 - 2x + 1) - 5) - 1)) + 3) - 5)^2 + 5(1(3((-3(-5(2x^2 - 2x + 1)^2 + 1(2x^2 - 2x + 1) - 5) + 3)/(3(-5(2x^2 - 2x + 1)^2 + 1(2x^2 - 2x + 1... | 5 | MATH |
[
{
"content": "How many solutions does the equation -3|((2(-1|(-1|(1(2sin(2πx) - 2)^2 - 2(2sin(2πx) - 2) - 3) + 3| + 3) + 1| + 1) + 1)/(2(-1|(-1|(1(2sin(2πx) - 2)^2 - 2(2sin(2πx) - 2) - 3) + 3| + 3) + 1| + 1) - 3)) + 2| + 3 = 4(1sin(1π(3(-1|(-3|x - 3|) + 3| + 1) + 2)) + 2)^2 - 1(1sin(1π(3(-1|(-3|x - 3|) + 3| + 1... | 18 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(3((-2(-3(-4(2cos(1πx + 1) + 3) - 4)^2 - 3(-4(2cos(1πx + 1) + 3) - 4)) + 1)/(3(-3(-4(2cos(1πx + 1) + 3) - 4)^2 - 3(-4(2cos(1πx + 1) + 3) - 4)) + 2))^2 - 2((-2(-3(-4(2cos(1πx + 1) + 3) - 4)^2 - 3(-4(2cos(1πx + 1) + 3) - 4)) +... | 118 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|((-3((-1((-3(4((-3x)/(-2x - 3))) + 2)/(-1(4((-3x)/(-2x - 3))) + 2)))/(2((-3(4((-3x)/(-2x - 3))) + 2)/(-1(4((-3x)/(-2x - 3))) + 2)) + 3)) - 1)/(3((-1((-3(4((-3x)/(-2x - 3))) + 2)/(-1(4((-3x)/(-2x - 3))) + 2)))/(2((-3(4((-3x)/(-2x... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(-5(2|(2(-3cos(3π(-1x^2 + 1) - 1) - 3) - 1) + 1|)^2 - 5(2|(2(-3cos(3π(-1x^2 + 1) - 1) - 3) - 1) + 1|) - 1) + 2| + 2 and g(x) = (-1(3sin(1π(-2(-1(-4x^2 - 1x)^2 - 3(-4x^2 - 1x) + 4) + 4) - 2) - 3) - 1)/(3(3sin(1π(-2(-1(-4x^2 - 1x)... | 388 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1cos(2π(-1(3((2((-3(2x^2 + 3x - 2))/(-2(2x^2 + 3x - 2) + 1)))/(-2((-3(2x^2 + 3x - 2))/(-2(2x^2 + 3x - 2) + 1)) + 3)) - 1)^2 + 1(3((2((-3(2x^2 + 3x - 2))/(-2(2x^2 + 3x - 2) + 1)))/(-2((-3(2x^2 + 3x - 2))/(-2(2x^2 + 3x - 2) + 1)) + 3... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(-3(-3((-1(-5(-2|x + 3| + 3)^2 + 2(-2|x + 3| + 3) + 3) - 2)/(-2(-5(-2|x + 3| + 3)^2 + 2(-2|x + 3| + 3) + 3) - 1))^2 - 1((-1(-5(-2|x + 3| + 3)^2 + 2(-2|x + 3| + 3) + 3) - 2)/(-2(-5(-2|x + 3| + 3)^2 + 2(-2|x + 3| + 3) + 3) - ... | 4 | MATH |
[
{
"content": "How many solutions does the equation (1((2(3(-1(-1|(2|x + 1| + 1) - 2| - 3) + 5) - 1))/(-3(3(-1(-1|(2|x + 1| + 1) - 2| - 3) + 5) - 1) - 3)) - 3)/(-2((2(3(-1(-1|(2|x + 1| + 1) - 2| - 3) + 5) - 1))/(-3(3(-1(-1|(2|x + 1| + 1) - 2| - 3) + 5) - 1) - 3)) - 1) = -2(1|((3(-5(1|x + 3| - 2)) + 2)/(3(-5(1|x ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(-1(1|(-3|(1|(5x + 4) - 1| - 1)|) + 2| - 2)))/(3(-1(1|(-3|(1|(5x + 4) - 1| - 1)|) + 2| - 2)) - 2) and g(x) = -1|((-3(2|((-2(-4x^2 - 4x + 4) - 2)/(-2(-4x^2 - 4x + 4) - 2)) - 1| + 2) - 3)/(2(2|((-2(-4x^2 - 4x + 4) - 2)/(-2(-4x^2 - 4x + 4) - 2)) - 1| +... | 6 | MATH |
[
{
"content": "How many solutions does the equation 2cos(3π((-1((1(-4(2((-3x - 1)/(2x + 1)) - 2)^2 - 2) - 3)/(-3(-4(2((-3x - 1)/(2x + 1)) - 2)^2 - 2) + 4)) + 3)/(1((1(-4(2((-3x - 1)/(2x + 1)) - 2)^2 - 2) - 3)/(-3(-4(2((-3x - 1)/(2x + 1)) - 2)^2 - 2) + 4)) + 1)) + 3) - 2 = 1((-2(-4(3cos(2π((3x)/(3x - 2)) - 1) - 3... | 24 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(-3(-1(-4(1(-3x^2) + 5)^2 + 3(1(-3x^2) + 5) + 5)^2 - 1(-4(1(-3x^2) + 5)^2 + 3(1(-3x^2) + 5) + 5) - 1) - 4) - 3 and g(x) = (-3(1(4(-1(-2|x + 2| + 3) + 3) - 5) + 2) + 1)/(2(1(4(-1(-2|x + 2| + 3) + 3) - 5) + 2) - 3) in the range -10 ... | 9 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((-1((-3((1(3(2sin(2πx - 3) - 1) - 1) - 2)/(2(3(2sin(2πx - 3) - 1) - 1) + 1)) - 2)/(-1((1(3(2sin(2πx - 3) - 1) - 1) - 2)/(2(3(2sin(2πx - 3) - 1) - 1) + 1)) + 3)) + 1)/(-2((-3((1(3(2sin(2πx - 3) - 1) - 1) - 2)/(2(3(2sin(2πx - 3) - 1) - 1) + 1)) - 2)/(... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(1sin(2π((3(-5((-1(-2|x| + 2) - 2)/(-3(-2|x| + 2)))^2 - 4((-1(-2|x| + 2) - 2)/(-3(-2|x| + 2))) - 4) + 1)/(3(-5((-1(-2|x| + 2) - 2)/(-3(-2|x| + 2)))^2 - 4((-1(-2|x| + 2) - 2)/(-3(-2|x| + 2))) - 4) + 1)) - 3))^2 + 1(1sin(2π((3... | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2((3(-1(-1(1((-3x + 1)/(2x - 1))^2 + 4)^2 - 5(1((-3x + 1)/(2x - 1))^2 + 4) - 4)^2 + 4(-1(1((-3x + 1)/(2x - 1))^2 + 4)^2 - 5(1((-3x + 1)/(2x - 1))^2 + 4) - 4) - 4) - 3)/(3(-1(-1(1((-3x + 1)/(2x - 1))^2 + 4)^2 - 5(1((-3x + 1)/(2x - 1))^2 + 4) - 4)^2 + 4... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(2(-2|(-1(2(-1sin(1πx)) + 4) + 1) + 3| + 2)^2 - 3(-2|(-1(2(-1sin(1πx)) + 4) + 1) + 3| + 2) + 1)^2 + 5(2(-2|(-1(2(-1sin(1πx)) + 4) + 1) + 3| + 2)^2 - 3(-2|(-1(2(-1sin(1πx)) + 4) + 1) + 3| + 2) + 1) - 3 and g(x) = 1cos(1π(-1((1(2|(2|x - 2| + 3) + 3| - 2... | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-1|(5(1e^(1(-3|((2x + 3)/(2x - 1)) - 3| - 1)) - 3) - 1) - 2| - 3) - 2| + 3 and g(x) = 3(3((-2(3|(-2sin(3πx + 1) + 1) - 2| - 1))/(1(3|(-2sin(3πx + 1) + 1) - 2| - 1) + 1)) - 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex fo... | 120 | MATH |
[
{
"content": "How many solutions does the equation (3(-3cos(2π((1(5(3(-1|x| - 3) - 2)^2 + 5(3(-1|x| - 3) - 2) + 1) + 2)/(-2(5(3(-1|x| - 3) - 2)^2 + 5(3(-1|x| - 3) - 2) + 1) + 3)) - 1) - 1) + 1)/(1(-3cos(2π((1(5(3(-1|x| - 3) - 2)^2 + 5(3(-1|x| - 3) - 2) + 1) + 2)/(-2(5(3(-1|x| - 3) - 2)^2 + 5(3(-1|x| - 3) - 2) +... | 4 | MATH |
[
{
"content": "How many solutions does the equation (2((1(5((-2(-2|(-3x^2 - 2x - 4) + 2| + 1) + 3)/(2(-2|(-3x^2 - 2x - 4) + 2| + 1) - 3)) - 1) - 3)/(-3(5((-2(-2|(-3x^2 - 2x - 4) + 2| + 1) + 3)/(2(-2|(-3x^2 - 2x - 4) + 2| + 1) - 3)) - 1))) - 3)/(-3((1(5((-2(-2|(-3x^2 - 2x - 4) + 2| + 1) + 3)/(2(-2|(-3x^2 - 2x - 4... | 268 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(3(1(4(-1(-2x - 4) - 4)^2 + 1(-1(-2x - 4) - 4) - 4)^2 - 5(4(-1(-2x - 4) - 4)^2 + 1(-1(-2x - 4) - 4) - 4) + 2)^2 + 5(1(4(-1(-2x - 4) - 4)^2 + 1(-1(-2x - 4) - 4) - 4)^2 - 5(4(-1(-2x - 4) - 4)^2 + 1(-1(-2x - 4) - 4) - 4) + 2) ... | 13 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(1(-2|((2(1(2|x + 1| + 1) + 4) + 1)/(1(1(2|x + 1| + 1) + 4) + 3)) - 2| + 1) + 4)^2 + 2(1(-2|((2(1(2|x + 1| + 1) + 4) + 1)/(1(1(2|x + 1| + 1) + 4) + 3)) - 2| + 1) + 4) - 1 and g(x) = 3|((1((-3(-1(1x^2 + 4x - 1) - 4) + 1)/(-2(... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(3|(3|((-2(1(-4x^2 + 2x - 4)^2 + 5(-4x^2 + 2x - 4) - 5) + 3)/(1(1(-4x^2 + 2x - 4)^2 + 5(-4x^2 + 2x - 4) - 5) + 1)) - 3| + 1)| - 2) and g(x) = 4(-1cos(1π((1((2((-1x - 1)/(1x - 3)) - 3)/(-3((-1x - 1)/(1x - 3)) - 3)) + 2)/(2((2((-1x ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(1e^(2(-3|(-1(5((-2x - 3)/(2x + 2)) + 5)^2 + 5(5((-2x - 3)/(2x + 2)) + 5) + 4) + 2| + 1)) + 2) - 1| + 1 and g(x) = -5(-2e^(1(-1(2(2x^2 + 4x - 3) - 4)^2 + 2(2(2x^2 + 4x - 3) - 4) + 5)) - 2) - 4 where -10 ≤ x ≤ 10.\n\nPresent... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(1|(5(-4((-2((1x)/(-2x + 1)) - 2)/(1((1x)/(-2x + 1)) + 3))^2 - 3((-2((1x)/(-2x + 1)) - 2)/(1((1x)/(-2x + 1)) + 3)) - 4) + 5)| - 3)^2 - 3(1|(5(-4((-2((1x)/(-2x + 1)) - 2)/(1((1x)/(-2x + 1)) + 3))^2 - 3((-2((1x)/(-2x + 1)) - 2... | 16 | MATH |
[
{
"content": "How many solutions does the equation -1sin(1π(-1|((-2(-5(-1(-1|x| - 3)^2 - 4(-1|x| - 3) + 3)^2 + 4(-1(-1|x| - 3)^2 - 4(-1|x| - 3) + 3)) - 3)/(3(-5(-1(-1|x| - 3)^2 - 4(-1|x| - 3) + 3)^2 + 4(-1(-1|x| - 3)^2 - 4(-1|x| - 3) + 3)))) + 1| + 1)) = 2|((1(3|(1(2x - 1)^2 + 4(2x - 1) - 3) + 3| - 3) + 2)/(3(3... | 25 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3((-3(-1|(-5((2((-1x - 2)/(1x - 2)) + 3)/(-3((-1x - 2)/(1x - 2)) + 1)) + 3) - 3| - 1) - 3)/(1(-1|(-5((2((-1x - 2)/(1x - 2)) + 3)/(-3((-1x - 2)/(1x - 2)) + 1)) + 3) - 3| - 1) + 2)) + 2)/(1((-3(-1|(-5((2((-1x - 2)/(1x - 2)) + 3)/(-... | 332 | MATH |
[
{
"content": "How many solutions does the equation -3|(2|((1(2((-2(-2e^(-2x) - 1) - 3)/(-3(-2e^(-2x) - 1))) + 4) - 3)/(2(2((-2(-2e^(-2x) - 1) - 3)/(-3(-2e^(-2x) - 1))) + 4) - 3)) - 3| + 1) + 3| + 1 = -1(5(5(3(-2x + 2) - 1)^2 - 3(3(-2x + 2) - 1) - 2)^2 - 2(5(3(-2x + 2) - 1)^2 - 3(3(-2x + 2) - 1) - 2) + 4)^2 - 1(... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1(-2|(4(-1sin(2π(-1(2x) - 4)) - 3)^2 - 3(-1sin(2π(-1(2x) - 4)) - 3) - 5) + 3| + 3) - 2 = 4(3|(-4(1sin(3π(-3|x - 2| + 3) + 2) - 2)^2 + 3) + 3| + 1)^2 + 2(3|(-4(1sin(3π(-3|x - 2| + 3) + 2) - 2)^2 + 3) + 3| + 1) - 5 have for x in the interval [-10, 10]?\n\nPresen... | 350 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(2π((1(-4((1(4((-1x + 1)/(1x - 3))) + 2)/(2(4((-1x + 1)/(1x - 3))) - 1))^2 + 1((1(4((-1x + 1)/(1x - 3))) + 2)/(2(4((-1x + 1)/(1x - 3))) - 1)) + 3) - 3)/(2(-4((1(4((-1x + 1)/(1x - 3))) + 2)/(2(4((-1x + 1)/(1x - 3))) - 1))^2 + 1((1(4((-1x + 1)/(1x - ... | 45 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(-3(-2(2cos(1π(-1|(2x + 2) - 3| - 2) - 2) - 1) - 5) + 3) + 1 and g(x) = -4(1((-2(-5(-2|x + 3| + 2)^2 + 4) + 3)/(2(-5(-2|x + 3| + 2)^2 + 4))) + 4) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 42 | MATH |
[
{
"content": "How many solutions does the equation -3(-1(3(-1sin(3π(-1|(-1x - 3) + 2| + 3) - 1) + 2)) + 5)^2 - 4(-1(3(-1sin(3π(-1|(-1x - 3) + 2| + 3) - 1) + 2)) + 5) + 5 = -2(2e^(1((-3(-1|((-2x + 2)/(-2x + 3)) - 3| + 2) + 2)/(1(-1|((-2x + 2)/(-2x + 3)) - 3| + 2) + 1))) - 1)^2 - 2(2e^(1((-3(-1|((-2x + 2)/(-2x + ... | 11 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1((-2((2((2((-2(1x^2 - 2x - 2) - 3)/(-1(1x^2 - 2x - 2) - 3)) + 3)/(-3((-2(1x^2 - 2x - 2) - 3)/(-1(1x^2 - 2x - 2) - 3)) + 4)) + 2)/(-1((2((-2(1x^2 - 2x - 2) - 3)/(-1(1x^2 - 2x - 2) - 3)) + 3)/(-3((-2(1x^2 - 2x - 2) - 3)/(-1(1x^2 -... | 1002 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4((-3((-2((1(-2sin(2π(-1x - 3)) - 3) - 1)/(-3(-2sin(2π(-1x - 3)) - 3))) - 3)/(1((1(-2sin(2π(-1x - 3)) - 3) - 1)/(-3(-2sin(2π(-1x - 3)) - 3))))) + 1)/(-2((-2((1(-2sin(2π(-1x - 3)) - 3) - 1)/(-3(-2sin(2π(-1x - 3)) - 3))) - 3)/(... | 118 | MATH |
[
{
"content": "How many solutions does the equation (1(-4(-1(5((-3(1x^2 - 1x + 4) + 3)/(-2(1x^2 - 1x + 4) + 3)) - 4)^2 + 1(5((-3(1x^2 - 1x + 4) + 3)/(-2(1x^2 - 1x + 4) + 3)) - 4) - 1) - 4) + 3)/(2(-4(-1(5((-3(1x^2 - 1x + 4) + 3)/(-2(1x^2 - 1x + 4) + 3)) - 4)^2 + 1(5((-3(1x^2 - 1x + 4) + 3)/(-2(1x^2 - 1x + 4) + 3... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-2(2(2|(-3|(-2sin(1πx + 3) + 2) - 2| + 2) + 1| - 3) - 1)^2 - 5(2(2|(-3|(-2sin(1πx + 3) + 2) - 2| + 2) + 1| - 3) - 1) - 1) + 3)/(-1(-2(2(2|(-3|(-2sin(1πx + 3) + 2) - 2| + 2) + 1| - 3) - 1)^2 - 5(2(2|(-3|(-2sin(1πx + 3) + 2) - 2| + 2) + 1| - 3) - 1) ... | 160 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2cos(1π(1(-3(-3(-4((-2x + 1)/(-1x + 2))^2 + 2((-2x + 1)/(-1x + 2)) - 4) - 3) + 2) - 1) + 1) and g(x) = -1sin(1π(-1(1|(1(-3x - 4) - 3) - 2| + 3)^2 + 4(1|(1(-3x - 4) - 3) - 2| + 3) + 5) + 2) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent... | 484 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(2|(-3(2sin(3π(1|(1|x - 1| + 1) + 1| + 1) - 3) + 2) + 3) + 2| - 3) + 1)/(-3(2|(-3(2sin(3π(1|(1|x - 1| + 1) + 1| + 1) - 3) + 2) + 3) + 2| - 3) - 3) and g(x) = -1(-2(-1cos(1π(-1((2x - 3)/(-3x - 3))^2 - 3((2x - 3)/(-3x - 3)) + 3) + 1))^2 - 4(-1cos(1π(-1... | 22 | MATH |
[
{
"content": "How many solutions does the equation (2(4(-2(1|((-3(-4x - 5) - 2)/(-2(-4x - 5) + 3)) - 3| - 2) - 2)^2 - 3(-2(1|((-3(-4x - 5) - 2)/(-2(-4x - 5) + 3)) - 3| - 2) - 2) - 4) + 1)/(3(4(-2(1|((-3(-4x - 5) - 2)/(-2(-4x - 5) + 3)) - 3| - 2) - 2)^2 - 3(-2(1|((-3(-4x - 5) - 2)/(-2(-4x - 5) + 3)) - 3| - 2) - ... | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-3|(-3sin(2π(2(-3(-2x^2 - 4x + 4)^2 + 2(-2x^2 - 4x + 4) + 4) + 1) + 1) + 2) + 1| + 1)^2 + 4(-3|(-3sin(2π(2(-3(-2x^2 - 4x + 4)^2 + 2(-2x^2 - 4x + 4) + 4) + 1) + 1) + 2) + 1| + 1) + 3 and g(x) = (-1((3(1(-3(-2e^(-2x) + 2)^2 + 1(-2e^(-2x) + 2) + 1)^2 + ... | 190 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(-3|(-1(3(-3(-5x + 3) - 1) - 1) - 2) - 1| + 1) - 1| - 2 and g(x) = 3|(-5((-1(-1cos(3π(4x^2 + 4x + 2) - 3) + 1))/(3(-1cos(3π(4x^2 + 4x + 2) - 3) + 1) - 2))^2 - 3((-1(-1cos(3π(4x^2 + 4x + 2) - 3) + 1))/(3(-1cos(3π(4x^2 + 4x + 2) - ... | 69 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3sin(2π(4((1(-2(1|((-3x + 1)/(3x + 3)) + 1| + 2)^2 - 5(1|((-3x + 1)/(3x + 3)) + 1| + 2)) - 3)/(1(-2(1|((-3x + 1)/(3x + 3)) + 1| + 2)^2 - 5(1|((-3x + 1)/(3x + 3)) + 1| + 2)))) - 2) + 1) + 2 and g(x) = -4(3|(-2|((2(3x^2 + 2x - 5) - 3)/(1(3x^2 + 2x - 5)))... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|((-2(1cos(3π(-3(5(-2x^2 + 5x - 3)^2 + 2(-2x^2 + 5x - 3) + 2)^2 + 1(5(-2x^2 + 5x - 3)^2 + 2(-2x^2 + 5x - 3) + 2) - 1) + 1) + 2) + 1)/(-1(1cos(3π(-3(5(-2x^2 + 5x - 3)^2 + 2(-2x^2 + 5x - 3) + 2)^2 + 1(5(-2x^2 + 5x - 3)^2 + 2(-2x^2 ... | 53 | MATH |
[
{
"content": "How many solutions does the equation -3|(1(-2|(1(-4(-3|x - 1| + 1))^2 + 5(-4(-3|x - 1| + 1)) + 2) + 1| - 1) + 2) - 2| = 5((1(1(-1|(4x - 4) + 2| - 2)^2 - 3(-1|(4x - 4) + 2| - 2) + 4) + 1)/(-1(1(-1|(4x - 4) + 2| - 2)^2 - 3(-1|(4x - 4) + 2| - 2) + 4))) + 1 have for x in the interval [-10, 10]?\n\nPre... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(4(-3(-3cos(2π(-1|((1x - 2)/(-1x - 1)) - 3| - 2) - 3) + 2)^2 - 2(-3cos(2π(-1|((1x - 2)/(-1x - 1)) - 3| - 2) - 3) + 2) - 4) - 5) and g(x) = -4(4(2|(-3|(-1x - 4) + 3| + 2)| + 2) - 1)^2 + 4(4(2|(-3|(-1x - 4) + 3| + 2)| + 2) - 1) - 2 intersect in the inte... | 59 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-1(5(1(-2|(4x - 5) - 2| - 2) + 5))^2 - 2(5(1(-2|(4x - 5) - 2| - 2) + 5)) - 3) - 3)/(2(-1(5(1(-2|(4x - 5) - 2| - 2) + 5))^2 - 2(5(1(-2|(4x - 5) - 2| - 2) + 5)) - 3)) and g(x) = 4((-3(-1|((2(-1|x + 1| + 3) - 1)/(1(-1|x + 1| + 3) +... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3((2((2((-1(-3(1x + 3) - 3) + 3)/(-3(-3(1x + 3) - 3) + 2)) - 2)/(3((-1(-3(1x + 3) - 3) + 3)/(-3(-3(1x + 3) - 3) + 2)))) - 2)/(2((2((-1(-3(1x + 3) - 3) + 3)/(-3(-3(1x + 3) - 3) + 2)) - 2)/(3((-1(-3(1x + 3) - 3) + 3)/(-3(-3(1x + 3) -... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3((-2(4(1(3|(-3sin(3πx - 2) - 2) + 3| - 3)^2 - 5(3|(-3sin(3πx - 2) - 2) + 3| - 3) - 1) + 3) + 3)/(-2(4(1(3|(-3sin(3πx - 2) - 2) + 3| - 3)^2 - 5(3|(-3sin(3πx - 2) - 2) + 3| - 3) - 1) + 3))) - 4 and g(x) = (-3(1|((1(-3|(-5x - 2) - 1... | 23 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(3cos(2π(2|((1(2(-4x + 4) + 5))/(-3(2(-4x + 4) + 5) + 4)) + 3| + 2) - 2) + 3) + 3 and g(x) = (-1(-2(5(3|(3|x - 1| - 1) + 1| - 3)^2 - 4(3|(3|x - 1| - 1) + 1| - 3) - 4) + 2) - 2)/(-1(-2(5(3|(3|x - 1| - 1) + 1| - 3)^2 - 4(3|(3|x - 1| - 1) + 1| - 3) - 4)... | 18 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-1(1sin(3π(-2|(-3((-2x + 1)/(3x + 3))^2 - 3((-2x + 1)/(3x + 3)) - 5) - 2|) - 3) - 3) - 3))/(-1(-1(1sin(3π(-2|(-3((-2x + 1)/(3x + 3))^2 - 3((-2x + 1)/(3x + 3)) - 5) - 2|) - 3) - 3) - 3)) and g(x) = -5(-3sin(2π(-2|(3|(2x^2 - 5x - 2) + 3|)| + 3) + 2) -... | 1305 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(1(-2cos(1π(3|(5((3x)/(-2x - 1))^2 - 5((3x)/(-2x - 1)) - 1) - 1| + 3) + 3)) + 2)^2 + 5(1(-2cos(1π(3|(5((3x)/(-2x - 1))^2 - 5((3x)/(-2x - 1)) - 1) - 1| + 3) + 3)) + 2) and g(x) = 1|(-2|(4(-1cos(3π(4x + 2) + 2) + 2)^2 - 5) - 1| + 3)... | 308 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-2(3((1(-2(3sin(3πx + 2) - 3) + 1) + 1)/(3(-2(3sin(3πx + 2) - 3) + 1) + 1)) - 1)^2 + 4(3((1(-2(3sin(3πx + 2) - 3) + 1) + 1)/(3(-2(3sin(3πx + 2) - 3) + 1) + 1)) - 1) - 2) - 2 and g(x) = -1((2(1cos(3π(-5(-3x^2 - 5x + 3)) + 2) + 1)... | 119 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(-5(-2|(2|(2cos(1π(-1x^2 - 1x - 2) - 3) + 1) + 3| - 1) - 2| + 3)^2 + 3(-2|(2|(2cos(1π(-1x^2 - 1x - 2) - 3) + 1) + 3| - 1) - 2| + 3) + 2) - 2| - 2 and g(x) = (3(5(-1(3((3x - 2)/(-2x - 3))^2 + 3((3x - 2)/(-2x - 3)) - 2) + 5)^2 + 3)... | 197 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(-1(3(-3((-1(2x + 5) - 2)/(-2(2x + 5) + 3))^2 + 4((-1(2x + 5) - 2)/(-2(2x + 5) + 3)) + 4) + 1))^2 - 5(-1(3(-3((-1(2x + 5) - 2)/(-2(2x + 5) + 3))^2 + 4((-1(2x + 5) - 2)/(-2(2x + 5) + 3)) + 4) + 1)) - 3 and g(x) = (-2((-2(2|((... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-4((2(-2|(4((3x + 1)/(2x + 1))^2 - 5((3x + 1)/(2x + 1)) - 5)|) - 2)/(1(-2|(4((3x + 1)/(2x + 1))^2 - 5((3x + 1)/(2x + 1)) - 5)|) + 3))^2 - 5((2(-2|(4((3x + 1)/(2x + 1))^2 - 5((3x + 1)/(2x + 1)) - 5)|) - 2)/(1(-2|(4((3x + 1)... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2((2(-1|(-1|((3(1|x + 2| + 2) + 3)/(-1(1|x + 2| + 2) + 3)) - 3| + 1) - 3| - 3) + 3)/(-3(-1|(-1|((3(1|x + 2| + 2) + 3)/(-1(1|x + 2| + 2) + 3)) - 3| + 1) - 3| - 3))) + 1)/(1((2(-1|(-1|((3(1|x + 2| + 2) + 3)/(-1(1|x + 2| + 2) ... | 6 | MATH |
[
{
"content": "How many solutions does the equation -1(-3|(5(-2sin(1π(4(3x - 1) + 2) + 1)) + 2) + 2| + 3) - 2 = 2((-3(-1|(-2(-5x^2) + 1) - 1| + 3) + 2)/(1(-1|(-2(-5x^2) + 1) - 1| + 3) + 3)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 32 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(3|(2|(3|(1sin(2π((2x)/(-3x + 1)) - 2) + 3) - 2| + 2) - 3| - 1) - 3| - 3) + 2| + 3 and g(x) = 2((-2((2(2|(-1|x + 2| - 3) + 2|) + 1)/(-1(2|(-1|x + 2| - 3) + 2|) + 3)) - 3)/(-3((2(2|(-1|x + 2| - 3) + 2|) + 1)/(-1(2|(-1|x + 2| - 3) + 2|) + 3))))^2 + 2(... | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(5(-1(2|(-3sin(2π(3|x - 3| + 1) + 2) - 3) - 1| + 3)^2 + 2(2|(-3sin(2π(3|x - 3| + 1) + 2) - 3) - 1| + 3) - 2) - 3) - 1| + 1 and g(x) = -4(3(-1(-4((-2x - 1)/(-3x + 1)) + 3)^2) - 4) where -10 ≤ x ≤ 10.\n\nPresent the answer in... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-1((-1(-3|(2(2e^(-2x) + 2)^2 + 3(2e^(-2x) + 2) + 3) + 2|) - 2)/(1(-3|(2(2e^(-2x) + 2)^2 + 3(2e^(-2x) + 2) + 3) + 2|) + 2))^2 + 1((-1(-3|(2(2e^(-2x) + 2)^2 + 3(2e^(-2x) + 2) + 3) + 2|) - 2)/(1(-3|(2(2e^(-2x) + 2)^2 + 3(2e^(... | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(2|((2(-5(-1((2x - 1)/(-1x + 2))^2 - 1((2x - 1)/(-1x + 2)) + 2) + 3) + 2)/(3(-5(-1((2x - 1)/(-1x + 2))^2 - 1((2x - 1)/(-1x + 2)) + 2) + 3) - 2)) + 2| - 3) - 4 and g(x) = (2(-3(1(2|((2x - 2)/(1x)) - 3|)^2 - 3(2|((2x - 2)/(1x)) - 3|... | 9 | MATH |
[
{
"content": "How many solutions does the equation (-3((-3(3((-1((2((-3x - 1)/(-1x + 3)) - 3)/(-2((-3x - 1)/(-1x + 3)) - 2)) + 3)/(-3((2((-3x - 1)/(-1x + 3)) - 3)/(-2((-3x - 1)/(-1x + 3)) - 2)) - 1))^2 - 5) - 1)/(-2(3((-1((2((-3x - 1)/(-1x + 3)) - 3)/(-2((-3x - 1)/(-1x + 3)) - 2)) + 3)/(-3((2((-3x - 1)/(-1x + 3... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-3(-1|(-3((3(-3x + 1))/(-1(-3x + 1) - 1)) + 2) - 1| + 2)^2 - 2(-1|(-3((3(-3x + 1))/(-1(-3x + 1) - 1)) + 2) - 1| + 2) + 3)^2 - 5(-3(-1|(-3((3(-3x + 1))/(-1(-3x + 1) - 1)) + 2) - 1| + 2)^2 - 2(-1|(-3((3(-3x + 1))/(-1(-3x + 1) - 1)... | 8 | MATH |
[
{
"content": "How many solutions does the equation 3|((3((1(-2sin(1π(-1|(-3x^2 - 2x + 4)| - 3) - 2) + 2) + 1)/(2(-2sin(1π(-1|(-3x^2 - 2x + 4)| - 3) - 2) + 2) + 1)) + 2)/(-3((1(-2sin(1π(-1|(-3x^2 - 2x + 4)| - 3) - 2) + 2) + 1)/(2(-2sin(1π(-1|(-3x^2 - 2x + 4)| - 3) - 2) + 2) + 1)) + 2)) + 3| - 1 = 3(-3(2(-3|(-3|x... | 37 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1((-2((2(-3|(2(-1x^2 - 4x + 2) + 4) + 1| - 2) - 1)/(-1(-3|(2(-1x^2 - 4x + 2) + 4) + 1| - 2) + 3)) - 1)/(-1((2(-3|(2(-1x^2 - 4x + 2) + 4) + 1| - 2) - 1)/(-1(-3|(2(-1x^2 - 4x + 2) + 4) + 1| - 2) + 3)) - 3)) - 2)/(-2((-2((2(-3|(2(-1x^2 - 4x + 2) + 4) + 1... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(5(2(-1|(-3cos(3π(-1|x + 2| + 2) + 3) + 1) + 2|) + 5) + 2) + 3)/(3(5(2(-1|(-3cos(3π(-1|x + 2| + 2) + 3) + 1) + 2|) + 5) + 2) - 2) and g(x) = -4((-1(-3(-2|(3|x + 2| - 1) + 1| + 2)^2 + 1(-2|(3|x + 2| - 1) + 1| + 2) + 1) + 3)/... | 18 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1((-2(3cos(3π(-1(-1(1x^2 - 1x - 4) + 2)^2 - 2(-1(1x^2 - 1x - 4) + 2) + 1) + 2) - 1) - 1)/(1(3cos(3π(-1(-1(1x^2 - 1x - 4) + 2)^2 - 2(-1(1x^2 - 1x - 4) + 2) + 1) + 2) - 1) + 3))^2 + 1((-2(3cos(3π(-1(-1(1x^2 - 1x - 4) + 2)^2 - 2... | 417 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(3(-2|(2|(1(-2e^(1x) + 1) - 2) - 1| - 3)| + 2) - 3) + 4 and g(x) = 2|(3(-4(-2|(-2|x + 1| + 3) - 2| + 1)^2 - 4(-2|(-2|x + 1| + 3) - 2| + 1) + 1)^2 + 5(-4(-2|(-2|x + 1| + 3) - 2| + 1)^2 - 4(-2|(-2|x + 1| + 3) - 2| + 1) + 1) - 2)| -... | 14 | MATH |
[
{
"content": "How many solutions does the equation -5(3|(4(-3sin(1π(3|(-5x^2 + 2x - 3) - 3| + 3) - 1) + 1) + 1)| + 1) + 3 = -2|(2(3cos(2π(-2(-3|x + 1|) + 4) + 3) + 3) + 1) + 1| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 236 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(3cos(3π((-2(-1(-2|(-1x^2 + 3x - 1) + 3| - 2) - 4) + 2)/(-2(-1(-2|(-1x^2 + 3x - 1) + 3| - 2) - 4))) - 3) - 2)^2 + 1(3cos(3π((-2(-1(-2|(-1x^2 + 3x - 1) + 3| - 2) - 4) + 2)/(-2(-1(-2|(-1x^2 + 3x - 1) + 3| - 2) - 4))) - 3) - 2) - 5 a... | 86 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1((1(-2((2(-1(3|x - 2| - 1) + 2) - 3)/(3(-1(3|x - 2| - 1) + 2) + 1)) + 1))/(1(-2((2(-1(3|x - 2| - 1) + 2) - 3)/(3(-1(3|x - 2| - 1) + 2) + 1)) + 1) + 3)) + 1)/(-3((1(-2((2(-1(3|x - 2| - 1) + 2) - 3)/(3(-1(3|x - 2| - 1) + 2) + 1)) ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(2(-2|(1(3((2x - 2)/(-2x)) - 1)^2 - 3(3((2x - 2)/(-2x)) - 1)) - 3| + 2)^2 - 1(-2|(1(3((2x - 2)/(-2x)) - 1)^2 - 3(3((2x - 2)/(-2x)) - 1)) - 3| + 2) - 4) + 2| + 2 and g(x) = (1((-1((1(1|(2|x| + 1) - 1| + 2) - 1)/(1(1|(2|x| + 1) - ... | 11 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(2|(-2cos(3π(-2((-3(-2|x + 1|) - 2)/(3(-2|x + 1|) - 2)) + 5) + 3) - 1) - 1| - 1) - 2)/(-2(2|(-2cos(3π(-2((-3(-2|x + 1|) - 2)/(3(-2|x + 1|) - 2)) + 5) + 3) - 1) - 1| - 1) + 3) and g(x) = 2(4((2(3|(-2cos(1πx - 1) - 3) + 2| +... | 32 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-3(5(-3(2|(-1x - 4) - 1| + 3)^2 - 4(2|(-1x - 4) - 1| + 3))^2 - 2(-3(2|(-1x - 4) - 1| + 3)^2 - 4(2|(-1x - 4) - 1| + 3)) + 4)^2 + 1(5(-3(2|(-1x - 4) - 1| + 3)^2 - 4(2|(-1x - 4) - 1| + 3))^2 - 2(-3(2|(-1x - 4) - 1| + 3)^2 - 4(2|(-1... | 160 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(4((1((1((3(3x^2 - 2x - 2) - 2)/(2(3x^2 - 2x - 2) + 2)))/(-3((3(3x^2 - 2x - 2) - 2)/(2(3x^2 - 2x - 2) + 2)) + 4)) - 2)/(2((1((3(3x^2 - 2x - 2) - 2)/(2(3x^2 - 2x - 2) + 2)))/(-3((3(3x^2 - 2x - 2) - 2)/(2(3x^2 - 2x - 2) + 2)) + 4)) + 1))^2 + 1((1((1((3... | 9 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2((-2(1(1(1cos(3π(-2x - 5) - 3) - 3)^2 + 3(1cos(3π(-2x - 5) - 3) - 3)) - 1))/(3(1(1(1cos(3π(-2x - 5) - 3) - 3)^2 + 3(1cos(3π(-2x - 5) - 3) - 3)) - 1) + 3)) + 3 and g(x) = -4(-2((1(-3|(-3x^2 + 2x + 5)| + 1) + 2)/(1(-3|(-3x^2 + 2x + ... | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-5((2(-2|((1(-2cos(3πx - 2) - 1) - 1)/(3(-2cos(3πx - 2) - 1) + 1)) - 2|) - 2)/(-1(-2|((1(-2cos(3πx - 2) - 1) - 1)/(3(-2cos(3πx - 2) - 1) + 1)) - 2|) - 3))^2 - 5((2(-2|((1(-2cos(3πx - 2) - 1) - 1)/(3(-2cos(3πx - 2) - 1) + 1)) - 2|) - 2)/(-1(-2|((1(-2c... | 41 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-2((-3(3((-2(4x - 3))/(2(4x - 3))) + 3) + 1)/(2(3((-2(4x - 3))/(2(4x - 3))) + 3) + 2))^2 + 2((-3(3((-2(4x - 3))/(2(4x - 3))) + 3) + 1)/(2(3((-2(4x - 3))/(2(4x - 3))) + 3) + 2)) + 5)| + 3 and g(x) = -3(4(-2(4(2x - 1) + 4))... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(1(-2cos(2π(2|(-5(1x - 3)^2 - 5(1x - 3) + 2) + 2| - 3) + 3) - 2) + 5)| - 2 and g(x) = 4(1(-1(1(1|x + 3| + 2)^2 - 4(1|x + 3| + 2) + 5) - 1) + 5)^2 + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your a... | 131 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(1(-2((1((-1(-2|x - 3| + 2) - 3)/(2(-2|x - 3| + 2) - 3)) - 1)/(-1((-1(-2|x - 3| + 2) - 3)/(2(-2|x - 3| + 2) - 3)) + 2))^2 + 3((1((-1(-2|x - 3| + 2) - 3)/(2(-2|x - 3| + 2) - 3)) - 1)/(-1((-1(-2|x - 3| + 2) - 3)/(2(-2|x - 3| + 2) - 3)) + 2)) + 5) - 1)... | 12 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2((-1((1(-1sin(3π(-2|(-3x - 5) - 3| + 2)) + 1) - 3)/(-2(-1sin(3π(-2|(-3x - 5) - 3| + 2)) + 1) + 3)) + 2)/(-1((1(-1sin(3π(-2|(-3x - 5) - 3| + 2)) + 1) - 3)/(-2(-1sin(3π(-2|(-3x - 5) - 3| + 2)) + 1) + 3)) - 3)) + 1)/(-1((-1((1(-1sin(3π(-2|(-3x - 5) - 3... | 23 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3((-2(3cos(2π(-2|(5(-2x + 3)^2 - 3(-2x + 3) + 2) + 3| - 3) - 3) - 2) - 1)/(-2(3cos(2π(-2|(5(-2x + 3)^2 - 3(-2x + 3) + 2) + 3| - 3) - 3) - 2) - 3))^2 - 1((-2(3cos(2π(-2|(5(-2x + 3)^2 - 3(-2x + 3) + 2) + 3| - 3) - 3) - 2) - 1)/... | 325 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-1(-2sin(1π((1(4(-2x + 1) - 5) + 2)/(2(4(-2x + 1) - 5))) - 3) - 3)^2 - 4(-2sin(1π((1(4(-2x + 1) - 5) + 2)/(2(4(-2x + 1) - 5))) - 3) - 3) - 1) + 2)/(-3(-1(-2sin(1π((1(4(-2x + 1) - 5) + 2)/(2(4(-2x + 1) - 5))) - 3) - 3)^2 - ... | 782 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(3(-4(-2(1(1|x + 2| - 3) + 2) + 2)^2 - 2(-2(1(1|x + 2| - 3) + 2) + 2) + 3)^2 + 2(-4(-2(1(1|x + 2| - 3) + 2) + 2)^2 - 2(-2(1(1|x + 2| - 3) + 2) + 2) + 3) - 1) - 3 and g(x) = 2((-1(-2sin(3π(-1|(-1|x - 3|)|) - 2) - 3) + 2)/(-3(-2sin... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(3((-1((1((-2((2x + 1)/(3x - 2)))/(2((2x + 1)/(3x - 2)) - 1)) - 1)/(2((-2((2x + 1)/(3x - 2)))/(2((2x + 1)/(3x - 2)) - 1)) - 1)) + 1)/(3((1((-2((2x + 1)/(3x - 2)))/(2((2x + 1)/(3x - 2)) - 1)) - 1)/(2((-2((2x + 1)/(3x - 2)))/(2((2... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-4((1(1(-5(-2x + 1)^2 - 4(-2x + 1) + 3)^2 - 2(-5(-2x + 1)^2 - 4(-2x + 1) + 3) + 4) - 1)/(-1(1(-5(-2x + 1)^2 - 4(-2x + 1) + 3)^2 - 2(-5(-2x + 1)^2 - 4(-2x + 1) + 3) + 4) - 1)) - 2) + 3)/(1(-4((1(1(-5(-2x + 1)^2 - 4(-2x + 1)... | 10 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4((1(2cos(1π(5(3(4x + 3)) - 3)) + 1) - 1)/(-1(2cos(1π(5(3(4x + 3)) - 3)) + 1) + 2)) - 5 and g(x) = (-2(-2|(-1|(1(1x^2 + 3x - 4) + 2) - 3| + 1) + 2| + 1) + 3)/(-3(-2|(-1|(1(1x^2 + 3x - 4) + 2) - 3| + 1) + 2| + 1) + 1) intersect in the interval [-10, 10]... | 401 | MATH |
[
{
"content": "How many solutions does the equation -1|(4(-1|(1(-4((2x - 3)/(-2x + 1)) - 1) - 2) + 2| + 2) - 3) + 2| - 3 = -1(-1(-3(4((3x + 3)/(-3x + 4)) - 4)^2 - 4(4((3x + 3)/(-3x + 4)) - 4) + 3)^2 + 2(-3(4((3x + 3)/(-3x + 4)) - 4)^2 - 4(4((3x + 3)/(-3x + 4)) - 4) + 3) - 2)^2 - 4(-1(-3(4((3x + 3)/(-3x + 4)) - 4... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3((-3(1|(2cos(1π(-1(4x^2 - 4)^2 - 3) - 1) + 2) - 3| + 2) + 2)/(-2(1|(2cos(1π(-1(4x^2 - 4)^2 - 3) - 1) + 2) - 3| + 2) + 3))^2 + 1((-3(1|(2cos(1π(-1(4x^2 - 4)^2 - 3) - 1) + 2) - 3| + 2) + 2)/(-2(1|(2cos(1π(-1(4x^2 - 4)^2 - 3) - 1) + ... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(3(-3((2(-3|((2x - 1)/(-2x - 3))| - 1) + 3)/(-1(-3|((2x - 1)/(-2x - 3))| - 1))) - 2)^2 + 3(-3((2(-3|((2x - 1)/(-2x - 3))| - 1) + 3)/(-1(-3|((2x - 1)/(-2x - 3))| - 1))) - 2) + 2) - 1| + 1 and g(x) = -4(1|((2(2|(-4x^2 - 2x + ... | 5 | MATH |
[
{
"content": "How many solutions does the equation 4((2(1((2(-1e^(1((-3x)/(1x + 1))) + 1) - 1)/(2(-1e^(1((-3x)/(1x + 1))) + 1) + 3))^2 + 2((2(-1e^(1((-3x)/(1x + 1))) + 1) - 1)/(2(-1e^(1((-3x)/(1x + 1))) + 1) + 3)) + 1))/(1(1((2(-1e^(1((-3x)/(1x + 1))) + 1) - 1)/(2(-1e^(1((-3x)/(1x + 1))) + 1) + 3))^2 + 2((2(-1e... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-2|(-5(-3|(-1|(-4x + 2) - 1| + 2) - 2| + 3) - 4)|) + 3)/(2(-2|(-5(-3|(-1|(-4x + 2) - 1| + 2) - 2| + 3) - 4)|) - 1) and g(x) = 3((3(-4(-3cos(2π((1x - 1)/(-1x + 2)) + 2) - 2)^2 + 4) + 3)/(-2(-4(-3cos(2π((1x - 1)/(-1x + 2)) +... | 44 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(5(5(-1((1(-2x) - 2)/(3(-2x) + 1)) - 2) + 4)^2 - 4(5(-1((1(-2x) - 2)/(3(-2x) + 1)) - 2) + 4) - 5)| - 2 and g(x) = (3(-1((-3(-2(-1|x| + 1)^2 - 4) - 2)/(2(-2(-1|x| + 1)^2 - 4) - 3)) - 5) + 3)/(-3(-1((-3(-2(-1|x| + 1)^2 - 4) - 2)/(2(-2(-1|x| + 1)^2 - 4)... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(1(1(1(1cos(3π(-2x^2 + 4x - 5) - 1) + 2) + 3)^2 - 5(1(1cos(3π(-2x^2 + 4x - 5) - 1) + 2) + 3) + 1)^2 + 4(1(1(1cos(3π(-2x^2 + 4x - 5) - 1) + 2) + 3)^2 - 5(1(1cos(3π(-2x^2 + 4x - 5) - 1) + 2) + 3) + 1) + 4)^2 + 5(1(1(1(1cos(3π... | 761 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-1e^(-1((-2(1((-3(-2x) - 2)/(1(-2x) + 1)) - 3) + 2)/(3(1((-3(-2x) - 2)/(1(-2x) + 1)) - 3) - 2))) - 2) + 2| + 2 and g(x) = -1|(1(-3|(-2sin(2π(-1x - 1) + 3) + 2) - 3| + 3) + 4) - 2| - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the a... | 15 | MATH |
[
{
"content": "How many solutions does the equation (3(-4(-1sin(3π(-1(-4((-1x - 1)/(-1x - 1)) + 1) + 3) + 3) + 2) - 5) - 1)/(2(-4(-1sin(3π(-1(-4((-1x - 1)/(-1x - 1)) + 1) + 3) + 3) + 2) - 5) - 1) = -5(-5((-1(2|(-2|x| + 2) - 1|) - 1)/(-2(2|(-2|x| + 2) - 1|) + 3))^2 - 2((-1(2|(-2|x| + 2) - 1|) - 1)/(-2(2|(-2|x| + ... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1((1(-2(4(3cos(2π(1x^2 + 2x) + 1) - 2)^2 - 2(3cos(2π(1x^2 + 2x) + 1) - 2) - 1) + 5))/(1(-2(4(3cos(2π(1x^2 + 2x) + 1) - 2)^2 - 2(3cos(2π(1x^2 + 2x) + 1) - 2) - 1) + 5) - 3)) and g(x) = -2(3(-1|(3(-2e^(2x))^2 - 2(-2e^(2x))) + ... | 144 | MATH |
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