messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation 1(-2((3(1(2|(-3|x - 3|)| - 1)^2 + 1(2|(-3|x - 3|)| - 1) - 3) + 2)/(3(1(2|(-3|x - 3|)| - 1)^2 + 1(2|(-3|x - 3|)| - 1) - 3) - 2))^2 + 1) - 5 = -3((3(-5(3|(1|x - 1| - 1) + 1| - 3) - 1))/(1(-5(3|(1|x - 1| - 1) + 1| - 3) - 1)))^2 + 1((3(-5(3|(1|x - 1| - 1) + 1| - 3)... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(4((2(3((1(2|x| + 2) - 2)/(1(2|x| + 2)))^2 + 4((1(2|x| + 2) - 2)/(1(2|x| + 2))) + 5) + 1)/(3(3((1(2|x| + 2) - 2)/(1(2|x| + 2)))^2 + 4((1(2|x| + 2) - 2)/(1(2|x| + 2))) + 5) - 2))^2 - 3((2(3((1(2|x| + 2) - 2)/(1(2|x| + 2)))^2... | 4 | MATH |
[
{
"content": "How many solutions does the equation 3(-1cos(2π(2(-2(-2(-4x + 5) + 1)^2 - 4(-2(-4x + 5) + 1) + 5) + 2) - 3) - 3)^2 - 2(-1cos(2π(2(-2(-2(-4x + 5) + 1)^2 - 4(-2(-4x + 5) + 1) + 5) + 2) - 3) - 3) = 1|(-1|(-4(3(-3|x - 2| + 3)^2 + 2(-3|x - 2| + 3) - 5) - 1) + 3| + 3) - 1| - 3 have for x in the interval... | 90 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(3|((1(-5(4(4x + 3)^2 - 2(4x + 3) - 3)^2 + 4(4(4x + 3)^2 - 2(4x + 3) - 3) + 3) + 1)/(-2(-5(4(4x + 3)^2 - 2(4x + 3) - 3)^2 + 4(4(4x + 3)^2 - 2(4x + 3) - 3) + 3))) + 1| + 1) + 1| - 2 and g(x) = -1|(-5(-1|(2(-2cos(2πx + 3) + 2)^2 - 4(-2cos(2πx + 3) + 2... | 161 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3sin(2π((2(-2|((3((-2(1x + 2))/(-1(1x + 2) + 2)) + 1)/(2((-2(1x + 2))/(-1(1x + 2) + 2)) + 3)) - 3| + 2) - 2)/(-3(-2|((3((-2(1x + 2))/(-1(1x + 2) + 2)) + 1)/(2((-2(1x + 2))/(-1(1x + 2) + 2)) + 3)) - 3| + 2) - 2)) - 1) + 3 and g(x) = -2|((-1(-2(1|(2|x +... | 33 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(-2|(3(1cos(2π(-1|((2x - 3)/(1x - 3)) + 2| + 1)) - 1)^2 - 4)| + 3)| + 3 and g(x) = 1|(1|(-1(-3cos(1π(-1x - 2) - 2) + 3)) - 3| + 1) - 3| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 42 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(1(-1|((3((-3(1x^2 - 4x - 1) + 3)/(3(1x^2 - 4x - 1) - 2)) - 1)/(-1((-3(1x^2 - 4x - 1) + 3)/(3(1x^2 - 4x - 1) - 2)) + 2)) + 3| + 3)^2 - 5(-1|((3((-3(1x^2 - 4x - 1) + 3)/(3(1x^2 - 4x - 1) - 2)) - 1)/(-1((-3(1x^2 - 4x - 1) + 3)/(3(1... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-1|((1(1|(3(-2x + 5)^2 - 2(-2x + 5) - 5) + 2| - 2) - 2)/(-2(1|(3(-2x + 5)^2 - 2(-2x + 5) - 5) + 2| - 2) - 1)) + 2|)^2 - 3(-1|((1(1|(3(-2x + 5)^2 - 2(-2x + 5) - 5) + 2| - 2) - 2)/(-2(1|(3(-2x + 5)^2 - 2(-2x + 5) - 5) + 2| -... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(3|(-2|((-2((2(-2e^(1x) + 2) - 2)/(-3(-2e^(1x) + 2))) + 2)/(1((2(-2e^(1x) + 2) - 2)/(-3(-2e^(1x) + 2)))))| + 3) + 1| + 3) + 3| + 3 and g(x) = -2|(-1e^(1((1(1((3x - 2)/(-2x + 1))^2 - 5((3x - 2)/(-2x + 1)) + 2) - 2)/(3(1((3x - 2)/(-2x + 1))^2 - 5((3x ... | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(3(-5((1((3(1|x - 2| - 1) + 1)/(-1(1|x - 2| - 1) + 2)) + 2)/(1((3(1|x - 2| - 1) + 1)/(-1(1|x - 2| - 1) + 2)) - 2))^2 + 4((1((3(1|x - 2| - 1) + 1)/(-1(1|x - 2| - 1) + 2)) + 2)/(1((3(1|x - 2| - 1) + 1)/(-1(1|x - 2| - 1) + 2)) - 2))... | 4 | MATH |
[
{
"content": "How many solutions does the equation -2|(1(-4((2(2(-1x^2 + 4x) - 3) + 2)/(-2(2(-1x^2 + 4x) - 3) + 1))^2 + 1((2(2(-1x^2 + 4x) - 3) + 2)/(-2(2(-1x^2 + 4x) - 3) + 1)) - 5)^2 - 5(-4((2(2(-1x^2 + 4x) - 3) + 2)/(-2(2(-1x^2 + 4x) - 3) + 1))^2 + 1((2(2(-1x^2 + 4x) - 3) + 2)/(-2(2(-1x^2 + 4x) - 3) + 1)) - ... | 4 | MATH |
[
{
"content": "How many solutions does the equation 4((2(-2|(-3(-1(3|x + 3|)^2 + 2) - 5) + 1| + 2) - 3)/(3(-2|(-3(-1(3|x + 3|)^2 + 2) - 5) + 1| + 2) + 3)) - 5 = 1(-1|((2(-2|((-2x + 1)/(3x)) - 1| - 1))/(1(-2|((-2x + 1)/(3x)) - 1| - 1) - 3)) - 2| - 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(4(2|(4(-3((1x + 1)/(-1x - 2))^2 - 4((1x + 1)/(-1x - 2)))^2 + 1(-3((1x + 1)/(-1x - 2))^2 - 4((1x + 1)/(-1x - 2))) + 1) - 1| - 2) - 3)| - 2 and g(x) = (1(-1((3(-2e^(1(2x^2 - 5x + 2)) + 3))/(1(-2e^(1(2x^2 - 5x + 2)) + 3) + 1)) + 5)... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1|(1(-3(-1|((-2(-5x^2 - 3x + 2) + 2)/(3(-5x^2 - 3x + 2) + 2))| - 1) - 3) - 3) + 1| - 1 and g(x) = (3(3(4(3sin(1π(4x^2 - 2x - 5) - 3)) + 1)^2 - 4) - 1)/(-1(3(4(3sin(1π(4x^2 - 2x - 5) - 3)) + 1)^2 - 4) + 3) in the range -10 ≤ x ≤ 10... | 97 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(2(-3sin(1π((1(-2|(-1x^2 - 4x - 3) + 3| + 3) - 3)/(-3(-2|(-1x^2 - 4x - 3) + 3| + 3) - 2)) + 1) - 3) + 5) + 2| + 1 and g(x) = (2(1|((-1(4(-3x^2 - 3x + 1) + 1) - 3)/(2(4(-3x^2 - 3x + 1) + 1) - 1))| - 2) - 3)/(1(1|((-1(4(-3x^2 - 3x + 1) + 1) - 3)/(2(4(-... | 49 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(1|((-2(4(2(-3x + 5) - 1) + 4) + 1)/(-3(4(2(-3x + 5) - 1) + 4) + 4)) + 3| - 3) - 3| + 3 and g(x) = 5(2|(2sin(2π(-2(2x - 3)^2 - 1(2x - 3) - 1) + 2) + 2) - 3| + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your... | 554 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(-2((3((2(-4(-1x^2 + 1x + 4)^2 - 5(-1x^2 + 1x + 4) + 2))/(-1(-4(-1x^2 + 1x + 4)^2 - 5(-1x^2 + 1x + 4) + 2) - 2)) + 2)/(-2((2(-4(-1x^2 + 1x + 4)^2 - 5(-1x^2 + 1x + 4) + 2))/(-1(-4(-1x^2 + 1x + 4)^2 - 5(-1x^2 + 1x + 4) + 2) -... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3((3(3|(2|(-1|(-1x^2 - 1x - 5) + 1| - 1) + 1| - 3)| - 2) - 3)/(3(3|(2|(-1|(-1x^2 - 1x - 5) + 1| - 1) + 1| - 3)| - 2) - 2)) + 3)/(2((3(3|(2|(-1|(-1x^2 - 1x - 5) + 1| - 1) + 1| - 3)| - 2) - 3)/(3(3|(2|(-1|(-1x^2 - 1x - 5) + 1| - 1) + 1| - 3)| - 2) - 2))... | 31 | MATH |
[
{
"content": "How many solutions does the equation 4(2|(-5(-3|(-3sin(3π(1x^2 + 1x) + 1) - 3)| + 2) + 5) + 1|) - 1 = 3(-1(-3|(-1|(-2cos(2πx + 2) - 2) + 3| - 2) + 3| - 1))^2 - 3(-1(-3|(-1|(-2cos(2πx + 2) - 2) + 3| - 2) + 3| - 1)) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\b... | 393 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|((-2(-2|(3|(1(-2|x - 3|) + 5) + 2| - 2) - 1| - 2) - 2)/(1(-2|(3|(1(-2|x - 3|) + 5) + 2| - 2) - 1| - 2) - 3)) + 2| - 3 and g(x) = -1(-1((1(-3sin(1π(2|x + 1| + 2)) - 2))/(-2(-3sin(1π(2|x + 1| + 2)) - 2) + 1)) - 3) - 3 intersect in the interval [-10, 1... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(-1|(1((2((-3(2|x| - 3) + 2)/(-2(2|x| - 3) - 3)))/(-2((-3(2|x| - 3) + 2)/(-2(2|x| - 3) - 3)) + 3))^2 - 5) - 2| + 1) - 5 and g(x) = -3(1((2((3((-3x)/(2x - 1)) + 2)/(2((-3x)/(2x - 1)) - 1)) - 1)/(-3((3((-3x)/(2x - 1)) + 2)/(2((-3x)/... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(3|((2(-3(2sin(1π(4x^2 - 4x - 5) + 2) - 2)^2 + 4) - 3)/(-1(-3(2sin(1π(4x^2 - 4x - 5) + 2) - 2)^2 + 4) - 2)) - 2| - 1)^2 - 1(3|((2(-3(2sin(1π(4x^2 - 4x - 5) + 2) - 2)^2 + 4) - 3)/(-1(-3(2sin(1π(4x^2 - 4x - 5) + 2) - 2)^2 + 4) - 2)) - 2| - 1) - 3 and g... | 105 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(1(4(-2(2(2x - 5) - 3)^2 - 5(2(2x - 5) - 3) + 3) - 1) + 1) - 1| + 1 and g(x) = (-3(-3|(1(-1e^(-2(1x - 2)))) + 2|) - 2)/(1(-3|(1(-1e^(-2(1x - 2)))) + 2|) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your an... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1(2(-3(-2|(5(3x^2 - 2x - 2) + 1) - 2| + 1) - 3) - 2)^2 + 3(2(-3(-2|(5(3x^2 - 2x - 2) + 1) - 2| + 1) - 3) - 2) + 2 = -1|(1(2((-1(-3x) + 3)/(2(-3x) + 2)) - 1)^2 + 3(2((-1(-3x) + 3)/(2(-3x) + 2)) - 1) + 5) - 1| + 3 have for x in the interval [-10, 10]?\n\nPresent ... | 4 | MATH |
[
{
"content": "How many solutions does the equation 5(3((-1(-2|(1|((-1x + 2)/(1x)) - 1| - 1) + 1|) + 1)/(-1(-2|(1|((-1x + 2)/(1x)) - 1| - 1) + 1|) - 3))^2 - 5((-1(-2|(1|((-1x + 2)/(1x)) - 1| - 1) + 1|) + 1)/(-1(-2|(1|((-1x + 2)/(1x)) - 1| - 1) + 1|) - 3)) + 3) + 1 = 1((2(3sin(3π(-3(5x)^2 + 3(5x) + 5) - 1)) - 3)/... | 12 | MATH |
[
{
"content": "How many solutions does the equation -2(-1cos(2π(1|((3(4(2|x| - 1) + 4) + 2)/(1(4(2|x| - 1) + 4) - 3)) - 2| + 1) - 2) - 1) - 4 = (-2(-1(4((-3(-2x^2 - 1x - 4) - 1)/(-1(-2x^2 - 1x - 4) - 1))) - 4) + 3)/(1(-1(4((-3(-2x^2 - 1x - 4) - 1)/(-1(-2x^2 - 1x - 4) - 1))) - 4) - 3) have for x in the interval [... | 112 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1e^(1(3(-2((-1(1(1|x - 3| + 1)^2 - 3(1|x - 3| + 1) - 4))/(-2(1(1|x - 3| + 1)^2 - 3(1|x - 3| + 1) - 4) + 3))^2 - 4((-1(1(1|x - 3| + 1)^2 - 3(1|x - 3| + 1) - 4))/(-2(1(1|x - 3| + 1)^2 - 3(1|x - 3| + 1) - 4) + 3)) - 1)^2 + 3(-2((-1(1(... | 4 | MATH |
[
{
"content": "How many solutions does the equation 2|(2(1|(2|(-1((1x + 2)/(3x - 2)) + 3) - 3| + 2)|) - 2) + 3| - 1 = 1(4(-3|(1(2x + 4)^2 + 3(2x + 4)) - 1| + 2)^2 - 5(-3|(1(2x + 4)^2 + 3(2x + 4)) - 1| + 2) + 2)^2 - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 8 | MATH |
[
{
"content": "How many solutions does the equation (-2(-3((1(-5((2(-3|x - 2| + 3) - 1)/(-2(-3|x - 2| + 3) - 1))^2 - 3((2(-3|x - 2| + 3) - 1)/(-2(-3|x - 2| + 3) - 1)) - 4))/(3(-5((2(-3|x - 2| + 3) - 1)/(-2(-3|x - 2| + 3) - 1))^2 - 3((2(-3|x - 2| + 3) - 1)/(-2(-3|x - 2| + 3) - 1)) - 4) - 2))^2 - 5((1(-5((2(-3|x -... | 920 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-1sin(2π((1((-3(-2|(1|x + 1| - 1) + 1| - 3) + 3)/(-2(-2|(1|x + 1| - 1) + 1| - 3) + 3)) - 2)/(-1((-3(-2|(1|x + 1| - 1) + 1| - 3) + 3)/(-2(-2|(1|x + 1| - 1) + 1| - 3) + 3)) - 3)) - 2) - 3) + 2| - 1 and g(x) = 1|((-3((-2(-1((3x - 3... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-2|(2|((3(3(3x + 2)^2 - 1(3x + 2) - 4) + 1)/(-1(3(3x + 2)^2 - 1(3x + 2) - 4))) - 2| - 2) - 2| - 3) + 2| + 1 and g(x) = -1(-3|(2cos(2π(-2|(-1x^2 - 1x + 5) - 1| - 3) - 1) - 2) + 1| + 1)^2 - 1(-3|(2cos(2π(-2|(-1x^2 - 1x + 5)... | 794 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(-1|((-2(3(-3|((3x)/(1x + 1)) + 2| + 3) - 4) - 1)/(1(3(-3|((3x)/(1x + 1)) + 2| + 3) - 4))) + 2| - 1)^2 - 1(-1|((-2(3(-3|((3x)/(1x + 1)) + 2| + 3) - 4) - 1)/(1(3(-3|((3x)/(1x + 1)) + 2| + 3) - 4))) + 2| - 1) + 5 and g(x) = (... | 22 | MATH |
[
{
"content": "How many solutions does the equation 4((-2((3(-5(1(1x + 1) + 1)^2 - 1(1(1x + 1) + 1)) + 3)/(-2(-5(1(1x + 1) + 1)^2 - 1(1(1x + 1) + 1)) + 3)))/(2((3(-5(1(1x + 1) + 1)^2 - 1(1(1x + 1) + 1)) + 3)/(-2(-5(1(1x + 1) + 1)^2 - 1(1(1x + 1) + 1)) + 3)) + 2))^2 - 3((-2((3(-5(1(1x + 1) + 1)^2 - 1(1(1x + 1) + ... | 47 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(5((3(-1cos(3π(1|(1x^2 + 3x - 4) - 1|) + 1) + 3))/(3(-1cos(3π(1|(1x^2 + 3x - 4) - 1|) + 1) + 3) - 1))^2 + 1((3(-1cos(3π(1|(1x^2 + 3x - 4) - 1|) + 1) + 3))/(3(-1cos(3π(1|(1x^2 + 3x - 4) - 1|) + 1) + 3) - 1)) + 1)^2 - 3 and g(x) = -3(-4(2(-2e^(-1(-3x^2... | 16 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(-3((3(2|(-2|(4x - 2) + 1| + 3) + 3| - 3) + 3)/(1(2|(-2|(4x - 2) + 1| + 3) + 3| - 3) - 2))^2 + 2((3(2|(-2|(4x - 2) + 1| + 3) + 3| - 3) + 3)/(1(2|(-2|(4x - 2) + 1| + 3) + 3| - 3) - 2)) + 5) + 4 and g(x) = (3(1|((3(2(-5x^2 - 2) - 4) + 1)/(-3(2(-5x^2 - ... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1(5(-3|(-1(2(-2|x - 1| - 1) + 4)^2 - 2(2(-2|x - 1| - 1) + 4) - 2) + 1| + 2)^2 - 3(-3|(-1(2(-2|x - 1| - 1) + 4)^2 - 2(2(-2|x - 1| - 1) + 4) - 2) + 1| + 2) - 4)^2 + 1(5(-3|(-1(2(-2|x - 1| - 1) + 4)^2 - 2(2(-2|x - 1| - 1) + 4) - 2) + 1| + 2)^2 - 3(-3|(-1(2(-2|x - ... | 16 | MATH |
[
{
"content": "How many solutions does the equation 3|(-4(-1sin(1π(5(-5((-3x + 2)/(-2x - 1)) - 3) - 4) + 1) + 1) - 1) - 1| + 2 = -5(3sin(2π((-1(-4(-5x^2 - 4x + 4)) - 3)/(-2(-4(-5x^2 - 4x + 4)) - 2)) + 3) + 1) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 19 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|(-5(4(3|(-3cos(1π(-2x + 2)) + 3) - 1| + 3)^2 + 5(3|(-3cos(1π(-2x + 2)) + 3) - 1| + 3) - 3) - 5) + 1| + 1 and g(x) = 5(5(1(-4(-1sin(2πx - 3) + 1) - 1) + 4))^2 + 5(5(1(-4(-1sin(2πx - 3) + 1) - 1) + 4)) - 5 where -10 ≤ x ≤ 10.... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5((-1(3(-1((-2(-3x + 1) - 3)/(-2(-3x + 1) - 3))^2 - 5((-2(-3x + 1) - 3)/(-2(-3x + 1) - 3)) + 2) - 1) - 2)/(-2(3(-1((-2(-3x + 1) - 3)/(-2(-3x + 1) - 3))^2 - 5((-2(-3x + 1) - 3)/(-2(-3x + 1) - 3)) + 2) - 1) - 3))^2 - 2((-1(3(-1((-2(... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1sin(2π(2|(-3|(1(2|(-1|x + 3| + 3) - 3| + 2)^2 + 2(2|(-1|x + 3| + 3) - 3| + 2) - 5) + 3| - 2)|) - 1) + 1 and g(x) = (-2((2(3|((2(1x + 2) + 3)/(-2(1x + 2) + 1)) - 2| + 2) - 1)/(-2(3|((2(1x + 2) + 3)/(-2(1x + 2) + 1)) - 2| + 2... | 1571 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(2((1(5((2(-3sin(2πx - 2) + 1) + 3)/(2(-3sin(2πx - 2) + 1) + 2))^2 - 5((2(-3sin(2πx - 2) + 1) + 3)/(2(-3sin(2πx - 2) + 1) + 2)) - 5) + 3)/(1(5((2(-3sin(2πx - 2) + 1) + 3)/(2(-3sin(2πx - 2) + 1) + 2))^2 - 5((2(-3sin(2πx - 2) + 1) + 3)/(2(-3sin(2πx - 2... | 98 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(2|(3|(1(-1(4x^2 - 5x - 3) + 5)^2 + 1(-1(4x^2 - 5x - 3) + 5) + 1) - 2| - 1) + 1| - 3) + 1)/(1(2|(3|(1(-1(4x^2 - 5x - 3) + 5)^2 + 1(-1(4x^2 - 5x - 3) + 5) + 1) - 2| - 1) + 1| - 3)) and g(x) = (1(-1(3e^(-1(2(4x - 1) + 5)) - 2)^2 -... | 9 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2((1(1|(-3(2|(1|x + 1| - 3) + 2| - 3)^2 - 1(2|(1|x + 1| - 3) + 2| - 3) - 2) + 3| + 2) - 3)/(-1(1|(-3(2|(1|x + 1| - 3) + 2| - 3)^2 - 1(2|(1|x + 1| - 3) + 2| - 3) - 2) + 3| + 2) + 2)))/(2((1(1|(-3(2|(1|x + 1| - 3) + 2| - 3)^2 ... | 8 | MATH |
[
{
"content": "How many solutions does the equation -5(1|(3(3|(2sin(3π(-1x - 3)) + 2) - 3| + 3)^2 - 4(3|(2sin(3π(-1x - 3)) + 2) - 3| + 3) - 5) + 2| - 3)^2 - 5(1|(3(3|(2sin(3π(-1x - 3)) + 2) - 3| + 3)^2 - 4(3|(2sin(3π(-1x - 3)) + 2) - 3| + 3) - 5) + 2| - 3) + 2 = -2(1|(1(-5((-2x - 2)/(2x + 1)) - 2) - 3) + 1| - 2)... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(2((2((3((2(-4x^2 - 1x - 5) + 2)/(-1(-4x^2 - 1x - 5) - 3)) + 1)/(-3((2(-4x^2 - 1x - 5) + 2)/(-1(-4x^2 - 1x - 5) - 3)) + 2)) + 2)/(1((3((2(-4x^2 - 1x - 5) + 2)/(-1(-4x^2 - 1x - 5) - 3)) + 1)/(-3((2(-4x^2 - 1x - 5) + 2)/(-1(-... | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(-5(-2sin(3π((2(1|(-1x - 5)| + 3) + 3)/(1(1|(-1x - 5)| + 3) - 2)) + 2) + 3)^2 - 4(-2sin(3π((2(1|(-1x - 5)| + 3) + 3)/(1(1|(-1x - 5)| + 3) - 2)) + 2) + 3) - 4) + 1| + 1 and g(x) = (-1(2(-3cos(3π(-1|((1x + 2)/(-1x + 3)) - 1| + 2) + 1) - 1) - 4) + 3)/(... | 28 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(3|((-1(-1|((-3((-2x - 3)/(2x)) - 2)/(-1((-2x - 3)/(2x)) + 2)) + 1| - 2) + 1)/(3(-1|((-3((-2x - 3)/(2x)) - 2)/(-1((-2x - 3)/(2x)) + 2)) + 1| - 2) - 1)) + 1| - 2) + 4 and g(x) = -3(-3(-2(3(5x^2 + 3x - 2) + 4) + 4) - 1)^2 + 5(-3(-2... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(5(2|(-1(-2e^(1(-4x + 3))) + 1) + 2| - 2)^2 + 5(2|(-1(-2e^(1(-4x + 3))) + 1) + 2| - 2) - 5) - 1)/(3(5(2|(-1(-2e^(1(-4x + 3))) + 1) + 2| - 2)^2 + 5(2|(-1(-2e^(1(-4x + 3))) + 1) + 2| - 2) - 5) + 1) and g(x) = 1|(3((2(3(5x^2 + 2x + 1) - 5))/(-2(3(5x^2 ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-3cos(1π(-5(-3(-4(1x - 1)^2 + 1(1x - 1) + 2)^2 - 5(-4(1x - 1)^2 + 1(1x - 1) + 2) + 4)^2 + 1(-3(-4(1x - 1)^2 + 1(1x - 1) + 2)^2 - 5(-4(1x - 1)^2 + 1(1x - 1) + 2) + 4) + 4) - 2) - 3) + 4 and g(x) = (2(-3(-3sin(3π(3(-5x^2 - 4x... | 187 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(-1((-1(3(-2(-1|x - 1| + 2) + 2) + 4) + 2)/(-1(3(-2(-1|x - 1| + 2) + 2) + 4) - 3))^2 + 2((-1(3(-2(-1|x - 1| + 2) + 2) + 4) + 2)/(-1(3(-2(-1|x - 1| + 2) + 2) + 4) - 3)) + 5)| and g(x) = 2sin(2π(-2|((1(1|(-3|x - 1| + 2) - 1| - 1) + 1)/(2(1|(-3|x - 1| +... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1(-4(-5((1(5(-3|x - 3| + 1)^2 + 2(-3|x - 3| + 1) + 2) + 3)/(-1(5(-3|x - 3| + 1)^2 + 2(-3|x - 3| + 1) + 2) + 3))^2 + 2((1(5(-3|x - 3| + 1)^2 + 2(-3|x - 3| + 1) + 2) + 3)/(-1(5(-3|x - 3| + 1)^2 + 2(-3|x - 3| + 1) + 2) + 3)) + 3)^2 + 3(-5((1(5(-3|x - 3| + 1)^2 + 2... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(2sin(2π(-2(5(-4(5x^2 + 4x + 2) - 3) + 1) + 2) - 2)) - 5 and g(x) = 2cos(2π(2|(2(-4((-3x + 1)/(-1x + 3))^2 + 2((-3x + 1)/(-1x + 3)) + 1)) + 3| + 2) - 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answ... | 1201 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(-2|(4(-2(1(3sin(1πx - 2)) - 1) + 1)^2 - 1(-2(1(3sin(1πx - 2)) - 1) + 1) + 5) + 3|) - 1)/(-2(-2|(4(-2(1(3sin(1πx - 2)) - 1) + 1)^2 - 1(-2(1(3sin(1πx - 2)) - 1) + 1) + 5) + 3|) + 3) and g(x) = 5(-3|(1((3(4x^2 - 2x - 3) + 3)/(-1(4x^2 - 2x - 3) - 3))^2... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(2cos(1π(-3|((2(2((-2x + 1)/(1x - 3))^2 + 2((-2x + 1)/(1x - 3)) - 3) - 2)/(-3(2((-2x + 1)/(1x - 3))^2 + 2((-2x + 1)/(1x - 3)) - 3) + 4)) + 3| + 2) + 3) - 2) - 2)/(-3(2cos(1π(-3|((2(2((-2x + 1)/(1x - 3))^2 + 2((-2x + 1)/(1x - 3)) ... | 12 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2((-2((2(-3(-1((1x + 1)/(-1x - 2)) + 1)^2 - 3(-1((1x + 1)/(-1x - 2)) + 1) - 1))/(-1(-3(-1((1x + 1)/(-1x - 2)) + 1)^2 - 3(-1((1x + 1)/(-1x - 2)) + 1) - 1) - 1)) + 1)/(-1((2(-3(-1((1x + 1)/(-1x - 2)) + 1)^2 - 3(-1((1x + 1)/(-1x - 2)... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(2((3((1(-1(-4x - 5)^2 - 2(-4x - 5) - 4))/(2(-1(-4x - 5)^2 - 2(-4x - 5) - 4) - 3)) - 3)/(-3((1(-1(-4x - 5)^2 - 2(-4x - 5) - 4))/(2(-1(-4x - 5)^2 - 2(-4x - 5) - 4) - 3)) + 2))^2 + 5((3((1(-1(-4x - 5)^2 - 2(-4x - 5) - 4))/(2(-1(-4x ... | 30 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(2(-3e^(1(-1(-3(-5x)^2 + 3(-5x) - 1)^2 + 5(-3(-5x)^2 + 3(-5x) - 1) - 4)) - 3) + 4) - 2| + 2 and g(x) = -1(1(5(4((1x + 1)/(3x - 2))^2 + 3((1x + 1)/(3x - 2)) - 4)^2 - 5(4((1x + 1)/(3x - 2))^2 + 3((1x + 1)/(3x - 2)) - 4) + 3)^2 - 5... | 4 | MATH |
[
{
"content": "How many solutions does the equation -2(-2|(1|(2cos(2π(-2|((2x - 3)/(2x)) + 3| - 3) - 3) + 2) - 2| + 3) - 3| + 1) + 1 = (3((2((1(1(-2|x - 1| + 1) + 3) + 3)/(2(1(-2|x - 1| + 1) + 3) - 1)))/(3((1(1(-2|x - 1| + 1) + 3) + 3)/(2(1(-2|x - 1| + 1) + 3) - 1)) - 2)) - 3)/(-2((2((1(1(-2|x - 1| + 1) + 3) + 3... | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-3sin(1π(1|(-2|(1|(-1x - 3) + 2| - 2) - 2|) - 3| - 3) - 3) - 2) - 1)/(2(-3sin(1π(1|(-2|(1|(-1x - 3) + 2| - 2) - 2|) - 3| - 3) - 3) - 2) - 1) and g(x) = 5(-1((-3(-1(1x^2 - 2x - 3)^2 - 4(1x^2 - 2x - 3) + 3) + 1)/(-3(-1(1x^2 - 2x ... | 43 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(-1(5(-4(-3cos(3π(-1x + 3)) - 3) - 2) - 4)^2 + 5(5(-4(-3cos(3π(-1x + 3)) - 3) - 2) - 4) + 3) + 1 and g(x) = -3(-1|((-2(2(-1e^(-2x) - 2) + 3) - 2)/(-2(2(-1e^(-2x) - 2) + 3) - 2)) + 2| - 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer... | 120 | MATH |
[
{
"content": "How many solutions does the equation (-2(1|(-5((3(3|(-1x^2 - 1x - 4) + 1| + 1))/(2(3|(-1x^2 - 1x - 4) + 1| + 1) + 1)) - 1) - 3| - 3) + 3)/(2(1|(-5((3(3|(-1x^2 - 1x - 4) + 1| + 1))/(2(3|(-1x^2 - 1x - 4) + 1| + 1) + 1)) - 1) - 3| - 3) + 3) = -1(-1|(4(-2cos(1π(-2x^2 - 4x + 3) - 1) + 3) - 5)| + 1) - 5... | 404 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5((-1((1((3((2(-1|x - 1| + 1) - 1)/(2(-1|x - 1| + 1) - 3)) - 3)/(-1((2(-1|x - 1| + 1) - 1)/(2(-1|x - 1| + 1) - 3)) - 2)) + 1)/(-2((3((2(-1|x - 1| + 1) - 1)/(2(-1|x - 1| + 1) - 3)) - 3)/(-1((2(-1|x - 1| + 1) - 1)/(2(-1|x - 1| ... | 154 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(-1((-2(2e^(-1(-3|(3|x + 1| + 3) + 2| - 1)) - 2) - 3)/(-2(2e^(-1(-3|(3|x + 1| + 3) + 2| - 1)) - 2) + 1))^2 + 5((-2(2e^(-1(-3|(3|x + 1| + 3) + 2| - 1)) - 2) - 3)/(-2(2e^(-1(-3|(3|x + 1| + 3) + 2| - 1)) - 2) + 1)) - 4))/(3(-... | 110 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(-1((-3(-5((-2((-3x + 1)/(2x + 2)) - 1)/(-2((-3x + 1)/(2x + 2)) + 1))^2 - 5((-2((-3x + 1)/(2x + 2)) - 1)/(-2((-3x + 1)/(2x + 2)) + 1)) + 4) + 3)/(-1(-5((-2((-3x + 1)/(2x + 2)) - 1)/(-2((-3x + 1)/(2x + 2)) + 1))^2 - 5((-2((-3x + 1)/(2x + 2)) - 1)/(-2(... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-3|(-1|(1(-1cos(2π(5x^2 - 1x + 1) - 3) + 2) - 3) - 2| + 2) + 3| - 1)^2 - 3(-3|(-1|(1(-1cos(2π(5x^2 - 1x + 1) - 3) + 2) - 3) - 2| + 2) + 3| - 1) + 3 and g(x) = -5(-2(5(-1(2sin(2πx - 2))^2 + 3(2sin(2πx - 2)) + 4) + 2) + 5) - 4 intersect in the interval... | 730 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(1|((2(-2|(4(2|x - 3| - 3) + 1) + 3| - 1) + 1)/(1(-2|(4(2|x - 3| - 3) + 1) + 3| - 1) + 2)) - 2| + 1) - 1)/(3(1|((2(-2|(4(2|x - 3| - 3) + 1) + 3| - 1) + 1)/(1(-2|(4(2|x - 3| - 3) + 1) + 3| - 1) + 2)) - 2| + 1) - 1) and g(x) = 1sin(2π(2|(4((3((-2x)/(1x... | 220 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(5(-1|(3(-2e^(1((-3x - 3)/(3x - 2))) + 3) + 5) - 2| + 1) - 3)^2 - 5(5(-1|(3(-2e^(1((-3x - 3)/(3x - 2))) + 3) + 5) - 2| + 1) - 3) + 1 and g(x) = -2cos(3π(1((1(3(-1x - 5) + 5) + 3)/(-2(3(-1x - 5) + 5) + 3)) - 5)) + 2 intersect in the interval [-10, 10]... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1(-4((-3(-1(-5((1x - 1)/(3x + 1))^2 + 1((1x - 1)/(3x + 1)) + 4)^2 - 5(-5((1x - 1)/(3x + 1))^2 + 1((1x - 1)/(3x + 1)) + 4) - 3) + 2)/(-2(-1(-5((1x - 1)/(3x + 1))^2 + 1((1x - 1)/(3x + 1)) + 4)^2 - 5(-5((1x - 1)/(3x + 1))^2 + 1((1x - 1)/(3x + 1)) + 4) - 3) - 2))^... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|((1((3(-4((2((-2x + 3)/(1x + 2)) - 1)/(3((-2x + 3)/(1x + 2))))^2 - 3((2((-2x + 3)/(1x + 2)) - 1)/(3((-2x + 3)/(1x + 2))))) + 3)/(-3(-4((2((-2x + 3)/(1x + 2)) - 1)/(3((-2x + 3)/(1x + 2))))^2 - 3((2((-2x + 3)/(1x + 2)) - 1)/(3((-2x + 3)/(1x + 2))))) +... | 656 | MATH |
[
{
"content": "How many solutions does the equation (2((-3(1(1(2|(1x + 1)| - 3) + 5)) - 1)/(2(1(1(2|(1x + 1)| - 3) + 5)))) - 1)/(3((-3(1(1(2|(1x + 1)| - 3) + 5)) - 1)/(2(1(1(2|(1x + 1)| - 3) + 5)))) - 1) = 1(-2cos(3π(-2|((1(-3x - 2))/(-1(-3x - 2) + 2)) + 3| + 1) - 3))^2 + 2(-2cos(3π(-2|((1(-3x - 2))/(-1(-3x - 2)... | 63 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-1|(1|(-3(2|(5x^2 + 3x - 3) - 2| - 2)^2 - 1(2|(5x^2 + 3x - 3) - 2| - 2) + 3) + 1| + 3) - 1| - 3) - 3| - 1 and g(x) = (-1(-2cos(1π(2(2(-4x^2 - 4) + 1)^2 + 3(2(-4x^2 - 4) + 1) + 3) + 3) - 1) - 3)/(-2(-2cos(1π(2(2(-4x^2 - 4) + 1)^2 + 3(2(-4x^2 - 4) + 1... | 9 | MATH |
[
{
"content": "How many solutions does the equation (-2(-1(3(5(-1(-2cos(1πx - 3))^2 - 1(-2cos(1πx - 3)))^2 + 5(-1(-2cos(1πx - 3))^2 - 1(-2cos(1πx - 3))) + 4)^2 - 2(5(-1(-2cos(1πx - 3))^2 - 1(-2cos(1πx - 3)))^2 + 5(-1(-2cos(1πx - 3))^2 - 1(-2cos(1πx - 3))) + 4) - 5)^2 + 5(3(5(-1(-2cos(1πx - 3))^2 - 1(-2cos(1πx - ... | 16 | MATH |
[
{
"content": "How many solutions does the equation (-1(-3cos(3π(5(1(4(-1x - 1) + 2))^2 - 4(1(4(-1x - 1) + 2)) + 4) - 3)) + 1)/(1(-3cos(3π(5(1(4(-1x - 1) + 2))^2 - 4(1(4(-1x - 1) + 2)) + 4) - 3)) + 2) = -5((2(-2|((-3(1e^(1x) + 2) + 3)/(1(1e^(1x) + 2)))| + 1) + 2)/(3(-2|((-3(1e^(1x) + 2) + 3)/(1(1e^(1x) + 2)))| +... | 314 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(5((-1(2(5(-2cos(3πx + 3) - 1)^2 - 4(-2cos(3πx + 3) - 1) - 5) - 1) - 2)/(-3(2(5(-2cos(3πx + 3) - 1)^2 - 4(-2cos(3πx + 3) - 1) - 5) - 1) - 2)) - 3) + 1)/(-3(5((-1(2(5(-2cos(3πx + 3) - 1)^2 - 4(-2cos(3πx + 3) - 1) - 5) - 1) ... | 38 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(2(-3(2|(2|(5x - 1) - 2| - 3) + 2| - 2) - 5)^2 - 4(-3(2|(2|(5x - 1) - 2| - 3) + 2| - 2) - 5) - 5) - 3| - 1 and g(x) = (1((3((3(3cos(3π((1x - 1)/(3x + 1)) + 2) - 3) - 1)/(1(3cos(3π((1x - 1)/(3x + 1)) + 2) - 3) - 3)))/(2((3(3cos(3π((1x - 1)/(3x + 1)) +... | 5 | MATH |
[
{
"content": "How many solutions does the equation -1(2sin(2π(-2|(-5(-5((1x)/(3x - 2)) - 1)^2 + 3(-5((1x)/(3x - 2)) - 1)) + 2| - 3) - 1)) - 1 = -3|(1(3|(-3|((-3x)/(-1x - 2)) - 2| - 1) + 1|)^2 - 4(3|(-3|((-3x)/(-1x - 2)) - 2| - 1) + 1|) - 4) + 2| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(5(-1cos(1π((1(-2((2x - 2)/(1x))^2 - 4((2x - 2)/(1x)) + 4))/(-1(-2((2x - 2)/(1x))^2 - 4((2x - 2)/(1x)) + 4) - 3)) - 1) - 2)^2 + 2(-1cos(1π((1(-2((2x - 2)/(1x))^2 - 4((2x - 2)/(1x)) + 4))/(-1(-2((2x - 2)/(1x))^2 - 4((2x - 2)/(1x)) + 4) - 3)) - 1) - 2)... | 17 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(2e^(2(3|(-2|(-5(-2|x - 3| + 1) + 3) + 3| - 2) + 2| - 2))) - 1)/(3(2e^(2(3|(-2|(-5(-2|x - 3| + 1) + 3) + 3| - 2) + 2| - 2))) - 2) and g(x) = -3sin(3π(-5(-1|(-5(3|x + 3|) - 2) + 1| + 1) + 2) + 1) + 1 intersect in the interval [-10, 10].\n\nPresent th... | 1763 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(1(1e^(-1((3(-5((-3x - 3)/(-1x + 2))^2 - 5((-3x - 3)/(-1x + 2)) - 1) + 2)/(2(-5((-3x - 3)/(-1x + 2))^2 - 5((-3x - 3)/(-1x + 2)) - 1) - 1))) + 1) + 4) - 1 and g(x) = -4(-3(1sin(1π(2(-2x^2 + 4x - 4) + 4) - 3) - 2) - 2)^2 - 4(-3(1sin(1π(2(-2x^2 + 4x - 4... | 713 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(-3(1sin(1π(-2|(-1(4x^2 - 1x - 2)^2 - 2(4x^2 - 1x - 2) - 2) + 1| + 1)) + 2)^2 + 4(1sin(1π(-2|(-1(4x^2 - 1x - 2)^2 - 2(4x^2 - 1x - 2) - 2) + 1| + 1)) + 2) - 2) - 2| - 2 and g(x) = -4(-2(-1|(2e^(-1(-3x - 5)) - 1) + 1| - 1)^2 - 4(-... | 9 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-1(2(1|((2(1|x - 1| + 1) + 3)/(-2(1|x - 1| + 1) + 3)) + 2| - 2)^2 - 3(1|((2(1|x - 1| + 1) + 3)/(-2(1|x - 1| + 1) + 3)) + 2| - 2) - 5) - 2) + 2)/(-3(-1(2(1|((2(1|x - 1| + 1) + 3)/(-2(1|x - 1| + 1) + 3)) + 2| - 2)^2 - 3(1|((2(1|x ... | 94 | MATH |
[
{
"content": "How many solutions does the equation -3(-2|((1(3(-2|(3x^2 - 3) + 3| - 1) + 4) + 1)/(-2(3(-2|(3x^2 - 3) + 3| - 1) + 4) + 3)) + 2| - 1) + 3 = (2(2(-4(1(1x^2 + 1x - 4) - 5) - 5)^2 + 4(-4(1(1x^2 + 1x - 4) - 5) - 5) - 1) + 2)/(-1(2(-4(1(1x^2 + 1x - 4) - 5) - 5)^2 + 4(-4(1(1x^2 + 1x - 4) - 5) - 5) - 1) ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(2(3(-2(-5(3cos(1πx - 1) - 2)) + 5)^2 + 1(-2(-5(3cos(1πx - 1) - 2)) + 5) - 2) + 2) and g(x) = 4((-2(1|(3e^(2(2|x + 1|))) + 1| - 3) + 3)/(2(1|(3e^(2(2|x + 1|))) + 1| - 3) - 1)) - 5 where -10 ≤ x ≤ 10.\n\nPresent the answer i... | 40 | MATH |
[
{
"content": "How many solutions does the equation 2(4(-1(2|(3|(4x - 2) + 1| - 1)| + 2) - 3) - 3)^2 + 5(4(-1(2|(3|(4x - 2) + 1| - 1)| + 2) - 3) - 3) + 2 = (2(1sin(1π(-1(-1|(-1x - 2) + 2| - 3)^2 - 3(-1|(-1x - 2) + 2| - 3) - 1) + 2) + 1) + 1)/(-1(1sin(1π(-1(-1|(-1x - 2) + 2| - 3)^2 - 3(-1|(-1x - 2) + 2| - 3) - 1)... | 8 | MATH |
[
{
"content": "How many solutions does the equation 4((-1(-1(1(-2e^(-1(1|x - 2| + 2)) - 1) + 5) - 1) - 1)/(2(-1(1(-2e^(-1(1|x - 2| + 2)) - 1) + 5) - 1)))^2 - 2((-1(-1(1(-2e^(-1(1|x - 2| + 2)) - 1) + 5) - 1) - 1)/(2(-1(1(-2e^(-1(1|x - 2| + 2)) - 1) + 5) - 1))) = (-3(2cos(1π(3(2(-3x + 1) + 3)^2 + 2(2(-3x + 1) + 3)... | 865 | MATH |
[
{
"content": "How many solutions does the equation -1|(3|(3|(3(-5((-1x + 3)/(-2x + 1)) + 5)) + 2| - 1) - 3| + 1)| - 2 = 4(4(-4(-1(1e^(1x) - 2)^2 - 2(1e^(1x) - 2) - 1)^2 - 2(-1(1e^(1x) - 2)^2 - 2(1e^(1x) - 2) - 1) - 1) + 4) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(-1|(1|(-5(3|(1x^2 + 2x - 3) - 2| - 1) - 4) - 1| - 3) - 1| + 2) + 3| - 3 and g(x) = 2(-2|((-1(-1(1x - 2) - 2) + 2)/(2(-1(1x - 2) - 2) + 1)) - 3| - 3)^2 + 1(-2|((-1(-1(1x - 2) - 2) + 2)/(2(-1(1x - 2) - 2) + 1)) - 3| - 3) + 1... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(-3(-3(3(2(1x)^2 - 1(1x)) - 5)^2 + 3(3(2(1x)^2 - 1(1x)) - 5) + 5) - 1) + 5 and g(x) = -1e^(1(-4(5((-3(-2|x - 2|) - 2)/(2(-2|x - 2|) + 1))^2 - 4((-3(-2|x - 2|) - 2)/(2(-2|x - 2|) + 1)) + 4))) - 1 intersect in the interval [-10, 10].\n\nPresent the ans... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2cos(2π((-2(3|((1(-1(1x + 2)^2 + 1(1x + 2) - 2) + 1)/(3(-1(1x + 2)^2 + 1(1x + 2) - 2) - 2)) + 1| + 3) + 2)/(-2(3|((1(-1(1x + 2)^2 + 1(1x + 2) - 2) + 1)/(3(-1(1x + 2)^2 + 1(1x + 2) - 2) - 2)) + 1| + 3) - 3)) + 1) - 2 and g(x) = (1((... | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(3cos(3π((-2(4(1(5x + 4) - 5)))/(-2(4(1(5x + 4) - 5)))) - 2)) - 3 and g(x) = -5(-1|(2cos(3π((1((-2x - 1)/(-3x - 2)) + 1)/(1((-2x - 1)/(-3x - 2)) - 2)) + 3) - 1)| + 3)^2 + 5(-1|(2cos(3π((1((-2x - 1)/(-3x - 2)) + 1)/(1((-2x -... | 22 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-5(-3(-3e^(-2(5(2x - 5)^2 - 3(2x - 5) + 2)) + 2)^2 - 4(-3e^(-2(5(2x - 5)^2 - 3(2x - 5) + 2)) + 2) - 5) + 5) + 4 and g(x) = 2(-2((1((-1(-2cos(2πx - 2) + 1) - 3)/(-2(-2cos(2πx - 2) + 1) + 1)) + 2)/(-1((-1(-2cos(2πx - 2) + 1)... | 37 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5((2(-1e^(2((2(1|(-1|x + 2| + 2)| - 3) + 3)/(-3(1|(-1|x + 2| + 2)| - 3)))) + 1) + 3)/(-1(-1e^(2((2(1|(-1|x + 2| + 2)| - 3) + 3)/(-3(1|(-1|x + 2| + 2)| - 3)))) + 1) - 2))^2 + 2((2(-1e^(2((2(1|(-1|x + 2| + 2)| - 3) + 3)/(-3(1|(-1|x +... | 14 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(-3cos(1π(-2|((3(-1((-1x + 1)/(-1x - 1)) - 3) - 1)/(-2(-1((-1x + 1)/(-1x - 1)) - 3) + 3)) + 1| - 3) - 2)) - 1 and g(x) = (2(1e^(-1(1(-3|(5x) + 2|)^2 - 1(-3|(5x) + 2|) + 4))))/(3(1e^(-1(1(-3|(5x) + 2|)^2 - 1(-3|(5x) + 2|) + 4))) - ... | 18 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-1(-1(-2(-1(-2x^2 - 3x + 3)^2 - 5(-2x^2 - 3x + 3) + 4) + 4)^2 - 3(-2(-1(-2x^2 - 3x + 3)^2 - 5(-2x^2 - 3x + 3) + 4) + 4) - 2) - 2) + 2| and g(x) = (-2(3|((-3(4(-1|x + 2| - 3)^2 - 3(-1|x + 2| - 3) + 2))/(1(4(-1|x + 2| - 3)^2 - 3(... | 8 | MATH |
[
{
"content": "How many solutions does the equation -1|((-2(-3cos(3π(3|(-1(2|x + 1| - 3)^2 - 2(2|x + 1| - 3) + 2) + 2|) - 2) - 1) - 3)/(-3(-3cos(3π(3|(-1(2|x + 1| - 3)^2 - 2(2|x + 1| - 3) + 2) + 2|) - 2) - 1) + 2))| + 2 = (-3((-3(3|((2(1|x + 2| - 1))/(3(1|x + 2| - 1) + 2))| - 3))/(1(3|((2(1|x + 2| - 1))/(3(1|x +... | 872 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(2(-1sin(3π(3|(-4(4x^2 - 5x + 5) - 1) + 2| + 3) + 3) - 1) - 1)^2 - 5(2(-1sin(3π(3|(-4(4x^2 - 5x + 5) - 1) + 2| + 3) + 3) - 1) - 1) + 5 and g(x) = 4(3(3sin(3π(3|((3x + 2)/(-3x - 3)) + 3| + 1) - 2) + 1)^2 + 3(3sin(3π(3|((3x + 2)/(-3x - 3)) + 3| + 1) - ... | 111 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(1(-2sin(1π(-1|(4((-2x + 1)/(1x + 3))^2 + 5((-2x + 1)/(1x + 3)) - 1)| - 3) - 2) - 3) + 3) + 1)/(-3(1(-2sin(1π(-1|(4((-2x + 1)/(1x + 3))^2 + 5((-2x + 1)/(1x + 3)) - 1)| - 3) - 2) - 3) + 3) + 2) and g(x) = -1|(2cos(2π(1((-2(... | 17 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1cos(2π(5(-2((-3(2(3|x + 1| - 2)^2 + 1))/(-3(2(3|x + 1| - 2)^2 + 1) + 2))^2 + 4((-3(2(3|x + 1| - 2)^2 + 1))/(-3(2(3|x + 1| - 2)^2 + 1) + 2)) + 2)^2 + 4(-2((-3(2(3|x + 1| - 2)^2 + 1))/(-3(2(3|x + 1| - 2)^2 + 1) + 2))^2 + 4((-... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2cos(1π((-2(2|(-3(-4(-2|x + 2| + 1)^2 + 4(-2|x + 2| + 1) + 1) + 1) + 2| + 2))/(-1(2|(-3(-4(-2|x + 2| + 1)^2 + 4(-2|x + 2| + 1) + 1) + 1) + 2| + 2) + 3)) + 2) + 1 and g(x) = (1((-1(-1(-3(-2|x - 2|) - 2)^2 - 3(-3(-2|x - 2|) - 2) + 5)... | 44 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1((-1((1(1(5(1cos(2πx + 3) - 2))^2 + 5(5(1cos(2πx + 3) - 2)) - 5) - 3)/(-3(1(5(1cos(2πx + 3) - 2))^2 + 5(5(1cos(2πx + 3) - 2)) - 5) + 1)))/(3((1(1(5(1cos(2πx + 3) - 2))^2 + 5(5(1cos(2πx + 3) - 2)) - 5) - 3)/(-3(1(5(1cos(2πx + 3) - 2))^2 + 5(5(1cos(2πx... | 10 | MATH |
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