messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|((-2(4(-1|(-1|(5x^2 + 5x + 5) + 1|) + 3| - 3) + 4) - 1)/(3(4(-1|(-1|(5x^2 + 5x + 5) + 1|) + 3| - 3) + 4) - 1)) + 3| - 2 and g(x) = 1|(2|(2|(-1(4x^2 - 4x - 2)) - 2| - 2) - 2| - 1) + 1| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-1((3(3(-1(-1x + 5)^2 - 1(-1x + 5) - 4)^2 + 5(-1(-1x + 5)^2 - 1(-1x + 5) - 4) - 3) + 3)/(1(3(-1(-1x + 5)^2 - 1(-1x + 5) - 4)^2 + 5(-1(-1x + 5)^2 - 1(-1x + 5) - 4) - 3) - 3)) + 1) - 3)/(2(-1((3(3(-1(-1x + 5)^2 - 1(-1x + 5) - 4)^2 + 5(-1(-1x + 5)^2 - ... | 80 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(5(-2|(3sin(2π(3(1x^2 - 2x - 4) + 1) - 3)) - 2| + 2) - 2) + 3| - 2 and g(x) = 2|(1cos(1π(5(3(1|x + 2|)^2 - 2(1|x + 2|)) + 4)) + 3) + 1| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 690 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5((-2((1(1|(-3|(-3|x + 1| - 3) - 1| - 3) + 1| + 2) + 3)/(-1(1|(-3|(-3|x + 1| - 3) - 1| - 3) + 1| + 2) + 2)) - 2)/(-1((1(1|(-3|(-3|x + 1| - 3) - 1| - 3) + 1| + 2) + 3)/(-1(1|(-3|(-3|x + 1| - 3) - 1| - 3) + 1| + 2) + 2)) - 2))^2 - 5((-2((1(1|(-3|(-3|x +... | 4 | MATH |
[
{
"content": "How many solutions does the equation (3(4(-4(1(4(3x - 5) - 4)^2 + 1(4(3x - 5) - 4) - 2)^2 + 1(1(4(3x - 5) - 4)^2 + 1(4(3x - 5) - 4) - 2) + 1) + 1) + 1)/(1(4(-4(1(4(3x - 5) - 4)^2 + 1(4(3x - 5) - 4) - 2)^2 + 1(1(4(3x - 5) - 4)^2 + 1(4(3x - 5) - 4) - 2) + 1) + 1) - 3) = 2sin(3π(2(-1(-2|(-1x^2 - 4x -... | 4 | MATH |
[
{
"content": "How many solutions does the equation -3(-3|((2(3|((2(3x^2 + 5x - 2) - 2)/(3(3x^2 + 5x - 2) + 2)) + 2| + 1) + 2)/(3(3|((2(3x^2 + 5x - 2) - 2)/(3(3x^2 + 5x - 2) + 2)) + 2| + 1) + 3)) - 2| + 1) + 4 = 2|((-2(-5((-3(-1x^2 - 5x + 2) - 2)/(-2(-1x^2 - 5x + 2)))^2 - 5((-3(-1x^2 - 5x + 2) - 2)/(-2(-1x^2 - 5... | 8 | MATH |
[
{
"content": "How many solutions does the equation 2((-1(4(-3sin(3π(-4(2|x - 2| - 1)^2 + 1(2|x - 2| - 1) - 5) + 1) - 3)^2 + 3(-3sin(3π(-4(2|x - 2| - 1)^2 + 1(2|x - 2| - 1) - 5) + 1) - 3) + 5) + 1)/(-3(4(-3sin(3π(-4(2|x - 2| - 1)^2 + 1(2|x - 2| - 1) - 5) + 1) - 3)^2 + 3(-3sin(3π(-4(2|x - 2| - 1)^2 + 1(2|x - 2| -... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2((1(-3|(-3|(-4(2|x + 1|) + 5) + 1| - 3) + 3| + 3) - 3)/(-3(-3|(-3|(-4(2|x + 1|) + 5) + 1| - 3) + 3| + 3) + 2))^2 - 5((1(-3|(-3|(-4(2|x + 1|) + 5) + 1| - 3) + 3| + 3) - 3)/(-3(-3|(-3|(-4(2|x + 1|) + 5) + 1| - 3) + 3| + 3) + 2)) - 4 and g(x) = 5(-1|(-1(... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|((2(2(-3|(1|(1x - 4) - 2| - 2) - 3| + 2) - 2) - 1)/(2(2(-3|(1|(1x - 4) - 2| - 2) - 3| + 2) - 2) + 3)) + 3| + 3 and g(x) = (1(-3(1(2sin(1π(-1|x - 1|) - 3) - 1) + 3) + 4) + 3)/(-2(-3(1(2sin(1π(-1|x - 1|) - 3) - 1) + 3) + 4) + 3) intersect in the interv... | 18 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|((1(-1|(2cos(3π((-3(5x) + 3)/(-3(5x) + 1)) - 2) - 3) - 3| - 1) + 2)/(-1(-1|(2cos(3π((-3(5x) + 3)/(-3(5x) + 1)) - 2) - 3) - 3| - 1) - 1)) + 3| + 3 and g(x) = -1(-1((2(-2|(3sin(3πx - 1)) + 3| + 1) + 3)/(1(-2|(3sin(3πx - 1)) + 3| +... | 120 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(1|(5(-4(-4(2x^2 - 5x - 1) + 2) - 5)^2 - 2(-4(-4(2x^2 - 5x - 1) + 2) - 5) + 3) + 3| + 1) + 3 and g(x) = -4(1(-1(-3((-3x - 2)/(2x - 1))^2 - 5((-3x - 2)/(2x - 1)) - 2)^2 - 1(-3((-3x - 2)/(2x - 1))^2 - 5((-3x - 2)/(2x - 1)) - ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(2|(-1|(-2|(4(1e^(-2x) + 3)^2 + 2(1e^(-2x) + 3) - 3) - 1| + 3) - 1| + 3) + 2| + 2) + 1)/(-1(2|(-1|(-2|(4(1e^(-2x) + 3)^2 + 2(1e^(-2x) + 3) - 3) - 1| + 3) - 1| + 3) + 2| + 2) + 2) and g(x) = (-1(-3((-2(4(1|x - 3| - 3)^2 + 1(1|x - 3| - 3) - 5) - 1)/(1... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|((3(1((2(3|(-5x^2 - 2x - 1) - 2| - 1) + 2)/(-1(3|(-5x^2 - 2x - 1) - 2| - 1) - 1))^2 + 5((2(3|(-5x^2 - 2x - 1) - 2| - 1) + 2)/(-1(3|(-5x^2 - 2x - 1) - 2| - 1) - 1)) + 1))/(-3(1((2(3|(-5x^2 - 2x - 1) - 2| - 1) + 2)/(-1(3|(-5... | 120 | MATH |
[
{
"content": "How many solutions does the equation -2|(4(-4((1(-5((3x + 1)/(3x)) + 5) + 3)/(1(-5((3x + 1)/(3x)) + 5)))^2 - 5((1(-5((3x + 1)/(3x)) + 5) + 3)/(1(-5((3x + 1)/(3x)) + 5))) + 2)^2 - 4(-4((1(-5((3x + 1)/(3x)) + 5) + 3)/(1(-5((3x + 1)/(3x)) + 5)))^2 - 5((1(-5((3x + 1)/(3x)) + 5) + 3)/(1(-5((3x + 1)/(3x... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|((-1(-1(-1|(3|((1x)/(-1x - 3))| + 3) - 2| + 3)^2 + 1))/(-2(-1(-1|(3|((1x)/(-1x - 3))| + 3) - 2| + 3)^2 + 1))) + 1| + 2 and g(x) = -1(-5(1|(-1(2x + 4)^2 + 2(2x + 4) + 2) + 1| - 3)^2 - 2(1|(-1(2x + 4)^2 + 2(2x + 4) + 2) + 1| - 3) -... | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(5(-4(1(-1cos(2π(5x^2 + 5x + 1) + 3) + 2)^2 - 1(-1cos(2π(5x^2 + 5x + 1) + 3) + 2) - 3)^2 + 2) + 3) - 2)/(-1(5(-4(1(-1cos(2π(5x^2 + 5x + 1) + 3) + 2)^2 - 1(-1cos(2π(5x^2 + 5x + 1) + 3) + 2) - 3)^2 + 2) + 3)) and g(x) = (2(-1(1(-2(5x + 2)^2 + 5(5x + 2)... | 7 | MATH |
[
{
"content": "How many solutions does the equation (-2(2(-1((-2(4(3x^2 + 4x)^2 - 4(3x^2 + 4x) - 5) + 1)/(-1(4(3x^2 + 4x)^2 - 4(3x^2 + 4x) - 5) - 2)) - 3) + 1) - 2)/(-3(2(-1((-2(4(3x^2 + 4x)^2 - 4(3x^2 + 4x) - 5) + 1)/(-1(4(3x^2 + 4x)^2 - 4(3x^2 + 4x) - 5) - 2)) - 3) + 1) - 1) = 1|(1((1(2|(-1|x + 2| + 1)| - 2) +... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-3cos(1π(1|(4(-2|((-1x + 2)/(-3x + 2)) + 3|)^2 - 4)| + 2) - 3) - 3)^2 + 1(-3cos(1π(1|(4(-2|((-1x + 2)/(-3x + 2)) + 3|)^2 - 4)| + 2) - 3) - 3) - 5 and g(x) = -2e^(1(-2(4(-2(5x + 5) + 5)^2 + 4(-2(5x + 5) + 5) - 2) - 5)) + 1 w... | 224 | MATH |
[
{
"content": "How many solutions does the equation 2(-1(-2e^(2((1(1|(3x^2 - 2x - 1) + 1| - 1) - 1)/(-3(1|(3x^2 - 2x - 1) + 1| - 1) + 4))) + 2)^2 + 1(-2e^(2((1(1|(3x^2 - 2x - 1) + 1| - 1) - 1)/(-3(1|(3x^2 - 2x - 1) + 1| - 1) + 4))) + 2) - 3)^2 - 1(-1(-2e^(2((1(1|(3x^2 - 2x - 1) + 1| - 1) - 1)/(-3(1|(3x^2 - 2x - ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(1|((3(2((3(3|x + 2| + 2) + 2)/(1(3|x + 2| + 2))) + 1))/(1(2((3(3|x + 2| + 2) + 2)/(1(3|x + 2| + 2))) + 1) - 3)) - 3| - 1)^2 - 5(1|((3(2((3(3|x + 2| + 2) + 2)/(1(3|x + 2| + 2))) + 1))/(1(2((3(3|x + 2| + 2) + 2)/(1(3|x + 2| ... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(1|(2|((-1(3(-2sin(1πx - 1) + 1) - 3) + 3)/(2(3(-2sin(1πx - 1) + 1) - 3))) + 3| - 3)| - 2) + 2 and g(x) = (-2(-3(-1(-3|((3x)/(-2x - 2)) - 2| - 2) - 1)^2 - 2(-1(-3|((3x)/(-2x - 2)) - 2| - 2) - 1)) - 1)/(3(-3(-1(-3|((3x)/(-2x ... | 57 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(3cos(3π(-1(-2|(-3|(-3|x - 2| + 3) + 3| + 3) + 1| - 1) + 4)) - 1)^2 - 1(3cos(3π(-1(-2|(-3|(-3|x - 2| + 3) + 3| + 3) + 1| - 1) + 4)) - 1) - 1 and g(x) = 2sin(1π(2((-2(-2|(2|x - 3| - 1) + 2| + 3) - 1)/(3(-2|(2|x - 3| - 1) + 2| + 3) ... | 8 | MATH |
[
{
"content": "How many solutions does the equation -2(-2((-2(1sin(2π(-3(3x^2 + 1x + 3) + 2) + 3)))/(-1(1sin(2π(-3(3x^2 + 1x + 3) + 2) + 3)) + 2))^2 + 4((-2(1sin(2π(-3(3x^2 + 1x + 3) + 2) + 3)))/(-1(1sin(2π(-3(3x^2 + 1x + 3) + 2) + 3)) + 2)) - 2) - 5 = -2|(-2((-3(1((2x + 3)/(-3x))^2 + 5((2x + 3)/(-3x)) + 1) - 1)... | 400 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(2|(2cos(1π(-5((-2((3x + 3)/(1x - 3)) - 1)/(-2((3x + 3)/(1x - 3)) - 3)) + 1)) + 1) + 3|) - 2| - 3 and g(x) = 1((3((1(1((2x + 1)/(-3x - 3))) - 1)/(-1(1((2x + 1)/(-3x - 3))) + 2)) + 1)/(3((1(1((2x + 1)/(-3x - 3))) - 1)/(-1(1((2x + 1)/(-3x - 3))) + 2)) ... | 62 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(3|(-2|(5(2|(-5x^2 - 2x - 3) + 3|)) - 3| + 2) + 1| - 2)^2 + 4(3|(-2|(5(2|(-5x^2 - 2x - 3) + 3|)) - 3| + 2) + 1| - 2) + 4 and g(x) = 1((3(-1sin(1π(5(-2|x - 1| + 1)^2 + 4(-2|x - 1| + 1) + 1) - 3) + 1) - 2)/(-2(-1sin(1π(5(-2|x - 1| +... | 6 | MATH |
[
{
"content": "How many solutions does the equation 5((3(3(-2|(2(-2sin(3πx - 1))^2 + 4(-2sin(3πx - 1)) + 2) - 1| + 3)^2 - 3(-2|(2(-2sin(3πx - 1))^2 + 4(-2sin(3πx - 1)) + 2) - 1| + 3) - 1) + 2)/(-1(3(-2|(2(-2sin(3πx - 1))^2 + 4(-2sin(3πx - 1)) + 2) - 1| + 3)^2 - 3(-2|(2(-2sin(3πx - 1))^2 + 4(-2sin(3πx - 1)) + 2) ... | 9 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(-1(1sin(1π(1(-3|(5x + 5) + 3| - 3)^2 + 3(-3|(5x + 5) + 3| - 3) - 2) - 3) - 3) - 3) - 3 and g(x) = (-2(2(1|(-2|(2|x - 2| + 2) - 2| - 2)| - 2)^2 + 4(1|(-2|(2|x - 2| + 2) - 2| - 2)| - 2) - 5) - 3)/(2(2(1|(-2|(2|x - 2| + 2) - 2| - 2)| - 2)^2 + 4(1|(-2|(... | 1494 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(4(2(2cos(3π(-1|(1|x - 3| + 2) - 1| - 3) + 1))^2 - 1(2cos(3π(-1|(1|x - 3| + 2) - 1| - 3) + 1)) - 1)^2 + 4(2(2cos(3π(-1|(1|x - 3| + 2) - 1| - 3) + 1))^2 - 1(2cos(3π(-1|(1|x - 3| + 2) - 1| - 3) + 1)) - 1) - 3))/(1(4(2(2cos(3π(-1|(1|x - 3| + 2) - 1| - ... | 160 | MATH |
[
{
"content": "How many solutions does the equation (1(-4(2|((3(1(3|x - 1| + 1) - 1) + 1)/(2(1(3|x - 1| + 1) - 1))) - 3| + 1)^2 - 4(2|((3(1(3|x - 1| + 1) - 1) + 1)/(2(1(3|x - 1| + 1) - 1))) - 3| + 1) + 5) - 2)/(3(-4(2|((3(1(3|x - 1| + 1) - 1) + 1)/(2(1(3|x - 1| + 1) - 1))) - 3| + 1)^2 - 4(2|((3(1(3|x - 1| + 1) -... | 22 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(-3|(-1(-1((2(-3|x - 2| + 1) - 1)/(-2(-3|x - 2| + 1) + 3))^2 - 5((2(-3|x - 2| + 1) - 1)/(-2(-3|x - 2| + 1) + 3)) - 2)^2 + 5(-1((2(-3|x - 2| + 1) - 1)/(-2(-3|x - 2| + 1) + 3))^2 - 5((2(-3|x - 2| + 1) - 1)/(-2(-3|x - 2| + 1)... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(1(1((-2(-3((2x + 1)/(-2x - 3)) - 1))/(2(-3((2x + 1)/(-2x - 3)) - 1) - 1)) + 3) - 4)^2 - 5(1(1((-2(-3((2x + 1)/(-2x - 3)) - 1))/(2(-3((2x + 1)/(-2x - 3)) - 1) - 1)) + 3) - 4) - 3 and g(x) = 3cos(2π(-2|(-5(3|(1|x - 1| - 3) + 1| - 3)^2 + 4(3|(1|x - 1| -... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(-1(-1|(4((2(1x^2 - 2x + 2) + 3)/(-3(1x^2 - 2x + 2) - 3)) + 3) + 3|)^2 - 2(-1|(4((2(1x^2 - 2x + 2) + 3)/(-3(1x^2 - 2x + 2) - 3)) + 3) + 3|) + 2) - 5 and g(x) = -2(1sin(3π(3|(-1|((-2x - 3)/(3x - 2)) - 2| + 2) - 1| - 2) + 3) +... | 15 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2((-3((-1(1|(-2(5x + 4)) + 2| + 1) + 3)/(1(1|(-2(5x + 4)) + 2| + 1) + 2)) - 3)/(-3((-1(1|(-2(5x + 4)) + 2| + 1) + 3)/(1(1|(-2(5x + 4)) + 2| + 1) + 2)) - 1)) - 5 and g(x) = 1(1(-2|(1cos(3π((2x - 3)/(3x + 3)) - 2)) + 1| + 3) - 1)^2 - 3(1(-2|(1cos(3π((2x ... | 33 | MATH |
[
{
"content": "How many solutions does the equation -1|(-3(3|((-2((-3(-4x^2 - 4x - 2) + 1)/(-1(-4x^2 - 4x - 2) - 2)) - 3)/(-1((-3(-4x^2 - 4x - 2) + 1)/(-1(-4x^2 - 4x - 2) - 2)))) - 1| + 3) - 2) + 2| + 1 = -2|(-3(5(2|(1x - 3) - 3| - 3) - 2) - 5) + 1| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer... | 4 | MATH |
[
{
"content": "How many solutions does the equation -4((2((3(2sin(3π(-1|((-1x - 1)/(-3x + 1)) + 1| - 1)) + 1) + 3)/(-3(2sin(3π(-1|((-1x - 1)/(-3x + 1)) + 1| - 1)) + 1) + 2)) + 3)/(-1((3(2sin(3π(-1|((-1x - 1)/(-3x + 1)) + 1| - 1)) + 1) + 3)/(-3(2sin(3π(-1|((-1x - 1)/(-3x + 1)) + 1| - 1)) + 1) + 2)) - 1))^2 + 3((2... | 257 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(3|(-2(5(-5((1x + 3)/(3x + 2)) - 2)^2 - 5(-5((1x + 3)/(3x + 2)) - 2) - 1)^2 + 3(5(-5((1x + 3)/(3x + 2)) - 2)^2 - 5(-5((1x + 3)/(3x + 2)) - 2) - 1) - 4)|) - 1| + 2 and g(x) = 3|(3(-2(-3|(-3cos(2πx + 3) + 1) + 3| + 2)^2 - 5(-3|(-3cos(2πx + 3) + 1) + 3|... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(1(4(-1(5(3x^2 + 1x - 3)^2 + 2(3x^2 + 1x - 3))^2 + 2(5(3x^2 + 1x - 3)^2 + 2(3x^2 + 1x - 3)) + 2)^2 - 2(-1(5(3x^2 + 1x - 3)^2 + 2(3x^2 + 1x - 3))^2 + 2(5(3x^2 + 1x - 3)^2 + 2(3x^2 + 1x - 3)) + 2) + 3)^2 + 5(4(-1(5(3x^2 + 1x - 3)^2 + 2(3x^2 + 1x - 3))... | 191 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(-3(2|(1((3(-1x) - 1)/(1(-1x) + 2)) + 2) - 1| - 3) + 2) + 2| + 2 and g(x) = 1(-2|(-5(3cos(1π(3x^2 - 2x - 3) + 2) - 1) - 3) - 1| - 2)^2 + 5(-2|(-5(3cos(1π(3x^2 - 2x - 3) + 2) - 1) - 3) - 1| - 2) - 3 where -10 ≤ x ≤ 10.\n\nPr... | 698 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-3(-1|(-2(-1|((1x - 2)/(-1x + 2)) + 2| + 3) + 5) - 2| + 3) + 2) - 2)/(1(-3(-1|(-2(-1|((1x - 2)/(-1x + 2)) + 2| + 3) + 5) - 2| + 3) + 2)) and g(x) = -1(-1sin(1π(-3(-3|(-3x + 1)|) - 4) + 2) + 2) + 3 where -10 ≤ x ≤ 10.\n\nPr... | 540 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(1(-4(4(1|(2|x + 2| - 1) - 3| - 2) - 1) + 1) + 2)^2 - 5(1(-4(4(1|(2|x + 2| - 1) - 3| - 2) - 1) + 1) + 2) + 2 and g(x) = -1|(1|((2(-4(1|x - 1| - 3) - 5))/(2(-4(1|x - 1| - 3) - 5) - 1)) + 3| + 1) - 2| + 1 where -10 ≤ x ≤ 10.\... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(4(3(-1((2(-4x + 2) + 2)/(2(-4x + 2) - 3))^2 + 1)^2 + 4(-1((2(-4x + 2) + 2)/(2(-4x + 2) - 3))^2 + 1) - 2)^2 - 4(3(-1((2(-4x + 2) + 2)/(2(-4x + 2) - 3))^2 + 1)^2 + 4(-1((2(-4x + 2) + 2)/(2(-4x + 2) - 3))^2 + 1) - 2) - 4) - 1... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-1|(1(-1(-3|(-1x^2 + 4x + 1) + 1| - 3) - 2)^2 + 4(-1(-3|(-1x^2 + 4x + 1) + 1| - 3) - 2) + 2) + 1|) + 1)/(-2(-1|(1(-1(-3|(-1x^2 + 4x + 1) + 1| - 3) - 2)^2 + 4(-1(-3|(-1x^2 + 4x + 1) + 1| - 3) - 2) + 2) + 1|) - 1) and g(x) =... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3((2(3sin(2π((1(-2|((-3x + 2)/(1x - 3))| - 1) - 2)/(1(-2|((-3x + 2)/(1x - 3))| - 1) + 2)) - 1)) - 3)/(-1(3sin(2π((1(-2|((-3x + 2)/(1x - 3))| - 1) - 2)/(1(-2|((-3x + 2)/(1x - 3))| - 1) + 2)) - 1)) + 3)) - 4 and g(x) = 1(3|(-3cos(1π(-1|(5x + 5) - 1| - 3... | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(3π((1(3|(-3|(3((-1x + 3)/(-3x + 4))) - 3| + 2)|) + 2)/(1(3|(-3|(3((-1x + 3)/(-3x + 4))) - 3| + 2)|) + 2)) - 1) + 1 and g(x) = (2(1|((-2(-2(5x - 2)^2 + 2(5x - 2)) - 2)/(1(-2(5x - 2)^2 + 2(5x - 2)) + 3)) - 1| - 1) - 2)/(-3(1|((-2(-2(5x - 2)^2 + 2(5x... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(-2sin(2π(-1|((-2(2((1x + 3)/(2x - 1))^2 + 5((1x + 3)/(2x - 1)) - 3))/(2(2((1x + 3)/(2x - 1))^2 + 5((1x + 3)/(2x - 1)) - 3) - 3)) + 3| - 1) - 1) - 2) - 3 and g(x) = 5(5((1(-2|((-2x - 2)/(1x + 1)) + 3| - 3) + 3)/(-2(-2|((-2x - 2)/(1x + 1)) + 3| - 3) -... | 70 | MATH |
[
{
"content": "How many solutions does the equation (2((1((2(-3|(-1(-3cos(1πx - 1) - 3)^2 + 4(-3cos(1πx - 1) - 3) + 1) - 1| - 3) - 2)/(1(-3|(-1(-3cos(1πx - 1) - 3)^2 + 4(-3cos(1πx - 1) - 3) + 1) - 1| - 3) + 3)))/(3((2(-3|(-1(-3cos(1πx - 1) - 3)^2 + 4(-3cos(1πx - 1) - 3) + 1) - 1| - 3) - 2)/(1(-3|(-1(-3cos(1πx - ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-2|(-3(-5(3(5x - 5) + 2)^2 - 3(3(5x - 5) + 2) + 2) + 3) + 3| + 3) - 3)/(-3(-2|(-3(-5(3(5x - 5) + 2)^2 - 3(3(5x - 5) + 2) + 2) + 3) + 3| + 3) + 1) and g(x) = 5(-2e^(-2(-3(4(-1|x - 2| + 2)^2 + 5(-1|x - 2| + 2) - 4) - 2)))^2 + 5(-... | 4 | MATH |
[
{
"content": "How many solutions does the equation 3(-1|(5(-1(2|((1x + 1)/(-1x + 2)) + 3| - 2)^2 - 5(2|((1x + 1)/(-1x + 2)) + 3| - 2) + 2)^2 + 4(-1(2|((1x + 1)/(-1x + 2)) + 3| - 2)^2 - 5(2|((1x + 1)/(-1x + 2)) + 3| - 2) + 2) - 5) - 2| - 1)^2 + 5(-1|(5(-1(2|((1x + 1)/(-1x + 2)) + 3| - 2)^2 - 5(2|((1x + 1)/(-1x +... | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(-4((-3((-2(4((-3x + 2)/(-3x))^2 + 5((-3x + 2)/(-3x)) + 1) + 2)/(1(4((-3x + 2)/(-3x))^2 + 5((-3x + 2)/(-3x)) + 1) + 1)) + 3)/(-2((-2(4((-3x + 2)/(-3x))^2 + 5((-3x + 2)/(-3x)) + 1) + 2)/(1(4((-3x + 2)/(-3x))^2 + 5((-3x + 2)/(-3x)) + 1) + 1)) + 3)) + ... | 1264 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(-3(4(-3|(-3|((2x - 2)/(-3x - 2)) - 1|) + 1| + 2)^2 + 3(-3|(-3|((2x - 2)/(-3x - 2)) - 1|) + 1| + 2) - 1)^2 - 5) + 4 and g(x) = -1|(-1(-1|(5(-2|x + 3| - 3) - 4) + 3| - 1) - 3)| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2((1(-2((2((2(5x^2 + 1x - 5) - 2)/(2(5x^2 + 1x - 5) - 3)) + 1)/(-2((2(5x^2 + 1x - 5) - 2)/(2(5x^2 + 1x - 5) - 3)) - 3))^2 + 2((2((2(5x^2 + 1x - 5) - 2)/(2(5x^2 + 1x - 5) - 3)) + 1)/(-2((2(5x^2 + 1x - 5) - 2)/(2(5x^2 + 1x - 5) - 3)... | 81 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-2cos(1π(4((-2(-4(-2|x + 1| + 3)^2 - 4(-2|x + 1| + 3) - 5))/(1(-4(-2|x + 1| + 3)^2 - 4(-2|x + 1| + 3) - 5) - 2))^2) + 3) - 3) + 1)/(2(-2cos(1π(4((-2(-4(-2|x + 1| + 3)^2 - 4(-2|x + 1| + 3) - 5))/(1(-4(-2|x + 1| + 3)^2 - 4(-... | 394 | MATH |
[
{
"content": "How many solutions does the equation 4((-1(1|(-4(-1(-2|x + 2| + 2)^2 + 2(-2|x + 2| + 2) + 2)^2 - 3(-1(-2|x + 2| + 2)^2 + 2(-2|x + 2| + 2) + 2)) + 2| + 1) + 1)/(3(1|(-4(-1(-2|x + 2| + 2)^2 + 2(-2|x + 2| + 2) + 2)^2 - 3(-1(-2|x + 2| + 2)^2 + 2(-2|x + 2| + 2) + 2)) + 2| + 1) + 1))^2 - 4((-1(1|(-4(-1(... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|((1(-3|(-1cos(1π(-2|(-2|x| - 1) + 3|) + 3) + 1) - 1| + 3) + 3)/(-1(-3|(-1cos(1π(-2|(-2|x| - 1) + 3|) + 3) + 1) - 1| + 3))) - 2| - 3 and g(x) = 1(2|(-2(-5(-1|x + 2| - 2) + 1)) + 2| + 2)^2 + 2(2|(-2(-5(-1|x + 2| - 2) + 1)) + ... | 32 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(1|(-2|((3(-5(1|x + 1| + 1)^2 - 1(1|x + 1| + 1) - 5) - 3)/(3(-5(1|x + 1| + 1)^2 - 1(1|x + 1| + 1) - 5) - 1)) - 3| - 1) - 3|) + 2)/(-3(1|(-2|((3(-5(1|x + 1| + 1)^2 - 1(1|x + 1| + 1) - 5) - 3)/(3(-5(1|x + 1| + 1)^2 - 1(1|x + ... | 1284 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2((-1((3((1(-3|(2|x - 3| - 3) + 3| - 1) + 3)/(1(-3|(2|x - 3| - 3) + 3| - 1))) + 2)/(-1((1(-3|(2|x - 3| - 3) + 3| - 1) + 3)/(1(-3|(2|x - 3| - 3) + 3| - 1))))) - 1)/(2((3((1(-3|(2|x - 3| - 3) + 3| - 1) + 3)/(1(-3|(2|x - 3| - 3) + 3|... | 4 | MATH |
[
{
"content": "How many solutions does the equation (1(5(-1e^(-2(-5((3(-2|x| + 1) + 2)/(1(-2|x| + 1) - 3))^2 + 3((3(-2|x| + 1) + 2)/(1(-2|x| + 1) - 3)) - 5)) + 2)^2 + 2(-1e^(-2(-5((3(-2|x| + 1) + 2)/(1(-2|x| + 1) - 3))^2 + 3((3(-2|x| + 1) + 2)/(1(-2|x| + 1) - 3)) - 5)) + 2)) - 2)/(-3(5(-1e^(-2(-5((3(-2|x| + 1) +... | 24 | MATH |
[
{
"content": "How many solutions does the equation (-2(1|(-2|(3((3(1x))/(2(1x) - 1))^2 - 4((3(1x))/(2(1x) - 1)) + 3) - 3| + 3) - 3| - 3) + 3)/(1(1|(-2|(3((3(1x))/(2(1x) - 1))^2 - 4((3(1x))/(2(1x) - 1)) + 3) - 3| + 3) - 3| - 3) + 2) = 2(1cos(2π((2(-1(-5x^2 + 5x + 3) - 4) + 2)/(-1(-1(-5x^2 + 5x + 3) - 4) + 2)) - ... | 38 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(3(-3e^(2((-2(3(-5x^2 - 4x - 1) - 4) + 1)/(2(3(-5x^2 - 4x - 1) - 4)))) + 1) - 1) - 3| + 3 and g(x) = -1(1|(-2(-4(-3x^2 + 1x + 3)^2 + 2(-3x^2 + 1x + 3) + 4)^2 - 5(-4(-3x^2 + 1x + 3)^2 + 2(-3x^2 + 1x + 3) + 4) - 5) - 3| - 3) in th... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1cos(1π(5(2|(-2(-5(-1x^2 + 3x - 4)^2 + 1)) - 1| + 2)^2 + 1(2|(-2(-5(-1x^2 + 3x - 4)^2 + 1)) - 1| + 2) + 4) + 2) - 2 and g(x) = 3sin(2π(1(-3((-3(1|x + 1|) - 2)/(2(1|x + 1|) - 3)) - 1)^2 + 2(-3((-3(1|x + 1|) - 2)/(2(1|x + 1|) - 3)) -... | 56 | MATH |
[
{
"content": "How many solutions does the equation (-2(-4(4((1(1e^(2(-5x^2 + 2x - 1)) + 3) + 3)/(3(1e^(2(-5x^2 + 2x - 1)) + 3) - 1))^2 + 4) - 3) + 3)/(1(-4(4((1(1e^(2(-5x^2 + 2x - 1)) + 3) + 3)/(3(1e^(2(-5x^2 + 2x - 1)) + 3) - 1))^2 + 4) - 3) - 2) = (-2(3sin(2π(1(1(2|x| - 1) + 5) - 4)) - 1) - 2)/(-3(3sin(2π(1(1... | 80 | MATH |
[
{
"content": "How many solutions does the equation 1(1(-4(-3(-4(2|x + 2| + 1)^2 - 5(2|x + 2| + 1) - 2) - 2) + 3) + 5) + 2 = -3|(-5(-4(-5(2cos(3πx + 2) - 2) - 4) + 5) - 1) - 1| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3e^(2(3(5(-2|(-5(4x^2 - 4x - 1) + 2) - 2| + 2) - 3)^2 - 5(5(-2|(-5(4x^2 - 4x - 1) + 2) - 2| + 2) - 3) + 1)) + 1 and g(x) = -2(-3(-4(-1(1cos(2πx + 3) + 2)^2 + 2(1cos(2πx + 3) + 2) - 3)^2 + 2(-1(1cos(2πx + 3) + 2)^2 + 2(1cos(2... | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-1(1(-1|(-1(-1x^2 + 1x + 4)) - 1|)^2 + 2(-1|(-1(-1x^2 + 1x + 4)) - 1|) - 2) + 2)^2 - 4(-1(1(-1|(-1(-1x^2 + 1x + 4)) - 1|)^2 + 2(-1|(-1(-1x^2 + 1x + 4)) - 1|) - 2) + 2) - 5 and g(x) = 1|((2(5(1(3|x| + 3)^2 - 1(3|x| + 3) + 1)^2 - 4(1(3|x| + 3)^2 - 1(3|... | 8 | MATH |
[
{
"content": "How many solutions does the equation -1(1|(-2|(-4(-2cos(3π(-5x + 5) - 2) - 2) - 2)| + 1) + 2|) + 1 = 2sin(3π(-1|((2(2(-3|x + 1|)^2 - 4(-3|x + 1|) - 3) + 1)/(-3(2(-3|x + 1|)^2 - 4(-3|x + 1|) - 3) + 1)) + 2| + 2) - 1) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \... | 301 | MATH |
[
{
"content": "How many solutions does the equation -2|(3cos(3π(-5(2(-1(4x^2 - 1x - 5) - 1)^2 - 3(-1(4x^2 - 1x - 5) - 1) + 1)^2 + 4(2(-1(4x^2 - 1x - 5) - 1)^2 - 3(-1(4x^2 - 1x - 5) - 1) + 1) + 2) + 1)) - 1| - 3 = 1(2((-3(-1((2x - 3)/(1x + 1))^2 + 4((2x - 3)/(1x + 1)) + 3))/(3(-1((2x - 3)/(1x + 1))^2 + 4((2x - 3)... | 13 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(5(2|(-2(2(2x - 2) + 1)^2 - 5(2(2x - 2) + 1) + 4) + 2| + 1) + 3) - 2)/(3(5(2|(-2(2(2x - 2) + 1)^2 - 5(2(2x - 2) + 1) + 4) + 2| + 1) + 3)) and g(x) = -1(5((-3(3(2|x - 1|)^2) + 3)/(3(3(2|x - 1|)^2) - 2))^2 + 5((-3(3(2|x - 1|)^2) + ... | 6 | MATH |
[
{
"content": "How many solutions does the equation 3|(-4(2((-3(1(-5x^2 + 5x + 4)^2 - 4(-5x^2 + 5x + 4) + 2) + 1)/(1(1(-5x^2 + 5x + 4)^2 - 4(-5x^2 + 5x + 4) + 2) + 1)) - 2) + 2) + 3| + 1 = (-3(1cos(1π(5((-1(-2x^2 + 5x + 3))/(1(-2x^2 + 5x + 3) - 3)) - 3) - 1) - 3))/(-1(1cos(1π(5((-1(-2x^2 + 5x + 3))/(1(-2x^2 + 5x... | 4 | MATH |
[
{
"content": "How many solutions does the equation (-2(2|(3|(1((1(-3x^2 - 4x - 2) - 1)/(-3(-3x^2 - 4x - 2) + 4)) - 4) + 3| + 1) + 1| - 1) + 3)/(2(2|(3|(1((1(-3x^2 - 4x - 2) - 1)/(-3(-3x^2 - 4x - 2) + 4)) - 4) + 3| + 1) + 1| - 1) - 1) = (-3(-3cos(2π(1|(3((1x + 1)/(2x - 3))^2 - 1((1x + 1)/(2x - 3)) + 5) + 2| + 2)... | 82 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(4(-1|(1(-1|(2|x + 1| + 2) - 3| + 3)^2 + 1(-1|(2|x + 1| + 2) - 3| + 3) + 3) - 2| + 3)^2 - 5(-1|(1(-1|(2|x + 1| + 2) - 3| + 3)^2 + 1(-1|(2|x + 1| + 2) - 3| + 3) + 3) - 2| + 3) + 5)^2 + 1(4(-1|(1(-1|(2|x + 1| + 2) - 3| + 3)^2 + 1(-1|(2|x + 1| + 2) - 3|... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-1(1(-1(2(-3|x - 1| + 1) + 5)^2 + 1(2(-3|x - 1| + 1) + 5) + 3) + 1) + 3) + 1 and g(x) = (1(-1|(1cos(3π((-2(-5x + 2))/(-2(-5x + 2) - 3))) - 3) - 3| - 2) - 3)/(-2(-1|(1cos(3π((-2(-5x + 2))/(-2(-5x + 2) - 3))) - 3) - 3| - 2) + 1) i... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(-4(-1e^(-2(2|(4(2|x| - 3)^2 + 4(2|x| - 3) - 5) + 2| - 3)) + 2)^2 - 2(-1e^(-2(2|(4(2|x| - 3)^2 + 4(2|x| - 3) - 5) + 2| - 3)) + 2))^2 + 3(-4(-1e^(-2(2|(4(2|x| - 3)^2 + 4(2|x| - 3) - 5) + 2| - 3)) + 2)^2 - 2(-1e^(-2(2|(4(2|x| - 3)^2 + 4(2|x| - 3) - 5) +... | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3cos(1π(-4(5(-4((-1((-3x - 3)/(2x - 1)) + 1)/(-2((-3x - 3)/(2x - 1)) + 3))^2 + 5((-1((-3x - 3)/(2x - 1)) + 1)/(-2((-3x - 3)/(2x - 1)) + 3)) + 3) + 5)^2 + 2(5(-4((-1((-3x - 3)/(2x - 1)) + 1)/(-2((-3x - 3)/(2x - 1)) + 3))^2 + ... | 328 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(3π((2(2(-4(-2|(-1|x - 3| + 2) + 1| + 2)^2 - 4(-2|(-1|x - 3| + 2) + 1| + 2) - 1)^2 - 1))/(-1(2(-4(-2|(-1|x - 3| + 2) + 1| + 2)^2 - 4(-2|(-1|x - 3| + 2) + 1| + 2) - 1)^2 - 1) - 1)) - 2) - 2 and g(x) = (1(-4(3sin(3π(-3|((3x... | 189 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(-1(5((-1((-3(2x - 5) + 2)/(-3(2x - 5) + 1)) + 1)/(2((-3(2x - 5) + 2)/(-3(2x - 5) + 1)) - 1))^2 - 5((-1((-3(2x - 5) + 2)/(-3(2x - 5) + 1)) + 1)/(2((-3(2x - 5) + 2)/(-3(2x - 5) + 1)) - 1)) + 4) + 4) and g(x) = 4(-2(1|(-1|((1x - 3)/(1x))|) + 3|) + 4) +... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(1(4(5(-2cos(1π(3x^2 - 4x - 4)) - 3) - 1) - 1) - 3)^2 and g(x) = -1(-2|(3(-1|(-3|x + 1| - 2) + 1| + 2) + 3) - 3| - 2)^2 + 3(-2|(3(-1|(-3|x + 1| - 2) + 1| + 2) + 3) - 3| - 2) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in... | 206 | MATH |
[
{
"content": "How many solutions does the equation -5(-2|(3|(-3e^(1((2((3x - 1)/(3x)) + 3)/(-3((3x - 1)/(3x)) - 1))) + 3)| - 3) + 1| + 2) - 4 = -1|(1|(3((3(-3|x - 2| + 3) + 1)/(2(-3|x - 2| + 3) + 1))) + 2| - 1)| - 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(-1|((2((2((-2(1x^2 + 5x + 1) - 2)/(3(1x^2 + 5x + 1) - 1)) - 1)/(-2((-2(1x^2 + 5x + 1) - 2)/(3(1x^2 + 5x + 1) - 1)) - 3)) - 1)/(-3((2((-2(1x^2 + 5x + 1) - 2)/(3(1x^2 + 5x + 1) - 1)) - 1)/(-2((-2(1x^2 + 5x + 1) - 2)/(3(1x^2 + 5x + 1) - 1)) - 3)) + 1))... | 1103 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(-2e^(2((-2((2(1(-1|x - 1|)^2 + 5(-1|x - 1|) + 1) + 1)/(-1(1(-1|x - 1|)^2 + 5(-1|x - 1|) + 1) - 2)) + 1)/(-1((2(1(-1|x - 1|)^2 + 5(-1|x - 1|) + 1) + 1)/(-1(1(-1|x - 1|)^2 + 5(-1|x - 1|) + 1) - 2)) + 3))) + 2) + 2 and g(x) = (2((2(-1|(2|(1x^2 - 2x + 2... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(5(-1|(-2|((-3((3x - 3)/(-1x - 1)) + 3)/(-1((3x - 3)/(-1x - 1)) - 1)) + 2|) + 1| + 3) + 5) - 2| + 3 and g(x) = (3(1|(5(-2(3e^(1x) + 1) + 3) + 3) - 3| - 1) + 1)/(-3(1|(5(-2(3e^(1x) + 1) + 3) + 3) - 3| - 1) + 4) intersect in the interval [-10, 10].\n\... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-4(1|((-3(2|(-5x^2 + 5x + 1) + 2|) - 2)/(-1(2|(-5x^2 + 5x + 1) + 2|) - 1)) + 1| + 1)^2 - 4)| + 3 and g(x) = 1((-2(1(1(3cos(2πx + 1) + 3) - 2) - 5) + 3)/(-1(1(1(3cos(2πx + 1) + 3) - 2) - 5) - 2))^2 + 3((-2(1(1(3cos(2πx + 1) + 3) - 2) - 5) + 3)/(-1(1(... | 80 | MATH |
[
{
"content": "How many solutions does the equation 3((3(-2|((-3((-3(-1sin(1πx + 3) + 1))/(1(-1sin(1πx + 3) + 1))) + 1)/(2((-3(-1sin(1πx + 3) + 1))/(1(-1sin(1πx + 3) + 1))) + 3)) - 3| + 2) + 1)/(-2(-2|((-3((-3(-1sin(1πx + 3) + 1))/(1(-1sin(1πx + 3) + 1))) + 1)/(2((-3(-1sin(1πx + 3) + 1))/(1(-1sin(1πx + 3) + 1)))... | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3((2(2((3((-2(-1x^2 + 4x + 5))/(-1(-1x^2 + 4x + 5) + 2)) - 3)/(-1((-2(-1x^2 + 4x + 5))/(-1(-1x^2 + 4x + 5) + 2)) - 1))^2 + 4((3((-2(-1x^2 + 4x + 5))/(-1(-1x^2 + 4x + 5) + 2)) - 3)/(-1((-2(-1x^2 + 4x + 5))/(-1(-1x^2 + 4x + 5) + 2)) ... | 4 | MATH |
[
{
"content": "How many solutions does the equation -4(-3e^(-1(-5(1((3(2|x + 3| - 2))/(1(2|x + 3| - 2) + 3)) + 5)^2 + 4(1((3(2|x + 3| - 2))/(1(2|x + 3| - 2) + 3)) + 5) + 1)) + 1) - 5 = -3e^(1(-2|(-1|(3|(5x + 4) + 1| - 1) + 1| + 2) + 3| - 3)) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTe... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|((-2(1e^(1(-2|(1((-1x - 2)/(-2x)) - 1) + 1| + 3)) - 1))/(2(1e^(1(-2|(1((-1x - 2)/(-2x)) - 1) + 1| + 3)) - 1) + 2)) + 3| + 3 and g(x) = 3(4((-3(-4(2|x - 3| - 1)^2 - 4(2|x - 3| - 1) + 1) - 2)/(1(-4(2|x - 3| - 1)^2 - 4(2|x - ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1((2(-3(-1(3|((-2x - 1)/(2x + 2))| + 2)^2 + 4(3|((-2x - 1)/(2x + 2))| + 2) + 2)^2 + 3(-1(3|((-2x - 1)/(2x + 2))| + 2)^2 + 4(3|((-2x - 1)/(2x + 2))| + 2) + 2) - 2) + 3)/(-1(-3(-1(3|((-2x - 1)/(2x + 2))| + 2)^2 + 4(3|((-2x - 1)/(2x + 2))| + 2) + 2)^2 + ... | 10 | MATH |
[
{
"content": "How many solutions does the equation (1(-3e^(2((2(3|(-1(1x^2 - 2x - 2) + 4) + 2| - 2) + 2)/(3(3|(-1(1x^2 - 2x - 2) + 4) + 2| - 2)))) - 1) - 1)/(3(-3e^(2((2(3|(-1(1x^2 - 2x - 2) + 4) + 2| - 2) + 2)/(3(3|(-1(1x^2 - 2x - 2) + 4) + 2| - 2)))) - 1) + 2) = -4((1(3cos(1π((2((2x - 1)/(-1x - 2)) - 1)/(-1((... | 27 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(2|(1(1(5(1x^2 + 5x - 2) - 5) + 4)^2 - 3(1(5(1x^2 + 5x - 2) - 5) + 4)) + 3| + 2)| + 1 and g(x) = 3(2|(5(1|(-3x^2 + 3x + 3) - 1| + 1) - 5) - 1| + 1)^2 + 1(2|(5(1|(-3x^2 + 3x + 3) - 1| + 1) - 5) - 1| + 1) - 5 in the range -10 ≤ x ≤... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(2sin(3π(2|(3((-3(-2|x - 2|) - 2)/(3(-2|x - 2|)))^2 - 5((-3(-2|x - 2|) - 2)/(3(-2|x - 2|))) + 3) + 1| + 1) - 1) - 1) - 2 and g(x) = (1(3|(3(2|(1x + 2) + 2| + 3) - 2) + 3| + 2) - 3)/(2(3|(3(2|(1x + 2) + 2| + 3) - 2) + 3| + 2) + 1)... | 124 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1((1(3sin(2π(-2(-5(-2|x| + 3)^2 - 1(-2|x| + 3) - 3)^2 - 3(-5(-2|x| + 3)^2 - 1(-2|x| + 3) - 3) + 2) + 1) + 1) - 3)/(3(3sin(2π(-2(-5(-2|x| + 3)^2 - 1(-2|x| + 3) - 3)^2 - 3(-5(-2|x| + 3)^2 - 1(-2|x| + 3) - 3) + 2) + 1) + 1) - 2)) - 1... | 55 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-3(-3(-2|((1(1x + 2) + 3)/(-3(1x + 2) - 1))| + 2) + 5)^2 + 5(-3(-2|((1(1x + 2) + 3)/(-3(1x + 2) - 1))| + 2) + 5) + 5) + 5 and g(x) = (1(-3|(-3(3((1x + 2)/(3x + 2))^2 + 5((1x + 2)/(3x + 2)) + 2)^2 - 5(3((1x + 2)/(3x + 2))^2 + 5((1x + 2)/(3x + 2)) + 2)... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1((-3(-2|(-4(1cos(2π((1x - 1)/(-1x - 3)) + 2) - 3) + 1)| + 1) - 1)/(3(-2|(-4(1cos(2π((1x - 1)/(-1x - 3)) + 2) - 3) + 1)| + 1) + 1))^2 + 3((-3(-2|(-4(1cos(2π((1x - 1)/(-1x - 3)) + 2) - 3) + 1)| + 1) - 1)/(3(-2|(-4(1cos(2π((1x... | 47 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(2π(-2(-4(1(1(1|x - 3| - 1) - 5) + 4) - 1)^2 + 3(-4(1(1(1|x - 3| - 1) - 5) + 4) - 1) + 2)) - 2 and g(x) = -2|(1((-1(2|(2|x - 3| - 1)| + 1) + 1)/(-1(2|(2|x - 3| - 1)| + 1) + 2))^2 - 3((-1(2|(2|x - 3| - 1)| + 1) + 1)/(-1(2|... | 1153 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(1(2cos(1π(1|(-3|(2|x - 3| - 1) - 1| + 2) + 3| + 3) - 1) - 1) - 3) + 1)/(3(1(2cos(1π(1|(-3|(2|x - 3| - 1) - 1| + 2) + 3| + 3) - 1) - 1) - 3) - 2) and g(x) = (-3((-1((3(3(1x - 5)^2 + 2(1x - 5) - 2) + 3)/(-1(3(1x - 5)^2 + 2(1... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-4(3(-3cos(3π(2|(-4x - 2) + 3|) - 2) - 2) - 1) + 2) - 1| and g(x) = -1((1(1(2(-1|x - 1| - 3) - 3)^2 + 5(2(-1|x - 1| - 3) - 3) - 1) + 1)/(-3(1(2(-1|x - 1| - 3) - 3)^2 + 5(2(-1|x - 1| - 3) - 3) - 1) - 3))^2 - 2((1(1(2(-1|x - 1| - ... | 228 | MATH |
[
{
"content": "How many solutions does the equation (2((2(-2(-3|(2((-1x + 3)/(3x - 2))^2 + 2((-1x + 3)/(3x - 2)) + 1) - 2|)^2 + 4(-3|(2((-1x + 3)/(3x - 2))^2 + 2((-1x + 3)/(3x - 2)) + 1) - 2|) + 5) - 1)/(2(-2(-3|(2((-1x + 3)/(3x - 2))^2 + 2((-1x + 3)/(3x - 2)) + 1) - 2|)^2 + 4(-3|(2((-1x + 3)/(3x - 2))^2 + 2((-1... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3sin(1π(4((2((-2(3(3|x - 1| + 1) - 4) + 2)/(-3(3(3|x - 1| + 1) - 4) - 2)) - 2)/(3((-2(3(3|x - 1| + 1) - 4) + 2)/(-3(3(3|x - 1| + 1) - 4) - 2)) - 2)) - 1) + 3) + 3 and g(x) = (3((-3((-1(-2cos(3π(5x - 3) + 2)) - 1)/(3(-2cos(3π... | 242 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1((2((-1(4(-2|(-2x^2 - 3x + 1) + 1| + 2)^2 + 3(-2|(-2x^2 - 3x + 1) + 1| + 2) + 1) - 3)/(1(4(-2|(-2x^2 - 3x + 1) + 1| + 2)^2 + 3(-2|(-2x^2 - 3x + 1) + 1| + 2) + 1))) - 1)/(3((-1(4(-2|(-2x^2 - 3x + 1) + 1| + 2)^2 + 3(-2|(-2x^2 - 3x... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(-1|(-2(-5(2e^(2((1x - 3)/(-3x + 4))) - 2)) + 1) + 3| - 1) - 1)/(2(-1|(-2(-5(2e^(2((1x - 3)/(-3x + 4))) - 2)) + 1) + 3| - 1) + 1) and g(x) = -3|(-5(-1(-2|(4x^2 - 3x - 2) + 3| + 3) - 1) - 1) + 1| + 2 where -10 ≤ x ≤ 10.\n\nP... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(1(1|(2(-2e^(1(-4x^2 + 5x - 4)) - 1)^2 + 5(-2e^(1(-4x^2 + 5x - 4)) - 1) + 5) - 1| + 2)) + 4 and g(x) = 4(-4(2((1(3sin(3πx - 1) + 2) + 3)/(2(3sin(3πx - 1) + 2) - 1)) + 4) - 5)^2 + 4(-4(2((1(3sin(3πx - 1) + 2) + 3)/(2(3sin(3πx - 1) + 2) - 1)) + 4) - 5) ... | 60 | MATH |
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