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) = -1sin(1π(2(4((2(1x - 2) - 2)/(3(1x - 2) - 2))^2 + 5((2(1x - 2) - 2)/(3(1x - 2) - 2)) + 3) - 4) + 1) + 2 and g(x) = (-1(-5(-3((-2x + 3)/(2x + 3)) - 3)^2 - 2(-3((-2x + 3)/(2x + 3)) - 3) - 3) - 1)/(-1(-5(-3((-2x + 3)/(2x + 3)) - 3)^2 ... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(1(2(-2(-1x - 4)^2 + 5(-1x - 4) + 4)^2 - 1(-2(-1x - 4)^2 + 5(-1x - 4) + 4) + 1)^2 + 4(2(-2(-1x - 4)^2 + 5(-1x - 4) + 4)^2 - 1(-2(-1x - 4)^2 + 5(-1x - 4) + 4) + 1) + 4) - 1 and g(x) = 3(-2|(-5(1x^2 - 3) + 2) - 3| - 1) + 5 wh... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(2π(-3|(2|((-3(-2x^2 + 5x + 3) - 2)/(-2(-2x^2 + 5x + 3) - 3)) + 3|)|) - 1) - 3 and g(x) = 2((-1(-5((-2x)/(1x))^2 + 1((-2x)/(1x)) + 4) - 2)/(-1(-5((-2x)/(1x))^2 + 1((-2x)/(1x)) + 4) + 2))^2 - 3((-1(-5((-2x)/(1x))^2 + 1((-2x)/(1... | 101 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(2π(3|(-3|((-1(3|x - 3| + 3) - 2)/(-1(3|x - 3| + 3) - 1))| - 3) - 2| + 1) - 1) and g(x) = (-3(1|(-2|(-2|x + 1| - 1) - 3| - 1) - 2| + 1) - 1)/(-2(1|(-2|(-2|x + 1| - 1) - 3| - 1) - 2| + 1) + 1) in the range -10 ≤ x ≤ 10?\n\nPrese... | 8 | MATH |
[
{
"content": "How many solutions does the equation -2|((3((3(2|(-2e^(-1x) - 3) - 2| - 3))/(-2(2|(-2e^(-1x) - 3) - 2| - 3) + 3)))/(-1((3(2|(-2e^(-1x) - 3) - 2| - 3))/(-2(2|(-2e^(-1x) - 3) - 2| - 3) + 3)) + 2)) + 3| + 2 = -4((1((-3(3cos(2πx + 2) + 2) - 2)/(2(3cos(2πx + 2) + 2) - 1)) - 3)/(-1((-3(3cos(2πx + 2) + 2... | 40 | MATH |
[
{
"content": "How many solutions does the equation -2(2cos(1π(-2|(-3(1|x + 2| + 2) + 1) + 3| - 2) + 2) - 2)^2 - 5(2cos(1π(-2|(-3(1|x + 2| + 2) + 1) + 3| - 2) + 2) - 2) + 4 = (3(3e^(1((1((-3x + 1)/(1x)) - 2)/(-3((-3x + 1)/(1x))))) + 2) - 2)/(-2(3e^(1((1((-3x + 1)/(1x)) - 2)/(-3((-3x + 1)/(1x))))) + 2) + 3) have ... | 120 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(1(4((2(-2cos(3πx - 3) - 1))/(-1(-2cos(3πx - 3) - 1) - 2))^2 - 3)^2 - 2(4((2(-2cos(3πx - 3) - 1))/(-1(-2cos(3πx - 3) - 1) - 2))^2 - 3)) - 5 and g(x) = -5(-5((2((-3x)/(-1x)))/(-3((-3x)/(-1x)) - 2)) - 2)^2 + 1(-5((2((-3x)/(-1x)))/(-... | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(1(-3(1(3x)^2 - 1(3x) + 4) - 3) - 5) - 2)/(-3(1(-3(1(3x)^2 - 1(3x) + 4) - 3) - 5)) and g(x) = 4(-2|(3cos(2π((-3x - 1)/(1x + 3)) - 3) - 2) + 2| + 2)^2 - 5(-2|(3cos(2π((-3x - 1)/(1x + 3)) - 3) - 2) + 2| + 2) intersect in the interval [-10, 10].\n\nPre... | 182 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-5(2(4(-1|x + 3| + 3) - 2)^2 - 2(4(-1|x + 3| + 3) - 2) - 5)^2 - 4(2(4(-1|x + 3| + 3) - 2)^2 - 2(4(-1|x + 3| + 3) - 2) - 5) + 1) + 3 and g(x) = (1(3cos(1π(-3|(-3|x - 3| - 3) - 2| + 2) + 3) - 2) - 2)/(-3(3cos(1π(-3|(-3|x - 3| - 3)... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3e^(-1(4(1(-1|(-2x + 4)| + 1) + 5) - 2)) + 2 and g(x) = -2(-3|(-2|(-4x - 5)| + 2) + 1| + 1)^2 + 5(-3|(-2|(-4x - 5)| + 2) + 1| + 1) + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"ro... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-3(1(-1(4x - 5) + 1) - 4) + 4) - 2 and g(x) = 1|(1(-4(-3sin(3πx) + 2)^2 + 1(-3sin(3πx) + 2) - 3) + 2) + 2| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 25 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2cos(3π(-5(-3((-3((-2x)/(2x - 1)) - 2)/(3((-2x)/(2x - 1)) - 1)) + 2) - 2) - 2) - 3 and g(x) = -1cos(3π(-2(-2|(3|x + 2|) - 1| + 1) + 3) - 2) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 278 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1sin(2π(3(1|(-1|((-2x - 1)/(3x - 2)) - 3| - 2) - 1| + 2)^2 - 4(1|(-1|((-2x - 1)/(3x - 2)) - 3| - 2) - 1| + 2) - 5) - 3) + 3 and g(x) = -3sin(1π((-3(2(1x)^2 + 4(1x) - 1) - 3)/(3(2(1x)^2 + 4(1x) - 1) - 1)) - 3) + 1 where -10 ≤ ... | 89 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(4(-2|(-1(-3sin(2πx - 2)) - 1) + 3| + 2) + 2) - 1)/(-1(4(-2|(-1(-3sin(2πx - 2)) - 1) + 3| + 2) + 2) - 2) and g(x) = -1(3e^(-1(1|((-3x + 2)/(2x + 3))| - 3)) + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your ... | 26 | MATH |
[
{
"content": "How many solutions does the equation 1e^(-1(2|(2((-1((2x + 1)/(1x + 2)) - 2)/(-2((2x + 1)/(1x + 2)) + 3)) - 4) + 1| + 1)) - 2 = -1|((3(-2cos(3π((-3x + 3)/(2x + 1)) + 2) - 1) + 2)/(-3(-2cos(3π((-3x + 3)/(2x + 1)) + 2) - 1) + 4)) - 1| have for x in the interval [-10, 10]?\n\nPresent the answer in La... | 50 | MATH |
[
{
"content": "How many solutions does the equation (-1(1e^(1(1(3((-3x + 2)/(-2x + 1)) + 3)^2 + 4)) - 3) - 2)/(-2(1e^(1(1(3((-3x + 2)/(-2x + 1)) + 3)^2 + 4)) - 3) - 1) = -2cos(3π(-1|(5(-3|x| - 1)^2 + 1) + 3|) + 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 1557 | MATH |
[
{
"content": "How many solutions does the equation (3(1cos(2π(-2((3(3|x + 3| + 1) - 1)/(3(3|x + 3| + 1) + 2)) - 1) - 2) - 3) - 1)/(1(1cos(2π(-2((3(3|x + 3| + 1) - 1)/(3(3|x + 3| + 1) + 2)) - 1) - 2) - 3) + 3) = 2|(5(-1(3x) - 4)^2 - 3(-1(3x) - 4) - 1) + 1| have for x in the interval [-10, 10]?\n\nPresent the ans... | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(2|((-3(1(-3cos(2πx) + 2) - 2) - 2)/(-3(1(-3cos(2πx) + 2) - 2))) + 2| - 3) + 1 and g(x) = 2|(1(5(-2e^(-2x)) + 4) - 4)| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 39 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4((-2(3cos(2π((1(-1|x + 2| - 2) - 3)/(1(-1|x + 2| - 2) + 2)) + 2) - 2) - 2)/(-3(3cos(2π((1(-1|x + 2| - 2) - 3)/(1(-1|x + 2| - 2) + 2)) + 2) - 2) - 3))^2 - 3((-2(3cos(2π((1(-1|x + 2| - 2) - 3)/(1(-1|x + 2| - 2) + 2)) + 2) - 2... | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1((3(3sin(2π(-5(-1x^2 - 4x + 3)^2 + 3(-1x^2 - 4x + 3) - 4)) + 3) + 2)/(2(3sin(2π(-5(-1x^2 - 4x + 3)^2 + 3(-1x^2 - 4x + 3) - 4)) + 3) + 1))^2 - 4((3(3sin(2π(-5(-1x^2 - 4x + 3)^2 + 3(-1x^2 - 4x + 3) - 4)) + 3) + 2)/(2(3sin(2π(... | 69 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2e^(2(-1|((-2(-4(-1|x + 2| - 3) - 4) - 3)/(-1(-4(-1|x + 2| - 3) - 4) - 3)) + 1| + 2)) + 1 and g(x) = 3|(-4(-4((1x + 1)/(-1x)) - 5)^2 - 3(-4((1x + 1)/(-1x)) - 5) + 2) + 2| + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(-3((-2(-2sin(1π((-2x - 2)/(-2x + 3)) + 1) + 2) + 2)/(-3(-2sin(1π((-2x - 2)/(-2x + 3)) + 1) + 2) + 2))^2 + 1((-2(-2sin(1π((-2x - 2)/(-2x + 3)) + 1) + 2) + 2)/(-3(-2sin(1π((-2x - 2)/(-2x + 3)) + 1) + 2) + 2)) + 3)^2 - 4(-3((-2(-2sin(1π((-2x - 2)/(-2x ... | 87 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2e^(1((-3(1|(5(-1|x - 3| + 3)^2 + 4(-1|x - 3| + 3) - 1) + 2|) + 3)/(3(1|(5(-1|x - 3| + 3)^2 + 4(-1|x - 3| + 3) - 1) + 2|) + 3))) + 1 and g(x) = (-3(1sin(3π(-4((-3x)/(-1x + 3))^2 + 3((-3x)/(-1x + 3)) - 1) - 2) - 1) - 2)/(-1(1s... | 365 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(3sin(3π(-5((-3(5x^2 + 5x + 2) - 1)/(-1(5x^2 + 5x + 2))) - 2) - 2) + 1)^2 + 5(3sin(3π(-5((-3(5x^2 + 5x + 2) - 1)/(-1(5x^2 + 5x + 2))) - 2) - 2) + 1) + 4 and g(x) = 2(-3cos(2π(2|(-3x^2 - 3x + 4)| - 1) - 2) + 2) + 5 intersect in the interval [-10, 10].\... | 132 | MATH |
[
{
"content": "How many solutions does the equation 2sin(3π(1|((-3((1(3|x + 3| - 1) - 2)/(-2(3|x + 3| - 1) - 1)) - 1)/(-3((1(3|x + 3| - 1) - 2)/(-2(3|x + 3| - 1) - 1)) + 2)) - 2|) - 3) - 1 = -1(-1|(-1(-1x^2 - 5x - 5)) + 2| + 3) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\bo... | 61 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-3sin(2π(1|(-3(-1x + 1)^2 + 3(-1x + 1) + 1) + 3| - 3) + 3) - 3)^2 + 5(-3sin(2π(1|(-3(-1x + 1)^2 + 3(-1x + 1) + 1) + 3| - 3) + 3) - 3) + 3 and g(x) = -2((1((2(-1cos(1πx) - 3) + 1)/(1(-1cos(1πx) - 3) + 2)) + 2)/(-3((2(-1cos(1... | 1222 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(3(-4(-2(3|x + 3|)^2 + 2(3|x + 3|) + 2) - 4)^2 + 3(-4(-2(3|x + 3|)^2 + 2(3|x + 3|) + 2) - 4) - 3) + 1| - 2 and g(x) = 2cos(3π((2(-2|(2|x + 2| - 2) + 3|) - 3)/(1(-2|(2|x + 2| - 2) + 3|))) - 3) intersect in the interval [-10, 10].\n\nPresent the answer... | 8 | MATH |
[
{
"content": "How many solutions does the equation -1(-5(-2(-1|(4x^2 + 5x - 5) - 2| + 1) + 1)^2 - 5(-2(-1|(4x^2 + 5x - 5) - 2| + 1) + 1) + 1) - 4 = (2(-1((-2((-3x - 1)/(-3x + 4)) + 2)/(3((-3x - 1)/(-3x + 4))))^2 + 1((-2((-3x - 1)/(-3x + 4)) + 2)/(3((-3x - 1)/(-3x + 4)))) - 4) + 2)/(-2(-1((-2((-3x - 1)/(-3x + 4)... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(3π(5(-2|(1(5x^2 - 5x + 5) - 2) + 3| - 1) - 5) - 2) and g(x) = (3(3((-3(-4x^2 - 3x - 4) - 2)/(3(-4x^2 - 3x - 4) + 2))^2 + 2((-3(-4x^2 - 3x - 4) - 2)/(3(-4x^2 - 3x - 4) + 2)) - 4) - 2)/(3(3((-3(-4x^2 - 3x - 4) - 2)/(3(-4x^... | 782 | MATH |
[
{
"content": "How many solutions does the equation 2((2((-1((1(2x) - 2)/(-3(2x) + 4)) + 2)/(-2((1(2x) - 2)/(-3(2x) + 4)) - 3)) + 3)/(-3((-1((1(2x) - 2)/(-3(2x) + 4)) + 2)/(-2((1(2x) - 2)/(-3(2x) + 4)) - 3)) + 4)) - 4 = (1(-3(2sin(3π(3x^2 - 3) - 2) + 1)^2 + 3(2sin(3π(3x^2 - 3) - 2) + 1) + 4) + 1)/(2(-3(2sin(3π(3... | 55 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(1(2|((-2(1x^2 - 4x - 3) - 3)/(3(1x^2 - 4x - 3))) + 3| - 1) + 1) - 3)/(-1(1(2|((-2(1x^2 - 4x - 3) - 3)/(3(1x^2 - 4x - 3))) + 3| - 1) + 1) + 2) and g(x) = -1((-2((-3(-1x^2 + 1) - 1)/(2(-1x^2 + 1))) - 2)/(1((-3(-1x^2 + 1) - 1)/(2(... | 4 | MATH |
[
{
"content": "How many solutions does the equation -4(-4(-3cos(3π(-5(-2x^2 + 5x - 3) + 3) - 1) - 3)^2 + 3(-3cos(3π(-5(-2x^2 + 5x - 3) + 3) - 1) - 3) - 1) + 3 = 4(3|((-2(-2x^2 + 4) - 1)/(-2(-2x^2 + 4) + 3)) - 3| + 2)^2 + 1(3|((-2(-2x^2 + 4) - 1)/(-2(-2x^2 + 4) + 3)) - 3| + 2) - 3 have for x in the interval [-10,... | 1325 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(3(-3(2|(-1e^(-1x) + 1) - 3| + 3)^2 + 1(2|(-1e^(-1x) + 1) - 3| + 3) - 1) - 4) + 2)/(-3(3(-3(2|(-1e^(-1x) + 1) - 3| + 3)^2 + 1(2|(-1e^(-1x) + 1) - 3| + 3) - 1) - 4) - 2) and g(x) = 5(2cos(2π(3(4x^2 + 5x + 1)^2 + 1) + 2) - 1)^2 - 4(2cos(2π(3(4x^2 + 5x... | 446 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1cos(1π(-5(3|(-5(1|x| + 3) - 5) + 1| + 2) + 3) - 2) - 3 and g(x) = (-3(-1((2(-3|x + 1| + 3))/(1(-3|x + 1| + 3))) + 1))/(-1(-1((2(-3|x + 1| + 3))/(1(-3|x + 1| + 3))) + 1) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaT... | 1500 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(1cos(2π(2(-3(-5x - 3) - 2)^2 + 2(-3(-5x - 3) - 2) + 4) + 3) - 2) + 2)/(2(1cos(2π(2(-3(-5x - 3) - 2)^2 + 2(-3(-5x - 3) - 2) + 4) + 3) - 2) + 3) and g(x) = -3sin(2π(-1(-5((-2x + 2)/(-2x))^2 - 2((-2x + 2)/(-2x))) + 3) - 3) + 1 intersect in the interval... | 267 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(2|(3cos(1π((1(-3x^2 + 3x + 1) - 2)/(-1(-3x^2 + 3x + 1) - 3)) + 3) - 3) + 1| + 3) - 3| + 2 and g(x) = -5(2cos(3π(-3(-1x^2 - 2x - 4)^2 - 4(-1x^2 - 2x - 4) + 1) - 3) + 1)^2 + 2(2cos(3π(-3(-1x^2 - 2x - 4)^2 - 4(-1x^2 - 2x - 4) + 1) - 3) + 1) + 4 interse... | 98 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(-1(1cos(1π(-2|((1x + 2)/(2x - 1)) + 2| + 3) + 3) + 2)^2 + 1(1cos(1π(-2|((1x + 2)/(2x - 1)) + 2| + 3) + 3) + 2) + 1) + 1)/(2(-1(1cos(1π(-2|((1x + 2)/(2x - 1)) + 2| + 3) + 3) + 2)^2 + 1(1cos(1π(-2|((1x + 2)/(2x - 1)) + 2| +... | 10 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-2|((3(-5(-3sin(2πx + 2) + 2)^2 + 1(-3sin(2πx + 2) + 2) + 2) + 3)/(1(-5(-3sin(2πx + 2) + 2)^2 + 1(-3sin(2πx + 2) + 2) + 2) - 2)) + 2| + 1)^2 + 4(-2|((3(-5(-3sin(2πx + 2) + 2)^2 + 1(-3sin(2πx + 2) + 2) + 2) + 3)/(1(-5(-3sin(2πx +... | 72 | MATH |
[
{
"content": "How many solutions does the equation -4((-1((-1(1|(3|x - 2|) + 3| - 1) + 3)/(2(1|(3|x - 2|) + 3| - 1) + 2)) - 1)/(2((-1(1|(3|x - 2|) + 3| - 1) + 3)/(2(1|(3|x - 2|) + 3| - 1) + 2)) - 1))^2 - 2 = 4(4(-1|(-2cos(2πx) - 2) + 2| + 2)^2 + 5(-1|(-2cos(2πx) - 2) + 2| + 2) - 3) - 4 have for x in the interva... | 80 | MATH |
[
{
"content": "How many solutions does the equation 2|(2|(4(3|((-3x - 2)/(3x))| - 2) - 5) - 1| + 3)| + 1 = 3|((2(2|(2x^2 - 1x - 5) + 2| - 1) - 1)/(-2(2|(2x^2 - 1x - 5) + 2| - 1) + 3)) - 3| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 11 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-2|((-2(3(3sin(3πx + 2) - 3)^2 + 4(3sin(3πx + 2) - 3) - 2))/(3(3(3sin(3πx + 2) - 3)^2 + 4(3sin(3πx + 2) - 3) - 2) + 3)) + 2| - 2) + 3| + 2 and g(x) = 2cos(2π((-2(3|(4x^2 - 2) + 1| - 2) + 1)/(-1(3|(4x^2 - 2) + 1| - 2) - 3)... | 141 | MATH |
[
{
"content": "How many solutions does the equation 1sin(3π(4(-3(-5(1x + 1) + 2) + 3) + 5)) - 1 = (-1(-5(-4(2sin(2πx + 3) + 2)^2 + 1(2sin(2πx + 3) + 2))^2 + 2(-4(2sin(2πx + 3) + 2)^2 + 1(2sin(2πx + 3) + 2)) + 1))/(-1(-5(-4(2sin(2πx + 3) + 2)^2 + 1(2sin(2πx + 3) + 2))^2 + 2(-4(2sin(2πx + 3) + 2)^2 + 1(2sin(2πx + ... | 425 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|(1((-3(2(1|x - 3| - 2)^2 + 5(1|x - 3| - 2) - 5) + 3)/(1(2(1|x - 3| - 2)^2 + 5(1|x - 3| - 2) - 5) + 1)) + 3) - 2| + 3 and g(x) = -5((2(1sin(1π(-3|x| - 2)) + 3) + 1)/(-1(1sin(1π(-3|x| - 2)) + 3) + 2)) - 5 where -10 ≤ x ≤ 10.\... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|((3((-1(-1|(-2|x - 1| + 3) - 1| - 3) + 3)/(1(-1|(-2|x - 1| + 3) - 1| - 3) + 1)) + 2)/(1((-1(-1|(-2|x - 1| + 3) - 1| - 3) + 3)/(1(-1|(-2|x - 1| + 3) - 1| - 3) + 1)) + 2))| - 3 and g(x) = 5(3|((2(3x^2 - 2x + 2) - 2)/(-1(3x^2 - 2x +... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(1(-1(3(-2e^(-1x) + 3)^2 - 3(-2e^(-1x) + 3) + 5) + 5) - 1))/(2(1(-1(3(-2e^(-1x) + 3)^2 - 3(-2e^(-1x) + 3) + 5) + 5) - 1) + 3) and g(x) = 2((2(-2|(-4x + 4) - 1| - 1) - 1)/(-3(-2|(-4x + 4) - 1| - 1) - 1))^2 + 2((2(-2|(-4x + 4) - 1|... | 4 | MATH |
[
{
"content": "How many solutions does the equation (2(-2sin(3π((-3(5(-4x^2 - 3x + 1)^2 + 3(-4x^2 - 3x + 1) - 4) + 2)/(2(5(-4x^2 - 3x + 1)^2 + 3(-4x^2 - 3x + 1) - 4) + 1)) - 1) + 1) + 2)/(-2(-2sin(3π((-3(5(-4x^2 - 3x + 1)^2 + 3(-4x^2 - 3x + 1) - 4) + 2)/(2(5(-4x^2 - 3x + 1)^2 + 3(-4x^2 - 3x + 1) - 4) + 1)) - 1) ... | 38 | MATH |
[
{
"content": "How many solutions does the equation -3|(5(-4((1(1|x + 2|) - 2)/(3(1|x + 2|) - 2))^2 - 2((1(1|x + 2|) - 2)/(3(1|x + 2|) - 2)) + 5) - 4) + 1| + 2 = -1|((1(3(-2x^2 + 4x - 4)^2 + 1))/(3(3(-2x^2 + 4x - 4)^2 + 1))) - 2| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\... | 4 | MATH |
[
{
"content": "How many solutions does the equation -2|(-3sin(3π(5((3(-2x^2 - 4x + 5) + 3)/(2(-2x^2 - 4x + 5))) + 2) - 2) + 3) - 3| + 3 = 1|(1cos(3π(-5((3x - 3)/(2x + 3))^2 + 2((3x - 3)/(2x + 3)) - 3) + 1) + 3) - 3| + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 135 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|((2(3(-2(-1|x| + 1) - 1) - 3) - 2)/(2(3(-2(-1|x| + 1) - 1) - 3) + 3)) - 1| - 2 and g(x) = 3|(4((-1(-2|x + 2| - 2) - 1)/(3(-2|x + 2| - 2) + 1)) - 5) + 3| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxe... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(-1(3(-2|(-1cos(2πx - 3) + 2) + 1|) + 2)^2 - 3(3(-2|(-1cos(2πx - 3) + 2) + 1|) + 2) + 4) - 2)/(1(-1(3(-2|(-1cos(2πx - 3) + 2) + 1|) + 2)^2 - 3(3(-2|(-1cos(2πx - 3) + 2) + 1|) + 2) + 4) + 2) and g(x) = -1|(-2sin(1π(-5((-3x + 2)/(... | 19 | MATH |
[
{
"content": "How many solutions does the equation 1|(2(-4(5(1x + 1) - 1) + 3)^2 - 1(-4(5(1x + 1) - 1) + 3) - 3) + 1| + 2 = 3|((2(2cos(3π(1|x - 1| - 3)) + 3) + 1)/(3(2cos(3π(1|x - 1| - 3)) + 3) + 1)) + 1| - 2 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) = -3cos(3π(-4(-3|(-1|((3x - 1)/(2x - 1)) - 3| + 2) + 3| + 2)^2 + 5(-3|(-1|((3x - 1)/(2x - 1)) - 3| + 2) + 3| + 2) - 4) + 3) - 3 and g(x) = (-1((2(3(-3cos(3πx + 2) + 1)^2 + 5(-3cos(3πx + 2) + 1) + 1) + 2)/(-3(3(-3cos(3πx + 2) + 1)^2 + 5(-3cos(3πx + 2) + 1... | 385 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-2sin(1π(-2((-2(-2x - 1) + 1)/(2(-2x - 1) - 1)) - 4) + 1) + 2)^2 - 4 and g(x) = 3((1(-3sin(3π(-3|x - 3| - 3) - 3) + 1))/(2(-3sin(3π(-3|x - 3| - 3) - 3) + 1) + 2)) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex forma... | 168 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(-2((1(1|(3|x - 2| + 2) + 3| + 2) - 1)/(-1(1|(3|x - 2| + 2) + 3| + 2) + 2))^2 + 5((1(1|(3|x - 2| + 2) + 3| + 2) - 1)/(-1(1|(3|x - 2| + 2) + 3| + 2) + 2)) + 3) + 1 and g(x) = -3(1((-2(-1x^2 + 3x + 5) - 2)/(1(-1x^2 + 3x + 5) - 3))^2... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-1sin(3π(-1|(1|(-4x - 2)| - 3) - 3| + 1) + 1) - 2) + 3)/(1(-1sin(3π(-1|(1|(-4x - 2)| - 3) - 3| + 1) + 1) - 2) - 2) and g(x) = -2(4(1(-1|x + 3| - 3)^2 + 4(-1|x + 3| - 3) - 2)^2 + 5(1(-1|x + 3| - 3)^2 + 4(-1|x + 3| - 3) - 2) - 2)^... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(3π(-5(-2(5((-3x - 3)/(1x + 3)) - 3) + 2) - 3) + 2) - 3 and g(x) = -1(-3cos(3π(2|((-1x - 2)/(2x + 2)) - 3| + 3) - 2)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 488 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(-1|(3(1((2x - 3)/(-3x))^2 - 4((2x - 3)/(-3x)))^2 - 1(1((2x - 3)/(-3x))^2 - 4((2x - 3)/(-3x))) - 2) - 1| + 3)^2 + 5(-1|(3(1((2x - 3)/(-3x))^2 - 4((2x - 3)/(-3x)))^2 - 1(1((2x - 3)/(-3x))^2 - 4((2x - 3)/(-3x))) - 2) - 1| + 3) - 4 and g(x) = 1(3(2(-1|x ... | 4 | MATH |
[
{
"content": "How many solutions does the equation 1(-2cos(1π(-1|(-5(-3|x + 3| - 2)^2 + 5(-3|x + 3| - 2) + 2) + 2| + 2) - 3) - 1)^2 + 4(-2cos(1π(-1|(-5(-3|x + 3| - 2)^2 + 5(-3|x + 3| - 2) + 2) + 2| + 2) - 3) - 1) - 5 = 3cos(3π((-3(-4(1x^2 - 1x - 4)^2 - 3(1x^2 - 1x - 4)))/(-2(-4(1x^2 - 1x - 4)^2 - 3(1x^2 - 1x - ... | 177 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(-1|(1e^(-1((-2(-3x^2 + 2x + 3) + 2)/(-3(-3x^2 + 2x + 3) - 3))) - 2) - 2| + 2) - 2 and g(x) = (-3(-1e^(-2((-1(-2|x + 2| + 3) - 3)/(2(-2|x + 2| + 3) - 3))) - 2) + 1)/(1(-1e^(-2((-1(-2|x + 2| + 3) - 3)/(2(-2|x + 2| + 3) - 3))) - 2)) intersect in the int... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(3|((3(-1|((-1x - 1)/(1x + 2)) - 1| + 1) - 3)/(3(-1|((-1x - 1)/(1x + 2)) - 1| + 1) - 1)) + 2| + 2) + 3| + 2 and g(x) = -5(-1(-3(2x^2 + 3x) + 2) - 3)^2 + 1(-1(-3(2x^2 + 3x) + 2) - 3) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the ... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3((-2((1((3(-1|x - 1| + 1) + 2)/(2(-1|x - 1| + 1) + 1)) - 2)/(-1((3(-1|x - 1| + 1) + 2)/(2(-1|x - 1| + 1) + 1)) - 2)) - 2)/(-1((1((3(-1|x - 1| + 1) + 2)/(2(-1|x - 1| + 1) + 1)) - 2)/(-1((3(-1|x - 1| + 1) + 2)/(2(-1|x - 1| + 1) + ... | 42 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(3(-5(-2|(4x^2 + 1x - 5) - 2| + 3)^2 - 4(-2|(4x^2 + 1x - 5) - 2| + 3))^2 - 4(-5(-2|(4x^2 + 1x - 5) - 2| + 3)^2 - 4(-2|(4x^2 + 1x - 5) - 2| + 3)) - 3) + 3 and g(x) = (-2(-5(-2(1|x - 2| + 2)^2 - 3(1|x - 2| + 2) - 5)^2 - 3(-2(1|x - 2| + 2)^2 - 3(1|x - 2|... | 8 | MATH |
[
{
"content": "How many solutions does the equation -2|(2((1(-5((1x - 1)/(2x + 2)) + 1) + 3)/(2(-5((1x - 1)/(2x + 2)) + 1) - 1))^2 - 4((1(-5((1x - 1)/(2x + 2)) + 1) + 3)/(2(-5((1x - 1)/(2x + 2)) + 1) - 1)) + 5) + 3| + 1 = -1(-1|(-4(-3cos(1πx) - 2) - 5) - 1| - 3)^2 + 3(-1|(-4(-3cos(1πx) - 2) - 5) - 1| - 3) - 5 ha... | 12 | MATH |
[
{
"content": "How many solutions does the equation -1|(-3|(-4((-2(5x - 3) - 3)/(2(5x - 3) - 1)) - 5) - 2| + 3) - 1| - 3 = (-3(-2|((1(3cos(2πx) + 3) - 3)/(3(3cos(2πx) + 3) - 1)) - 1| - 1) - 2)/(1(-2|((1(3cos(2πx) + 3) - 3)/(3(3cos(2πx) + 3) - 1)) - 1| - 1) + 1) have for x in the interval [-10, 10]?\n\nPresent th... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-5(2cos(3π((1(-1|x + 3| + 2) + 3)/(1(-1|x + 3| + 2))))) + 1) - 2)/(-3(-5(2cos(3π((1(-1|x + 3| + 2) + 3)/(1(-1|x + 3| + 2))))) + 1) - 1) and g(x) = (-1(1(3|(2x + 4)| + 3)^2 - 4(3|(2x + 4)| + 3) + 3))/(2(1(3|(2x + 4)| + 3)^2 - 4(3|(2x + 4)| + 3) + 3) ... | 144 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(-2(1((-3((-3x + 2)/(-1x + 3)) + 2)/(-2((-3x + 2)/(-1x + 3)) - 3))^2 - 5((-3((-3x + 2)/(-1x + 3)) + 2)/(-2((-3x + 2)/(-1x + 3)) - 3)) - 4) + 4) - 1)/(3(-2(1((-3((-3x + 2)/(-1x + 3)) + 2)/(-2((-3x + 2)/(-1x + 3)) - 3))^2 - 5((-3((... | 4 | MATH |
[
{
"content": "How many solutions does the equation -2(-4(3(-1(-5x + 2)^2 - 4(-5x + 2) + 1) + 3)^2 + 5(3(-1(-5x + 2)^2 - 4(-5x + 2) + 1) + 3) - 4) - 1 = 3|(-4(4(-1x + 3) - 2) + 5) - 2| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation 1e^(-1((1(-3|(3(-1x + 3)^2 - 5(-1x + 3) + 5) - 1| - 2) - 1)/(-2(-3|(3(-1x + 3)^2 - 5(-1x + 3) + 5) - 1| - 2) - 1))) + 1 = (3((3(-3|(-1|x + 2| + 1) + 1| + 2) + 1)/(-2(-3|(-1|x + 2| + 1) + 1| + 2) + 3)))/(1((3(-3|(-1|x + 2| + 1) + 1| + 2) + 1)/(-2(-3|(-1|x + 2| +... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3((2((-3((2(-1x + 1) - 2)/(3(-1x + 1) + 3)) + 3)/(3((2(-1x + 1) - 2)/(3(-1x + 1) + 3)) + 2)) - 2)/(-2((-3((2(-1x + 1) - 2)/(3(-1x + 1) + 3)) + 3)/(3((2(-1x + 1) - 2)/(3(-1x + 1) + 3)) + 2)) + 3))^2 + 5((2((-3((2(-1x + 1) - 2)/(3(-1... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4((2(-1(3|(1x - 5) - 3| - 2) - 2))/(-3(-1(3|(1x - 5) - 3| - 2) - 2) - 1)) - 4 and g(x) = 2|(-5((3(-3sin(3πx) + 3) - 3)/(3(-3sin(3πx) + 3) + 3)) - 1) - 3| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxe... | 7 | MATH |
[
{
"content": "How many solutions does the equation 4(-4(2|(2|(1cos(3πx - 1) - 1) + 2|)| + 3) - 5) - 3 = 5(3(-2|(-2|x - 1| + 3) + 1| + 1) - 4) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 20 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2((-1(2sin(1π(2|((-3x - 2)/(2x - 1)) + 2|) - 2) + 3) + 1)/(2(2sin(1π(2|((-3x - 2)/(2x - 1)) + 2|) - 2) + 3)))^2 + 5((-1(2sin(1π(2|((-3x - 2)/(2x - 1)) + 2|) - 2) + 3) + 1)/(2(2sin(1π(2|((-3x - 2)/(2x - 1)) + 2|) - 2) + 3))) ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|((1(3(-3((-2x)/(-3x + 2)) - 4)^2 - 4(-3((-2x)/(-3x + 2)) - 4) - 3) + 1)/(-1(3(-3((-2x)/(-3x + 2)) - 4)^2 - 4(-3((-2x)/(-3x + 2)) - 4) - 3) + 2)) - 1| + 1 and g(x) = 2sin(1π(4(2(1|x + 3| + 3) - 2)^2 + 1) - 2) + 3 in the range -10 ... | 1834 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(1(3|(2(-1sin(2πx + 1) + 1)^2 - 2(-1sin(2πx + 1) + 1) - 3) + 3| + 2) - 3)^2 - 3(1(3|(2(-1sin(2πx + 1) + 1)^2 - 2(-1sin(2πx + 1) + 1) - 3) + 3| + 2) - 3) - 2 and g(x) = 1e^(-1(-3((1((-1x - 1)/(1x + 1)))/(-3((-1x - 1)/(1x + 1)... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(4(-1|(-1((-2x - 1)/(-1x))^2 + 5((-2x - 1)/(-1x)) + 4) - 2| - 1)^2 + 4(-1|(-1((-2x - 1)/(-1x))^2 + 5((-2x - 1)/(-1x)) + 4) - 2| - 1) - 3) - 1| + 2 and g(x) = (2(-2(-2|(-4x^2 + 5) - 3| - 1) - 2))/(3(-2(-2|(-4x^2 + 5) - 3| - 1) - 2) - 2) intersect in ... | 7 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(-2(-2|(3e^(-1(-4x^2 + 2x - 4)) + 1) - 1|) - 5) + 1)/(2(-2(-2|(3e^(-1(-4x^2 + 2x - 4)) + 1) - 1|) - 5) - 1) and g(x) = 3(-3(3e^(2(4x^2 + 3x - 3)) - 3) - 3)^2 - 4(-3(3e^(2(4x^2 + 3x - 3)) - 3) - 3) + 1 in the range -10 ≤ x ≤ 10?\n... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1e^(-2(5((2(3((-3x + 1)/(-2x + 1)) + 5) - 3)/(-3(3((-3x + 1)/(-2x + 1)) + 5) + 4))^2 - 2((2(3((-3x + 1)/(-2x + 1)) + 5) - 3)/(-3(3((-3x + 1)/(-2x + 1)) + 5) + 4)))) - 2 and g(x) = 2(-3((-3(-2sin(2πx - 2) - 2) - 3)/(2(-2sin(2πx - 2) - 2) + 1)) + 3) + 5 ... | 40 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(3π((1(-1(5(1x^2 - 2x - 5)^2 + 4(1x^2 - 2x - 5) + 3) - 5))/(-2(-1(5(1x^2 - 2x - 5)^2 + 4(1x^2 - 2x - 5) + 3) - 5) - 3)) + 2) + 1 and g(x) = (1(-1(1|(1cos(3πx - 1)) + 2| - 1)^2 - 5(1|(1cos(3πx - 1)) + 2| - 1) + 2) + 1)/(-1... | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2((2(-3|(-1|(-1|x - 1| - 2) + 1| + 2) + 1|) + 1)/(3(-3|(-1|(-1|x - 1| - 2) + 1| + 2) + 1|) - 2)) + 3)/(-2((2(-3|(-1|(-1|x - 1| - 2) + 1| + 2) + 1|) + 1)/(3(-3|(-1|(-1|x - 1| - 2) + 1| + 2) + 1|) - 2)) - 1) and g(x) = -3|(-2sin(3π(-2|((-2x - 3)/(-1x - ... | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(3π(-2(2(-1((2x - 3)/(-3x - 2)) - 4)^2 - 2(-1((2x - 3)/(-3x - 2)) - 4)) + 2) - 3) - 2 and g(x) = 1|(-1((-1(1x - 1) + 1)/(-1(1x - 1) - 3)) - 4) + 3| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Yo... | 255 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3e^(2(3(5((1(-3|x + 3| + 1) + 3)/(-3(-3|x + 3| + 1) - 1)))^2 - 3(5((1(-3|x + 3| + 1) + 3)/(-3(-3|x + 3| + 1) - 1))) - 2)) and g(x) = 2((-3(1cos(2π(2|x + 2| - 1)) - 2) - 1)/(-3(1cos(2π(2|x + 2| - 1)) - 2) - 2))^2 - 5((-3(1cos(2π(2|x + 2| - 1)) - 2) - 1... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(5(2|((-2(1x^2 + 4x - 5))/(2(1x^2 + 4x - 5) + 3)) - 3| - 3)^2 + 1(2|((-2(1x^2 + 4x - 5))/(2(1x^2 + 4x - 5) + 3)) - 3| - 3) + 3) + 1)/(-3(5(2|((-2(1x^2 + 4x - 5))/(2(1x^2 + 4x - 5) + 3)) - 3| - 3)^2 + 1(2|((-2(1x^2 + 4x - 5))/(2(1x^2 + 4x - 5) + 3)) ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(1(-2cos(2π((-1((-2x - 1)/(2x - 3)) - 1)/(1((-2x - 1)/(2x - 3))))) + 3) - 1) + 1)/(3(1(-2cos(2π((-1((-2x - 1)/(2x - 3)) - 1)/(1((-2x - 1)/(2x - 3))))) + 3) - 1)) and g(x) = -3(-2cos(2π(2|(5x^2 + 2x)| - 1)) + 1)^2 - 5(-2cos(2π(2|(5x^2 + 2x)| - 1)) + ... | 1643 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(1(1(1|(-1x^2 + 5x + 1) + 1| - 1) + 3) - 5) - 2| + 2 and g(x) = 3((3((-3(1cos(3πx + 1)) - 2)/(2(1cos(3πx + 1)) - 1)))/(-3((-3(1cos(3πx + 1)) - 2)/(2(1cos(3πx + 1)) - 1)) + 1))^2 - 5((3((-3(1cos(3πx + 1)) - 2)/(2(1cos(3πx + 1)) - ... | 103 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(1(3sin(3π(-5(2|x - 2| + 2)^2 + 4(2|x - 2| + 2) + 4) + 3))^2 - 3(3sin(3π(-5(2|x - 2| + 2)^2 + 4(2|x - 2| + 2) + 4) + 3)) - 3) - 5 and g(x) = -1(-2((-3(3|x| + 2) - 3)/(2(3|x| + 2) + 2))^2 + 1((-3(3|x| + 2) - 3)/(2(3|x| + 2) +... | 1329 | MATH |
[
{
"content": "How many solutions does the equation 5((-2(-4(-5((3x + 1)/(-2x + 1))^2 + 5((3x + 1)/(-2x + 1)) + 2) - 2) + 1)/(2(-4(-5((3x + 1)/(-2x + 1))^2 + 5((3x + 1)/(-2x + 1)) + 2) - 2) + 3)) + 2 = -5(-3|((2(1x - 2))/(3(1x - 2) - 2)) + 1| + 3)^2 + 3(-3|((2(1x - 2))/(3(1x - 2) - 2)) + 1| + 3) - 1 have for x i... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(-3(3(-1cos(3π((-1x - 3)/(-3x + 1)) + 1) - 1)^2 - 4)^2 + 5(3(-1cos(3π((-1x - 3)/(-3x + 1)) + 1) - 1)^2 - 4) + 4) - 5 and g(x) = 1|(4(1|((-3x - 1)/(3x + 2)) + 1| + 1)^2 - 3(1|((-3x - 1)/(3x + 2)) + 1| + 1) + 5) - 1| where -10... | 46 | MATH |
[
{
"content": "How many solutions does the equation 5(4(3sin(2π(1|(2|x| + 2)| + 2) - 3) - 2) + 5)^2 + 2(4(3sin(2π(1|(2|x| + 2)| + 2) - 3) - 2) + 5) - 2 = (-3(-1(-1e^(1(1x + 2)) - 3)^2 + 4(-1e^(1(1x + 2)) - 3) + 5) - 1)/(2(-1(-1e^(1(1x + 2)) - 3)^2 + 4(-1e^(1(1x + 2)) - 3) + 5) + 1) have for x in the interval [-1... | 120 | MATH |
[
{
"content": "How many solutions does the equation (2(-5(3((-1(-1sin(1πx - 2)) - 2)/(3(-1sin(1πx - 2)) - 1)) + 4)^2 - 3(3((-1(-1sin(1πx - 2)) - 2)/(3(-1sin(1πx - 2)) - 1)) + 4) + 1) - 2)/(-3(-5(3((-1(-1sin(1πx - 2)) - 2)/(3(-1sin(1πx - 2)) - 1)) + 4)^2 - 3(3((-1(-1sin(1πx - 2)) - 2)/(3(-1sin(1πx - 2)) - 1)) + 4... | 19 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-3((-1(-3|(-2|x - 1| + 1)| + 3))/(3(-3|(-2|x - 1| + 1)| + 3) + 3))^2 - 1((-1(-3|(-2|x - 1| + 1)| + 3))/(3(-3|(-2|x - 1| + 1)| + 3) + 3)) + 5) - 2)/(2(-3((-1(-3|(-2|x - 1| + 1)| + 3))/(3(-3|(-2|x - 1| + 1)| + 3) + 3))^2 - 1((-1(... | 1526 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1cos(3π(5(4(1|(1x + 4) - 1|) - 5) - 1) - 1) and g(x) = 5(2|(3|(1|x - 2| + 2) - 2| - 3) + 1|) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 15 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((-1((-2(1(-4x - 1) - 4) + 2)/(2(1(-4x - 1) - 4) + 3)) + 3)/(-1((-2(1(-4x - 1) - 4) + 2)/(2(1(-4x - 1) - 4) + 3)) + 3))^2 + 3 and g(x) = 5((3(2e^(-2(-4x^2 + 1x + 5)) - 2) + 3)/(-1(2e^(-2(-4x^2 + 1x + 5)) - 2) - 2))^2 + 1((3(2e^(-2(-4x^2 + 1x + 5)) - 2... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-2|(5(-1(-3|x + 3| + 1)^2 - 1(-3|x + 3| + 1) + 4) - 3) + 2|) + 1| + 2 and g(x) = -2e^(-1(1(-5(-1|x + 3| - 2) - 4) + 2)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation 2(3((3(5(2e^(-2x) + 3)^2 + 4(2e^(-2x) + 3) - 2) - 1)/(1(5(2e^(-2x) + 3)^2 + 4(2e^(-2x) + 3) - 2) + 3)) - 3)^2 - 4(3((3(5(2e^(-2x) + 3)^2 + 4(2e^(-2x) + 3) - 2) - 1)/(1(5(2e^(-2x) + 3)^2 + 4(2e^(-2x) + 3) - 2) + 3)) - 3) + 3 = -1(-3(5(-3|x| + 3)^2 - 3(-3|x| + 3)... | 4 | MATH |
[
{
"content": "How many solutions does the equation 3|(1(-2e^(2(1(3|x + 1| - 2) - 1)) + 3) - 4) + 3| + 2 = 2(1|(-2(3sin(2πx + 1) + 2) + 1) + 1| - 2)^2 - 4(1|(-2(3sin(2πx + 1) + 2) + 1) + 1| - 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2((-2(3|(1|(5x - 2) + 1| - 1) - 1| - 2) + 2)/(-1(3|(1|(5x - 2) + 1| - 1) - 1| - 2) + 2)) - 2)/(3((-2(3|(1|(5x - 2) + 1| - 1) - 1| - 2) + 2)/(-1(3|(1|(5x - 2) + 1| - 1) - 1| - 2) + 2))) and g(x) = (-1(-1((-1(-4x^2 - 4x - 3) ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2e^(-2(1(2|((-3(-1x^2 - 2x + 3) - 3)/(-1(-1x^2 - 2x + 3))) + 3|)^2 - 4(2|((-3(-1x^2 - 2x + 3) - 3)/(-1(-1x^2 - 2x + 3))) + 3|) + 4)) and g(x) = 2|(-1((-3(-2|x - 1| + 2) + 3)/(-2(-2|x - 1| + 2) - 2)) + 1) - 3| - 1 in the range -10 ≤... | 4 | MATH |
[
{
"content": "How many solutions does the equation 4((1((3(5((-1x - 1)/(-1x)) - 4) - 2)/(1(5((-1x - 1)/(-1x)) - 4) + 3)) - 2)/(-2((3(5((-1x - 1)/(-1x)) - 4) - 2)/(1(5((-1x - 1)/(-1x)) - 4) + 3)) + 1)) + 4 = 2(-1(-2cos(3π(4x^2 - 3x - 1)) - 3) - 4) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer i... | 799 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1cos(2π(-3|(1|(1(-3|x| + 1)^2 - 3(-3|x| + 1) + 1) + 1| - 1) - 1| + 3) - 3) + 3 and g(x) = (2(2|(2sin(2π(-3x^2 - 3x - 4) - 1) + 1) - 2| - 1))/(-1(2|(2sin(2π(-3x^2 - 3x - 4) - 1) + 1) - 2| - 1) + 3) in the range -10 ≤ x ≤ 10?\n\nPre... | 399 | MATH |
[
{
"content": "How many solutions does the equation -3(4((3((-1(1x^2 + 4x - 2) - 3)/(1(1x^2 + 4x - 2) + 3)) + 2)/(-2((-1(1x^2 + 4x - 2) - 3)/(1(1x^2 + 4x - 2) + 3))))^2 + 2((3((-1(1x^2 + 4x - 2) - 3)/(1(1x^2 + 4x - 2) + 3)) + 2)/(-2((-1(1x^2 + 4x - 2) - 3)/(1(1x^2 + 4x - 2) + 3)))) + 4)^2 - 4 = -2((-1(-1((3x - 2... | 4 | MATH |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.