messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Find the number of times the graphs of f(x) = 1(1|(-1(1cos(1π(-5x^2 + 1x - 1) + 2) - 1) - 2) + 3| - 3)^2 - 5(1|(-1(1cos(1π(-5x^2 + 1x - 1) + 2) - 1) - 2) + 3| - 3) and g(x) = -2(5(3(-1x + 2) + 2)^2 - 4(3(-1x + 2) + 2) - 3) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex for... | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(5(-1((1(3x) - 2)/(1(3x)))^2 + 4((1(3x) - 2)/(1(3x))) + 4)^2 + 4(-1((1(3x) - 2)/(1(3x)))^2 + 4((1(3x) - 2)/(1(3x))) + 4) - 5) and g(x) = -1|((1(-1(2|x| + 3)^2 - 4(2|x| + 3) + 1))/(1(-1(2|x| + 3)^2 - 4(2|x| + 3) + 1) + 2)) - ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(3(-4(-1e^(-1(-1|x + 2|)) + 2) - 3) - 4)^2 - 2(3(-4(-1e^(-1(-1|x + 2|)) + 2) - 3) - 4) - 5 and g(x) = -1|((1(-5(-4x + 3) + 3) + 3)/(-2(-5(-4x + 3) + 3))) + 3| - 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3((1(-3|(3(-1sin(2πx + 2) + 3) + 1) - 3| + 1) + 2)/(2(-3|(3(-1sin(2πx + 2) + 3) + 1) - 3| + 1) - 1))^2 - 5((1(-3|(3(-1sin(2πx + 2) + 3) + 1) - 3| + 1) + 2)/(2(-3|(3(-1sin(2πx + 2) + 3) + 1) - 3| + 1) - 1)) + 1 and g(x) = (3(-2(3co... | 844 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(1(2(-1|(-2x^2 - 5x - 3) + 3| - 1) - 4)^2 - 4(2(-1|(-2x^2 - 5x - 3) + 3| - 1) - 4) + 4) + 1 and g(x) = 3|(-1(-1(5x - 1)^2 - 4(5x - 1) + 3)^2 + 2(-1(5x - 1)^2 - 4(5x - 1) + 3) - 4) - 3| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(3((-2(2(2sin(1πx + 1) + 2)^2 - 1(2sin(1πx + 1) + 2) - 3) - 1)/(2(2(2sin(1πx + 1) + 2)^2 - 1(2sin(1πx + 1) + 2) - 3) + 1)) - 2) - 1)/(3(3((-2(2(2sin(1πx + 1) + 2)^2 - 1(2sin(1πx + 1) + 2) - 3) - 1)/(2(2(2sin(1πx + 1) + 2)^2 - 1(2sin(1πx + 1) + 2) - ... | 15 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(2|((-3(1|(1|x + 2| - 1) + 3| - 3) - 2)/(-2(1|(1|x + 2| - 1) + 3| - 3) - 1)) + 1| - 2) + 1)/(-2(2|((-3(1|(1|x + 2| - 1) + 3| - 3) - 2)/(-2(1|(1|x + 2| - 1) + 3| - 3) - 1)) + 1| - 2) - 3) and g(x) = 3|((-3((-1(-2|x + 3|) + 1)/(1(-... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1(1cos(1π(3(-3(-3x^2 - 4x + 3) + 3) - 3) - 1) - 1)^2 + 2(1cos(1π(3(-3(-3x^2 - 4x + 3) + 3) - 3) - 1) - 1) - 5 = -2|((2((-1(-2|x + 3| - 2) + 1)/(2(-2|x + 3| - 2) - 1)) + 3)/(-1((-1(-2|x + 3| - 2) + 1)/(2(-2|x + 3| - 2) - 1)) - 2)) - 2| - 1 have for x in the int... | 1507 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((-1(1|(3sin(1π(4x + 1) - 2) + 2) - 1| - 2) + 2)/(3(1|(3sin(1π(4x + 1) - 2) + 2) - 1| - 2) - 2)) - 4 and g(x) = (2((1(1|(2cos(2πx - 2) - 2) + 3|) + 2)/(-3(1|(2cos(2πx - 2) - 2) + 3|) + 2)) - 1)/(1((1(1|(2cos(2πx - 2) - 2) + 3|) + 2)/(-3(1|(2cos(2πx - ... | 43 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1((3((3((1(-3x^2 - 3x - 5) + 2)/(1(-3x^2 - 3x - 5) - 2)) - 3)/(2((1(-3x^2 - 3x - 5) + 2)/(1(-3x^2 - 3x - 5) - 2)) - 1)) + 1)/(-2((3((1(-3x^2 - 3x - 5) + 2)/(1(-3x^2 - 3x - 5) - 2)) - 3)/(2((1(-3x^2 - 3x - 5) + 2)/(1(-3x^2 -... | 1047 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(1|(-1(1(1x + 1)^2 - 4)^2 + 1(1(1x + 1)^2 - 4)) - 1| + 1) + 3| - 1 and g(x) = -2(3sin(3π((3(-4x^2 + 3x + 4) + 3)/(-2(-4x^2 + 3x + 4) - 3)) - 1) + 2)^2 - 5(3sin(3π((3(-4x^2 + 3x + 4) + 3)/(-2(-4x^2 + 3x + 4) - 3)) - 1) + 2)... | 34 | MATH |
[
{
"content": "How many solutions does the equation (2(-1(1(-1|(3x^2 + 1x - 4) - 2|)^2 + 4(-1|(3x^2 + 1x - 4) - 2|) - 2) - 5) - 1)/(1(-1(1(-1|(3x^2 + 1x - 4) - 2|)^2 + 4(-1|(3x^2 + 1x - 4) - 2|) - 2) - 5)) = (-3(3e^(-1(2(-2|x| + 1)^2 - 5(-2|x| + 1) - 2)) + 3) + 3)/(3(3e^(-1(2(-2|x| + 1)^2 - 5(-2|x| + 1) - 2)) + ... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3((1(-1|(-3|((-3x + 2)/(2x - 1)) + 1| + 2) - 2| + 1) - 1)/(2(-1|(-3|((-3x + 2)/(2x - 1)) + 1| + 2) - 2| + 1) + 2))^2 + 5((1(-1|(-3|((-3x + 2)/(2x - 1)) + 1| + 2) - 2| + 1) - 1)/(2(-1|(-3|((-3x + 2)/(2x - 1)) + 1| + 2) - 2| + 1) + 2... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-1(1sin(3π(-3(4x^2 - 4x - 5)^2 + 1(4x^2 - 4x - 5) - 2) + 2))^2 + 4(1sin(3π(-3(4x^2 - 4x - 5)^2 + 1(4x^2 - 4x - 5) - 2) + 2)) - 2)^2 - 1(-1(1sin(3π(-3(4x^2 - 4x - 5)^2 + 1(4x^2 - 4x - 5) - 2) + 2))^2 + 4(1sin(3π(-3(4x^2 - 4x - 5)^... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2cos(1π((-3(-3(5((2x + 2)/(2x - 1))^2 + 3) + 3) - 1)/(3(-3(5((2x + 2)/(2x - 1))^2 + 3) + 3)))) + 3 and g(x) = -2cos(2π((2(-3((-2x + 1)/(1x))^2 - 4((-2x + 1)/(1x)) + 2) - 2)/(2(-3((-2x + 1)/(1x))^2 - 4((-2x + 1)/(1x)) + 2) + 1... | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3((3((1((1(-1e^(1x) + 2) + 2)/(-2(-1e^(1x) + 2) + 3)) + 2)/(3((1(-1e^(1x) + 2) + 2)/(-2(-1e^(1x) + 2) + 3)) - 2)) + 3)/(1((1((1(-1e^(1x) + 2) + 2)/(-2(-1e^(1x) + 2) + 3)) + 2)/(3((1(-1e^(1x) + 2) + 2)/(-2(-1e^(1x) + 2) + 3)) - 2)) - 3)) - 1 and g(x) =... | 12 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(2(-1cos(2π(1(1x - 3)^2 - 2(1x - 3) - 1)) + 1)^2 - 3(-1cos(2π(1(1x - 3)^2 - 2(1x - 3) - 1)) + 1) + 3) - 2| - 3 and g(x) = 2|(3((2(3cos(2πx - 3)) + 1)/(1(3cos(2πx - 3)) + 3))^2 - 2((2(3cos(2πx - 3)) + 1)/(1(3cos(2πx - 3)) + 3)) + ... | 50 | MATH |
[
{
"content": "How many solutions does the equation 4(1(-2|(2cos(3π(-1x^2 + 4x + 2) + 3) - 1) - 2| + 2)^2 + 1(-2|(2cos(3π(-1x^2 + 4x + 2) + 3) - 1) - 2| + 2)) + 5 = 5(-1|(-2|(-3cos(1πx - 2) + 2) + 3| - 2) - 2| + 1)^2 + 4(-1|(-2|(-3cos(1πx - 2) + 2) + 3| - 2) - 2| + 1) - 4 have for x in the interval [-10, 10]?\n\... | 36 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-3(-1|((3(1cos(3πx - 3) - 1) + 1)/(1(1cos(3πx - 3) - 1))) - 2| + 1) + 2))/(2(-3(-1|((3(1cos(3πx - 3) - 1) + 1)/(1(1cos(3πx - 3) - 1))) - 2| + 1) + 2) - 1) and g(x) = -3cos(2π(3(-3(4x - 1)^2 - 1(4x - 1) - 4)) + 1) + 1 in the ran... | 1097 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3cos(1π(3|(-2|((-1((3x + 1)/(2x)) - 1)/(-3((3x + 1)/(2x)) + 1)) - 1| - 2) + 2| + 2) + 2) + 1 and g(x) = 2|(-3cos(1π(-3((-2x + 1)/(1x + 3))^2 + 2) + 2))| - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed... | 181 | MATH |
[
{
"content": "How many solutions does the equation -2(2cos(3π(-2((-1((-1x + 2)/(-3x)) - 2)/(3((-1x + 2)/(-3x)))) + 4) + 1) - 3)^2 + 1(2cos(3π(-2((-1((-1x + 2)/(-3x)) - 2)/(3((-1x + 2)/(-3x)))) + 4) + 1) - 3) - 4 = -4(-2((1((1x - 2)/(-2x + 3)) + 1)/(2((1x - 2)/(-2x + 3)) - 1))^2 + 1((1((1x - 2)/(-2x + 3)) + 1)/(... | 104 | MATH |
[
{
"content": "How many solutions does the equation 3sin(1π(1|(-1|(1|(3|x| - 1) - 1| - 2) + 3| + 3) + 2| + 2) - 1) = 2((1(-3|(2|x - 3| + 2) - 3| + 2) - 3)/(-3(-3|(2|x - 3| + 2) - 3| + 2) - 1)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 54 | MATH |
[
{
"content": "How many solutions does the equation (-3(-3sin(3π(2(-1|(3x^2 - 5x - 4) - 2| - 2)^2 + 1(-1|(3x^2 - 5x - 4) - 2| - 2)) + 3)) - 2)/(-3(-3sin(3π(2(-1|(3x^2 - 5x - 4) - 2| - 2)^2 + 1(-1|(3x^2 - 5x - 4) - 2| - 2)) + 3))) = -3|(-2|((-1(4x + 5) - 2)/(3(4x + 5) + 3)) - 1| + 1) - 1| - 1 have for x in the in... | 13 | MATH |
[
{
"content": "How many solutions does the equation (1(1cos(1π(4(3|(-5x^2 + 5x - 3) - 3|)^2 - 3(3|(-5x^2 + 5x - 3) - 3|) + 2) + 3) + 2))/(-2(1cos(1π(4(3|(-5x^2 + 5x - 3) - 3|)^2 - 3(3|(-5x^2 + 5x - 3) - 3|) + 2) + 3) + 2) + 3) = -3(-5(-2(2x^2 - 1x)^2 + 2(2x^2 - 1x) - 1) - 4)^2 + 1(-5(-2(2x^2 - 1x)^2 + 2(2x^2 - 1... | 37 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(-2(-3(3e^(-1((-2x + 2)/(-3x))) - 3)^2 - 1(3e^(-1((-2x + 2)/(-3x))) - 3) + 3)^2 + 3(-3(3e^(-1((-2x + 2)/(-3x))) - 3)^2 - 1(3e^(-1((-2x + 2)/(-3x))) - 3) + 3) - 3)| + 2 and g(x) = -3(-4((1(-2|x + 1| + 2) + 2)/(-1(-2|x + 1| ... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|((-2(-3cos(3π((3(-5x + 5) + 3)/(2(-5x + 5) + 1)) + 2)) + 2)/(2(-3cos(3π((3(-5x + 5) + 3)/(2(-5x + 5) + 1)) + 2)) - 1)) - 2| - 1 and g(x) = (2((2(-3(-1x)^2 - 1(-1x) - 3) - 3)/(-1(-3(-1x)^2 - 1(-1x) - 3) - 2)) - 2)/(1((2(-3(-1x)^2 ... | 17 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-3cos(1π(-5(-3|((1x + 3)/(-2x - 3)) + 3| + 2)^2 - 2(-3|((1x + 3)/(-2x - 3)) + 3| + 2) + 4) - 3) - 3)^2 - 5(-3cos(1π(-5(-3|((1x + 3)/(-2x - 3)) + 3| + 2)^2 - 2(-3|((1x + 3)/(-2x - 3)) + 3| + 2) + 4) - 3) - 3) - 5 and g(x) = 1|(-3e^(-1((3(-2x + 3) + 1)... | 332 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(-2|(5(-3(1x^2 - 3x - 1)^2 + 4(1x^2 - 3x - 1))^2) - 1|) + 1)/(-2(-2|(5(-3(1x^2 - 3x - 1)^2 + 4(1x^2 - 3x - 1))^2) - 1|) - 1) and g(x) = (-1((2(-2|(-1x^2 - 3x + 3)| - 3) - 1)/(1(-2|(-1x^2 - 3x + 3)| - 3))) - 1)/(-1((2(-2|(-... | 5 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3e^(-2((-3(3|((2(-1x + 5))/(-1(-1x + 5) - 1)) + 2| + 3))/(-2(3|((2(-1x + 5))/(-1(-1x + 5) - 1)) + 2| + 3) - 2))) + 3 and g(x) = -3sin(3π((3(-5((3x + 2)/(-2x + 3)) + 3) - 2)/(-2(-5((3x + 2)/(-2x + 3)) + 3) - 1))) + 1 intersect in the interval [-10, 10]... | 31 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5((3((-2(-1|(2sin(2πx - 3) - 2) + 2| + 1) - 2)/(-2(-1|(2sin(2πx - 3) - 2) + 2| + 1) - 1)) - 1)/(-2((-2(-1|(2sin(2πx - 3) - 2) + 2| + 1) - 2)/(-2(-1|(2sin(2πx - 3) - 2) + 2| + 1) - 1)) + 3))^2 - 1((3((-2(-1|(2sin(2πx - 3) - 2... | 547 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2e^(1(-2((1((3(-2|x + 1| + 3))/(2(-2|x + 1| + 3) - 3)))/(3((3(-2|x + 1| + 3))/(2(-2|x + 1| + 3) - 3)) - 2)) - 3)) + 2 and g(x) = 3(1((-2(-2x^2 + 2x + 2))/(-2(-2x^2 + 2x + 2) - 1))^2 + 4((-2(-2x^2 + 2x + 2))/(-2(-2x^2 + 2x + 2) - 1)) - 5) - 1 intersect ... | 4 | MATH |
[
{
"content": "How many solutions does the equation (2((-2((1(2(1sin(2πx - 2) - 3)^2 - 1(1sin(2πx - 2) - 3) + 4) + 3)/(2(2(1sin(2πx - 2) - 3)^2 - 1(1sin(2πx - 2) - 3) + 4) + 2)) - 3)/(1((1(2(1sin(2πx - 2) - 3)^2 - 1(1sin(2πx - 2) - 3) + 4) + 3)/(2(2(1sin(2πx - 2) - 3)^2 - 1(1sin(2πx - 2) - 3) + 4) + 2)) + 3)) + ... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1((3(-1cos(2π(2(5x - 4) + 2) - 1) + 3) - 3)/(1(-1cos(2π(2(5x - 4) + 2) - 1) + 3) + 2))^2 + 3((3(-1cos(2π(2(5x - 4) + 2) - 1) + 3) - 3)/(1(-1cos(2π(2(5x - 4) + 2) - 1) + 3) + 2)) + 5 and g(x) = 1(-2((-3(-2|x - 3| + 2) - 3)/(3(... | 12 | MATH |
[
{
"content": "How many solutions does the equation -3e^(1(2|(4(-5(-3|x + 3|)) - 1) - 1|)) + 1 = -3(-2|(-2(1|x - 1| - 3) + 5) - 2| - 1)^2 + 4(-2|(-2(1|x - 1| - 3) + 5) - 2| - 1) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((-2(-1cos(2π(-5(2x^2 - 1x + 3) - 4) + 1)) - 3)/(-1(-1cos(2π(-5(2x^2 - 1x + 3) - 4) + 1))))^2 - 3((-2(-1cos(2π(-5(2x^2 - 1x + 3) - 4) + 1)) - 3)/(-1(-1cos(2π(-5(2x^2 - 1x + 3) - 4) + 1)))) + 2 and g(x) = 2(-5(1|(2e^(2x) - 3) + 3| + 2)^2 + 4(1|(2e^(2x... | 433 | MATH |
[
{
"content": "How many solutions does the equation 2|(2(-1|(-1((-3x + 3)/(-2x - 1))^2 - 5((-3x + 3)/(-2x - 1)) + 4) - 2|)^2 - 4(-1|(-1((-3x + 3)/(-2x - 1))^2 - 5((-3x + 3)/(-2x - 1)) + 4) - 2|) + 3) - 1| - 2 = 2|(4(5(-2sin(2πx - 3) + 1) - 1) - 3)| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer ... | 13 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(-3cos(1π(-5(-4(-1|x + 1| + 1) - 4)) + 3)) - 2)/(3(-3cos(1π(-5(-4(-1|x + 1| + 1) - 4)) + 3)) + 3) and g(x) = 1(3(4(1x - 4))^2 - 1(4(1x - 4)) - 2)^2 - 4(3(4(1x - 4))^2 - 1(4(1x - 4)) - 2) - 3 where -10 ≤ x ≤ 10.\n\nPresent t... | 11 | MATH |
[
{
"content": "How many solutions does the equation (2(2(-5(1cos(1π(-1|x + 2| - 1) - 2) + 1)^2 + 3(1cos(1π(-1|x + 2| - 1) - 2) + 1))^2 + 1(-5(1cos(1π(-1|x + 2| - 1) - 2) + 1)^2 + 3(1cos(1π(-1|x + 2| - 1) - 2) + 1)) - 3) + 3)/(-2(2(-5(1cos(1π(-1|x + 2| - 1) - 2) + 1)^2 + 3(1cos(1π(-1|x + 2| - 1) - 2) + 1))^2 + 1(... | 600 | MATH |
[
{
"content": "How many solutions does the equation -2((-3(-2(1|(1sin(2πx))|)^2 - 1(1|(1sin(2πx))|) + 3) + 3)/(-1(-2(1|(1sin(2πx))|)^2 - 1(1|(1sin(2πx))|) + 3) + 2)) + 4 = (-1(-1sin(2π(1(-1x^2 - 2x + 5) - 5) + 1) - 3) + 3)/(-2(-1sin(2π(1(-1x^2 - 2x + 5) - 5) + 1) - 3)) have for x in the interval [-10, 10]?\n\nPr... | 78 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3sin(3π((-1((-2((-1((-2x + 1)/(1x + 2)) - 1)/(1((-2x + 1)/(1x + 2)))) + 1)/(-3((-1((-2x + 1)/(1x + 2)) - 1)/(1((-2x + 1)/(1x + 2)))) - 2)) - 3)/(-3((-2((-1((-2x + 1)/(1x + 2)) - 1)/(1((-2x + 1)/(1x + 2)))) + 1)/(-3((-1((-2x + 1)/(1... | 1186 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(-1((-2(-3e^(-1(-2x^2 + 2)) + 3) + 3)/(1(-3e^(-1(-2x^2 + 2)) + 3))) + 2) + 2)/(-2(-1((-2(-3e^(-1(-2x^2 + 2)) + 3) + 3)/(1(-3e^(-1(-2x^2 + 2)) + 3))) + 2) - 2) and g(x) = 1(2|(-1(2e^(1x) - 2)) + 1| + 1) - 5 where -10 ≤ x ≤ ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1((-2(1(-1|((1x)/(-3x - 2)) + 2| - 1)^2 + 5(-1|((1x)/(-3x - 2)) + 2| - 1) + 3) - 1)/(1(1(-1|((1x)/(-3x - 2)) + 2| - 1)^2 + 5(-1|((1x)/(-3x - 2)) + 2| - 1) + 3))) + 1)/(1((-2(1(-1|((1x)/(-3x - 2)) + 2| - 1)^2 + 5(-1|((1x)/(-... | 8 | MATH |
[
{
"content": "How many solutions does the equation 3(-1cos(3π(-5(-3(4x^2 + 3x + 5) - 4) + 4) - 2)) - 2 = -5((3(-1(3sin(3πx - 2) - 1)^2 - 4(3sin(3πx - 2) - 1) + 1) + 2)/(-3(-1(3sin(3πx - 2) - 1)^2 - 4(3sin(3πx - 2) - 1) + 1) + 2)) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \... | 53 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(-5(4((-3(3x^2 + 3x) - 1)/(1(3x^2 + 3x))) - 1)^2 + 4(4((-3(3x^2 + 3x) - 1)/(1(3x^2 + 3x))) - 1) - 1)^2 + 2(-5(4((-3(3x^2 + 3x) - 1)/(1(3x^2 + 3x))) - 1)^2 + 4(4((-3(3x^2 + 3x) - 1)/(1(3x^2 + 3x))) - 1) - 1) - 4 and g(x) = -1e^(2(-... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(1((3(-5(-1x - 2)^2 - 4(-1x - 2) + 5) + 1)/(-1(-5(-1x - 2)^2 - 4(-1x - 2) + 5) - 2)) + 2)| - 2 and g(x) = -3cos(1π((-3((2(-4x^2 - 5x - 1) - 2)/(-3(-4x^2 - 5x - 1) + 1)) - 1)/(-1((2(-4x^2 - 5x - 1) - 2)/(-3(-4x^2 - 5x - 1) + 1)) - 2)) + 3) - 3 inters... | 26 | MATH |
[
{
"content": "How many solutions does the equation -3(-1(2sin(1π(1|(-1|x + 1| + 2) - 3| + 3) - 2) - 1)^2 - 3(2sin(1π(1|(-1|x + 1| + 2) - 3| + 3) - 2) - 1) - 5) - 3 = 1|(-2|((-3(2x + 5) - 3)/(-3(2x + 5) + 4)) - 1| - 3) + 1| + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{... | 34 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1((-3(-1|(-3cos(3π(-2x - 4) + 1) + 1) + 1| - 1) - 1)/(-2(-1|(-3cos(3π(-2x - 4) + 1) + 1) + 1| - 1) + 3)) - 1 and g(x) = (1(-3e^(2(4(1|x + 2| - 2) - 5)) + 1) - 1)/(-3(-3e^(2(4(1|x + 2| - 2) - 5)) + 1) + 4) in the range -10 ≤ x ≤ 10?... | 202 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1cos(2π(3(1(-3|(4x) - 2| + 1) + 1)^2 - 4(1(-3|(4x) - 2| + 1) + 1) - 4) - 3) + 3 and g(x) = -1cos(3π(3(2(-2|x - 3| - 1) - 1)^2 - 5(2(-2|x - 3| - 1) - 1) - 5) + 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: ... | 661 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(2(-1((-3((3x - 2)/(-2x)) - 1)/(-1((3x - 2)/(-2x)) - 1))^2 - 5((-3((3x - 2)/(-2x)) - 1)/(-1((3x - 2)/(-2x)) - 1)) - 3) - 1) + 3)/(-2(2(-1((-3((3x - 2)/(-2x)) - 1)/(-1((3x - 2)/(-2x)) - 1))^2 - 5((-3((3x - 2)/(-2x)) - 1)/(-1((3x - 2)/(-2x)) - 1)) - 3... | 14 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2((-1(-1|(1(1x^2 + 5x + 2) - 2) - 2| + 1) - 2)/(1(-1|(1(1x^2 + 5x + 2) - 2) - 2| + 1))) - 2)/(1((-1(-1|(1(1x^2 + 5x + 2) - 2) - 2| + 1) - 2)/(1(-1|(1(1x^2 + 5x + 2) - 2) - 2| + 1)))) and g(x) = 1sin(2π(1|((2((-3x - 2)/(2x - 1)))/(2((-3x - 2)/(2x - 1))... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(3((-3(-1(2x^2 + 5x - 3) - 3) + 3)/(-3(-1(2x^2 + 5x - 3) - 3) + 1))) + 2| - 3 and g(x) = 4(-1|(4((-2x + 1)/(2x - 1))^2 - 4((-2x + 1)/(2x - 1)) - 2) + 2| + 3)^2 - 1(-1|(4((-2x + 1)/(2x - 1))^2 - 4((-2x + 1)/(2x - 1)) - 2) + 2| + 3... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(1(1(-4(1|x + 2| - 2) + 2)^2 - 4(-4(1|x + 2| - 2) + 2) - 2)^2 - 2(1(-4(1|x + 2| - 2) + 2)^2 - 4(-4(1|x + 2| - 2) + 2) - 2) - 4) - 1)/(1(1(1(-4(1|x + 2| - 2) + 2)^2 - 4(-4(1|x + 2| - 2) + 2) - 2)^2 - 2(1(-4(1|x + 2| - 2) + 2)^2 -... | 6 | MATH |
[
{
"content": "How many solutions does the equation (-3(4(-2(-3|(1x - 1) - 2| - 2)^2 + 2(-3|(1x - 1) - 2| - 2) - 2) - 2) + 2)/(-1(4(-2(-3|(1x - 1) - 2| - 2)^2 + 2(-3|(1x - 1) - 2| - 2) - 2) - 2) - 1) = (1(-2(-3|(4x^2 + 3x + 1) - 2| + 2)^2 + 2(-3|(4x^2 + 3x + 1) - 2| + 2) + 2) + 3)/(-3(-2(-3|(4x^2 + 3x + 1) - 2| ... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-1(1|(-4(-1(-2x + 2)^2 + 4(-2x + 2) + 1)^2 - 3(-1(-2x + 2)^2 + 4(-2x + 2) + 1) + 1) - 3| - 3) + 3)/(2(1|(-4(-1(-2x + 2)^2 + 4(-2x + 2) + 1)^2 - 3(-1(-2x + 2)^2 + 4(-2x + 2) + 1) + 1) - 3| - 3) + 1) and g(x) = (-3((-2(1(3cos(2πx + 3) + 3) + 4) + 2)/(1(... | 4 | MATH |
[
{
"content": "How many solutions does the equation (1(-2|((-3(2(1x^2 - 5x - 5) + 4) + 1)/(-1(2(1x^2 - 5x - 5) + 4) + 2)) + 2| - 1) + 2)/(2(-2|((-3(2(1x^2 - 5x - 5) + 4) + 1)/(-1(2(1x^2 - 5x - 5) + 4) + 2)) + 2| - 1) + 2) = (1(4(3sin(2π(2|x - 3| - 3) + 3) + 1) + 2) + 3)/(-3(4(3sin(2π(2|x - 3| - 3) + 3) + 1) + 2)... | 84 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(2|(-3sin(2π(5(-4x^2 - 4x - 4)^2 + 1(-4x^2 - 4x - 4) + 2) - 3) - 2) + 3| - 3) and g(x) = 3((3(1|(2cos(1πx + 2) - 2) - 2| - 2))/(-1(1|(2cos(1πx + 2) - 2) - 2| - 2) - 1))^2 - 1((3(1|(2cos(1πx + 2) - 2) - 2| - 2))/(-1(1|(2cos(1πx + 2... | 100 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|((-3(3|(3(-2e^(-1x) - 1)) - 1|) + 1)/(1(3|(3(-2e^(-1x) - 1)) - 1|)))| - 1 and g(x) = -4((2(3(-2|x - 3| + 1)^2 + 4(-2|x - 3| + 1) - 2) - 3)/(-3(3(-2|x - 3| + 1)^2 + 4(-2|x - 3| + 1) - 2))) - 3 where -10 ≤ x ≤ 10.\n\nPresent ... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1((2(-3(-2|(-2x^2 - 4x - 2) + 1|)^2 - 4(-2|(-2x^2 - 4x - 2) + 1|) - 3))/(-1(-3(-2|(-2x^2 - 4x - 2) + 1|)^2 - 4(-2|(-2x^2 - 4x - 2) + 1|) - 3) - 3))^2 - 5((2(-3(-2|(-2x^2 - 4x - 2) + 1|)^2 - 4(-2|(-2x^2 - 4x - 2) + 1|) - 3))/(-1(-3(-2|(-2x^2 - 4x - 2) + 1|)^2 -... | 4 | MATH |
[
{
"content": "How many solutions does the equation -3(-2(2sin(2π(3(2x - 1) - 4)) - 1) - 4) - 4 = -1((-3((-1(-1cos(2πx + 2)) + 3)/(-2(-1cos(2πx + 2)) + 3)))/(3((-1(-1cos(2πx + 2)) + 3)/(-2(-1cos(2πx + 2)) + 3)) + 1))^2 - 4((-3((-1(-1cos(2πx + 2)) + 3)/(-2(-1cos(2πx + 2)) + 3)))/(3((-1(-1cos(2πx + 2)) + 3)/(-2(-1... | 240 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(-4(1((2(-1|x| + 2) - 1)/(-1(-1|x| + 2) - 3))^2 + 4((2(-1|x| + 2) - 1)/(-1(-1|x| + 2) - 3)) + 1)^2 + 5(1((2(-1|x| + 2) - 1)/(-1(-1|x| + 2) - 3))^2 + 4((2(-1|x| + 2) - 1)/(-1(-1|x| + 2) - 3)) + 1) - 1) - 2)/(1(-4(1((2(-1|x| + 2) -... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(-2e^(-2((2(3((1x - 2)/(-1x - 1))^2 + 4((1x - 2)/(-1x - 1)) - 4) + 2)/(-3(3((1x - 2)/(-1x - 1))^2 + 4((1x - 2)/(-1x - 1)) - 4) + 4))) + 3) - 1)/(2(-2e^(-2((2(3((1x - 2)/(-1x - 1))^2 + 4((1x - 2)/(-1x - 1)) - 4) + 2)/(-3(3((1x - 2)/(-1x - 1))^2 + 4((... | 87 | MATH |
[
{
"content": "How many solutions does the equation -1(3((1(-2|(2x^2 - 5x - 2) - 3| + 3) - 2)/(-2(-2|(2x^2 - 5x - 2) - 3| + 3) + 1))^2 + 5((1(-2|(2x^2 - 5x - 2) - 3| + 3) - 2)/(-2(-2|(2x^2 - 5x - 2) - 3| + 3) + 1)) + 2) = (3((2(-2|(-1x^2 + 1x - 1)| + 1))/(2(-2|(-1x^2 + 1x - 1)| + 1) + 3)) + 1)/(1((2(-2|(-1x^2 + ... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-4(2(4(2x^2 - 5x - 2) - 1) - 2)^2 - 3(2(4(2x^2 - 5x - 2) - 1) - 2) + 5) + 4 and g(x) = -2|(-1((2(-3|x + 1| + 1) + 3)/(2(-3|x + 1| + 1)))^2 - 5((2(-3|x + 1| + 1) + 3)/(2(-3|x + 1| + 1))) - 5) - 1| where -10 ≤ x ≤ 10.\n\nPres... | 8 | MATH |
[
{
"content": "How many solutions does the equation -2(1(-3e^(-1((-2((-1x + 1)/(-3x + 2)))/(2((-1x + 1)/(-3x + 2)) - 1))))^2 + 4(-3e^(-1((-2((-1x + 1)/(-3x + 2)))/(2((-1x + 1)/(-3x + 2)) - 1)))) - 1) - 3 = -2cos(2π(1(5(4x^2 + 5x - 1) - 2)^2 + 4) - 1) - 1 have for x in the interval [-10, 10]?\n\nPresent the answe... | 236 | MATH |
[
{
"content": "How many solutions does the equation 4(2(-3sin(2π(-1(1x + 1)^2 + 2(1x + 1) + 5) - 1) + 2) - 4)^2 + 4(2(-3sin(2π(-1(1x + 1)^2 + 2(1x + 1) + 5) - 1) + 2) - 4) + 1 = 2sin(3π(3((-2(2x - 5) - 2)/(-3(2x - 5) + 2))^2 - 5((-2(2x - 5) - 2)/(-3(2x - 5) + 2)) - 3) + 1) have for x in the interval [-10, 10]?\n... | 32 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1((-3(1(-2|((-1x + 1)/(-1x - 3)) + 2| - 1) + 2) - 1)/(2(1(-2|((-1x + 1)/(-1x - 3)) + 2| - 1) + 2) - 1))^2 + 5((-3(1(-2|((-1x + 1)/(-1x - 3)) + 2| - 1) + 2) - 1)/(2(1(-2|((-1x + 1)/(-1x - 3)) + 2| - 1) + 2) - 1)) - 1 and g(x) = 2cos... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(5(-3e^(2(3|(-2|x|) + 2| - 3)) - 2)^2 + 5(-3e^(2(3|(-2|x|) + 2| - 3)) - 2) - 4) and g(x) = 3|(3(-2e^(-2((-3x + 1)/(2x))) - 1) + 1) - 3| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1sin(2π(-2|(-2|((3(-1x + 5) + 1)/(-1(-1x + 5) - 3)) - 1| - 2) - 3| - 2) - 3) and g(x) = (-2(2((-1(-5x + 1) - 3)/(1(-5x + 1))) - 2) - 3)/(-2(2((-1(-5x + 1) - 3)/(1(-5x + 1))) - 2)) in the range -10 ≤ x ≤ 10?\n\nPresent the answer i... | 226 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(2((2(1((2x)/(-2x - 3)) - 3) - 1)/(2(1((2x)/(-2x - 3)) - 3) + 1))^2 - 3((2(1((2x)/(-2x - 3)) - 3) - 1)/(2(1((2x)/(-2x - 3)) - 3) + 1)) - 3) - 2)/(-2(2((2(1((2x)/(-2x - 3)) - 3) - 1)/(2(1((2x)/(-2x - 3)) - 3) + 1))^2 - 3((2(1((2x)... | 88 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2sin(2π(-3(-2((1((-1x + 1)/(-2x + 1)) - 3)/(-1((-1x + 1)/(-2x + 1)) - 1)) + 5) + 3) + 1) - 1 and g(x) = (-3((1((-1(1sin(3πx + 2) + 1) + 1)/(1(1sin(3πx + 2) + 1) - 2)))/(-2((-1(1sin(3πx + 2) + 1) + 1)/(1(1sin(3πx + 2) + 1) - ... | 71 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(5(3(5(3x^2 + 3x)^2 - 4)^2 + 4(5(3x^2 + 3x)^2 - 4))^2 - 2) - 1 and g(x) = (-2(-3(3(-3x - 2))^2 - 4(3(-3x - 2)) - 1) - 3)/(2(-3(3(-3x - 2))^2 - 4(3(-3x - 2)) - 1) - 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex forma... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1((-1(1((-3(-2x^2 + 4))/(2(-2x^2 + 4) - 3))^2 + 2((-3(-2x^2 + 4))/(2(-2x^2 + 4) - 3)) - 4) - 3)/(-1(1((-3(-2x^2 + 4))/(2(-2x^2 + 4) - 3))^2 + 2((-3(-2x^2 + 4))/(2(-2x^2 + 4) - 3)) - 4) - 2))^2 - 3((-1(1((-3(-2x^2 + 4))/(2(-2x^2 + 4... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(1(2|((-1(-1x - 5))/(-2(-1x - 5) - 3)) + 2|) - 4)^2 + 3(1(2|((-1(-1x - 5))/(-2(-1x - 5) - 3)) + 2|) - 4) - 3 and g(x) = (2(-2cos(2π(2(1x + 4) - 4) - 1) - 1) - 1)/(3(-2cos(2π(2(1x + 4) - 4) - 1) - 1) - 2) intersect in the interval [-10, 10].\n\nPresen... | 74 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 4(-1sin(3π(-3|(-5(-3|x - 2| - 2) - 5) + 3| + 2) + 2) + 1)^2 - 4(-1sin(3π(-3|(-5(-3|x - 2| - 2) - 5) + 3| + 2) + 2) + 1) - 4 and g(x) = 1(-1((-2((3x + 3)/(2x + 2)) - 1)/(1((3x + 3)/(2x + 2)) + 2))^2 - 4((-2((3x + 3)/(2x + 2)) - 1)/(1((3x + 3)/(2x + 2)) ... | 1989 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(2(2|((3(-1x^2 - 2x + 5))/(3(-1x^2 - 2x + 5) - 1)) - 1| + 1) - 5) + 1| - 1 and g(x) = -2|(-1(-2sin(3π(3|x + 1| + 3) + 2) + 3)^2 - 1(-2sin(3π(3|x + 1| + 3) + 2) + 3) + 4) + 3| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex fo... | 358 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(-2(-3|(-2|(2x + 2) - 2|) - 3| - 2)^2 - 2(-3|(-2|(2x + 2) - 2|) - 3| - 2) + 1) + 3)/(-3(-2(-3|(-2|(2x + 2) - 2|) - 3| - 2)^2 - 2(-3|(-2|(2x + 2) - 2|) - 3| - 2) + 1)) and g(x) = 1cos(1π(2(-2|(5x^2 - 5x - 3) - 2| - 3)^2 - 5) - 3) - 1 intersect in the ... | 1097 | MATH |
[
{
"content": "How many solutions does the equation -1((-2((1((2(-2|x + 1| + 1) - 2)/(3(-2|x + 1| + 1) - 2)) - 3)/(-3((2(-2|x + 1| + 1) - 2)/(3(-2|x + 1| + 1) - 2)) - 3)) + 3)/(2((1((2(-2|x + 1| + 1) - 2)/(3(-2|x + 1| + 1) - 2)) - 3)/(-3((2(-2|x + 1| + 1) - 2)/(3(-2|x + 1| + 1) - 2)) - 3)))) + 5 = 4(2cos(2π(4((2... | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1sin(3π(-2(-3(-2(1x - 2) - 5)^2 + 4(-2(1x - 2) - 5) + 2)^2 + 4(-3(-2(1x - 2) - 5)^2 + 4(-2(1x - 2) - 5) + 2) + 4) + 1) + 2 and g(x) = 4(-2cos(1π(-3(-2|x - 3| + 1) - 4) + 1) - 2)^2 + 5(-2cos(1π(-3(-2|x - 3| + 1) - 4) + 1) - 2... | 31 | MATH |
[
{
"content": "How many solutions does the equation (1(-3|(-4(-2(1x^2 - 2x + 4)) - 2) + 2| + 1) - 1)/(3(-3|(-4(-2(1x^2 - 2x + 4)) - 2) + 2| + 1) + 1) = -2e^(2(-1(-3|(-2x^2 - 2x + 5) + 2| + 3) - 3)) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(2|(2sin(1π(2(-1x + 4) - 1) + 1))| - 3) + 1)/(3(2|(2sin(1π(2(-1x + 4) - 1) + 1))| - 3) - 2) and g(x) = (2((2(2|(1sin(1πx + 1) + 3)| + 1) + 3)/(-3(2|(1sin(1πx + 1) + 3)| + 1) + 1)) + 3)/(3((2(2|(1sin(1πx + 1) + 3)| + 1) + 3)... | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-1((2((-1(3sin(2πx) + 1) + 1)/(3(3sin(2πx) + 1) - 2)) + 1)/(2((-1(3sin(2πx) + 1) + 1)/(3(3sin(2πx) + 1) - 2)) + 2))^2 - 3((2((-1(3sin(2πx) + 1) + 1)/(3(3sin(2πx) + 1) - 2)) + 1)/(2((-1(3sin(2πx) + 1) + 1)/(3(3sin(2πx) + 1) - 2)... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(-5(-5(1|(-2|x - 3| + 2) + 3|) + 1) - 2) + 1 and g(x) = 1(3(3sin(3π(-2x^2 + 2x - 1) - 1))^2 + 3(3sin(3π(-2x^2 + 2x - 1) - 1)) + 4) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 21 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(-1|(-5(3sin(1π(-2x + 4) - 3) + 1)^2 + 1(3sin(1π(-2x + 4) - 3) + 1) + 5) - 1| + 2)^2 - 3(-1|(-5(3sin(1π(-2x + 4) - 3) + 1)^2 + 1(3sin(1π(-2x + 4) - 3) + 1) + 5) - 1| + 2) + 3 and g(x) = 1sin(2π(-4(-1(1x - 2) + 3) + 3) + 1) ... | 300 | MATH |
[
{
"content": "How many solutions does the equation (3(4(2|(2(-2|x + 1| + 3) - 2) - 2| - 1)^2 + 4(2|(2(-2|x + 1| + 3) - 2) - 2| - 1) + 3) - 1)/(-3(4(2|(2(-2|x + 1| + 3) - 2) - 2| - 1)^2 + 4(2|(2(-2|x + 1| + 3) - 2) - 2| - 1) + 3) - 2) = -1|(4(2e^(-2(-1x^2 - 1x + 4)) - 3) - 4) - 3| + 2 have for x in the interval ... | 4 | MATH |
[
{
"content": "How many solutions does the equation 3sin(3π(-2(-1(-1|((-2x + 2)/(-3x)) - 1|)) - 1) + 3) + 2 = -1sin(2π(-2|(4(5x - 1)^2 + 5(5x - 1) - 5) - 2| + 2) - 2) + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 225 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|((-1(-1((1(-2e^(-1x) + 3) + 3)/(1(-2e^(-1x) + 3))) - 4) - 2)/(-1(-1((1(-2e^(-1x) + 3) + 3)/(1(-2e^(-1x) + 3))) - 4) - 2)) + 2| and g(x) = -1((2(2sin(2π(3x + 3) + 3) + 3) - 2)/(-1(2sin(2π(3x + 3) + 3) + 3) + 2)) - 1 where -1... | 120 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-1|(-2cos(1π(3|(2x - 1) - 3|))) + 2|) and g(x) = (2((2(-5(1|x - 1| + 1)^2 - 5(1|x - 1| + 1) - 4) + 1)/(3(-5(1|x - 1| + 1)^2 - 5(1|x - 1| + 1) - 4) + 3)) + 3)/(2((2(-5(1|x - 1| + 1)^2 - 5(1|x - 1| + 1) - 4) + 1)/(3(-5(1|x - 1| + ... | 120 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(2π(-4(-4((3((-2x + 1)/(3x)) + 1)/(2((-2x + 1)/(3x)) + 1))^2 + 1((3((-2x + 1)/(3x)) + 1)/(2((-2x + 1)/(3x)) + 1)) + 2)^2 + 1(-4((3((-2x + 1)/(3x)) + 1)/(2((-2x + 1)/(3x)) + 1))^2 + 1((3((-2x + 1)/(3x)) + 1)/(2((-2x + 1)/(3x)) ... | 165 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(-3cos(1π((3(2|(1|x - 1| - 3) + 3| - 3) + 2)/(2(2|(1|x - 1| - 3) + 3| - 3) - 1))) + 1)^2 - 1(-3cos(1π((3(2|(1|x - 1| - 3) + 3| - 3) + 2)/(2(2|(1|x - 1| - 3) + 3| - 3) - 1))) + 1) - 4 and g(x) = -1(1|(1|(-2e^(1x) + 1) + 2| + ... | 48 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1cos(1π(3((-1(-2(-1x^2 + 5x + 1)^2 + 4(-1x^2 + 5x + 1) - 1) - 3)/(2(-2(-1x^2 + 5x + 1)^2 + 4(-1x^2 + 5x + 1) - 1) - 1))^2 + 3) - 2) + 3 and g(x) = -2e^(-2((1(1(4x^2 - 4x - 4)^2 - 3(4x^2 - 4x - 4) - 1) + 2)/(-2(1(4x^2 - 4x - 4... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5(3(-3cos(2π(-2(-1x^2 + 2)^2 + 1(-1x^2 + 2) - 1) + 1) - 2)^2 + 1(-3cos(2π(-2(-1x^2 + 2)^2 + 1(-1x^2 + 2) - 1) + 1) - 2) - 1)^2 - 5(3(-3cos(2π(-2(-1x^2 + 2)^2 + 1(-1x^2 + 2) - 1) + 1) - 2)^2 + 1(-3cos(2π(-2(-1x^2 + 2)^2 + 1(-1... | 406 | MATH |
[
{
"content": "How many solutions does the equation (-3(-5((1(-2cos(1π(1|x| + 2) - 2) + 1) - 1)/(-3(-2cos(1π(1|x| + 2) - 2) + 1) + 4)) + 5))/(1(-5((1(-2cos(1π(1|x| + 2) - 2) + 1) - 1)/(-3(-2cos(1π(1|x| + 2) - 2) + 1) + 4)) + 5)) = -3(5(1(2e^(-2x) - 2)^2 - 3(2e^(-2x) - 2) - 2)^2 + 3(1(2e^(-2x) - 2)^2 - 3(2e^(-2x)... | 7 | MATH |
[
{
"content": "How many solutions does the equation -2((-2(-1(4(-1|x - 1| + 3) + 4) + 4) + 3)/(-1(-1(4(-1|x - 1| + 3) + 4) + 4) + 2)) - 4 = 1sin(3π(3|(-3|(-1x^2 + 3x)| - 3) - 1|)) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 5 | MATH |
[
{
"content": "How many solutions does the equation -2|((3(2|(-2|(3x + 3) + 1|) - 1| + 3) - 2)/(2(2|(-2|(3x + 3) + 1|) - 1| + 3) - 1)) - 1| = 3sin(3π(1|(4(5x - 3)^2 - 2(5x - 3) + 5) - 1| + 1) + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 1255 | MATH |
[
{
"content": "How many solutions does the equation (-1(-1|(1|(2|(-3|x + 3| - 3) - 2| - 1) - 1| - 1) + 1| - 1) - 1)/(3(-1|(1|(2|(-3|x + 3| - 3) - 2| - 1) - 1| - 1) + 1| - 1)) = -1|(3sin(3π(5(1|x + 3| - 1)^2 - 5(1|x + 3| - 1) - 4) + 2)) + 2| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex... | 1387 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(-3e^(-1(-5((-1((-3x + 1)/(-2x + 1)) + 2)/(3((-3x + 1)/(-2x + 1)) - 2)) - 4)) + 1))/(-3(-3e^(-1(-5((-1((-3x + 1)/(-2x + 1)) + 2)/(3((-3x + 1)/(-2x + 1)) - 2)) - 4)) + 1) + 1) and g(x) = 3sin(1π(-1(-1|(-1x^2 + 1x + 2) + 3|)^2 - 1(-1|(-1x^2 + 1x + 2) ... | 688 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3|(2cos(3π(2(-2|(3|x + 2| - 2) - 3|) + 4) - 1) - 1) + 1| + 3 and g(x) = (2(4(2|(-3x - 1) - 2| + 3) + 3))/(1(4(2|(-3x - 1) - 2| + 3) + 3) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 681 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1e^(-1(-1(-4(5((1x + 2)/(1x)) - 3) + 1) + 4)) - 3 and g(x) = -3cos(2π(3|(-1|(-2|x + 1| - 2) + 1| + 1) + 1| - 2) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 58 | MATH |
[
{
"content": "How many solutions does the equation -3(5(-1cos(2π(4(1|x - 3| - 3)^2 - 4(1|x - 3| - 3) - 3) - 2)) + 3)^2 - 4(5(-1cos(2π(4(1|x - 3| - 3)^2 - 4(1|x - 3| - 3) - 3) - 2)) + 3) - 5 = (2(2((1((-2x + 1)/(-3x)) - 2)/(-1((-2x + 1)/(-3x))))^2 - 5((1((-2x + 1)/(-3x)) - 2)/(-1((-2x + 1)/(-3x))))) + 3)/(-1(2((... | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(1((-2(2(-3cos(1πx + 3))^2 - 4(-3cos(1πx + 3))) + 2)/(-1(2(-3cos(1πx + 3))^2 - 4(-3cos(1πx + 3))) + 2)) + 2) - 5 and g(x) = (1(2(5(-2|x + 3| - 3)^2 + 3(-2|x + 3| - 3) - 2)^2 + 4(5(-2|x + 3| - 3)^2 + 3(-2|x + 3| - 3) - 2) - 5))/(1... | 39 | MATH |
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