messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "Find the number of times the graphs of f(x) = 2|(1|(-1((1(2x - 2) - 1)/(3(2x - 2))) + 5) - 1| - 3) + 3| - 3 and g(x) = (-2((1(-3cos(2π((-1x - 2)/(-2x + 3)) + 2) + 3) + 3)/(-1(-3cos(2π((-1x - 2)/(-2x + 3)) + 2) + 3) + 3)) + 3)/(-1((1(-3cos(2π((-1x - 2)/(-2x + 3)) + 2) + 3) + 3)/(-1(-3cos(2π((-1x - ... | 64 | MATH |
[
{
"content": "How many solutions does the equation -1e^(1(1(1(2(5x + 4) + 4)^2 + 4(2(5x + 4) + 4) - 2)^2 + 5(1(2(5x + 4) + 4)^2 + 4(2(5x + 4) + 4) - 2) + 1)) - 1 = -3sin(3π((-2(-1(1|x + 1| - 1)^2 + 3(1|x + 1| - 1) + 1) - 1)/(1(-1(1|x + 1| - 1)^2 + 3(1|x + 1| - 1) + 1))) - 2) - 2 have for x in the interval [-10,... | 11 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(1π(-3|(2|(-4(-1x^2 + 2x + 1)^2 + 4(-1x^2 + 2x + 1) + 3) - 3| + 1) + 2| - 1)) - 1 and g(x) = 3cos(2π((1(-1(-3x^2 - 1x - 2) - 1) - 3)/(3(-1(-3x^2 - 1x - 2) - 1) + 3)) + 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex forma... | 410 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|((-2((-1(-5(2x^2 + 2x - 3) + 2) + 3)/(-1(-5(2x^2 + 2x - 3) + 2))) + 3)/(2((-1(-5(2x^2 + 2x - 3) + 2) + 3)/(-1(-5(2x^2 + 2x - 3) + 2))) + 2))| and g(x) = 4((2((-2(-3|x| - 2) + 1)/(-2(-3|x| - 2) + 3)) + 3)/(1((-2(-3|x| - 2) + 1)/(-2(-3|x| - 2) + 3)) -... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|((2(1(-2(-2|x + 1| + 2) + 2) - 1) + 3)/(2(1(-2(-2|x + 1| + 2) + 2) - 1) - 1)) + 1| - 2 and g(x) = -1((-1(-5(3e^(1x) + 3)^2 + 2(3e^(1x) + 3) - 5) - 3)/(1(-5(3e^(1x) + 3)^2 + 2(3e^(1x) + 3) - 5) - 3)) - 1 intersect in the interval [-10, 10].\n\nPresent... | 4 | MATH |
[
{
"content": "How many solutions does the equation (2((-1(2|(-4(2x^2 - 5x - 3) + 5) + 3| - 2) - 2)/(1(2|(-4(2x^2 - 5x - 3) + 5) + 3| - 2) + 3)) - 2)/(2((-1(2|(-4(2x^2 - 5x - 3) + 5) + 3| - 2) - 2)/(1(2|(-4(2x^2 - 5x - 3) + 5) + 3| - 2) + 3))) = 1|(5(4(3|x + 3| + 1) - 3)^2 - 3(4(3|x + 3| + 1) - 3) - 5) - 1| have... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(3sin(1π(-1(5(-2|x| - 2) - 2) - 3))) - 3 and g(x) = -2|(2(1e^(1(-1x + 1)) + 3) + 1) + 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 86 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-1(-1(2(-3x^2 - 4x - 1)^2 - 3) + 3) - 2) + 1 and g(x) = -5(-1sin(2π(-1|(2x^2 - 5x + 5) + 3|) + 2) + 1) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many solutions does the equation 1(1|(2sin(2π(-2(1x)^2 - 2(1x) - 5) + 3) - 2) - 3| + 2) - 1 = 3|((2(-1((1x + 2)/(-2x - 1))^2 - 3((1x + 2)/(-2x - 1)) + 2) + 3)/(-2(-1((1x + 2)/(-2x - 1))^2 - 3((1x + 2)/(-2x - 1)) + 2) + 1)) + 1| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in... | 402 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|((-3(1(-1(1x - 2) - 5)^2 + 5(-1(1x - 2) - 5) + 5))/(3(1(-1(1x - 2) - 5)^2 + 5(-1(1x - 2) - 5) + 5) + 2)) + 1| - 1 and g(x) = -3e^(-2((3(1|((1x)/(-1x + 3)) - 1|) - 2)/(-2(1|((1x)/(-1x + 3)) - 1|) - 1))) - 3 intersect in the interval [-10, 10].\n\nPre... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2((3(-5(-2|(4x^2 + 4x) + 1| + 1)^2 - 5(-2|(4x^2 + 4x) + 1| + 1) - 5) + 3)/(1(-5(-2|(4x^2 + 4x) + 1| + 1)^2 - 5(-2|(4x^2 + 4x) + 1| + 1) - 5)))^2 - 4((3(-5(-2|(4x^2 + 4x) + 1| + 1)^2 - 5(-2|(4x^2 + 4x) + 1| + 1) - 5) + 3)/(1(-5(-2|... | 168 | MATH |
[
{
"content": "How many solutions does the equation 1cos(1π(-3|(2|(-3((-2x - 3)/(-3x + 1)) - 1) + 1| + 3) - 3| + 1) - 2) - 1 = -1|(-1((2(-3sin(2πx + 2) - 1) - 3)/(-1(-3sin(2πx + 2) - 1) - 3))^2 + 3((2(-3sin(2πx + 2) - 1) - 3)/(-1(-3sin(2πx + 2) - 1) - 3))) - 1| + 3 have for x in the interval [-10, 10]?\n\nPresen... | 39 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3cos(3π(5(-1(-4(-2|x - 1| + 1)^2 + 1(-2|x - 1| + 1) - 1)^2 - 2(-4(-2|x - 1| + 1)^2 + 1(-2|x - 1| + 1) - 1) - 2)^2 + 2(-1(-4(-2|x - 1| + 1)^2 + 1(-2|x - 1| + 1) - 1)^2 - 2(-4(-2|x - 1| + 1)^2 + 1(-2|x - 1| + 1) - 1) - 2) - 2) - 3) ... | 168 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|((3(1|(4(-5x - 5) - 4)| - 3) + 1)/(2(1|(4(-5x - 5) - 4)| - 3) + 3)) + 1| + 1 and g(x) = -3((2(-2|((3x + 3)/(1x)) + 3| + 3) + 1)/(-1(-2|((3x + 3)/(1x)) + 3| + 3) - 1))^2 + 5((2(-2|((3x + 3)/(1x)) + 3| + 3) + 1)/(-1(-2|((3x ... | 8 | MATH |
[
{
"content": "How many solutions does the equation 5(-2sin(2π(-1|((-1(-4x^2 + 1x) - 2)/(1(-4x^2 + 1x))) + 3| + 2) - 2) - 2) + 5 = 1(4(-2(5x + 1)^2 - 1(5x + 1))^2 + 1(-2(5x + 1)^2 - 1(5x + 1)) - 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 22 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-1|(-3|(2(-2|x - 1| + 3) + 4) + 1| + 1) + 3| + 2) + 2)/(2(-1|(-3|(2(-2|x - 1| + 3) + 4) + 1| + 1) + 3| + 2)) and g(x) = 2sin(1π((-3(-1(-1|x| - 1) - 4) - 3)/(-2(-1(-1|x| - 1) - 4) - 3)) - 3) + 2 intersect in the interval [-10, 10].\n\nPresent the ans... | 62 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(5(1(-1(5x^2 - 5) + 2)^2 - 5(-1(5x^2 - 5) + 2) + 4) + 2)^2 + 5(5(1(-1(5x^2 - 5) + 2)^2 - 5(-1(5x^2 - 5) + 2) + 4) + 2) - 4 and g(x) = 1((2(3|(-1x^2 - 4x + 5) - 2| + 2) + 2)/(-3(3|(-1x^2 - 4x + 5) - 2| + 2) + 2))^2 + 4((2(3|(-1x^2... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(-5(-2((3((-2x + 1)/(-2x - 3)) + 2)/(-2((-2x + 1)/(-2x - 3)) + 3))^2 + 4((3((-2x + 1)/(-2x - 3)) + 2)/(-2((-2x + 1)/(-2x - 3)) + 3)) + 1) + 3)^2 + 3(-5(-2((3((-2x + 1)/(-2x - 3)) + 2)/(-2((-2x + 1)/(-2x - 3)) + 3))^2 + 4((3... | 10 | MATH |
[
{
"content": "How many solutions does the equation -3|(3(1sin(1π(-3(5x^2 + 1) + 1) - 2) - 2)^2 - 2(1sin(1π(-3(5x^2 + 1) + 1) - 2) - 2) - 2) + 2| - 1 = -5(3((3(2x - 2))/(1(2x - 2) + 3))^2 + 3((3(2x - 2))/(1(2x - 2) + 3)) + 4) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxe... | 129 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-5(2cos(1π(2(2x + 1) - 1) + 3) + 2)^2 - 2(2cos(1π(2(2x + 1) - 1) + 3) + 2)) - 3)/(1(-5(2cos(1π(2(2x + 1) - 1) + 3) + 2)^2 - 2(2cos(1π(2(2x + 1) - 1) + 3) + 2))) and g(x) = 2(-3|(3|(3x^2 + 4x) + 1| - 3) + 1| + 3)^2 + 4(-3|(3|(3x^2 + 4x) + 1| - 3) + 1... | 18 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(2|(5((-1(-3x - 3) - 1)/(-2(-3x - 3) - 1))^2 - 4((-1(-3x - 3) - 1)/(-2(-3x - 3) - 1)) - 4) + 2|)^2 - 4(2|(5((-1(-3x - 3) - 1)/(-2(-3x - 3) - 1))^2 - 4((-1(-3x - 3) - 1)/(-2(-3x - 3) - 1)) - 4) + 2|) + 5 and g(x) = -3|(1(-3|... | 6 | MATH |
[
{
"content": "How many solutions does the equation -2cos(3π(-2(3|(1(3|x - 3|) - 2) + 3| + 2)^2 + 1(3|(1(3|x - 3|) - 2) + 3| + 2) - 1) + 2) + 1 = -1|(-2((-3(1x - 1) - 2)/(1(1x - 1) + 3)) - 3)| + 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 719 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1((1(-2((1(3sin(1πx + 2) + 1) + 3)/(-3(3sin(1πx + 2) + 1) - 1))^2 + 2((1(3sin(1πx + 2) + 1) + 3)/(-3(3sin(1πx + 2) + 1) - 1)) + 2) + 1)/(2(-2((1(3sin(1πx + 2) + 1) + 3)/(-3(3sin(1πx + 2) + 1) - 1))^2 + 2((1(3sin(1πx + 2) + 1) + 3)... | 13 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(5(1(-3sin(1π(4x^2 - 5) - 3) + 3)^2 + 4(-3sin(1π(4x^2 - 5) - 3) + 3) - 2)) - 4 and g(x) = -2sin(2π(-2((1(1x) - 1)/(-1(1x) - 2))^2 + 1((1(1x) - 1)/(-1(1x) - 2)) + 1) + 2) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: ... | 631 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(3(2(4(3x + 4) - 2)^2 - 5(4(3x + 4) - 2) - 2) + 2) + 3)/(-1(3(2(4(3x + 4) - 2)^2 - 5(4(3x + 4) - 2) - 2) + 2) + 2) and g(x) = -3((1(3cos(2π(2|x - 2| - 3) - 1) + 1))/(-1(3cos(2π(2|x - 2| - 3) - 1) + 1) + 3))^2 + 2 intersect in the interval [-10, 10].... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(1|((-2(-3((1x - 2)/(-1x + 2)) + 5) + 1)/(-3(-3((1x - 2)/(-1x + 2)) + 5) - 2)) - 2| + 3) + 4 and g(x) = -1(5(1sin(2π(-1x^2 - 2x - 1)) + 3)^2 - 4(1sin(2π(-1x^2 - 2x - 1)) + 3) - 3) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answ... | 404 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1((-1(1(3sin(1π(2x + 3) + 3) + 2)^2 + 5(3sin(1π(2x + 3) + 3) + 2) - 3) + 3)/(2(1(3sin(1π(2x + 3) + 3) + 2)^2 + 5(3sin(1π(2x + 3) + 3) + 2) - 3) + 1))^2 + 3 and g(x) = 2(3e^(2((-1(-1x + 4))/(2(-1x + 4) + 2))) - 1)^2 + 1(3e^(2... | 60 | MATH |
[
{
"content": "How many solutions does the equation (-2(-3|(2sin(2π(-3((-3x + 3)/(-1x - 2))^2 - 5((-3x + 3)/(-1x - 2))) + 1) + 1) + 2|) - 3)/(2(-3|(2sin(2π(-3((-3x + 3)/(-1x - 2))^2 - 5((-3x + 3)/(-1x - 2))) + 1) + 1) + 2|) + 1) = (2(-3(-1|(3|x - 2| - 3)| + 1)^2 + 3(-1|(3|x - 2| - 3)| + 1) + 1) - 3)/(1(-3(-1|(3|... | 4 | MATH |
[
{
"content": "How many solutions does the equation 5((-1(4(2|(-1e^(-1x) - 2) - 1| - 2)^2 - 3(2|(-1e^(-1x) - 2) - 1| - 2) - 1) + 3)/(2(4(2|(-1e^(-1x) - 2) - 1| - 2)^2 - 3(2|(-1e^(-1x) - 2) - 1| - 2) - 1) + 2))^2 - 5((-1(4(2|(-1e^(-1x) - 2) - 1| - 2)^2 - 3(2|(-1e^(-1x) - 2) - 1| - 2) - 1) + 3)/(2(4(2|(-1e^(-1x) -... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(3|(2|(2sin(3π(2x^2 - 1x + 1)) - 2) + 3|) - 2|) and g(x) = 1(1|(1|(-5x)| + 1) + 3|)^2 - 2(1|(1|(-5x)| + 1) + 3|) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-2(2sin(3π(4(3|(-1x^2 - 4x - 4) - 3| - 3)^2 + 2(3|(-1x^2 - 4x - 4) - 3| - 3) + 4) - 2) + 1) + 2)/(-1(2sin(3π(4(3|(-1x^2 - 4x - 4) - 3| - 3)^2 + 2(3|(-1x^2 - 4x - 4) - 3| - 3) + 4) - 2) + 1) - 1) and g(x) = 3|(2(-2((-3x + 2)/... | 25 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|((2((3(-3|(-3cos(2πx - 1) + 2) + 3|) + 2)/(-2(-3|(-3cos(2πx - 1) + 2) + 3|) + 1)) + 1)/(-2((3(-3|(-3cos(2πx - 1) + 2) + 3|) + 2)/(-2(-3|(-3cos(2πx - 1) + 2) + 3|) + 1)) - 3))| - 2 and g(x) = -1(-1(-2sin(2π(4x^2 + 2x + 2) + 3) - 2... | 1231 | MATH |
[
{
"content": "How many solutions does the equation 2(-4(-3|(-1(2sin(2πx + 1)) + 5) + 1| - 1) + 3) - 2 = -4(-1((3(3|x - 1| - 1) - 1)/(-2(3|x - 1| - 1) - 1))^2 - 5((3(3|x - 1| - 1) - 1)/(-2(3|x - 1| - 1) - 1)) - 3) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 4(4(-4(5((-1x + 2)/(2x))^2 + 1((-1x + 2)/(2x)) - 2) - 4) + 2) - 3 and g(x) = -4(-1e^(-1(2(-3|x - 1| + 1) - 1)) - 1)^2 + 3(-1e^(-1(2(-3|x - 1| + 1) - 1)) - 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\bo... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3((2((2(-2|(1x + 4) + 3|))/(3(-2|(1x + 4) + 3|) + 1)) - 3)/(1((2(-2|(1x + 4) + 3|))/(3(-2|(1x + 4) + 3|) + 1)) + 1)) - 2)/(-3((2((2(-2|(1x + 4) + 3|))/(3(-2|(1x + 4) + 3|) + 1)) - 3)/(1((2(-2|(1x + 4) + 3|))/(3(-2|(1x + 4) +... | 813 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2(2cos(3π(1(3((2x - 3)/(-1x + 3)) + 3)^2 + 5(3((2x - 3)/(-1x + 3)) + 3) + 4) + 1) - 1) and g(x) = -1|((3(-5((-1x + 3)/(-2x - 1)) - 2) - 1)/(-1(-5((-1x + 3)/(-2x - 1)) - 2) + 2)) + 3| + 3 where -10 ≤ x ≤ 10.\n\nPresent the ans... | 162 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4((2(-2|((2(-5x^2 + 5x - 4) + 2)/(1(-5x^2 + 5x - 4) + 3)) + 2| - 1) - 1)/(-3(-2|((2(-5x^2 + 5x - 4) + 2)/(1(-5x^2 + 5x - 4) + 3)) + 2| - 1) - 2)) - 2 and g(x) = (-3(3sin(2π(-3(-2|x + 1|)^2 - 3(-2|x + 1|) + 1) - 2) - 1) + 3)/(-3(3s... | 75 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3|(-2|((-2(-3|(2|x - 2| + 1) - 3| + 3) - 3)/(3(-3|(2|x - 2| + 1) - 3| + 3) - 2)) - 1|) + 3| and g(x) = (-2(3|(-1cos(2π(-3x^2 - 3x - 3) - 1)) - 2| + 3) + 1)/(-1(3|(-1cos(2π(-3x^2 - 3x - 3) - 1)) - 2| + 3) + 3) intersect in the interval [-10, 10].\n\nPre... | 20 | MATH |
[
{
"content": "How many solutions does the equation 4(-1|(-1(-5(2|x + 2| - 2) - 3) - 4) - 3| + 2)^2 + 3(-1|(-1(-5(2|x + 2| - 2) - 3) - 4) - 3| + 2) - 1 = (-1(-4(-1(1|x - 2| + 3) + 2)^2 + 1(-1(1|x - 2| + 3) + 2) + 2) + 2)/(1(-4(-1(1|x - 2| + 3) + 2)^2 + 1(-1(1|x - 2| + 3) + 2) + 2) + 3) have for x in the interval... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-1e^(-1((-2(1((-2x + 2)/(3x + 2))^2 - 3))/(-2(1((-2x + 2)/(3x + 2))^2 - 3) - 1))) - 3)^2 - 3(-1e^(-1((-2(1((-2x + 2)/(3x + 2))^2 - 3))/(-2(1((-2x + 2)/(3x + 2))^2 - 3) - 1))) - 3) + 4 and g(x) = -1sin(2π(-1(-1(4x^2 + 1x + 3) - 1... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|((-1((2(-1cos(1π((2x - 3)/(-3x - 1)) - 1) - 2) - 3)/(2(-1cos(1π((2x - 3)/(-3x - 1)) - 1) - 2) - 3)) - 2)/(1((2(-1cos(1π((2x - 3)/(-3x - 1)) - 1) - 2) - 3)/(2(-1cos(1π((2x - 3)/(-3x - 1)) - 1) - 2) - 3)) + 1)) - 2| - 3 and g(x) = -3|(-4(-3|(1x - 3) -... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(1|(-3(5(1x^2 + 1x + 1)) - 3) - 1|) - 2)/(2(1|(-3(5(1x^2 + 1x + 1)) - 3) - 1|) - 1) and g(x) = 4((3(1(-3|x + 2| - 1)^2 + 5(-3|x + 2| - 1) + 5) + 1)/(1(1(-3|x + 2| - 1)^2 + 5(-3|x + 2| - 1) + 5) - 2)) + 4 where -10 ≤ x ≤ 10... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(1sin(3π((-1((-1(-2|x + 2| - 1) - 3)/(-1(-2|x + 2| - 1) + 3)) - 2)/(-2((-1(-2|x + 2| - 1) - 3)/(-1(-2|x + 2| - 1) + 3)))) + 3) + 2) and g(x) = -2|(-1sin(3π(1|(4x + 5) - 1|) + 2) - 1) + 1| intersect in the interval [-10, 10].\n\nPresent the answer in ... | 283 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3(3|((2(-2|(-4x^2 + 5x + 5) - 3| + 2) - 3)/(2(-2|(-4x^2 + 5x + 5) - 3| + 2) + 3)) - 3|)^2 + 3(3|((2(-2|(-4x^2 + 5x + 5) - 3| + 2) - 3)/(2(-2|(-4x^2 + 5x + 5) - 3| + 2) + 3)) - 3|) + 4 and g(x) = -1(-1|((-2((2x + 2)/(3x - 2)) - 3)/(... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(-5(-1(3|(-2|x + 3|) + 3| + 2) + 1) - 4)^2 + 4(-5(-1(3|(-2|x + 3|) + 3| + 2) + 1) - 4) + 4 and g(x) = -5(4((2(-1x^2 - 4x - 4) - 3)/(3(-1x^2 - 4x - 4) - 2))^2 + 2((2(-1x^2 - 4x - 4) - 3)/(3(-1x^2 - 4x - 4) - 2)) - 2)^2 - 2(4((2(-1... | 4 | MATH |
[
{
"content": "How many solutions does the equation -4((2(3|(3(4x + 4)^2 + 3(4x + 4) - 2) + 1| + 2) - 3)/(1(3|(3(4x + 4)^2 + 3(4x + 4) - 2) + 1| + 2) + 1)) + 2 = (1(-1|((1((-2x)/(-2x - 1)) - 2)/(3((-2x)/(-2x - 1)))) + 1| - 1) - 1)/(-3(-1|((1((-2x)/(-2x - 1)) - 2)/(3((-2x)/(-2x - 1)))) + 1| - 1) + 4) have for x i... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(-2cos(1π(-1((-1(-1x^2 - 1x - 5) - 1)/(-1(-1x^2 - 1x - 5) + 2))^2 - 5((-1(-1x^2 - 1x - 5) - 1)/(-1(-1x^2 - 1x - 5) + 2)) + 2) + 3) - 3) - 3 and g(x) = 2|(-2(1((3x + 1)/(3x + 2))^2 - 4((3x + 1)/(3x + 2)) - 4)^2 + 4(1((3x + 1)/(3x ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(-1|(-2(-4(-4x + 3) + 5) + 1) + 3|) + 3| - 3 and g(x) = -2sin(2π(5(-4(-1x)^2 + 5(-1x) + 3)^2 + 3(-4(-1x)^2 + 5(-1x) + 3) + 4)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 12 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(2π(-3|((-1((-2(1x^2 - 5x - 5) - 3)/(2(1x^2 - 5x - 5) + 1)) + 1)/(-1((-2(1x^2 - 5x - 5) - 3)/(2(1x^2 - 5x - 5) + 1))))| - 3) - 1) - 1 and g(x) = (3(-2|(-3(-4x^2 - 2x - 3) + 5)|) - 1)/(-3(-2|(-3(-4x^2 - 2x - 3) + 5)|) + 1) in t... | 71 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(5(3(-1(5x - 5) - 1) + 1) - 5) - 3)/(-3(5(3(-1(5x - 5) - 1) + 1) - 5)) and g(x) = -2sin(3π(3((-3((-1x + 2)/(3x - 2)) + 1)/(1((-1x + 2)/(3x - 2)) + 2))^2 + 5((-3((-1x + 2)/(3x - 2)) + 1)/(1((-1x + 2)/(3x - 2)) + 2)) + 4)) + 2 in ... | 191 | MATH |
[
{
"content": "How many solutions does the equation (-3(1|(3|((-3((-2x - 2)/(1x + 1)) + 1)/(-1((-2x - 2)/(1x + 1)) - 3)) - 2|) + 2| - 1) + 3)/(-1(1|(3|((-3((-2x - 2)/(1x + 1)) + 1)/(-1((-2x - 2)/(1x + 1)) - 3)) - 2|) + 2| - 1) - 1) = -3(-4((-2(-1|x + 1| + 1) - 2)/(1(-1|x + 1| + 1)))^2 + 4((-2(-1|x + 1| + 1) - 2)... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3|((-2(3((1(3cos(2πx + 1) + 2) + 3)/(2(3cos(2πx + 1) + 2) - 1))^2 - 1((1(3cos(2πx + 1) + 2) + 3)/(2(3cos(2πx + 1) + 2) - 1)) + 2))/(-1(3((1(3cos(2πx + 1) + 2) + 3)/(2(3cos(2πx + 1) + 2) - 1))^2 - 1((1(3cos(2πx + 1) + 2) + 3)/... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-4(1|(1|((3x - 3)/(-2x + 3))| - 2) - 1| + 1)^2 + 3(1|(1|((3x - 3)/(-2x + 3))| - 2) - 1| + 1) + 4) + 1)/(1(-4(1|(1|((3x - 3)/(-2x + 3))| - 2) - 1| + 1)^2 + 3(1|(1|((3x - 3)/(-2x + 3))| - 2) - 1| + 1) + 4) + 3) and g(x) = (1(-2sin... | 42 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(1π(3|(-2(-1|(-3x^2 + 3x + 3) + 1| + 1)^2 + 2) + 1| - 2) - 1) + 1 and g(x) = -3e^(2((-2(5((3x)/(-2x - 2))^2 - 2((3x)/(-2x - 2)) - 2) + 1)/(3(5((3x)/(-2x - 2))^2 - 2((3x)/(-2x - 2)) - 2)))) + 2 intersect in the interval [-10, 10].\n\nPresent the ans... | 1299 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1(2sin(3π(-1((-1(1x - 1) + 3)/(1(1x - 1) + 2))^2 - 5((-1(1x - 1) + 3)/(1(1x - 1) + 2)) + 3) - 3) + 2) - 1)/(-3(2sin(3π(-1((-1(1x - 1) + 3)/(1(1x - 1) + 2))^2 - 5((-1(1x - 1) + 3)/(1(1x - 1) + 2)) + 3) - 3) + 2) + 2) and g(x) = 2co... | 105 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2(-1(-2(-2|(1x) + 2| - 1) - 5)^2 - 5(-2(-2|(1x) + 2| - 1) - 5) + 1)^2 - 3(-1(-2(-2|(1x) + 2| - 1) - 5)^2 - 5(-2(-2|(1x) + 2| - 1) - 5) + 1) and g(x) = -1|(5(-1e^(-1(-5x - 2)) - 1) - 4) + 1| - 1 where -10 ≤ x ≤ 10.\n\nPresent... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(-3(-2(-3(2|x + 3| - 3)^2 - 5(2|x + 3| - 3) + 3)^2 + 3(-3(2|x + 3| - 3)^2 - 5(2|x + 3| - 3) + 3) + 4) - 4))/(-2(-3(-2(-3(2|x + 3| - 3)^2 - 5(2|x + 3| - 3) + 3)^2 + 3(-3(2|x + 3| - 3)^2 - 5(2|x + 3| - 3) + 3) + 4) - 4) - 1)... | 7 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((2((2(-4(-1x^2 - 3x + 5)^2 - 4(-1x^2 - 3x + 5) + 1) - 2)/(-2(-4(-1x^2 - 3x + 5)^2 - 4(-1x^2 - 3x + 5) + 1) - 1)) - 3)/(-3((2(-4(-1x^2 - 3x + 5)^2 - 4(-1x^2 - 3x + 5) + 1) - 2)/(-2(-4(-1x^2 - 3x + 5)^2 - 4(-1x^2 - 3x + 5) + 1) - 1))))^2 - 2((2((2(-4(-... | 8 | MATH |
[
{
"content": "How many solutions does the equation 1|((3(-1|(-2((-1x - 3)/(3x)) - 4)|))/(-3(-1|(-2((-1x - 3)/(3x)) - 4)|) - 1)) - 3| + 3 = 1(-5(3cos(3π(-2|x| - 1) - 2) + 2) - 4)^2 + 4(-5(3cos(3π(-2|x| - 1) - 2) + 2) - 4) + 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Yo... | 240 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(2π(-5(-5(-2|((-3x + 3)/(2x - 1)) + 2| + 3) + 5)) + 3) and g(x) = -1(1|(4(3|x + 2| - 3) - 1) - 2|) + 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 56 | MATH |
[
{
"content": "How many solutions does the equation -2|(1(-3|(3((1x - 1)/(2x - 1))^2 - 5((1x - 1)/(2x - 1)) - 1) + 3| - 1)^2 - 2) - 1| - 1 = -3(-2((1(-3cos(2πx + 3)) - 3)/(3(-3cos(2πx + 3)) + 1)) - 4)^2 - 1(-2((1(-3cos(2πx + 3)) - 3)/(3(-3cos(2πx + 3)) + 1)) - 4) + 1 have for x in the interval [-10, 10]?\n\nPres... | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(2(5(2e^(2((3x + 3)/(-2x - 2))) + 2)^2 + 5(2e^(2((3x + 3)/(-2x - 2))) + 2) + 1) - 2))/(1(2(5(2e^(2((3x + 3)/(-2x - 2))) + 2)^2 + 5(2e^(2((3x + 3)/(-2x - 2))) + 2) + 1) - 2)) and g(x) = -2|((1(1sin(1π(5x + 2) - 1) + 1) - 3)/... | 50 | MATH |
[
{
"content": "How many solutions does the equation 3cos(1π((3(-3|(3|(2|x - 2| - 2) - 2|) + 3| + 1) + 1)/(3(-3|(3|(2|x - 2| - 2) - 2|) + 3| + 1) - 2)) + 1) + 3 = (3(-2|(-3cos(2π(3|x - 1|) + 2) + 1) + 1| + 2) + 1)/(2(-2|(-3cos(2π(3|x - 1|) + 2) + 1) + 1| + 2) - 3) have for x in the interval [-10, 10]?\n\nPresent ... | 52 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(3(-1((1(1|x - 1| + 3) + 2)/(3(1|x - 1| + 3) + 1)) + 5) + 4) + 3)/(-2(3(-1((1(1|x - 1| + 3) + 2)/(3(1|x - 1| + 3) + 1)) + 5) + 4) + 1) and g(x) = 3|(-5(-3|((-1x - 1)/(-1x + 2)) + 3| + 3)^2 + 5(-3|((-1x - 1)/(-1x + 2)) + 3| ... | 7 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-2((3(-3sin(3π(4x - 4) + 3) - 3) - 1)/(1(-3sin(3π(4x - 4) + 3) - 3) + 1))^2 + 2((3(-3sin(3π(4x - 4) + 3) - 3) - 1)/(1(-3sin(3π(4x - 4) + 3) - 3) + 1)) + 2)^2 + 1 and g(x) = -5(-2|(1((1x + 3)/(2x - 1)) + 3) + 3| + 2)^2 + 1(-2|(1(... | 264 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1((3(2|((-1(4x^2 - 5) + 2)/(1(4x^2 - 5) + 3)) - 3| + 2))/(2(2|((-1(4x^2 - 5) + 2)/(1(4x^2 - 5) + 3)) - 3| + 2) + 3)) - 1)/(2((3(2|((-1(4x^2 - 5) + 2)/(1(4x^2 - 5) + 3)) - 3| + 2))/(2(2|((-1(4x^2 - 5) + 2)/(1(4x^2 - 5) + 3)) - 3| + 2) + 3)) - 3) and g(... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(1|(-1(-3(1|x + 1| - 1)^2 + 5(1|x + 1| - 1) + 5) + 4)| + 1)^2 - 2(1|(-1(-3(1|x + 1| - 1)^2 + 5(1|x + 1| - 1) + 5) + 4)| + 1) and g(x) = (-2(3|(1|(-3|x - 3| - 2) + 1| - 1) + 3| + 2) + 3)/(3(3|(1|(-3|x - 3| - 2) + 1| - 1) + 3| + 2) - 2) intersect in the... | 8 | MATH |
[
{
"content": "How many solutions does the equation 2(2cos(2π(-3|(-2|(4x - 3)| - 3) - 3| + 2) + 3) - 1) + 2 = -3(-5(-3sin(2π(-1x - 4) + 3) + 3)^2 - 3(-3sin(2π(-1x - 4) + 3) + 3) + 2)^2 + 4(-5(-3sin(2π(-1x - 4) + 3) + 3)^2 - 3(-3sin(2π(-1x - 4) + 3) + 3) + 2) - 2 have for x in the interval [-10, 10]?\n\nPresent t... | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1((-2(3(3(1x + 3)^2 - 5(1x + 3) - 3) - 4) - 1)/(1(3(3(1x + 3)^2 - 5(1x + 3) - 3) - 4)))^2 - 3((-2(3(3(1x + 3)^2 - 5(1x + 3) - 3) - 4) - 1)/(1(3(3(1x + 3)^2 - 5(1x + 3) - 3) - 4))) and g(x) = 3(2(-5(-2cos(2πx + 2) - 2) - 1)^2 - 4(-... | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(2e^(-1(-2(-5(1|x - 1| - 2))^2 + 5(-5(1|x - 1| - 2)) - 2)) - 3) + 1)/(-1(2e^(-1(-2(-5(1|x - 1| - 2))^2 + 5(-5(1|x - 1| - 2)) - 2)) - 3) + 3) and g(x) = (2(1|((2(-3|x + 1| + 2) - 2)/(-3(-3|x + 1| + 2))) - 2| - 1) - 1)/(3(1|((2(-3... | 7 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3(2(-5(1|(3x + 2) - 2| - 2)^2 - 5(1|(3x + 2) - 2| - 2))) and g(x) = 1((1(4(1x - 1)^2 + 5(1x - 1) - 3) + 2)/(1(4(1x - 1)^2 + 5(1x - 1) - 3))) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 5 | MATH |
[
{
"content": "How many solutions does the equation -3|(-1e^(-1(1((-3((-1x)/(-1x - 2)) + 1)/(-2((-1x)/(-1x - 2)) - 2)) + 1)) + 2) + 2| - 1 = -2(4((2(1|x + 3| - 1) - 1)/(3(1|x + 3| - 1) - 2)) - 4)^2 - 3(4((2(1|x + 3| - 1) - 1)/(3(1|x + 3| - 1) - 2)) - 4) - 3 have for x in the interval [-10, 10]?\n\nPresent the an... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -5(-2|(-2(-2|(-2|x - 3| - 1) + 1| + 2)^2 - 4(-2|(-2|x - 3| - 1) + 1| + 2) + 5) + 1| + 3)^2 - 3(-2|(-2(-2|(-2|x - 3| - 1) + 1| + 2)^2 - 4(-2|(-2|x - 3| - 1) + 1| + 2) + 5) + 1| + 3) + 5 and g(x) = -3cos(2π(-2|(1|(2|x + 1| + 3) - 2| ... | 16 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(1cos(3π((-3((2(-1x + 3) - 3)/(1(-1x + 3) - 2)) - 1)/(3((2(-1x + 3) - 3)/(1(-1x + 3) - 2)) + 2)) + 2)) - 2 and g(x) = -2sin(3π((-1((-2(1x + 3) + 1)/(-2(1x + 3) - 3)) - 1)/(1((-2(1x + 3) + 1)/(-2(1x + 3) - 3)))) + 2) - 3 in the ra... | 95 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(5((-1(3((2x - 3)/(-3x + 1)) - 1) - 1)/(3(3((2x - 3)/(-3x + 1)) - 1) + 2)) - 5) - 2)/(-1(5((-1(3((2x - 3)/(-3x + 1)) - 1) - 1)/(3(3((2x - 3)/(-3x + 1)) - 1) + 2)) - 5) + 3) and g(x) = 3|(-3e^(-1((1(-2|x - 3| + 3) - 1)/(-1(-2|x - 3| + 3) - 2))) + 3) -... | 5 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(5((3(-3|((2x + 3)/(-2x + 1)) + 3| + 3))/(-1(-3|((2x + 3)/(-2x + 1)) + 3| + 3) + 2))^2 - 3((3(-3|((2x + 3)/(-2x + 1)) + 3| + 3))/(-1(-3|((2x + 3)/(-2x + 1)) + 3| + 3) + 2)))^2 - 1(5((3(-3|((2x + 3)/(-2x + 1)) + 3| + 3))/(-1... | 20 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-5((-1(1cos(2π((-2x - 3)/(-3x + 1)) - 1)) + 3)/(2(1cos(2π((-2x - 3)/(-3x + 1)) - 1)) + 1))^2 - 4((-1(1cos(2π((-2x - 3)/(-3x + 1)) - 1)) + 3)/(2(1cos(2π((-2x - 3)/(-3x + 1)) - 1)) + 1)) + 2) + 2| + 3 and g(x) = -4(-3cos(2π(1(-1x^2 + 4x + 1)^2 + 5(-1x... | 213 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(-3sin(1π(-2(-1(-1|x + 2| - 1)^2 + 3(-1|x + 2| - 1) - 4) + 2)) + 3) + 1)/(3(-3sin(1π(-2(-1(-1|x + 2| - 1)^2 + 3(-1|x + 2| - 1) - 4) + 2)) + 3) + 1) and g(x) = -3sin(3π(-1(-2(-5x + 4)^2 + 2) - 1) - 3) - 1 where -10 ≤ x ≤ 10.... | 1274 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5((1((3(-1(5x^2 + 5x - 5)^2 - 5(5x^2 + 5x - 5) - 4) + 1)/(-3(-1(5x^2 + 5x - 5)^2 - 5(5x^2 + 5x - 5) - 4) + 2)) + 1)/(1((3(-1(5x^2 + 5x - 5)^2 - 5(5x^2 + 5x - 5) - 4) + 1)/(-3(-1(5x^2 + 5x - 5)^2 - 5(5x^2 + 5x - 5) - 4) + 2)) ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-1sin(2π(-2(2(-4x - 5)^2 + 5(-4x - 5) - 5) - 3) - 2)) - 3)/(3(-1sin(2π(-2(2(-4x - 5)^2 + 5(-4x - 5) - 5) - 3) - 2)) - 2) and g(x) = -2(1cos(2π(1|(1|x + 1| - 3) + 1| + 1) + 1) + 3)^2 - 5 intersect in the interval [-10, 10].\n\nPresent the answer in L... | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(3(-1(2|(3x + 4) - 3| - 2)^2 + 5(2|(3x + 4) - 3| - 2) + 1) + 1)^2 + 1(3(-1(2|(3x + 4) - 3| - 2)^2 + 5(2|(3x + 4) - 3| - 2) + 1) + 1) + 1 and g(x) = 3(2(-1(-3x^2 + 2x + 2)^2 - 4(-3x^2 + 2x + 2) - 5)^2 - 3(-1(-3x^2 + 2x + 2)^2 - 4(-... | 4 | MATH |
[
{
"content": "How many solutions does the equation -2|(2(-1(-5(-2|x - 1| + 3) + 1))^2 - 5(-1(-5(-2|x - 1| + 3) + 1)) - 5) + 1| = (-2(-1(5(-2e^(-1x) - 3) + 4) + 2) - 1)/(2(-1(5(-2e^(-1x) - 3) + 4) + 2) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(4(-3((-1(-4x - 3))/(-3(-4x - 3) + 2))^2)^2 + 1) + 2| + 3 and g(x) = (-2(-1sin(1π(1(-3x)^2 - 5(-3x) - 2) + 2) + 1))/(3(-1sin(1π(1(-3x)^2 - 5(-3x) - 2) + 2) + 1) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{... | 170 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1((1(-1(2(2e^(-1x) - 3) - 4)^2 + 5(2(2e^(-1x) - 3) - 4) - 4))/(2(-1(2(2e^(-1x) - 3) - 4)^2 + 5(2(2e^(-1x) - 3) - 4) - 4) - 3)) - 1 and g(x) = 2(1(-4(-2x^2 + 1x + 1)^2 - 4(-2x^2 + 1x + 1) + 2)^2 - 4(-4(-2x^2 + 1x + 1)^2 - 4(-2x^2 + 1x + 1) + 2) - 1) - 5... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1|((-1(1cos(1π((1(3x^2 + 5x - 4) + 3)/(2(3x^2 + 5x - 4) - 1)) - 3) + 2) - 1)/(3(1cos(1π((1(3x^2 + 5x - 4) + 3)/(2(3x^2 + 5x - 4) - 1)) - 3) + 2) - 2)) - 3| = (2(4(-2(-3cos(3πx) - 1)^2 + 4(-3cos(3πx) - 1) + 3)^2 - 1) + 1)/(2(4(-2(-3cos(3πx) - 1)^2 + 4(-3cos(3πx... | 13 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(-2|(1|((2(1|x + 2|))/(2(1|x + 2|) - 1)) - 3| - 3) + 3| - 1) + 2| - 1 and g(x) = (2(-2sin(1π(2|(2|x + 1|) - 1| - 2)) + 2) + 2)/(-2(-2sin(1π(2|(2|x + 1|) - 1| - 2)) + 2) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaT... | 77 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-1(-3(1(3x^2 - 4x) - 1) + 1) - 2)^2 + 5(-1(-3(1(3x^2 - 4x) - 1) + 1) - 2) + 1 and g(x) = 3((-2(1((-1x - 3)/(2x - 3)) - 4) - 3)/(3(1((-1x - 3)/(2x - 3)) - 4) - 1))^2 + 4((-2(1((-1x - 3)/(2x - 3)) - 4) - 3)/(3(1((-1x - 3)/(2x - 3))... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(3((-1((3((-2x + 1)/(2x + 2)))/(-2((-2x + 1)/(2x + 2)) - 3)) - 2)/(1((3((-2x + 1)/(2x + 2)))/(-2((-2x + 1)/(2x + 2)) - 3)) - 3))^2 - 4)^2 + 3(3((-1((3((-2x + 1)/(2x + 2)))/(-2((-2x + 1)/(2x + 2)) - 3)) - 2)/(1((3((-2x + 1)/(2x + 2)))/(-2((-2x + 1)/(2x... | 36 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|(5(-1sin(3π(-4((1x + 3)/(-1x))^2 - 1((1x + 3)/(-1x)) + 5) - 1) - 2) - 1) + 3| - 3 and g(x) = -2(-5(1|(2e^(1x) + 1) - 2| + 2) + 2) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 56 | MATH |
[
{
"content": "How many solutions does the equation -3cos(1π(-3(1|(2|(5x^2 + 1x + 4) + 2| + 1) - 3| + 2)^2 - 4(1|(2|(5x^2 + 1x + 4) + 2| + 1) - 3| + 2) + 1) + 3) + 1 = -5(3(3|((2x + 2)/(-2x + 3)) - 2| - 3)^2 - 2(3|((2x + 2)/(-2x + 3)) - 2| - 3) - 5)^2 - 5(3(3|((2x + 2)/(-2x + 3)) - 2| - 3)^2 - 2(3|((2x + 2)/(-2x... | 13 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(3sin(3π(-3|(2(1|x + 3| + 1)^2)| - 2) + 1) + 2) + 2)/(-3(3sin(3π(-3|(2(1|x + 3| + 1)^2)| - 2) + 1) + 2) + 4) and g(x) = (2((1(3((2x)/(-3x + 1)) + 3) + 1)/(-1(3((2x)/(-3x + 1)) + 3) - 1)) - 2)/(-2((1(3((2x)/(-3x + 1)) + 3) + 1)/(... | 50 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2cos(1π(-2(1(4(4x^2 - 4x + 5)^2 + 2(4x^2 - 4x + 5)) + 3) - 3) - 1) - 1 and g(x) = -3(-1|(1(-1x + 3))| + 3) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 102 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(1π(-3|(2|(2(3x + 3)^2 - 5(3x + 3))| + 1) + 2| + 3) - 1) and g(x) = 1|(-3sin(1π(-5(-3x^2 + 2x + 3) + 3) + 1) + 1) - 3| in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"... | 990 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(1(1cos(2π(-4(-3x^2 + 3x)^2 + 2(-3x^2 + 3x) + 5) - 1))^2 + 1(1cos(2π(-4(-3x^2 + 3x)^2 + 2(-3x^2 + 3x) + 5) - 1)) + 4) - 3)/(1(1(1cos(2π(-4(-3x^2 + 3x)^2 + 2(-3x^2 + 3x) + 5) - 1))^2 + 1(1cos(2π(-4(-3x^2 + 3x)^2 + 2(-3x^2 + 3x) + ... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2(-2cos(3π(2|(2(3x + 5)^2 + 1(3x + 5) + 1) + 2| - 1) + 3) - 2)^2 - 3(-2cos(3π(2|(2(3x + 5)^2 + 1(3x + 5) + 1) + 2| - 1) + 3) - 2) - 2 and g(x) = 5(1cos(1π(-4((2x + 2)/(-2x)) - 4) + 3) + 1) - 3 in the range -10 ≤ x ≤ 10?\n\nPresent... | 143 | MATH |
[
{
"content": "How many solutions does the equation -4(-5((-2(1(-5x + 4)^2 + 5(-5x + 4) + 3) - 2)/(2(1(-5x + 4)^2 + 5(-5x + 4) + 3) + 2)) - 4) - 2 = -1((1(3(-1x - 3)^2 + 5(-1x - 3) + 2) + 3)/(-2(3(-1x - 3)^2 + 5(-1x - 3) + 2) + 3))^2 + 3((1(3(-1x - 3)^2 + 5(-1x - 3) + 2) + 3)/(-2(3(-1x - 3)^2 + 5(-1x - 3) + 2) +... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(4(1(-2(-4x^2 + 3x + 3) - 2)^2 + 3) - 1) + 3)/(-2(4(1(-2(-4x^2 + 3x + 3) - 2)^2 + 3) - 1) + 3) and g(x) = 2|(-3|((2(1|x + 1| - 2))/(-3(1|x + 1| - 2) - 1)) - 1| + 3) + 3| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: ... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(1(-3((-2(-4x^2 - 4x + 1) - 1)/(2(-4x^2 - 4x + 1) - 1)) + 2) - 4) + 4 and g(x) = 1|((2(-2|((1x - 1)/(2x - 1)) + 3|) - 3)/(-2(-2|((1x - 1)/(2x - 1)) + 3|) - 1)) + 1| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\box... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|((1(-3|(2|(-5x^2 + 4x + 3) - 2|) - 3| + 1) + 2)/(2(-3|(2|(-5x^2 + 4x + 3) - 2|) - 3| + 1) + 1)) - 1| + 1 and g(x) = (2((-2(-4(-2e^(2x) - 3)^2 - 3(-2e^(2x) - 3) + 5) + 2)/(-1(-4(-2e^(2x) - 3)^2 - 3(-2e^(2x) - 3) + 5) + 2)) - 2)/(1((-2(-4(-2e^(2x) - 3... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(1(4(-3cos(1π(-2x^2 + 5x + 5) - 1) + 3) - 1)^2 + 2(4(-3cos(1π(-2x^2 + 5x + 5) - 1) + 3) - 1) - 5) - 3| + 3 and g(x) = -2((-2((-2(5x + 1) - 1)/(1(5x + 1) - 3)) + 1)/(-2((-2(5x + 1) - 1)/(1(5x + 1) - 3)) - 1)) - 5 where -10 ... | 456 | MATH |
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