messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
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[
{
"content": "How many points of intersection exist between the functions f(x) = -5(-2sin(1π(4(3((2x + 3)/(1x))^2 + 3((2x + 3)/(1x)) + 2)) - 2) + 3) - 1 and g(x) = 3((2(1e^(-1((2x + 2)/(2x - 3))) - 3) + 1)/(3(1e^(-1((2x + 2)/(2x - 3))) - 3) - 1)) - 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex for... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|((-3(2e^(1(-3((2x - 3)/(-3x - 1)) - 5)) - 3) - 1)/(-3(2e^(1(-3((2x - 3)/(-3x - 1)) - 5)) - 3) - 1)) - 2| + 1 and g(x) = -1sin(1π(-2|(3(-2|x + 2| + 3)^2 - 3(-2|x + 2| + 3) - 3) - 3| - 2) - 3) - 1 intersect in the interval [-10, 10].\n\nPresent the an... | 949 | MATH |
[
{
"content": "How many solutions does the equation -1(1|(1(2|(3e^(-2x) - 1) + 2|)^2) - 3| - 3)^2 + 2(1|(1(2|(3e^(-2x) - 1) + 2|)^2) - 3| - 3) + 1 = (-2(3|(-1cos(1π(4x^2 + 1x + 1) - 1)) + 1| - 2))/(3(3|(-1cos(1π(4x^2 + 1x + 1) - 1)) + 1| - 2) + 1) have for x in the interval [-10, 10]?\n\nPresent the answer in La... | 19 | MATH |
[
{
"content": "How many solutions does the equation 3(5(4(-1|(2cos(2πx + 2) - 2) + 2| + 1)^2 + 1(-1|(2cos(2πx + 2) - 2) + 2| + 1) - 3) + 4)^2 - 2 = 5(5((-2(-1cos(2πx - 3) - 1) - 2)/(2(-1cos(2πx - 3) - 1)))^2 + 1((-2(-1cos(2πx - 3) - 1) - 2)/(2(-1cos(2πx - 3) - 1))) - 2)^2 + 3(5((-2(-1cos(2πx - 3) - 1) - 2)/(2(-1... | 220 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(-4(-5(-2|(1|x - 2|) - 2| + 1) + 1)^2 + 1(-5(-2|(1|x - 2|) - 2| + 1) + 1) + 3)^2 - 1(-4(-5(-2|(1|x - 2|) - 2| + 1) + 1)^2 + 1(-5(-2|(1|x - 2|) - 2| + 1) + 1) + 3) + 4 and g(x) = 2((-3(-1(1x - 4)^2 + 5(1x - 4) + 3) - 3)/(-1(-1(1x - 4)^2 + 5(1x - 4) + ... | 8 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1((-3((-1(3(1x + 5)^2 - 2(1x + 5) - 2) - 2)/(1(3(1x + 5)^2 - 2(1x + 5) - 2) - 3)) - 3)/(-2((-1(3(1x + 5)^2 - 2(1x + 5) - 2) - 2)/(1(3(1x + 5)^2 - 2(1x + 5) - 2) - 3)) - 2)) - 4 and g(x) = -1(-2|((3(2|x - 1| - 3) + 3)/(-3(2|x... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2((-3(-1((3(-1x - 5) + 1)/(1(-1x - 5) + 3))^2 - 4((3(-1x - 5) + 1)/(1(-1x - 5) + 3)) - 5) + 3)/(2(-1((3(-1x - 5) + 1)/(1(-1x - 5) + 3))^2 - 4((3(-1x - 5) + 1)/(1(-1x - 5) + 3)) - 5) + 1)) - 5 and g(x) = -3cos(2π(5(2|((-3x - 1)/(1x... | 1154 | MATH |
[
{
"content": "How many solutions does the equation -2(1|(3(-2|(-4x + 4)|)^2 + 2) - 3|) + 4 = -2sin(3π(4(5(-1x^2 + 4x - 4)^2 + 1(-1x^2 + 4x - 4) - 4)^2 - 1(5(-1x^2 + 4x - 4)^2 + 1(-1x^2 + 4x - 4) - 4) - 3) - 2) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}"... | 5 | MATH |
[
{
"content": "How many solutions does the equation (-1(2e^(-2(-3(2(1|x + 2| - 3) + 1)^2 - 2(2(1|x + 2| - 3) + 1) + 3)) - 3) + 3)/(3(2e^(-2(-3(2(1|x + 2| - 3) + 1)^2 - 2(2(1|x + 2| - 3) + 1) + 3)) - 3) + 2) = -1(-3sin(3π(-3(-3x^2 - 3x + 3) - 1) - 1))^2 + 3(-3sin(3π(-3(-3x^2 - 3x + 3) - 1) - 1)) - 2 have for x in... | 1049 | MATH |
[
{
"content": "How many solutions does the equation -1|(-4(-4(-1((-3x + 2)/(3x + 1))^2 + 2((-3x + 2)/(3x + 1)) + 4)^2 - 2(-1((-3x + 2)/(3x + 1))^2 + 2((-3x + 2)/(3x + 1)) + 4))^2 - 4(-4(-1((-3x + 2)/(3x + 1))^2 + 2((-3x + 2)/(3x + 1)) + 4)^2 - 2(-1((-3x + 2)/(3x + 1))^2 + 2((-3x + 2)/(3x + 1)) + 4)) - 2) - 2| - ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(-1(-5(-1|(3cos(2πx + 1) + 1) - 2| + 3) + 3)^2 - 3(-5(-1|(3cos(2πx + 1) + 1) - 2| + 3) + 3) + 4) + 1 and g(x) = -5((-2(1|(-1x + 3) - 3| + 1))/(-3(1|(-1x + 3) - 3| + 1) + 2))^2 + 1((-2(1|(-1x + 3) - 3| + 1))/(-3(1|(-1x + 3) ... | 120 | MATH |
[
{
"content": "How many solutions does the equation 4(-3sin(2π(2(4(1x^2 - 3x + 4) - 4) - 3) - 1) - 2)^2 - 3(-3sin(2π(2(4(1x^2 - 3x + 4) - 4) - 3) - 1) - 2) - 3 = -2cos(2π(2|(-1((1x + 3)/(-2x - 3))^2 + 4((1x + 3)/(-2x - 3)) - 3) - 1| - 2) - 2) - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaT... | 743 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1((-3(-2|(-1(1x^2 - 1x + 2) - 2) - 3| + 2) + 1)/(-1(-2|(-1(1x^2 - 1x + 2) - 2) - 3| + 2) + 3))^2 + 5((-3(-2|(-1(1x^2 - 1x + 2) - 2) - 3| + 2) + 1)/(-1(-2|(-1(1x^2 - 1x + 2) - 2) - 3| + 2) + 3)) + 4 and g(x) = (-3(5(-3(2sin(2πx - 3) - 1)^2 + 5(2sin(2πx... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(-4((-1(1sin(1π(-1|x + 3|) + 1)))/(1(1sin(1π(-1|x + 3|) + 1))))^2 - 3((-1(1sin(1π(-1|x + 3|) + 1)))/(1(1sin(1π(-1|x + 3|) + 1)))) + 5) - 3| and g(x) = -3((-1((-1(-2cos(3πx - 2) + 2) + 2)/(-3(-2cos(3πx - 2) + 2) + 1)) - 1)/(2((-1... | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-5(3|((2((-2x + 3)/(-2x - 1)) + 1)/(3((-2x + 3)/(-2x - 1)) + 2)) - 2| + 2) - 1)| + 2 and g(x) = -2(2(5((3x + 1)/(-1x + 2)) - 5) - 5)^2 - 2(2(5((3x + 1)/(-1x + 2)) - 5) - 5) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex for... | 4 | MATH |
[
{
"content": "How many solutions does the equation 2|(1(2cos(2π((2(2x^2 + 5x - 3) - 3)/(1(2x^2 + 5x - 3) + 2)) - 1) - 3) + 2) + 3| = (3((3(-3cos(1π(3x^2 + 1x - 2) - 3) - 2) + 3)/(-2(-3cos(1π(3x^2 + 1x - 2) - 3) - 2) - 3)) + 3)/(-3((3(-3cos(1π(3x^2 + 1x - 2) - 3) - 2) + 3)/(-2(-3cos(1π(3x^2 + 1x - 2) - 3) - 2) -... | 91 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(2|(-5(2cos(2π(-3|x| + 3) - 1) + 1) - 4) + 1| + 1) + 1 and g(x) = -2|(1|(-3|(-5x - 2) + 2| + 3) + 1| - 2) - 3| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 22 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(2((-1((-3(2|x - 2| + 3) - 3)/(3(2|x - 2| + 3) + 1)) - 2)/(-1((-3(2|x - 2| + 3) - 3)/(3(2|x - 2| + 3) + 1)) - 2))^2 + 5((-1((-3(2|x - 2| + 3) - 3)/(3(2|x - 2| + 3) + 1)) - 2)/(-1((-3(2|x - 2| + 3) - 3)/(3(2|x - 2| + 3) + 1)) - 2)) + 3) + 3| + 3 and ... | 154 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(5(4(2cos(2π((3x + 1)/(3x + 3)) - 3) + 2) + 4) + 2))/(-1(5(4(2cos(2π((3x + 1)/(3x + 3)) - 3) + 2) + 4) + 2) + 2) and g(x) = 3(5(-3|(2x^2 + 1x - 1) - 1| + 1) + 1) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed... | 4 | MATH |
[
{
"content": "How many solutions does the equation (2(-1|(1|(4(-4x - 1)^2 + 1(-4x - 1) + 2) + 3| + 3)| - 2) + 1)/(2(-1|(1|(4(-4x - 1)^2 + 1(-4x - 1) + 2) + 3| + 3)| - 2) + 3) = 3|(2|((2(-2|x + 3|) - 2)/(3(-2|x + 3|) + 2)) + 1| - 2) - 1| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex fo... | 6 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3(3|(1(5(-2|x + 1|) - 2)^2 + 4(5(-2|x + 1|) - 2) - 2) + 3| - 3) - 3)/(2(3|(1(5(-2|x + 1|) - 2)^2 + 4(5(-2|x + 1|) - 2) - 2) + 3| - 3) - 3) and g(x) = -2(-2e^(2(-1(3x^2 - 4x - 5)^2 + 3(3x^2 - 4x - 5) - 5)) + 2)^2 - 1 in the range ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5((-2(-2(-1cos(3π(3x^2 + 2x + 2) + 1) + 1) + 1))/(1(-2(-1cos(3π(3x^2 + 2x + 2) + 1) + 1) + 1) + 1))^2 + 5((-2(-2(-1cos(3π(3x^2 + 2x + 2) + 1) + 1) + 1))/(1(-2(-1cos(3π(3x^2 + 2x + 2) + 1) + 1) + 1) + 1)) and g(x) = -2((2(-2si... | 941 | MATH |
[
{
"content": "How many solutions does the equation 1(-4(3cos(2π(4(2x - 1)^2 + 4(2x - 1) - 2) + 1) + 1) - 3)^2 + 4(-4(3cos(2π(4(2x - 1)^2 + 4(2x - 1) - 2) + 1) + 1) - 3) + 2 = (-3(3(-5(1x^2 + 1x - 1)^2 + 3(1x^2 + 1x - 1) - 2)^2 + 3) - 1)/(-3(3(-5(1x^2 + 1x - 1)^2 + 3(1x^2 + 1x - 1) - 2)^2 + 3) - 3) have for x in... | 616 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3cos(1π(3|(1|((1(3x^2 + 4x - 2) - 3)/(3(3x^2 + 4x - 2) - 2)) - 3| - 1)| - 1) + 1) + 1 and g(x) = -5((2(-2|(-1x + 3) + 3| + 3) - 1)/(3(-2|(-1x + 3) + 3| + 3) - 2)) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Y... | 41 | MATH |
[
{
"content": "How many solutions does the equation -2|((-3((-3(-3|((3x - 2)/(-1x + 3)) - 2| + 1))/(-2(-3|((3x - 2)/(-1x + 3)) - 2| + 1) - 3)) + 3)/(-1((-3(-3|((3x - 2)/(-1x + 3)) - 2| + 1))/(-2(-3|((3x - 2)/(-1x + 3)) - 2| + 1) - 3)) - 3)) - 3| - 1 = -4(2cos(2π((-1((2x + 3)/(1x)) - 2)/(1((2x + 3)/(1x)) - 3))) +... | 188 | MATH |
[
{
"content": "How many solutions does the equation 3(1(2(-3(2cos(3πx) + 2) + 1) - 2)^2 + 1(2(-3(2cos(3πx) + 2) + 1) - 2) - 1)^2 - 2 = 3(5((1(1sin(2πx - 3) + 1) - 1)/(-2(1sin(2πx - 3) + 1) - 2))^2 + 1((1(1sin(2πx - 3) + 1) - 1)/(-2(1sin(2πx - 3) + 1) - 2)) + 4)^2 - 2(5((1(1sin(2πx - 3) + 1) - 1)/(-2(1sin(2πx - 3... | 60 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (1((-3(2sin(1π((1(3|x - 3| - 2) + 1)/(-1(3|x - 3| - 2) - 3)) - 1) - 1) + 2)/(-1(2sin(1π((1(3|x - 3| - 2) + 1)/(-1(3|x - 3| - 2) - 3)) - 1) - 1) + 2)) + 3)/(1((-3(2sin(1π((1(3|x - 3| - 2) + 1)/(-1(3|x - 3| - 2) - 3)) - 1) - 1) + 2)/... | 80 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(-2((-3(2cos(3π(1x - 3) - 3) + 2) + 2)/(1(2cos(3π(1x - 3) - 3) + 2)))^2 + 4((-3(2cos(3π(1x - 3) - 3) + 2) + 2)/(1(2cos(3π(1x - 3) - 3) + 2))) - 3)^2 - 4(-2((-3(2cos(3π(1x - 3) - 3) + 2) + 2)/(1(2cos(3π(1x - 3) - 3) + 2)))^2... | 120 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(-1|((3(-5(5x^2 - 5x - 1) + 4) + 2)/(3(-5(5x^2 - 5x - 1) + 4) - 2))| + 2) + 5 and g(x) = -2((1(-2(3x - 3)^2 + 1(3x - 3) + 3) - 3)/(-3(-2(3x - 3)^2 + 1(3x - 3) + 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) = -2|((-2(-3|(-1|(-4x^2 - 2x - 2)| - 3) - 1| - 1) - 2)/(-3(-3|(-1|(-4x^2 - 2x - 2)| - 3) - 1| - 1) - 3)) - 1| + 2 and g(x) = -4(3cos(3π(2|(3x - 2) - 2| - 3)) + 1)^2 + 5(3cos(3π(2|(3x - 2) - 2| - 3)) + 1) + 1 intersect in the interval [-10, 10].\n\nPresen... | 600 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(1|(4(-2|(-3|x - 1| + 3) - 1| + 3)^2 + 5(-2|(-3|x - 1| + 3) - 1| + 3) + 4) + 3|) + 2| + 2 and g(x) = 1(5(2sin(2π(2x - 4) - 2) + 3)^2 - 3(2sin(2π(2x - 4) - 2) + 3) - 4)^2 - 2(5(2sin(2π(2x - 4) - 2) + 3)^2 - 3(2sin(2π(2x - 4)... | 76 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(2(-3|(-2sin(1π(-3|x + 1|) + 1) - 2) - 3| + 3)^2 + 4(-3|(-2sin(1π(-3|x + 1|) + 1) - 2) - 3| + 3) + 1)^2 + 5(2(-3|(-2sin(1π(-3|x + 1|) + 1) - 2) - 3| + 3)^2 + 4(-3|(-2sin(1π(-3|x + 1|) + 1) - 2) - 3| + 3) + 1) and g(x) = 4(-5... | 5 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|((3(5(-2sin(2π(3|x - 1| + 3) - 2) - 1) + 5) - 2)/(-1(5(-2sin(2π(3|x - 1| + 3) - 2) - 1) + 5) - 3)) - 2| - 2 and g(x) = 2(-2cos(1π(3(3|x + 1| + 2)^2 + 3(3|x + 1| + 2) + 4) - 1) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answer... | 47 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-2(-1|(-2(-2x + 1) - 1) - 2| - 1) + 5) - 1)/(2(-2(-1|(-2(-2x + 1) - 1) - 2| - 1) + 5) + 2) and g(x) = 2sin(3π(-4(1((2x - 2)/(-2x - 2))^2 + 5((2x - 2)/(-2x - 2)) + 1) + 2) - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex forma... | 274 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(-3(-3|((2(-1x^2 - 5x - 2))/(2(-1x^2 - 5x - 2) - 1)) + 1| + 3)^2 - 5(-3|((2(-1x^2 - 5x - 2))/(2(-1x^2 - 5x - 2) - 1)) + 1| + 3) + 2) - 1)/(-3(-3(-3|((2(-1x^2 - 5x - 2))/(2(-1x^2 - 5x - 2) - 1)) + 1| + 3)^2 - 5(-3|((2(-1x^2 - 5x -... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-2((-1(-2|(-1|x - 2| + 1) + 2| + 2) + 2)/(-1(-2|(-1|x - 2| + 1) + 2| + 2) - 1)) + 3) + 1| + 2 and g(x) = 2|(-5(-3(-3x^2 - 2x + 5)^2 + 1(-3x^2 - 2x + 5) + 1)^2 + 5(-3(-3x^2 - 2x + 5)^2 + 1(-3x^2 - 2x + 5) + 1) + 5) - 3| - 3 in th... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(1((2(-1sin(3π(-5x + 1) - 3) + 1) + 3)/(-3(-1sin(3π(-5x + 1) - 3) + 1) - 3)) - 4) + 5 and g(x) = -4(2|(-3(3x + 5) - 3) + 1| - 2)^2 + 1(2|(-3(3x + 5) - 3) + 1| - 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed... | 4 | MATH |
[
{
"content": "How many solutions does the equation (1((-1(2|(-2(1x^2 + 1x - 3)^2 + 3(1x^2 + 1x - 3) + 3) - 2|))/(2(2|(-2(1x^2 + 1x - 3)^2 + 3(1x^2 + 1x - 3) + 3) - 2|) + 1)) + 3)/(-2((-1(2|(-2(1x^2 + 1x - 3)^2 + 3(1x^2 + 1x - 3) + 3) - 2|))/(2(2|(-2(1x^2 + 1x - 3)^2 + 3(1x^2 + 1x - 3) + 3) - 2|) + 1)) - 3) = -2... | 247 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|((-2(-3(5(4x + 2)^2 + 5(4x + 2) - 3) - 1) - 2)/(1(-3(5(4x + 2)^2 + 5(4x + 2) - 3) - 1))) - 1| + 3 and g(x) = -4(2|(3|(-2cos(3πx + 2) + 1) + 1|) - 2| - 3) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\box... | 120 | MATH |
[
{
"content": "How many solutions does the equation (-1((2(4(1(1x - 3)^2 - 1(1x - 3) - 2)) + 2)/(1(4(1(1x - 3)^2 - 1(1x - 3) - 2)))) + 2)/(-2((2(4(1(1x - 3)^2 - 1(1x - 3) - 2)) + 2)/(1(4(1(1x - 3)^2 - 1(1x - 3) - 2))))) = (1(-1cos(2π(-4(4x + 3)^2 + 2(4x + 3) + 2) - 3) - 2) + 2)/(2(-1cos(2π(-4(4x + 3)^2 + 2(4x + ... | 1502 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(-1|((2((-1(2|x| - 3) + 3)/(-1(2|x| - 3) - 2)) - 3)/(3((-1(2|x| - 3) + 3)/(-1(2|x| - 3) - 2)) + 1)) - 3| + 1) - 3)/(2(-1|((2((-1(2|x| - 3) + 3)/(-1(2|x| - 3) - 2)) - 3)/(3((-1(2|x| - 3) + 3)/(-1(2|x| - 3) - 2)) + 1)) - 3| +... | 134 | MATH |
[
{
"content": "How many solutions does the equation 1(1(2((2((-2x + 2)/(-3x + 4)) - 1)/(2((-2x + 2)/(-3x + 4)) + 1))^2 - 3((2((-2x + 2)/(-3x + 4)) - 1)/(2((-2x + 2)/(-3x + 4)) + 1)) - 3)^2 + 2(2((2((-2x + 2)/(-3x + 4)) - 1)/(2((-2x + 2)/(-3x + 4)) + 1))^2 - 3((2((-2x + 2)/(-3x + 4)) - 1)/(2((-2x + 2)/(-3x + 4)) ... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3((2(3(-1(2x + 4)^2 + 2(2x + 4)) - 4))/(-2(3(-1(2x + 4)^2 + 2(2x + 4)) - 4) - 2))^2 - 1((2(3(-1(2x + 4)^2 + 2(2x + 4)) - 4))/(-2(3(-1(2x + 4)^2 + 2(2x + 4)) - 4) - 2)) + 5 and g(x) = (-3(-3sin(2π((3(-2x + 5) + 2)/(-1(-2x + 5))) - 3... | 14 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2|((2(1|(4((1x - 3)/(2x + 2)) - 3) + 1| - 1) - 3)/(-1(1|(4((1x - 3)/(2x + 2)) - 3) + 1| - 1) - 3)) - 3| - 1 and g(x) = 3|(-1|(-2(2e^(-1x) - 1) + 1) + 1| + 2) + 1| + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format:... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5((2(-5(5((1x - 3)/(-2x + 3))^2 - 5((1x - 3)/(-2x + 3)) + 5)^2 + 2(5((1x - 3)/(-2x + 3))^2 - 5((1x - 3)/(-2x + 3)) + 5) + 5) - 2)/(3(-5(5((1x - 3)/(-2x + 3))^2 - 5((1x - 3)/(-2x + 3)) + 5)^2 + 2(5((1x - 3)/(-2x + 3))^2 - 5((1... | 58 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(2((3((-2(5x^2 + 1x - 4) + 1)/(-2(5x^2 + 1x - 4))) + 1)/(-2((-2(5x^2 + 1x - 4) + 1)/(-2(5x^2 + 1x - 4))) + 1))^2 + 2((3((-2(5x^2 + 1x - 4) + 1)/(-2(5x^2 + 1x - 4))) + 1)/(-2((-2(5x^2 + 1x - 4) + 1)/(-2(5x^2 + 1x - 4))) + 1)) + 2)... | 6 | MATH |
[
{
"content": "How many solutions does the equation (3(-1(1|(-1|(-2cos(3πx + 3) - 3) + 2|) + 2| - 1)^2 + 2(1|(-1|(-2cos(3πx + 3) - 3) + 2|) + 2| - 1) + 3))/(1(-1(1|(-1|(-2cos(3πx + 3) - 3) + 2|) + 2| - 1)^2 + 2(1|(-1|(-2cos(3πx + 3) - 3) + 2|) + 2| - 1) + 3) + 1) = (-3(-3(4(-3|x + 2| + 2) - 3)^2 - 4(4(-3|x + 2| ... | 120 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-5(-5(1(3x - 2)^2 + 3(3x - 2) + 5) + 1)^2 - 4(-5(1(3x - 2)^2 + 3(3x - 2) + 5) + 1) - 1))/(1(-5(-5(1(3x - 2)^2 + 3(3x - 2) + 5) + 1)^2 - 4(-5(1(3x - 2)^2 + 3(3x - 2) + 5) + 1) - 1) - 2) and g(x) = -5((-3(1sin(2π(-1|x - 2| + 2) +... | 16 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(4((2(3cos(1π(-4x + 1) + 3) + 1))/(-3(3cos(1π(-4x + 1) + 3) + 1) - 1)) - 3)^2 + 2(4((2(3cos(1π(-4x + 1) + 3) + 1))/(-3(3cos(1π(-4x + 1) + 3) + 1) - 1)) - 3) and g(x) = (-2((1(3((3x - 1)/(-2x - 1))^2 + 2((3x - 1)/(-2x - 1)) - 2))/(-2(3((3x - 1)/(-2x -... | 6 | MATH |
[
{
"content": "How many solutions does the equation (-2(-1|((2((-3((-1x - 3)/(1x + 3)) + 2)/(-1((-1x - 3)/(1x + 3)) + 3)) - 3)/(-3((-3((-1x - 3)/(1x + 3)) + 2)/(-1((-1x - 3)/(1x + 3)) + 3)) + 2)) + 2|) - 3)/(2(-1|((2((-3((-1x - 3)/(1x + 3)) + 2)/(-1((-1x - 3)/(1x + 3)) + 3)) - 3)/(-3((-3((-1x - 3)/(1x + 3)) + 2)... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -5(3sin(3π(3|(-1|(2|x + 3| - 1) + 1| + 1)| - 3) - 1) + 2) - 1 and g(x) = -4(-1|(4(-1sin(1πx - 3) + 1) - 3) - 3| - 1)^2 - 3(-1|(4(-1sin(1πx - 3) + 1) - 3) - 3| - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your... | 157 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3cos(1π(-3(1(1|(2|x - 2| + 3) - 2| - 1)^2 - 2(1|(2|x - 2| + 3) - 2| - 1)) - 5)) and g(x) = 2cos(3π(3(3(-3|x + 1| + 1)^2 + 5(-3|x + 1| + 1) + 1))) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your ans... | 1322 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (2(1e^(-2(5(-1|(2|x - 1| + 2) + 3| + 1)^2 + 1)) - 1) - 3)/(2(1e^(-2(5(-1|(2|x - 1| + 2) + 3| + 1)^2 + 1)) - 1) + 1) and g(x) = -3(-2|((1(1x^2 + 2x - 3) - 3)/(-3(1x^2 + 2x - 3))) - 2|)^2 - 5(-2|((1(1x^2 + 2x - 3) - 3)/(-3(1x^2 + 2x ... | 5 | MATH |
[
{
"content": "How many solutions does the equation 1(1(-3|(2(-3cos(3πx - 1) + 3) - 4)| - 1) - 5) + 1 = (-2(-2cos(1π(2(5x + 5)^2 - 2(5x + 5) + 4) + 2) - 3) - 1)/(1(-2cos(1π(2(5x + 5)^2 - 2(5x + 5) + 4) + 2) - 3) + 3) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 380 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1sin(3π(-1(-2|(-1(-1x + 4) + 4) + 3|)^2 - 4(-2|(-1(-1x + 4) + 4) + 3|) + 2) + 2) and g(x) = (1((2(4(-3x + 3) + 2) - 2)/(-1(4(-3x + 3) + 2) - 3)) + 1)/(3((2(4(-3x + 3) + 2) - 2)/(-1(4(-3x + 3) + 2) - 3))) intersect in the interval [-10, 10].\n\nPresent ... | 1373 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(1(-1((-2(-3cos(1πx - 2) - 2) - 3)/(-2(-3cos(1πx - 2) - 2) - 2))^2 - 5((-2(-3cos(1πx - 2) - 2) - 3)/(-2(-3cos(1πx - 2) - 2) - 2))) + 2) - 4 and g(x) = 3cos(3π((3((3((1x + 1)/(2x - 3)) - 1)/(-3((1x + 1)/(2x - 3)) + 1)) + 2)/(-2((3(... | 40 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (3(-3sin(1π((-3(5(1|x + 3| - 2)^2 + 1(1|x + 3| - 2) - 1) - 3)/(3(5(1|x + 3| - 2)^2 + 1(1|x + 3| - 2) - 1))) - 2)) + 3)/(1(-3sin(1π((-3(5(1|x + 3| - 2)^2 + 1(1|x + 3| - 2) - 1) - 3)/(3(5(1|x + 3| - 2)^2 + 1(1|x + 3| - 2) - 1))) - 2)... | 58 | MATH |
[
{
"content": "How many solutions does the equation (1(4(-1cos(3π(-3|(-1x - 5) + 1|)))^2 - 1(-1cos(3π(-3|(-1x - 5) + 1|))) + 3) + 1)/(-1(4(-1cos(3π(-3|(-1x - 5) + 1|)))^2 - 1(-1cos(3π(-3|(-1x - 5) + 1|))) + 3) + 2) = -1cos(3π(-4(2|(-2|x + 3| - 3) + 3| + 3) + 2)) - 1 have for x in the interval [-10, 10]?\n\nPrese... | 450 | MATH |
[
{
"content": "How many solutions does the equation -3|((1((-1(3(-2|x + 1| - 1)^2 + 5(-2|x + 1| - 1) - 4) + 3)/(-1(3(-2|x + 1| - 1)^2 + 5(-2|x + 1| - 1) - 4))) - 2)/(-3((-1(3(-2|x + 1| - 1)^2 + 5(-2|x + 1| - 1) - 4) + 3)/(-1(3(-2|x + 1| - 1)^2 + 5(-2|x + 1| - 1) - 4))) + 2))| - 2 = 2((1((-1(1x + 1))/(-2(1x + 1) ... | 4 | MATH |
[
{
"content": "How many solutions does the equation -1((1((2((-1(4x - 2) + 2)/(-2(4x - 2))) + 3)/(-3((-1(4x - 2) + 2)/(-2(4x - 2))) + 2)))/(-1((2((-1(4x - 2) + 2)/(-2(4x - 2))) + 3)/(-3((-1(4x - 2) + 2)/(-2(4x - 2))) + 2)) - 1))^2 + 4((1((2((-1(4x - 2) + 2)/(-2(4x - 2))) + 3)/(-3((-1(4x - 2) + 2)/(-2(4x - 2))) +... | 14 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-2(2cos(2π(-3|(2|((1x - 2)/(-1x - 1)) - 1| + 1)| + 3) - 3) + 3) - 2)/(-2(2cos(2π(-3|(2|((1x - 2)/(-1x - 1)) - 1| + 1)| + 3) - 3) + 3) + 3) and g(x) = -3(2(2e^(1(-1x^2 + 3x - 3)) + 3) - 4)^2 + 1(2(2e^(1(-1x^2 + 3x - 3)) + 3) - 4) + 4 intersect in the i... | 100 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-1(-1(-2|(-3x^2 - 3x) - 1|) - 1)^2 + 5(-1(-2|(-3x^2 - 3x) - 1|) - 1) + 3) + 3)/(-2(-1(-1(-2|(-3x^2 - 3x) - 1|) - 1)^2 + 5(-1(-2|(-3x^2 - 3x) - 1|) - 1) + 3) + 3) and g(x) = (-3(1((3((3x - 3)/(-3x - 1)) - 2)/(-2((3x - 3)/(-3x - ... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (3(5(3|(4(1x)^2 + 5(1x) + 5) - 2| - 3)^2 + 1(3|(4(1x)^2 + 5(1x) + 5) - 2| - 3) - 3))/(1(5(3|(4(1x)^2 + 5(1x) + 5) - 2| - 3)^2 + 1(3|(4(1x)^2 + 5(1x) + 5) - 2| - 3) - 3)) and g(x) = 1((-2(2sin(3π(2|x + 2| - 3) - 1)))/(-2(2sin(... | 120 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1((-1((-1(-4(-2e^(-1x) + 3) - 5) - 3)/(-2(-4(-2e^(-1x) + 3) - 5) + 3)) + 2)/(1((-1(-4(-2e^(-1x) + 3) - 5) - 3)/(-2(-4(-2e^(-1x) + 3) - 5) + 3)) + 3)) + 1)/(-1((-1((-1(-4(-2e^(-1x) + 3) - 5) - 3)/(-2(-4(-2e^(-1x) + 3) - 5) + 3)) +... | 179 | MATH |
[
{
"content": "How many solutions does the equation -5(-2|(2cos(3π(3((3x + 3)/(2x + 3)) + 2) - 1) - 3) + 2| + 3) - 1 = 5((1(1(-5x^2 + 1x - 4)^2 + 1(-5x^2 + 1x - 4) - 1))/(-3(1(-5x^2 + 1x - 4)^2 + 1(-5x^2 + 1x - 4) - 1) - 3))^2 + 3((1(1(-5x^2 + 1x - 4)^2 + 1(-5x^2 + 1x - 4) - 1))/(-3(1(-5x^2 + 1x - 4)^2 + 1(-5x^2... | 92 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2(-2|(3(3(-2|x - 2| + 3)^2 - 5(-2|x - 2| + 3)) + 3)| + 2) + 5 and g(x) = -2|(2|(2|(-5x - 1) + 2| - 1) - 3| + 2) - 1| - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3sin(1π(3(2|(-1|(-1|x - 1| + 2)|) - 1| + 1)^2 - 3(2|(-1|(-1|x - 1| + 2)|) - 1| + 1) + 1) - 3) and g(x) = (-3(-5((3(-1x) + 1)/(3(-1x) + 1))^2 + 4((3(-1x) + 1)/(3(-1x) + 1)) + 2) + 3)/(1(-5((3(-1x) + 1)/(3(-1x) + 1))^2 + 4((3(-1x) +... | 1610 | MATH |
[
{
"content": "How many solutions does the equation 1cos(2π(1((-3(5(-5x^2 - 4x)^2 - 3(-5x^2 - 4x) - 3) + 3)/(-2(5(-5x^2 - 4x)^2 - 3(-5x^2 - 4x) - 3) - 2)) + 3) - 3) - 3 = -3|(4(1cos(3π(-3x + 5) + 1) + 1)^2 + 4(1cos(3π(-3x + 5) + 1) + 1) - 2) - 2| have for x in the interval [-10, 10]?\n\nPresent the answer in LaT... | 283 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4((-1((-1(1((-3x - 2)/(-3x - 3))^2 - 4((-3x - 2)/(-3x - 3)) + 2) - 3)/(-2(1((-3x - 2)/(-3x - 3))^2 - 4((-3x - 2)/(-3x - 3)) + 2))) - 1)/(3((-1(1((-3x - 2)/(-3x - 3))^2 - 4((-3x - 2)/(-3x - 3)) + 2) - 3)/(-2(1((-3x - 2)/(-3x - 3))^2... | 4 | MATH |
[
{
"content": "How many solutions does the equation -3|((2(2|(1(3|x + 1| - 2) - 2) - 2| - 1) + 1)/(1(2|(1(3|x + 1| - 2) - 2) - 2| - 1)))| - 3 = -3((1(-2|(2x^2 + 2x - 5) + 2| + 1) + 1)/(1(-2|(2x^2 + 2x - 5) + 2| + 1))) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your a... | 4 | MATH |
[
{
"content": "How many solutions does the equation (1(-2|(-2(5(1e^(2x) + 1)^2 + 4)^2 - 3(5(1e^(2x) + 1)^2 + 4) - 4) + 1| + 3))/(-3(-2|(-2(5(1e^(2x) + 1)^2 + 4)^2 - 3(5(1e^(2x) + 1)^2 + 4) - 4) + 1| + 3) + 1) = 1((-2(1sin(1π((2x - 1)/(2x + 3)) + 3) - 3) - 2)/(2(1sin(1π((2x - 1)/(2x + 3)) + 3) - 3) + 3))^2 - 1((-... | 34 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3cos(1π(3(-1((-3((-2x + 2)/(1x + 2)) + 2)/(1((-2x + 2)/(1x + 2)) - 2))^2 + 1((-3((-2x + 2)/(1x + 2)) + 2)/(1((-2x + 2)/(1x + 2)) - 2)) - 2) + 4) - 2) + 3 and g(x) = 3(-3|(1|(4x - 2)| - 3) + 1|)^2 + 1(-3|(1|(4x - 2)| - 3) + 1|) - 3 intersect in the int... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1|(-3sin(1π(-4(3|(1x^2 + 3x - 5) + 1| - 2) + 1) + 2) - 2) + 2| - 2 and g(x) = (1(-1|(3sin(2π(3|x - 2| + 2) - 3) + 1) + 1| - 1))/(-2(-1|(3sin(2π(3|x - 2| + 2) - 3) + 1) + 1| - 1) + 3) in the range -10 ≤ x ≤ 10?\n\nPresent the answer... | 1467 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (3(1|(-1|(2sin(3π(1x^2 + 1x + 4) + 1) + 3) + 2| + 1) + 2| + 2) + 1)/(-1(1|(-1|(2sin(3π(1x^2 + 1x + 4) + 1) + 3) + 2| + 1) + 2| + 2)) and g(x) = -4(1cos(3π(5(-4x^2 + 5x - 2)^2 - 3(-4x^2 + 5x - 2) + 4) + 3)) - 3 intersect in the interval [-10, 10].\n\nPr... | 910 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2|(-2(-2cos(2π(5(2|x - 2| - 1)^2 + 2)) - 2) + 5) - 3| + 3 and g(x) = 2((1(-2(-5x^2 - 1x)) - 1)/(2(-2(-5x^2 - 1x)) + 1)) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 203 | MATH |
[
{
"content": "How many solutions does the equation 3(1(-3cos(3π(-3(2x^2 - 5x - 1) - 1) - 2) - 1)^2 - 1(-3cos(3π(-3(2x^2 - 5x - 1) - 1) - 2) - 1) + 1) + 4 = 3|(-4(2e^(2(-1x^2 - 4x - 4)) + 1)) - 1| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 1545 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 2|((1(-1(1((3x + 3)/(3x - 1))^2 + 4((3x + 3)/(3x - 1)) - 5) - 1) + 2)/(-1(-1(1((3x + 3)/(3x - 1))^2 + 4((3x + 3)/(3x - 1)) - 5) - 1) + 2))| and g(x) = 2|((-1(-2sin(2π((1x + 3)/(-3x + 2)) + 1) + 1) - 1)/(1(-2sin(2π((1x + 3)/(-3x + 2... | 66 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-2|(1|((2((-1x + 1)/(3x + 1)) + 3)/(-1((-1x + 1)/(3x + 1)) - 2)) + 3|) - 2| + 1) + 1| - 2 and g(x) = (-2(-4(1sin(3π(-2x - 2) - 3) + 2)^2 - 2(1sin(3π(-2x - 2) - 3) + 2) + 3) + 3)/(3(-4(1sin(3π(-2x - 2) - 3) + 2)^2 - 2(1sin(3π(-2x - 2) - 3) + 2) + 3))... | 100 | MATH |
[
{
"content": "How many solutions does the equation -2|(-1|(-5(-3sin(1π(-1|x - 1| + 3) + 1) + 2) - 4) - 3| - 2)| - 1 = -3|(3(2cos(2π(-3x + 5)) - 2) + 2) + 1| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 62 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-4((1(-1(-4x^2 + 1x + 5) - 4) + 2)/(2(-1(-4x^2 + 1x + 5) - 4) - 1)) - 1)^2 - 1(-4((1(-1(-4x^2 + 1x + 5) - 4) + 2)/(2(-1(-4x^2 + 1x + 5) - 4) - 1)) - 1) - 1 and g(x) = 1|((1(-2|(-5x + 5)| + 2))/(-2(-2|(-5x + 5)| + 2) - 3)) - 2| + 3 intersect in the in... | 8 | MATH |
[
{
"content": "How many solutions does the equation (-2((-3(-4(1(2x^2 - 3)^2 - 3(2x^2 - 3) - 2)^2 + 1(1(2x^2 - 3)^2 - 3(2x^2 - 3) - 2) - 2))/(-3(-4(1(2x^2 - 3)^2 - 3(2x^2 - 3) - 2)^2 + 1(1(2x^2 - 3)^2 - 3(2x^2 - 3) - 2) - 2) + 4)) + 2)/(-2((-3(-4(1(2x^2 - 3)^2 - 3(2x^2 - 3) - 2)^2 + 1(1(2x^2 - 3)^2 - 3(2x^2 - 3)... | 1536 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 5((-3(2(-2(3|x - 2| - 2) - 2) - 5))/(-3(2(-2(3|x - 2| - 2) - 2) - 5) + 2))^2 + 3 and g(x) = 3|(-2sin(1π((1(5x - 5))/(-2(5x - 5) - 2)) - 1) - 2) + 3| + 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your ... | 11 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(3|(-1(2|((-1x + 3)/(2x - 1)) - 2| - 1)) - 1| - 1))/(3(3|(-1(2|((-1x + 3)/(2x - 1)) - 2| - 1)) - 1| - 1)) and g(x) = (3(-1sin(1π(5(-4x^2 + 4x + 1)^2 - 3(-4x^2 + 4x + 1) - 3) + 1)) + 1)/(1(-1sin(1π(5(-4x^2 + 4x + 1)^2 - 3(-4x^2 + 4x + 1) - 3) + 1)) + ... | 770 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(1sin(3π((2(-1|(-1x + 3) - 1| - 2) + 2)/(2(-1|(-1x + 3) - 1| - 2)))) + 3) - 1| and g(x) = 5((-1(1(5x^2 - 1x)))/(2(1(5x^2 - 1x)) - 1))^2 - 1((-1(1(5x^2 - 1x)))/(2(1(5x^2 - 1x)) - 1)) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in L... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(3π(-3(-1|(-1(-5x^2 + 1x - 2)^2 + 1(-5x^2 + 1x - 2) + 3) + 2| + 2) - 1)) - 1 and g(x) = (-2((3(1sin(1π(2x^2 + 5x + 4) - 3) + 2) - 1)/(3(1sin(1π(2x^2 + 5x + 4) - 3) + 2) + 3)))/(2((3(1sin(1π(2x^2 + 5x + 4) - 3) + 2) - 1)/(... | 677 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(-2((-1(-2sin(3π(-2|x + 1| - 2) - 1) + 2) + 3)/(3(-2sin(3π(-2|x + 1| - 2) - 1) + 2) + 1)) + 2) + 3)/(3(-2((-1(-2sin(3π(-2|x + 1| - 2) - 1) + 2) + 3)/(3(-2sin(3π(-2|x + 1| - 2) - 1) + 2) + 1)) + 2) + 3) and g(x) = 3|(-2cos(3π(-3|... | 51 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (-3(1(-1|(5(3x - 5)^2 + 5(3x - 5) + 1) + 1| + 3)^2 - 1(-1|(5(3x - 5)^2 + 5(3x - 5) + 1) + 1| + 3) - 3) - 1)/(-1(1(-1|(5(3x - 5)^2 + 5(3x - 5) + 1) + 1| + 3)^2 - 1(-1|(5(3x - 5)^2 + 5(3x - 5) + 1) + 1| + 3) - 3)) and g(x) = 1|(2cos(3π(3|((-2x + 2)/(-2x ... | 68 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1e^(1((-1(-1(5(1x - 1) + 5)^2 - 5(5(1x - 1) + 5) - 1) + 3)/(-1(-1(5(1x - 1) + 5)^2 - 5(5(1x - 1) + 5) - 1) + 2))) and g(x) = -5(-3(3(-2x^2 + 3x + 2)^2 - 5(-2x^2 + 3x + 2))^2 + 3(3(-2x^2 + 3x + 2)^2 - 5(-2x^2 + 3x + 2)) + 4)^... | 17 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(-4(2|((1(-1|x + 2| + 3))/(2(-1|x + 2| + 3) + 2))| - 3)^2 - 2(2|((1(-1|x + 2| + 3))/(2(-1|x + 2| + 3) + 2))| - 3) + 3) - 1)/(2(-4(2|((1(-1|x + 2| + 3))/(2(-1|x + 2| + 3) + 2))| - 3)^2 - 2(2|((1(-1|x + 2| + 3))/(2(-1|x + 2| + 3) ... | 64 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(1(-1(-1|(3x^2 + 3x) - 2|)^2 + 3(-1|(3x^2 + 3x) - 2|) - 5) + 1) + 3| + 2 and g(x) = -2|(1(1cos(2π((2x - 2)/(-3x)) - 3))^2 + 5(1cos(2π((2x - 2)/(-3x)) - 3)) - 4) + 3| - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex f... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1e^(2((-2(3|(3|(5x^2 + 5x - 2) - 2| + 1) - 1| - 2) - 3)/(-2(3|(3|(5x^2 + 5x - 2) - 2| + 1) - 1| - 2) - 1))) + 3 and g(x) = -5((-1(-1(-3x^2 - 2x - 4) + 5) + 1)/(-2(-1(-3x^2 - 2x - 4) + 5) + 3)) + 4 where -10 ≤ x ≤ 10.\n\nPres... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1|(2|(-1|(1|(-3cos(2πx - 2) + 2) - 2| - 3) - 1| - 1)| - 2) - 3| - 2 and g(x) = 2((-3((3(1|x + 3| + 1) - 1)/(-2(1|x + 3| + 1) - 3)))/(3((3(1|x + 3| + 1) - 1)/(-2(1|x + 3| + 1) - 3)) - 1)) - 3 where -10 ≤ x ≤ 10.\n\nPresent th... | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3((3(-1(2sin(3π(-4x - 1) + 3) - 3) - 5) + 1)/(3(-1(2sin(3π(-4x - 1) + 3) - 3) - 5))) - 2 and g(x) = (-3(-1(3((3x)/(1x + 3)) - 3)^2 - 5(3((3x)/(1x + 3)) - 3) - 3) - 3)/(-1(-1(3((3x)/(1x + 3)) - 3)^2 - 5(3((3x)/(1x + 3)) - 3) - 3) - ... | 4 | MATH |
[
{
"content": "How many solutions does the equation -2((-3(2(-2|(-4x + 5) + 3| + 2)^2 - 2) + 2)/(3(2(-2|(-4x + 5) + 3| + 2)^2 - 2) + 3)) + 1 = (2(-3(-3(3|x - 3| + 2) + 1) - 4) + 2)/(-3(-3(-3(3|x - 3| + 2) + 1) - 4)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-3((1(1(3e^(1(2x^2 + 3x - 1)) - 2)^2 - 1(3e^(1(2x^2 + 3x - 1)) - 2) - 2) - 2)/(-2(1(3e^(1(2x^2 + 3x - 1)) - 2)^2 - 1(3e^(1(2x^2 + 3x - 1)) - 2) - 2) + 3)) - 1)/(2((1(1(3e^(1(2x^2 + 3x - 1)) - 2)^2 - 1(3e^(1(2x^2 + 3x - 1)) - 2) - ... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3((3(-3sin(3π(1|((1x + 3)/(2x - 1)) - 1| - 3) + 2) - 1) + 3)/(1(-3sin(3π(1|((1x + 3)/(2x - 1)) - 1| - 3) + 2) - 1) - 3)) - 1 and g(x) = 3sin(2π((1(-5(-2|x| + 3) - 1) + 2)/(2(-5(-2|x| + 3) - 1) - 1)) - 1) + 2 in the range -10 ≤ x ≤... | 84 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(2(5(1|(5x + 1) - 3| + 1) - 1) - 4) + 5 and g(x) = 2(-1((-2(2cos(1πx + 2) + 2) - 1)/(-3(2cos(1πx + 2) + 2) + 4)) + 2) - 3 in the range -10 ≤ x ≤ 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) = -3|(-5(1((-1(5x^2 + 4x - 1) - 3)/(-2(5x^2 + 4x - 1))) + 3)^2 + 2(1((-1(5x^2 + 4x - 1) - 3)/(-2(5x^2 + 4x - 1))) + 3) + 3) + 1| - 2 and g(x) = -3|(-1|(-2sin(2π((-2x + 2)/(1x - 3)) + 1)) - 2| + 1) - 2| - 1 in the range -10 ≤ x ≤ 10?\... | 6 | MATH |
[
{
"content": "How many solutions does the equation (2((1((1(-2|(1x + 3) + 1| + 2) + 3)/(-3(-2|(1x + 3) + 1| + 2))) - 3)/(3((1(-2|(1x + 3) + 1| + 2) + 3)/(-3(-2|(1x + 3) + 1| + 2))) - 2)) + 3)/(3((1((1(-2|(1x + 3) + 1| + 2) + 3)/(-3(-2|(1x + 3) + 1| + 2))) - 3)/(3((1(-2|(1x + 3) + 1| + 2) + 3)/(-3(-2|(1x + 3) + ... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-1|((3(-3|(1cos(1πx - 3)) - 2| - 1) - 1)/(-1(-3|(1cos(1πx - 3)) - 2| - 1) + 2)) - 1| + 3) - 1| + 2 and g(x) = 5(-1|(5(2|x + 2| - 2) + 4) - 3|)^2 + 2(-1|(5(2|x + 2| - 2) + 4) - 3|) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in La... | 4 | MATH |
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