\(x^3+y^3=x-y\)

\(\Rightarrow x^3+y^3-x+y=0\)

\(\Rightarrow x^3-x+y^3+y=0\)

\(\Rightarrow x\left(x^2-1\right)+y\left(y^2+1\right)=0\)

But \(x,y\ge0\)

\(\Rightarrow x^2,y^2\ge0\Leftrightarrow x\left(x^2-1\right);y\left(y^2+1\right)\ge0\)

=> \(x\left(x^2-1\right)andy\left(y^2+1\right)\) aren't two numbers opposite

\(\Rightarrow x\left(x^2-1\right)+y\left(y^2+1\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}x\left(x^2-1\right)=0\\y\left(y^2+1\right)=0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}\left[{}\begin{matrix}x=0\\x^2-1=0\end{matrix}\right.\\\left[{}\begin{matrix}y=0\\y^2+1=0\end{matrix}\right.\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}\left[{}\begin{matrix}x=0\\x=\pm1\end{matrix}\right.\\\left[{}\begin{matrix}y=0\\y\in\phi\end{matrix}\right.\end{matrix}\right.\)

But \(x\ge0\Rightarrow\left[{}\begin{matrix}x=0\\x=1\end{matrix}\right.\)

\(\Rightarrow MaxA=x^2+y^2=1^2+0^2=1+0=1\)

We have:

\(4a^2+3ab-11b^2⋮5\)

But again, we have:

\(5\left(a^2+ab-2b^2\right)⋮5\)

\(\Rightarrow\left(5a^2+5ab-10b^2\right)-\left(4a^2+3ab-11b^2\right)⋮5\)

\(\Rightarrow5a^2-4a^2+5ab-3ab-10b^2+11b^2\)

\(\Rightarrow a^2+2ab+b^2=\left(a+b\right)^2⋮5\)

\(\Rightarrow\left(a^2+b^2\right)\left(a^2-b^2\right)=a^4-\left(ab\right)^2+\left(ab\right)^2-b^4\)

\(\Rightarrow a^4-b^4⋮5\)

Applying the properties of the final digit of a power, we have:

3⁴ⁿ⁺¹ \(\left(n\in N\right)\) = ...3

Thence inferred:

3⁴ⁿ⁺² = ...3 * 3 = ...9

3⁴ⁿ⁺³ = ...3 * ...9 = ...27 = ...7

3⁴ⁿ = ...3 : 3 = ...1

Last powers of 4n + 3 is 3⁹⁹ and first powers of 4n+3 is 3³

So the power of 3 when rased to 4n + 3 will have the last digit of 7. And in the above sequence there are:

\(\left(99-3\right):4+1=25\left(powers\right)\)

Answer: 25 powers

Everyday he read 10 words more than he read during the lesson before

So, after 30 lessons, the number of words he read more are:

10 . 30 = 300 words

And if we do not count the number of word he read more than, the number of words he read in 30 lessons is:

50 . 30 = 1500 words

So the total number of words he read in 30 lessons are:

1500 + 300 = 1800 [words]

I only know in near time.

\(f\left(x\right)=3x-4\)

\(\Rightarrow f\left(10\right)=3\cdot10-4=30-4=26\)

\(g\left(x\right)=x^2-2x+5\)

\(\Rightarrow g\left(-4\right)=\left(-4\right)^2-2\cdot\left(-4\right)+5=16-\left(-8\right)+5=16+8+5=24+5=29\)

\(\Rightarrow g\left(-4\right)+f\left(10\right)=29+26=55\)

x.3 + x.2 = 50

=> x.\(\left(3+2\right)\) = 50

=> x . 5 = 50

=> x = 50/5

=> x = 10

x . 5 + x . 3 = 120

=> x.\(\left(5+3\right)\) = 120

=> x . 8 = 120

=> x = 120/8

=> x = 15

When a = 80; b = 10; c = 30

=> a : b : c = 80 : 10 : 30 = 8: 1: 3

21 > 3

=> 21/3 > 1

5 < 9

=> 5/9 < 1

==> 21/3 > 5/9

The time from 11:15 to 11:30 is: 11:30 - 11:15 = 0:15 = 15 minutes

The number % of downloads from 11:15 - 11:30 are: 80% - 35% = 45%

So load speed is: 45% : 15 = 3%/minute

The number of remaining % is: 100% - 80% = 20%

So it will complete download after time is: 20% : 3% = 6minutes 40seconds

So a x b = 6 x 40 = 240

Put A = \(\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+...+\dfrac{1}{512}\)

\(\Leftrightarrow A=\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^9}\)

\(\Rightarrow2A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^8}\)

\(\Rightarrow2A-A=\left(1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^8}\right)-\left(\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^9}\right)\)

\(\Rightarrow A=1-\dfrac{1}{2^9}=1-\dfrac{1}{512}=\dfrac{511}{512}\)

Volume this soil cube  of is:

10 x 10 x 10 = 1000 [cm³]

The volume of each rectangular prism is:

2 x 4 x 5 = 40 [cm³]

So the number of rectangular prisms size of 2x4x5 made from the above ground is:

1000 : 40 = 25 [rectangular prisms]

30% of 80% of 20 is 30% of 16 is 16*30% = 4,8

=> x% of 60% of 40 is the same as 4,8

=> x% of 24 is 4,8

=> x% = 4,8 : 24

=> x% = 1/5

=> x = 20

So it is 20

This number is:

\(3^2\left(2\cdot5\right)=9.10=90\)

The number of pages in her notes is: 

40 . 240 = 9600 [word]

The total speed of writing of Chanel and Marcus is:

36 + 54 = 90 [word/minute]

Time for Chanel and Marcus to finish writing the book is:

9600 : 90 = 106m40s 

Answer: 106minutes 40seconds

Everyday have 24h; a hour have 60minutes

=> Everyday have:

24 . 60 = 1440 [minutes]

So everyday, there are more people on Earth are:

1440 . 155 = 223 200 [people]

So everyday, there are more 223 200 people on Earth

20 + 30 + 40 + 50 + 60

= 50 + 40 + 50 + 60

= 50 + 50 + 40 + 60

= 100 + 100

= 200

One-half of rectangular sports field is:

880: 2 = 440 [m]

The length of rectangular sports field is:

440 : \(\left(3+1\right)\cdot3=330\left(m\right)\)

The width of rectangular sports field is:

440 - 330 = 110 [m]

So the area of the field is:

330.110 = 36300 [m²]

Digits thousands have 2 choices are 1 or 2

Digits hundreds have 2 choices are 1 or 2 too.

Digits tens and Digits units have 2 choices too.

So The numbers of four digits containing two digits 1 and 2 are:

2.2.2.2 = 16 

The radius of this circle is:

\(\dfrac{\dfrac{24\pi}{3,14}}{2}=\dfrac{24}{2}=12\left(cm\right)\)

4 + 22 + 23 + 24 + ... + 220

we have:

22 + 23 + 24 + ... + 220 have 220 - 22 + 1 = 199

And the number of even number are:

\(\left(\text{220 - 20}\right):2+1=102\) [even number]

So the number of odd are:

199 - 102 = 97 [odd]

Because the number of odd is a odd 

=> Their sum are a odd

So 22 + 23 + ... + 220 is a odd

P is a odd

So P can't written as power of 2