\(\dfrac{x+y}{2}=12\Leftrightarrow x+y=24\Rightarrow x=24-y\)

\(\dfrac{y+12}{2}=\dfrac{z}{2}\Leftrightarrow y+12=z\)

=> \(\dfrac{x+z}{2}=\dfrac{24-y+y+12}{2}=\dfrac{36}{2}=18\)

So the mean of x and z is 18

\(\left\{{}\begin{matrix}a\le a^2\\b\le b^2\\c\le c^2\end{matrix}\right.\Rightarrow a^2+b^2+c^2\ge a+b+c\ge abc\)

So \(a^2+b^2+c^2\ge abc\)

The answer is: \(\left|\left(4x+9\right)-\left(4x-3\right)\right|=\left|4x+9-4x+3\right|=\left|12\right|=12\)

\(\dfrac{a+b}{2}=8\Leftrightarrow a+b=16\)

\(\dfrac{b+c}{2}=16\Leftrightarrow b+c=32\)

\(\dfrac{a+c}{2}=14\Leftrightarrow a+c=28\)

\(\Rightarrow a+b+b+c+a+c=16+32+28=76\)

\(\Leftrightarrow2\left(a+b+c\right)=76\Leftrightarrow a+b+c=38\)

\(40\%\cdot\dfrac{1}{3}\cdot24:0,8\)

\(=\dfrac{2}{5}\cdot\dfrac{1}{3}\cdot24:0,8\)

\(=\dfrac{2}{15}\cdot24:0,8\)

\(=\dfrac{48}{15}:0,8=4\)

99999 times 55555 is 99999.55555 = 5555444445

The sum marbles now is:

20 + 40 = 60 [marbles]

The sum marbles after add purple marbles is:

40 : 5 . 12 = 96 [green marbles]

The number of purple marbles after add purple marbles is:

96 - 40 = 56 [marbles]

So we need add:

56 - 20 = 36 [purple] to 5/12 of the marbles will be green

64,8 pounds = 1036,8 ounces

=> the edge length is: \(\dfrac{1036,8}{0,6}\cdot1=1728\) inch

The answer: \(\dfrac{2}{3}\left(3\dfrac{1}{8}-\left(-1,3\right)\right)=\dfrac{59}{20}=2,95\)

5 ziggles + 4 zoggles = 30 zaggles

2 ziggles + 3 zoggles = 19 zaggles

=> 7 ziggles + 7 zoggles = 49 zaggles

=> \(7\left(ziggles+zoggles\right)=49zaggles\)

=> 1 ziggles + 1 zoggles = 7 zaggles

Answer: 7 zaggles

4! + 5! = n!3!

=> 24 + 120 = n!.6

=> 144 = n!.6

=> n! = 24

=> n = 4

Each jump the dog closer the rabbit is:

0.9 - 0.7 = 0.2 [m]

So the dog will catched the rabbit after:

150 : 0.2 = 750 [steps]

Answer: 750 steps

If Jesse doesn't let out any number, their sum is:

1+2+3+4+5+6+7+8+9+10 = 55

The biggest and nearest square number compare to 55 is 49

but 55 - 49 = 6

So this number is 6

The area of rectangle is: 5.4 = 20 [cm²]

The area of each small square is: \(\left(5-4\right)\left(5-4\right)=1\) [cm²]

So the area of the dodecagon is: 20 - 1.4 = 16 [cm²]

123,456 is a integer. The answer must be:

\(\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{123456-6}{10}-5}{10}-4}{10}-3}{10}-2}{10}=\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{123450}{10}-5}{10}-4}{10}-3}{10}-2}{10}\)\(=\dfrac{\dfrac{\dfrac{\dfrac{12345-5}{10}-4}{10}-3}{10}-2}{10}=\dfrac{\dfrac{\dfrac{\dfrac{12340}{10}-4}{10}-3}{10}-2}{10}=\dfrac{\dfrac{\dfrac{1234-4}{10}-3}{10}-2}{10}=\dfrac{\dfrac{\dfrac{1230}{10}-3}{10}-2}{10}=\dfrac{\dfrac{123-3}{10}-2}{10}=\dfrac{\dfrac{120}{10}-2}{10}=\dfrac{12-2}{10}=\dfrac{10}{10}=1\)

2ⁿ = 2.2.2....2 with n factor 2

we see: 

2 split 3 residual 2

2.2 = 4 split 3 residual 1

2.2.2 = 8 split 3 residual 2

.........................................

So if n is odd, 2ⁿ will split 3 residual 2

=> 2ⁿ - 2 is a multiple of 3

So 2ⁿ - 2 is a multiple of 3 then 2\(\text{|}̸\)n

2ⁿ = 2.2.2....2 with n factor 2

we see: 

2 split 3 residual 2

2.2 = 4 split 3 residual 1

2.2.2 = 8 split 3 residual 2

.........................................

So if n is even number, 2ⁿ will split 3 residual 1

=> 2ⁿ - 1 is a multiple of 3

So 2ⁿ - 1 is a multiple of 3 then 2 | n

The volume of sugar syrup that Ms.Maryna made was:

10: 25% = 40 [lbs]

If the sugar soup have 40% sugar, the volume of sugar is:

40.40% = 16 [lbs]

So she need add:

16 - 10 = 6 [lbs]

The answer is: \(\left|\dfrac{1}{2}-\dfrac{1}{3}\right|=\left|\dfrac{1}{3}-\dfrac{1}{2}\right|=\left|\dfrac{1}{6}\right|=\dfrac{1}{6}\)

Tell a,b sequence is the length of small and large paper.

=> 5a = 2b

=> 8a = 2b : 5 . 8 = \(\dfrac{16}{5}b=3\dfrac{1}{5}b\)

So the answer is \(3\dfrac{1}{5}\)