Hey Dao Trong Luan, if you see more strictly, you will see the number 4 appear 3 times, do the same with all the number left, then it must be 3 ( 2+3+...+40) 

We have:

The sum of all the angle in a convex polygon is 180 . ( n - 2 )(degrees) with n = the number of sides

=>180 . ( n - 2 ) = 9720

=> n - 2 = 54

So this is a 52-side polygon.

Is it answered?

Call the number is a, we have:

a = 8k + 7

a = 9k + 8

a = 12k + 11

=> a + 1 \(⋮8;9;12\)

and a is the smallest

So a + 1 = 72

=> a = 71

=> The value of the sum is:

(1+2+3+39+40+41) x 39 / 2 = 2457

We have: 

\(1342_8=1.8^3+3.8^2+4.8+2=738_{10}\)

We have: 

\(2007^{2008}=2007^{4.502}=\left(\overline{...1}\right)^{502}=\overline{...1}\)

\(2008^{2007}=2008^{4.501+3}=\left(\overline{...6}\right)^{501}+2008^3=\overline{...6}+\overline{...2}=\overline{...8}\)

\(\Rightarrow2007^{2008}+2008^{2007}=\overline{...1}+\overline{...8}=\overline{...9}\)

So the ones digit is 9

We have: 2,6,10,14,18,22,26,...

=> x = 18

y = 22

=> x + y = 18 + 22 = 40

We have:

x + 2x + 3x + ... + 99x + 100x = 100

=> 5050x = 100

=> x = \(\dfrac{100}{5050}=\dfrac{2}{101}\)

We have:

\(\dfrac{1}{a}.4=a\)

\(\Rightarrow a^2=4\Rightarrow a\in\left\{2;-2\right\}\)

\(\dfrac{1}{b}.9=b\Rightarrow b\in\left\{3;-3\right\}\)

\(\Rightarrow a+b\in\left\{5;-1;1;-5\right\}\)

We have:

The sum of four interior angles of the convex quadrilateral is \(360^0\)

\(\Rightarrow4x+3x+20^0+2x+40^0+x+80^0=360^0\)

\(\Rightarrow10x+140^0=360^0\)

\(\Rightarrow10x=220^0\)

\(\Rightarrow x=22^0\)

So the smallest angle is: \(2x+40^0=44^0+40^0=84^0\)

We have:

\(\overline{abc}+\overline{acb}=\overline{ccc}\)

\(\Rightarrow100a+10b+c+100a+10c+b=111c\)

\(\Leftrightarrow200a+11b=100c\)

We see: a different than 0

=> 200a different than 0

c different than 0

=> 100c different than 0

We also have: \(\left\{{}\begin{matrix}200a⋮100\\100c⋮100\end{matrix}\right.\Rightarrow11b⋮100\)

But 0\(\le\)b\(\le\)9

=> \(0\le11b\le99\)

=> 11b \(⋮\) 100 <=> b = 0

=> 200a = 100c

=> 2a = c

Because c<10

=> 2a < 10

=> a < 5 

\(\Rightarrow\left[{}\begin{matrix}a=1\Rightarrow c=2\\a=2\Rightarrow c=4\\a=3\Rightarrow c=6\\a=4\Rightarrow c=8\end{matrix}\right.\)

\(\Rightarrow\overline{abc}\in\left\{102;204;306;408\right\}\)

The number of alcohol in the first is:

400 x 64% = 256 (kg) 

=> There are 144 kg of water

When drained 100kg of the mixture or drained 25% of the mixture and replaced by 100kg of water, the total kg of the tank don't change.

The alcohol left is: 256 - 256 x 25% = 192 (kg)

The water is: 144 - 144 x 25% + 100 = 208 (kg)

After two steps, the water is: 208 - 208 x 25% + 100 = 256 (kg)

After three steps, the water is: 256 - 256 x 25% + 100 = 292 (kg)

=> The alcohol is: 400 - 292 = 108 (kg)

=> The percent is: 108 / 400 = 27% of the tank

We have: S(3;4) = \(3^2-4^2=9-16=-7\)

=> S (3;S(3;4)) = S (3;-7) = \(3^2-\left(-7\right)^2=9-49=-40\)

We have:

12 = 3 x 4 

So T \(⋮12\) <=> T \(⋮3and4\)

T \(⋮3\Leftrightarrow\)The sum of all the number of T \(⋮3\)

\(\Leftrightarrow\left(1+1+1+...+1+0+...+0\right)⋮3\)

But T's sum must be different than 0 

=> The smallest sum must be 3 = 1 + 1 + 1 + 0 + 0 + ... + 0

T divided by 4 <=> Two-last digit must be divided by 4

=> Two last digits must be 00

=> T = 11100

It's too big :) 

In the sequence, a must be the first number in the sequence and d must be the distance

We have: 10 = 1+9 = 2+8 = 3+7 = 4+6 = 5+5

So there are 10 pair of (a;d) counting the conversion 

It's not sure yet ^^

Call \(AD\perp BC=\left\{E\right\}\)

We have: \(a^2+b^2=13^2=169\)(Pythagorean theorem in triangle EAB)

We also have: 

\(\left(a+10\right)^2+\left(b+24\right)^2=39^2=1521\)

\(\Rightarrow a^2+20a+100+b^2+48b+576=1521\)

\(\Rightarrow a^2+20a+b^2+48b=1521-576-100=845\)

Because \(a^2+b^2=169\)

\(\Rightarrow20a+48b=845-169=676\)

\(\Rightarrow5a+12b=169\)

\(a^2+b^2=169\)

=> \(\left\{{}\begin{matrix}a^2=5a\\b^2=12b\end{matrix}\right.\)

\(\Rightarrow a=5;b=12\)

=> DE = 15

BE = 12 

=> \(S_{DBE}=\dfrac{15.12}{2}=90\left(units^2\right)\)
\(S_{ABE}=\dfrac{5.12}{2}=30\left(units^2\right)\)

\(\Rightarrow S_{ABCD}=90-30=60\left(units^2\right)\)

The number of male students is:

250 x 20% + 250 = 300 (students)

=> The number of male students tall over 6 feet is: 

300 x 10% = 30 ( students)

=> The number of female tall over 6 feet is

30 x 10% = 3 ( students) 

Call the number of sides of the polygon is a

Each polygon has the sum of all the interior angles is  \(180^0.\left(a-2\right)\)

We have: \(180^0.\left(n-2\right)=9720^0\)

\(\Rightarrow a-2=54\)

=> a = 52 

So there are 52-side polygon =]]

To have the smallest number of people, the 55% and 60% must be in 90% who eat hotdogs

=> The smallest is: 800 x 90% = 720 (people)

Call the hexagon's side is x

the circle's radius is y

=> The hexagon's perimeter is 6x

The circle's perimeter is 6,28y

If 6x = 6,28y

=> \(\dfrac{x}{y}=\dfrac{6,28}{6}=\dfrac{3,14}{3}=\dfrac{\Pi}{3}\)