We have : 2³⁰⁰ = (2³)¹⁰⁰ = 8¹⁰⁰

                 3²⁰⁰ = (3²)¹⁰⁰ = 9¹⁰⁰ 
For 9¹⁰⁰ > 8¹⁰⁰ then 3¹⁰⁰ > 2³⁰⁰

We have : \(5^{30}=\left(5^3\right)^{10}=125^{10}\left(1\right)\)

                 \(3^{50}=\left(3^5\right)^{10}=243^{10}\left(2\right)\)

From (1) and (2) we have 125¹⁰ < 243¹⁰ then 5³⁰ < 3⁵⁰

The term for P is

         (500 - 1) + 1 = 500 (skull)

Total P is : 

         (500 + 1) x 500 : 2 = 125250

                    Answer : 125250

The term for S is : 

         (50 - 1) + 1 = 50 (skull)

General S is 

          (50 + 1) x 50 : 2 = 1275 

               Answer : 1275

We have : \(\left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)\left(1-\dfrac{1}{4}\right)......\left(1-\dfrac{1}{2017}\right)\)

\(=\dfrac{1}{2}.\dfrac{2}{3}.\dfrac{3}{4}......\dfrac{2016}{2017}\)

\(=\dfrac{1}{2017}\)

\(\text{After 20 people arrive, the unit has all:}\)
                \(\text{ 50 + 20 = 70 (people) }\)

70 people are 50 times as many:
                 70: 50 = 7/5 (times)
70 people eat that rice in the number of days are:
                10 x 7/5 = 14 (day) (corresponding to 14 rice grades)
The additional rice capacity is:
                 14 - 10 = 4 (productivity)
                    \(\text{Answer: 4 rates}\)

After 20 people arrive, the unit has all:
                 50 + 20 = 70 (people)
70 people are 50 times as many:
                 70: 50 = 7/5 (times)
70 people eat that rice in the number of days are:
                10 x 7/5 = 14 (day) (corresponding to 14 rice grades)
The additional rice capacity is:
                 14 - 10 = 4 (productivity)
                    Answer: 4 rates

We have the following formula: If a number n divided by a, surplus b then (n - b) is divisible by a.
Example: 14 divide 3 by 2, and 14 - 2 = 12 divide by 3
Return to the problem Call the number to find n.
We have n - 7 divisible by 11, 13, 17, which is divisible by 11 x 13 x 17 = 2431
Since 2431 is not the 4 digit number, we increase n - 7 by 9999 9999: 2431 = 4 by 275.
Let n - 7 = 2431 x 4 = 9724.
Vâỵn = 9724 + 7 = 9731

Motorcycles ahead of time cars are:

                    9 hours - 8 hours 24 minutes = 36 minutes

          Change: 36 minutes = 0.6 hours

When cars start to go, two cars are separated:

                   36 x 0.6 = 21.6 (km)

The velocity of the two vehicles is:

                   54 - 36 = 18 (km / h)

Automobile catch up motorcycles after the time is:

                   21.6: 18 = 1.2 (hour)

The long AB distance is:

                   54 x 1.2 = 64.8 (km)

                      Answer ; 64,8 km

Motorcycles ahead of time cars are:

                    9 hours - 8 hours 24 minutes = 36 minutes

          Change: 36 minutes = 0.6 hours

When cars start to go, two cars are separated:

                   36 x 0.6 = 21.6 (km)

The velocity of the two vehicles is:

                   54 - 36 = 18 (km / h)

Automobile catch up motorcycles after the time is:

                   21.6: 18 = 1.2 (hour)

The long AB distance is:

                   54 x 1.2 = 64.8 (km)

                      Answer ; 64,8 km

Motorcycles ahead of time cars are:

                    9 hours - 8 hours 24 minutes = 36 minutes

          Change: 36 minutes = 0.6 hours

When cars start to go, two cars are separated:

                   36 x 0.6 = 21.6 (km)

The velocity of the two vehicles is:

                   54 - 36 = 18 (km / h)

Automobile catch up motorcycles after the time is:

                   21.6: 18 = 1.2 (hour)

The long AB distance is:

                   54 x 1.2 = 64.8 (km)

                      Answer ; 64,8 km

We have\(\dfrac{1}{2.4}+\dfrac{1}{4.6}+\dfrac{1}{6.8}+......+\dfrac{1}{98.100}\)

\(=\dfrac{1}{2}\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+\dfrac{2}{6.8}+......+\dfrac{2}{98.100}\right)\)

\(=\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+......+\dfrac{1}{98}-\dfrac{1}{100}\right)\)

\(=\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{100}\right)\)

\(=\dfrac{1}{2}.\dfrac{49}{100}=\dfrac{49}{200}\)

We have : \(\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+.....+\dfrac{1}{9900}\)

\(=\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+.....+\dfrac{1}{99.100}\)

\(=\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+.....+\dfrac{1}{99}-\dfrac{1}{100}\)

\(=\dfrac{1}{2}-\dfrac{1}{100}\)

\(=\dfrac{49}{100}\)

Land k = \(\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{6}\)

Then  \(k^3=\dfrac{x}{2}.\dfrac{y}{3}.\dfrac{z}{6}=\dfrac{xyz}{2.3.6}=\dfrac{36}{36}=1\)

Candle : \(k=1\)

Deduced \(\dfrac{x}{2}=1=>x=2\)

               \(\dfrac{y}{3}=1\Rightarrow y=3\)

               \(\dfrac{z}{6}=1\Rightarrow z=6\)

Dress x = 2;y=3;z=6

\(\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+......+\dfrac{1}{1999.2001}\)

\(=\dfrac{1}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+.......+\dfrac{1}{1999}-\dfrac{1}{2001}\right)\)

\(=\dfrac{1}{2}\left(1-\dfrac{1}{2001}\right)\)

\(=\dfrac{1}{2}.\dfrac{2000}{2001}\)

\(=\dfrac{1000}{2001}\)

\(\dfrac{1}{1.5}+\dfrac{1}{5.9}+\dfrac{1}{9.13}+.......+\dfrac{1}{1991.1995}\)

\(=\dfrac{1}{4}\left(\dfrac{4}{1.5}+\dfrac{4}{5.9}+\dfrac{4}{9.13}+.......+\dfrac{4}{1991.1995}\right)\)

\(=\dfrac{1}{4}\left(1-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{9}+.......+\dfrac{1}{1991}-\dfrac{1}{1995}\right)\)

\(=\dfrac{1}{4}\left(1-\dfrac{1}{1995}\right)\)

\(=\dfrac{1}{4}.\dfrac{1994}{1995}\)

\(=\dfrac{997}{3990}\)

We have : \(\dfrac{1}{1.4}+\dfrac{1}{4.7}+\dfrac{1}{7.10}+.......+\dfrac{1}{100.103}\)

\(\dfrac{1}{3}\left(\dfrac{3}{1.4}+\dfrac{3}{4.7}+\dfrac{3}{7.10}+.......+\dfrac{3}{100.103}\right)\)

\(=\dfrac{1}{3}\left(1-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+.......+\dfrac{1}{100}-\dfrac{1}{103}\right)\)

\(=\dfrac{1}{3}\left(1-\dfrac{1}{103}\right)\)

\(=\dfrac{1}{3}.\dfrac{102}{103}=\dfrac{34}{103}\)

The volume of a cubic cubic by 1cm is:
         1 x 1 x 1 = 1 (cm3)
The volume of the large cube is:
         1 x 2197 = 2197 (cm3)
Because 13 x 13 x 13 = 2197 => the large cubic side is 13 cm
                 Answer: 13 cm

Edge is:
           24: 4 = 6 (dm)
The oil level of the tank is:
          6 x 3/4 = 4.5 (dm)
The tank volume of the tank is:
          6 x 6 x 4.5 = 162 (dm2) = 162 liters
Weight is:
         0.9 x 162 = 145.8 (kg)

The circumference of that wheel is:
         650 x 3.14 = 2041 (mm) = 0.002041 km.
The wheel roll number to go all the way is:
         2,041: 0.002041 = 1000 (round)
                   Answer: 1000 rounds.