The topic of drawer principle 1. Proof: in any five integers, three numbers must be taken out, so that their sum can be divided by three 2. A school sent 204 students to plant 15301 trees on the mountain, among which at least one person planted 50 trees and at most one person planted 100 trees, proving that trees planted by at least five people were the same

The topic of drawer principle 1. Proof: in any five integers, three numbers must be taken out, so that their sum can be divided by three 2. A school sent 204 students to plant 15301 trees on the mountain, among which at least one person planted 50 trees and at most one person planted 100 trees, proving that trees planted by at least five people were the same

1. Certification:
There are only three possibilities for the remainder of any integer divided by 3: or divisible, the remainder is 0, or divisible, the remainder is 1 or 2. Therefore, we construct three drawers and place the numbers in the form of 3M, 3M + 1 and 3M + 2 respectively, where m is an integer. These three types of numbers can also be called remainder 0, remainder 1 and remainder 2
According to the drawer principle, there must be one box with [5 / 3] + 1 = 2 numbers with the same remainder about 3, then the other three boxes have 3 numbers, or belong to the same class, then the conclusion is obviously true. If two belong to one class and the other belongs to another class, then take one number from each of the three boxes with different remainder, then the sum of the three numbers must be a multiple of 3
The proposition is proved
2. Certification:
According to the number of trees planted 50,51,..., 100, 51 boxes are constructed
If there are exactly 4 students in each box, the total number of trees planted is
4*(50+51+...+100)=4*150*51/2=4*3825=15300

Drawer principle topic
Proof: in any 11 infinite decimals, there must be two decimals whose difference contains infinite numbers of 0 or 9

If all the decimals are the same from a certain place, then the difference between any two decimals is finite, and there can be no infinite number of zeros or nines

The principle of drawer
There are 20 balls with the same texture and size in one pocket, including 4 red balls, 6 yellow balls and 10 blue balls. How many balls can we take to ensure at least 5 balls of the same color?
There are 50 identical balls with editor's number in one pocket, 10 of which are labeled 1.2.3.4.5
1. At least how many can guarantee that there are two numbers with the same number?
2. At least how many can guarantee that there are four numbers with the same number?
3. At least how many can guarantee that there are 5 numbers with the same number?

Hello, wonderful apple
At least four red balls, four yellow balls and four blue balls will be taken. If you take any other ball, you will have at least five of the same color
So at least: 4 + 4 + 4 + 1 = 13
There are 50 identical balls with editor's number in one pocket, 10 of which are labeled 1.2.3.4.5
1. At least how many can guarantee that there are two numbers with the same number?
In the same way, take one number for each number, and then make sure that the two numbers are the same, so at least take 5 + 1 = 6
2. At least how many can guarantee that there are four numbers with the same number?
In the same way, take 3 numbers for each number, and then make sure that the 4 numbers are the same, so at least take 5x3 + 1 = 16
3. At least how many can guarantee that there are 5 numbers with the same number?
In the same way, take 4 numbers for each number, and then make sure that the 4 numbers are the same, so at least take 5X4 + 1 = 21
I wish you progress in your study! Don't forget to adopt the answer!