8

Part of 2024 Assara - South Russian Girl's MO

Problems(2)

Chosing any 10 (9) we can add 1 such a their sum is divisible by 10

Source: Assara 2024 J8

S

2024

natural numbers satisfying the following condition: if you select any

10

(different) numbers from

S

, then you can select another number from

S

so that the sum of all

11

selected numbers is divisible by

10

. Prove that one of the numbers can be thrown out of

S

so that the resulting set

S'

2023

numbers satisfies the condition: if you choose any

9

(different) numbers from

S'

, then you can choose another number from

S'

so that the sum of all

10

selected numbers is divisible by

10

Setscombinatorics

Friendly classmates

Source: Assara 2024 S8

15

15

girls in the class. The first girl is friends with

4

boys, the second with

5

, the third with

6

11

14

, and each of the other four girls is friends with all the boys. It turned out that there are exactly

3 \cdot 2^{25}

ways to split the entire class into pairs, so that each pair has a boy and a girl who are friends. Prove that any of the friends of the first girl are friends with all the other girls too. G.M.Sharafetdinova