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#### Extremal principle (other) , Partitions into pairs and groups bijections , Pigeonhole principle (other)

Ben noticed that all 25 of his classmates have a different number of friends in this class. How many friends does Ben have?

#### Pigeonhole principle (other) , Proof by contradiction

10 friends sent one another greetings cards; each sent 5 cards. Prove that there will be two friends who sent cards to one another.

#### Arithmetic of remainders , Pigeonhole principle (other)

Prove that in any group of 7 natural numbers – not necessarily consecutive – it is possible to choose three numbers such that their sum is divisible by 3.

#### Pigeonhole principle (other)

A class contains 38 pupils. Prove that within the class there will be at least 4 pupils born in the same month.

#### Pigeonhole principle (other) , Proof by contradiction

Is it possible to arrange 44 marbles into 9 piles, so that the number of marbles in each pile is different?

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