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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.

#### Examples and counterexamples. Constructive proofs , Pigeonhole principle (other) , Theory of algorithms (other)

The function F is given on the whole real axis, and for each x the equality holds: F $(x + 1)$ F $(x)$ + F $(x + 1)$ + 1 = 0.
Prove that the function F can not be continuous.

#### 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)

Identical to 98634.

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