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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.

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)

Identical to 98634.

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