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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?

#### Division with remainder , Periodicity and aperiodicity , Probability theory (other) , Proof by exhaustion

A numerical sequence is defined by the following conditions:
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How many complete squares are found among the first members of this sequence, not exceeding 1,000,000?

#### Division with remainder , Periodicity and aperiodicity , Probability theory (other) , Proof by exhaustion

A numerical sequence is defined by the following conditions:

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Prove that among the terms of this sequence there are an infinite number of complete squares.

#### 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.

#### Equations in integer numbers , Integer and fractional parts. Archimedean property

PFind the number of solutions in natural numbers of the equation [x / 10] = [x / 11] + 1.

#### 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.

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