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

A numerical sequence is defined by the following conditions:
$a_1 = 1, \,\,\, a_{n+1} = a_n + ⌊\sqrt{a_n}⌋$
$\\$
Prove that among the terms of this sequence there are an infinite number of complete squares.

#### Decimal number system , Proof by exhaustion

Write the first 10 prime numbers in a line. How can you remove 6 digits to get the largest possible number?

#### Division with remainder , Proof by exhaustion

Find all of the natural numbers that, when divided by 7, have the same remainder and quotient.

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

Is it possible to place the numbers $1, 2,…12$ around a circle so that the difference between any two adjacent numbers is 3, 4, or 5?

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

What has a greater value: 300! or $100^{300}$?

#### Proof by exhaustion , Puzzles

Specify any solution of the puzzle: 2014 + YES =BEAR.

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

Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.

#### Partitions into pairs and groups bijections , Pigeonhole principle (other) , Proof by exhaustion

In any group of 10 children, out of a total of 60 pupils, there will be three who are in the same class. Will it always be the case that amongst the 60 pupils there will be:

1) 15 classmates?

2) 16 classmates?

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