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#### Divisibility of a number. General properties , Examples and counterexamples. Constructive proofs , Integer and fractional parts. Archimedean property

Does there exist a number $h$ such that for any natural number n the number [$h \times 2021^n$] is not divisible by [$h \times 2021^{n-1}$]?

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

What is the largest amount of numbers that can be selected from the set 1, 2, …, 1963 so that the sum of any two numbers is not divisible by their difference?

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

The triangle $C_1C_2O$ is given. Within it the bisector $C_2C_3$ is drawn, then in the triangle $C_2C_3O$ – bisector $C_3C_4$ and so on. Prove that the sequence of angles $γ_n$ = $C_{n + 1}C_nO$ tends to a limit, and find this limit if $C_1OC_2$ = α.

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

In each cell of a board of size $5\times5$ a cross or a nought is placed, and no three crosses are positioned in a row, either horizontally, vertically or diagonally. What is the largest number of crosses on the board?

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