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#### Examples and counterexamples. Constructive proofs , Mathematical logic (other)

Among 4 people there are no three with the same name, the same middle name and the same surname, but any two people have either the same first name, middle name or surname. Can this be so?

#### 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.

#### Examples and counterexamples. Constructive proofs , Pigeonhole principle (finite number of poits, lines etc.)

The smell of a flowering lavender plant diffuses through a radius of 20m around it. How many lavender plants must be planted along a straight 400m path so that the smell of the lavender reaches every point on the path.

#### 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 [$h \times 1969^n$] is not divisible by [$h \times 1969^{n-1}$]?

#### 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 , Number tables and its properties , 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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