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#### 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.

#### 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) , 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) , Theory of algorithms (other)

Cut the interval [-1, 1] into black and white segments so that the integrals of any a$)$ linear function; b$)$ a square trinomial in white and black segments are equal.

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

a$)$ Using geometric considerations, prove that the base and the side of an isosceles triangle with an angle of 36 $^{\circ}$ at the vertex are incommensurable.

b$)$ Invent a geometric proof of the irrationality of $\sqrt{2}$.

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

Let f $(x)$ be a polynomial of degree n with roots $α_1$, …, $α_n$. We define the polygon M as the convex hull of the points $α_1$, …, $α_n$ on the complex plane. Prove that the roots of the derivative of this polynomial lie inside the polygon M.

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

Let f $(x)$ = $(x – a)$ $(x – b)$ $(x – c)$ be a polynomial of the third degree with complex roots a, b, c.
Prove that the roots of the derivative of this polynomial lie inside the triangle with vertices at the points a, b, c.

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

Using the theorem on rational roots of a polynomial $($ see Problem 61013 $)$, prove that if p / q is rational and cos $(p / q)$ $^{\circ}$ ≠ 0, ± 1, ± 1, then
cos $(p / q)$ $^{\circ}$ is irrational.

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

It is known that cos α$^{\circ}$ = 1/3. Is α a rational number?

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

Prove that there are infinitely many composite numbers among the numbers [$2^k$ $\sqrt{2}$] $(k = 0, 1, …)$.

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

$x_1$ is the real root of the equation $x^2$ + ax + b = 0, $x_2$ is the real root of the equation $x^2$ – ax – b = 0.
Prove that the equation $x^2$ + 2ax + 2b = 0 has a real root, enclosed between $x_1$ and $x_2$. $($ a and b are real numbers $)$.

#### Boundedness, monotonicity , Divisibility of a number. General properties , Examples and counterexamples. Constructive proofs , Identical transformations , Sequnces

Prove that for any natural number $a_1> 1$ there exists an increasing sequence of natural numbers $a_1, a_2, a_3$, …, for which $a_1^2+ a_2^2 +…+ a_k^2$ is divisible by $a_1+ a_2+…+ a_k$ for all k ≥ 1.

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