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

We are given a polynomial P $(x)$ and numbers $a_1$, $a_2$, $a_3$, $b_1$, $b_2$, $b_3$ such that $a_1a_2a_3$ ≠ 0. It turned out that P $(a_1x + b_1)$ + P $(a_2x + b_2)$ = P $(a_3x + b_3)$ for any real x. Prove that P $(x)$ has at least one real root.

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

Given a square trinomial f $(x)$ = $x^2$ + ax + b. It is known that for any real x there exists a real number y such that f $(y)$ = f $(x)$ + y. Find the greatest possible value of a.

#### Iterations , Mathematical induction (other) , Processes and operations , Rational and irrational numbers , Theory of algorithms (other)

With a non-zero number, the following operations are allowed: $x \rightarrow \frac{1+x}{x}, x \rightarrow \frac{1-x}{x}$. Is it true that from every non-zero rational number one can obtain each rational number with the help of a finite number of such operations?

#### 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 $)$.

#### Theory of algorithms (other)

Solve problem number 108736 for the inscription A, BC, DEF, CGH, CBE, EKG.

#### 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.

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