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#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

Is there a line on the coordinate plane relative to which the graph of the function $y = 2^x$ is symmetric?

#### Measurement of segments and angles. Adjacent angles. , Pigeonhole principle (angles and lengths)

You are given 7 straight lines on a plane, no two of which are parallel. Prove that there will be two lines such that the angle between them is less than $26^{\circ}$ .

#### Mean values , Pigeonhole principle (angles and lengths) , Quadratic inequalites (several variables) , Statistics

Condition
The point O, lying inside the triangle ABC, is connected by segments with the vertices of the triangle. Prove that the variance of the set of angles AOB, AOC and BOC is less than

a) $10π^2/27$;

b) $2π^2/9$.

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

Find the general formula for the coefficients of the series
$(1 – 4x)^{ ½} = 1 + 2x + 6x^2 + 20x^3 + … + a_nx^n + …$

#### Geometric interpretations in algebra , Pigeonhole principle (angles and lengths) , Trigonometric substitutions

Prove that amongst any 7 different numbers it is always possible to choose two of them, $x$ and $y$, so that the following inequality was true:
$$0 < \frac{x-y}{1+xy} < \frac{1}{\sqrt3}$$

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

For what values of n does the polynomial $(x+1)^n$ – $x^n$ – 1 divide by:

a$)$ $x^2$ + x + 1; b$)$ $(x^2 + x + 1)^2$; c$)$ $(x^2 + x + 1)^3$?

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

Find the rational roots of the following polynomials:

a$)$ $x^5$ – $2x^4$ – $4x^3$ + $4x^2$ – 5x + 6

b$)$ $x^5$ + $x^4$ – $6x^3$ – $14x^2$ – 11x – 3

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

Is it possible to draw from some point on a plane n tangents to a polynomial of n-th power?

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

Could it be that a) $σ(n) > 3n;$ b) $σ(n) > 100n?$

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