Problems

Let $f(x)$ be a polynomial of degree $n$ with roots ${\alpha }_1,\dots ,{\alpha }_n$. We define the polygon $M$ as the convex hull of the points ${\alpha }_1,\dots ,{\alpha }_n$ on the complex plane. Prove that the roots of the derivative of this polynomial lie inside the polygon $M$.

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) $(x^2+x+1{)}^3$?

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 $\sqrt{2}$.

Find the largest and smallest values of the functions

a) $f_1(x)=a\cos x+b\sin x$; b) $f_2(x)=a{\cos }^{2}x+b\cos x\sin x+c{\sin }^{2}x$.

Prove that the function $\cos \sqrt{x}$ is not periodic.

Prove the formulae: $\arcsin (-x)=-\arcsin x$, $\arccos (-x)=\pi -\arccos x$.

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: $$0<\frac{x-y}{1+xy}<\frac{1}{\sqrt{3}}.$$

Old calculator I.

a) Suppose that we want to find $\sqrt[3]{x}$ ($x>0$) on a calculator that can find $\sqrt{x}$ in addition to four ordinary arithmetic operations. Consider the following algorithm. A sequence of numbers $\{y_n\}$ is constructed, in which $y_0$ is an arbitrary positive number, for example, $y_0=\sqrt{\sqrt{x}}$, and the remaining elements are defined by $y_{n+1}=\sqrt{\sqrt{x{y}_n}}$ ($n\ge 0$).

Prove that $\underset{n\to \mathrm{\infty }}{lim}{y}_n=\sqrt[3]{x}$.

b) Construct a similar algorithm to calculate the fifth root.

An iterative polyline serves as a geometric interpretation of the iteration process. To construct it, on the $Oxy$ plane, the graph of the function $f(x)$ is drawn and the bisector of the coordinate angle is drawn, as is the straight line $y=x$. Then on the graph of the function the points $$A_0(x_0,f(x_0)),A_1(x_1,f(x_1)),\dots ,A_n(x_n,f(x_n)),\dots$$ are noted and on the bisector of the coordinate angle – the points $$B_0(x_0,x_0),B_1(x_1,x_1),\dots ,B_n(x_n,x_n),\dots .$$ The polygonal line $B_0{A}_0{B}_1{A}_1\dots B_n{A}_n\dots$ is called iterative.

Construct an iterative polyline from the following information:

a) $f(x)=1+x/2$, $x_0=0$, $x_0=8$;

b) $f(x)=1/x$, $x_0=2$;

c) $f(x)=2x-1$, $x_0=0$, $x_0=\mathrm{1,125}$;

d) $f(x)=-3x/2+6$, $x_0=5/2$;

e) $f(x)=x^2+3x-3$, $x_0=1$, $x_0=0,99$, $x_0=1,01$;

f) $f(x)=\sqrt{1+x}$, $x_0=0$, $x_0=8$;

g) $f(x)=x^3/3-5x^2/x+25x/6+3$, $x_0=3$.

The sequence of numbers $a_n$ is given by the conditions $a_1=1$, $a_{n+1}=a_n+1/a_n^2$ ($n\ge 1$).

Is it true that this sequence is limited?