Problems

With a non-zero number, the following operations are allowed: $x\to \frac{1+x}{x}$, $x\to \frac{1-x}{x}$. 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?

Let $f(x)$ be a polynomial about which it is known that the equation $f(x)=x$ has no roots. Prove that then the equation $f(f(x))=x$ does not have any roots.

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

Prove that for a monotonically increasing function $f(x)$ the equations $x=f(f(x))$ and $x=f(x)$ are equivalent.

Author: A.K. Tolpygo

An irrational number $\alpha$, where $0<\alpha <\frac{1}{2}$, is given. It defines a new number ${\alpha }_1$ as the smaller of the two numbers $2\alpha$ and $1-2\alpha$. For this number, ${\alpha }_2$ is determined similarly, and so on.

a) Prove that for some $n$ the inequality ${\alpha }_n<3/16$ holds.

b) Can it be that ${\alpha }_n>7/40$ for all positive integers $n$?