Problems

We consider a function $y=f(x)$ defined on the whole set of real numbers and satisfying $f(x+k)\times (1-f(x))=1+f(x)$ for some number $k\ne 0$. Prove that $f(x)$ is a periodic function.

The function $f(x)$ for each real value of $x\in (-\mathrm{\infty },+\mathrm{\infty })$ satisfies the equality $f(x)+(x+1/2)\times f(1-x)=1$.

a) Find $f(0)$ and $f(1)$. b) Find all such functions $f(x)$.

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.

Let $z_1$ and $z_2$ be fixed points of a complex plane. Give a geometric description of the sets of all points $z$ that satisfy the conditions:

a) $\arg \frac{z-z_1}{z-z_2}=0$;

b) $\arg \frac{z_1-z}{z-z_2}=0$.

The numerical function $f$ is such that for any $x$ and $y$ the equality $f(x+y)=f(x)+f(y)+80xy$ holds. Find $f(1)$ if $f(0.25)=2$.