Problems

Can there exist two functions $f$ and $g$ that take only integer values such that for any integer $x$ the following relations hold:

a) $f(f(x))=x$, $g(g(x))=x$, $f(g(x))>x$, $g(f(x))>x$?

b) $f(f(x))<x$, $g(g(x))<x$, $f(g(x))>x$, $g(f(x))>x$?

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.

Peter has 28 classmates. Each 2 out of these 28 have a different number of friends in the class. How many friends does Peter have?

To each pair of numbers $x$ and $y$ some number $x\ast y$ is placed in correspondence. Find $1993\ast 1935$ if it is known that for any three numbers $x,y,z$, the following identities hold: $x\ast x=0$ and $x\ast (y\ast z)=(x\ast y)+z$.

$x_1$ 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).

In the number $a=0.12457\dots$ the $n$th digit after the decimal point is equal to the digit to the left of the decimal point in the number. Prove that $\alpha$ is an irrational number.

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?

Find all functions $f(x)$ defined for all positive $x$, taking positive values and satisfying the equality $f(x^y)=f(x{)}^f(y)$ for any positive $x$ and $y$.

The function $f(x)$ is defined and satisfies the relationship $(x-1)f((x=1)/(x-1))-f(x)=x$ for all $x\ne 1$. Find all such functions.

At all rational points of the real line, integers are arranged. Prove that there is a segment such that the sum of the numbers at its ends does not exceed twice the number on its middle.