Problems

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?

$x_1$ 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).

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?

A number set $M$ contains $2003$ distinct positive numbers, such that for any three distinct elements $a,b,c$ in $M$, the number $a^2+bc$ is rational. Prove that we can choose a natural number $n$ such that for any $a$ in $M$ the number $a\sqrt{n}$ is rational.

Given a square trinomial $f(x)=x^2+ax+b$. It is known that for any real $x$ there exists a real number $y$ such that $f(y)=f(x)+y$. Find the greatest possible value of $a$.

We are given a polynomial $P(x)$ and numbers $a_1$, $a_2$, $a_3$, $b_1$, $b_2$, $b_3$ such that $a_1{a}_2{a}_3\ne 0$. It turned out that $P(a_{1}x+b_1)+P(a_{2}x+b_2)=P(a_{3}x+b_3)$ for any real $x$. Prove that $P(x)$ has at least one real root.

It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.

Prove that multiplying the polynomial $(x+1{)}^{n-1}$ by any polynomial different from zero, we obtain a polynomial having at least $n$ nonzero coefficients.

Prove that there are infinitely many composite numbers among the numbers $\lfloor 2^k\sqrt{2}\rfloor$ ($k=0,1,\dots$).