Problems

What is the maximum number of kings, that cannot capture each other, which can be placed on a chessboard of size $8\times 8$ cells?

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.

• Eight schoolchildren solved $8$ tasks. It turned out that $5$ schoolchildren solved each problem. Prove that there are two schoolchildren, who solved every problem at least once.

• If each problem is solved by $4$ pupils, prove that it is not necessary to have two schoolchildren who would solve each problem.

$x_1$ 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).

Is there a sequence of natural numbers in which every natural number occurs exactly once, and for any $k=1,2,3,\dots$ the sum of the first $k$ terms of the sequence is divisible by $k$?

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 $a$.

Two ants crawled along their own closed route on a $7\times 7$ board. Each ant crawled only on the sides of the cells of the board and visited each of the 64 vertices of the cells exactly once. What is the smallest possible number of cell edges, along which both the first and second ants crawled?

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.

Is it possible to draw five lines from one point on a plane so that there are exactly four acute angles among the angles formed by them? Angles between not only neighboring rays, but between any two rays, can be considered.

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