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?

On a table, there are five coins lying in a row: the middle one lies with a head facing upwards, and the rest lie with the tails side up. It is allowed to simultaneously flip three adjacent coins. Is it possible to make all five coins positioned with the heads side facing upwards with the help of several such overturns?

Three hedgehogs divided three pieces of cheese of mass of 5g, 8g and 11g. The fox began to help them. It can cut off and eat 1 gram of cheese from any two pieces at the same time. Can the fox leave the hedgehogs equal pieces of cheese?

A rectangle is cut into several smaller rectangles, the perimeter of each of which is an integer number of meters. Is it true that the perimeter of the original rectangle is also an integer number of meters?

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?