Solve the equation \(x^3 - \lfloor x\rfloor = 3\).
A table of \(4\times4\) cells is given, in some cells of which a star is placed. Show that you can arrange seven stars so that when you remove any two rows and any two columns of this table, there will always be at least one star in the remaining cells. Prove that if there are fewer than seven stars, you can always remove two rows and two columns so that all the remaining cells are empty.
Prove that in a group of 11 arbitrary infinitely long decimal numbers, it is possible to choose two whose difference contains either, in decimal form, an infinite number of zeroes or an infinite number of nines.
At what value of \(k\) is the quantity \(A_k = (19^k + 66^k)/k!\) at its maximum?
In draughts, the king attacks by jumping over another draughts-piece. What is the maximum number of draughts kings we can place on the black squares of a standard \(8\times 8\) draughts board, so that each king is attacking at least one other?
The number \(n\) has the property that when it is divided by \(q^2\) the remainder is smaller than \(q^2 / 2\), whatever the value of \(q\). List all numbers that have this property.
Does there exist a number \(h\) such that for any natural number \(n\) the number \(\lfloor h \times 2021^n\rfloor\) is not divisible by \(\lfloor h \times 2021^{n-1}\rfloor\)?
Prove that for every convex polyhedron there are two faces with the same number of sides.
A spherical sun is observed to have a finite number of circular sunspots, each of which covers less than half of the sun’s surface. These sunspots are said to be enclosed, that is no two sunspots can touch, and they do not overlap with one another. Prove that the sun will have two diametrically opposite points that are not covered by sunspots.
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.