a) A square of area 6 contains three polygons, each of area 3. Prove that among them there are two polygons that have an overlap of area no less than 1.
b) A square of area 5 contains nine polygons of area 1. Prove that among them there are two polygons that have an overlap of area no less than \(\frac{1}{9}\).
How many are there six-digit numbers that are divisible by \(5\)?
In a volleyball tournament teams play each other once. A win gives the team 1 point, a loss 0 points. It is known that at one point in the tournament all of the teams had different numbers of points. How many points did the team in second last place have at the end of the tournament, and what was the result of its match against the eventually winning team?
An endless board is painted in three colours (each cell is painted in one of the colours). Prove that there are four cells of the same colour, located at the vertices of the rectangle with sides parallel to the side of one cell.
On a plane, six points are given so that no three of them lie on the same line. Each pair of points is connected by a blue or red segment.
Prove that among these points three such points can be chosen so that all sides of the triangle formed by them will be of the same colour.
Out of two mathematicians and ten economists, it is necessary to form a committee made up of eight people. In how many ways can a committee be formed if it has to include at least one mathematician?
On two parallel lines \(a\) and \(b\), the points \(A_1, A_2, \dots , A_m\) and \(B_1, B_2, \dots , B_n\) are chosen, respectively, and all of the segments of the form \(A_iB_j\), where \(1 \leq i \leq m\), \(1 \leq j \leq n\). How many intersection points will there be if it is known that no three of these segments intersect at one point?
Find \(m\) and \(n\) knowing the relation \(\binom{n+1}{m+1}: \binom{n+1}{m}:\binom{n+1}{m-1} = 5:5:3\).
Write at random a two-digit number. What is the probability that the sum of the digits of this number is 5?
A player in the card game Preferans has 4 trumps, and the other 4 are in the hands of his two opponents. What is the probability that the trump cards are distributed a) \(2: 2\); b) \(3: 1\); c) \(4: 0\)?