In the cabinet of Anchuria there are 100 ministers. Among them there are honest and dishonest ministers. It is known that out of any ten ministers, at least one minister is dishonest. What is the smallest number of dishonest ministers there could be in the cabinet?
An ant goes out of the origin along a line and makes \(a\) steps of one unit to the right, \(b\) steps of one unit to the left in some order, where \(a > b\). The wandering span of the ant is the difference between the largest and smallest coordinates of the ant for the entire length of its journey.
a) Find the largest possible wandering range.
b) Find the smallest possible range.
c) How many different sequences of motion of the ant are there, where the wandering range is the greatest possible?
A square is divided into triangles (see the figure). How many ways are there to paint exactly one third of the square? Small triangles cannot be painted partially.
We will assume that the birth of a girl and a boy is equally probable. It is known that in some family there are two children.
a) What is the probability that one of them is a boy and one a girl?
b) Additionally, it is known that one of the children is a boy. What is the probability that there is one boy and one girl in the family now?
c) Additionally, it is known that the boy was born on a Monday. What is the probability that there is one boy and one girl in the family now?
Every day, Patrick the dog chews one slipper from the available stock in the house. Strictly with a probability of 0.5 Patrick wants to chew the left slipper, and with a probability of 0.5 – the right one. If the desired slippers are not present, Patrick becomes upset. How many pairs of the same slippers need to be bought, so that with a probability of not less than 0.8 Patrick does not get upset for an entire week (7 days)?
Find the probability that heads will fall an even number of times, in an experiment in which:
a) a symmetrical coin is thrown \(n\) times;
b) a coin is thrown \(n\) times, for which the probability of getting heads in one throw is \(p(0 < p < 1)\).
In Anchuria, presidential elections are being prepared, in which President Miraflores wants to win. Exactly half of the voters support Miraflores, and the other half support Dick Maloney. Miraflores is also a voter. According to the law, he has the right to divide all of the voters into two constituencies at his own discretion. In each of the districts, the voting is conducted as follows: each voter marks the name of their candidate on the ballot; all ballots are placed in the ballot box. Then one random ballot is chosen from the ballot box, and the one whose name is marked on it will win in this district. The candidate wins the election only if he wins in both districts. If the winner does not appear, the next round of voting is appointed according to the same rules. How should Miraflores divide the electorate in order to maximize the probability of his victory on the first round?
What is the minimum number of \(1\times 1\) squares that need to be drawn in order to get an image of a \(25\times 25\) square divided into 625 smaller 1x1 squares?
What is the smallest number of cells that can be chosen on a \(15\times15\) board so that a mouse positioned on any cell on the board touches at least two marked cells? (The mouse also touches the cell on which it stands.)
a) There are three identical large vessels. In one there are 3 litres of syrup, in the other – 20 litres of water, and the third is empty. You can pour all the liquid from one vessel into another or into a sink. You can choose two vessels and pour into one of them liquid from the third, until the liquid levels in the selected vessels are equal. How can you get 10 litres of diluted 30% syrup?
b) The same, but there is \(N\) l of water. At what integer values of \(N\) can you get 10 liters of diluted 30% syrup?