In a one-on-one tournament 10 chess players participate. What is the least number of rounds after which the single winner could have already been determined? (In each round, the participants are broken up into pairs. Win – 1 point, draw – 0.5 points, defeat – 0).
16 teams took part in a handball tournament where a victory was worth 2 points, a draw – 1 point and a defeat – 0 points. All teams scored a different number of points, and the team that ranked seventh, scored 21 points. Prove that the winning team drew at least once.
In a school football tournament, 8 teams participate, each of which plays equally well in football. Each game ends with the victory of one of the teams. A randomly chosen by a draw number determines the position of the teams in the table:
What is the probability that teams \(A\) and \(B\):
a) will meet in the semifinals;
b) will meet in the finals.
In the magical land of Anchuria there is a drafts championship made up of several rounds. The days and cities in which the rounds are carried out are determined by a draw. According to the rules of the championship, no two rounds can take place in one city, and no two rounds can take place on one day. Among the fans, a lottery is arranged: the main prize is given to those who correctly guess, before the start of the championship, in which cities and on which days all of the round will take place. If no one guesses, then the main prize will go to the organising committee of the championship. In total, there are eight cities in Anchuria, and the championship is only allotted eight days. How many rounds should there be in the championship, so that the organising committee is most likely to receive the main prize?
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?
Monica is in a broken space buggy at a distance of 18 km from the Lunar base, in which Rachel sits. There is a stable radio communication system between them. The air reserve in the space buggy is enough for 3 hours, in addition, Monica has an air cylinder for the spacesuit, with an air reserve of 1 hour. Rachel has a lot of cylinders with an air supply of 2 hours each. Rachel can not carry more than two cylinders at the same time (one of them she uses herself). The speed of movement on the Moon in the suit is 6 km/h. Could Rachel save Monica and not die herself?
301 schoolchildren came to the school’s New Year’s party in the city of Moscow. Some of them always tell the truth, and the rest always lie. Each of some 200 students said: “If I leave the hall, then among the remaining students, the majority will be liars.” Each of the other schoolchildren said: “If I leave the room, then among the remaining students, there will be twice as many liars as those who speak the truth.” How many liars were at the party?
The pupils of class 5A had a total of 2015 pencils. One of them lost a box with five pencils, and instead bought a box with 50 pencils. How many pencils do the pupils of class 5A now have?
What is the largest number of horses that can be placed on an \(8\times8\) chessboard so that no horse touches more than seven of the others?