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?
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?
Two play the following game. There is a pile of stones. The first takes either 1 stone or 10 stones with each turn. The second takes either m or n stones with every turn. They take turns, beginning with the first player. He who can not make a move, loses. It is known that for any initial quantity of stones, the first one can always play in such a way as to win (for any strategy of the second player). What values can m and n take?
On the left bank of the river, there were 5 physicists and 5 chemists. All of them need to cross to the right bank. There is a two-seater boat. On the right bank at any time there can not be exactly three chemists or exactly three physicists. How do they all cross over by making 9 trips to the right side?
A group of children from two classes came to an after school club: Jack, Ben, Fred, Louis, Claudia, Janine and Charlie. To the question: “How many of your classmates are here?” everyone honestly answered with either “Two” or “Three”. But the boys thought that they were only being asked about the boy classmates, and the girls correctly understood that they were asking about everyone. Is Charlie a boy or a girl?
Hannah recorded the equality \(MA \times TE \times MA \times TI \times CA = 2016000\) and suggested that Charlie replace the same letters with the same numbers, and different letters with different digits, so that the equality becomes true. Does Charlie have the possibility of fulfilling the task?
Catherine laid out 2016 matches on a table and invited Anna and Natasha to play a game which involves taking turns to remove matches from a table: Anna can take 5 matches or 26 matches in her turn, and Natasha can take either 9 or 23. Without waiting for the start of the game, Catherine left, and when she returned, the game was already over. On the table there are two matches, and the one who can not make another turn loses. After a good reflection, Catherine realised which person went first and who won. Figure it out for yourself now.
At a round table, there are 10 people, each of whom is either a knight who always speaks the truth, or a liar who always lies. Two of them said: “Both my neighbors are liars,” and the remaining eight stated: “Both my neighbors are knights.” How many knights could there be among these 10 people?
There are 23 students in a class. During the year, each student of this class celebrated their birthday once, which was attended by some (at least one, but not all) of their classmates. Could it happen that every two pupils of this class met each other the same number of times at such celebrations? (It is believed that at every party every two guests met, and also the birthday person met all the guests.)
Chess board fields are numbered in rows from top to bottom by the numbers from 1 to 64. 6 rooks are randomly assigned to the board, which do not capture each other (one of the possible arrangements is shown in the figure). Find the mathematical expectation of the sum of the numbers of fields occupied by the rooks.