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.
A tennis tournament takes place in a sports club. The rules of this tournament are as follows. The loser of the tennis match is eliminated (there are no draws in tennis). The pair of players for the next match is determined by a coin toss. The first match is judged by an external judge, and every other match must be judged by a member of the club who did not participate in the match and did not judge earlier. Could it be that there is no one to judge the next match?
On the school board a chairman is chosen. There are four candidates: \(A\), \(B\), \(C\) and \(D\). A special procedure is proposed – each member of the council writes down on a special sheet of candidates the order of his preferences. For example, the sequence \(ACDB\) means that the councilor puts \(A\) in the first place, does not object very much to \(C\), and believes that he is better than \(D\), but least of all would like to see \(B\). Being placed in first place gives the candidate 3 points, the second – 2 points, the third – 1 point, and the fourth - 0 points. After collecting all the sheets, the election commission summarizes the points for each candidate. The winner is the one who has the most points.
After the vote, \(C\) (who scored fewer points than everyone) withdrew his candidacy in connection with his transition to another school. They did not vote again, but simply crossed out \(B\) from all the leaflets. In each sheet there are three candidates left. Therefore, first place was worth 2 points, the second – 1 point, and the third – 0 points. The points were summed up anew.
Could it be that the candidate who previously had the most points, after the self-withdrawal of \(B\) received the fewest points?
Four outwardly identical coins weigh 1, 2, 3 and 4 grams respectively.
Is it possible to find out in four weighings on a set of scales without weights, which one weighs how much?
Three cyclists travel in one direction along a circular track that is 300 meters long. Each of them moves with a constant speed, with all of their speeds being different. A photographer will be able to make a successful photograph of the cyclists, if all of them are on some part of the track which has a length of \(d\) meters. What is the smallest value of \(d\) for which the photographer will be able to make a successful photograph sooner or later?
There was a football match of 10 versus 10 players between a team of liars (who always lie) and a team of truth-tellers (who always tell the truth). After the match, each player was asked: “How many goals did you score?” Some participants answered “one”, Callum said “two”, some answered “three”, and the rest said “five”. Is Callum lying if it is known that the truth-tellers won with a score of 20:17?
There are 13 weights, each weighing an integer number of grams. It is known that any 12 of them can be divided into two cups of weights, six weights on each one, which will come to equilibrium. Prove that all the weights have the same weight.