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.)
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?
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.
The tracks in a zoo form an equilateral triangle, in which the middle lines are drawn. A monkey ran away from its cage. Two guards try to catch the monkey. Will they be able to catch the monkey if all three of them can run only along the tracks, and the speed of the monkey and the speed of the guards are equal and they can always see each other?
The judges of an Olympiad decided to denote each participant with a natural number in such a way that it would be possible to unambiguously reconstruct the number of points received by each participant in each task, and that from each two participants the one with the greater number would be the participant which received a higher score. Help the judges solve this problem!
Is it possible for the mean of some 35 whole numbers to equal \(6.35\)?
Does there exist a natural number which, when divided by the sum of its digits, gives a quotient and remainder both equal to the number 2011?
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 \(\frac{1}{9}\).