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).
A castle is surrounded by a circular wall with nine towers, at which there are knights on duty. At the end of each hour, they all move to the neighbouring towers, each knight moving either clockwise or counter-clockwise. During the night, each knight stands for some time at each tower. It is known that there was an hour when at least two knights were on duty at each tower, and there was an hour when there was precisely one knight on duty on each of exactly five towers. Prove that there was an hour when there were no knights on duty on one of the towers.
A chequered strip of \(1 \times N\) is given. Two players play the game. The first player puts a cross into one of the free cells on his turn, and subsequently the second player puts a nought in another one of the cells. It is not allowed for there to be two crosses or two noughts in two neighbouring cells. The player who is unable to make a move loses.
Which of the players can always win (no matter how their opponent played)?
On a ring road at regular intervals there are 25 posts, each with a policeman. The police are numbered in some order from 1 to 25. It is required that they cross the road so that there is a policeman on each post, but so that number 2 was clockwise behind number 1, number 3 was clockwise behind number 2, and so on. Prove that if you organised the transition so that the total distance travelled was the smallest, then one of the policemen will remain at his original post.
A pack of 36 cards was placed in front of a psychic face down. He calls the suit of the top card, after which the card is opened, shown to him and put aside. After this, the psychic calls out the suit of the next card, etc. The task of the psychic is to guess the suit as many times as possible. However, the card backs are in fact asymmetrical, and the psychic can see in which of the two positions the top card lies. The deck is prepared by a bribed employee. The clerk knows the order of the cards in the deck, and although he cannot change it, he can prompt the psychic by having the card backs arranged in a way according to a specific arrangement. Can the psychic, with the help of such a clue, ensure the guessing of the suit of
a) more than half of the cards;
b) no less than 20 cards?
Every day, James bakes a square cake size \(3\times3\). Jack immediately cuts out for himself four square pieces of size \(1\times1\) with sides parallel to the sides of the cake (not necessarily along the \(3\times3\) grid lines). After that, Sarah cuts out from the rest of the cake a square piece with sides, also parallel to the sides of the cake. What is the largest piece of cake that Sarah can count on, regardless of Jack’s actions?
A unit square is divided into \(n\) triangles. Prove that one of the triangles can be used to completely cover a square with side length \(\frac{1}{n}\).
Carry out the following experiment 10 times: first, toss a coin 10 times in a row and record the number of heads, then toss the coin 9 times in a row and again, record the number of heads. We call the experiment successful, if, in the first case, the number of heads is greater than in the second case. After conducting a series of 10 such experiments, record the number of successful and unsuccessful experiments. Collect the statistics in the form of a table.
a) Anton throws a coin 3 times, and Tina throws it two times. What is the probability that Anton gets more heads than Tina?
b) Anton throws a coin \(n + 1\) times, and Tanya throws it \(n\) times. What is the probability that Anton gets more heads than Tina?
Is it possible to:
a) load two coins so that the probability of “heads” and “tails” were different, and the probability of getting any of the combinations “tails, tails,” “heads, tails”, “heads, heads” be the same?
b) load two dice so that the probability of getting any amount from 2 to 12 would be the same?
Anna, Boris and Fred decided to go to a children’s Christmas party. They agreed to meet at the bus stop, but they do not know who will come to what time. Each of them can come at a random time from 15:00 to 16:00. Fred is the most patient of them all: if he comes and finds that neither Anna nor Boris are at the bus stop, then Fred will wait for one of them for 15 minutes, and if he waits for more than 15 minutes and no one arrives he will go to the Christmas party by himself. Boris is less patient: he will only wait for 10 minutes. Anna is very impatient: she will not wait at all. However, if Boris and Fred meet, they will wait for Anna until 16:00. What is the probability that all of them will go to the Christmas party?