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 the isosceles triangle \(ABC\), the angle \(B\) is equal to \(30^{\circ}\), and \(AB = BC = 6\). The height \(CD\) of the triangle \(ABC\) and the height \(DE\) of the triangle \(BDC\) are drawn. Find the length \(BE\).
Three players are playing knockout table tennis – that is, the player who loses a game swaps places with the player who did not take part in that game and the winner stays on. In total Andrew played 10 games, Ben played 15, and Charlotte played 17. Which player lost the second game 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.
Valentina added a number (not equal to 0) taken to the power of four and the same number to the power two and reported the result to Peter. Can Peter determine the unique number that Valentina chose?
In a row there are 20 different natural numbers. The product of every two of them standing next to one another is the square of a natural number. The first number is 42. Prove that at least one of the numbers is greater than 16,000.
The numbers \(1, 2, 3,\dots , 10\) are written around a circle in a particular order. Peter calculated the sum of each of the 10 possible groups of three adjacent numbers around the circle and wrote down the smallest value he had calculated. What is the largest possible value he could have written down?
In a group of six people, any five can sit down at a round table so that every two neighbours know each other.
Prove that the entire group can be seated at the round table so that every two neighbours will know each other.
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?
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?