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 \(BE\).
Authors: B.R. Frenkin, T.V. Kazitcina
On the tree sat 100 parrots of three kinds: green, yellow, multi-coloured. A crow flew past and croaked: “Among you, there are more green parrots than multi-coloured ones!” – “Yes!” – agreed 50 parrots, and the others shouted “No!”. Glad to the dialogue, the crow again croaked: “Among you, there are more multi-coloured parrots than yellow ones!” Again, half of the parrots shouted “Yes!”, and the rest – “No!”. The green parrots both told the truth, the yellow ones lied both times, and each of the multi-coloured ones lied once, and once told the truth. Could there be more yellow than green parrots?
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?
Author: E.V. Bakaev
From the beginning of the academic year, Andrew wrote down his marks for mathematics. When he received another evaluation (2, 3, 4 or 5), he called it unexpected, if before that time this mark was met less often than each of the other possible marks. (For example, if he had received the following marks: 3, 4, 2, 5, 5, 5, 2, 3, 4, 3 from the beginning of the year, the first five and the second four would have been unexpected). For the whole academic year, Andrew received 40 marks - 10 fives, fours, threes and twos (it is not known in which order). Is it possible to say exactly how many marks were unexpected?
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.
In a line 40 signs are written out: 20 crosses and 20 zeros. In one move, you can swap any two adjacent signs. What is the least number of moves in which it is guaranteed that you can ensure that some 20 consecutive signs are crosses?
To a certain number, we add the sum of its digits and the answer we get is 2014. Give an example of such a number.
Author: E.V. Bakaev
After a hockey match Anthony said that he scored 3 goals, and Ilya only one. Ilya said that he scored 4 goals, and Serge scored 5 goals. Serge said that he scored 6 goals, and Anthony only two. Could it be that the three of them scored 10 goals, if it is known that each of them once told the truth, and once lied?
Find all of the solutions of the puzzle: \(ARKA + RKA + KA + A = 2014\). (Different letters correspond to different numbers, and the same letters correspond to the same numbers.)
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?