Author: A.V. Shapovalov
To the cabin of the cable car leading up to the mountain, four people arrived who weigh 50, 60, 70 and 90 kg. A supervisor does not exist, but the cable car travels back and forth in automatic mode only with a load from 100 to 250 kg (in particular, it does not go anywhere when the cable car is empty), provided that passengers can be seated on two benches so that the weights on the benches differ by no more than 25 kg. How can they all climb the mountain?
Author: N. Medved
Peter and Victoria are playing on a board measuring \(7 \times 7\). They take turns putting the numbers from 1 to 7 in the board cells so that the same number does not appear in one line nor in one column. Peter goes first. The player who loses is the one who cannot make a move. Who of them can win, no matter how the opponent plays?
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.
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?
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?
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?
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?
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.