Problems

Author: A.V. Khachaturyan

Replace the letters of the word $MATEMATIKA$ with numbers and signs of addition and subtraction so that a numeric expression equal to 2014 is obtained.

(The same letters denote the same numbers or signs, different letters denote different numbers or signs. Note that it is enough to give an example.)

Author: N.K. Agakhanov

On the board, the equation $x{p}^3+\ast x^2+\ast x+\ast =0$ is written. Peter and Vlad take turns to replace the asterisks with rational numbers: first, Peter replaces any of the asterisks, then Vlad – any of the two remaining ones, and then Peter replaces the remaining asterisk. Is it true that for any of Vlad’s actions, Peter can get an equation in which the difference of some two roots is equal to 2014?

Author: M.A. Khachaturyan

Mum baked identical pies with the same appearance: 7 with cabbage, 7 with meat and one with cherries, and laid them out in a circle on a round dish in this order. Then she put the dish into a microwave and to warm up the pies. Olga knows how she originally arranged the pies, but she does not know the dish turned in the microwave. She wants to eat a pie with cherries, and she thinks that the rest are tasteless. How does Olga surely achieve this, after biting into no more than three tasteless pies?

Ali Baba followed by 40 robbers lined up on the crossing across the Bosporus Strait. There is only one boat and in it there can be either two or three people (there cannot be one person in the boat). Among those in the boat there should not be people who are not friends with each other. Will all of them be able to cross, if every two people standing next to each other in the queue are friends, while Ali Baba is also friends with the robber standing behind the person next to him?

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?

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?

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.)