Author: A.V. Khachaturyan
The mum baked some pies – three with peach, three with kiwi and one with blackberries – and laid them on the dish in a circle (see the picture). Then she put the dish in a microwave to warm it up. All of the pies look the same. Maria knows how they lie on the dish but does not know how the dish turned in the microwave. She wants to eat a pie with blackberries, but she doesn’t want any of the others because she doesn’t like their taste. How can Maria surely achieve this by biting as few tasteless pies as possible?
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.)
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)?
Author: N.K. Agakhanov
On the board, the equation \(xp^3 + * x^2 + * x + * = 0\) 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?
Peter marks several cells on a \(5 \times 5\) board. His friend, Richard, will win if he can cover all of these cells with non-overlapping corners of three squares, that do not overlap with the border of the square (you can only place the corners on the squares). What is the smallest number of cells that Peter should mark so that Richard cannot win?
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?