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?
The numbers \(x\), \(y\) and \(z\) are such that all three numbers \(x + yz\), \(y + zx\) and \(z + xy\) are rational, and \(x^2 + y^2 = 1\). Prove that the number \(xyz^2\) is also rational.
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?
A group of several friends was in correspondence in such a way that each letter was received by everyone except for the sender. Each person wrote the same number of letters, as a result of which all together the friends received 440 letters. How many people could be in this group of friends?
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.
In the Republic of mathematicians, the number \(\alpha > 2\) was chosen and coins were issued with denominations of 1 pound, as well as in \(\alpha^k\) pounds for every natural \(k\). In this case \(\alpha\) was chosen so that the value of all the coins, except for the smallest, was irrational. Could it be that any amount of a natural number of pounds can be made with these coins, using coins of each denomination no more than 6 times?