Hannah placed 101 counters in a row which had values of 1, 2 and 3 points. It turned out that there was at least one counter between every two one point counters, at least two counters lie between every two two point counters, and at least three counters lie between every two three point counters. How many three point counters could Hannah have?
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)?
A polynomial of degree \(n > 1\) has \(n\) distinct roots \(x_1, x_2, \dots , x_n\). Its derivative has the roots \(y_1, y_2, \dots , y_{n-1}\). Prove the inequality \[\frac{x_1^2 + \dots + x_n^2}{n}> \frac{y_1^2 + \dots + y_n^2}{n}.\]
We are given 111 different natural numbers that do not exceed 500. Could it be that for each of these numbers, its last digit coincides with the last digit of the sum of all of the remaining numbers?
The number \(x\) is such a number that exactly one of the four numbers \(a = x - \sqrt{2}\), \(b = x-1/x\), \(c = x + 1/x\), \(d = x^2 + 2\sqrt{2}\) is not an integer. Find all such \(x\).
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?