Problems

Author: D.V. Baranov

Vlad and Peter are playing the following game. On the board two numbers written are: $1/2009$ and $1/2008$. At each turn, Vlad calls any number $x$, and Peter increases one of the numbers on the board (whichever he wants) by $x$. Vlad wins if at some point one of the numbers on the board becomes equal to 1. Will Vlad win, no matter how Peter acts?

Find the largest value of the expression $a+b+c+d-ab-bc-cd-da$, if each of the numbers $a$, $b$, $c$ and $d$ belongs to the interval $[0,1]$.

A disk contains 2013 files of 1 MB, 2 MB, 3 MB, ..., 2012 MB, 2013 MB. Can I distribute them in three folders so that each folder has the same number of files and all three folders have the same size (in MB)?

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 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 $$\frac{x_1^2+\cdots +x_n^2}{n}>\frac{y_1^2+\cdots +y_n^2}{n}.$$

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?

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?