Two chess players play each other at chess using clocks (when a player makes a move , the player stops his clock and starts the clock of his opponent) . It is known that when both players have just completed their $40$th move , both of their clocks read exactly $2$ hr $30$ min . Prove that there was a moment in the game when the clock of one player registered $1$ min $51$ sec less than that of the other . Furthermore , can one assert that the difference between the two clock readings was ever equal to $2$ minutes? (S . Fomin , Leningrad)

Prove that total number of ones which is showed in all nonrestricted partitions of natural number $n$ is equal to sum of numbers of distinct elements in that partitions.

A regular octagon is drawn on the patio floor. Emiliano writes in the vertices the numbers from $1$ to $8$ in any order. Put a stone at point $1$. He walks towards point $2$, having traveled $1/2$ of the way he stops and leaves the second stone. From there he walks to point $3$, having traveled $1/3$ of the way, he stops and leaves the third stone. From there he walks to point $4$, having traveled $1/4$ of the way, he stops and leaves the fourth stone. This goes on until, after leaving the seventh stone, he walks towards point 8 and having traveled $1/8$ of the way, he leaves the eighth stone. The number of stones left in the center of the octagon depends on the order in which you wrote the numbers on the vertices. What is the greatest number of stones that can remain in that center?

A boy goes $n$ times at a merry-go-round with $n$ seats. After every time he moves in the clockwise direction and takes another seat, not making a full circle. The number of seats he passes by at each move is called the length of the move. For which $n$ can he sit at every seat, if the lengths of all the $n-1$ moves he makes have different lengths? [i]V. New[/i]

On the table there are $2n$ coins that look the same. It is known that $n$ of them weigh 9 g. each, while the remaining $n$ weigh 10 g. each. It is required to split the coins into $n$ pairs with total weight of each pair 19 g. Prove that this can be done in less than $n$ weighings using a balance without additional weights (the balance shows which pan is heavier or that their weight is equal).

Given seven points in the plane, some of them are connected by segments such that: [b](i)[/b] among any three of the given points, two are connected by a segment; [b](ii)[/b] the number of segments is minimal. How many segments does a figure satisfying [b](i)[/b] and [b](ii)[/b] have? Give an example of such a figure.

Two persons are playing the following game. They have a pile of stones and take turns removing stones from it, with the first player taking the first turn. At each turn, the first player removes either 1 or 10 stones from the pile, and the second player removes either m or n stones. The player who can not make his move loses. It is known that for any number of stones in the pile, the first player can always win (regardless of the second player's moves). What are the possible values of m and n?

Every point of a plane is colored red or blue, not all with the same color. Can this be done in such a way that, on every circumference of radius 1, (a) there is exactly one blue point; (b) there are exactly two blue points?

Let $n\ge 3$. Suppose $a_1, a_2, ... , a_n$ are $n$ distinct in pairs real numbers. In terms of $n$, find the smallest possible number of different assumed values by the following $n$ numbers: $$a_1 + a_2, a_2 + a_3,..., a_{n- 1} + a_n, a_n + a_1$$

Is it possible to divide a polygon with $21$ sides into $2022$ triangles in such a way that among all the vertices there are not three collinear?

In a group of $n$ people, each one had a different ball. They performed a sequence of swaps, in each swap, two people swapped the ball they had at that moment. Each pair of people performed at least one swap. In the end each person had the ball he/she had at the start. Find the least possible number of swaps, if: a) $n = 5$, b) $n = 6$.

A [i]mixing[/i] of the sequence $a_1,a_2,\dots ,a_{3n}$ is called the following sequence: $a_3,a_6,\dots ,a_{3n},a_2,a_5,\dots ,a_{3n-1},a_1,a_4,\dots ,a_{3n-2}$. Is it possible after finite amount of [i]mixings[/i] to reach the sequence $192,191,\dots ,1$ from $1,2,\dots ,192$?

Consider a $m*n$ board. On each box there's a non-negative integrer number assigned. An operation consists on choosing any two boxes with $1$ side in common, and add to this $2$ numbers the same integrer number (it can be negative), so that both results are non-negatives. What conditions must be satisfied initially on the assignment of the boxes, in order to have, after some operations, the number $0$ on every box?.

There are $ n \geq 5$ pairwise different points in the plane. For every point, there are just four points whose distance from which is $ 1$. Find the maximum value of $ n$.

Let $n$ be a positive integer. Find the number of sequences $a_0,a_1,a_2,\dots,a_{2n}$ of integers in the range $[0,n]$ such that for all integers $0\leq k\leq n$ and all nonnegative integers $m$, there exists an integer $k\leq i\leq 2k$ such that $\lfloor k/2^m\rfloor=a_i.$ [i]Andrew Carratu[/i]

Let $b_0, b_1, b_2, \ldots$ be a sequence of pairwise distinct nonnegative integers such that $b_0=0$ and $b_n<2n$ for all positive integers $n$. Prove that for each nonnegative integer $m$ there exist nonnegative integers $k, \ell$ such that \begin{align*} b_k+b_{\ell}=m. \end{align*}

There are three boxes, one blue, one white and one red, and $8$ balls. Each of the balls has a number from $1$ to $8$ written on it, without repetitions. The $8$ balls are distributed in the boxes, so that there are at least two balls in each box. Then, in each box, add up all the numbers written on the balls it contains. The three outcomes are called the blue sum, the white sum, and the red sum, depending on the color of the corresponding box. Find all possible distributions of the balls such that the red sum equals twice the blue sum, and the red sum minus the white sum equals the white sum minus the blue sum.

Each deputy of the Academic Duma quarreled with exactly three other deputies. The President ordered the Speaker to divide the deputies into n factions so that agreement reigned within one faction. For what smallest $n$ is this always possible? (This means that there is such $n$ that deputies could always be divided into $n$ factions, but not always into $(n- 1)$ factions.)

For non-empty number sets $S, T$, define the sets $S+T=\{s+t\mid s\in S, t\in T\}$ and $2S=\{2s\mid s\in S\}$. Let $n$ be a positive integer, and $A, B$ be two non-empty subsets of $\{1,2\ldots,n\}$. Show that there exists a subset $D$ of $A+B$ such that 1) $D+D\subseteq 2(A+B)$, 2) $|D|\geq\frac{|A|\cdot|B|}{2n}$, where $|X|$ is the number of elements of the finite set $X$.

Determine the maximal size of the set $S$ such that: i) all elements of $S$ are natural numbers not exceeding $100$; ii) for any two elements $a,b$ in $S$, there exists $c$ in $S$ such that $(a,c)=(b,c)=1$; iii) for any two elements $a,b$ in $S$, there exists $d$ in $S$ such that $(a,d)>1,(b,d)>1$. [i]Yao Jiangang[/i]

$12$ schoolchildren are engaged in a circle of patriotic songs, each of them knows a few songs (maybe none). We will say that a group of schoolchildren can sing a song if at least one member of the group knows it. Supervisor the circle noticed that any group of $10$ circle members can sing exactly $20$ songs, and any group of $8$ circle members - exactly $16$ songs. Prove that the group of all $12$ circle members can sing exactly $24$ songs.

Alice and Bob play the following game. To start, Alice arranges the numbers $1,2,\ldots,n$ in some order in a row and then Bob chooses one of the numbers and places a pebble on it. A player's [i]turn[/i] consists of picking up and placing the pebble on an adjacent number under the restriction that the pebble can be placed on the number $k$ at most $k$ times. The two players alternate taking turns beginning with Alice. The first player who cannot make a move loses. For each positive integer $n$, determine who has a winning strategy.

A square with side $2008$ is broken into regions that are all squares with side $1$. In every region, either $0$ or $1$ is written, and the number of $1$'s and $0$'s is the same. The border between two of the regions is removed, and the numbers in each of them are also removed, while in the new region, their arithmetic mean is recorded. After several of those operations, there is only one square left, which is the big square itself. Prove that it is possible to perform these operations in such a way, that the final number in the big square is less than $\frac{1}{2^{10^6}}$.

Alice and Bob play the following game. They write some fractions of the form $1/n$, where $n{}$ is positive integer, onto the blackboard. The first move is made by Alice. Alice writes only one fraction in each her turn and Bob writes one fraction in his first turn, two fractions in his second turn, three fractions in his third turn and so on. Bob wants to make the sum of all the fractions on the board to be an integer number after some turn. Can Alice prevent this? [i]Andrey Arzhantsev[/i]

Maria and Skyler have a square-shaped cookie with a side length of $1$ inch. They split the cookie by choosing two points on distinct sides of the cookie uniformly at random and cutting across the line segment formed by connecting the two points. If Maria always gets the larger piece, what is the expected amount of extra cookie in Maria’s piece compared to Skyler’s, in square inches?