A square of side $N=n^2+1$, $n\in \mathbb{N}^*$, is partitioned in unit squares (of side $1$), along $N$ rows and $N$ columns. The $N^2$ unit squares are colored using $N$ colors, $N$ squares with each color. Prove that for any coloring there exists a row or a column containing unit squares of at least $n+1$ colors.

We call a square table of a binary, if at each cell is written a single number 0 or 1. The binary table is called regular if each row and each column exactly two units. Determine the number of regular size tables $n\times n$ ($n> 1$ - given a fixed positive integer). (We can assume that the rows and columns of the tables are numbered: the cases of coincidence in turn, reflect, and so considered different).

Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board. [list=i] [*] If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$. [*] If no such pair exists, we write two times the number $0$. [/list] Prove that, no matter how we make the choices in $(i)$, operation $(ii)$ will be performed only finitely many times. Proposed by [I]Serbia[/I].

In Miss Lies' class there are only students who never lie and students who always lie. All students know which category they belong to. During the day in a class discussion, every student in the class says about every other student or he or she a liar or not. In total, it is said $320$ times that someone is not lying. The next day, one of the students who always lies is sick. There will be one again organize such a class discussion in which no mention is made of the sick pupil. Now it is said $300$ times that someone does lie. How many liars are there in the Miss Lies' class ?

Given $n$ vectors of unit length in the plane. The length of their total sum is less than one. Prove that you can rearrange them to provide the property: [i]for every[/i] $k, k\le n$[i], the length of the sum of the first[/i] $k$ [i]vectors is less than[/i] $2$.

You have a collection of at least two tokens where each one has a number less than or equal to 10 written on it. The sum of the numbers on the tokens is \( S \). Find all possible values of \( S \) that guarantee that the tokens can be separated into two groups such that the sum of each group does not exceed 80.

Six boys and six girls are participating at a tango course. They meet every evening for three weeks (a total of 21 times). Each evening, at least one boy-girl pair is selected to dance in front of the others. At the end of the three weeks, every boy-girl pair has been selected at least once. Prove that there exists a person who has been selected on at least 5 distinct evenings. [i]Note:[/i] a person can be selected twice on the same evening.

Ati has $ 7$ pots of flower, ordered in $ P_1,P_2,P_3,P_4,P_5,P_6,P_7$. She wants to rearrange the position of those pots to $ B_1,B_2,B_2,B_3,B_4,B_5,B_6,B_7$ such that for every positive integer $ n<7$, $ B_1,B_2,\dots,B_n$ is not the permutation of $ P_1,P_2,\dots,P_7$. In how many ways can Ati do this?

In how many ways rooks can be placed on a $8$ by $8$ chess board such that every row and every column has at least one rook? (Any number of rooks are available,each square can have at most one rook and there is no relation of attacking between them)

The squares of an n*n chessboard (n >1) are filled with 1s and -1s. A series of steps are performed. For each step, the number in each square is replaced with the product of the numbers that were in the squares adjacent. Find all values of n for which, starting from an arbitrary collection of numbers, after finitely many steps one obtains a board filled with 1s.

Two players $A$ and $B$ play a game with a ball and $n$ boxes placed onto the vertices of a regular $n$-gon where $n$ is a positive integer. Initially, the ball is hidden in a box by player $A$. At each step, $B$ chooses a box, then player $A$ says the distance of the ball to the selected box to player $B$ and moves the ball to an adjacent box. If $B$ finds the ball, then $B$ wins. Find the least number of steps for which $B$ can guarantee to win.

Let $n>2$ be an even integer. In the squares of a board of $n \times n$, pieces must be placed so that in each column the number of chips is even and different from zero, and in each row the number of chips is odd. Determine the fewest number of checkers to place on the board to satisfy this rule. To show a configuration with that number of tokens and explain why with fewer tokens the rule.

Determine alle positive integers $n>1$ with the following property: For each colouring of the lattice points in the plane with $n$ colours, there are three lattice points of the same colour forming an isosceles right triangle with legs parallel to the coordinate axes.

Given the integer $n\geq 2$ and a integer ${a}$, which is coprime with ${n}$. A country has ${n}$ islands $D_1$, $D_2$, $\cdots$, $D_n$. For any $1\leq i\neq j\leq n$, there is a one-way ferry $D_i$ to $D_j$ if and only if $ij\equiv ia\pmod n$. A tourist can initially fly to any of the islands, and then he can only take a one-way ferry. What is the maximum number of islands he can visit? [i]Created by Zhenhua Qu[/i]

In a hat are $1992$ notes with all numbers from $1$ to $1992$. At random way, two bills are drawn simultaneously from the hat; the difference between the numbers on the two notes are written on a new note, which is placed in the hat, while the two drawn notes thrown away. It continues in this way until there is only one note left in it the hat. Show that there is an even number on this slip of paper.

A game has two players. In the game there is a rectangular chocolate bar, with $60$ pieces, arranged in a $6 \times 1 0$ formation , which can be broken only along the lines dividing the pieces. The first player breaks the bar along one line, discarding one section . The second player then breaks the remaining section, discarding one section. The first player repeats this process with the remaining section , and so on. The game is won by the player who leaves a single piece. In a perfect game which player wins? {S. Fomin , Leningrad)

In a game, two players control an adventurer in a dungeon. The adventurer starts with $H{}$ hit points, an integer greater than one. The dungeon consists of several chambers. There are some passageways in the dungeon, each leading from a chamber to a chamber. These passageways are one-way, and a passageway may return to its starting chamber. Every chamber can be exited through at least one passageway. There are 5 types of chambers: [list] [*]Entrance: the adventurer starts here, no passageway comes in here; [*]Hollow: nothing happens; [*]Spike: the adventurer loses a hit point; [*]Trap: the adventurer gets shot by an arrow; [*]Catacomb: the adventurer loses hit points equal to the total number of times they have been hit by an arrow. [/list] The two players control the adventurer alternatively. At a turn, a player can move him through one passageway. A player loses if the adventurer’s hit points fall below zero due to their action (at 0 hit points, the character is alive). Construct a dungeon map, which has at most 20 chambers in total and exactly one entrance, with the following condition: the first player has a winning strategy if $H{}$ is a prime, and the second player has a winning strategy if $H{}$ is composite. [i]Note: If the game doesn’t end after a finite number of moves, neither player wins.[/i]

N numbers are marked in the set $\{1,2,...,2000\}$ so that any pair of the numbers $(1,2),(2,4),...,(1000,2000)$ contains at least one marked number. Find the least possible value of $N$. I.Gorodnin

Let $S = \{1,2,3,\ldots,n\}$. Consider a function $f\colon S\to S$. A subset $D$ of $S$ is said to be invariant if for all $x\in D$ we have $f(x)\in D$. The empty set and $S$ are also considered as invariant subsets. By $\deg (f)$ we define the number of invariant subsets $D$ of $S$ for the function $f$. [b]i)[/b] Show that there exists a function $f\colon S\to S$ such that $\deg (f)=2$. [b]ii)[/b] Show that for every $1\leq k\leq n$ there exists a function $f\colon S\to S$ such that $\deg (f)=2^{k}$.

The mayor of a city wishes to establish a transport system with at least one bus line, in which: - each line passes exactly three stops, - every two different lines have exactly one stop in common, - for each two different bus stops there is exactly one line that passes through both. Determine the number of bus stops in the city.

Let $T$ be a tree. Prove that there is a constant $c>0$ (independent of $n$) such that every graph with $n$ vertices that does not contain a subgraph isomorphic to $T$ has at most $cn$ edges. [i]David Yang.[/i]

Let $S$ be a set consisting of $m$ pairs $(a,b)$ of positive integers with the property that $1 \leq a < b \leq n$. Show that there are at least \[ 4m \cdot \dfrac{(m - \dfrac{n^2}{4})}{3n} \] triples $(a,b,c)$ such that $(a,b)$, $(a,c)$, and $(b,c)$ belong to $S$.

In the plane we have $16$ lines(not parallel and not concurrents), we have $120$ point(s) of intersections of this lines. Sebastian has to paint this $120$ points such that in each line all the painted points are with colour differents, find the minimum(quantity) of colour(s) that Sebastian needs to paint this points. If we have have $15$ lines(in this situation we have $105$ points), what's the minimum(quantity) of colour(s)?

Show that every subset of {1,2,...,99,100} with 55 elements contains at least 2 numbers with a difference of 9.

If $A=\{1,2,...,4s-1,4s\}$ and $S \subseteq A$ such that $\mid S \mid =2s+2$, prove that in $S$ we can find three distinct numbers $x$, $y$ and $z$ such that $x+y=2z$