In a classroom of at least four students, when any four of them take seats around a round table, there is always someone who either knows both of his neighbors, or does not know either of his neighbors. Prove that it is possible to divide the students into two groups so that in one of them, all students knows one another, and in the other, none of the students know each other. [i]Note: If $A$ knows $B$, then $B$ knows $A$ as well.[/i]

Let $S = \{1,2,3,4,5,6,7\}$. Compute the number of sets of subsets $T = \{A, B, C\}$ with $A, B, C \in S$ such that $A \cup B \cup C = S$, $(A \cap C) \cup (B \cap C) = \emptyset$, and no subset contains two consecutive integers.

On a whiteboard, the numbers $1$ to $9$ are written. Players $A$ and $B$ take turns, and $A$ is first. Each player in turn chooses one of the numbers on the whiteboard and removes it, along with all multiples (if any). The player who removes the last number loses. Determine whether any of the players has a winning strategy, and explain why.

Let $ ABC$ be an equilateral triangle with side length equal to $ N \in \mathbb{N}.$ Consider the set $ S$ of all points $ M$ inside the triangle $ ABC$ satisfying \[ \overrightarrow{AM} \equal{} \frac{1}{N} \cdot \left(n \cdot \overrightarrow{AB} \plus{} m \cdot \overrightarrow{AC} \right)\] with $ m, n$ integers, $ 0 \leq n \leq N,$ $ 0 \leq m \leq N$ and $ n \plus{} m \leq N.$ Every point of S is colored in one of the three colors blue, white, red such that [b](i) [/b]no point of $ S \cap [AB]$ is coloured blue [b](ii)[/b] no point of $ S \cap [AC]$ is coloured white [b](iii)[/b] no point of $ S \cap [BC]$ is coloured red Prove that there exists an equilateral triangle the following properties: [b](1)[/b] the three vertices of the triangle are points of $ S$ and coloured blue, white and red, respectively. [b](2)[/b] the length of the sides of the triangle is equal to 1. [i]Variant:[/i] Same problem but with a regular tetrahedron and four different colors used.

The freshman class of a school consists of $229$ boys and $271$ girls, and is divided into $10$ rooms of $50$ students each, the students in each room are numbered from $1$ to $50$. The physical education teacher wants to select a relay running team consisting of $1$ boy and $3$ girls or $1$ girl and $3$ boys, so that the four students must be two pairs of students with the same number from two rooms. Show that the number of possible teams is odd.

We have a bipartite graph $G$ (with parts $X$ and $Y$). We orient each edge arbitrarily. [i]Hessam[/i] chooses a vertex at each turn and reverse the orientation of all edges that $v$ is one of their endpoint. Prove that with these steps we can reach to a graph that for each vertex $v$ in part $X$, $\deg^{+}(v)\geq \deg^{-}(v)$ and for each vertex in part $Y$, $\deg^{+}v\leq \deg^{-}v$

[b]7.1 / 6.1[/b] There are $8$ rooks on the chessboard such that no two of them they don't hit each other. Prove that the black squares contain an even number of rooks. [b]7.2[/b] The sides of triangle $ABC$ are extended as shown in the figure. At this $AA' = 3 AB$,, $BB' = 5BC$ , $CC'= 8 CA$. How many times is the area of the triangle $ABC$ less than the area of the triangle $A'B'C' $? [img]https://cdn.artofproblemsolving.com/attachments/9/f/06795292291cd234bf2469e8311f55897552f6.png[/img] [url=https://artofproblemsolving.com/community/c893771h1860178p12579333]7.3[/url] Prove the equality $$\frac{2}{x^2-1}+\frac{4}{x^2-4} +\frac{6}{x^2-9}+...+\frac{20}{x^2-100} =\frac{11}{(x-1)(x+10)}+\frac{11}{(x-2)(x+9)}+...+\frac{11}{(x-10)(x+1)}$$ [url=https://artofproblemsolving.com/community/c893771h1861966p12597273]7.4* / 8.4 *[/url] (asterisk problems in separate posts) [b]7.5 [/b]. The collective farm consists of $4$ villages located in the peaks of square with side $10$ km. It has the means to conctruct 28 kilometers of roads . Can a collective farm build such a road system so that was it possible to get from any village to any other? [b]7.6 / 6.6[/b] Two brilliant mathematicians were told in natural terms number and were told that these numbers differ by one. After that they take turns asking each other the same question: “Do you know my number?" Prove that sooner or later one of them will answer positively. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988085_1969_leningrad_math_olympiad]here[/url].

How many $4-$digit numbers $\overline{abcd}$ are there such that $a<b<c<d$ and $b-a<c-b<d-c$ ?

The squares of a rectangular chessboard with 10 rows and 14 columns are colored alternatingly black and white in the usual manner. Some stones are placed the board (possibly more than one on the same square) so that there are an odd number of stones in each row and each column. Show that the total number of stones on black squares is even.

There are $2022$ users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.) Starting now, Mathbook will only allow a new friendship to be formed between two users if they have [i]at least two[/i] friends in common. What is the minimum number of friendships that must already exist so that every user could eventually become friends with every other user?

$9004$ lemmings, including an equal number of both sexes, cross in rank and file the new bridge from Eyjafjallajokull to Katla. The entire column therefore moves equidistantly and at a constant speed about the bridge, whereby this is able to hold exactly half of the lemmings for the present distance. The bridge should fulfill as many lemming dreams as possible and at the same time preserve the species be opened briefly at some point in order to halve the total population. However, the law to prevent gender discrimination requires that exact half is female. Show that these sufficient claims are also can be done. [hide=original wording]9004 Lemminge, davon gleich viele von beiden Geschlechtern, uberqueren in Reih und Glied die neue Brucke vom Eyjafjallajokull zum Katla. Die gesammte Kolonne bewegt sich also aquidistant und mit konstanter Geschwindigkeit uber die Brucke, wobei diese fur den vorliegenden Abstand genau die Halfte der Lemminge zu fassen vermag. Zur Erfullung moglichst vieler Lemmingtraume und gleichzeitiger Arterhaltung soll die Brucke irgendwann einmalig kurzzeitig aufgeklappt werden, um die Gesamtpopulation zu halbieren. Das Gesetz zur Verhinderung geschlechtsspezifischer Diskriminierung erfordert jedoch, dass davon ex akt die Halfte weiblich ist. Man zeige, dass diesen Anspruchen auch Genuge getan werden kann.[/hide] [img]https://cdn.artofproblemsolving.com/attachments/f/8/3bf1ef0f90d3eb3761ca3db04ed48480c8aab5.png[/img]

Let $ n, k \in \mathbb{N}$ with $ 1 \leq k \leq \frac {n}{2} - 1.$ There are $ n$ points given on a circle. Arbitrarily we select $ nk + 1$ chords among the points on the circle. Prove that of these chords there are at least $ k + 1$ chords which pairwise do not have a point in common.

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

Find all positive integers $k$ such that there exists some permutation of $(1, 2,...,1000)$ namely $(a_1, a_2,..., a_{1000}) $ and satisfy $|a_i - i| = k$ for all $i = 1,1000$.

Let $P_1,P_2,...,P_{2556}$ be distinct points inside a regular hexagon $ABCDEF$ of side $1$. If any three points from the set $S=\{A,B,C,D,E,F,P_1,P_2...,P_{2556}\}$ aren't collinear, prove that there exists a triangle with area smaller than $\frac{1}{1700}$, with vertices from the set $S$.

Given $m+n$ numbers: $a_i,$ $i = 1,2, \ldots, m,$ $b_j$, $j = 1,2, \ldots, n,$ determine the number of pairs $(a_i,b_j)$ for which $|i-j| \geq k,$ where $k$ is a non-negative integer.

Consider the set $A=\{1,2,...,120\}$ and $M$ a subset of $A$ such that $|M|=30$.Prove that there are $5$ different subsets of $M$,each of them having two elements,such that the absolute value of the difference of the elements of each subset is the same.

Let be given three-element subsets $A_1,A_2,\ldots,A_6$ of a six-element set $X$. Prove that the elements of $X$ can be colored with two colors in such a way that none of the given subsets are monochromatic.

A field has a shape of checkboard $\text{41x41}$ square. A tank concealed in one of the cells of the field. By one shot, a fighter airplane fires one of the cells. If a shot hits the tank, then the tank moves to a neighboring cell of the field, otherwise it stays in its cell (the cells are neighbours if they share a side). A pilot has no information about the tank , one needs to hit it twice. Find the least number of shots sufficient to destroy the tank for sure. [i](S.Berlov,A.Magazinov)[/i]

Let $n$ and $k$ be positive integers, with $n\geq k+1$. There are $n$ countries on a planet, with some pairs of countries establishing diplomatic relations between them, such that each country has diplomatic relations with at least $k$ other countries. An evil villain wants to divide the countries, so he executes the following plan: (1) First, he selects two countries $A$ and $B$, and let them lead two allies, $\mathcal{A}$ and $\mathcal{B}$, respectively (so that $A\in \mathcal{A}$ and $B\in\mathcal{B}$). (2) Each other country individually decides wether it wants to join ally $\mathcal{A}$ or $\mathcal{B}$. (3) After all countries made their decisions, for any two countries with $X\in\mathcal{A}$ and $Y\in\mathcal{B}$, eliminate any diplomatic relations between them. Prove that, regardless of the initial diplomatic relations among the countries, the villain can always select two countries $A$ and $B$ so that, no matter how the countries choose their allies, there are at least $k$ diplomatic relations eliminated. [i]Proposed by YaWNeeT.[/i]

On a table are given $n$ markers, each of which is denoted by an integer. At any time, if some two markers are denoted with the same number, say $k$, we can redenote one of them with $k +1$ and the other one with $k -1$. Prove that after a finite number of moves all the markers will be denoted with different numbers.

Five men play several sets of dominoes (two against two) so that each player has each other player once as a partner and two times as an opponent. Find the number of sets and all ways to arrange the players.

A building has some rooms and there is one or more than one doors between the rooms. We know that we can go from each room to another one. Two rooms $E,S$ has been labeled, and the room $S$ has exactly one door. Someone is in the room $S$ and wants to move to the room $E$. [list][list][list][list][list][list][img]http://s1.picofile.com/file/6475095570/image005.jpg[/img][/list][/list][/list][/list][/list][/list] A "[i]way[/i]" $P$ for moving between the rooms is an infinite sequence of $L$ and $R$. We say that someone moves according to the "[i]way[/i]" $P$, if he start moving from the room $S$, and after passing the $n$'th door, if $P_n$ is $R$, then he goes to the first door which is in the right side, and if $P_n$ is $L$, then he goes to the first door which is in the left side (obviously, if some room has exactly one door, then there is no difference between $L$ and $R$), and when he arrives to the room $E$, he stops moving. Prove that there exists a "[i]way[/i]" such that if the person move according to it, then he can arrive to the room $E$ of any building.

$x_1, x_2, ... , x_8$ is a permutation of $1, 2, ..., 8$. A move is to take $x_3$ or $x_8$ and place it at the start to from a new sequence. Show that by a sequence of moves we can always arrive at $1, 2, ..., 8$.

Write the natural numbers from left to right in ascending order. Every minute, we perform an operation. After $m$ minutes, we divide the entire available series into consecutive blocks of $m$ numbers. We leave the first block unchanged and in each of the other blocks we move all the numbers except the first one one place to the left, and move the first one to the end of the block. Prove that throughout the process, each natural number will only move a finite number of times.