At the meeting, each person is familiar with 22 people. If two persons $A$ and $B$ know each with one another, among the remaining people they do not have a common friend. For each pair individuals $A$ and $B$ are not familiar with each other, there are among the remaining six common acquaintances. How many people were at the meeting?

Let [n] be the set {1, 2,. . . , n}. For any $a, b \in N$, denote $G (a, b)$ by a graph (not directed) defined by the following rule: the vertices have the form (i, f), where $i \in [a]$, and $f: [a] \to [b]$. A vertex (i, f) and a vertex (j, g) are connected if $i \neq j$, and $f (k) \neq g (k)$ holds exactly for k strictly between i and j. Prove that for any $c \in N$ there is $a, b \in N$ such that the vertices of G (a, b) cannot be well-colored with $c$ colors.

On a table there are 2004 boxes, and in each box a ball lies. I know that some the balls are white and that the number of white balls is even. In each case I may point to two arbitrary boxes and ask whether in the box contains at least a white ball lies. After which minimum number of questions I can indicate two boxes for sure, in which white balls lie?

In the county some pairs of towns connected by two-way non-stop flight. From any town we can flight to any other (may be not on one flight). Gives, that if we consider any cyclic (i.e. beginning and finish towns match) route, consisting odd number of flights, and close all flights of this route, then we can found two towns, such that we can't fly from one to other. Proved, that we can divided all country on $4$ regions, such that any flight connected towns from other regions.

Given positive integers $n,k$, $n \ge 2$. Find the minimum constant $c$ satisfies the following assertion: For any positive integer $m$ and a $kn$-regular graph $G$ with $m$ vertices, one could color the vertices of $G$ with $n$ different colors, such that the number of monochrome edges is at most $cm$.

Given $S$ be the finite lattice (with integer coordinate) set in the $xy$-plane. $A$ is the subset of $S$ with most elements such that the line connecting any two points in $A$ is not parallel to $x$-axis or $y$-axis. $B$ is the subset of integer with least elements such that for any $(x,y)\in S$, $x \in B$ or $y \in B$ holds. Prove that $|A| \geq |B|$.

Let $S$ be a set of $N \ge 3$ points in the plane. Assume that no $3$ points in $S$ are collinear. The segments with both endpoints in $S$ are colored in two colors. Prove that there is a set of $N - 1$ segments of the same color which don't intersect except in their endpoints such that no subset of them forms a polygon with positive area.

Prove that we can color every subset with $n$ element of a set with $3n$ elements with $8$ colors . In such a way that there are no $3$ subsets $A,B,C$ with the same color where : $$|A \cap B| \le 1,|A \cap C| \le 1,|B \cap C| \le 1$$ Proposed by [i]Morteza Saghafian[/i] and [i]Amir Jafari[/i]

Prove that if $n$ is large enough, then for each coloring of the subsets of the set $\{1,2,...,n\}$ with $1391$ colors, two non-empty disjoint subsets $A$ and $B$ exist such that $A$, $B$ and $A\cup B$ are of the same color.

At a certain orphanage, every pair of orphans are either friends or enemies. For every three of an orphan's friends, an even number of pairs of them are enemies. Prove that it's possible to assign each orphan two parents such that every pair of friends shares exactly one parent, but no pair of enemies does, and no three parents are in a love triangle (where each pair of them has a child).

In a group of nine persons it is not possible to choose four persons such that every one knows the three others. Prove that this group of nine persons can be partitioned into four groups such that nobody knows anyone from his or her group.

Graph $G$ becomes planar when any vertex is removed. Prove that its vertices can be properly colored with 5 colors. (Using the four-color theorem without proof is not allowed!) [i]Proposed by D. Karpov[/i]

There are 10,000 vertices in a graph, with at least one edge coming out of each vertex. Call a set $S{}$ of vertices [i]delightful[/i] if no two of its vertices are connected by an edge, but any vertex not from $S{}$ is connected to at least one vertex from $S{}$. For which smallest $m$ is there necessarily a delightful set of at most $m$ vertices?

In a mathematical competition, in which $6$ problems were posed to the participants, every two of these problems were solved by more than $\frac 25$ of the contestants. Moreover, no contestant solved all the $6$ problems. Show that there are at least $2$ contestants who solved exactly $5$ problems each. [i]Radu Gologan and Dan Schwartz[/i]

There are $2024$ cities in a country, some pairs of which are connected by bidirectional flights. For any distinct cities $A, B, C, X, Y, Z$, it is possible to fly directly from some of the cities $A, B, C$ to some of the cities $X, Y, Z$. Prove that it is possible to plan a route $T_1\to T_2 \to \ldots \to T_{2022}$ that passes through $2022$ distinct cities. [i]Proposed by Lior Shayn[/i]

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

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?

Armadillo and Badger are playing a game. Armadillo chooses a nonempty tree (a simple acyclic graph) and places apples at some of its vertices (there may be several apples on a single vertex). First, Badger picks a vertex $v_0$ and eats all its apples. Next, Armadillo and Badger take turns alternatingly, with Armadillo starting. On the $i$-th turn the animal whose turn it is picks a vertex $v_i$ adjacent to $v_{i-1}$ that hasn't been picked before and eats all its apples. The game ends when there is no such vertex $v_i$. Armadillo's goal is to have eaten more apples than Badger once the game ends. Can she fulfill her wish?

In a certain country there are $n$ cities. Some pairs of cities are connected by highways in such a way that for each two cities there is at most one highway connecting them. Assume that for a certain positive integer $k$, the total number of highways is greater than $\frac{nk}{2}$. Show that there exist $k+2$ distinct cities $C_1, C_2, \dots, C_{k+2}$ such that $C_i$ and $C_{i+1}$ are connected by a highway for $i=1, 2, \dots, k+1$. [i]Proposed by Oriol Solé[/i]

In connection with the formation of a stable government, the President invited all $240$ Members of Parliament to three separate consultations, where each member participated in exactly one consultation, and at each consultation there has been at least one member present. Discussions between pairs of members are to take place to discuss the consultations. Is it possible for these discussions to occur in such a way that there exists a non-negative integer $k$, such that for every two members who participated in different consultations, there are exactly $k$ members who participated in the remaining consultation, with whom each of the two members has a conversation, and exactly $k$ members who participated in the remaining consultation, with whom neither of the two has a conversation? If yes, then find all possible values of $k$.

Let $K=(V, E)$ be a finite, simple, complete graph. Let $\phi: E \to \mathbb{R}^2$ be a map from the edge set to the plane, such that the preimage of any point in the range defines a connected graph on the entire vertex set $V$, and the points assigned to the edges of any triangle are collinear. Show that the range of $\phi$ is contained in a line.

What is the greatest possible number of edges in a planar graph with $12$ vertices? A planar graph is one that can be drawn in a plane with none of the edges crossing (they intersect only at vertices).

Let $G$ be a simple $k$ edge-connected graph on $n$ vertices and let $u$ and $v$ be different vertices of $G$. Prove that there are $k$ edge-disjoint paths from $u$ to $v$ each having at most $\frac{20n}{k}$ edges.

Is it possible to place the numbers $0,1,2,\dots,9$ on a circle so that the sum of any three consecutive numbers is a) 13, b) 14, c) 15?

10000 children came to a camp; every of them is friend of exactly eleven other children in the camp (friendship is mutual). Every child wears T-shirt of one of seven rainbow's colours; every two friends' colours are different. Leaders demanded that some children (at least one) wear T-shirts of other colours (from those seven colours). Survey pointed that 100 children didn't want to change their colours [translator's comment: it means that any of these 100 children (and only them) can't change his (her) colour such that still every two friends' colours will be different]. Prove that some of other children can change colours of their T-shirts such that as before every two friends' colours will be different.