In a mathematical competition, some competitors are friends; friendship is mutual, that is, when $A$ is a friend of $B$, then $B$ is also a friend of $A$. We say that $n \geq 3$ different competitors $A_1, A_2, \ldots, A_n$ form a [i]weakly-friendly cycle [/i]if $A_i$ is not a friend of $A_{i+1}$ for $1 \leq i \leq n$ (where $A_{n+1} = A_1$), and there are no other pairs of non-friends among the components of the cycle. The following property is satisfied: "for every competitor $C$ and every weakly-friendly cycle $\mathcal{S}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{S}$ which are not friends of $C$ has at most one element" Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors in the same room are friends. ([i]Serbia[/i])

There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn.

Let $A_n$ be the set of all sequences with length $n$ and members of the set $\{1,2…q\}$. We denote with $B_n$ a subset of $A_n$ with a minimal number of elements with the following property: For each sequence $a_1,a_2,...,a_n$ from $A_n$ there exist a sequence $b_1,b_2,...,b_n$ from $B_n$ such that $a_i\neq b_i$ for each $i=1,2,....,n$. Prove that, if $q>n$, then $|B_n |=n+1$.

Can there be a continuum set of continuum sets such that (i) the intersection of any two is finite, and (ii) every set that intersects all sets intersects any in an infinite set? note: a continuum set is a set that can be put into a 1-to-1 bijection with the reals.

For each subset $S$ of $\mathbb{N}$, with $r_S (n)$ we denote the number of ordered pairs $(a,b)$, $a,b\in S$, $a\neq b$, for which $a+b=n$. Prove that $\mathbb{N}$ can be partitioned into two subsets $A$ and $B$, so that $r_A(n)=r_B(n)$ for $\forall$ $n\in \mathbb{N}$.

On the VI-th International Festival of Young Mathematicians in Sozopol $n$ teams were participating, each of which was with $k$ participants ($n>k>1$). The organizers of the competition separated the $nk$ participants into $n$ groups, each with $k$ people, in such way that no two teammates are in the same group. Prove that there can be found $n$ participants no two of which are in the same team or group.

Given a collection of sets $X = \{A_1, A_2, ..., A_n\}$. A set $\{a_1, a_2, ..., a_n\}$ is called a single representation of $X$ if $a_i \in A_i$ for all i. Let $|S| = mn$, $S = A_1\cup A_2 \cup ... \cup A_n = B_1 \cup B_2 \cup ... \cup B_n$ with $|A_i| = |B_i| = m$ for all $i$. Prove that $S = C_1 \cup C_2 \cup ... \cup C_n$ where for every $i, C_i $ is a single represenation for $\{A_j\}_{j=1}^n $and $\{B_j\}_{j=1}^n$.

In a mathematical competition, some competitors are friends; friendship is mutual, that is, when $A$ is a friend of $B$, then $B$ is also a friend of $A$. We say that $n \geq 3$ different competitors $A_1, A_2, \ldots, A_n$ form a [i]weakly-friendly cycle [/i]if $A_i$ is not a friend of $A_{i+1}$ for $1 \leq i \leq n$ (where $A_{n+1} = A_1$), and there are no other pairs of non-friends among the components of the cycle. The following property is satisfied: "for every competitor $C$ and every weakly-friendly cycle $\mathcal{S}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{S}$ which are not friends of $C$ has at most one element" Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors in the same room are friends. ([i]Serbia[/i])

For two different prime numbers $p, q$, define $S_{p,q} = \{p,q,pq\}$. If two elements in $S_{p,q}$ are numbers in the form of $x^2 + 2005y^2, (x, y \in Z)$, prove that all three elements in $S_{p,q}$ are in such form.

Pupils of a school are divided into $112$ groups, of $11$ members each. Any two groups have exactly one common pupil. Prove that: a) there is a pupil that belongs to at least $12$ groups. b) there is a pupil that belongs to all the groups.

Let $n$ be an integer greater than $1$ and let $X$ be an $n$-element set. A non-empty collection of subsets $A_1, ..., A_k$ of $X$ is tight if the union $A_1 \cup \cdots \cup A_k$ is a proper subset of $X$ and no element of $X$ lies in exactly one of the $A_i$s. Find the largest cardinality of a collection of proper non-empty subsets of $X$, no non-empty subcollection of which is tight. [i]Note[/i]. A subset $A$ of $X$ is proper if $A\neq X$. The sets in a collection are assumed to be distinct. The whole collection is assumed to be a subcollection.

Let $x_1,\dots ,x_n$ be real numbers. We look at all the $2^{n-1}$ possible sums between some of the numbers. If the number of different sums is at least $1.8^n$, prove that the number of sums equal to $2022$ is no more than $1.67^n$.

Find the biggest natural number $m$ that has the following property: among any five 500-element subsets of $\{ 1,2,\dots, 1000\}$ there exist two sets, whose intersection contains at least $m$ numbers.

Find the numbers of ordered array $(x_1,...,x_{100})$ that satisfies the following conditions: ($i$)$x_1,...,x_{100}\in\{1,2,..,2017\}$; ($ii$)$2017|x_1+...+x_{100}$; ($iii$)$2017|x_1^2+...+x_{100}^2$.

Givan the set $S = \{1,2,3,....,n\}$. We want to partition the set $S$ into three subsets $A,B,C$ disjoint (to each other) with $A\cup B\cup C=S$ , such that the sums of their elements $S_{A} S_{B} S_{C}$ to be equal .Examine if this is possible when: a) $n=2014$ b) $n=2015 $ c) $n=2018$

Construct a set $A\subset[0,1]\times[0,1]$ such that $A$ is dense in $[0,1]\times[0,1]$ and every vertical and every horizontal line intersects $A$ in at most one point.

Let $n{}$ be a positive integer, and let $\mathcal{C}$ be a collection of subsets of $\{1,2,\ldots,2^n\}$ satisfying both of the following conditions:[list=1] [*]Every $(2^n-1)$-element subset of $\{1,2,\ldots,2^n\}$ is a member of $\mathcal{C}$, and [*]Every non-empty member $C$ of $\mathcal{C}$ contains an element $c$ such that $C\setminus\{c\}$ is again a member of $\mathcal{C}$. [/list]Determine the smallest size $\mathcal{C}$ may have. [i]Serbia, Pavle Martinovic ́[/i]

There are $n \geq 2$ classes organized $m \geq 1$ extracurricular groups for students. Every class has students participating in at least one extracurricular group. Every extracurricular group has exactly $a$ classes that the students in this group participate in. For any two extracurricular groups, there are no more than $b$ classes with students participating in both groups simultaneously. a) Find $m$ when $n = 8, a = 4 , b = 1$ . b) Prove that $n \geq 20$ when $m = 6 , a = 10 , b = 4$. c) Find the minimum value of $n$ when $m = 20 , a = 4 , b = 1$.

Determine the greatest natural number $n$, such that for each set $S$ of 2015 different integers there exist 2 subsets of $S$ (possible to be with 1 element and not necessarily non-intersecting) each of which has a sum of its elements divisible by $n$.

Each student at a school divided 18 subjects into six disjoint triples. Could it happen that every triple of subjects is among the triples of exactly one student?

Suppose $A_1,A_2,\cdots ,A_n \subseteq \left \{ 1,2,\cdots ,2018 \right \}$ and $\left | A_i \right |=2, i=1,2,\cdots ,n$, satisfying that $$A_i + A_j, \; 1 \le i \le j \le n ,$$ are distinct from each other. $A + B = \left \{ a+b|a\in A,\,b\in B \right \}$. Determine the maximal value of $n$.

King Radford of Peiza is hosting a banquet in his palace. The King has an enormous circular table with $2021$ chairs around it. At The King's birthday celebration, he is sitting in his throne (one of the $2021$ chairs) and the other $2020$ chairs are filled with guests, with the shortest guest sitting to the King's left and the remaining guests seated in increasing order of height from there around the table. The King announces that everybody else must get up from their chairs, run around the table, and sit back down in some chair. After doing this, The King notices that the person seated to his left is different from the person who was previously seated to his left. Each other person at the table also notices that the person sitting to their left is different. Find a closed form expression for the number of ways the people could be sitting around the table at the end. You may use the notation $D_{n},$ the number of derangements of a set of size $n$, as part of your expression.

Let $t\ge 3$ be an integer, and for $1\le i <j\le t$ let $A_{ij}=A_{ji}$ be an arbitrary subset of an $n$-element set $X$. Prove that there exist $1\le i < j\le t$ for which $$\left| \left( X\,\backslash\, A_{ij}\right) \cup \bigcup_{k\neq i,j}\left( A_{ik}\cap A_{jk}\right) \right| \ge \frac{t-2}{2t-2}n$$ (translated by Miklós Maróti)

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]

A tournament is held among $n$ teams, following such rules: a) every team plays all others once at home and once away.(i.e. double round-robin schedule) b) each team may participate in several away games in a week(from Sunday to Saturday). c) there is no away game arrangement for a team, if it has a home game in the same week. If the tournament finishes in 4 weeks, determine the maximum value of $n$.