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) $n=2018$

Let $n$ be positive integer. Let $S_1,S_2,\cdots,S_k$ be a collection of $2n$-element subsets of $\{1,2,3,4,...,4n-1,4n\}$ so that $S_{i}\cap S_{j}$ contains at most $n$ elements for all $1\leq i<j\leq k$. Show that $$k\leq 6^{(n+1)/2}$$

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])

Let $k$ be a positive integer and $S$ be a set of sets which have $k$ elements. For every $A,B \in S$ and $A\neq B$ we have $A \Delta B \in S$. Find all values of $k$ when $|S|=1023$ and $|S|=2023$. Note:$A \Delta B = (A \setminus B) \cup (B \setminus A)$

Let $I_1, I_2, I_3$ be three open intervals of $\mathbb{R}$ such that none is contained in another. If $I_1\cap I_2 \cap I_3$ is non-empty, then show that at least one of these intervals is contained in the union of the other two.

Let $A,B,C$ be nonempty sets in $\mathbb R^n$. Suppose that $A$ is bounded, $C$ is closed and convex, and $A+B\subseteq A+C$. Prove that $B\subseteq C$. Recall that $E+F=\{e+f:e\in E,f\in F\}$ and $D\subseteq\mathbb R^n$ is convex iff $tx+(1-t)y\in D$ for all $x,y\in D$ and any $t\in[0,1]$.

Let $S$ be a set of integers which has the following properties: 1) There exists $x,y\in S$ such that $(x,y)=(x-2,y-2)=1$; 2) For $\forall$ $x,y\in S, x^2-y\in S$. Prove that $S\equiv \mathbb{Z}$ .

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 $t\ge 3$ 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)

[b]4.[/b] Let $\left (H_{\alpha} \right ) $ be a system of sets of integers having the property that for any $\alpha _1 \neq \alpha _2 , H_{\alpha _1}\cap H_{\alpha _2}$ is a finite set and $H_{{\alpha} _1} \neq H_{{\alpha} _2}$. Prove that there exists a system $\left (H_{\alpha} \right )$ of this kind whose cardinality is that of the continuum. Prove further that if none of the intersections of two sets $H_\alpha$ contains more than $K$ elements, then the system $\left (H_{\alpha} \right ) $ is countable ($K$ is an arbitrary fixed integer). [b](St. 4)[/b]

Let $\mathcal{F}$ be a finite family of subsets of some set $X{}$. It is known that for any two elements $x,y\in X$ there exists a permutation $\pi$ of the set $X$ such that $\pi(x)=y$, and for any $A\in\mathcal{F}$ \[\pi(A):=\{\pi(a):a\in A\}\in\mathcal{F}.\]A bear and crocodile play a game. At a move, a player paints one or more elements of the set $X$ in his own color: brown for the bear, green for the crocodile. The first player to fully paint one of the sets in $\mathcal{F}$ in his own color loses. If this does not happen and all the elements of $X$ have been painted, it is a draw. The bear goes first. Prove that he doesn't have a winning strategy.

Let $ A_i,i=1,2,\dots,t$ be distinct subsets of the base set $\{1,2,\dots,n\}$ complying to the following condition $$ \displaystyle A_ {i} \cap A_ {k} \subseteq A_ {j}$$for any $1 \leq i <j <k \leq t.$ Find the maximum value of $t.$ Thanks @dgrozev

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 $n$.

Let $A_1 , A_2 ,\ldots, A_{1066}$ be subsets of a finite set $X$ such that $|A_i | > \frac{1}{2} |X|$ for $1\leq i \leq 1066.$ Prove that there exist ten elements $x_1 ,x_2 ,\ldots , x_{10}$ of $X$ such that every $A_i $ contains at least one of $x_1 , x_2 ,\ldots, x_{10}.$

Let $f:[0,1]\to\mathbb R$ be a continuous function such that $f(0)=f(1)=0$. Prove that the set $$A:=\{h\in[0,1]:f(x+h)=f(x)\text{ for some }x\in[0,1]\}$$is Lebesgue measureable and has Lebesgue measure at least $\frac12$.

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.

The sets $A_1,A_2,...,A_n$ are finite. With $d$ we denote the number of elements in $\bigcup_{i=1}^n A_i$ which are in odd number of the sets $A_i$. Prove that the number: $D(k)=d-\sum_{i=1}^n|A_i|+2\sum_{i<j}|A_i\cap A_j |+...+(-1)^k2^{k-1}\sum_{i_1<i_2<...<i_k}|A_{i_1}\cap A_{i_2}\cap ...\cap A_{i_k}|$ is divisible by $2^k$.

$24$ students attend a mathematical circle. For any team consisting of $6$ students, the teacher considers it to be either [b]GOOD [/b] or [b]OK[/b]. For the tournament of mathematical battles, the teacher wants to partition all the students into $4$ teams of $6$ students each. May it happen that every such partition contains either $3$ [b]GOOD[/b] teams or exactly one [b]GOOD[/b] team and both options are present?

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.

Show that there exists a proper non-empty subset $S$ of the set of real numbers such that, for every real number $x$, the set $\{nx + S : n \in N\}$ is finite, where $nx + S =\{nx + s : s \in S\}$

For $S_n=\{1, 2, ..., n\}$, find the maximum value of $m$ that makes the following proposition true. [b]Proposition[/b] There exists $m$ different subsets of $S$, say $A_1, A_2, ..., A_m$, such that for every $i, j=1, 2, ..., m$, the set $A_i \cup A_j$ is not $S$.

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 $n$.

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 $|B_n |=n+1$.

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 $m = 20 , a = 4 , b = 1$.