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.

Consider a finite set $ A, $ and two functions $ f,g: A\longrightarrow A. $ Prove that: $$ |\{ x\in A| g(f(x))\neq x \} | =|\{ x\in A| f(g(x))\neq x \} | $$ [i]Cristinel Mortici[/i]

Let $(X, <)$ be an arbitrary ordered set. Show that the elements of $X$ can be coloured by two colours in such a way that between any two points of the same colour there is a point of the opposite colour. (translated by L. Erdős)

Let $S=\{A_1,A_2,\ldots ,A_n\}$, where $A_1,A_2,\ldots ,A_n$ are $n$ pairwise distinct finite sets $(n\ge 2)$, such that for any $A_i,A_j\in S$, $A_i\cup A_j\in S$. If $k= \min_{1\le i\le n}|A_i|\ge 2$, prove that there exist $x\in \bigcup_{i=1}^n A_i$, such that $x$ is in at least $\frac{n}{k}$ of the sets $A_1,A_2,\ldots ,A_n$ (Here $|X|$ denotes the number of elements in finite set $X$).

Let $A_1, A_2, \ldots , A_m$ be subsets of the set $\{ 1,2, \ldots , n \}$, such that the cardinal of each subset $A_i$, such $1 \le i \le m$ is not divisible by $30$, while the cardinal of each of the subsets $A_i \cap A_j$ for $1 \le i,j \le m$, $i \neq j$ is divisible by $30$. Prove \begin{align*} 2m - \left \lfloor \frac{m}{30} \right \rfloor \le 3n \end{align*}

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 $ 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

We partition the $n$-element subsets of an $n^2+n-1$-element set into two classes. Prove that one of the classes contains $n$-many pairwise disjunct sets. (translated by Miklós Maróti)

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 $S$.

Let $S$ be a set of $n$ distinct real numbers. Let $A_{S}$ be the set of numbers that occur as averages of two distinct elements of $S$. For a given $n \geq 2$, what is the smallest possible number of elements in $A_{S}$?

Let $S$ be the set $\{1, 2, ..., 10\}$. Let $A$ be a subset of $S$. We arrange the elements of $A$ in increasing order, that is, $A = \{a_1, a_2, ...., a_k\}$ with $a_1 < a_2 < ... < a_k$. Define [i]WSUM [/i] for this subset as $3(a_1 + a_3 +..) + 2(a_2 + a_4 +...)$ where the first term contains the odd numbered terms and the second the even numbered terms. (For example, if $A = \{2, 5, 7, 8\}$, [i]WSUM [/i] is $3(2 + 7) + 2(5 + 8)$.) Find the sum of [i]WSUMs[/i] over all the subsets of S. (Assume that WSUM for the null set is $0$.)

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 $S$ be a set of $n>0$ elements, and let $A_1 , A_2 , \ldots A_k$ be a family of distinct subsets such that any two have a non-empty intersection. Assume that no other subset of $S$ intersects all of the $A_i.$ Prove that $ k=2^{n-1}.$

A subset of the set $A={1,2,\dots ,n}$ is called [i]connected[/i], if it consists of one number or a certain amount of consecutive numbers. Find the greatest $k$ (defined as a function of $n$) for which there exists $k$ different subsets $A_1,A_2,…,A_k$ of $A$ the intersection of each two of which is a [i]connected[/i] set.

[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]

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) $n=2018$

The set of all positive real numbers is partitioned into three mutually disjoint non-empty subsets: $\mathbb R^+ = A \cup B\cup C$ and $A \cap B = B \cap C = C \cap A = \emptyset$ whereas none of $A, B, C$ is empty. [list=a][*] Show that one can choose $a \in A, b \in B$ and $c \in C$ such that $a,b, c$ are the sides of a triangle. [*] Is it always possible to choose three numbers from three different sets $A,B,C$ such that these three numbers are the sides of a right-angled triangle?[/list]

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.

Suppose $S=\{1,2,3,...,2017\}$,for every subset $A$ of $S$,define a real number $f(A)\geq 0$ such that: $(1)$ For any $A,B\subset S$,$f(A\cup B)+f(A\cap B)\leq f(A)+f(B)$; $(2)$ For any $A\subset B\subset S$, $f(A)\leq f(B)$; $(3)$ For any $k,j\in S$,$$f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\});$$ $(4)$ For the empty set $\varnothing$, $f(\varnothing)=0$. Confirm that for any three-element subset $T$ of $S$,the inequality $$f(T)\leq \frac{27}{19}f(\{1,2,3\})$$ holds.

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

Let $T$ be a set of natural numbers, each of which is greater than 1. A subset $S$ of $T$ is called “good”, if for each $t\in T$ there exists $s\in S$, for which $gcd(t,s)>1$. Prove that the number of "good" subsets of $T$ is odd.

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 $X$ be a finite set with $n\geqslant 3$ elements and let $A_1,A_2,\ldots, A_p$ be $3$-element subsets of $X$ satisfying $|A_i\cap A_j|\leqslant 1$ for all indices $i,j$. Show that there exists a subset $A{}$ of $X$ so that none of $A_1,A_2,\ldots, A_p$ is included in $A{}$ and $|A|\geqslant\lfloor\sqrt{2n}\rfloor$.

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 $\frac12$.