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)

In a city's bus route system, any two routes share exactly one stop, and every route includes at least four stops. Prove that the stops can be classified into two groups such that each route includes stops from each group.

If for an arbitrary permutation $(a_1,a_2,...,a_n)$ of set ${1,2,...,n}$ holds $\frac{{a_k}^2}{a_{k+1}}\leq k+2$, $k=1,2,...,n-1$, prove that $a_k=k$ for $k=1,2,...,n$

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]

Let $n\geqslant 3$ be a positive integer and $N=\{1,2,\ldots,n\}$ and let $k>0$ be a real number. Let's associate each non-empty of $N{}$ with a point in the plane, such that any two distinct subsets correspond to different points. If the absolute value of the difference between the arithmetic means of the elements of two distinct non-empty subsets of $N{}$ is at most $k{}$ we connect the points associated with these subsets with a segment. Determine the minimum value of $k{}$ such that the points associated with any two distinct non-empty subsets of $N{}$ are connected by a segment or a broken line. [i]Cristi Săvescu[/i]

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.

2. Let $A$ be a set of $2020$ distinct real numbers. Call a number [i]scarily epic[/i] if it can be expressed as the product of two (not necessarily distinct) numbers from $A$. What is the minimum possible number of distinct [i]scarily epic[/i] numbers? [i]Proposed by Monkey_king1[/i]

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 $x_1,\dots ,x_n$ 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 $1.67^n$.

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.

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 $A_1, A_2, ..., A_k$ be $5$-element subsets of set $\{1, 2, ..., 23\}$ such that, for all $1 \le i < j \le k$ set $A_i \cap A_j$ has at most three elements. Show that $k \le 2018$.

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

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]

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 $\forall$ $n\in \mathbb{N}$.

Let $ p $ be a natural number and let two partitions $ \mathcal{A} =\left\{ A_1,A_2,...,A_p\right\} ,\mathcal{B}=\left\{ B_1,B_2,...B_p\right\} $ of a finite set $ \mathcal{M} . $ Knowing that, whenever an element of $ \mathcal{A} $ doesn´t have any elements in common with another of $ \mathcal{B} , $ it holds that the number of elements of these two is greater than $ p, $ prove that $ \big| \mathcal{M}\big|\ge\frac{1}{2}\left( 1+p^2\right) . $ Can equality hold?

Show that there exists a tournament $ (T,\rightarrow)$ of cardinality $ \aleph_1$ containing no transitive subtournament of size $ \aleph_1$. ( A structure $ (T,\rightarrow)$ is a $ \textit{tournament}$ if $ \rightarrow$ is a binary, irreflexive, asymmetric and trichotomic relation. The tournament $ (T,\rightarrow)$ is transitive if $ \rightarrow$ is transitive, that is, if it orders $ T$.) [i]A. Hajnal[/i]

Let $A$ be a non empty subset of positive reals such that for every $a,b,c \in A$, the number $ab+bc+ca$ is rational. Prove that $\frac{a}{b}$ is a rational for every $a,b \in A$.

Given are twenty-two different five-element sets, such that any two of them have exactly two elements in common. Prove that they all have two elements in common.

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 $n\ge k$ be positive integers, and let $\mathcal{F}$ be a family of finite sets with the following properties: (i) $\mathcal{F}$ contains at least $\binom{n}{k}+1$ distinct sets containing exactly $k$ elements; (ii) for any two sets $A, B\in \mathcal{F}$, their union $A\cup B$ also belongs to $\mathcal{F}$. Prove that $\mathcal{F}$ contains at least three sets with at least $n$ elements. (Proposed by Fedor Petrov, St. Petersburg State University)

Having three sets $ A,B\subset C, $ solve the set equation $ (X\cup (C\setminus A))\cap ((C\setminus X)\cup A)=B. $

The points $A_1,A_...,A_n$ lie on a circle with radius 1. The points $B_1,B_2,…,B_n$ are such that $B_i B_j<A_i A_j$ for $i\neq j$. Is it always true that the points $B_1,B_2,...,B_n$ lie on a circle with radius lesser than 1?

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*}

The elements of the set $F$ are some subsets of $\left\{1,2,\ldots ,n\right\}$ and satisfy the conditions: i) if $A$ belongs to $F$, then $A$ has three elements; ii)if $A$ and $B$ are distinct elements of $F$ , then $A$ and $B$ have at most one common element. Let $f(n)$ be the greatest possible number of elements of $F$. Prove that $\frac{n^{2}-4n}{6}\leq f(n) \leq \frac{n^{2}-n}{6}$