In one of the hotels of the wellness planet Oxys, there are $2019$ saunas. The managers have decided to accommodate $k$ couples for the upcoming long weekend. We know the following about the guests: if two women know each other then their husbands also know each other, and vice versa. There are several restrictions on the usage of saunas. Each sauna can be used by either men only, or women only (but there is no limit on the number of people using a sauna at once, as long as they are of a single gender). Each woman is only willing to share a sauna with women whom she knows, and each man is only willing to share a sauna with men whom he does not know. What is the greatest possible $k$ for which we can guarantee, without knowing the exact relationships between the couples, that all the guests can use the saunas simultaneously while respecting the restrictions above?

Cyclic quadrilateral $ABCD$ is inscribed in circle $k$ with center $O$. The angle bisector of $ABD$ intersects $AD$ and $k$ in $K,M$ respectively, and the angle bisector of $CBD$ intersects $CD$ and $k$ in $L,N$ respectively. If $KL\parallel MN$ prove that the circumcircle of triangle $MON$ bisects segment $BD$.

Find three different positive integers whose sum is minimum than meet the condition that the sum of each pair of them is a perfect square.

Let $\mathbf T$ be a triangle with $[\mathbf T]=1.$ Show that there is a right triangle $\mathbf R$ such that $[\mathbf R]\le\sqrt3$ and $\mathbf T\subseteq\mathbf R.$ ($[-]$ denotes area of a triangle.)

Triangle $ABC$ has side lengths $AB = 5$, $BC = 9$, and $AC = 6$. Define the incircle of $ABC$ to be $C_1$. Then, define $C_i$ for $i > 1$ to be externally tangent to $C_{i-1}$ and tangent to $AB$ and $BC$. Compute the sum of the areas of all circles $C_n$.

Let $ABC$ be a triangle with $AB \ne AC$ and $ I$ its incenter. Let $M$ be the midpoint of the side $BC$ and $D$ the projection of $I$ on $BC.$ The line $AI$ intersects the circle with center $M$ and radius $MD$ at $P$ and $Q.$ Prove that $\angle BAC + \angle PMQ = 180^{\circ}.$

Consider a $12$-gon with sidelengths $1$, $2$, $3$, $4$, ..., $12$. Prove that there are three consecutive sides in this $12$-gon, whose lengths have a sum $> 20$.

Set $X$ is called [i]fancy[/i] if it satisfies all of the following conditions: [list] [*]The number of elements of $X$ is $2022$. [*]Each element of $X$ is a closed interval contained in $[0, 1]$. [*]For any real number $r \in [0, 1]$, the number of elements of $X$ containing $r$ is less than or equal to $1011$. [/list] For [i]fancy[/i] sets $A, B$, and intervals $I \in A, J \in B$, denote by $n(A, B)$ the number of pairs $(I, J)$ such that $I \cap J \neq \emptyset$. Determine the maximum value of $n(A, B)$.

Find all positive integers $n$ that can be represented in the form $$n=\mathrm{lcm}(a,b)+\mathrm{lcm}(b,c)+\mathrm{lcm}(c,a)$$ where $a,b,c$ are positive integers.

There are nine cards that have the digits $1, 2, 3, 4, 5, 6, 7, 8$ and $9$ written on them, with one digit on each card. Using all the cards, some numbers are formed (for example, the numbers $8$, $213$, $94$, $65$ and $7$). a) If all the numbers formed are prime, determine the smallest possible value of their sum. b) If all formed numbers are composite, determine the smallest possible value of their sum. Note: A number $p$ is prime if its only divisors are $1$ and $p$. A number is composite if it has more than two dividers. The number $1$ is neither prime nor composite.

The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$, such that $CD=BC$. The side $CA$ is extended beyond $A$ to $E$, such that $AE=2CA$. Prove that if $AD=BE$, then the triangle $ABC$ is right.

Prove that there are infinitely many different natural numbers of the form $k^2 + 1$, $k \in N$ that have no real divisor of this form.

Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have: \[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\] Show that this sequence has a finite limit. Determine this limit.

Let $a_1<a_2$ be two given integers. For any integer $n\ge 3$, let $a_n$ be the smallest integer which is larger than $a_{n-1}$ and can be uniquely represented as $a_i+a_j$, where $1\le i<j\le n-1$. Given that there are only a finite number of even numbers in $\{a_n\}$, prove that the sequence $\{a_{n+1}-a_{n}\}$ is eventually periodic, i.e. that there exist positive integers $T,N$ such that for all integers $n>N$, we have \[a_{T+n+1}-a_{T+n}=a_{n+1}-a_{n}.\]

Is there a power of $2$ such that it is possible to rearrange the digits, giving another power of $2$?

Let $X$ denotes the set of integers from $1$ to $239$. A magician with an assistant perform a trick. The magician leaves the hall and the spectator writes a sequence of $10$ elements on the board from the set $X$. The magician’s assistant looks at them and adds $k$ more elements from $X$ to the existing sequence. After that the spectator replaces three of these $k+10$ numbers by random elements of $X$ (it is permitted to change them by themselves, that is to not change anything at all, for example). The magician enters and looks at the resulting row of $k+10$ numbers and without error names the original $10$ numbers written by the spectator. Find the minimal possible $k$ for which the trick is possible.