Let $a, b$ and $n$ be positive integers with $a>b$ such that all of the following hold: i. $a^{2021}$ divides $n$, ii. $b^{2021}$ divides $n$, iii. 2022 divides $a-b$. Prove that there is a subset $T$ of the set of positive divisors of the number $n$ such that the sum of the elements of $T$ is divisible by 2022 but not divisible by $2022^2$. [i]Proposed by Silouanos Brazitikos, Greece[/i]

Let $ACE$ be a triangle with $\angle ECA=60^{\circ}, \angle AEC=90^{\circ}$. Let $B$ and $D$ be points on the sides $AC$ and $CE$ respectively such that the $\triangle BCD$ is equilateral. Now suppose $BD \cap AE=F$. Find $\angle EAC+\angle EFD$.

Find all functions $f : R \to R$ satisfying $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.

[b]6.[/b] Let $T$ be a one-to-one mapping of the unit square $E$ of the plane into itself. Suppose that $T$ and $T^{-1}$ are measure-preserving (i.e. if $M \subseteq E$ is a measurable set, then $TM$ and $T^{-1}M$ are also measurable and $\mu (M)= \mu (TM)= \mu (T^{-1}M)$, where $\mu$ denotes the Lebesgue measure) and, furthermore, that if $Tx \in N$ for almost all points $x$ of a measurable set $N \subseteq E$, then either $n$ or $ E \setminus N$ is of measure 0. Prove that, for any measurable set $A \subseteq E$, with $\mu (A)>0$, the function $n(x)$ defined by $$n(x)=\begin{cases} 0, \mbox{if} \quad T^k x \notin A \quad (k=1, 2, \dots),\\ \min (k: T^k x \in A; k=1,2, \dots ) &\mbox{otherwise} \end{cases} $$ is measurable and $\int_{A}n(x) d\mu(x) =1$ [b](R. 18)[/b]

Find the number of ordered palindromic partitions of an integer $n$.

Let $f(x) = x^{2}(1-x)^{2}$. What is the value of the sum \begin{align*} f\left(\frac{1}{2019}\right)-f\left(\frac{2}{2019}\right)+f\left(\frac{3}{2019}\right)-&f\left(\frac{4}{2019}\right)+\cdots\\ &\,+f\left(\frac{2017}{2019}\right) - f\left(\frac{2018}{2019}\right)? \end{align*} $\textbf{(A) }0\qquad\textbf{(B) }\frac{1}{2019^{4}}\qquad\textbf{(C) }\frac{2018^{2}}{2019^{4}}\qquad\textbf{(D) }\frac{2020^{2}}{2019^{4}}\qquad\textbf{(E) }1$

Five people are sitting at a round table. Let $f \ge 0$ be the number of people sitting next to at least one female and $m \ge 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11 $

Parallelogram ${ABCD}$with obtuse angle $\angle ABC$ is given. After rotation of the triangle ${ACD}$ around the vertex ${C}$, we get a triangle ${CD'A'}$, such that points $B,C$ and ${D'}$are collinear. Extensions of median of triangle ${CD'A'}$ that passes through ${D'}$intersects the straight line ${BD}$ at point ${P}$. Prove that ${PC}$is the bisector of the angle $\angle BP{D}'$.

Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

Define $a_0 = 2$ and $a_{n+1} = a^2_n + a_n -1$ for $n \ge 0$. Prove that $a_n$ is coprime to $2n + 1$ for all $n \in N$.

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous, strictly decreasing function such that $f([0,1])\subseteq[0,1]$. [list=i] [*]For all positive integers $n{}$ prove that there exists a unique $a_n\in(0,1)$, solution of the equation $f(x)=x^n$. Moreover, if $(a_n){}$ is the sequence defined as above, prove that $\lim_{n\to\infty}a_n=1$. [*]Suppose $f$ has a continuous derivative, with $f(1)=0$ and $f'(1)<0$. For any $x\in\mathbb{R}$ we define \[F(x)=\int_x^1f(t) \ dt.\]Let $\alpha{}$ be a real number. Study the convergence of the series \[\sum_{n=1}^\infty F(a_n)^\alpha.\] [/list]

Given a board $3 \times 2021$, all cells of which are white. Two players in turns colour two white cells, which are either in the same row or column, in black. A player, which can not make a move, loses. Which of the player can guarantee his win regardless of the moves of his opponent?

Compute the product of all positive integers $n$ for which $3(n!+1)$ is divisible by $2n - 5$.

Given circle $O$ with radius $R$, the inscribed triangle $ABC$ is an acute scalene triangle, where $AB$ is the largest side. $AH_A, BH_B,CH_C$ are heights on $BC,CA,AB$. Let $D$ be the symmetric point of $H_A$ with respect to $H_BH_C$, $E$ be the symmetric point of $H_B$ with respect to $H_AH_C$. $P$ is the intersection of $AD,BE$, $H$ is the orthocentre of $\triangle ABC$. Prove: $OP\cdot OH$ is fixed, and find this value in terms of $R$. (Edited)

Let $ABC$ be an acute triangle with $AC > AB$. Let $\Gamma$ be the circle circumscribed to the triangle $ABC$ and $D$ the midpoint of the smaller arc $BC$ of this circle. Let $I$ be the incenter of $ABC$ and let $E$ and $F$ be points on sides $AB$ and $AC$, respectively, such that $AE = AF$ and $I$ lies on the segment $EF$. Let $P$ be the second intersection point of the circumcircle of the triangle $AEF$ with $\Gamma$ with $P \ne A$. Let $G$ and $H$ be the intersection points of the lines $PE$ and $PF$ with $\Gamma$ different from $P$, respectively. Let $J$ and $K$ be the intersection points of lines $DG$ and $DH$ with lines AB and $AC$, respectively. Show that the line $JK$ passes through the midpoint of $BC$.

Check and justify , if in every tetrahedron are concurrent: a) The perpendiculars to the faces at their circumcenters. b) The perpendiculars to the faces at their orthocenters. c) The perpendiculars to the faces at their incenters. If so, characterize with some simple geometric property the point in that attend If not, show an example that clearly shows the not concurrency.

Let $f(x)$ be a polynomial with integer coefficients, for which there exist $a,b\in \mathbb{Z}$ ($a\neq b$), such that $f(a)$ and $f(b)$ are coprime. Prove that there exist infinitely many values for $x$, such that each $f(x)$ is coprime with any other.

Let $0<\alpha<\beta <\gamma\in \mathbb{R}$. Let $K=\{(x,y,z)\in \mathbb{R}^{3}\;|\; x,y,z\geq 0\; \text{and}\; x^{\beta}+y^{\beta}+z^{\beta}=1\}$. Define $f:K\rightarrow \mathbb{R},\; (x,y,z)\mapsto x^{\alpha}+y^{\beta}+z^{\gamma}$. At what points of $K$ does $f$ assume its minimal and maximal values?

Let $A$ be an integer and $A>1$. Let $a_{1}=A^{A}$, $a_{n+1}=A^{a_{n}}$ and $b_{1}=A^{A+1}$, $b_{n+1}=2^{b_{n}}$, $n=1, 2, 3, ...$. Prove that $a_{n}<b_{n}$ for each $n$.

Three persons $A,B,C$, are playing the following game: A $k$-element subset of the set $\{1, . . . , 1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to $1986$. The winner is $A,B$ or $C$, respectively, if the sum of the chosen numbers leaves a remainder of $0, 1$, or $2$ when divided by $3$. For what values of $k$ is this game a fair one? (A game is fair if the three outcomes are equally probable.)

Planet Tarator is a planet in the Yoghurty way galaxy. This planet has a shape of convex $1392$-hedron. On earth we don't have any other information about sides of planet tarator. We have discovered that each side of the planet is a country, and has it's own currency. Each two neighbour countries have their own constant exchange rate, regardless of other exchange rates. Anybody who travels on land and crosses the border must change all his money to the currency of the destination country, and there's no other way to change the money. Incredibly, a person's money may change after crossing some borders and getting back to the point he started, but it's guaranteed that crossing a border and then coming back doesn't change the money. On a research project a group of tourists were chosen and given same amount of money to travel around the Tarator planet and come back to the point they started. They always travel on land and their path is a nonplanar polygon which doesn't intersect itself. What is the maximum number of tourists that may have a pairwise different final amount of money? [b]Note 1:[/b] Tourists spend no money during travel! [b]Note 2:[/b] The only constant of the problem is 1392, the number of the sides. The exchange rates and the way the sides are arranged are unknown. Answer must be a constant number, regardless of the variables. [b]Note 3:[/b] The maximum must be among all possible polyhedras. Time allowed for this problem was 90 minutes.

There are $n$ lamps $L_1, L_2, \dots, L_n$ arranged in a circle in that order. At any given time, each lamp is either [i]on[/i] or [i]off[/i]. Every second, each lamp undergoes a change according to the following rule: (a) For each lamp $L_i$, if $L_{i-1}, L_i, L_{i+1}$ have the same state in the previous second, then $L_i$ is [i]off[/i] right now. (Indices taken mod $n$.) (b) Otherwise, $L_i$ is [i]on[/i] right now. Initially, all the lamps are [i]off[/i], except for $L_1$ which is [i]on[/i]. Prove that for infinitely many integers $n$ all the lamps will be [i]off[/i] eventually, after a finite amount of time.

Let $(a_{n})_{n\ge 0}$ and $(b_{n})_{n\ge 0}$ be two sequences with arbitrary real values $a_0, a_1, b_0, b_1$. For $n\ge 1$, let $a_{n+1}, b_{n+1}$ be defined in this way: $$a_{n+1}=\dfrac{b_{n-1}+b_{n}}{2}, b_{n+1}=\dfrac{a_{n-1}+a_{n}}{2}$$ Prove that for any constant $c>0$ there exists a positive integer $N$ s.t. for all $n>N$, $|a_{n}-b_{n}|<c$.

Show that there are infinitely many pairs $(m, n)$ of natural numbers $m, n \ge 2$, for $m^m- 1$ is divisible by $n$ and $n^n- 1$ is divisible by $m$.

Let $ABC$ be an equilateral triangle with $AB=a$ and $M\in(AB)$ a fixed point. Points $N\in(AC)$ and $P\in(BC)$ are taken such that the perimeter of $MNP$ is minimal. If the ratio between the areas of triangles $MNP$ and $ABC$ is $\textstyle\frac{7}{30},$ find the perimeter of triangle $MNP.$