In a football tournament with $n\geq 2$ teams each two played a match. For a won match the victor gets 2 points and for a draw each one gets 1 point. In the final results there weren’t two teams with equal amount of points. It turned out that because of a mistake each match that was written in the results as won was actually a draw and each one that was written as draw was actually won. In the new ranking there were also no two teams with the same amount of points. Find all n for which it is possible for the two rankings to be opposite of each other, that is the first team in the first ranking is actually the last one, the second team is pre-last and so on.

Does there exist a solution in integers for the equation $a^2+b^2+c^2+d^2+e^2=abcde-78$ where $a,b,c,d,e>2022$?

Let $p_1,p_2,\dots ,p_n$ be all prime numbers lesser than $2^{100}$. Prove that $\frac{1}{p_1} +\frac{1}{p_2} +\dots +\frac{1}{p_n} <10$.

Does there exist a surjective function $f:\mathbb{R} \rightarrow \mathbb{R}$ for which $f(x+y)-f(x)-f(y)$ takes only 0 and 1 for values for random $x$ and $y$?

Let $\Delta ABC$ be an acute scalene triangle with $AC<BC$, an orthocenter $H$ and altitudes $AE$, $BF$. The points $E'$ and $F'$ are symmetrical to $E$ and $F$ with respect to $A$ and $B$ respectively. Point $O$ is the center of the circumscribed circle of $ABC$ and $M$ is the midpoint of $AB$. Let $N$ be the midpoint of $OM$. Prove that the tangent through $H$ to the circumscribed circle of $\Delta E'HF'$ is perpendicular to line $CN$.

Let $k$ be a fixed circle in a given plane and a point $C$ out of the plane. Let $A$ be a random point from $k$ and $B$ be its diametrically opposite one in $k$. Find the geometric place of the center of the circumscribed circle of $ABC$.

Let’s note the set of all integers $n>1$ which are not divisible by a square of a prime number. We define the number $f(n)$ as the greatest amount of divisors of $n$ which could be chosen in such way so that for each two chosen $a$ and $b$, not necessarily different, the number $a^2+ab+b^2+n$ is not a square. Find all $m$ for which there exists $n$ so that $f(n)=m$.

A subset of the set $A={1,2,\dots ,n}$ is called connected, 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 connected set.

Let $ABC$ be a triangle for which the shortest side is $AC$. Its inscribed circle with center $I$ touches sides $AB$ and $BC$ in points $D$ and $E$ respectively. Point $M$ is the midpoint of $AC$. Points $F$ and $G$ lie on sides $BC$ and $AB$ respectively so that $FC=CA=AG$. The line through $I$ perpendicular to $MI$ intersects the line segments $AF$ and $CG$ in $P$ and $Q$ respectively. Prove that $AB=BC\Leftrightarrow PD=QE$.

Let $ABC$ be a triangle with $\angle BAC=40^\circ $, $O$ be the center of its circumscribed circle and $G$ is its centroid. Point $D$ of line $BC$ is such that $CD=AC$ and $C$ is between $B$ and $D$. If $AD\parallel OG$, find $\angle ACB$.

The set of quadruples $(a,b,c,d)$ where each of $a,b,c,d$ is either $0$ or $1$ is called vertices of the four dimensional unit cube or 4-cube for short. Two vertices are called adjacent, if their respective quadruples differ by one variable only. Each two adjacent vertices are connected by an edge. A robot is moving through the edges of the 4-cube starting from $(0,0,0,0)$ and each turn consists of passing an edge and moving to adjacent vertex. In how many ways can the robot go back to $(0,0,0,0)$ after $4042$ turns? Note that it is NOT forbidden for the robot to pass through $(0,0,0,0)$ before the $4042$-nd turn.

a) Prove that for each positive integer $n$ the number or ordered pairs of integers $(x,y)$ for which $x^2-xy+y^2=n$ is finite and is multiple of 6. b) Find all ordered pairs of integers $(x,y)$ for which $x^2-xy+y^2=727$.

Let $a$, $b$ and $c$ be given positive integers which are two by two coprime. A positive integer $n$ is called sozopolian, if it can’t be written as $n=bcx+cay+abz$ where $x$, $y$, $z$ are also positive integers. Find the number of sozopolian numbers as a function of $a$, $b$ and $c$.

Let $D$ be an infinite in both sides sequence of $0$s and $1$s. For each positive integer $n$ we denote with $a_n$ the number of different subsequences of $0$s and $1$s in $D$ of length $n$. Does there exist a sequence $D$ for which for each $n\geq 22$ the number $a_n$ is equal to the $n$-th prime number?

Find the least possible value of the following expression $\lfloor \frac{a+b}{c+d}\rfloor +\lfloor \frac{a+c}{b+d}\rfloor +\lfloor \frac{a+d}{b+c}\rfloor + \lfloor \frac{c+d}{a+b}\rfloor +\lfloor \frac{b+d}{a+c}\rfloor +\lfloor \frac{b+c}{a+d}\rfloor$ where $a$, $b$, $c$ and $d$ are positive real numbers.

Let $x$ be a real number. Find the greatest possible value of the following expression: $\frac{47^x}{\sqrt{43}}+\frac{43^x}{\sqrt{47}}-2021^x$.

Let $p$ and $q$ be given prime numbers and $S$ be a subset of ${1,2,3,\dots ,p-2,p-1}$. Prove that the number of elements in the set $A=\{ (x_1,x_2,…,x_q ):x_i\in S,\sum_{i=1}^q x_i \equiv 0(mod\: p)\}$ is multiple of $q$.

We say that a rectangle and a triangle are similar, if they have the same area and the same perimeter. Let $P$ be a rectangle for which the ratio of the longer to the shorter side is at least $\lambda -1+\sqrt{\lambda (\lambda -2)}$ where $\lambda =\frac{3\sqrt{3}}{2}$. Prove that there exists a tringle that is similar to $P$.

The positive integers $p$, $q$ are such that for each real number $x$ $(x+1)^p (x-3)^q=x^n+a_1 x^{n-1}+a_2 x^{n-2}+\dots +a_{n-1} x+a_n$ where $n=p+q$ and $a_1,\dots ,a_n$ are real numbers. Prove that there exists infinitely many pairs $(p,q)$ for which $a_1=a_2$.

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 $1.67^n$.

Prove that $\sum_{n=1}^{2022^{2022}} \frac{1}{\sqrt{n^3+2n^2+n}}<\frac{19}{10}$.

A mixing of the sequence $a_1,a_2,\dots ,a_{3n}$ is called the following sequence: $a_3,a_6,\dots ,a_{3n},a_2,a_5,\dots ,a_{3n-1},a_1,a_4,\dots ,a_{3n-2}$. Is it possible after finite amount of mixings to reach the sequence $192,191,\dots ,1$ from $1,2,\dots ,192$?

Points $M$, $N$, $P$ and $Q$ are midpoints of the sides $AB$, $BC$, $CD$ and $DA$ of the inscribed quadrilateral $ABCD$ with intersection point $O$ of its diagonals. Let $K$ be the second intersection point of the circumscribed circles of $MOQ$ and $NOP$. Prove that $OK\perp AC$.

Determine the number of ordered quadruples of integers $(a,b,c,d)$ for which $0\leq a,b,c,d\leq 36$ and $37|a^2+b^2-c^3-d^3$.

Are there natural numbers $n$ and $N$ such that $n > 10^{10}$, $$n^n < 2^{2^{\frac{8N}{\omega (N)}}}$$and $n$ is divisible by $p^{2022(v_p(N)-1)}(p-1)$ for every prime divisor $p$ of $N$? (For a natural number $N$, we denote by $\omega (N)$ the number of its different prime divisors and with $v_p(N)$ the power of the prime number $p$ in its canonical representation.)

Finding all quads of integers $(a, b, c, p)$ where $p \ge 5$ is prime number such that the remainders of the numbers $am^3 + bm^2 + cm$, $m = 0, 1, . . . , p - 1$, upon division of $p$ are two by two different..

Given an acute-angled $\vartriangle AB$C with altitude $AH$ ( $\angle BAC > 45^o > \angle AB$C). The perpendicular bisector of $AB$ intersects $BC$ at point $D$. Let $K$ be the midpoint of $BF$, where $F$ is the foot of the perpendicular from $C$ on $AD$. Point $H'$ is the symmetric to $H$ wrt $K$. Point $P$ lies on the line $AD$, such that $H'P \perp AB$. Prove that $AK = KP$.

Let $n$ be a natural number. To prove that the value of the expression $$\prod^n_{i=0}\frac{x^{n-1}_i}{\prod_{j \ne i}(x_i - x_j)}$$does not depend on the choice of the different real numbers $x_0, x_1, ... , x_n$.

Find the number of subsets of $\{1, 2,... , 2100\}$ such that each has sum of the elements giving a remainder of $3$ when divided by $7$.

For the function $f : Z^2_{\ge0} \to Z_{\ge 0}$ it is known that $$f(0, j) = f(i, 0) = 1, \,\,\,\,\, \forall i, j \in N_0$$$$f(i, j) = if (i, j - 1) + jf(i - 1, j),\,\,\,\,\, \forall i, j \in N$$Prove that for every natural number $n$ the following inequality holds: $$\sum_{0\le i+j\le n+1} f(i, j) \le 2 \left(\sum^n_{k=0}\frac{1}{k!}\right)\left(\sum^n_{p=1}p!\right)+ 3$$

Given an acute-angled $\vartriangle ABC$ with orthocenter $H$ and altitude $CC_1$. Points $D, E$ and $F$ lie on the segments $AC$, $BC$ and $AB$ respectively, so that $DE \parallel AB$ and $EF \parallel AC$. Denote by $Q$ the symmetric point of $H$ wrt to the midpoint of $DE$. Let $BD \cap CF = P$. If $HP \parallel AB$, prove that the points $C_1, D, Q$ and $E$ lie on a circle.

A magician wants to demonstrate the following trick to an audience of $n \ge 16$ people. He gives them $15$ hats and after giving instructions to his assistant (which the audience does not hear), leaves the hall. Some $15$ people in the audience put on one of the hats. The assistant tags in front of everyone, one of the hats with a marker and then the person with an unmarked hat takes it off. The magician then returns back to the hall and after surveying the situation, knows who in the audience has taken off his hat. For what $n$ is this possible? original wordingМагьосник иска да покаже следния фокус пред публика от $n \ge 16$ души. Той им дава $15$ шапки и след като даде инструкции на помощника си (които публиката не чува), напуска залата. Някои $15$ души от публиката си слагат по една от шапките. Асистентът маркира пред всички една от шапките с маркер и след това човек с немаркирана шапка си я сваля. След това магьосникът се връща обратно в залата и след оглед на ситуацията познава кой от публиката си е свалил шапката. За кои $n$ е възможно това?

Find all triples of complex numbers $(x, y, z)$ for which $$(x + y)^3 + (y + z)^3 + (z + x)^3 - 3(x + y)(y + z)(z + x) = x^2(y + z) + y^2(z + x ) + z^2(x + y) = 0$$

Let $k$ be the circumcircle of the acute triangle $ABC$. Its inscribed circle touches sides $BC$, $CA$ and $AB$ at points $D, E$ and $F$ respectively. The line $ED$ intersects $k$ at the points $M$ and $N$, so that $E$ lies between $M$ and $D$. Let $K$ and $L$ be the second intersection points of the lines $NF$ and $MF$ respectively with $k$. Let $AK \cap BL = Q$. Prove that the lines $AL$, $BK$ and $QF$ intersect at a point.

Quadrilateral $ABCD$ is circumscribed around circle $k$. Gind the smallest possible value of $$\frac{AB + BC + CD + DA}{AC + BD}$$, as well as all quadrilaterals with the above property where it is reached.

A natural number $x$ is written on the board. In one move, we can take the number on the board and between any two of its digits in its decimal notation we can we put a sign $+$, or we may not put it, then we calculate the obtained result and we write it on the board in place of $x$. For example, from the number $819$. we can get $18$ by $8 + 1 + 9$, $90$ by $81 + 9$, and $27$ by $8 + 19$. Prove that no matter what $x$ is, we can reach a single digit number with at most $4$ moves.

Find all functions $f : N \to N$ such that $f(p)$ divides $f(n)^p -n$ by any natural number $n$ and prime number $p$.

Let $n$ be a natural number and $P_1, P_2, ... , P_n$ are polynomials with integer coefficients, each of degree at least $2$. Let $S$ be the set of all natural numbers $N$ for which there exists a natural number $a$ and an index $1 \le i \le n$ such that $P_i(a) = N$. Prove, that there are infinitely many primes that do not belong to $S$.

A graph $ G$ with $ n$ vertices is given. Some $ x$ of its edges are colored red so that each triangle has at most one red edge. The maximum number of vertices in $ G$ that induce a bipartite graph equals $ y.$ Prove that $ n\ge 4x/y.$

Let $p$ and $q$ be mutually prime natural numbers greater than $1$. Starting with the permutation $(1, 2, . . . , n)$, in one move we can switch the places of two numbers if their difference is $p$ or $q$. Prove that with such moves we can get any another permutation if and only if $n \ge p + q - 1$.