A number is written on the board. Petya can change the number on the board to the sum of the squares of digits of the number on the board. A number is called interesting if Petya, when starting from this number, will not ever get the number on the board to be $1$. Prove that there infinitely many interesting numbers.

Integers $a,b$ and $c$ satisfy the equality $a+b+c=0$. Denote $S=ab+bc+ac$, $A=a^2+a+1$, $B=b^2+b+1$ and $C=c^2+c+1$. Prove that the number $(S+A)(S+B)(S+C)$ is a perfect square.

Inside a square $ABCD$ point $P$ is marked, and on the sides $AB$, $BC$, $CD$ and $DA$ points $K,L,M$ and $N$ are chosen respectively. Lines $KP,LP,MP$ and $NP$ intersect sides $CD,DA,AB$ and $BC$ at points $K_1, L_1, M_1$ and $N_1$ respectively. It turned out that $$\frac{KP}{PK_1}+\frac{LP}{PL_1}+\frac{MP}{PM_1}+\frac{NP}{PN_1}=4$$Prove that $KP+LP+MP+NP=K_1P+L_1P+M_1P+N_1P$.

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?

Inside a triangle $ABC$ three circles with radius $1$ are drawn. (Circles can be tangent to each other and to the sides of the triangle, but can not have any common internal points.) Find the biggest value of $r$ for which one can state that he can always draw a fourth circle inside the triangle of radius $r$, which does not intersect three already drawn circles.

A table $2022 \times 2022$ is divided onto the tiles of two types: $L$-tetromino and $Z$-tetromino. Determine the least amount of $Z$-tetromino one needs to use.

A polynomial $p(x)$ with integer coefficients satisfies the equality $$p(\sqrt{2}+\sqrt{3})=\sqrt{2}-\sqrt{3}$$a) Find all possible values of $p(\sqrt{2}-\sqrt{3})$ b) Find an example of any polynomial $p(x)$ which satisfies the condition.

Vitya and Masha are playing a game. At first, Vitya thinks of three different integers. In one move Masha can ask one of the following three numbers: the sum of the numbers, the product of the numbers or the sum of pairwise products of the numbers. Masha asks questions and Vitya immediately answers before Masha asks the next question. a) Prove that Masha can always guess Vitya's numbers. b) What is the least amount of questions Masha needs to ask to guaranteely guess them?

Given an isosceles triangle $ABC$ with base $BC$. On the sides $BC$, $AC$ and $AB$ points $X,Y$ and $Z$ are chosen respectively such that triangles $ABC$ and $YXZ$ are similar. Point $W$ is symmetric to point $X$ with respect to the midpoint of $BC$. Prove that points $X,Y,Z$ and $W$ lie on a circle.

Prove the inequality $$\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\ldots+\frac{1}{2022!}>\frac{1^2}{2!}+\frac{2^2}{3!}+\frac{3^2}{4!}+\ldots+\frac{2022^2}{2023!}$$.

Positive integers $a$ and $b$ satisfy the equality $a+d(a)=b^2+2$ where $d(n)$ denotes the number of divisors of $n$. Prove that $a+b$ is even.

Numbers $1,2,\ldots,50$ are written on the board. Anya does the following operation: removes the numbers $a$ and $b$ from the board and writes their sum - $a+b$, after which also notes down the number $ab(a+b)$. After $49$ of this operations only one number was left on the board. Anya summed up all the $49$ numbers in her notes and got $S$. a) Prove that $S$ does not depend on the order of Anya's actions. b) Calculate $S$.

Given $n \geq 2$ distinct integers, which are bigger than $-10$. It turned out that the amount of odd numbers among them is equal to the biggest even number, and the amount of even to the biggest of odd. a) Find the smallest $n$ possible b) Find the biggest $n$ possible

Given triangle $ABC$ in which $\angle CAB= 30^{\circ}$ and $\angle ACB=60^{\circ}$. On the ray $AB$ a point $D$ is chosen, and on the ray $CB$ a point $E$ is chosen such that $\angle BDE=60^{\circ}$. Lines $AC$ and $DE$ intersect at $F$. Prove that the circumcircle of $AEF$ passes through a fixed point, which is different from $A$ and does not depend on $D$.

Prove that for any positive integer $n$ there exist coprime numbers $a$ and $b$ such that for all $1 \leq k \leq n$ numbers $a+k$ and $b+k$ are not coprime.

Does there exist a polynomial $p(x)$ with integer coefficients for which $$p(\sqrt{2})=\sqrt{2}$$$$p(2\sqrt{2})=2\sqrt{2}+2$$

Prove that for any positive integer one can place all it's divisor on a circle such that among any two neighbours one is a multiple of the other

A positive integer $n$ is given. On the segment $[0,n]$ of the real line $m$ distinct segments whose endpoints have integer coordinates are chosen. It turned out that it is impossible to choose some of thos segments such that their total length is $n$ and their union is $[0,n]$ Find the maximum possible value of $m$

Through the point $F(0,\frac{1}{4})$ of the coordinate plane two perpendicular lines pass, that intersect parabola $y=x^2$ at points $A,B,C,D$ ($A_x<B_x<C_x<D_x$) The difference of projections of segments $AD$ and $BC$ onto the $Ox$ line is $m$ Find the area of $ABCD$

On the semicircle with diameter $AB$ and center $O$ point $D$ is marked. Points $E$ and $F$ are the midpoints of minor arcs $AD$ and $BD$ respectively. It turned out that the line connecting orthocenters of $ADF$ and $BDE$ passes through $O$ Find $\angle AOD$

$n$ distinct integers are given, all of which are bigger than $-a$, where $a$ is a positive integer. It turned out that the amount of odd numbers among them is equal to the biggest even number, and the amount of even numbers is equal to the biggest odd numbers a) Find the least possible value of $n$ for all $a$ b) For each $a \geq 2$ find the maximum possible value of $n$

Circles $\omega_1$ and $\omega_2$ intersect at $X$ and $Y$. Through point $Y$ two lines pass, one of which intersects $\omega_1$ and $\omega_2$ for the second time at $A$ and $B$, and the other at $C$ and $D$. Line $AD$ intersects for the second time circles $\omega_1$ and $\omega_2$ at $P$ and $Q$. It turned out that $YP=YQ$ Prove that the circumcircles of triangles $BCY$ and $PQY$ are tangent to each other.

Find all positive integers $a$ for which there exists a polynomial $p(x)$ with integer coefficients such that $p(\sqrt{2}+1)=2-\sqrt{2}$ and $p(\sqrt{2}+2)=a$

A sequence $a_1,\ldots,a_n$ of positive integers is given. For each $l$ from $1$ to $n-1$ the array $(gcd(a_1,a_{1+l}),\ldots,gcd(a_n,a_{n+l}))$ is considered, where indices are taken modulo $n$. It turned out that all this arrays consist of the same $n$ pairwise distinct numbers and differ only,possibly, by their order. Can $n$ be a) $21$ b) $2021$

A sequence of positive integer numbers $a_1,a_2,\ldots$ for $i \geq 3$ satisfies $$a_{i+1}=a_i+gcd(a_{i-1},a_{i-2})$$Prove that there exist two positive integer numbers $N, M$, such that $a_{n+1}-a_n=M$ for all $n \geq N$

Two perpendicular lines pass through the point $F(1;1)$ of coordinate plane. One of them intersects hyperbola $y=\frac{1}{2x}$ at $A$ and $C$ ($C_x>A_x$), and the other one intersects the left part of hyperbola at $B$ and the right at $D$. Let $m=(C_x-A_x)(D_x-B_x)$ Find the area of non-convex quadraliteral $ABCD$ (in terms of $m$)

$2021$ points are marked on a circle. $2021$ segments with marked endpoints are drawn. After that one counts the number of different points where some $2$ drawn segments intersect(endpoints of segments do not count as intersections) Find the maximum number one can get.

On plane circles $\omega_1, \omega_2, \omega_3$ with centers $O_1,O_2,O_3$ are given such that $\omega_1$ is externally tangent $\omega_2$ and $\omega_3$ at points $P, Q$ respectively. On $\omega_1$ point $C$ is chosen arbitrarily. Line $CP$ intersects $\omega_2$ at $B$, line $CQ$ intersects $\omega_3$ at $A$. Point $O$ is the circumcenter of $ABC$. Prove that the locus of points $O$ (when $C$ changes) is a circle, the center of which lies on the circumcircle of $O_1O_2O_3$

In cells of a $2022 \times 2022$ table numbers from $1$ to $2022^2$ are written, in each cell exactly one number, all numbers are used once. For every row Vlad marks the second biggest number in it, Dima does the same for every column. It turned out that boys marked $4044$ pairwise distinct numbers, and there are $k$ numbers marked by Vlad, each of which is less than all numbers marked by Dima. Find the maximum possible value of $k$

The incircle of a right-angled triangle $ABC$ touches hypotenus $AB$ at $P$, $BC$ and $AC$ at $R$ and $Q$ respectively. $C_1$ and $C_2$ are reflections of $C$ in $PQ$ and $PR$. Find the angle $C_1IC_2$, where $I$ is the incenter of $ABC$.

Numbers $-1011, -1010, \ldots, -1, 1, \ldots, 1011$ in some order form the sequence $a_1,a_2,\ldots, a_{2022}$. Find the maximum possible value of the sum $$|a_1|+|a_1+a_2|+\ldots+|a_1+\ldots+a_{2022}|$$

A polynomial $P(x,y)$ with integer coefficients satisfies two following conditions: 1. for every integer $a$ there exists exactly one integer $y$, such that $P(a,y)=0$ 2. for every integer $b$ there exists exactly one integer $x$, such that $P(x,b)=0$ a) Prove that if the degree of $P$ is $2$, then it is divisible by either $x-y+C$ for some integer $C$, or $x+y+C$ for some integer $C$ b) Is there a polynomial $P$ that isn't divisible by any of $x-y+C$ or $x+y+C$ for integers $C$?