Prove that there exists a $2021$-digit positive integer $\overline{a_1a_2\ldots a_{2021}}$, with all its digits being non-zero, such that for every $1 \leq n \leq 2020$ the following equality holds $$\overline{a_1a_2\ldots a_n} \cdot \overline{a_{n+1}a_{n+2}\ldots a_{2021}}=\overline{a_1a_2\ldots a_{2021-n}} \cdot \overline{a_{2022-n}a_{2023-n}\ldots a_{2021}}$$and all four numbers in the equality are pairwise different.

Given quadratic trinomials $P(x)=x^2+ax+b$ and $Q(x)=x^2+cx+d$, where $a>c$. It is known that for every real $t$ and $s$ with $t+s=1$ the polynomial $B(x)=tP(x)+sQ(x)$ has at least one real root. Prove that $bc \geq ad$.

The incircle of the triangle $ABC$ is tangent to $BC$,$CA$ and $AB$ at $A_1$,$B_1$ and $C_1$ respectively. In triangles $AB_1C_1$, $BC_1A_1$ and $CB_1A_1$ points $H_1$,$H_2$ and $H_3$ are orthocenters. Prove that the triangles $A_1B_1C_1$ and $H_1H_2H_3$ are equal.

Several soldiers are standing in a row. After a command each of them turned their head either to the left or to the right. After that every second every soldier performs the following operation simultaneously: 1) if the soldier is facing right and the majority of soldiers to the right of him are facing left, he starts facing left; 2) if the soldier is facing left and the majority of soldiers to the left of him are facing right, he starts facing right; 3) otherwise he does nothing. Prove that at some point the process will stop.

Let $f(x)$ be a linear function and $k,l,m$ - pairwise different real numbers. It is known that $f(k)=l^3+m^3$, $f(l)=m^3+k^3$ and $f(m)=k^3+l^3$. Find the value of $k+l+m$.

For four pairwise different positive integers $a,b,c$ and $d$ six numbers are calculated: $ab+10$,$ac+10$,$ad+10$,$bc+10$,$bd+10$ and $cd+10$. Find the maximum amount of them which can be perfect squares.

The sequence $n_1<n_2<\ldots < n_k$ consists of all positive integers $n$ for which in a square $n \times n$ one can mark $10$ cells such that in any square $3 \times 3$ an odd amount of cells are marked. Find $n_{k-2}$.

On the sides $AB,BC,CD$ and $DA$ of a unit square $ABCD$ points $P,Q,R$ and $S$ are chosen respectively. It turned out that the perimeter of $PQRS$ is $2\sqrt{2}$. Find the sum of perpendiculars from $A,B,C,D$ to $SP,PQ,QR,RS$ respectively.

Given triangle $ABC$. A circle passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at points $C_1$ and $B_1$ respectively. The line $B_1C_1$ intersects the circle $\omega$, which is the circumcircle of $ABC$, at points $X$ and $Y$. Lines $BB_1$ and $CC_1$ intersect $\omega$ at points $P$ and $Q$ respectively ($P \neq B$ and $Q \neq C$). Prove that $QX=PY$.

A bug is walking on the surface of a Rubik's cube(cube $3 \times 3 \times 3$). It can go to the adjacent cell on the same face or on the adjacent face. One day the bug started walking from some cell and returned to it, and visited all other cells exactly once. Prove that he made an even amount of moves that changed the face he is on.

Find all positive integers $n$ for which $$S(n^2)+S(n)^2=n$$where $S(m)$ denotes the sum of digits of $m$.

In the table $n \times n$ numbers from $1$ to $n$ are written in a spiral way. For which $n$ all the numbers on the main diagonal are distinct?

Prove that for some positive integer $n$ there exist positive integers $a$,$b$ and $c$ such that $a^2-n=xy$, $b^2-n=yz$ and $c^2-n=xz$ where $x,y$ and $z$ - some pairwise different positive integers.

The medians of a right triangle $ABC$ ($\angle C = 90^{\circ}$) intersect at $M$. Point $L$ lies on the $AC$ such that $\angle ABL=\angle CBL$. It turned out that $\angle BML = 90^{\circ}$. Find the ration $AB : BC$.

It is known that $(x-y)^3 \vdots 6x^2-2y^2$, where $x,y$ are some integers. Prove that then also $(x+y)^3 \vdots 6x^2-2y^2$.

Given a positive integer $n$. An inversion of a permutation is the amount of pairs $(i,j)$ such that $i<j$ and the $i$-th number is smaller than $j$-th number in the permutation. Prove that for every positive integer $k \leq n$ there exist exactly $\frac{n!}{k}$ permutations in which the inversion is divisible by $k$.

An arbitrary positive number $a$ is given. A sequence ${a_n}$ is defined by equalities $a_1=\frac{a}{a+1}$ and $a_{n+1}=\frac{aa_n}{a^2+a_n-aa_n}$ for all $n \geq 1$ Find the minimal constant $C$ such that inequality $$a_1+a_1a_2+\ldots+a_1\ldots a_m<C$$holds for all positive integers $m$ regardless of $a$

In a triangle $ABC$ equality $2BC=AB+AC$ holds. The angle bisector of $\angle BAC$ inteesects $BC$ at $L$. A circle, that is tangent to $AL$ at $L$ and passes through $B$ intersects $AB$ for the second time at $X$. A circle, that is tangent to $AL$ at $L$ and passes through $C$ intersects $AC$ for the second time at $Y$ Find all possible values of $XY:BC$

Odd numbers $x,y,z$ such that $gcd(x,y,z)=1$ are given. It turned out that $x^2+y^2+z^2 \vdots x+y+z$ Prove that $x+y+z-2$ is not divisible by $3$

Quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$, both of which have real roots, are called friendly if for all $t \in [0,1]$ quadratic polynomial $tP(x)+(1-t)Q(x)$ also has real roots. a) Provide an example of quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ and which have real roots, that are not friendly. b) Prove that for any two quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ that have real roots, there is a quadratic polynomial $R(x)$ which has a leading coefficient $1$ and which is friendly to both $P$ and $Q$

Prove that for any positive integer $n$ there exist infinitely many triples $(a,b,c)$ of pairwise distinct positive integers such that $ab+n,bc+n,ac+n$ are all perfect squares

In a $10 \times 10$ table some cells(at least one) are marked such that in every $3 \times 3$ subtable an even number of cells are marked. What is the minimal possible amount of marked cells?

An inscribed into a circle quadraliteral $ABCD$ is given. Points $M$ and $N$ lie on sides $AB$ and $CD$ such that $AK:KB=DM:MC$ and points $L$ and $N$ lie on sides $BC$ and $DA$ such that $BL:LC=AN:ND$. The circumcircle of the triangle $CML$ intersects diagonal $AC$ for the second time in point $P$. The circumcircle of triangle $DNM$ intersects diagonal $BD$ for the second time in point $Q$. Circumcircles of triangles $AKN$ and $BLK$ intersect for the second time in point $R$. Prove that the circumcircle of $PQR$ passes through the intersection of $AC$ and $BD$

Two numbers $1+\sqrt[3]{2}+\sqrt[3]{4}$ and $1+2\sqrt[3]{2}+3\sqrt[3]{4}$ are given. In one move you can do one of the following operations: 1. Replace one of the numbers $a$ with either $a-\sqrt[3]{2}$ or $-2a$ 2. Replace both numbers $a$ and $b$ with $a-b$ and $a+b$ (you can choose the order of $a$ and $b$ yourself) Prove that the obtained numbers are always non-zero

Find all functions $f: \mathbb{R} \to \mathbb{R}$, such that for all real $x,y$ the following equation holds:$$f(x-0.25)+f(y-0.25)=f(x+\lfloor y+0.25 \rfloor - 0.25)$$

Points $A_1,B_1,C_1$ lie on sides $BC, CA, AB$ of an acute-angled triangle, respectively. Denote by $P, Q, R$ the intersections of $BB_1$ and $CC_1$, $CC_1$ and $AA_1$, $AA_1$ and $BB_1$. If triangle $PQR$ is similar to $ABC$ and $\angle AB_1C_1 = \angle BC_1A_1 = \angle CA_1B_1$, prove that $ABC$ is equilateral.

A polynomial $P(x)$ with real coefficients and degree $2021$ is given. For any real $a$ polynomial $x^{2022}+aP(x)$ has at least one real root. Find all possible values of $P(0)$

State consists of $2021$ cities, between some of them there are direct flights. Each pair of cities has not more than one flight, every flight belongs to one of $2021$ companies. Call a group of cities incomplete, if at least one company doesn't have any flights between cities of the group. Find the maximum positive integer $m$, so that one can always find an incomplete group of $m$ cities.

$n_1<n_2<\ldots<n_k$ are all positive integer numbers $n$, that have the following property: In a square $n \times n$ one can mark $50$ cells so that in any square $3 \times 3$ an odd number of cells are marked. Find $n_{k-2}$

A convex quadrilateral $ABCD$ is given. $\omega_1$ is a circle with diameter $BC$, $\omega_2$ is a circle with diameter $AD$. $AC$ meets $\omega_1$ and $\omega_2$ for the second time at $B_1$ and $D_1$. $BD$ meets $\omega_1$ and $\omega_2$ for the second time at $C_1$ and $A_1$. $AA_1$ meets $DD_1$ at $X$, $BB_1$ meets $CC_1$ at $Y$. $\omega_1$ intersects $\omega_2$ at $P$ and $Q$. $XY$ meets $PQ$ at $N$. Prove that $XN=NY$.

Prove that for any positive integer $n$, there exist pairwise distinct positive integers $a,b,c$, not equal to $n$, such that $ab+n, ac+n, bc+n$ are all perfect squares.

Watermelon(a sphere) with radius $R$ lies on a table. $n$ flies fly above the table, each at distance $\sqrt{2}R$ from the center of the watermelon. At some moment any fly couldn't see any of the other flies. (Flies can't see each other, if the segment connecting them intersects or touches watermelon). Find the maximum possible value of $n$