It is known that for some integers $a_{2021}, a_{2020}, \ldots , a_1 , a_0$ and any positive integer $n$, the expression $$a_{2021}n^{2021} + a_{2020}n^{2020} +\ldots +a_1n + a_0$$is divisible by $10$. Is it necessary that each of the numbers $a_{2021}, a_{2020}, \ldots , a_1 , a_0$ is also divisible by $10$?

The Kotoras alphabet has exactly $11$ letters. It is known that for any positive integer $k$, there are exactly five $k$-letter bad words in the language of this tribe. Prove that it is possible to come up with a word in the Kotoras language with any number of letters that cannot be cut into two smaller words, of which at least one is considered bad. Proposed by Arseniy Nikolaiev

The numbers $(a_1 , a_2 , \ldots, a_{2021})$ and $(b_1 , b_2 , \ldots, b_{2021})$ are some different permutations of the numbers $(1, 2, \ldots, 2021)$, and the numbers $(c_1 , c_2 , \ldots, c_{2021})$ are some permutation of the numbers $(2, 4, \ldots, 4042)$. Prove that the given number D is positive: $$D = \frac{c_1^2 -4 a_1b_1}{a_1 + b_1 + c_1} + \frac{c_2^2 - 4a_2b_2}{a_2 + b_2 + c_2} + \ldots + \frac{c_{2021}^2 - 4a_{2021}b_{2021}}{a_{2021} + b_{2021} + c_{2021}}$$ Proposed by Bogdan Rublov

Let $ABC$ be an isosceles triangle with $AB = AC$. Points $P$ and $Q$ inside $\triangle ABC$ are such that $\angle BPC = \frac{3}{2} \angle BAC$, $BP = AQ$ and $AP = CQ$. Prove that $AP = PQ$. Proposed by Fedir Yudin

Find all positive integers that can be represented as $$\frac{abc+ab+a}{abc+bc+c}$$ for some positive integers $a, b, c$. Proposed by Oleksii Masalitin

You are given a $2021 \times 2021$ square, its inner border is a mirror. $2020$ of its squares have mirror borders, and all other squares have transparent borders. You are given an angle $\alpha \in (0^\circ, 90^\circ)$. A ray of light passes without change of direction through the transparent boundary, and is reflected according to the law of reflection when hitting the mirror boundary (if the ray hits exactly the corner of the mirror square, it is reflected from the horizontal side of the square, see the figure below for examples). Prove that from some point on the bottom side of the square you can release a ray of light forming the chosen angle $\alpha$ with this side (there are two such rays) so that the light reaches the top side of the square. Proposed by Arsenii Nikolaiev

Andriy and Bogdan take turns putting integers from $1$ to $n$ inclusive in the cells of an $n \times n$ board in such a way that no row or column contains two equal numbers. Andriy begins, and the player who cannot make a move loses. Depending on $n$, which player has a winning strategy? Proposed by Fedir Yudin

You are given $k$ positive integers $a_1, a_2, \ldots, a_k$. Prove that there exists a positive integer $N$, such that sums of digits of numbers $Na_1, Na_2, \ldots, Na_k$ differ in less than $\frac{2021}{2020}$ times. That is, if $S_{max}$ is the largest sum of digits of any of these numbers, and $S_{min}$ is the smallest, then $$\frac{S_{max}}{S_{min}} < \frac{2021}{2020}$$ Proposed by Arsenii Nikolaiev

Alexey and Bogdan play a game with two piles of stones. In the beginning, one of the piles contains $2021$ stones, and the second is empty. In one move, each of the guys has to pick up an even number of stones (more than zero) from an arbitrary pile, then transfer half of the stones taken to another pile, and the other half - to remove from the game. Loses the one who cannot make a move. Who will win this game if both strive to win, and Bogdan begins? (Oleksii Masalitin)

Prove that there exist distinct positive integers $a$ and $b$ greater than $1000000$ such that $(a^b + 1)$ is divisible by $(b^a + 1)$. Proposed by Arsenii Nikolaiev

Nonnegative integers $x_1, x_2 , \ldots, x_n$ are such that $x_1^2 + x_2^2 + \ldots +x_n^2 = n$, $n > 1$. Prove that for any nonnegative real numbers $y_1, y_2 , \ldots, y_n$ there exist indices $i, j \in \{1, 2, \ldots, n\}$, not necessarily different, for which the following inequality holds: $$x_i + y_j - x_{j+1}y_{i+1} \geq 1$$ (Here index $n+1 = $ index $1$). Proposed by Nazar Serdyuk

Let $ABC$ be a triangle with $\angle A = 60^\circ$. Points $P$ and $Q$ are chosen on sides $AB$ and $AC$, respectively, such that $BP = PQ = QC$. Prove that the circumcircle of $\triangle APQ$ passes through the projection of the orthocenter of $ABC$ onto its $A$-median. Proposed by Fedir Yudin

Let $M$ denote the set consisting of positive integers that can be represented as $(a + b)(a^2 + b^2)$, where $a, b$ are some distinct positive integers. Prove that for any positive integer $n > k$, the number $n^4 - k^4$ is the sum of some $n - k$ distinct elements of the set $M$. Proposed by Oleksii Masalitin

Circles $w_1$ and $w_2$ intersect at points $P$ and $Q$ and touch a circle $w$ with center at point $O$ internally at points $A$ and $B$, respectively. It is known that the points $A,B$ and $Q$ lie on one line. Prove that the point $O$ lies on the external bisector $\angle APB$. (Nazar Serdyuk)

The All-Ukrainian Mathematical Olympiad was attended by $100$ children from $10$ regions of Ukraine, $10$ children from each region. At the opening ceremony, they stood in one row on the stage, but not all children from the same region stood consecutively. Two neighboring children from different regions can be swapped in $1$ second. (You can only swap one pair of children per second). For which shortest time $T$ is it always possible to make sure that the children are lined up by the regions? The regions can go in any order. Proposed by Anton Trygub

For any positive integer $a > 1$, we define the sequence $(a_n)$ as follows: $a_{n+1} = a_n + d(a_n) - 1$, $n \in \mathbb{N}$, $a_1 = a$, where $d(b)$ denotes the smallest prime divisor of $b$. Prove that for any positive integer $k$, the sequence $d(a_n)$ for $n \geq k$ is not increasing, i.e. the condition $d(a_{n+1}) > d(a_n)$ is not true for at least one $n \geq k$. Proposed by Vadym Koval

Denote by $P^{(n)}$ the set of all polynomials of degree $n$ the coefficients of which is a permutation of the set of numbers $\{2^0, 2^1,..., 2^n\}$. Find all pairs of natural numbers $(k,d)$ for which there exists a $n$ such that for any polynomial $p \in P^{(n)}$, number $P(k)$ is divisible by the number $d$. (Oleksii Masalitin)

For arbitrary positive reals $a\ge b \ge c$ prove the inequality: $$\frac{a^2+b^2}{a+b}+\frac{a^2+c^2}{a+c}+\frac{c^2+b^2}{c+b}\ge (a+b+c)+ \frac{(a-c)^2}{a+b+c}$$(Anton Trygub)

Let $O, I, H$ be the circumcenter, the incenter, and the orthocenter of $\triangle ABC$. The lines $AI$ and $AH$ intersect the circumcircle of $\triangle ABC$ for the second time at $D$ and $E$, respectively. Prove that if $OI \parallel BC$, then the circumcenter of $\triangle OIH$ lies on $DE$. (Fedir Yudin)

Find all sets of $n\ge 2$ consecutive integers $\{a+1,a+2,...,a+n\}$ where $a\in Z$, in which one of the numbers is equal to the sum of all the others. (Bogdan Rublev)

The sequence $a_1,a_2, ..., a_{2n}$ of integers is such that each number occurs in no more than $n$ times. Prove that there are two strictly increasing sequences of indices $b_1,b_2, ..., b_{n}$ and $c_1,c_2, ..., c_{n}$ are such that every positive integer from the set $\{1,2,...,2n\}$ occurs exactly in one of these two sequences, and for each $1\le i \le n$ is true the condition $a_{b_i} \ne a_{c_i}$ . (Anton Trygub)

Given a natural number $n$. Prove that you can choose $ \phi (n)+1 $ (not necessarily different) divisors $n$ with the sum $n$. Here $ \phi (n)$ denotes the number of natural numbers less than $n$ that are coprime with $n$. (Fedir Yudin)

It is known that for some integers $a_{2021},a_{2020},...,a_1,a_0$ the expression $$a_{2021}n^{2021}+a_{2020}n^{2020}+...+a_1n+a_0$$is divisible by $2021$ for any arbitrary integer $n$. Is it required that each of the numbers $a_{2021},a_{2020},...,a_1,a_0$ also divisible by $2021$?

Find all natural numbers $n \ge 3$ for which in an arbitrary $n$-gon one can choose $3$ vertices dividing its boundary into three parts, the lengths of which can be the lengths of the sides of some triangle. (Fedir Yudin)

Find all the following functions $f:R\to R$ , which for arbitrary valid $x,y$ holds equality: $$f(xf(x+y))+f((x+y)f(y))=(x+y)^2$$(Vadym Koval)

Are there natural numbers $(m,n,k)$ that satisfy the equation $m^m+ n^n=k^k$ ?

The altitudes $AA_1, BB_1$ and $CC_1$ were drawn in the triangle $ABC$. Point $K$ is a projection of point $B$ on $A_1C_1$. Prove that the symmmedian $\vartriangle ABC$ from the vertex $B$ divides the segment $B_1K$ in half. (Anton Trygub)

There are $101$ not necessarily different weights, each of which weighs an integer number of grams from $1$ g to $2020$ g. It is known that at any division of these weights into two heaps, the total weight of at least one of the piles is no more than $2020$. What is the largest number of grams can weigh all $101$ weights? (Bogdan Rublev)