Oleksiy wrote several distinct positive integers on the board and calculated all their pairwise sums. It turned out that all digits from $0$ to $9$ appear among the last digits of these sums. What could be the smallest number of integers that Oleksiy wrote? Proposed by Oleksiy Masalitin

There is a table with $n > 2$ cells in the first row, $n-1$ cells in the second row is a cell, $n-2$ in the third row, $\ldots$, $1$ cell in the $n$-th row. The cells are arranged as shown below. In each cell of the top row Petryk writes a number from $1$ to $n$, so that each number is written exactly once. For each other cell, if the cells directly above it contains numbers $a, b$, it contains number $|a-b|$. What is the largest number that can be written in a single cell of the bottom row? Proposed by Bogdan Rublov

Points $X$ and $Y$ are chosen inside an acute triangle $ABC$ so that: $$\angle AXB = \angle CYB = 180^\circ - \angle ABC, \text{ } \angle ABX = \angle CBY$$ Show that the points $X$ and $Y$ are equidistant from the center of the circumscribed circle of $\triangle ABC$. Proposed by Anton Trygub

The board contains $20$ non-constant linear functions, not necessarily distinct. For each pair $(f, g)$ of these functions ($190$ pairs in total), Victor writes on the board a quadratic function $f(x)\cdot g(x) - 2$, and Solomiya writes on the board a quadratic function $f(x)g(x)-1$. Victor calculated that exactly $V$ of his quadratic functions have a root, and Solomiya calculated that exactly $S$ of her quadratic functions have a root. Find the largest possible value of $S-V$. Remarks. A linear function $y = kx+b$ is called non-constant if $k\neq 0$. Proposed by Oleksiy Masalitin

Real numbers $a, b, c$ are such that $$a^2+c-bc = b^2+a-ca = c^2+b-ab$$ Does it follow that $a=b=c$? Proposed by Mykhailo Shtandenko

Cyclic quadrilateral $ABCD$ is such that $\angle BAD = 2\angle ADC$ and $CD = 2BC$. Let $H$ be the projection of $C$ onto $AD$. Prove that $BH \parallel CD$. Proposed by Fedir Yudin, Anton Trygub

Prove that there exist infinitely many positive integers that can't be represented in form $a^{bc} - b^{ad}$, where $a, b, c, d$ are positive integers and $a, b>1$. Proposed by Anton Trygub, Oleksii Masalitin

Oleksii and Solomiya play the following game on a square $6n\times 6n$, where $n$ is a positive integer. Oleksii in his turn places a piece of type $F$, consisting of three cells, on the board. Solomia, in turn, after each move of Oleksii, places the numbers $0, 1, 2$ in the cells of the figure that Oleksii has just placed, using each of the numbers exactly once. If two of Oleksii's pieces intersect at any moment (have a common square), he immediately loses. Once the square is completely filled with numbers, the game stops. In this case, if the sum of the numbers in each row and each column is divisible by $3$, Solomiya wins, and otherwise Oleksii wins. Who can win this game if the figure of type $F$ is: a) a rectangle ; b) a corner of three cells? Proposed by Oleksii Masalitin

For some positive integer $n$, consider the board $n\times n$. On this board you can put any rectangles with sides along the sides of the grid. What is the smallest number of such rectangles that must be placed so that all the cells of the board are covered by distinct numbers of rectangles (possibly $0$)? The rectangles are allowed to have the same sizes. Proposed by Anton Trygub

$2024$ positive real numbers with sum $1$ are arranged on a circle. It is known that any two adjacent numbers differ at least in $2$ times. For each pair of adjacent numbers, the smaller one was subtracted from the larger one, and then all these differences were added together. What is the smallest possible value of this resulting sum? Proposed by Oleksiy Masalitin

Points $E, F$ are selected on sides $AC, AB$ respectively of triangle $ABC$ with $AC=AB$ so that $AE = BF$. Point $D$ is chosen so that $D, A$ are in the same halfplane with respect to line $EF$, and $\triangle DFE \sim \triangle ABC$. Lines $EF, BC$ intersect at point $K$. Prove that the line $DK$ is tangent to the circumscribed circle of $\triangle ABC$. Proposed by Fedir Yudin

For real numbers $a, b, c, d \in [0, 1]$, find the largest possible value of the following expression: $$a^2+b^2+c^2+d^2-ab-bc-cd-da$$ Proposed by Mykhailo Shtandenko

You are given a convex hexagon with parallel opposite sides. For each pair of opposite sides, a line is drawn parallel to these sides and equidistant from them. Prove that the three lines thus obtained intersect at one point if and only if the lengths of the opposite sides are equal. Proposed by Nazar Serdyuk

Find all composite odd positive integers, all divisors of which can be divided into pairs so that the sum of the numbers in each pair is a power of two, and each divisor belongs to exactly one such pair. Proposed by Anton Trygub

Solomiya wrote the numbers $1, 2, \ldots, 2024$ on the board. In one move, she can erase any two numbers $a, b$ from the board and write the sum $a+b$ instead of each of them. After some time, all the numbers on the board became equal. What is the minimum number of moves Solomiya could make to achieve this? Proposed by Oleksiy Masalitin

You are given a positive integer $n$. Find the smallest positive integer $k$, for which there exist integers $a_1, a_2, \ldots, a_k$, for which the following equality holds: $$2^{a_1} + 2^{a_2} + \ldots + 2^{a_k} = 2^n - n + k$$ Proposed by Mykhailo Shtandenko

Altitudes $AH_A, BH_B, CH_C$ of triangle $ABC$ intersect at $H$, and let $M$ be the midpoint of the side $AC$. The bisector $BL$ of $\triangle ABC$ intersects $H_AH_C$ at point $K$. The line through $L$ parallel to $HM$ intersects $BH_B$ in point $T$. Prove that $TK = TL$. Proposed by Anton Trygub

Find all functions $f:\mathbb{R} \to \mathbb{R}$, such that for any $x, y \in \mathbb{R}$ holds the following: $$f(x)f(yf(x)) + yf(xy) = xf(xy) + y^2f(x)$$ Proposed by Mykhailo Shtandenko

Inside a quadrilateral $ABCD$ with $AB=BC=CD$, the points $P$ and $Q$ are chosen so that $AP=PB=CQ=QD$. The line through the point $P$ parallel to the diagonal $AC$ intersects the line through the point $Q$ parallel to the diagonal $BD$ at the point $T$. Prove that $BT=CT$. Proposed by Mykhailo Shtandenko

Find all polynomials $P(x)$ with integer coefficients, such that for any positive integer $n$ number $P(n)$ is a positive integer and a divisor of $n!$. Proposed by Mykyta Kharin

There are $2024$ cities in a country, some pairs of which are connected by bidirectional flights. For any distinct cities $A, B, C, X, Y, Z$, it is possible to fly directly from some of the cities $A, B, C$ to some of the cities $X, Y, Z$. Prove that it is possible to plan a route $T_1\to T_2 \to \ldots \to T_{2022}$ that passes through $2022$ distinct cities. Proposed by Lior Shayn

Find all pairs $a, b$ of positive integers, for which $$(a, b) + 3[a, b] = a^3 - b^3$$ Here $(a, b)$ denotes the greatest common divisor of $a, b$, and $[a, b]$ denotes the least common multiple of $a, b$. Proposed by Oleksiy Masalitin

You are given positive integers $m, n>1$. Vasyl and Petryk play the following game: they take turns marking on the coordinate plane yet unmarked points of the form $(x, y)$, where $x, y$ are positive integers with $1 \leq x \leq m, 1 \leq y \leq n$. The player loses if after his move there are two marked points, the distance between which is not a positive integer. Who will win this game if Vasyl moves first and each player wants to win? Proposed by Mykyta Kharin

Let's define almost mean of numbers $a_1, a_2, \ldots, a_n$ as $\frac{a_1 + a_2 + \ldots + a_n}{n+1}$. Oleksiy has positive real numbers $b_1, b_2, \ldots, b_{2023}$, not necessarily distinct. For each pair $(i, j)$ with $1 \leq i, j \leq 2023$, Oleksiy wrote on a board almost mean of numbers $b_i, b_{i+1}, \ldots, b_j$. Prove that there are at least $45$ distinct numbers on the board. Proposed by Anton Trygub

Point $X$ is chosen inside a convex $ABCD$ so that $\angle XBC = \angle XAD, \angle XCB = \angle XDA$. Rays $AB, DC$ intersect at point $O$, circumcircles of triangles $BCO, ADO$ intersect at point $T$. Prove that line $TX$ and the line through $O$ perpendicular to $BC$ intersect on the circumcircle of $\triangle AOD$. Proposed by Anton Trygub

You are given some $12$ non-zero, not necessarily distinct real numbers. Find all positive integers $k$ from $1$ to $12$, such that among these numbers you can always choose $k$ numbers whose sum has the same sign as their product, that is, either both the sum and the product are positive, or both are negative. Proposed by Anton Trygub

The points $A, B, C, D$ lie on the line $\ell$ in this order. The points $P$ and $Q$ are chosen on one side of the line $\ell$, and the point $R$ is chosen on the other side so that: $$\angle APB = \angle CPD = \angle QBC = \angle QCB = \angle RAD = \angle RDA$$ Prove that the points $P, Q, R$ lie on the same line. Proposed by Mykhailo Shtandenko, Fedir Yudin

You are given $2024$ yellow and $2024$ blue points on the plane, and no three of the points are on the same line. We call a pair of nonnegative integers $(a, b)$ good if there exists a half-plane with exactly $a$ yellow and $b$ blue points. Find the smallest possible number of good pairs. The points that lie on the line that is the boundary of the half-plane are considered to be outside the half-plane. Proposed by Anton Trygub

Find all polynomials $P(x)$ with integer coefficients, such that for each of them there exists a positive integer $N$, such that for any positive integer $n\geq N$, number $P(n)$ is a positive integer and a divisor of $n!$. Proposed by Mykyta Kharin