Prove that for any real numbers $x, y, z$ at least one of numbers $x^2 + y + \frac{1}{4}, y^2 + z + \frac{1}{4}, z^2 + x + \frac{1}{4}$ is nonnegative. Proposed by Oleksii Masalitin

You are given a positive integer $n$. What is the largest possible number of numbers that can be chosen from the set $\{1, 2, \ldots, 2n\}$ so that there are no two chosen numbers $x > y$ for which $x - y = (x, y)$? Here $(x, y)$ denotes the greatest common divisor of $x, y$. Proposed by Anton Trygub

$2024$ ones and $2024$ twos are arranged in a circle in some order. Is it always possible to divide the circle into a) two (contiguous) parts with equal sums? b) three (contiguous) parts with equal sums? Proposed by Fedir Yudin

Points $X$ and $Y$ are chosen inside an acute-angled triangle $ABC$ with altitude $AD$ so that $\angle BXA + \angle ACB = 180^\circ , \angle CYA + \angle ABC = 180^\circ$, and $CD + AY = BD + AX$. Point $M$ is chosen on the ray $BX$ so that $X$ lies on segment $BM$ and $XM = AC$, and point $N$ is chosen on the ray $CY$ so that $Y$ lies on segment $CN$ and $YN = AB$. Prove that $AM = AN$. Proposed by Mykhailo Shtandenko

Find the smallest positive integer $n$ for which one can select $n$ distinct real numbers such that each of them is equal to the sum of some two other selected numbers. Proposed by Anton Trygub

Let $\omega$ denote the circumscribed circle of an acute-angled $\triangle ABC$ with $AB \neq BC$. Let $A'$ be the point symmetric to the point $A$ with respect to the line $BC$. The lines $AA'$ and $A'C$ intersect $\omega$ for the second time at points $D$ and $E$, respectively. Let the lines $AE$ and $BD$ intersect at point $P$. Prove that the line $A'P$ is tangent to the circumscribed circle of $\triangle A'BC$. Proposed by Oleksii Masalitin

In a certain magical country, there are banknotes in denominations of $2^0, 2^1, 2^2, \ldots$ UAH. Businessman Victor has to make cash payments to $44$ different companies totaling $44000$ UAH, but he does not remember how much he has to pay to each company. What is the smallest number of banknotes Victor should withdraw from an ATM (totaling exactly $44000$ UAH) to guarantee that he would be able to pay all the companies without leaving any change? Proposed by Oleksii Masalitin

Solve the following system of equations in real numbers: $$\left\{\begin{array}{l}x^2=y^2+z^2,\\x^{2023}=y^{2023}+z^{2023},\\x^{2025}=y^{2025}+z^{2025}.\end{array}\right.$$Proposed by Mykhailo Shtandenko, Anton Trygub

You are given a positive integer $n > 1$. What is the largest possible number of integers that can be chosen from the set $\{1, 2, 3, \ldots, 2^n\}$ so that for any two different chosen integers $a, b$, the value $a^k + b^k$ is not divisible by $2^n$ for any positive integer $k$? Proposed by Oleksii Masalitin

Let $BD$ be an altitude of $\triangle ABC$ with $AB < BC$ and $\angle B > 90^\circ$. Let $M$ be the midpoint of $AC$, and point $K$ be symmetric to point $D$ with respect to point $M$. A perpendicular drawn from point $M$ to the line $BC$ intersects line $AB$ at point $L$. Prove that $\angle MBL = \angle MKL$. Proposed by Oleksandra Yakovenko

For some positive integer $n$, Katya wrote on the board next to each other numbers $2^n$ and $14^n$ (in this order), thus forming a new number $A$. Can the number $A - 1$ be prime? Proposed by Oleksii Masalitin

For any positive real numbers $a, b, c, d$, prove the following inequality: $$(a^2+b^2)(b^2+c^2)(c^2+d^2)(d^2+a^2) \geq 64abcd|(a-b)(b-c)(c-d)(d-a)|$$Proposed by Anton Trygub

Let $AH_A, BH_B, CH_C$ be the altitudes of the triangle $ABC$. Points $A_1$ and $C_1$ are the projections of the point $H_B$ onto the sides $AB$ and $BC$, respectively. $B_1$ is the projection of $B$ onto $H_AH_C$. Prove that the diameter of the circumscribed circle of $\triangle A_1B_1C_1$ is equal to $BH_B$. Proposed by Anton Trygub

There are $n \geq 1$ notebooks, numbered from $1$ to $n$, stacked in a pile. Zahar repeats the following operation: he randomly chooses a notebook whose number $k$ does not correspond to its location in this stack, counting from top to bottom, and returns it to the $k$th position, counting from the top, without changing the location of the other notebooks. If there is no such notebook, he stops. Is it guaranteed that Zahar will arrange all the notebooks in ascending order of numbers in a finite number of operations? Proposed by Zahar Naumets

Solve the following system of equations in real numbers: $$\left\{\begin{array}{l}x^2=y^2+z^2,\\x^{2024}=y^{2024}+z^{2024},\\x^{2025}=y^{2025}+z^{2025}.\end{array}\right.$$Proposed by Mykhailo Shtandenko, Anton Trygub, Bogdan Rublov

Mykhailo wants to arrange all positive integers from $1$ to $2024$ in a circle so that each number is used exactly once and for any three consecutive numbers $a, b, c$ the number $a + c$ is divisible by $b + 1$. Can he do it? Proposed by Fedir Yudin

For a given positive integer $n$, we consider the set $M$ of all intervals of the form $[l, r]$, where the integers $l$ and $r$ satisfy the condition $0 \leq l < r \leq n$. What largest number of elements of $M$ can be chosen so that each chosen interval completely contains at most one other selected interval? Proposed by Anton Trygub

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Rays $AO$, $CO$ intersect sides $BC, BA$ in points $A_1, C_1$ respectively, $K$ is the projection of $O$ onto the segment $A_1C_1$, $M$ is the midpoint of $AC$. Prove that $\angle HMA = \angle BKC_1$. Proposed by Anton Trygub