Square $ABCD$ is cut by a line segment $EF$ into two rectangles $AEFD$ and $BCFE$. The lengths of the sides of each of these rectangles are positive integers. It is known that the area of the rectangle $AEFD$ is $30$ and it is larger than the area of the rectangle $BCFE$. Find the area of square $ABCD$. Proposed by Bogdan Rublov

Is it possible to write the numbers from $1$ to $100$ in the cells of a of a $10 \times 10$ square so that: 1. Each cell contains exactly one number; 2. Each number is written exactly once; 3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the original $10 \times 10$ square, the numbers in them must have the same parity. The figure below shows examples of such pairs of cells, in which the numbers must have the same parity. Proposed by Mykhailo Shtandenko

Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $2024$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $2024$ loses. Who wins if every player wants to win? Proposed by Mykhailo Shtandenko

For real numbers $a_1, a_2, \ldots, a_{200}$, we consider the value $S = a_1a_2 + a_2a_3 + \ldots + a_{199}a_{200} + a_{200}a_1$. In one operation, you can change the sign of any number (that is, change $a_i$ to $-a_i$), and then calculate the value of $S$ for the new numbers again. What is the smallest number of operations needed to always be able to make $S$ nonnegative? Proposed by Oleksii Masalitin

Find the number of positive integers for which the product of digits and the sum of digits are the same and equal to $8$.

Write the numbers from $1$ to $16$ in the cells of a of a $4 \times 4$ square so that: 1. Each cell contains exactly one number; 2. Each number is written exactly once; 3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the original $4 \times 4$ square, the sum of numbers in them is a prime number The figure below shows examples of such pairs of cells, sums of numbers in which have to be prime. Proposed by Mykhailo Shtandenko

The circle $\gamma$ passing through the vertex $A$ of triangle $ABC$ intersects its sides $AB$ and $AC$ for the second time at points $X$ and $Y$, respectively. Also, the circle $\gamma$ intersects side $BC$ at points $D$ and $E$ so that $AD = AE$. Prove that the points $B, X, Y, C$ lie on the same circle. Proposed by Mykhailo Shtandenko

Positive real numbers $a_1, a_2, \ldots, a_{2024}$ are arranged in a circle. It turned out that for any $i = 1, 2, \ldots, 2024$, the following condition holds: $a_ia_{i+1} < a_{i+2}$. (Here we assume that $a_{2025} = a_1$ and $a_{2026} = a_2$). What largest number of positive integers could there be among these numbers $a_1, a_2, \ldots, a_{2024}$? Proposed by Mykhailo Shtandenko

Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$, $k \geq 7$, and for which the following equalities hold: $$d_7 = 2d_5 + 1\text{ and }d_7 = 3d_4 - 1$$ Proposed by Mykyta Kharin

The difference of fractions $\frac{2024}{2023} - \frac{2023}{2024}$ was represented as an irreducible fraction $\frac{p}{q}$. Find the value of $p$.

Let $BL, AD$ be the bisector and the altitude correspondingly of an acute triangle ABC. They intersect at point $T$. It turned out that the altitude $LK$ of $\triangle ALB$ is divided in half by the line $AD$. Prove that $KT \perp BL$. Proposed by Mariia Rozhkova

Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $2023$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $2023$ loses. Who wins if every player wants to win? Proposed by Mykhailo Shtandenko

Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$. Proposed by Oleksii Masalitin

Find all pairs of positive integers $(a, b)$ such that $4b - 1$ is divisible by $3a + 1$ and $3a - 1$ is divisible by $2b + 1$.

There are $2025$ people living on the island, each of whom is either a knight, i.e. always tells the truth, or a liar, which means they always lie. Some of the inhabitants of the island know each other, and everyone has at least one acquaintance, but no more than three. Each inhabitant of the island claims that there are exactly two liars among his acquaintances. a) What is the smallest possible number of knights among the inhabitants of the island? b) What is the largest possible number of knights among the inhabitants of the island? Proposed by Oleksii Masalitin

For a positive integer $n$, does there exist a permutation of all its positive integer divisors $(d_1 , d_2 , \ldots, d_k)$ such that the equation $d_kx^{k-1} + \ldots + d_2x + d_1 = 0$ has a rational root, if: a) $n = 2024$; b) $n = 2025$? Proposed by Mykyta Kharin

Find the smallest real number $M$, for which $\{a\}+\{b\}+\{c\}\leq M$ for any real positive numbers $a, b, c$ with $abc = 2024$. Here $\{a\}$ denotes the fractional part of number $a$. Proposed by Fedir Yudin, Anton Trygub

Four positive integers $a, b, c, d$ satisfy the condition: $a < b < c < d$. For what smallest possible value of $d$ could the following condition be true: the arithmetic mean of numbers $a, b, c$ is twice smaller than the arithmetic mean of numbers $a, b, c, d$?

$ABCD$ is a trapezoid with $BC\parallel AD$ and $BC = 2AD$. Point $M$ is chosen on the side $CD$ such that $AB = AM$. Prove that $BM \perp CD$. Proposed by Bogdan Rublov

Let $n>1$ be a given positive integer. Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $n$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $n$ loses. Who wins if every player wants to win? Find answer for each $n>1$. Proposed by Mykhailo Shtandenko, Anton Trygub

Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for any positive integers $m, n$ the number $$(f(m))^2+ 2mf(n) + f(n^2)$$is the square of an integer. Proposed by Fedir Yudin