Lines \( FD \) and \( BE \) intersect at point \( O \). Rays \( OA \) and \( OC \) are drawn from point \( O \). You are given the following information about the angles: \[ \angle DOC = 36^\circ, \quad \angle AOC = 90^\circ, \quad \angle AOB = 4x, \quad \angle FOE = 5x, \]as shown in the figure below. What is the degree measure of \( x \)?

Prove that the number \[ 3 \underbrace{99\ldots9}_{2025} \underbrace{60\ldots01}_{2025} \]is a square of a positive integer.

In the Faculty of Cybernetics football championship, \( n \geq 3 \) teams participated. The competition was held in a round-robin format, meaning that each team played against every other team exactly once. For a win, a team earns 3 points, for a loss no points are awarded, and for a draw, both teams receive 1 point each. It turned out that the winning team scored strictly more points than any other team and had at most as many wins as losses. What is the smallest \( n \) for which this could happen? Proposed by Bogdan Rublov

Oleksii wrote some \( 2n \) (\( n > 1 \)) consecutive positive integers on the board. After that, he grouped these numbers into pairs in some way, and within each pair, he multiplied the two numbers together. He then wrote the resulting \( n \) products on the board instead of the original numbers. Afterward, Anton wrote down the difference between the largest and the smallest of the numbers Oleksii wrote. Oleksii wants Anton to write the smallest possible number. What is the smallest number that can be written? Proposed by Oleksii Masalitin, Anton Trygub

Find all triples of positive integers \( a, b, c \) that satisfy the equation: \[ a + \frac{1}{b + \frac{1}{c}} = 20.25. \]

Is it possible to write positive integers from $1$ to $2025$ in the cells of a \( 45 \times 45 \) grid such that each number is used exactly once, and at the same time, each written number is either greater than all the numbers located in its side-adjacent cells or smaller than all the numbers located in its side-adjacent cells? Proposed by Anton Trygub

What's the smallest positive integer \( n > 3 \), for which there does not exist a (not necessarily convex) \( n \)-gon such that all its diagonals have equal lengths? A diagonal of any polygon is defined as a segment connecting any two non-adjacent vertices of the polygon. Proposed by Anton Trygub

Find all quadruples of positive integers \( (a, p, q, r) \), where \( p, q, r \) are prime numbers, such that the following equation holds: \[ p^2q^2 + q^2r^2 + r^2p^2 + 3 = 4 \cdot 13^a. \] Proposed by Oleksii Masalitin

How many three-digit numbers are there, which do not have a zero in their decimal representation and whose sum of digits is $7$?

Can the numbers from \( 1 \) to \( 2025 \) be arranged in a circle such that the difference between any two adjacent numbers has the form \( 2^k \) for some non-negative integer \( k \)? For different adjacent pairs of numbers, the values of \( k \) may be different. Proposed by Anton Trygub

Point \( H \) is the orthocenter of the acute triangle \( ABC \), and \( AD \) is its altitude. Tangents are drawn from points \( B \) and \( C \) to the circle with center \( A \) and radius \( AD \), which do not coincide with the line \( BC \). These tangents intersect at point \( P \). Prove that the radius of the incircle of \( \triangle BCP \) is equal to \( HD \). Proposed by Danylo Khilko

Distinct real numbers \( a, b, c \) satisfy the following condition: \[ \frac{a - b}{a^3b^3} + \frac{b - c}{b^3c^3} + \frac{c - a}{c^3a^3} = 0. \]What are the possible values of the expression \[ \frac{a^4 + b^4 + c^4}{a^2b^2 + b^2c^2 + c^2a^2}? \] Proposed by Vadym Solomka

Some positive integer has an even number of divisors. Anya wants to split these divisors into pairs so that the products of the numbers in each pair have the same number of divisors. Prove that she can do this in exactly one way. Proposed by Oleksii Masalitin

You are given \( 11 \) numbers with an arithmetic mean of \( 10 \). Each of the first \( 4 \) numbers is increased by \( 20 \), and each of the last \( 7 \) numbers is decreased by \( 24 \). What is the arithmetic mean of the new \( 11 \) numbers?

The diameter \( AD \) of the circumcircle of triangle \( ABC \) intersects line \( BC \) at point \( K \). Point \( D \) is reflected symmetrically with respect to point \( K \), resulting in point \( L \). On line \( AB \), a point \( F \) is chosen such that \( FL \perp AC \). Prove that \( FK \perp AD \). Proposed by Matthew Kurskyi

Find all functions \( f : \mathbb{N} \to \mathbb{N} \) that satisfy the following condition: for any positive integers \( m \) and \( n \) such that \( m > n \) and \( m \) is not divisible by \( n \), if we denote by \( r \) the remainder of the division of \( m \) by \( n \), then the remainder of the division of \( f(m) \) by \( n \) is \( f(r) \). Proposed by Mykyta Kharin

Real numbers \( a, b, c \) satisfy the following conditions: \[ 1000 < |a| < 2000, \quad 1000 < |b| < 2000, \quad 1000 < |c| < 2000, \]and \[ \frac{ab^2}{a+b} + \frac{bc^2}{b+c} + \frac{ca^2}{c+a} = 0. \]What are the possible values of the expression \[ \frac{a}{b} + \frac{b}{c} + \frac{c}{a}? \] Proposed by Vadym Solomka

Find all three-digit numbers that are \( 5 \) times greater than the product of their digits.

All positive integers from \( 1 \) to \( 2025 \) are written on a board. Mykhailo and Oleksii play the following game. They take turns, starting with Mykhailo, erasing one of the numbers written on the board. The game ends when exactly two numbers remain on the board. If their sum is a perfect square of an integer, Mykhailo wins; otherwise, Oleksii wins. Who wins if both players play optimally? Proposed by Fedir Yudin

Determine the largest possible constant \( C \) such that for any positive real numbers \( x, y, z \), which are the sides of a triangle, the following inequality holds: \[ \frac{xy}{x^2 + y^2 + xz} + \frac{yz}{y^2 + z^2 + yx} + \frac{zx}{z^2 + x^2 + zy} \geq C. \] Proposed by Vadym Solomka