2025 Kyiv City MO Round 2

Grade 7

Problem 1

Mykhailo drew a triangular grid with side \( n \) for \( n \geq 2 \). It is formed from an equilateral triangle \( T \) with side length \( n \), by dividing each side into \( n \) equal parts. Then lines are drawn parallel to the sides of triangle \( T \), dividing it into \( n^2 \) equilateral triangles with side length \( 1 \), which we will call \textbf{cells}. Next, Oleksii writes some positive integer into each cell. Mykhailo receives 1 candy for each cell, where the number written is equal to the sum of all the numbers in the adjacent cells. Oleksii wants to arrange the numbers in such a way that Mykhailo receives the maximum number of candies. How many candies can Mykhailo receive under such conditions? In the figure below, an example is shown for \( n = 4 \) with 16 cells and numbers written inside them. For the numbers arranged as in the figure, Mykhailo receives 5 candies for the numbers \( 2 \) (the topmost cell), \( 8 \), \( 13 \), \( 12 \), and \( 11 \). Proposed by Mykhailo Shtandenko

Problem 2

Mykhailo chose three distinct positive real numbers \( a, b, c \) and wrote the following numbers on the board: \[ a + b, \quad b + c, \quad c + a, \quad ab, \quad bc, \quad ca. \]What is the minimum possible number of distinct numbers that can be written on the board? Proposed by Anton Trygub

Problem 3

A positive integer \( n \), which has at least one proper divisor, is divisible by the arithmetic mean of the smallest and largest of its proper divisors (which may coincide). What can be the number of divisors of \( n \)? A proper divisor of a positive integer \( n \) is any of its divisors other than \( 1 \) and \( n \). Proposed by Mykhailo Shtandenko

Problem 4

Let \( BE \) and \( CF \) be the medians of \( \triangle ABC \), and \( G \) be their intersection point. On segments \( GF \) and \( GE \), points \( K \) and \( L \), respectively, are chosen such that \( BK = CL = AG \). Prove that \[ \angle BKF + \angle CLE = \angle BGC. \] Proposed by Vadym Solomka

Grade 8

Problem 1

Mykhailo chose three distinct real numbers \( a, b, c \) and wrote the following numbers on the board: \[ a + b, \quad b + c, \quad c + a, \quad ab, \quad bc, \quad ca. \]What is the minimum possible number of distinct numbers that can be written on the board? Proposed by Anton Trygub

Problem 2

Find all pairs of positive integers \( a, b \) such that one of the two numbers \( 2(a^2 + b^2) \) and \( (a + b)^2 + 4 \) is divisible by the other. Proposed by Oleksii Masalitin

Problem 3

In a school, \( n \) different languages are taught. It is known that for any subset of these languages (including the empty set), there is exactly one student who knows these and only these languages (there are \( 2^n \) students in total). Each day, the students are divided into pairs and teach each other the languages that only one of them knows. If students are not allowed to be in the same pair twice, what is the minimum number of days the school administration needs to guarantee that all their students know all \( n \) languages? Proposed by Oleksii Masalitin

Problem 4

Inside a convex quadrilateral \( ABCD \), a point \( P \) is chosen such that \[ \angle PAD = \angle PAB = \angle PBC = \angle PCB = \angle PDA = 30^\circ. \]Prove that \( \angle CDP = 30^\circ \). Proposed by Vadym Solomka

Grade 9

Problem 1

Find the largest possible value of the expression \( y - x \), if the non-negative real numbers \( x, y \) satisfy the equation: \[ x^4 = y(y - 2025)^3. \] Proposed by Mykhailo Shtandenko, Anton Trygub

Problem 2

A positive integer \( n \) satisfies the following conditions: The number \( n \) has exactly \( 60 \) divisors: \( 1 = a_1 < a_2 < \cdots < a_{60} = n \); The number \( n+1 \) also has exactly \( 60 \) divisors: \( 1 = b_1 < b_2 < \cdots < b_{60} = n+1 \). Let \( k \) be the number of indices \( i \) such that \( a_i < b_i \). Find all possible values of \( k \). Note: Such numbers exist, for example, the numbers \( 4388175 \) and \( 4388176 \) both have \( 60 \) divisors. Proposed by Anton Trygub

Problem 3

Does there exist a sequence of positive integers \( a_1, a_2, \ldots, a_{100} \) such that every number from \( 1 \) to \( 100 \) appears exactly once, and for each \( 1 \leq i \leq 100 \), the condition \[ a_{a_i + i} = i \]holds? Here it is assumed that \( a_{k+100} = a_k \) for each \( 1 \leq k \leq 100 \). Proposed by Mykhailo Shtandenko

Problem 4

Let \( H \) be the orthocenter, and \( O \) be the circumcenter of \( \triangle ABC \). The line \( AH \) intersects the circumcircle of \( \triangle ABC \) at point \( N \) for the second time. The circumcircle of \( \triangle BOC \), with center at point \( Q \), intersects the line \( OH \) at point \( X \) for the second time. Prove that the points \( O, Q, N, X \) lie on the same circle. Proposed by Matthew Kurskyi

Grade 10

Same as 7.2 - Problem 1

Same as 9.2 - Problem 2

Problem 3

A sequence \( a_1, a_2, \ldots \) of real numbers satisfies the following condition: for every positive integer \( k \geq 2 \), there exists a positive integer \( i < k \) such that \( a_i + a_k = k \). It is known that for some \( j \), the fractional parts of the numbers \( a_j \) and \( a_{j+1} \) are equal. Prove that for some positive integers \( x \neq y \), the equality \[ a_x - a_y = x - y \]holds. The fractional part of a real number \( a \) is defined as the number \( \{a\} \in [0, 1) \), which satisfies the condition \( a = n + \{a\} \), where \( n \) is an integer. For example, \( \{-3\} = 0 \), \( \{3.14\} = 0.14 \), and \( \{-3.14\} = 0.86 \). Proposed by Mykhailo Shtandenko

Problem 4

Point \( A_1 \) inside the acute-angled triangle \( ABC \) is such that \[ \angle ACB = 2\angle A_1BC \quad \text{and} \quad \angle ABC = 2\angle A_1CB. \]Point \( A_2 \) is chosen so that points \( A \) and \( A_2 \) lie on opposite sides of line \( BC \), \( AA_2 \perp BC \), and the perpendicular bisector of \( AA_2 \) is tangent to the circumcircle of \( \triangle ABC \). Define points \( B_1, B_2, C_1, C_2 \) analogously. Prove that the circumcircles of \( \triangle AA_1A_2 \), \( \triangle BB_1B_2 \), and \( \triangle CC_1C_2 \) intersect at exactly two common points. Proposed by Vadym Solomka

Grade 11

Same as 8.2 - Problem 1

Problem 2

For some positive integer \( n \), Katya wrote the numbers from \( 1 \) to \( 2^n \) in a row in increasing order. Oleksii rearranged Katya's numbers and wrote the new sequence directly below the first row. Then, they calculated the sum of the two numbers in each column. Katya calculated \( N \), the number of powers of two among the results, while Oleksii calculated \( K \), the number of distinct powers of two among the results. What is the maximum possible value of \( N + K \)? Proposed by Oleksii Masalitin

Problem 3

On sides \( AB \) and \( AC \) of an acute-angled, non-isosceles triangle \( ABC \), points \( P \) and \( Q \) are chosen such that the center \( O_9 \) of the nine-point circle of \( \triangle ABC \) is the midpoint of segment \( PQ \). Let \( O \) be the circumcenter of \( \triangle ABC \). On the ray \( OP \) beyond \( P \), segment \( PX \) is marked such that \( PX = AQ \). On the ray \( OQ \) beyond \( Q \), segment \( QY \) is marked such that \( QY = AP \). Prove that the midpoint of side \( BC \), the midpoint of segment \( XY \), and the point \( O_9 \) are collinear. The nine-point circle or the Euler circle of \( \triangle ABC \) is the circle passing through nine significant points of the triangle — the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments connecting the orthocenter with the vertices of \( \triangle ABC \). Proposed by Danylo Khilko

Problem 4

A square \( K = 2025 \times 2025 \) is given. We define a stick as a rectangle where one of its sides is \( 1 \), and the other side is a positive integer from \( 1 \) to \( 2025 \). Find the largest positive integer \( C \) such that the following condition holds: If several sticks with a total area not exceeding \( C \) are taken, it is always possible to place them inside the square \( K \) so that each stick fully completely covers an integer number of \( 1 \times 1 \) squares, and no \( 1 \times 1 \) square is covered by more than one stick. (Basically, you can rotate sticks, but they have to be aligned by lines of the grid) Proposed by Anton Trygub