[b](a)[/b] A $2 \times n$-table (with $n > 2$) is filled with numbers so that the sums in all the columns are different. Prove that it is possible to permute the numbers in the table so that the sums in the columns would still be different and the sums in the rows would also be different. [i]($2$ points)[/i] [b](b)[/b] A $100 \times 100$-table is filled with numbers such that the sums in all the columns are different. Is it always possible to permute the numbers in the table so that the sums in the columns would still be different and the sums in the rows would also be different? [i]($6$ points)[/i]

On a line, $44$ points are marked and numbered $1, 2, 3,…,44$ from left to right. Various crickets jump around the line. Each starts at point $1$, jumping on the marked points and ending up at point $44$. In addition, each cricket jumps from a marked point to another marked point with a greater number. When all the crickets have finished jumping, it turns out that for pair $i, j$ with ${1}\leq{i}<{j}\leq{44}$, there was a cricket that jumped directly from point $i$ to point $j$, without visiting any of the points in between the two. Determine the smallest number of crickets such that this is possible.

A regular $2018$-gon is inscribed in a circle. The numbers $1, 2, ..., 2018$ are arranged on the vertices of the $2018$-gon, with each vertex having one number on it, such that the sum of any $2$ neighboring numbers ($2$ numbers are neighboring if the vertices they are on lie on a side of the polygon) equals the sum of the $2$ numbers that are on the antipodes of those $2$ vertices (with respect to the given circle). Determine the number of different arrangements of the numbers. (Two arrangements are identical if you can get from one of them to the other by rotating around the center of the circle).

A finite set of integers is called [i]bad[/i] if its elements add up to $2010$. A finite set of integers is a [i]Benelux-set[/i] if none of its subsets is bad. Determine the smallest positive integer $n$ such that the set $\{502, 503, 504, . . . , 2009\}$ can be partitioned into $n$ Benelux-sets. (A partition of a set $S$ into $n$ subsets is a collection of $n$ pairwise disjoint subsets of $S$, the union of which equals $S$.) [i](2nd Benelux Mathematical Olympiad 2010, Problem 1)[/i]

Let $n$ be a positive integer and $M = \{1, 2, 3, 4, 5, 6\}$. Two persons $A$ and $B$ play in the following Way: $A$ writes down a digit from $M$, $B$ appends a digit from $M$, and so it becomes alternately one digit from $M$ is appended until the $2n$-digit decimal representation of a number has been created. If this number is divisible by $9$, $B$ wins, otherwise $A$ wins. For which $n$ can $A$ and for which $n$ can $B$ force the win?

There is a pile of $1000$ matches. Two players each take turns and can take $1$ to $5$ matches. It is also allowed at most $10$ times during the whole game to take $6$ matches, for example $7$ exceptional moves can be done by the first player and $3$ moves by the second and then no more exceptional moves are allowed. Whoever takes the last match wins. Determine which player has a winning strategy.

$2017$ chess players participated in the chess tournament, each of them played exactly one chess game with each other. Let's call a trio of chess players $A, B, C$ a [i]principled [/i]one, if $A$ defeated $B$, $B$ defeated $C$, and $C$ defeated $A$. What is the largest possible number of threes of principled chess players?

There are $2019$ coins on a table. Some are placed with head up and others tail up. A group of $2019$ persons perform the following operations: the first person chooses any one coin and then turns it over, the second person choses any two coins and turns them over and so on and the $2019$-th person turns over all the coins. Prove that no matter which sides the coins are up initially, the $2019$ persons can come up with a procedure for turning the coins such that all the coins have smae side up at the end of the operations.

Consider an equilateral triangle with every side divided by $n$ points into $n+1$ equal parts. We put a marker on every of the $3n$ division points. We draw lines parallel to the sides of the triangle through the division points, and this way divide the triangle into $(n+1)^2$ smaller ones. Consider the following game: if there is a small triangle with exactly one vertex unoccupied, we put a marker on it and simultaneously take markers from the two its occupied vertices. We repeat this operation as long as it is possible. (a) If $n\equiv1\pmod3$, show that we cannot manage that only one marker remains. (b) If $n\equiv0$ or $n\equiv2\pmod3$, prove that we can finish the game leaving exactly one marker on the triangle.

In a $5 \times 9$ checkered rectangle, the middle row and middle column are colored gray. You leave the corner cell and move to the cell adjacent to the side with each move. For each transition from a gray cell to a gray one you need to pay a ruble. What is the smallest number of rubles you need to pay to go around all the squares of the board exactly once (it is not necessary to return to the starting square)?

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 \( 11 \). [img]https://i.ibb.co/LrLks9q/Kyiv-MO-2025-R2-7-1.png[/img] [i]Proposed by Mykhailo Shtandenko[/i]

Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$. (a) If $G_{n}$ does not contain $K_{2,3}$ , prove that $e(G_{n}) \leq\frac{n\sqrt{n}}{\sqrt{2}}+n$. (b) Given $n \geq 16$ distinct points $P_{1}, . . . , P_{n}$ in the plane, prove that at most $n\sqrt{n}$ of the segments $P_{i}P_{j}$ have unit length.

[b]Problem 1. [/b]In the cells of square table are written the numbers $1$, $0$ or $-1$ so that in every line there is exactly one $1$, amd exactly one $-1$. Each turn we change the places of two columns or two rows. Is it possible, from any such table, after finite number of turns to obtain its opposite table (two tables are opposite if the sum of the numbers written in any two corresponding squares is zero)? [i] Emil Kolev[/i]

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Consider $n$ segments in the plane which no two intersect and between their $2n$ endpoints no three are collinear. Is the following statement true? Statement: There exists a simple $2n$-gon such that it's vertices are the $2n$ endpoints of the segments and each segment is either completely inside the polygon or an edge of the polygon.

First $20$ positive integers are written on a board. It is known that, after you erase a number from the board, there exists a number that is equal to the arithmetic mean of the rest of the numbers left on the board. Find all the numbers that could've been erased.

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?

Is it possible to draw a closed $20$-link polyline on the plane and number its links with the numbers $1, 2, 3, \ldots, 20$ in the order of traversal so that for each natural $i = 1, 2, 3, \ldots, 10$ the links numbered $i$ and $10+i$ intersect each other and do not intersect the other links? [i] I. Efremov[/i]

Compute the number of ways of tiling the $2\times 10$ grid below with the three tiles shown. There is an infinite supply of each tile, and rotating or reflecting the tiles is not allowed. [img]https://cdn.artofproblemsolving.com/attachments/5/a/bb279c486fc85509aa1bcabcda66a8ea3faff8.png[/img]

We call a graph with n vertices $k-flowing-chromatic$ if: 1. we can place a chess on each vertex and any two neighboring (connected by an edge) chesses have different colors. 2. we can choose a hamilton cycle $v_1,v_2,\cdots , v_n$, and move the chess on $v_i$ to $v_{i+1}$ with $i=1,2,\cdots ,n$ and $v_{n+1}=v_1$, such that any two neighboring chess also have different colors. 3. after some action of step 2 we can make all the chess reach each of the n vertices. Let T(G) denote the least number k such that G is k-flowing-chromatic. If such k does not exist, denote T(G)=0. denote $\chi (G)$ the chromatic number of G. Find all the positive number m such that there is a graph G with $\chi (G)\le m$ and $T(G)\ge 2^m$ without a cycle of length small than 2017.

Xenia and Yagve take turns in playing the following game: A coin is placed on the first box in a row of nine cells. At each turn the player may choose to move the coin forward one step, move the coin forward four steps, or move coin back two steps. For a move to be allowed, the coin must land on one of them of nine cells. The winner is one who gets to move the coin to the last ninth cell. Who wins, given that Xenia makes the first move, and both players play optimally?

There are $1988$ towns and $4000$ roads in a certain country (each road connects two towns) . Prove that there is a closed path passing through no more than $20$ towns. (A. Razborov , Moscow)

[b]O[/b]n a blackboard a stranger writes the values of $s_7(n)^2$ for $n=0,1,...,7^{20}-1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board.

Find the number of functions $ A\stackrel{f}{\longrightarrow } A $ for which there exist two functions $ A\stackrel{g}{\longrightarrow } B\stackrel{h}{\longrightarrow } A $ having the properties that $ g\circ h =\text{id.} $ and $ h\circ g=f, $ where $ B $ and $ A $ are two finite sets.

An urn contains $n$ balls labeled $1, 2, ... , n$. We draw the balls out one by one (without replacing them) until we obtain a ball whose number is divisible by $k$. Find all $k$ such that the expected number of balls removed is $k$.