The 16 fields of a $4 \times 4$ checker board can be arranged in 18 lines as follows: the four lines, the four columns, the five diagonals from north west to south east and the five diagonals from north east to south west. These diagonals consists of 2,3 or 4 edge-adjacent fields of same colour; the corner fields of the chess board alone do not form a diagonal. Now, we put a token in 10 of the 16 fields. Each of the 18 lines contains an even number of tokens contains a point. What is the highest possible point number when can be achieved by optimal placing of the 10 tokens. Explain your answer.

There are $36$ participants at a BdMO event. Some of the participants shook hands with each other. But no two participants shook hands with each other more than once. Each participant recorded the number of handshakes they made. It was found that no two participants with the same number of handshakes made, had shaken hands with each other. Find the maximum possible number of handshakes at the party with proof. (When two participants shake hands with each other, this will be counted as one handshake.)

There are three empty jugs on a table. Winnie the Pooh, Rabbit, and Piglet put walnuts in the jugs one by one. They play successively, with the initial determined by a draw. Thereby Winnie the Pooh plays either in the first or second jug, Rabbit in the second or third, and Piglet in the first or third. The player after whose move there are exactly 1999 walnuts loses the games. Show that Winnie the Pooh and Piglet can cooperate so as to make Rabbit lose.

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

Denote by $S_{n}$ the set of all permutations of the set $\{1,2,\dots, n\}$. Let $\sigma \in S_{n}$ be a permutation. We define the $\textit{displacement}$ of $\sigma$ to be the number $d(\sigma)=\sum_{i=1}^{n} \vert \sigma(i)-i \vert$. We saw that $\sigma$ is $\textit{maximally}$ $\textit{displacing}$ if $d(\sigma)$ is the largest possible, i.e. if $d(\sigma) \geq d({\pi})$, for all $\pi \in S_{n}$. $\text{a)}$ Suppose $\sigma$ is a maximally displacing permutation of $\{1,2, \dots, 2024\}$. Prove that $\sigma(i)\neq i$, for all $i \in \{1,2, \dots, 2024.\}$ $\text{b)}$ Does the statement of part a) hold for permutations of $\{1,2, \dots, 2025\}$?

There are $100$ people in the group. Is it possible that for each pair of people exist at least $50$ others, so every in that group knows exactly one person from the pair?

Each cell of the infinite checkered plane contains one from the numbers $1, 2, 3, 4$ so that each number appears at least once. Let's call a cell [i]correct [/i] if the number of distinct numbers written in four adjacent (side) cells to it, equal to the number written in this cell. Can all the cells of the plane be [i]correct[/i]?

Consider $n$ lines in the plane in general position, that is, there are not three of the $n$ lines that pass through the same point. Determine if it is possible to label the $k$ points where these lines are inserted with the numbers $1$ through $k$ (using each number exactly once), so that on each line, the labels of the $n-1$ points of that line are arranged in increasing order (in one of the two directions in which they can be traversed).

An [i]animal[/i] with $n$ [i]cells[/i] is a connected figure consisting of $n$ equal-sized cells[1]. A [i]dinosaur[/i] is an animal with at least $2007$ cells. It is said to be [i]primitive[/i] it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur. (1) Animals are also called [i]polyominoes[/i]. They can be defined inductively. Two cells are [i]adjacent[/i] if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.

A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following two conditions are satisfied: i) No line passes through any point of the configuration. ii) No region contains points of both colors. Find the least value of $k$ such that for any Colombian configuration of $4027$ points, there is a good arrangement of $k$ lines. Proposed by [i]Ivan Guo[/i] from [i]Australia.[/i]

Suppose that zeros and ones are written in the cells of an $n\times n$ board, in such a way that the four cells in the intersection of any two rows and any two columns contain at least one zero. Prove that the number of ones does not exceed $\frac n2\left(1+\sqrt{4n-3}\right)$.

There are $n -1$ red balls, $n$ green balls and $n + 1$ blue balls in a bag. The number of ways of choosing two balls from the bag that have different colours is $299$. What is the value of $n$?

Niek has $16$ square cards that are yellow on one side and red on the other. He puts down the cards to form a $4 \times 4$-square. Some of the cards show their yellow side and some show their red side. For a colour pattern he calculates the [i]monochromaticity [/i] as follows. For every pair of adjacent cards that share a side he counts $+1$ or $-1$ according to the following rule: $+1$ if the adjacent cards show the same colour, and $-1$ if the adjacent cards show different colours. Adding this all together gives the monochromaticity (which might be negative). For example, if he lays down the cards as below, there are $15$ pairs of adjacent cards showing the same colour, and $9$ such pairs showing different colours. [asy] unitsize(1 cm); int i; fill(shift((0,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((1,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((2,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((3,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((0,1))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((1,1))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); fill(shift((2,1))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((3,1))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((0,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((1,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((2,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((3,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); fill(shift((0,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); fill(shift((1,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((2,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((3,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); for (i = 0; i <= 4; ++i) { draw((i,0)--(i,4)); draw((0,i)--(4,i)); } [/asy] The monochromaticity of this pattern is thus $15 \cdot (+1) + 9 \cdot (-1) = 6$. Niek investigates all possible colour patterns and makes a list of all possible numbers that appear at least once as a value of the monochromaticity. That is, Niek makes a list with all numbers such that there exists a colour pattern that has this number as its monochromaticity. (a) What are the three largest numbers on his list? ([i]Explain your answer. If your answer is, for example, $ 12$, $9$ and $6$, then you have to show that these numbers do in fact appear on the list by giving a colouring for each of these numbers, and furthermore prove that the numbers $7$, $ 8$, $10$, $11$ and all numbers bigger than $ 12$ do not appear.[/i]) (b) What are the three smallest (most negative) numbers on his list? (c) What is the smallest positive number (so, greater than $0$) on his list?

Determine all natural numbers $n$ for which it is possible to construct a rectangle of sides $15$ and $n$, with pieces congruent to: [asy] unitsize(0.6 cm); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(1,2)); draw((2,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); draw((5,-0.5)--(6,-0.5)); draw((4,0.5)--(7,0.5)); draw((4,1.5)--(7,1.5)); draw((5,2.5)--(6,2.5)); draw((4,0.5)--(4,1.5)); draw((5,-0.5)--(5,2.5)); draw((6,-0.5)--(6,2.5)); draw((7,0.5)--(7,1.5)); [/asy] The squares of the pieces have side $1$ and the pieces cannot overlap or leave free spaces

In a plane are given $n > 2$ distinct points. Some pairs of these points are connected by segments so that no two of the segments intersect. Prove that there are at most $3n-6$ segments.

We call a table of size $n \times n$ self-describing if each cell of the table contains the total number of even numbers in its row and column other than itself. How many self-describing tables of size a) $3 \times 3$ exist? b) $4 \times 4$ exist? c) $5 \times 5$ exist? Two tables are different if they differ in at least one cell.

2500 chess kings have to be placed on a $100 \times 100$ chessboard so that [b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); [b](ii)[/b] each row and each column contains exactly 25 kings. Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.) [i]Proposed by Sergei Berlov, Russia[/i]

A set consisting of $n$ points of a plane is called an isosceles $n$-point if any three of its points are located in vertices of an isosceles triangle. Find all natural numbers for which there exist isosceles $n$-points.

Several knights are arranged on an infinite chessboard. No square is attacked by more than one knight (in particular, a square occupied by a knight can be attacked by one knight but not by two). Sasha outlined a $ 14\times 16$ rectangle. What maximum number of knights can this rectangle contain?

There is an $n*n$ table with some unit cells colored black and the others are white. In each step , Amin takes a $row$ with exactly one black cell in it , and color all cells in that black cell's $column$ red. While Ali , takes a $column$ with exactly one black cell in it , and color all cells in that black cell's $row$ red. Prove that Amin can color all the cells red , iff Ali can do so.

Let $n$ be a positive integer. The numbers $\{1, 2, ..., n^2\}$ are placed in an $n \times n$ grid, each exactly once. The grid is said to be [i]Muirhead-able[/i] if the sum of the entries in each column is the same, but for every $1 \le i,k \le n-1$, the sum of the first $k$ entries in column $i$ is at least the sum of the first $k$ entries in column $i+1$. For which $n$ can one construct a Muirhead-able array such that the entries in each column are decreasing? [i]Proposed by Evan Chen[/i]

Allen and Brian are playing a game in which they roll a $6$-sided die until one of them wins. Allen wins if two consecutive rolls are equal and at most 3. Brian wins if two consecutive rolls add up to $7$ and the latter is at most $3$. What is the probability that Allen wins

We call a collection of weights (each weighing an integer value) basic if their total weight equals $200$ and each object of integer weight not greater than $200$ can be balanced exactly with a uniquely determined set of weights from the collection. (Uniquely means that we are not concerned with order or which weights of equalc value are chosen to balance against a particular object, if in fact there is a choice.) (a) Find an example of a basic collection other than the collection of $200$ weights each of value $1$. (b) How many different basic collections are there? (D. Fomin, Leningrad)

For a set $A$ of integers, define $f(A)=\{x^2+xy+y^2: x,y\in A\}$. Is there a constant $c$ such that for all positive integers $n$, there exists a set $A$ of size $n$ such that $|f(A)|\le cn$? [i]David Yang.[/i]

In Sikinia we only pay with coins that have a value of either $11$ or $12$ Kulotnik. In a burglary in one of Sikinia's banks, $11$ bandits cracked the safe and could get away with $5940$ Kulotnik. They tried to split up the money equally - so that everyone gets the same amount - but it just doesn't worked. After a while their leader claimed that it actually isn't possible. Prove that they didn't get any coin with the value $12$ Kulotnik.