The numbers $1, 2,..., 2020$ are written on the blackboard. Venus and Serena play the following game. First, Venus connects by a line segment two numbers such that one of them divides the other. Then Serena connects by a line segment two numbers which has not been connected and such that one of them divides the other. Then Venus again and they continue until there is a triangle with one vertex in $2020$, i.e. $2020$ is connected to two numbers that are connected with each other. The girl that has drawn the last line segment (completed the triangle) is the winner. Which of the girls has a winning strategy? (Tomáš Bárta, Czech Republic)

In a circle there are $2019$ plates, on each lies one cake. Petya and Vasya are playing a game. In one move, Petya points at a cake and calls number from $1$ to $16$, and Vasya moves the specified cake to the specified number of check clockwise or counterclockwise (Vasya chooses the direction each time). Petya wants at least some $k$ pastries to accumulate on one of the plates and Vasya wants to stop him. What is the largest $k$ Petya can succeed?

On a round table, $n$ bowls are arranged in a circle. Anja walks around the table clockwise, placing marbles in the bowls according to the following rule: She places a marble in any first bowl, then goes one bowl further and puts a marble in there. Then she goes two shells before putting another marble, then she goes three shells, etc. If there is at least one marble in each shell, she stops. For which $n$ does this occur?

Antonio and Beltran have impeccable logical reasoning, they put on a hat with a integer between $0$ and $19$ (including both) so that each of them sees the number that has the other (but cannot see his own number), and they must try to guess the number that have on their hat. They have a timer that a bell rings every minute and the moment it rings. This is when they must say if they know the number on their hat. A third person tells them: ''the sum of the numbers is $6$ or $11$ or $19$''. At that moment it begins to run time. After a minute the bell rings and neither of them says anything. The second minute passes , the doorbell rings and neither of us says anything. Time continues to pass and when the bell rings for the tenth time Antonio says that he already knows what is his number. Just determine the number each has in his hat.

For integers $m,n \ge 3$ we consider a $m \times n$ rectangular frame, consisting of the $2m+2n-4$ boundary squares of a $m \times n$ rectangle. Renate and Erhard play the following game on this frame, with Renate to start the game. In a move, a player colours a rectangular area consisting of a single or several white squares. If there are any more white squares, they have to form a connected region. The player who moves last wins the game. Determine all pairs $(m,n)$ for which Renate has a winning strategy.

Kolya and Dima play a game on an $8\times 8$ board, making moves in turn. During his turn, Kolya must put one cross in any empty cell (i.e., in a cell in which a cross has not yet been drawn and which has not yet been covered with a domino). Dima must cover two adjacent cells with a domino (which are not yet covered with other dominoes), in which there are an even number of crosses in total (0 or 2). The one who can't make a move loses. Which of does the player have a winning strategy, if [list=a] [*]Dima makes the first move? [*]Kolya makes the first move? [/list] [i]Proposed by M. Didin[/i]

Simon plays a game on an $n\times n$ grid of cells. Initially, each cell is filled with an integer. Every minute, Simon picks a cell satisfying the following: [list] [*] The magnitude of the integer in the chosen cell is less than $n^{n^n}$ [*] The sum of all the integers in the neighboring cells (sharing one side with the chosen cell) is non-zero [/list] Simon then adds each integer in a neighboring cell to the chosen cell. Show that Simon will eventually not be able to make any valid moves.

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

An investigator works out that he needs to ask at most $91$ questions on the basis that all the answers will be yes or no and all will be true. The questions may depend upon the earlier answers. Show that he can make do with $105$ questions if at most one answer could be a lie.

In a table consisting of $2021\times 2021$ unit squares, some unit squares are colored black in such a way that if we place a mouse in the center of any square on the table it can walk in a straight line (up, down, left or right along a column or row) and leave the table without walking on any black square (other than the initial one if it is black). What is the maximum number of squares that can be colored black?

There are $n{}$ stones in a heap. Two players play the game by alternatively taking either 1 stone from the heap or a prime number of stones which divides the current number of stones in the heap. The player who takes the last stone wins. For which $n{}$ does the first player have a strategy so that he wins no matter how the other player plays? [i]Fedor Ivlev[/i]

Let $m$ be a positive integer with $m \leq 2024$. Ana and Banana play a game alternately on a $1\times2024$ board, with squares initially painted white. Ana starts the game. Each move by Ana consists of choosing any $k \leq m$ white squares on the board and painting them all green. Each Banana play consists of choosing any sequence of consecutive green squares and painting them all white. What is the smallest value of $m$ for which Ana can guarantee that, after one of her moves, the entire board will be painted green?

Amy and Bec play the following game. Initially, there are three piles, each containing $2020$ stones. The players take turns to make a move, with Amy going first. Each move consists of choosing one of the piles available, removing the unchosen pile(s) from the game, and then dividing the chosen pile into $2$ or $3$ non-empty piles. A player loses the game if he/she is unable to make a move. Prove that Bec can always win the game, no matter how Amy plays.

The number 7 is written on a board. Alice and Bob in turn (Alice begins) write an additional digit in the number on the board: it is allowed to write the digit at the beginning (provided the digit is nonzero), between any two digits or at the end. If after someone’s turn the number on the board is a perfect square then this person wins. Is it possible for a player to guarantee the win? [i]Alexandr Gribalko[/i]

Nils is playing a game with a bag originally containing $n$ red and one black marble. He begins with a fortune equal to $1$. In each move he picks a real number $x$ with $0 \le x \le y$, where his present fortune is $y$. Then he draws a marble from the bag. If the marble is red, his fortune increases by $x$, but if it is black, it decreases by $x$. The game is over after $n$ moves when there is only a single marble left. In each move Nils chooses $x$ so that he ensures a final fortune greater or equal to $Y$ . What is the largest possible value of $Y$?

Players $A$ and $B$ play a "paintful" game on the real line. Player $A$ has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player $A$ picks some positive integer $m$ and provides $1/2^m $ units of ink from the pot. Player $B$ then picks an integer $k$ and blackens the interval from $k/2^m$ to $(k+1)/2^m$ (some parts of this interval may have been blackened before). The goal of player $A$ is to reach a situation where the pot is empty and the interval $[0,1]$ is not completely blackened. Decide whether there exists a strategy for player $A$ to win in a finite number of moves.

Let $m$ and $n$ be natural numbers with $mn$ even. Jetze is going to cover an $m \times n$ board (consisting of $m$ rows and $n$ columns) with dominoes, so that every domino covers exactly two squares, dominos do not protrude or overlap, and all squares are covered by a domino. Merlin then moves all the dominoe color red or blue on the board. Find the smallest non-negative integer $V$ (in terms of $m$ and $n$) so that Merlin can always ensure that in each row the number squares covered by a red domino and the number of squares covered by a blue one dominoes are not more than $V$, no matter how Jetze covers the board.

Alice and Bob play the following game. They write some fractions of the form $1/n$, where $n{}$ is positive integer, onto the blackboard. The first move is made by Alice. Alice writes only one fraction in each her turn and Bob writes one fraction in his first turn, two fractions in his second turn, three fractions in his third turn and so on. Bob wants to make the sum of all the fractions on the board to be an integer number after some turn. Can Alice prevent this? [i]Andrey Arzhantsev[/i]

Mathematicians $M$ and $N$ each have their own favorite collection of manuals on the book, which he often uses in his work. Once they decided to make a statement in which each mathematician proves at each turn any theorem from his handbook which neither has yet been proven. Everything is done in turn, the mathematician starts $M$. The theorems of the handbook can win first all proven; if the theorems of both manuals can proved at once, wins the last theorem proved by a mathematician. Let $m$ be a theorem in the mathematician's handbook $M$. Find all values of $m$ for which the mathematician $M$ has a winning strategy if is It is known that there are $222$ theorems in the mathematician's handbook $N$ and $101$ of them also appears in the mathematician's $M$ handbook.

Yuri is looking at the great Mayan table. The table has $200$ columns and $2^{200}$ rows. Yuri knows that each cell of the table depicts the sun or the moon, and any two rows are different (i.e. differ in at least one column). Each cell of the table is covered with a sheet. The wind has blown aways exactly two sheets from each row. Could it happen that now Yuri can find out for at least $10000$ rows what is depicted in each of them (in each of the columns)? [i]Proposed by I. Bogdanov, K. Knop[/i]

Alina and Bogdan play the following game. They have a heap and $330$ stones in it. They take turns. In one turn it is allowed to take from the heap exactly $1$, exactly $n$ or exactly $m$ stones. The player who takes the last stone wins. Before the beginning Alina says the number $n$, ($1 < n < 10$). After that Bogdan says the number $m$, ($m \ne n, 1 < m < 10$). Alina goes first. Which of the two players has a winning strategy? What if initially there are 2018 stones in the heap? adapted from a Belarus Olympiad problem

Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are replaced by $ a\minus{}b,b\minus{}c,c\minus{}d,d\minus{}a.$ Is it possible after 1996 such to have numbers $ a,b,c,d$ such the numbers $ |bc\minus{}ad|, |ac \minus{} bd|, |ab \minus{} cd|$ are primes?

The teacher wrote on a blackboard: $$x^2 + 10x + 20$$ Then all the pupils in the class came up in turn and either decreased or increased by $1$ either the free coefficient or the coefficient at $x$, but not both. Finally they have obtained: $$x^2 + 20x + 10$$ Is it true that some time during the process there was written the square polynomial with the integer roots?

The natural numbers $1$ through $2003$ are arranged in a sequence. We repeatedly perform the following operation: If the first number in the sequence is $k$, the order of the first $k$ terms is reversed. Prove that after several operations number $1$ will occur on the first place.

Consider the set of all integer points in $Z^3$. Sasha and Masha play such a game. At first, Masha marks an arbitrary point. After that, Sasha marks all the points on some a plane perpendicular to one of the coordinate axes and at no point, which Masha noted. Next, they continue to take turns (Masha can't to select previously marked points, Sasha cannot choose the planes on which there are points said Masha). Masha wants to mark $n$ consecutive points on some line that parallel to one of the coordinate axes, and Sasha seeks to interfere with it. Find all $n$, in which Masha can achieve the desired result.