Two players, Agon and Besa, choose a number from the set $\{1,2,3,4,5,6,7,8\}$, in turns, until no number is left. Then, each player sums all the numbers that he has chosen. We say that a player wins if the sum of his chosen numbers is a prime and the sum of the numbers that his opponent has chosen is composite. In the contrary, the game ends in a draw. Agon starts first. Does there exist a winning strategy for any of the players?

On a table, there is an empty bag and a chessboard containing exactly one token on each square. Next to the table is a large pile that contains an unlimited supply of tokens. Using only the following types of moves what is the maximum possible number of tokens that can be in the bag? $\bullet$ Type 1: Choose a non-empty square on the chessboard that is not in the rightmost column. Take a token from this square and place it, along with one token from the pile, on the square immediately to its right. $\bullet$ Type 2: Choose a non-empty square on the chessboard that is not in the bottommost row. Take a token from this square and place it, along with one token from the pile, on the square immediately below it. $\bullet$ Type 3: Choose two adjacent non-empty squares. Remove a token from each and put them both into the bag.

The vertices of a regular polygon with $N$ sides are marked on the blackboard. Ana and Beto play alternately, Ana begins. Each player, in turn, must do the following: $\bullet$ join two vertices with a segment, without cutting another already marked segment; or $\bullet$ delete a vertex that does not belong to any marked segment. The player who cannot take any action on his turn loses the game. Determine which of the two players can guarantee victory: a) if $N=28$ b) if $N=29$

Three gamblers play against each other for money. They each start by placing a pile of one-krone coins on the table, and from this point on the total number of coins on the table does not change. The ratio between the number of coins they start with is $6 : 5 : 4$. At the end of the game, the ratio of the number of coins they have is $7 : 6 : 5$ in some order. At the end of the game, one of the gamblers has three coins more than at the beginning. How many coins does this gambler have at the end?

On the table are $300$ coins. Petya, Vasya and Tolya play the next game. They go in turn in the following order: Petya, Vasya, Tolya, Petya. Vasya, Tolya, etc. In one move, Petya can take $1, 2, 3$, or $4$ coins from the table, Vasya, $1$ or $2$ coins, and Tolya, too, $1$ or $2$ coins. Can Vasya and Tolya agree so that, as if Petya were playing, one of them two will take the last coin off the table?

The pupil is training in the square equation solution. Having the recurrent equation solved, he stops, if it doesn't have two roots, or solves the next equation, with the free coefficient equal to the greatest root, the coefficient at $x$ equal to the least root, and the coefficient at $x^2$ equal to $1$. Prove that the process cannot be infinite. What maximal number of the equations he will have to solve?

A ladder is a non-decreasing sequence $a_1, a_2, \dots, a_{2020}$ of non-negative integers. Diego and Pablo play by turns with the ladder $1, 2, \dots, 2020$, starting with Diego. In each turn, the player replaces an entry $a_i$ by $a_i'<a_i$, with the condition that the sequence remains a ladder. The player who gets $(0, 0, \dots, 0)$ wins. Who has a winning strategy? [i]Proposed by Violeta Hernández[/i]

There is a set of cards with numbers from $1$ to $30$ (which may be repeated) . Each student takes one such card. The teacher can perform the following operation: He reads a list of such numbers (possibly only one) and then asks the students to raise an arm if their number was in this list. How many times must he perform such an operation in order to determine the number on each student 's card? (Indicate the number of operations and prove that it is minimal . Note that there are not necessarily 30 students.)

On a square of a chessboard there is a pawn . Two players take turns to move it to another square, subject to the rule that , at each move the distance moved is strictly greater than that of the previous move. A player loses when unable to make a move on his turn. Who wins if the players always choose the best strategy? (The pawn is always placed in the centre of its square. ) ( F . L . Nazarov)

A dragon gave a captured knight $100$ coins. Half of them are magical, but only dragon knows which are. Each day, the knight should divide the coins into two piles (not necessarily equal in size). The day when either magic coins or usual coins are spread equally between the piles, the dragon set the knight free. Can the knight guarantee himself a freedom in at most (a) $50$ days? (b) $25$ days?

Two players Arnaldo and Betania play alternately, with Arnaldo being the first to play. Initially there are two piles of stones containing $x$ and $y$ stones respectively. In each play, it is possible to perform one of the following operations: 1. Choose two non-empty piles and take one stone from each pile. 2. Choose a pile with an odd amount of stones, take one of their stones and, if possible, split into two piles with the same amount of stones. The player who cannot perform either of operations 1 and 2 loses. Determine who has the winning strategy based on $x$ and $y$.

A and B play a game on a square board consisting of $n \times n$ white tiles, where $n \ge 2$. A moves first, and the players alternate taking turns. A move consists of picking a square consisting of $2\times 2$ or $3\times 3$ white tiles and colouring all these tiles black. The first player who cannot find any such squares has lost. Show that A can always win the game if A plays the game right.

We are given a row of $n\geq7$ tiles. In the leftmost 3 tiles, there is a white piece each, and in the rightmost 3 tiles, there is a black piece each. The white and black players play in turns (the white starts). In each move, a player may take a piece of their color, and move it to an adjacent tile, so long as it's not occupied by a piece of the [u]same color[/u]. If the new tile is empty, nothing happens. If the tile is occupied by a piece of the [u]opposite color[/u], both pieces are destroyed (both white and black). The player who destroys the last two pieces wins the game. Which player has a winning strategy, and what is it? (The answer may depend on $n$)

(Game) At the beginning of the game, the organisers place paper disks on the table, grouped into piles which may contain various numbers of disks. The two players take turns. On a player’s turn, their opponent selects two piles (one if there is only one pile left), and the player must remove some number of disks from one of the piles selected. This means that at least one disk has to be removed, and removing all disks in the pile is also permitted. The player removing the last disk from the table wins. [i]Defeat the organisers in this game twice in a row! A starting position will be given and then you can decide whether you want to go first or second.[/i]

A grasshopper is sitting at an integer point in the Euclidean plane. Each second it jumps to another integer point in such a way that the jump vector is constant. A hunter that knows neither the starting point of the grasshopper nor the jump vector (but knows that the jump vector for each second is constant) wants to catch the grasshopper. Each second the hunter can choose one integer point in the plane and, if the grasshopper is there, he catches it. Can the hunter always catch the grasshopper in a finite amount of time?

General Tilly and the Duke of Wallenstein play "Divide and rule!" (Divide et impera!). To this end, they arrange $N$ tin soldiers in $M$ companies and command them by turns. Both of them must give a command and execute it in their turn. Only two commands are possible: The command "[i]Divide![/i]" chooses one company and divides it into two companies, where the commander is free to choose their size, the only condition being that both companies must contain at least one tin soldier. On the other hand, the command "[i]Rule![/i]" removes exactly one tin soldier from each company. The game is lost if in your turn you can't give a command without losing a company. Wallenstein starts to command. a) Can he force Tilly to lose if they start with $7$ companies of $7$ tin soldiers each? b) Who loses if they start with $M \ge 1$ companies consisting of $n_1 \ge 1, n_2 \ge 1, \dotsc, n_M \ge 1$ $(n_1+n_2+\dotsc+n_M=N)$ tin soldiers?

(a)The game of "super- chess" is played on a $30 \times 30$ board and involves $20$ different pieces. Each piece moves according to its own rules , but cannot move from any square to more than $20$ other squares . A piece "captures" another piece which is on a square to which it has moved. A permitted move (e.g. $m$ squares forward and $n$ squares to the right) does not depend on the piece 's starting square . Prove that (i) A piece cannot cap ture a piece on a given square from more than $20$ starting squares. (ii) It is possible to arrange all $20$ pieces on the board in such a way that not one of them can capture any of the others in one move. (b) The game of "super-chess" is played on a $100 \times 100$ board and involves $20$ different pieces. Each piece moves according to its own rules , but cannot move from any square to more than $20$ other squares. A piece "captures" another piece which is on a square to which it has moved. It is possible that a permitted move (e.g. $m$ squares forward and $n$ squares to the right) may vary, depending on the piece's position . Prove that one can arrange all $20$ pieces on the board in such a way that not one of them can capture any of the others in one move. ( A . K . Tolpygo, Kiev) PS. (a) for Juniors , (b) for Seniors

Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector). At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy? [i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]

All the natural numbers from $1$ to $1982$ are gathered in an array in an arbitrary order in computer's memory. The program looks through all the sequent pairs (first and second, second and third,...) and exchanges numbers in the pair, if the number on the lower place is greater than another. Then the program repeats the process, but moves from another end of the array. The number, that stand initially on the $100$-th place reserved its place. Find that number.

Mari and Yuri play the next play. At first, there are two piles on the table, with $m$ and $n$ candies, respectively. At each turn, players eats one pile of candy from the table and distribute another pile of candy into two non-empty parts ,. Everything is done in turn and wins the player who can no longer share the pile (when there is only one candy left). Which player will win if both use the optimal strategy and Mari makes the first move?

A set $A$ of squares is given on a chessboard which is infinite in all directions. On each square of this chessboard which does not belong to $A$ there is a king. On a command all kings may be moved in such a way that each king either remains on its square or is moved to an adjacent square, which may have been occupied by another king before the command. Each square may be occupied by at most one king. Does there exist such a number $k$ and such a way of moving the kings that after $k$ moves the kings will occupy all squares of the chessboard? Consider the following cases: (a) $A$ is the set of all squares, both of whose coordinates are multiples of $100$. (There is a horizontal line numbered by the integers from $-\infty$ to $+\infty$, and a similar vertical line. Each square of the chessboard may be denoted by two numbers, its coordinates with respect to these axes.) (b) $A$ is the set of all squares which are covered by $100$ fixed arbitrary queens (i.e. each square covered by at least one queen). Remark: If $A$ consists of just one square, then $k = 1$ and the required way is the following: all kings to the left of the square of $A$ make one move to the right.

A 2-player game is played on $n\geq 3$ points, where no 3 points are collinear. Each move consists of selecting 2 of the points and drawing a new line segment connecting them. The first player to draw a line segment that creates an odd cycle loses. (An odd cycle must have all its vertices among the $n$ points from the start, so the vertices of the cycle cannot be the intersections of the lines drawn.) Find all $n$ such that the player to move first wins.

In the game "cat and mouse" the cat chases the mouse in either labyrinth $A, B$ or $C$ . [img]https://cdn.artofproblemsolving.com/attachments/4/5/429d106736946011f4607cf95956dcb0937c84.png[/img] The cat makes the first move starting at the point marked "$K$" , moving along a marked line to an adjacent point . The mouse then moves , under the same rules, starting from the point marked "$M$" . Then the cat moves again, and so on . If, at a point of time , the cat and mouse are at the same point the cat eats the mouse. Is there available to the cat a strategy which would enable it to catch the mouse , in cases $A, B$ and $C$? (A. Sosinskiy, Moscow)

Wiebke and Stefan play the following game on a rectangular sheet of paper. They start with a rectangle with $60$ rows and $40$ columns and cut it in turns into smaller rectangles. The cuttings must be made along the gridlines, and a player in turn may cut only one smaller rectangle. By that, Stefan makes only vertical cuts, while Wiebke makes only horizontal cuts. A player who cannot make a regular move loses the game. (a) Who has a winning strategy if Stefan makes the first move? (b) Who has a winning strategy if Wiebke makes the first move?

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)