A game consists of pieces of the shape of a regular tetrahedron of side $1$. Each face of each piece is painted in one of $n$ colors, and by this, the faces of one piece are not necessarily painted in different colors. Determine the maximum possible number of pieces, no two of which are identical.

A game has two players. In the game there is a rectangular chocolate bar, with $60$ pieces, arranged in a $6 \times 1 0$ formation , which can be broken only along the lines dividing the pieces. The first player breaks the bar along one line, discarding one section . The second player then breaks the remaining section, discarding one section. The first player repeats this process with the remaining section , and so on. The game is won by the player who leaves a single piece. In a perfect game which player wins? {S. Fomin , Leningrad)

Two players in turn put two or three coins into their own hats (before the game starts, the hats are empty). Each time, after both players made five moves, they exchange hats.The player wins, if after his move his hat contains one hundred or more coins. Which player has a winning strategy?

Anna and Ben are playing with a permutation $p$ of length $2020$, initially $p_i = 2021 - i$ for $1\le i \le 2020$. Anna has power $A$, and Ben has power $B$. Players are moving in turns, with Anna moving first. In his turn player with power $P$ can choose any $P$ elements of the permutation and rearrange them in the way he/she wants. Ben wants to sort the permutation, and Anna wants to not let this happen. Determine if Ben can make sure that the permutation will be sorted (of form $p_i = i$ for $1\le i \le 2020$) in finitely many turns, if a) $A = 1000, B = 1000$ b) $A = 1000, B = 1001$ c) $A = 1000, B = 1002$ [i]Anton Trygub[/i]

Egor and Igor take turns (Igor starts) replacing the coefficients of the polynomial \[a_{99}x^{99} + \cdots + a_1x + a_0\]with non-zero integers. Egor wants the polynomial to have as many different integer roots as possible. What is the largest number of roots he can always achieve?

Let $n$ be a positive integer. A \emph{pseudo-Gangnam Style} is a dance competition between players $A$ and $B$. At time $0$, both players face to the north. For every $k\ge 1$, at time $2k-1$, player $A$ can either choose to stay stationary, or turn $90^{\circ}$ clockwise, and player $B$ is forced to follow him; at time $2k$, player $B$ can either choose to stay stationary, or turn $90^{\circ}$ clockwise, and player $A$ is forced to follow him. After time $n$, the music stops and the competition is over. If the final position of both players is north or east, $A$ wins. If the final position of both players is south or west, $B$ wins. Determine who has a winning strategy when: (a) $n=2013^{2012}$ (b) $n=2013^{2013}$

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?

Two players $A$ and $B$ take stones one after the other from a heap with $n \ge 2$ stones. $A$ begins the game and takes at least one stone, but no more than $n -1$ stones. Thereafter, a player on turn takes at least one, but no more than the other player has taken before him. The player who takes the last stone wins. Who of the players has a winning strategy?

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 $x$. )

Alice and Bob are playing the following game: They take turns writing on the board natural numbers not exceeding $2018$ (to write the number twice is forbidden). Alice begins. A player wins if after his or her move there appear three numbers on the board which are in arithmetic progression. Which player has a winning strategy?

There are three piles of matches on the table: one with $100$ matches, one with $200$, and one with $300$. Two players play the following game. They play alternatively, and a player on turn removes one of the piles and divides one of the remaining piles into two nonempty piles. The player who cannot make a legal move loses. Who has a winning strategy? (K. Kokhas’)

Define $f : \mathbb{Z}_+ \to \mathbb{Z}_+$ such that $f(1) = 1$ and $f(n) $ is the greatest prime divisor of $n$ for $n > 1$. Aino and Väinö play a game, where each player has a pile of stones. On each turn the player to turn with $m$ stones in his pile may remove at most $f(m)$ stones from the opponent's pile, but must remove at least one stone. (The own pile stays unchanged.) The first player to clear the opponent's pile wins the game. Prove that there exists a positive integer $n$ such that Aino loses, when both players play optimally, Aino starts, and initially both players have $n$ stones.

In an $8$-square board -like the one in the figure- there is initially one checker in each square. $ \begin{tabular}{ | l | c | c |c | c| c | c | c | r| } \hline & & & & & & & \\ \hline \end{tabular} $ A move consists of choosing two tokens and moving one of them one square to the right and the other one one square to the left. If after $4$ moves the $8$ checkers are distributed in only $2$ boxes, determine what those boxes can be and how many checkers are in each one.

On an infinite grid, a square with four vertices lie at $(m, n)$, $(m-1, n)$, $(m,n-1)$, $(m-1, n-1)$ is denoted as cell $(m,n)$ $(m, n \in Z)$. Some marbles are dropped on some cell. Each cell may have more than one marble or have no marble at all. Consider a "move" can be conducted in one of two following ways: i) Remove one marble from cell $(m,n)$ (if there is marble at that cell), then add one marble to each of cell $(m - 1, n- 2)$ and cell $(m -2, n - 1)$. ii) Remove two marbles from cell $(m,n)$ (if there is marble at that cell), then add one marble to each of cell $(m +1, n - 2)$ and cell $(m - 2, n +1)$. Assume that initially, there are $n$ marbles at the cell $(1,n), (2,n - 1),..., (n, 1)$ (each cell contains one marble). Can we conduct an finite amount of moves such that both cells $(n + 1, n)$ and $(n, n + 1)$ have marbles?

There is a $2021 \times 2023$ board that has a white piece in the central square, on which Mich and Moka are going to play in turns. First Mich places a green token on any free space so that it is not in the same row or column as the white token, then Moka places a red token on any free space so that it is not in the same row or column as the white token. white or green. From now on, Mich will place green tokens and Moka will place red tokens alternately according to the following rules: $\bullet$ For the placed piece there must be another piece of the same color in its row or column, such that there is no other piece between both pieces. $\bullet$ If there is at least one box that meets the previous rule, then it is mandatory to place a token. When a token is placed, it changes all the tokens that are on squares adjacent to it to the same color. The game ends when one of the players can no longer place tiles. If when the game ends the board has more green tiles then Mich wins, and if it has more red tiles then Moka wins. Determine if either player has a winning strategy.

Chris and Michael play a game on a $5 \times 5$ board, initially containing some black and white counters as shown below: [img]https://cdn.artofproblemsolving.com/attachments/8/0/42e1a64b3524a0db722c007b8d6b8eddf2d9e5.png[/img] Chris begins by removing any black counter, and sliding a white counter from an adjacent square onto the empty square. From that point on, the players take turns. Michael slides a black counter onto an adjacent empty square, and Chris does the same with white counters (no more counters are removed). If a player has no legal move, then he loses. (a) Show that, even if Chris and Michael play cooperatively, the game will come to an end. (b) Which player has a winning strategy?

In a game, a player can level up to 16 levels. In each level, the player can upgrade an ability spending that level on it. There are three kinds of abilities, however, one ability can not be upgraded before level 6 for the first time. And that special ability can not be upgraded before level 11. Other abilities can be upgraded at any level, any times (possibly 0), but the special ability needs to be upgraded exactly twice. In how many ways can these abilities be upgraded?

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 $\frac n2\left(1+\sqrt{4n-3}\right)$.

You are given positive integers $m, n>1$. Vasyl and Petryk play the following game: they take turns marking on the coordinate plane yet unmarked points of the form $(x, y)$, where $x, y$ are positive integers with $1 \leq x \leq m, 1 \leq y \leq n$. The player loses if after his move there are two marked points, the distance between which is not a positive integer. Who will win this game if Vasyl moves first and each player wants to win? [i]Proposed by Mykyta Kharin[/i]

Let there be $2004$ be bicolor tiles, white on one side and black on the other, placed in a circle. A move consists of choosing a black piece and turning over three pieces: the chosen one, the one on its left and the one on its right. If at the beginning there is only one black piece, will it be possible, repeating the movement described, to make all the pieces have the white face up?

Two players play alternatively on an infinite square grid. The first player puts an $X$ in an empty cell and the second player puts an $O$ in an empty cell. The first player wins if he gets $11$ adjacent $X$'s in a line - horizontally, vertically or diagonally. Show that the second player can always prevent the first player from winning.

The "Sea battle" game. a) You are trying to find the $4$-field ship -- a rectangle $1x4$, situated on the $7x7$ playing board. You are allowed to ask a question, whether it occupies the particular field or not. How many questions is it necessary to ask to find that ship surely? b) The same question, but the ship is a connected (i.e. its fields have common sides) set of $4$ fields.

A table $10 \times 10$ was filled according to the rules of the game “Bomb Squad”: several cells contain bombs (one bomb per cell) while each of the remaining cells contains a number, equal to the number of bombs in all cells adjacent to it by side or by vertex. Then the table is rearranged in the “reverse” order: bombs are placed in all cells previously occupied with numbers and the remaining cells are filled with numbers according to the same rule. Can it happen that the total sum of the numbers in the table will increase in a result?

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.

Players $A$ and $B$ play a game on a blackboard that initially contains 2020 copies of the number 1 . In every round, player $A$ erases two numbers $x$ and $y$ from the blackboard, and then player $B$ writes one of the numbers $x+y$ and $|x-y|$ on the blackboard. The game terminates as soon as, at the end of some round, one of the following holds: $(1)$ one of the numbers on the blackboard is larger than the sum of all other numbers; $(2)$ there are only zeros on the blackboard. Player $B$ has to pay to player $A$ an amount in reais equivalent to the quantity of numbers left on the blackboard after the game ends. Show that player $A$ can earn at least 8 reais regardless of the moves taken by $B$ Ps.: Easier version of [url = https://artofproblemsolving.com/community/c6h2625868p22698110]ISL 2020 C8[/url]