Batman, Robin, and The Joker are in three of the vertex cells in a square $2025 \times 2025$ board, such that Batman and Robin are on the same diagonal (picture). In each round, first The Joker moves to an adjacent cell (having a common side), without exiting the board. Then in the same round Batman and Robin move to an adjacent cell. The Joker wins if he reaches the fourth "target" vertex cell (marked T). Batman and Robin win if they catch The Joker i.e. at least one of them is on the same cell as The Joker. If in each move all three can see where the others moved, who has a winning strategy, The Joker, or Batman and Robin? Explain the answer. [b]Comment.[/b] Batman and Robin decide their common strategy at the beginning. [img]https://i.imgur.com/PeLBQNt.png[/img]

A lottery ticket has $50$ cells into which one must put a permutation of $1, 2, 3, ... , 50$. Any ticket with at least one cell matching the winning permutation wins a prize. How many tickets are needed to be sure of winning a prize?

Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$; Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$. Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins. [i]Proposed by Hans Zantema, Netherlands[/i]

Alice and Bob determine a number with $2018$ digits in the decimal system by choosing digits from left to right. Alice starts and then they each choose a digit in turn. They have to observe the rule that each digit must differ from the previously chosen digit modulo $3$. Since Bob will make the last move, he bets that he can make sure that the final number is divisible by $3$. Can Alice avoid that? [i](Proposed by Richard Henner)[/i]

LeBron James and Carmelo Anthony play a game of one-on-one basketball where the first player to $3$ points or more wins. LeBron James has a $20\%$ chance of making a $3$-point shot; Carmelo has a $10\%$ chance of making a $3$-pointer. LeBron has a $40\%$ chance of making a $2$-point shot from anywhere inside the $3$-point line (excluding dunks, which are also worth $2$ points); Carmelo has a $52\%$ chance of making a $ 2$-point shot from anywhere inside the 3-point line (excluding dunks). LeBron has a $90\%$ chance of dunking on Carmelo; Carmelo has a $95\%$ chance of dunking on LeBron. If each player has $3$ possessions to try to win, LeBron James goes first, and both players follow a rational strategy to try to win, what is the probability that Carmelo Anthony wins the game?

Ana & Bruno decide to play a game with the following rules.: a) Ana has cards $1, 3, 5,7,..., 2n-1$ b) Bruno has cards $2, 4,6, 8,...,2n$ During the first turn and all odd turns afterwards, Bruno chooses one of his cards first and reveals it to Ana, and Ana chooses one of her cards second. Whoever's card is higher gains a point. During the second turn and all even turns afterwards, Ana chooses one of her cards first and reveals it to Bruno, and Bruno chooses one of his cards second. Similarly, whoever's card is higher gains a point. During each turn, neither player can use a card they have already used on a previous turn. The game ends when all cards have been used after $n$ turns. Determine the highest number of points Ana can earn, and how she manages to do this.

Three players $A,B$ and $C$ take turns removing stones from a pile of $N$ stones. They move in the order $A,B,C,A,B,C,…A$. The game begins, and the one who takes out the last stone loses the game. The players $A$ and $C$ team up against $B$ , they agree on a joint strategy. $B$ can take in each play $1,2,3,4$ or $5$ stones, while $A$ and $C$, they can each get $1,2$ or $3$ stones each turn. Determine for what values ​​of $N$ have winning strategy $A$ and $C$, and for what values ​​the winning strategy is from $B$. .

$2019$ point grasshoppers sit on a line. At each move one of the grasshoppers jumps over another one and lands at the point the same distance away from it. Jumping only to the right, the grasshoppers are able to position themselves so that some two of them are exactly $1$ mm apart. Prove that the grasshoppers can achieve the same, jumping only to the left and starting from the initial position. (Sergey Dorichenko)

We’re playing a game with a sequence of $2008$ non-negative integers. A move consists of picking a integer $b$ from that sequence, of which the neighbours $a$ and $c$ are positive. We then replace $a, b$ and $c$ by $a - 1, b + 7$ and $c - 1$ respectively. It is not allowed to pick the first or the last integer in the sequence, since they only have one neighbour. If there is no integer left such that both of its neighbours are positive, then there is no move left, and the game ends. Prove that the game always ends, regardless of the sequence of integers we begin with, and regardless of the moves we make.

Two player are playing in an $100 \times 100$ grid. Initially the whole board is black. On $A$'s move, he selects $4 \times 4$ subgrid and color it white. On $B$'s move, he selects a $3 \times 3$ subgrid and colors it black. $A$ wants to make the whole board white. Can he do it? [i]Proposed by S M A Nahian[/i]

Give and Take divide $100$ coins between themselves as follows. In each step, Give chooses a handful of coins from the heap, and Take decides who gets this handful. This is repeated until all coins have been taken, or one of them has $9$ handfuls. In the latter case, the other gets all the remaining coins. What is the largest number of coins that Give can be sure of getting no matter what Take does? (A Shapovalov)

The maze is an $8 \times 8 $square, each cell contains $1 \times 1$ which has one of four arrows drawn (up, down, right, left). The upper side of the upper right cell is the exit from the maze.In the lower left cell there is a chip that, with each move, moves one square in the direction indicated by the arrow. After each move, the shooter in the cell in which there was just a chip rotates $90^o$ clockwise. If a chip must move, taking it outside the $8 \times 8$ square, it remains in place, and the arrow also rotates $90^o$ clockwise. Prove that sooner or later, the chip will come out of the maze.

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?

Rodolfo and Gabriela have $9$ chips numbered from $1$ to $9$ and they have fun with the following game: They remove the chips one by one and alternately (until they have $3$ chips each), with the following rules: $\bullet$ Rodolfo begins the game, choosing a chip and in the following moves he must remove, each time, a chip three units greater than the last chip drawn by Gabriela. $\bullet$ Gabriela, on her turn, chooses a first chip and in the following times she must draw, each time, a chip two units smaller than the last chip that she herself drew. $\bullet$ The game is won by whoever gets the highest number by adding up their three tokens. $\bullet$ If the game cannot be completed, a tie is declared. If they play without making mistakes, how should Rodolfo play to be sure he doesn't lose?

There are $68$ coins , each coin having a different weight than that of each other . Show how to find the heaviest and lightest coin in $100$ weighings on a balance beam. (S. Fomin, Leningrad)

The game [i]Clobber [/i] is played by two on a strip of $2k$ squares. At the beginning there is a piece on each square, the pieces of both players stand alternatingly. At each move the player shifts one of his pieces to the neighbouring square that holds a piece of his opponent and removes his opponent’s piece from the table. The moves are made in turn, the player whose opponent cannot move anymore is the winner. Prove that if for some $k$ the player who does not start the game has the winning strategy, then for $k + 1$ and $k + 2$ the player who makes the first move has the winning strategy.

Kostya and Sergey play a game on a white strip of length 2016 cells. Kostya (he plays first) in one move should paint black over two neighboring white cells. Sergey should paint either one white cell either three neighboring white cells. It is forbidden to make a move, after which a white cell is formed the doesn't having any white neighbors. Loses the one that can make no other move. However, if all cells are painted, then Kostya wins. Who will win if he plays the right game (has a winning strategy)?

$A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$, but his number cannot be bigger than $N$. The player who writes $N$ wins. For which values of $N$ does $B$ win? [i]Proposed by A. Slinko & S. Marshall, New Zealand[/i]

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/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 $x$. )

On a rectangular board $100 \times 300$, two people take turns coloring the cells that have not yet been colored. The first one colors cells in yellow, and the second one in blue. Coloring is completed when every cell of the board is colored. A [i]connected sequence[/i] of cells is a sequence of cells in which every two consecutive cells share a common side (and all cells in the sequence are different). Consider all possible connected sequences of yellow cells. The result of the first player is the number of cells in the connected sequence of yellow cells of maximum length. The first player's goal is to maximize the result, and the second player's goal is to make the first player's result as small as possible. Prove that if each player tries to achieve his goal, the result of the first player will be no more than $200$. [i]Proposed by Mykhailo Shtandenko and Fedir Yudin[/i]

Lamija and Faris are playing the following game. Cards, which are numerated from $1$ to $100$, are placed one next to other, starting from $1$ to $100$. Now Faris picks every $7$th card, and after that every card which contains number $7$. After that Lamija picks from remaining cards ones divisible with $5$, and after that cards which contain number $5$. Who will have more cards and how many ? How would game end, if Lamija started with "$5$ rule" and Faris continues with "$7$ rule"?

A fraction with $1010$ squares in the numerator and $1011$ squares in the denominator serves as a game board for a two player game. $$\frac{\square + \square +...+ \square}{\square + \square +...+ \square+ \square}$$ Players take turns in moves. In each turn, the player chooses one of the numbers $1, 2,. . . , 2021$ and inserts it in any empty field. Each number can only be used once. The starting player wins if the value of the fraction after all the fields is filled differs from number $1$ by less than $10^{-6}$. Otherwise, the other player wins. Decide which of the players has a winning strategy. (Pavel Šalom)

Leo and Smilla find $2020$ gold nuggets with masses $1,2,\dots,2020$ gram, which they distribute to a red and a blue treasure chest according to the following rule: First, Leo chooses one of the chests and tells its colour to Smilla. Then Smilla chooses one of the not yet distributed nuggets and puts it into this chest. This is repeated until all the nuggets are distributed. Finally, Smilla chooses one of the chests and wins all the nuggets from this chest. How many gram of gold can Smilla make sure to win?

A magician and his assistant are performing the following trick. There is a row of $13$ empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of his choice, so that the assistant knows which boxes contain coins. The magician returns and the assistant is allowed to open one box that does not contain a coin. Next, the magician selects four boxes, which are then simultaneously opened. The goal of the magician is to open both boxes that contain coins. Devise a method that will allow the magician and his assistant to always successfully perform the trick. (Igor Zhizhilkin) [url=https://artofproblemsolving.com/community/c6h1801447p11962869]junior version posted here[/url]