Two teams $A$ and $B$ play a school ping pong tournament. The team $A$ consists of $m$ students, and the team $B$ consists of $n$ students where $m \ne n$. There is only one ping pong table to play and the tournament is organized as follows: Two students from different teams start to play while other players form a line waiting for their turn to play. After each game the first player in the line replaces the member of the same team at the table and plays with the remaining player. The replaced player then goes to the end of the line. Prove that every two players from the opposite teams will eventually play against each other.

There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game. In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps: (a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$. (b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group. Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning. [i]Czech Republic[/i]

Monica and Bogdan are playing a game, depending on given integers $n, k$. First, Monica writes some $k$ positive numbers. Bogdan wins, if he is able to find $n$ points on the plane with the following property: for any number $m$ written by Monica, there are some two points chosen by Bogdan with distance exactly $m$ between them. Otherwise, Monica wins. Determine who has a winning strategy depending on $n, k$. [i](Proposed by Fedir Yudin)[/i]

A table with three rows and 100 columns is given. Initially, in the left cell of each row there are $400\cdot 3^{100}$ chips. At one move, Petya marks some (but at least one) chips on the table, and then Vasya chooses one of the three rows. After that, all marked chips in the selected row are shifted to the right by a cell, and all marked chips in the other rows are removed from the table. Petya wins if one of the chips goes beyond the right edge of the table; Vasya wins if all the chips are removed. Who has a winning strategy? [i]Proposed by P. Svyatokum, A. Khuzieva and D. Shabanov[/i]

If there are four numbers $(a,b,c,d)$ in four registers of the calculating machine, they turn into $(a-b,b-c,c-d,d-a)$ numbers whenever you press the button. Prove that if not all the initial numbers are equal, machine will obtain at least one number more than $1985$ after some number of the operations.

$16$ chess players held a tournament among themselves: every two chess players played exactly one game. For victory in the party was given $1$ point, for a draw $0.5$ points, for defeat $0$ points. It turned out that exactly 15 chess players shared the first place. How many points could the sixteenth chess player score?

A game is played in three moves. The first player picks any real number, then the second player makes it the coefficient of a cubic, except that the coefficient of $x^3$ is already fixed at $1$. Can the first player make his choices so that the final cubic has three distinct integer roots?

There is an empty table with $2^{100}$ rows and $100$ columns. Alice and Eva take turns filling the empty cells of the first row of the table, Alice plays first. In each move, Alice chooses an empty cell and puts a cross in it; Eva in each move chooses an empty cell and puts a zero. When no empty cells remain in the first row, the players move on to the second row, and so on (in each new row Alice plays first). The game ends when all the rows are filled. Alice wants to make as many different rows in the table as possible, while Eva wants to make as few as possible. How many different rows will be there in the table if both follow their best strategies? Proposed by Denis Afrizonov

Let it \(k\) be a fixed positive integer. Alberto and Beralto play the following game: Given an initial number \(N_0\) and starting with Alberto, they alternately do the following operation: change the number \(n\) for a number \(m\) such that \(m < n\) and \(m\) and \(n\) differ, in its base-2 representation, in exactly \(l\) consecutive digits for some \(l\) such that \(1 \leq l \leq k\). If someone can't play, he loses. We say a non-negative integer \(t\) is a [i]winner[/i] number when the gamer who receives the number \(t\) has a winning strategy, that is, he can choose the next numbers in order to guarrantee his own victory, regardless the options of the other player. Else, we call it [i]loser[/i]. Prove that, for every positive integer \(N\), the total of non-negative loser integers lesser than \(2^N\) is \(2^{N-\lfloor \frac{log(min\{N,k\})}{log 2} \rfloor}\)

A sequence of positive integers is constructed as follows. If the last digit of $a_n$ is greater than $5$, then $a_{n+1}$ is $9a_n$. If the last digit of $a_n$ is $5$ or less and an has more than one digit, then $a_{n+1}$ is obtained from $a_n$ by deleting the last digit. If $a_n$ has only one digit, which is $5$ or less, then the sequence terminates. Can we choose the first member of the sequence so that it does not terminate?

The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer ${n\ge 4}$, find the least number of edges of a graph that can be obtained by repeated applications of this operation from the complete graph on $n$ vertices (where each pair of vertices are joined by an edge). [i]Proposed by Norman Do, Australia[/i]

A board consists of $2021 \times 2021$ squares all of which are white, except for one corner square which is black. Alma and Bertha play the following game. At the beginning, there is a piece on the black square. In each turn, the player must move the piece to a new square in the same row or column as the one in which the piece is currently. All squares that the piece moves across, including the ending square but excluding the starting square, must be white, and all squares that the piece moves across, including the ending square, become black by this move. Alma begins, and the first player unable to move loses. Which player may prepare a strategy which secures her the victory? [img]https://cdn.artofproblemsolving.com/attachments/a/7/270d82f37b729bfe661f8a92cea8be67e5625c.png[/img]

Let $n \ge 2$. Ana and Beto play the following game: Ana chooses $2n$ non-negative real numbers $x_1, x_2,\ldots , x_{2n}$ (not necessarily different) whose total sum is $1$, and shows them to Beto. Then Beto arranges these numbers in a circle in the way she sees fit, calculates the product of each pair of adjacent numbers, and writes the maximum value of these products. Ana wants to maximize the number written by Beto, while Beto wants to minimize it. What number will be written if both play optimally?

Two players play the following game on a square of $N \times N$ squares. They color one square in turn so that no two colored squares are on the same diagonal. A player who cannot make a move loses. For what values of $N$ does the first player have a winning strategy?

In the beginning there are $100$ integers in a row on the blackboard. Kain and Abel then play the following game: A [i]move[/i] consists in Kain choosing a chain of consecutive numbers; the length of the chain can be any of the numbers $1,2,\dots,100$ and in particular it is allowed that Kain only chooses a single number. After Kain has chosen his chain of numbers, Abel has to decide whether he wants to add $1$ to each of the chosen numbers or instead subtract $1$ from of the numbers. After that the next move begins, and so on. If there are at least $98$ numbers on the blackboard that are divisible by $4$ after a move, then Kain has won. Prove that Kain can force a win in a finite number of moves.

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?

Chip and Dale play the following game. Chip starts by splitting $222$ nuts between two piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $222$. Then Chip moves nuts from the piles he prepared to a new (third) pile until there will be exactly $N$ nuts in any one or two piles. When Chip accomplishes his task, Dale gets an exact amount of nuts that Chip moved. What is the maximal number of nuts that Dale can get for sure, no matter how Chip acts? (Naturally, Dale wants to get as many nuts as possible, while Chip wants to lose as little as possible).

There are $64$ booths around a circular table and on each one there is a chip. The chips and the corresponding booths are numbered $1$ to $64$ in this order. At the center of the table there are $1996$ light bulbs which are all turned off. Every minute the chips move simultaneously in a circular way (following the numbering sense) as follows: chip $1$ moves one booth, chip $2$ moves two booths, etc., so that more than one chip can be in the same booth. At any minute, for each chip sharing a booth with chip $1$ a bulb is lit. Where is chip $1$ on the first minute in which all bulbs are lit?

Initially, the number $10$ is written on the board. In each subsequent moves, you can either (i) erase the number $1$ and replace it with a $10$, or (ii) erase the number $10$ and replace it with a $1$ and a $25$ or (iii) erase a $25$ and replace it with two $10$. After sometime, you notice that there are exactly one hundred copies of $1$ on the board. What is the least possible sum of all the numbers on the board at that moment?

Fedir and Mykhailo have three piles of stones: the first contains $100$ stones, the second $101$, the third $102$. They are playing a game, going in turns, Fedir makes the first move. In one move player can select any two piles of stones, let's say they have $a$ and $b$ stones left correspondently, and remove $gcd(a, b)$ stones from each of them. The player after whose move some pile becomes empty for the first time wins. Who has a winning strategy? As a reminder, $gcd(a, b)$ denotes the greatest common divisor of $a, b$. [i](Proposed by Oleksii Masalitin)[/i]

On a board $4\times 4$ the numbers from $1$ to $16$ are written, one in each box. Andres and Pablo choose four numbers each. Andrés chooses the biggest of each row and Pablo, the biggest of each column. The same number can be chosen by both. Then they are removed from the board all chosen numbers. What is the greatest value that the sum of the numbers can have what are left on the board?

Consider a game played on the integers in the closed interval $[1, n]$. The game begins with some tokens placed in $[1, n]$. At each turn, tokens are added or removed from$ [1, n]$ using the following rule: For each integer $k \in [1, n]$, if exactly one of $k - 1$ and $k + 1$ has a token, place a token at $k$ for the next turn, otherwise leave k blank for the next turn. We call a position [i]static [/i] if no changes to the interval occur after one turn. For instance, the trivial position with no tokens is static because no tokens are added or removed after a turn (because there are no tokens). Find all non-trivial static positions.

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?

Julian and Johan are playing a game with an even number of cards, say $2n$ cards, ($n \in Z_{>0}$). Every card is marked with a positive integer. The cards are shuffled and are arranged in a row, in such a way that the numbers are visible. The two players take turns picking cards. During a turn, a player can pick either the rightmost or the leftmost card. Johan is the first player to pick a card (meaning Julian will have to take the last card). Now, a player’s score is the sum of the numbers on the cards that player acquired during the game. Prove that Johan can always get a score that is at least as high as Julian’s.

Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn. During their turn, a player must pick two numbers a and b on the sheet such that $gcd(a, b) = 1$, erase these numbers from the sheet, and write the number $a + b$ on the sheet. The first player who is not able to do so, loses. Determine which player can always win this game.