Form $10A$ has $29$ students who are listed in order on its duty roster. Form $10B$ has $32$ students who are listed in order on its duty roster. Every day two students are on duty, one from form $10A$ and one from form $10B$. Each day just one of the students on duty changes and is replaced by the following student on the relevant roster (when the last student on a roster is replaced he is replaced by the first). On two particular days the same two students were on duty. Is it possible that starting on the first of these days and ending the day before the second, every pair of students (one from $10A$ and one from $10B$) shared duty exactly once?

For positive integers $t,a$, and $b$, Lucy and Windy play the $(t,a,b)$- [i]game [/i] defined by the following rules. Initially, the number $t$ is written on a blackboard. On her turn, a player erases the number on the board and writes either the number $t - a$ or $t - b$ on the board. Lucy goes first and then the players alternate. The player who first reaches a negative losses the game. Prove that there exist infinitely many values of $t$ in which Lucy has a winning strategy for all pairs $(a, b)$ with $a + b = 2005$.

Two players play a game as follows: there $n > 1$ rounds and $d \geq 1$ is fixed. In the first round A picks a positive integer $m_1$, then B picks a positive integer $n_1 \not = m_1$. In round $k$ (for $k = 2, \ldots , n$), A picks an integer $m_k$ such that $m_{k-1} < m_k \leq m_{k-1} + d$. Then B picks an integer $n_k$ such that $n_{k-1} < n_k \leq n_{k-1} + d$. A gets $\gcd(m_k,n_{k-1})$ points and B gets $\gcd(m_k,n_k)$ points. After $n$ rounds, A wins if he has at least as many points as B, otherwise he loses. For each $(n, d)$ which player 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 $x$. )

Let $k$ be a positive real. $A$ and $B$ play the following game: at the start, there are $80$ zeroes arrange around a circle. Each turn, $A$ increases some of these $80$ numbers, such that the total sum added is $1$. Next, $B$ selects ten consecutive numbers with the largest sum, and reduces them all to $0$. $A$ then wins the game if he/she can ensure that at least one of the number is $\geq k$ at some finite point of time. Determine all $k$ such that $A$ can always win the game.

(a) An infinite sheet is divided into squares by two sets of parallel lines. Two players play the following game: the first player chooses a square and colours it red, the second player chooses a non-coloured square and colours it blue, the first player chooses a non-coloured square and colours it red, the second player chooses a non-coloured square and colours it blue, and so on. The goal of the first player is to colour four squares whose vertices form a square with sides parallel to the lines of the two parallel sets. The goal of the second player is to prevent him. Can the first player win? (b) What is the answer to this question if the second player is permitted to colour two squares at once? (DG Azov) PS. (a) for Juniors, (a),(b) for Seniors

Andrej and Barbara play the following game with two strips of newspaper of length $a$ and $b$. They alternately cut from any end of any of the strips a piece of length $d$. The player who cannot cut such a piece loses the game. Andrej allows Barbara to start the game. Find out how the lengths of the strips determine the winner.

Nickolas and Peter divide $2n+1$ nuts amongst each other. Both of them want to get as many as possible. Three methods are suggested to them for doing so, each consisting of three stages. The first two stages are the same in all three methods: [i]Stage 1:[/i] Peter divides the nuts into 2 heaps, each containing at least 2 nuts. [i]Stage 2:[/i] Nickolas divides both heaps into 2 heaps, each containing at least 1 nut. Finally, stage 3 varies among the three methods as follows: [i]Method 1:[/i] Nickolas takes the smallest and largest of the heaps. [i]Method 2:[/i] Nickolas takes the two middle size heaps. [i]Method 3:[/i] Nickolas chooses between taking the biggest and the smallest heap or the two middle size heaps, but gives one nut to Peter for the right of choice. Determine the most and the least profitable method for Nickolas.

Albert and Brita play a game with a bar of $19$ adjacent squares. Initially, there is a button on the middle square of the bar. At every turn Albert mentions one positive integer less than $5$, and Brita moves button a number of squares in the direction of her choice - while doing so however, Brita must not move the button more than twice in one direction order. Prove that Albert can choose the numbers so that by the $19$th turn, Brita to be forced to move the button out of the bar.

Two players play a game as follows. The first player chooses two non-zero integers A and B. The second player forms a quadratic with A, B and 1998 as coefficients (in any order). The first player wins iff the equation has two distinct rational roots. Show that the first player can always win.

A regular icosahedron is a regular solid of $20$ faces, each in the form of an equilateral triangle, with $12$ vertices, so that each vertex is in $5$ edges. Twelve indistinguishable candies are glued to the vertices of a regular icosahedron (one at each vertex), and four of these twelve candies are special. André and Lucas want to together create a strategy for the following game: • First, André is told which are the four special sweets and he must remove exactly four sweets that are not special from the icosahedron and leave the solid on a table, leaving afterwards without communicating with Lucas. • Later, Sponchi, who wants to prevent Lucas from discovering the special sweets, can pick up the icosahedron from the table and rotate it however he wants. • After Sponchi makes his move, he leaves the room, Lucas enters and he must determine the four special candies out of the eight that remain in the icosahedron. Determine if there is a strategy for which Lucas can always properly discover the four special sweets.

Agustín and Lucas, by turns, each time mark a box that has not yet been marked on a $101\times 101$ grid board. Augustine starts the game. You cannot check a box that already has two checked boxes in its row or column. The one who can't make his move loses. Decide which of the two players has a winning strategy.

Alice and Bob play the following game: Alice picks a set $A = \{1, 2, ..., n \}$ for some natural number $n \ge 2$. Then, starting from Bob, they alternatively choose one number from the set $A$, according to the following conditions: initially Bob chooses any number he wants, afterwards the number chosen at each step should be distinct from all the already chosen numbers and should differ by $1$ from an already chosen number. The game ends when all numbers from the set $A$ are chosen. Alice wins if the sum of all the numbers that she has chosen is composite. Otherwise Bob wins. Decide which player has a winning strategy. Proposed by [i]Demetres Christofides, Cyprus[/i]

There is a row of $100N$ sandwiches with ham. A boy and his cat play a game. In one action the boy eats the first sandwich from any end of the row. In one action the cat either eats the ham from one sandwich or does nothing. The boy performs 100 actions in each of his turns, and the cat makes only 1 action each turn; the boy starts first. The boy wins if the last sandwich he eats contains ham. Is it true that he can win for any positive integer $N{}$ no matter how the cat plays? [i]Ivan Mitrofanov[/i]

There is a piece on each square of the solitaire board shown except for the central square. A move can be made when there are three adjacent squares in a horizontal or vertical line with two adjacent squares occupied and the third square vacant. The move is to remove the two pieces from the occupied squares and to place a piece on the third square. (One can regard one of the pieces as hopping over the other and taking it.) Is it possible to end up with a single piece on the board, on the square marked $X$?

Odin and Evelyn are playing a game, Odin going first. There are initially $3k$ empty boxes, for some given positive integer $k$. On each player’s turn, they can write a non-negative integer in an empty box, or erase a number in a box and replace it with a strictly smaller non-negative integer. However, Odin is only ever allowed to write odd numbers, and Evelyn is only allowed to write even numbers. The game ends when either one of the players cannot move, in which case the other player wins; or there are exactly $k$ boxes with the number $0$, in which case Evelyn wins if all other boxes contain the number $1$, and Odin wins otherwise. Who has a winning strategy? $Agnijo \ Banerjee \ , United \ Kingdom$

In an infinite grid of regular triangles, Niels and Henrik are playing a game they made up. Every other time, Niels picks a triangle and writes $\times$ in it, and every other time, Henrik picks a triangle where he writes a $o$. If one of the players gets four in a row in some direction (see figure), he wins the game. Determine whether one of the players can force a victory. [img]https://cdn.artofproblemsolving.com/attachments/6/e/5e80f60f110a81a74268fded7fd75a71e07d3a.png[/img]

SQUARING OFF: Master Chief and Samus Aran take turns firing rockets at one another from across the Cartesian plane. Master Chief’s movement is restricted to lattice points within the $10\times 10$ square with vertices $(0,0)$, $(0,10)$, $(10,0)$, and $(10,10)$, while Samus Aran’s movement is restricted to lattice points inside the $10\times 10$ square with vertices $(0,0)$, $(-10,0)$, $(0,-10)$, and $(-10,-10)$. Neither player can be located on or beyond the border of his or her square. Both players randomly choose a lattice point at which they begin the game, and do not move the rest of the game (until either they are killed or kill the other player). Each player’s turn consists of firing a rocket, targeted at a specific undestroyed lattice point inside the border of the opponent’s movement square, which hits immediately. When a rocket hits its intended lattice point, it explodes, destroying the surrounding $3\times 3$ square ($8$ additional adjacent lattice points). The game ends when one player is hit by a rocket (when the player is located within the $3\times 3$ grid hit by a rocket). If the highest possible probability that Samus Aran wins the game in three turns or less, assuming Master Chief goes first, is expressed as $a/b$, where $a$ and $b$ are relatively prime integers, find $a+b$.

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.

Players $A$ and $B$ play the following game: $A$ tosses a coin $n$ times, and $B$ does $n+1$ times. The player who obtains more ”heads” wins; or in the case of equal balances, $A$ is assigned victory. Find the values of $n$ for which this game is fair (i.e. both players have equal chances for victory).

A solitaire game is played on an $m\times n$ rectangular board, using $mn$ markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs $(m,n)$ of positive integers such that all markers can be removed from the board.

On competition which has $16$ teams, it is played $55$ games. Prove that among them exists $3$ teams such that they have not played any matches between themselves.

Positive integers $n$, $k>1$ are given. Pasha and Vova play a game on a board $n\times k$. Pasha begins, and further they alternate the following moves. On each move a player should place a border of length 1 between two adjacent cells. The player loses if after his move there is no way from the bottom left cell to the top right without crossing any order. Determine who of the players has a winning strategy.

Player Zero and Player One play a game on a $n \times n$ board ($n \ge 1$). The columns of this $n \times n$ board are numbered $1,2,4,\dots,2^{n-1}$. Turn my turn, the players put their own number in one of the free cells (thus Player Zero puts a $0$ and Player One puts a $1$). Player Zero begins. When the board is filled, the game ends and each row yields a (reverse binary) number obtained by adding the values of the columns with a $1$ in that row. For instance, when $n=4$, a row with $0101$ yields the number $0 \cdot1+1 \cdot 2+0 \cdot 4+1 \cdot 8=10$. a) For which natural numbers $n$ can Player One always ensure that at least one of the row numbers is divisible by $4$? b) For which natural numbers $n$ can Player One always ensure that at least one of the row numbers is divisible by $3$?

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]