Chip and Dale play the following game. Chip starts by splitting $1001$ nuts between three piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $1001$. Then Chip moves nuts from the piles he prepared to a new (fourth) pile until there will be exactly $N$ nuts in any one or more 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 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]

Let $m$ and $n$ be fixed positive integers. Tsvety and Freyja play a game on an infinite grid of unit square cells. Tsvety has secretly written a real number inside of each cell so that the sum of the numbers within every rectangle of size either $m$ by $n$ or $n$ by $m$ is zero. Freyja wants to learn all of these numbers. One by one, Freyja asks Tsvety about some cell in the grid, and Tsvety truthfully reveals what number is written in it. Freyja wins if, at any point, Freyja can simultaneously deduce the number written in every cell of the entire infinite grid (If this never occurs, Freyja has lost the game and Tsvety wins). In terms of $m$ and $n$, find the smallest number of questions that Freyja must ask to win, or show that no finite number of questions suffice. [i]Nikolai Beluhov[/i]

Players $A$ and $B$ play a game with $N \geq 2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B,A,B,A,\ldots $ by the following rules: [b](a)[/b] On every move of his $B$ passes $1$ coin from every box to an adjacent box. [b](b)[/b] On every move of hers $A$ chooses several coins that were [i]not[/i] involved in $B$'s previous move and are in different boxes. She passes every coin to an adjacent box. Player $A$'s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.

Ana and Bogdan play the following turn based game: Ana starts with a pile of $n$ ($n \ge 3$) stones. At his turn each player has to split one pile. The winner is the player who can make at his turn all the piles to have at most two stones. Depending on $n$, determine which player has a winning strategy.

On a blackboard are written all the integers from $1$ to $2008$ inclusive. Two numbers are deleted and their difference is written. For example, if you erase $5$ and $241$, you write $236$. This continues, erasing two numbers and writing their difference, until only one number remains. Determine if the number left at the end can be $2008$. What about $2007$? In each case, if the answer is affirmative, indicate a sequence with that final number, and if it is negative, explain why.

Jesse and Tjeerd are playing a game. Jesse has access to $n\ge 2$ stones. There are two boxes: in the black box there is room for half of the stones (rounded down) and in the white box there is room for half of the stones (rounded up). Jesse and Tjeerd take turns, with Jesse starting. Jesse grabs in his turn, always one new stone, writes a positive real number on the stone and places put him in one of the boxes that isn't full yet. Tjeerd sees all these numbers on the stones in the boxes and on his turn may move any stone from one box to the other box if it is not yet full, but he may also choose to do nothing. The game stops when both boxes are full. If then the total value of the stones in the black box is greater than the total value of the stones in the white box, Jesse wins; otherwise win Tjeerd. For every $n \ge 2$, determine who can definitely win (and give a corresponding winning strategy).

Mathematicians $M$ and $N$ each have their own favorite collection of manuals on the book, which he often uses in his work. Once they decided to make a statement in which each mathematician proves at each turn any theorem from his handbook which neither has yet been proven. Everything is done in turn, the mathematician starts $M$. The theorems of the handbook can win first all proven; if the theorems of both manuals can proved at once, wins the last theorem proved by a mathematician. Let $m$ be a theorem in the mathematician's handbook $M$. Find all values of $m$ for which the mathematician $M$ has a winning strategy if is It is known that there are $222$ theorems in the mathematician's handbook $N$ and $101$ of them also appears in the mathematician's $M$ handbook.

In cells of a $2022 \times 2022$ table numbers from $1$ to $2022^2$ are written, in each cell exactly one number, all numbers are used once. For every row Vlad marks the second biggest number in it, Dima does the same for every column. It turned out that boys marked $4044$ pairwise distinct numbers, and there are $k$ numbers marked by Vlad, each of which is less than all numbers marked by Dima. Find the maximum possible value of $k$

There are red and blue balls in an urn : $1024$ in total. In one round, we do the following: we draw the balls from the urn two by two. After all balls have been drawn, we put a new ball back into the urn for each pair of drawn balls: the colour of the new ball depends on that of the drawn pair. For two red balls drawn, we put back a red ball. For two blue balls, we put back a blue ball. For a red and a blue ball, we put back a black ball. For a red and a black ball, we put back a red ball. For a blue and a black ball, we put back a blue ball. Finally, for two black balls we put back a black ball. Then the next round begins. After $10$ rounds, a single ball remains in the urn, which is red. What is the maximal number of blue balls that might have been in the urn at the very beginning?

Two players $A$ and $B$ take turns taking stones from a pile of $N$ stones. They play in the order $A$, $B$, $A$, $B$, $A$, $....$, $A$ starts the game and the one who takes out the last stone loses.$ B$ can serve on each play $1$, $2$ or 3 stones, while$ A$ can draw $2, 3, 4$ stones or $1$ stone in each turn f it is the last one in the pile. Determine for what values of $N$ does $A$ have a winning strategy, and for what values the winning strategy is $B$'s.

Let $n > 1$ be a fixed integer. Alberto and Barbara play the following game: (i) Alberto chooses a positive integer, (ii) Barbara chooses an integer greater than $1$ which is a multiple or submultiple of the number Alberto chose (including itself), (iii) Alberto increases or decreases the Barbara’s number by $1$. Steps (ii) and (iii) are alternatively repeated. Barbara wins if she succeeds to reach the number $n$ in at most $50$ moves. For which values of $n$ can she win, no matter how Alberto plays?

Players $A$ and $B$ play a game with $N \geq 2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B,A,B,A,\ldots $ by the following rules: [b](a)[/b] On every move of his $B$ passes $1$ coin from every box to an adjacent box. [b](b)[/b] On every move of hers $A$ chooses several coins that were [i]not[/i] involved in $B$'s previous move and are in different boxes. She passes every coin to an adjacent box. Player $A$'s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.

The King gives the following task to his two wizards. The First Wizard should choose $7$ distinct positive integers with total sum $100$ and secretly submit them to the King. To the Second Wizard he should tell only the fourth largest number. The Second Wizard must figure out all the chosen numbers. Can the wizards succeed for sure? The wizards cannot discuss their strategy beforehand. (Mikhail Evdokimov)

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 $x$. )

There are seven piles with $2014$ pebbles each and a pile with $2008$ pebbles. Ana and Beto play in turns and Ana always plays first. One move consists of removing pebbles from all the piles. From each pile is removed a different amount of pebbles, between $1$ and $8$ pebbles. The first player who cannot make a move loses. a) Who has a winning strategy? b) If there were seven piles with $2015$ pebbles each and a pile with $2008$ pebbles, who has a winning strategy?

Mr. Ganzgenau would like to take his tea mug out of the microwave right at the front. But Mr. Ganzgenau's microwave doesn't really want to be very precise play along. To be precise, the two of them play the following game: Let $n$ be a positive integer. The turntable of the microwave makes one in $n$ seconds full turn. Each time the microwave is switched on, an integer number of seconds turned either clockwise or counterclockwise so that there are n possible positions in which the tea mug can remain. One of these positions is right up front. At the beginning, the microwave turns the tea mug to one of the $n$ possible positions. After that Mr. Ganzgenau enters an integer number of seconds in each move, and the microwave decides either clockwise or counterclockwise this number of spin for seconds. For which $n$ can Mr. Ganzgenau force the tea cup after a finite number of puffs to be able to take it out of the microwave right up front? (Birgit Vera Schmidt) [hide=original wording, in case it doesn't make much sense]Herr Ganzgenau möchte sein Teehäferl ganz genau vorne aus der Mikrowelle herausnehmen. Die Mikrowelle von Herrn Ganzgenau möchte da aber so ganz genau gar nicht mitspielen. Ganz genau gesagt spielen die beiden das folgende Spiel: Sei n eine positive ganze Zahl. In n Sekunden macht der Drehteller der Mikrowelle eine vollständige Umdrehung. Bei jedem Einschalten der Mikrowelle wird eine ganzzahlige Anzahl von Sekunden entweder im oder gegen den Uhrzeigersinn gedreht, sodass es n mögliche Positionen gibt, auf denen das Teehäferl stehen bleiben kann. Eine dieser Positionen ist ganz genau vorne. Zu Beginn dreht die Mikrowelle das Teehäferl auf eine der n möglichen Positionen. Danach gibt Herr Ganzgenau in jedem Zug eine ganzzahlige Anzahl von Sekunden ein, und die Mikrowelle entscheidet, entweder im oder gegen den Uhrzeigersinn diese Anzahl von Sekunden lang zu drehen. Für welche n kann Herr Ganzgenau erzwingen, das Teehäferl nach endlich vielen Zügen ganz genau vorne aus der Mikrowelle nehmen zu können? (Birgit Vera Schmidt) [/hide]

Players $A$ and $B$ play a "paintful" game on the real line. Player $A$ has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player $A$ picks some positive integer $m$ and provides $1/2^m $ units of ink from the pot. Player $B$ then picks an integer $k$ and blackens the interval from $k/2^m$ to $(k+1)/2^m$ (some parts of this interval may have been blackened before). The goal of player $A$ is to reach a situation where the pot is empty and the interval $[0,1]$ is not completely blackened. Decide whether there exists a strategy for player $A$ to win in a finite number of moves.

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?

(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

The five sides and five diagonals of a regular pentagon are drawn on a piece of paper. Two people play a game, in which they take turns to colour one of these ten line segments. The first player colours line segments blue, while the second player colours line segments red. A player cannot colour a line segment that has already been coloured. A player wins if they are the first to create a triangle in their own colour, whose three vertices are also vertices of the regular pentagon. The game is declared a draw if all ten line segments have been coloured without a player winning. Determine whether the first player, the second player, or neither player can force a win.

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)?

On the board are written numbers $1, 2,. . . , 33$. In one step we select two numbers written on the product of which is the square of the natural number, we wipe off the two chosen numbers and write the square root of their product on the board. This way we continue to the board only the numbers remain so that the product of neither of them is a square. (In one we can also wipe out two identical numbers and replace them with the same number.) Prove that at least $16$ numbers remain on the board.

(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]

$200$ soldiers occupy in a rectangle (military call it a square and educated military a carree): $20$ men (per row) times $10$ men (per column). In each row, we consider the tallest man (if some are of equal height, choose any of them) and of the $10$ men considered we select the shortest (if some are of equal height, choose any of them). Call him $A$. Next the soldiers assume their initial positions and in each column the shortest soldier is selected, of these $20$, the tallest is chosen. Call him $B$. Two colonels bet on which of the two soldiers chosen by these two distinct procedures is taller: $A$ or $B$. Which colonel wins the bet?