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]

A participant is given a deck of thirteen cards numerated from 1 to 13, from which he chooses seven and gives them to the assistant. Then the assistant chooses three of these seven cards and the participant – one of the remaining six in his hand. The magician then takes the chosen four cards (arranged by the participant) and guesses which one is chosen from the participant. What should the magician and assistant do so that the magic trick always happens?

Nikolai and Peter are dividing a cake in the shape of a triangle. Firstly, Nikolai chooses one point $P$ inside the triangle and after that Peter cuts the cake by any line he chooses through $P$, then takes one of the pieces and leaves the other one for Nikolai. What’s the greatest portion of the cake Nikolai can be sure he could take, if he chooses $P$ in the best way possible?

Vaggelis has a box that contains $2015$ white and $2015$ black balls. In every step, he follows the procedure below: He choses randomly two balls from the box. If they are both blacks, he paints one white and he keeps it in the box, and throw the other one out of the box. If they are both white, he keeps one in the box and throws the other out. If they are one white and one black, he throws the white out, and keeps the black in the box. He continues this procedure, until three balls remain in the box. He then looks inside and he sees that there are balls of both colors. How many white balls does he see then, and how many black?

Given an endless supply of white, blue and red cubes. In a circle arrange any $N$ of them. The robot, standing in any place of the circle, goes clockwise and, until one cube remains, constantly repeats this operation: destroys the two closest cubes in front of him and puts a new one behind him a cube of the same color if the destroyed ones are the same, and the third color if the destroyed two are different colors. We will call the arrangement of the cubes [i]good [/i] if the color of the cube remaining at the very end does not depends on where the robot started. We call $N$ [i]successful [/i] if for any choice of $N$ cubes all their arrangements are good. Find all successful $N$. I. Bogdanov

Three persons $A,B,C$, are playing the following game: A $k$-element subset of the set $\{1, . . . , 1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to $1986$. The winner is $A,B$ or $C$, respectively, if the sum of the chosen numbers leaves a remainder of $0, 1$, or $2$ when divided by $3$. For what values of $k$ is this game a fair one? (A game is fair if the three outcomes are equally probable.)

a) A game for two. The first player writes two rows of ten numbers each, the second under the first. He should provide the following property: if number $b$ is written under $a$, and $d$ -- under $c$, then $a + d = b + c$. The second player has to determine all the numbers. He is allowed to ask the questions like "What number is written in the $x$ place in the $y$ row?" What is the minimal number of the questions asked by the second player before he founds out all the numbers? b) There was a table $m\times n$ on the blackboard with the property: if You chose two rows and two columns, then the sum of the numbers in the two opposite vertices of the rectangles formed by those lines equals the sum of the numbers in two another vertices. Some of the numbers are cleaned but it is still possible to restore all the table. What is the minimal possible number of the remaining numbers?

A circle is divided by $2019$ points into equal parts. Two players delete these points in turns. A player loses, if after his turn it is possible to draw a diameter of the circle such that there are no undeleted points on one side of it. Which player has a winning strategy?

Given a cube, a cubic box, that exactly suits for the cube, and six colours. First man paints each side of the cube with its (side's) unique colour. Another man does the same with the box. Prove that the third man can put the cube in the box in such a way, that every cube side will touch the box side of different colour.

A point in the cartesian plane with integer coordinates is called a lattice point. Consider the following one player game. A finite set of selected lattice points and finite set of selected segments is called a position in this game if the following hold: (i) The endpoints of each selected segment are lattice points; (ii) Each selected segment is parallel to a coordinate axis or to one of the lines $y = \pm x$, (iii) Each selected segment contains exactly five lattice points, all of which are selected, (iv) Every two selected segments have at most one common point. A move in this game consists of selecting a lattice point and a segment such that the new set of selected lattice points and segments is a position. Prove or disprove that there exists an initial position such that the game can have infinitely many moves.

A circle is divided by $2019$ points into equal parts. Two players delete these points in turns. A player wins, if after his turn it is possible to draw a diameter of the circle such that there are no undeleted points on one side of it. 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’)

Nico picks $13$ pairwise distinct $3-$digit positive integers. Ian then selects several of these 13 numbers, the ones he wants, and using only once each selected number and some of the operations addition, subtraction, multiplication and division ($+,-,\times ,:$) must get an expression whose value is greater than $3$ and less than $4$. If he succeeds, Ian wins; otherwise, Nico wins. Which of the two has a winning strategy?

$200$ 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?

A cube 10x10x10 is constructed from 1000 white unit cubes. Polly and Velly play the following game: Velly chooses a certain amount of parallelepipeds 1x1x10, no two of which have a common vertex or an edge, and repaints them in black. Polly can choose an arbitrary number of unit cubes and ask Velly for their color. What’s the least amount of unit cubes she has to choose so that she can determine the color of each unit cube?

Two students $ A$ and $ B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $ A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $ B.$ If $ B$ answers “no,” the referee puts the question back to $ A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”

There are $36$ cards in a deck arranged in the sequence spades, clubs, hearts, diamonds, spades, clubs, hearts, diamonds, etc. Somebody took part of this deck off the top, turned it upside down, and cut this part into the remaining part of the deck (i.e. inserted it between two consecutive cards). Then four cards were taken off the top, then another four, etc. Prove that in any of these sets of four cards, all the cards are of different suits. (A Merkov, Moscow)

Thomas and Nils are playing a game. They have a number of cards, numbered $1, 2, 3$, et cetera. At the start, all cards are lying face up on the table. They take alternate turns. The person whose turn it is, chooses a card that is still lying on the table and decides to either keep the card himself or to give it to the other player. When all cards are gone, each of them calculates the sum of the numbers on his own cards. If the difference between these two outcomes is divisible by $3$, then Thomas wins. If not, then Nils wins. (a) Suppose they are playing with $2018$ cards (numbered from $1$ to $2018$) and that Thomas starts. Prove that Nils can play in such a way that he will win the game with certainty. (b) Suppose they are playing with $2020 $cards (numbered from $1$ to $2020$) and that Nils starts. Which of the two players can play in such a way that he wins with certainty?

Ana plays a game on a $100\times 100$ chessboard. Initially, there is a white pawn on each square of the bottom row and a black pawn on each square of the top row, and no other pawns anywhere else.\\ Each white pawn moves toward the top row and each black pawn moves toward the bottom row in one of the following ways: [list] [*] it moves to the square directly in front of it if there is no other pawn on it; [*] it [b]captures[/b] a pawn on one of the diagonally adjacent squares in the row immediately in front of it if there is a pawn of the opposite color on it. [/list] (We say a pawn $P$ [b]captures[/b] a pawn $Q$ of the opposite color if we remove $Q$ from the board and move $P$ to the square that $Q$ was previously on.)\\ \\ Ana can move any pawn (not necessarily alternating between black and white) according to those rules. What is the smallest number of pawns that can remain on the board after no more moves can be made? [i]Proposed by José Alejandro Reyes González, Mexico[/i]

Two students $ A$ and $ B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $ A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $ B.$ If $ B$ answers “no,” the referee puts the question back to $ A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”

Suppose you are playing a game against Daniel. There are $2017$ chips on a table. During your turn, if you can write the number of chips on the table as a sum of two cubes of not necessarily distinct, nonnegative integers, then you win. Otherwise, you can take some number of chips between $1$ and $6$ inclusive off the table. (You may not leave fewer than $0$ chips on the table.) Daniel can also do the same on his turn. You make the first move, and you and Daniel always make the optimal move during turns. Who should win the game? Explain.

Chris and Michael play a game on a board which is a rhombus of side length $n$ (a positive integer) consisting of two equilateral triangles, each of which has been divided into equilateral triangles of side length $ 1$. Each has a single token, initially on the leftmost and rightmost squares of the board, called the “home” squares (the illustration shows the case $n = 4$). [img]https://cdn.artofproblemsolving.com/attachments/e/b/8135203c22ce77c03c144850099ad1c575edb8.png[/img] A move consists of moving your token to an adjacent triangle (two triangles are adjacent only if they share a side). To win the game, you must either capture your opponent’s token (by moving to the triangle it occupies), or move on to your opponent’s home square. Supposing that Chris moves first, which, if any, player has a winning strategy?

Daniel chooses a positive integer $n$ and tells Ana. With this information, Ana chooses a positive integer $k$ and tells Daniel. Daniel draws $n$ circles on a piece of paper and chooses $k$ different points on the condition that each of them belongs to one of the circles he drew. Then he deletes the circles, and only the $k$ points marked are visible. From these points, Ana must reconstruct at least one of the circumferences that Daniel drew. Determine which is the lowest value of $k$ that allows Ana to achieve her goal regardless of how Daniel chose the $n$ circumferences and the $k$ points.

Two people play a game on a $9 \times 9$ board. They move alternately. On each move, the first player draws a cross in an empty cell, and the second player draws a nought in an empty cell. When all $81$ cells are filled, the number $K$ of rows and columns in which there are more crosses and the number $H$ of rows and columns in which there are more noughts are counted. The score for the first player is the difference $B = K- H$. Find a value of $B$ such that the first player can guarantee a score of at least $B$, while the second player can hold the first player's score to at most B, regardless how the opponent plays. (A Kanel)

( "Sisyphian Labour" ) There are $1001$ steps going up a hill , with rocks on some of them {no more than 1 rock on each step ) . Sisyphus may pick up any rock and raise it one or more steps up to the nearest empty step . Then his opponent Aid rolls a rock (with an empty step directly below it) down one step . There are $500$ rocks, originally located on the first $500$ steps. Sisyphus and Aid move rocks in turn , Sisyphus making the first move . His goal is to place a rock on the top step. Can Aid stop him? ( S . Yeliseyev)