Amber and Brian are playing a game using $2010$ coins. Throughout the game, the coins are divided into a number of piles of at least 1 coin each. A move consists of choosing one or more piles and dividing each of them into two smaller piles. (So piles consisting of only $1$ coin cannot be chosen.) Initially, there is only one pile containing all $2010$ coins. Amber and Brian alternatingly take turns to make a move, starting with Amber. The winner is the one achieving the situation where all piles have only one coin. Show that Amber can win the game, no matter which moves Brian makes.

The figure shows a game board with $16$ squares. At the start of the game, two cars are placed in different squares. Two players $A$ and $B$ alternately take turns, and A starts. In each turn, the player chooses one of the cars and moves it one or more squares to the right. The left-most car may never overtake or land on the same square as the right-most car. The first player which is unable to move loses. [img]https://cdn.artofproblemsolving.com/attachments/1/b/8d6f40fac4983d6aa9bd076392c91a6d200f6a.png[/img] (a) Prove that A can win regardless of how $B$ plays, if the two cars start as shown in the figure. (b) Determine all starting positions in which $B$ can win regardless of how $A$ plays.

Among a group of programmers, every two either know each other or do not know each other. Eleven of them are geniuses. Two companies hire them one at a time, alternately, and may not hire someone already hired by the other company. There are no conditions on which programmer a company may hire in the first round. Thereafter, a company may only hire a programmer who knows another programmer already hired by that company. Is it possible for the company which hires second to hire ten of the geniuses, no matter what the hiring strategy of the other company may be?

A set $A$ of squares is given on a chessboard which is infinite in all directions. On each square of this chessboard which does not belong to $A$ there is a king. On a command all kings may be moved in such a way that each king either remains on its square or is moved to an adjacent square, which may have been occupied by another king before the command. Each square may be occupied by at most one king. Does there exist such a number $k$ and such a way of moving the kings that after $k$ moves the kings will occupy all squares of the chessboard? Consider the following cases: (a) $A$ is the set of all squares, both of whose coordinates are multiples of $100$. (There is a horizontal line numbered by the integers from $-\infty$ to $+\infty$, and a similar vertical line. Each square of the chessboard may be denoted by two numbers, its coordinates with respect to these axes.) (b) $A$ is the set of all squares which are covered by $100$ fixed arbitrary queens (i.e. each square covered by at least one queen). Remark: If $A$ consists of just one square, then $k = 1$ and the required way is the following: all kings to the left of the square of $A$ make one move to the right.

Ari and Beri play a game using a deck of $2020$ cards with exactly one card with each number from $1$ to $2020$. Ari gets a card with a number $a$ and removes it from the deck. Beri sees the card, chooses another card from the deck with a number $b$ and removes it from the deck. Then Beri writes on the board exactly one of the trinomials $x^2-ax+b$ or $x^2-bx+a$ from his choice. This process continues until no cards are left on the deck. If at the end of the game every trinomial written on the board has integer solutions, Beri wins. Otherwise, Ari wins. Prove that Beri can always win, no matter how Ari plays.

Let $n$ be a natural number. At first the cells of a table $2n$ x $2n$ are colored in white. Two players $A$ and $B$ play the following game. First is $A$ who has to color $m$ arbitrary cells in red and after that $B$ chooses $n$ rows and $n$ columns and color their cells in black. Player $A$ wins, if there is at least one red cell on the board. Find the least value of $m$ for which $A$ wins no matter how $B$ plays.

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.”

Two players $A$ and $B$ play alternatively in a convex polygon with $n \geq 5$ sides. In each turn, the corresponding player has to draw a diagonal that does not cut inside the polygon previously drawn diagonals. A player loses if after his turn, one quadrilateral is formed such that its two diagonals are not drawn. $A$ starts the game. For each positive integer $n$, find a winning strategy for one of the players.

Anna and Orjan play the following game: they start with a positive integer $n>1$, Anna writes it as the sum of two other positive integers, $n = n_1+n_2$. Orjan deletes one of them, $n_1$ or $n_2$. If the remaining number is larger than $1$, the process is repeated, i.e. Anna writes it as the sum of two positive integers, $ n_3+n_4$, Orjan deletes one of them etc. The game ends when the last number is $1$. Orjan is the winner if there are two equal numbers among the numbers he has deleted, otherwise Anna wins. Who is winning the game if n = 2008 and they both play optimally?

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.

Two players play a game on an infinite board that consists of unit squares. Player $I$ chooses a square and marks it with $O$. Then player $II$ chooses another square and marks it with $X$. They play until one of the players marks a whole row or a whole column of five consecutive squares, and this player wins the game. If no player can achieve this, the result of the game is a tie. Show that player $II$ can prevent player $I$ from winning.

Veronica, Ana and Gabriela are forming a round and have fun with the following game. One of them chooses a number and says out loud, the one to its left divides it by its largest prime divisor and says the result out loud and so on. The one who says the number out loud $1$ wins , at which point the game ends. Ana chose a number greater than $50$ and less than $100$ and won. Veronica chose the number following the one chosen by Ana and also won. Determine all the numbers that could have been chosen by Ana.

A game consists of a grid of $4\times 4$ and tiles of two colors (Yellow and White). A player chooses a type of token and gives it to the second player who places it where he wants, then the second player chooses a type of token and gives it to the first who places it where he wants, They continue in this way and the one who manages to form a line with three tiles of the same color wins (horizontal, vertical or diagonal and regardless of whether it is the tile you started with or not). Before starting the game, two yellow and two white pieces are already placed as shows the figure below. [img]https://cdn.artofproblemsolving.com/attachments/b/5/ba11377252c278c4154a8c3257faf363430ef7.png[/img] Yolanda and Xinia play a game. If Yolanda starts (choosing the token and giving it to Xinia for this to place) indicate if there is a winning strategy for either of the two players and, if any, describe the strategy.

Two are playing the game "cats and rats" on the chess-board $8\times 8$. The first has one piece -- a rat, the second -- several pieces -- cats. All the pieces have four available moves -- up, down, left, right -- to the neighbour field, but the rat can also escape from the board if it is on the boarder of the chess-board. If they appear on the same field -- the rat is eaten. The players move in turn, but the second can move all the cats in independent directions. a) Let there be two cats. The rat is on the interior field. Is it possible to put the cats on such a fields on the border that they will be able to catch the rat? b) Let there be three cats, but the rat moves twice during the first turn. Prove that the rat can escape.

There are $n$ coins in a row. Two players take turns picking a coin and flipping it. The location of the heads and tails should not repeat. Loses the one who can not make a move. Which of player can always win, no matter how his opponent plays?

The vertices of a regular polygon with $N$ sides are marked on the blackboard. Ana and Beto play alternately, Ana begins. Each player, in turn, must do the following: $\bullet$ join two vertices with a segment, without cutting another already marked segment; or $\bullet$ delete a vertex that does not belong to any marked segment. The player who cannot take any action on his turn loses the game. Determine which of the two players can guarantee victory: a) if $N=28$ b) if $N=29$

On a blackboard, there are $17$ integers not divisible by $17$. Alice and Bob play a game. Alice starts and they alternately play the following moves: $\bullet$ Alice chooses a number $a$ on the blackboard and replaces it with $a^2$ $\bullet$ Bob chooses a number $b$ on the blackboard and replaces it with $b^3$. Alice wins if the sum of the numbers on the blackboard is a multiple of $17$ after a finite number of steps. Prove that Alice has a winning strategy. (Daniel Holmes)

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).

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.

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?

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.

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?

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)

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?