A game starts with four heaps of beans, containing 3, 4, 5 and 6 beans. The two players move alternately. A move consists of taking [list] (a) $\text{either}$ one bean from a heap, provided at least two beans are left behind in that heap, (b) $\text{or}$ a complete heap of two or three beans.[/list] The player who takes the last heap wins. To win the game, do you want to move first or second? Give a winning strategy.

Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following: The last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) becomes four times the number deleted number added. The number on the board is now replaced by the result of this calculation. Lisa repeats this until she gets a number for the first time was on the board. (a) Show that the sequence of moves always ends. (b) If Lisa begins with the number $53^{2022} - 1$, what is the last number on the board? Example: If Lisa starts with the number $2022$, she gets $202 + 4\cdot 2 = 210$ in the first move and overall the result $$2022 \to 210 \to 21 \to 6 \to 24 \to 18 \to 33 \to 15 \to 21$$. Since Lisa gets $21$ for the second time, the turn order ends. [i](Stephan Pfannerer)[/i]

Two players take turns placing the numbers $1, 2, 3,. . . , 24$, in each of the $24$ squares on the surface of a $2 \times 2 \times 2$ cube (each number can be placed once). The second player wants the sum of the numbers in each cell the rings of $8$ cells encircling the cube were identical. Will he be able to the first player to stop him?

The King decided to reduce his Council consisting of thousand wizards. He placed them in a line and placed hats with numbers from $1$ to $1001$ on their heads not necessarily in this order (one hat was hidden). Each wizard can see the numbers on the hats of all those before him but not on himself or on anyone who stayed behind him. By King's command, starting from the end of the line each wizard calls one integer from $1$ to $1001$ so that every wizard in the line can hear it. No number can be repeated twice. In the end each wizard who fails to call the number on his hat is removed from the Council. The wizards knew the conditions of testing and could work out their strategy prior to it. (a) Can the wizards work out a strategy which guarantees that more than $500$ of them remain in the Council? (b) Can the wizards work out a strategy which guarantees that at least $999$ of them remain in the Council?

Two players in turn play a game. First Player has cards with numbers $2, 4, \ldots, 2000$ while Second Player has cards with numbers $1, 3, \ldots, 2001$. In each his turn, a player chooses one of his cards and puts it on a table; the opponent sees it and puts his card next to the first one. Player, who put the card with a larger number, scores 1 point. Then both cards are discarded. First Player starts. After $1000$ turns the game is over; First Player has used all his cards and Second Player used all but one. What are the maximal scores, that players could guarantee for themselves, no matter how the opponent would play?

Let $(m,n)$ be pair of positive integers. Julia has carefully planted $m$ rows of $n$ dandelions in an $m \times n$ array in her back garden. Now, Jana un Viviane decides to play a game with a lawnmower they just found. Taking alternating turns and starting with Jana, they can now mow down all the dandelions in a straight horizontal or vertical line (and they must mow down at least one dandelion ). The winner is the player who mows down the final dandelion. Determine all pairs of $(m,n)$ for which Jana has a winning strategy.

We have a regular polygon $P$ with 2019 vertices, and in each vertex there is a coin. Two players [i]Azul[/i] and [i]Rojo[/i] take turns alternately, beginning with Azul, in the following way: first, Azul chooses a triangle with vertices in $P$ and colors its interior with blue, then Rojo selects a triangle with vertices in $P$ and colors its interior with red, so that the triangles formed in each move don't intersect internally the previous colored triangles. They continue playing until it's not possible to choose another triangle to be colored. Then, a player wins the coin of a vertex if he colored the greater quantity of triangles incident to that vertex (if the quantities of triangles colored with blue or red incident to the vertex are the same, then no one wins that coin and the coin is deleted). The player with the greater quantity of coins wins the game. Find a winning strategy for one of the players. [i]Note.[/i] Two triangles can share vertices or sides.

Let $P$ be a cyclic polygon with circumcenter $O$ that does not lie on any diagonal, and let $S$ be the set of points on 2D plane containing $P$ and $O$. The $\textit{Matcha Sweep Game}$ is a game between two players $A$ and $B$, with $A$ going first, such that each choosing a nonempty subset $T$ of points in $S$ that has not been previously chosen, and such that if $T$ has at least $3$ vertices then $T$ forms a convex polygon. The game ends with all points have been chosen, with the player picking the last point wins. For which polygons $P$ can $A$ guarantee a win? [i]Proposed by Anzo Teh Zhao Yang[/i]

Wiebke and Stefan play the following game on a rectangular sheet of paper. They start with a rectangle with $60$ rows and $40$ columns and cut it in turns into smaller rectangles. The cuttings must be made along the gridlines, and a player in turn may cut only one smaller rectangle. By that, Stefan makes only vertical cuts, while Wiebke makes only horizontal cuts. A player who cannot make a regular move loses the game. (a) Who has a winning strategy if Stefan makes the first move? (b) Who has a winning strategy if Wiebke makes the first move?

Ana and Beta play a turn-based game on a $m \times n$ board. Ana begins. At the beginning, there is a stone in the lower left square and the objective is to move it to the upper right corner. A move consists of the player moving the stone to the right or up as many squares as the player wants. Find all the values ​​of $(m, n)$ for which Ana can guarantee victory.

In a chess tournament , each of two players have only one game played. After 2 rounds 5 players left the tournament. At the final of tournament was found that the number of total games played is 100. How many players were at the start of the tournament?

Joe and Penny play a game. Initially there are $5000$ stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the $k$-th move, any number of stones between $1$ and $k$ inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?

Two players play a card game. They have a deck of $n$ distinct cards. About any two cards from the deck know which of them has a different (in this case, if $A$ beats $B$, and $B$ beats $C$, then it may be that $C$ beats $A$). The deck is split between players in an arbitrary manner. In each turn the players over the top card from his deck and one whose card has a card from another player takes both cards and puts them to the bottom of your deck in any order of their discretion. Prove that for any initial distribution of cards, the players can with knowing the location agree and act so that one of the players left without a card. [i]E. Lakshtanov[/i]

In a video game, there is a board divided into squares, with $27$ rows and $27$ columns. The squares are painted alternately in black, gray and white as follows: $\bullet$ in the first row, the first square is black, the next is gray, the next is white, the next is black, and so on; $\bullet$ in the second row, the first is white, the next is black, the next is gray, the next is white, and so on; $\bullet$ in the third row, the order would be gray-white-black-gray and so on; $\bullet$ the fourth row is painted the same as the first, the fifth the same as the second, $\bullet$ the sixth the same as the third, and so on. In the box in row $i$ and column $j$, there are $ij$ coins. For example, in the box in row $15$ and column $20$ there are $(15) (20) = 300$ coins. Verify that in total there are, in the black squares, $9^2 (13^2 + 14^2 + 15^2)$ coins.

Let $n$ and $k$ be given integers such that $n\ge k\ge 2$. Alice and Bob play a game on an $n$ by $n$ table with white cells. They take turns to pick a white cell and color it black. Alice moves first. The game ends as soon as there is at least one black cell in every $k$ by $k$ square after a player moves, who is declared the winner of the game. Who has the winning strategy?

A 2-player game is played on $n\geq 3$ points, where no 3 points are collinear. Each move consists of selecting 2 of the points and drawing a new line segment connecting them. The first player to draw a line segment that creates an odd cycle loses. (An odd cycle must have all its vertices among the $n$ points from the start, so the vertices of the cycle cannot be the intersections of the lines drawn.) Find all $n$ such that the player to move first wins.