A natural number $n$ is written on a board. On every step, Neneng and Asep changes the number on the board with the following rule: Suppose the number on the board is $X$. Initially, Neneng chooses the sign up or down. Then, Asep will pick a positive divisor $d$ of $X$, and replace $X$ with $X+d$ if Neneng chose the sign "up" or $X-d$ if Neneng chose "down". This procedure is then repeated. Asep wins if the number on the board is a nonzero perfect square, and loses if at any point he writes zero. Prove that if $n \geq 14$, Asep can win in at most $(n-5)/4$ steps.

Andy, Bella, and Chris are playing in a trivia contest. Andy has $21,200$ points, Bella has $23,600$ points, and Chris has $11,200$ points. They have reached the final round, which works as follows: [list] [*] they are given a hint as to what the only question of the round will be about; [*] then, each of them must bet some amount of their points---this bet must be a nonnegative integer (a player does not know any of the other players' bets, and this bet cannot be changed later on); [*] then, they will be shown the question, where they will have $30$ seconds to individually submit a response (a player does not know any of the other players' answers); [*] finally, once time runs out, their responses and bets are revealed---if a player's response is correct, then the number of points they bet will be added to their score, otherwise, it will be subtracted from their score, and whoever ends up having the most points wins the contest (if there is a tie for the win, then the winner is randomly decided). [/list] Suppose that the contestants are currently deciding their bets based on the hint that the question will be about history. Bella knows that she will likely get the question wrong, but she also knows that Andy, who dislikes history, will definitely get it wrong. Knowing this, Bella wagers an amount that will guarantee her a win. Find the maximum number of points Bella could have ended up with.

A pile of $2000$ coins is given on a table. In each step, we choose a pile with at least three coins, remove one coin from it, and divide the rest of this pile into two piles (not necessarily of the same size). Is it possible that after several steps each pile on the table has exactly three coins?

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.

Six teams participate in a hockey tournament. Each team plays once against every other team. In each game, a team is awarded $3$ points for a win, $1$ point for a draw, and none for a loss. After the tournament the teams are ranked by total points. No two teams have the same total points. Each team (except the bottom team) has $2$ points more than the team ranking one place lower. Prove that the team that finished fourth has won two games and lost three games.

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.

We are given a natural number $k$. Let us consider the following game on an infinite onedimensional board. At the start of the game, we distrubute $n$ coins on the fields of the given board (one field can have multiple coins on itself). After that, we have two choices for the following moves: $(i)$ We choose two nonempty fields next to each other, and we transfer all the coins from one of the fields to the other. $(ii)$ We choose a field with at least $2$ coins on it, and we transfer one coin from the chosen field to the $k-\mathrm{th}$ field on the left , and one coin from the chosen field to the $k-\mathrm{th}$ field on the right. $\mathbf{(a)}$ If $n\leq k+1$, prove that we can play only finitely many moves. $\mathbf{(b)}$ For which values of $k$ we can choose a natural number $n$ and distribute $n$ coins on the given board such that we can play infinitely many moves.

All the cells of a $10\times10$ board are colored white initially. Two players are playing a game with alternating moves. A move consists of coloring any un-colored cell black. A player is considered to loose, if after his move no white domino is left. Which of the players has a winning strategy? [I]Proposed by A. Khrabrov[/i]

Kunal and Arnab play a game as follows. Initially there are $2$ piles of coins with $x$ and $y$ coins respectively. The game starts with Kunal. In each turn a player chooses one pile and removes as many coins as he wants from that pile. The game goes on and the last one to remove a coin loses. Determine all possible values of $(x,y)$ which ensure Kunal's victory against Arnab given both os them play optimally. \\ [i]You are required to find an exhaustive set of solutions[/i]

Two intelligent people playing a game on the $1403 \times 1403$ table with $1403^2$ cells. The first one in each turn chooses a cell that didn't select before and draws a vertical line segment from the top to the bottom of the cell. The second person in each turn chooses a cell that didn't select before and draws a horizontal line segment from the left to the right of the cell. After $1403^2$ steps the game will be over. The first person gets points equal to the longest verticals line segment and analogously the second person gets point equal to the longest horizonal line segment. At the end the person who gets the more point will win the game. What will be the result of the game?

Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector). At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy? [i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]

A game is played with two players and an initial stack of $n$ pennies $(n \geq 3)$. The players take turns choosing one of the stacks of pennies on the table and splitting it into two stacks. The winner is the player who makes a move that causes all stacks to be of height $1$ or $2.$ For which starting values of n does the player who goes first win, assuming best play by both players?

Ada and Charles play the following game:at the beginning, an integer n>1 is written on the blackboard.In turn, Ada and Charles remove the number k that they find on the blackboard.In turn Ad and Charles remove the number k that they find on the blackboard and they replace it : 1 -either with a positive divisor k different from 1 and k 2- or with k+1 At the beginning each players have a thousand points each.When a player choses move 1, he/she gains one point;when a player choses move 2, he/she loses one point.The game ends when one of the tho players is left with zero points and this player loses the game.Ada moves first.For what values Chares has a winning strategy?

On the occasion of the 47th Mathematical Olympiad 2016 the numbers 47 and 2016 are written on the blackboard. Alice and Bob play the following game: Alice begins and in turns they choose two numbers $a$ and $b$ with $a > b$ written on the blackboard, whose difference $a-b$ is not yet written on the blackboard and write this difference additionally on the board. The game ends when no further move is possible. The winner is the player who made the last move. Prove that Bob wins, no matter how they play. (Richard Henner)

Let $a$, $b$, $c$ be fixed positive integers. There are $a+b+c$ ducks sitting in a circle, one behind the other. Each duck picks either rock, paper, or scissors, with $a$ ducks picking rock, $b$ ducks picking paper, and $c$ ducks picking scissors. A move consists of an operation of one of the following three forms: [list] [*] If a duck picking rock sits behind a duck picking scissors, they switch places. [*] If a duck picking paper sits behind a duck picking rock, they switch places. [*] If a duck picking scissors sits behind a duck picking paper, they switch places. [/list] Determine, in terms of $a$, $b$, and $c$, the maximum number of moves which could take place, over all possible initial configurations.

Let $n$ be a positive integer. Aslı and Zehra are playing a game on an $n\times n$ grid. Initially, $10n^2$ stones are placed on some of the unit squares of this grid. On each move (starting with Aslı), Aslı chooses a row or a column that contains at least two squares with different numbers of stones, and Zehra redistributes the stones in that row or column so that after redistribution, the difference in the number of stones between any two squares in that row or column is at most one. Furthermore, this move must change the number of stones in at least one square. For which values of $n$, regardless of the initial placement of the stones, can Aslı guarantee that every square ends up with the same number of stones?

A deck of $52$ cards is given. There are four suites each having cards numbered $1,2,\dots, 13$. The audience chooses some five cards with distinct numbers written on them. The assistant of the magician comes by, looks at the five cards and turns exactly one of them face down and arranges all five cards in some order. Then the magician enters and with an agreement made beforehand with the assistant, he has to determine the face down card (both suite and number). Explain how the trick can be completed.

The vertices of the regular $n$-gon are marked. Two players play the following game: they, in turn, select a vertex and connect it by a segment to either the adjacent vertex or the center of the $n$-gon. The winner is a player if after his move it is possible to get any vertex from any other vertex moving along segments. For each integer $n\geqslant 3$ determine who has a winning strategy.

Ana and Bob are playing the following game. [list] [*] First, Bob draws triangle $ABC$ and a point $P$ inside it. [*] Then Ana and Bob alternate, starting with Ana, choosing three different permutations $\sigma_1$, $\sigma_2$ and $\sigma_3$ of $\{A, B, C\}$. [*] Finally, Ana draw a triangle $V_1V_2V_3$. [/list] For $i=1,2,3$, let $\psi_i$ be the similarity transformation which takes $\sigma_i(A), \sigma_i(B)$ and $\sigma_i(C)$ to $V_i, V_{i+1}$ and $ X_i$ respectively (here $V_4=V_1$) where triangle $\Delta V_iV_{i+1}X_i$ lies on the outside of triangle $V_1V_2V_3$. Finally, let $Q_i=\psi_i(P)$. Ana wins if triangles $Q_1Q_2Q_3$ and $ABC$ are similar (in some order of vertices) and Bob wins otherwise. Determine who has the winning strategy.

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.

A $100 \times 100$ table is given. At the beginning, every unit square has number $"0"$ written in them. Two players playing a game and the game stops after $200$ steps (each player plays $100$ steps). In every step, one can choose a row or a column and add $1$ to the written number in all of it's squares $\pmod 3.$ First player is the winner if more than half of the squares ($5000$ squares) have the number $"1"$ written in them, Second player is the winner if more than half of the squares ($5000$ squares) have the number $"0"$ written in them. Otherwise, the game is draw. Assume that both players play at their best. What will be the result of the game ? [i]Proposed by Mahyar Sefidgaran[/i]

A triangle $ABC$ with area $1$ is given. Anja and Bernd are playing the following game: Anja chooses a point $X$ on side $BC$. Then Bernd chooses a point $Y$ on side $CA$ und at last Anja chooses a point $Z$ on side $AB$. Also, $X,Y$ and $Z$ cannot be a vertex of triangle $ABC$. Anja wants to maximize the area of triangle $XYZ$ and Bernd wants to minimize that area. What is the area of triangle $XYZ$ at the end of the game, if both play optimally?

Let $N$ a positive integer. In a spaceship there are $2 \cdot N$ people, and each two of them are friends or foes (both relationships are symmetric). Two aliens play a game as follows: 1) The first alien chooses any person as she wishes. 2) Thenceforth, alternately, each alien chooses one person not chosen before such that the person chosen on each turn be a friend of the person chosen on the previous turn. 3) The alien that can't play in her turn loses. Prove that second player has a winning strategy [i]if, and only if[/i], the $2 \cdot N$ people can be divided in $N$ pairs in such a way that two people in the same pair are friends.

A magician has $300$ cards with numbers from $1$ to $300$ written on them, each number on exactly one card. The magician then lays these cards on a $3 \times 100$ rectangle in the following way - one card in each unit square so that the number cannot be seen and cards with consecutive numbers are in neighbouring squares. Afterwards, the magician turns over $k$ cards of his choice. What is the smallest value of $k$ for which it can happen that the opened cards definitely determine the exact positions of all other cards?

Alice and Bob play the following game. To start, Alice arranges the numbers $1,2,\ldots,n$ in some order in a row and then Bob chooses one of the numbers and places a pebble on it. A player's [i]turn[/i] consists of picking up and placing the pebble on an adjacent number under the restriction that the pebble can be placed on the number $k$ at most $k$ times. The two players alternate taking turns beginning with Alice. The first player who cannot make a move loses. For each positive integer $n$, determine who has a winning strategy.