Xenia and Yagve take turns in playing the following game: A coin is placed on the first box in a row of nine cells. At each turn the player may choose to move the coin forward one step, move the coin forward four steps, or move coin back two steps. For a move to be allowed, the coin must land on one of them of nine cells. The winner is one who gets to move the coin to the last ninth cell. Who wins, given that Xenia makes the first move, and both players play optimally?

Numbers $1,\frac12,\frac13,\ldots,\frac1{2001}$ are written on a blackboard. A student erases two numbers $x,y$ and writes down the number $x+y+xy$ instead. Determine the number that will be written on the board after $2000$ such operations.

A solitaire game is played on an $m\times n$ rectangular board, using $mn$ markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs $(m,n)$ of positive integers such that all markers can be removed from the board.

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

There is a set of cards with numbers from $1$ to $30$ (which may be repeated) . Each student takes one such card. The teacher can perform the following operation: He reads a list of such numbers (possibly only one) and then asks the students to raise an arm if their number was in this list. How many times must he perform such an operation in order to determine the number on each student 's card? (Indicate the number of operations and prove that it is minimal . Note that there are not necessarily 30 students.)

Fix an integer $k>2$. Two players, called Ana and Banana, play the following game of numbers. Initially, some integer $n \ge k$ gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number $m$ just written on the blackboard and replaces it by some number $m'$ with $k \le m' < m$ that is coprime to $m$. The first player who cannot move anymore loses. An integer $n \ge k $ is called good if Banana has a winning strategy when the initial number is $n$, and bad otherwise. Consider two integers $n,n' \ge k$ with the property that each prime number $p \le k$ divides $n$ if and only if it divides $n'$. Prove that either both $n$ and $n'$ are good or both are bad.

Let $(m,n,N)$ be a triple of positive integers. Bruce and Duncan play a game on an m\times n array, where the entries are all initially zeroes. The game has the following rules. $\bullet$ The players alternate turns, with Bruce going first. $\bullet$ On Bruce's turn, he picks a row and either adds $1$ to all of the entries in the row or subtracts $1$ from all the entries in the row. $\bullet$ On Duncan's turn, he picks a column and either adds $1$ to all of the entries in the column or subtracts $1$ from all of the entries in the column. $\bullet$ Bruce wins if at some point there is an entry $x$ with $|x|\ge N$. Find all triples $(m, n,N)$ such that no matter how Duncan plays, Bruce has a winning strategy.

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

Ana and Beto play against each other. Initially, Ana chooses a non-negative integer $N$ and announces it to Beto. Next Beto writes a succession of $2016$ numbers, $1008$ of them equal to $1$ and $1008$ of them equal to $-1$. Once this is done, Ana must split the succession into several blocks of consecutive terms (each term belonging to exactly one block), and calculate the sum of the numbers of each block. Finally, add the squares of the calculated numbers. If this sum is equal to $N$, Ana wins. If not, Beto wins. Determine all values of $N$ for which Ana can ensure victory, no matter how Beto plays.

On a board, Ana and Bob start writing $0$s and $1$s alternatively until each of them has written $2021$ digits. Ana starts this procedure and each of them always adds a digit to the right of the already existing ones. Ana wins the game if, after they stop writing, the resulting number (in binary) can be written as the sum of two squares. Otherwise, Bob wins. Determine who has a winning strategy.

Anna and Basilis play a game writing numbers on a board as follows: The two players play in turns and if in the board is written the positive integer $n$, the player whose turn is chooses a prime divisor $p$ of $n$ and writes the numbers $n+p$. In the board, is written at the start number $2$ and Anna plays first. The game is won by whom who shall be first able to write a number bigger or equal to $31$. Find who player has a winning strategy, that is who may writing the appropriate numbers may win the game no matter how the other player plays.

We have $105$ coins, among which we know that there are three fake ones. Authentic coins have all the same weight, which is greater than that of the false ones, which also have the same weight. Determine from can $26$ authentic coins be selected by weighing only two in one two pan balance.

Let $n\geq 2$ be an integer. Alice and Bob play a game concerning a country made of $n$ islands. Exactly two of those $n$ islands have a factory. Initially there is no bridge in the country. Alice and Bob take turns in the following way. In each turn, the player must build a bridge between two different islands $I_1$ and $I_2$ such that: $\bullet$ $I_1$ and $I_2$ are not already connected by a bridge. $\bullet$ at least one of the two islands $I_1$ and $I_2$ is connected by a series of bridges to an island with a factory (or has a factory itself). (Indeed, access to a factory is needed for the construction.) As soon as a player builds a bridge that makes it possible to go from one factory to the other, this player loses the game. (Indeed, it triggers an industrial battle between both factories.) If Alice starts, then determine (for each $n\geq 2$) who has a winning strategy. ([i]Note:[/i] It is allowed to construct a bridge passing above another bridge.)

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 $x$. )

In the middle cell of the $1 \times 2005$ strip there is a chip. Two players each queues move it: first, the first player moves the piece one cell in any direction, then the second one moves it $2$ cells, the $1$st - by $4$ cells, the 2nd by $8$, etc. (the $k$-th shift occurs by $2^{k-1}$ cells). That, whoever cannot make another move loses. Who can win regardless of the opponent's play?

Nicu plays the Next game on the computer. Initially the number $S$ in the computer has the value $S = 0$. At each step Nicu chooses a certain number $a$ ($0 <a <1$) and enters it in computer. The computer arbitrarily either adds this number $a$ to the number $S$ or it subtracts from $S$ and displays on the screen the new result for $S$. After that Nicu does Next step. It is known that among any $100$ consecutive operations the computer the at least once apply the assembly. Give an arbitrary number $M> 0$. Show that there is a strategy for Nicu that will always allow him after a finite number of steps to get a result $S> M$. [hide=original wording]Nicu joacă la calculator următorul joc. Iniţial numărul S din calculator are valoarea S = 0. La fiecare pas Nicu alege un număr oarecare a (0 < a < 1) şi îl introduce în calculator. Calculatorul, în mod arbitrar, sau adună acest număr a la numărul S sau îl scade din S şi afişează pe ecran rezultatul nou pentru S. După aceasta Nicu face următorul pas. Se ştie că printre oricare 100 de operaţii consecutive calculatorul cel puţin o dată aplică adunarea. Fie dat un număr arbitrar M > 0. Să se arate că există o strategie pentru Nicu care oricând îi va permite lui după un număr finit de paşi să obţină un rezulat S > M.[/hide]

Let \( n \) be a positive integer. A function \( f : \{0, 1, \dots, n\} \to \{0, 1, \dots, n\} \) is called \( n \)-Bolivian if it satisfies the following conditions: • \( f(0) = 0 \); • \( f(t) \in \{ t-1, f(t-1), f(f(t-1)), \dots \} \) for all \( t = 1, 2, \dots, n \). For example, if \( n = 3 \), then the function defined by \( f(0) = f(1) = 0 \), \( f(2) = f(3) = 1 \) is 3-Bolivian, but the function defined by \( f(0) = f(1) = f(2) = 0 \), \( f(3) = 1 \) is not 3-Bolivian. For a fixed positive integer \( n \), Gollum selects an \( n \)-Bolivian function. Smeagol, knowing that \( f \) is \( n \)-Bolivian, tries to figure out which function was chosen by asking questions of the type: \[ \text{How many integers } a \text{ are there such that } f(a) = b? \] given a \( b \) of his choice. Show that if Gollum always answers correctly, Smeagol can determine \( f \) and find the minimum number of questions he needs to ask, considering all possible choices of \( f \).

Petya and Kolya play the following game: they take turns changing one of the coefficients $a$ or $b$ of the quadratic trinomial $f = x^2 + ax + b$: Petya is on $1$, Kolya is on $1$ or $3$. Kolya wins if after the move of one of the players a trinomial is obtained that has whole roots. Is it true that Kolya can win for any initial integer odds $a$ and $b$ regardless of Petya's game? [hide=original wording]Петя и Коля играют в следующую игру: они по очереди изменяют один из коэффициентов a или b квадратного трехчлена f = x^2 + ax + b: Петя на 1, Коля- на 1 или на 3. Коля выигрывает, если после хода одного из игроков получается трехчлен, имеющий целые корни. Верно ли, что Коля может выигратьпр и любых начальных целых коэффициентах a и b независимо от игры Пети?[/hide]

Two players take turns playing on a $3\times1001$ board whose squares are initially all white. Each player, in his turn, paints two squares located in the same row or column black, not necessarily adjacent. The player who cannot make his move loses the game. Determine which of the two players has a strategy that allows them to win, no matter how well his opponent plays.

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 $x$. )

Two players in turn put two or three coins into their own hats (before the game starts, the hats are empty). Each time, after the second player duplicated the move of the first player, they exchange hats. The player wins, if after his move his hat contains one hundred or more coins. Which player has a winning strategy?

Alina and Bogdan play a game on a $2\times n$ rectangular grid ($n\ge 2$) whose sides of length $2$ are glued together to form a cylinder. Alternating moves, each player cuts out a unit square of the grid. A player loses if his/her move causes the grid to lose circular connection (two unit squares that only touch at a corner are considered to be disconnected). Suppose Alina makes the first move. Which player has a winning strategy?

There are $68$ coins , each coin having a different weight than that of each other . Show how to find the heaviest and lightest coin in $100$ weighings on a balance beam. (S. Fomin, Leningrad)

Players $A$ and $B$ play the following game: $A$ tosses a coin $n$ times, and $B$ does $n+1$ times. The player who obtains more ”heads” wins; or in the case of equal balances, $A$ is assigned victory. Find the values of $n$ for which this game is fair (i.e. both players have equal chances for victory).

Let $m$ and $n$ be positive integers. Player $A$ has a field of $m \times n$, and player $B$ has a $1 \times n$ field (the first is the number of rows). On the first move, each player places on each square of his field white or black chip as he pleases. At each next on the move, each player can change the color of randomly chosen pieces on your field to the opposite, provided that in no row for this move will not change more than one chip (it is allowed not to change not a single chip). The moves are made in turn, player $A$ starts. Player $A$ wins if there is such a position that in the only row player $B$'s squares, from left to right, are the same as in some row of player's field $A$. Prove that player $A$ has the ability to win for any game of player $B$ if and only if $n <2m$.