Found problems: 1385
Tanya and Serezha take turns putting chips in empty squares of a chessboard. Tanya starts with a chip in an arbitrary square. At every next move, Serezha must put a chip in the column where Tanya put her last chip, while Tanya must put a chip in the row where Serezha put his last chip. The player who cannot make a move loses. Which of the players has a winning strategy?
[i]Proposed by A. Golovanov[/i]
Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).
At least $3$ players take part in a tennis tournament. Each participant plays exactly one match against each other participant. After the tournament has ended, we find out that each player has won at least one match. (There are no ties in tennis).
Show that in the tournament, there was at least one trio of players $A,B,C$ such that $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$.
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]
$n$ players took part in badminton tournament, where $n$ is positive and odd integer. Each two players played two matches with each other. There were no draws. Each player has won as many matches as he has lost. Prove that you can cancel half of the matches s.t. each player still has won as many matches as he has lost.
The game involves two players $A$ and $B$. Player $A$ sets the value of one of the coefficients $a, b$ or $c$ of the polynomial $$x^3 + ax^2 + bx + c.$$
Player $B$ indicates the value of any of the two remaining coefficients . Player $A$ then sets the value of the last coefficients. Is there a strategy for player A such that no matter how player $B$ plays, the equation $$x^3 + ax^2 + bx + c = 0$$ to have three different (real) solutions?
On the board are written $n$ natural numbers, $n\in \mathbb{N}$. In one move it is possible to choose two
equal written numbers and increase one by $1$ and decrease the other by $1$. Prove that in this
the game cannot be played more than $\frac{n^3}{6}$ moves.
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?
Helena and Luis are going to play a game with two bags with marbles. They play alternately and on each turn they can do one and only one of the following moves:
[list]
Take out a marble from one bag.
Take out a marble from each bag.
Take out a marble from one bag and then put it into the other bag.
[/list]
The player who leaves both bags empty wins the game.
Before starting the game, Helena counted out the marbles of each bag and said to Luis: "You may start!", while she thought "I will certainly win...". What are the possible distributions of the marbles in the bags?
Elza draws $2013$ cities on the map and connects some of them with $N$ roads. Then Elza and Susy erase cities in turns until just two cities left (first city is to be erased by Elza). If these cities are connected with a road then Elza wins, otherwise Susy wins. Find the smallest $N$ for which Elza has a winning strategy.
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.
For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A [i]tromino[/i] is an $L$-shape formed by three connected unit squares.
$(a)$ For which values of $n$ is it possible to cover all the black squares with non-overlapping trominoes lying entirely on the chessboard?
$(b)$ When it is possible, find the minimum number of trominoes needed.
The number $328$ is written on the board. Two players alternate writing positive divisors of $328$ on the board, subject to the following rules:
$\bullet$ No divisor of a previously written number may be written.
$\bullet$ The player who writes 328 loses.
Who has a winning strategy, the first player or the second player?
Let ${n}$ be a fixed positive integer. ${A}$ and ${B}$ play the following game: $2023$ coins marked $1, 2, \dots, 2023$ lie on a circle (the marks are considered in module $2023$) and each coin has two sides. Initially, all coins are head up and ${A}$'s goal is to make as many coins with tail up. In each operation, ${A}$ choose two coins marked ${k}$ and $k+3$ with head up (if ${A}$ can't choose, the game ends) and ${B}$ choose a coin marked $k+1$ or $k+2$ and flip it. If at some moment there are ${n}$ coins with tail up, ${A}$ wins. Find the largest ${n}$ such that ${A}$ has a winning strategy.
Anana has an ordered $n$-tuple $(a_1,a_2,...,a_n)$ if integers. Banana may make a guess on Anana's ordered integer $n$-tuple $(x_1,x_2,...,x_n)$, upon which Anana will reveal the product of differences $(a_1-x_1)(a_2-x_2)...(a_n-x_n)$. How many guesses does Banana need to figure out Anana's $n$-tuple for certain?
A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or
[*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter.
[i]Proposed by Aron Thomas[/i]
For integers $m,n \ge 3$ we consider a $m \times n$ rectangular frame, consisting of the $2m+2n-4$ boundary squares of a $m \times n$ rectangle.
Renate and Erhard play the following game on this frame, with Renate to start the game. In a move, a player colours a rectangular area consisting of a single or several white squares. If there are any more white squares, they have to form a connected region. The player who moves last wins the game.
Determine all pairs $(m,n)$ for which Renate has a winning strategy.
Anne and Bill decide to play a game together. At the beginning, they chose a positive integer $n$; then, starting from a positive integer $\mathcal{N}_0$, Anne subtracts to $\mathcal{N}_0$ an integer $k$-th power (possibly $0$) of $n$ less than or equal to $\mathcal{N}_0$. The resulting number $\mathcal{N}_1=\mathcal{N}_0-n^k$ is then passed to Bill, who repeats the same process starting from $\mathcal{N}_1$: he subtracts to $\mathcal{N}_1$ an integer $j$-th power of $n$ less than or equal to $\mathcal{N}_1$, and he then gives the resulting number $\mathcal{N}_2=\mathcal{N}_1-n^j$ to Anne. The game continues like that until one player gets $0$ as the result of his operation, winning the game. For each $1\leq n \leq 1000$, let $f(n)$ be the number of integers $1\leq \mathcal{N}_0\leq 5000$ such that Anne has a winning strategy starting from them. For how many values of $n$ we have that $f(n)\geq 2520$?
[I]Proposed by [b]FedeX333X[/b][/I]
Alice and Bob play a game where they start from a complete graph with $n$ vertices and take turns removing a single edge from the graph, with Alice taking the fi rst turn. The first player to disconnect the graph loses. Compute the sum of all $n$ between $2$ and $100$ inclusive such that Alice has a winning strategy. (A complete graph is one where there is an edge between every pair of vertices.)
Martin and Chayo have an bag with $2016$ chocolates each one. Both empty his bag on a table making a pile of chocolates. They decide to make a competence to see who gets the chocolates, as follows: A movement consist that a player take two chocolates of his pile, keep a chocolate in his bag and put the other chocolate in the pile of the other player, in his turn the player needs to make at least one movement and he can repeat as many times as he wish before passing his turn. Lost the player that can not make at least one movement in his turn. If Martin starts the game, who can ensure the victory and keep all the chocolates?
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.
In the country of Sikinia there are finitely many cities. From each city, exactly three roads go out and each road goes to another Sikinian city. A tourist starts a trip from city $A$ and drives according to the following rule: he turns left at the first city, then right at the next city, and so on, alternately. Show that he will eventually return to $A.$
Arnulfo and Berenice play the following game: One of the two starts by writing a number from $ 1$ to $30$, the other chooses a number from $ 1$ to $30$ and adds it to the initial number, the first player chooses a number from $ 1$ to $30$ and adds it to the previous result, they continue doing the same until someone manages to add $2018$. When Arnulfo was about to start, Berenice told him that it was unfair, because whoever started had a winning strategy, so the numbers had better change. So they asked the following question:
Adding chosen numbers from $1 $ to $a$, until reaching the number $ b$, what conditions must meet $a$ and $ b$ so that the first player does not have a winning strategy?
Indicate if Arnulfo and Berenice are right and answer the question asked by them.
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 $N=29$
$n(n>3)$ ping-pong players have played a few ping-pong games. The set of players that player A has played with is $A$, The set of players that player B has played with is $B$. for any two players, $A\neq B$. Prove that we can delete a player, so that this character remains.