Two players play a game. They have $n > 2$ piles containing $n^{10}+1$ stones each. A move consists of removing all the piles but one and dividing the remaining pile into $n$ nonempty piles. The player that cannot move loses. Who has a winning strategy, the player that moves first or his adversary?

A pile of $2020$ stones is given. Arnaldo and Bernaldo play the following game: In each move, it is allowed to remove $1, 4, 16, 64, ...$ (any power of $4$) stones from the pile. They make their moves alternately, and the player who can no longer play loses. If Arnaldo is the first to play, who has the winning strategy?

Let $f(x)$ be a polynomial, such that $f(x)=x^{2015}+a_1 x^{2014}+...+a_{2014} x+a_{2015}$. Velly and Polly are taking turns, starting from Velly changing the coefficients $a_i$ with real numbers , where each coefficient is changed exactly once. After 2015 turns they calculate the number of real roots of the created polynomial and if the root is only one, then Velly wins, and if it’s not – Polly wins. Which one has a winning strategy?

Ana and Beto play on a grid of $2022 \times 2022$. Ana colors the sides of some squares on the board red, so that no square has two red sides that share a vertex. Next, Bob must color a blue path that connects two of the four corners of the board, following the sides of the squares and not using any red segments. If Beto succeeds, he is the winner, otherwise Ana wins. Who has a winning strategy?

Let \( m \) and \( n \) be positive integers. Kellem and Carmen play the following game: initially, the number $0$ is on the board. Starting with Kellem and alternating turns, they add powers of \( m \) to the previous number on the board, such that the new value on the board does not exceed \( n \). The player who writes \( n \) wins. Determine, for each pair \( (m, n) \), who has the winning strategy. [b]Note:[/b] A power of \( m \) is a number of the form \( m^k \), where \( k \) is a non-negative integer.

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)

Ali and Naqi are playing a game. At first, they have Polynomial $P(x) = 1+x^{1398}$. Naqi starts. In each turn one can choice natural number $k \in [0,1398]$ in his trun, and add $x^k$ to the polynomial. For example after 2 moves $P$ can be : $P(x) = x^{1398} + x^{300} + x^{100} +1$. If after Ali's turn, there exist $t \in R$ such that $P(t)<0$ then Ali loses the game. Prove that Ali can play forever somehow he never loses the game!

The figure below shows a hexagonal net formed by many congruent equilateral triangles. Taking turns, Gabriel and Arnaldo play a game as follows. On his turn, the player colors in a segment, including the endpoints, following these three rules: i) The endpoints must coincide with vertices of the marked equilateral triangles. ii) The segment must be made up of one or more of the sides of the triangles. iii) The segment cannot contain any point (endpoints included) of a previously colored segment. Gabriel plays first, and the player that cannot make a legal move loses. Find a winning strategy and describe it.

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]

Susana and Brenda play a game writing polynomials on the board. Susana starts and they play taking turns. 1) On the preparatory turn (turn 0), Susana choose a positive integer $n_0$ and writes the polynomial $P_0(x)=n_0$. 2) On turn 1, Brenda choose a positive integer $n_1$, different from $n_0$, and either writes the polynomial $$P_1(x)=n_1x+P_0(x) \textup{ or } P_1(x)=n_1x-P_0(x)$$ 3) In general, on turn $k$, the respective player chooses an integer $n_k$, different from $n_0, n_1, \ldots, n_{k-1}$, and either writes the polynomial $$P_k(x)=n_kx^k+P_{k-1}(x) \textup{ or } P_k(x)=n_kx^k-P_{k-1}(x)$$ The first player to write a polynomial with at least one whole whole number root wins. Find and describe a winning strategy.

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.

There are $50$ coins in a row; each coin has a value. Two people are playing a game alternating moves. In one move a player can take either the leftmost or the rightmost coin. Who can always accumulate coins whose total value is at least the value of the coins of the opponent?

A person is locked in a room with a password-protected computer. If they enter the correct password, the door opens and they are freed. However, the password changes every time it is entered incorrectly. The person knows that the password is always a 10-digit number, and they also know that the password change follows a fixed pattern. This means that if the current password is \( b \) and \( a \) is entered, the new password is \( c \), which is determined by \( b \) and \( a \) (naturally, the person does not know \( c \) or \( b \)). Prove that regardless of the characteristics of this computer, the prisoner can free themselves. Proposed by Reza Tahernejad Karizi

There are 2019 cards in a box. Each card has a number written on one of its sides and a letter on the other side. Amy and Ben play the following game: in the beginning Amy takes all the cards, places them on a line and then she flips as many cards as she wishes. Each time Ben touches a card he has to flip it and its neighboring cards. Ben is allowed to have as many as 2019 touches. Ben wins if all the cards are on the numbers' side, otherwise Amy wins. Determine who has a winning strategy.

Two players, Agon and Besa, choose a number from the set $\{1,2,3,4,5,6,7,8\}$, in turns, until no number is left. Then, each player sums all the numbers that he has chosen. We say that a player wins if the sum of his chosen numbers is a prime and the sum of the numbers that his opponent has chosen is composite. In the contrary, the game ends in a draw. Agon starts first. Does there exist a winning strategy for any of the players?

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.