Kostya and Sergey play a game on a white strip of length 2016 cells. Kostya (he plays first) in one move should paint black over two neighboring white cells. Sergey should paint either one white cell either three neighboring white cells. It is forbidden to make a move, after which a white cell is formed the doesn't having any white neighbors. Loses the one that can make no other move. However, if all cells are painted, then Kostya wins. Who will win if he plays the right game (has a winning strategy)?

Let ${k}$ be a fixed positive integer. A finite sequence of integers ${x_1,x_2, ..., x_n}$ is written on a blackboard. Pepa and Geoff are playing a game that proceeds in rounds as follows. - In each round, Pepa first partitions the sequence that is currently on the blackboard into two or more contiguous subsequences (that is, consisting of numbers appearing consecutively). However, if the number of these subsequences is larger than ${2}$, then the sum of numbers in each of them has to be divisible by ${k}$. - Then Geoff selects one of the subsequences that Pepa has formed and wipes all the other subsequences from the blackboard. The game finishes once there is only one number left on the board. Prove that Pepa may choose his moves so that independently of the moves of Geoff, the game finishes after at most ${3k}$ rounds. (Poland)

Two players play the following game. At the outset there are two piles, containing $10,000$ and $20,000$ tokens,respectively . A move consists of removing any positive number of tokens from a single pile $or$ removing $x>0$ tokens from one pile and $y>0$ tokens from the other , where $x+y$ is divisible by $2015$. The player who can not make a move loses. Which player has a winning strategy

A game is played on an ${n \times n}$ chessboard. At the beginning there are ${99}$ stones on each square. Two players ${A}$ and ${B}$ take turns, where in each turn the player chooses either a row or a column and removes one stone from each square in the chosen row or column. They are only allowed to choose a row or a column, if it has least one stone on each square. The first player who cannot move, looses the game. Player ${A}$ takes the first turn. Determine all n for which player ${A}$ has a winning strategy.

Thomas and Nils are playing a game. They have a number of cards, numbered $1, 2, 3$, et cetera. At the start, all cards are lying face up on the table. They take alternate turns. The person whose turn it is, chooses a card that is still lying on the table and decides to either keep the card himself or to give it to the other player. When all cards are gone, each of them calculates the sum of the numbers on his own cards. If the difference between these two outcomes is divisible by $3$, then Thomas wins. If not, then Nils wins. (a) Suppose they are playing with $2018$ cards (numbered from $1$ to $2018$) and that Thomas starts. Prove that Nils can play in such a way that he will win the game with certainty. (b) Suppose they are playing with $2020 $cards (numbered from $1$ to $2020$) and that Nils starts. Which of the two players can play in such a way that he wins with certainty?

A wizard kidnaps $31$ members from party $A$, $28$ members from party $B$, $23$ members from party $C$, and $19$ members from party $D$, keeping them isolated in individual rooms in his castle, where he forces them to work. Every day, after work, the kidnapped people can walk in the park and talk with each other. However, when three members of three different parties start talking with each other, the wizard reconverts them to the fourth party (there are no conversations with $4$ or more people involved). a) Find out whether it is possible that, after some time, all of the kidnapped people belong to the same party. If the answer is yes, determine to which party they will belong. b) Find all quartets of positive integers that add up to $101$ that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.