An infinite road has traffic lights at intervals of 1500m. The lights are all synchronised and are alternately green for $\frac 32$ minutes and red for 1 minute. For which $v$ can a car travel at a constant speed of $v$m/s without ever going through a red light?

Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow? [i]Proposed by Gerhard Woeginger, Netherlands[/i]

There are $2003$ pieces of candy on a table. Two players alternately make moves. A move consists of eating one candy or half of the candies on the table (the “lesser half” if there are an odd number of candies). At least one candy must be eaten at each move. The loser is the one who eats the last candy. Which player has a winning strategy?

Rațiu and Horațiu are playing a game on a $100\times 100$ grid. They make moves alternatively, starting with Rațiu. At a move, a player places a token on an empty cell of the grid. If a player places a token on a cell which is adjacent to another cell with a token, he loses. Determine who has a winning strategy.

On a rectangular board $100 \times 300$, two people take turns coloring the cells that have not yet been colored. The first one colors cells in yellow, and the second one in blue. Coloring is completed when every cell of the board is colored. A [i]connected sequence[/i] of cells is a sequence of cells in which every two consecutive cells share a common side (and all cells in the sequence are different). Consider all possible connected sequences of yellow cells. The result of the first player is the number of cells in the connected sequence of yellow cells of maximum length. The first player's goal is to maximize the result, and the second player's goal is to make the first player's result as small as possible. Prove that if each player tries to achieve his goal, the result of the first player will be no more than $200$. [i]Proposed by Mykhailo Shtandenko and Fedir Yudin[/i]

There are $2022$ stones on a table. At the start of the game, Teacher Tseng will choose a positive integer $m$ and let Ming and LTF play a game. LTF is the first to move, and he can remove at most $m$ stones on his round. Then the two people take turns removing stone, each round they must remove at least one stone, and they cannot remove more than twice the amount of stones the last person removed. The player unable to move loses. Find the smallest positive integer $m$ such that LTF has a winning strategy. [i]Proposed by ltf0501[/i]

Consider the equilateral triangular lattice in the complex plane defined by the Eisenstein integers; let the ordered pair $(x,y)$ denote the complex number $x+y\omega$ for $\omega=e^{2\pi i/3}$. We define an $\omega$-chessboard polygon to be a (non self-intersecting) polygon whose sides are situated along lines of the form $x=a$ or $y=b$, where $a$ and $b$ are integers. These lines divide the interior into unit triangles, which are shaded alternately black and white so that adjacent triangles have different colors. To tile an $\omega$-chessboard polygon by lozenges is to exactly cover the polygon by non-overlapping rhombuses consisting of two bordering triangles. Finally, a [i]tasteful tiling[/i] is one such that for every unit hexagon tiled by three lozenges, each lozenge has a black triangle on its left (defined by clockwise orientation) and a white triangle on its right (so the lozenges are BW, BW, BW in clockwise order). a) Prove that if an $\omega$-chessboard polygon can be tiled by lozenges, then it can be done so tastefully. b) Prove that such a tasteful tiling is unique. [i]Victor Wang.[/i]

Consider $2n$ points in the plane. Two players $A$ and $B$ alternately choose a point on each move. After $2n$ moves, there are no points left to choose from and the game ends. Add up all the distances between the points chosen by $A$ and add up all the distances between the points chosen by $B$. The one with the highest sum wins. If $A$ starts the game, describe the winner's strategy. Clarification: Consider that all the partial sums of distances between points give different numbers.

Given a $10 \times 10$ chessboard colored as black-and-white alternately. Prove that for any $46$ unit squares without common edges, there are at least $30$ unit squares with the same color.

Mr. Ganzgenau would like to take his tea mug out of the microwave right at the front. But Mr. Ganzgenau's microwave doesn't really want to be very precise play along. To be precise, the two of them play the following game: Let $n$ be a positive integer. The turntable of the microwave makes one in $n$ seconds full turn. Each time the microwave is switched on, an integer number of seconds turned either clockwise or counterclockwise so that there are n possible positions in which the tea mug can remain. One of these positions is right up front. At the beginning, the microwave turns the tea mug to one of the $n$ possible positions. After that Mr. Ganzgenau enters an integer number of seconds in each move, and the microwave decides either clockwise or counterclockwise this number of spin for seconds. For which $n$ can Mr. Ganzgenau force the tea cup after a finite number of puffs to be able to take it out of the microwave right up front? (Birgit Vera Schmidt) [hide=original wording, in case it doesn't make much sense]Herr Ganzgenau möchte sein Teehäferl ganz genau vorne aus der Mikrowelle herausnehmen. Die Mikrowelle von Herrn Ganzgenau möchte da aber so ganz genau gar nicht mitspielen. Ganz genau gesagt spielen die beiden das folgende Spiel: Sei n eine positive ganze Zahl. In n Sekunden macht der Drehteller der Mikrowelle eine vollständige Umdrehung. Bei jedem Einschalten der Mikrowelle wird eine ganzzahlige Anzahl von Sekunden entweder im oder gegen den Uhrzeigersinn gedreht, sodass es n mögliche Positionen gibt, auf denen das Teehäferl stehen bleiben kann. Eine dieser Positionen ist ganz genau vorne. Zu Beginn dreht die Mikrowelle das Teehäferl auf eine der n möglichen Positionen. Danach gibt Herr Ganzgenau in jedem Zug eine ganzzahlige Anzahl von Sekunden ein, und die Mikrowelle entscheidet, entweder im oder gegen den Uhrzeigersinn diese Anzahl von Sekunden lang zu drehen. Für welche n kann Herr Ganzgenau erzwingen, das Teehäferl nach endlich vielen Zügen ganz genau vorne aus der Mikrowelle nehmen zu können? (Birgit Vera Schmidt) [/hide]

Given an $1$ x $n$ table ($n\geq 2$), two players alternate the moves in which they write the signs + and - in the cells of the table. The first player always writes +, while the second always writes -. It is not allowed for two equal signs to appear in the adjacent cells. The player who can’t make a move loses the game. Which of the players has a winning strategy?

Let $n$ be a positive integer. Anna and Beatrice play a game with a deck of $n$ cards labelled with the numbers $1, 2,...,n$. Initially, the deck is shuffled. The players take turns, starting with Anna. At each turn, if $k$ denotes the number written on the topmost card, then the player first looks at all the cards and then rearranges the $k$ topmost cards. If, after rearranging, the topmost card shows the number k again, then the player has lost and the game ends. Otherwise, the turn of the other player begins. Determine, depending on the initial shuffle, if either player has a winning strategy, and if so, who does.

There are $4$ piles of stones with the following quantities: $1004$, $1005$, $2009$ and $2010$. A legitimate move is to remove a stone from each from $3$ different piles. Two players $A$ and $B$ play in turns. $A$ begins the game . The player who, on his turn, cannot make a legitimate move, loses. Determine which of the players has a winning strategy and give a strategy for that player.

Arne and Bertil play a game on an $11 \times 11$ grid. Arne starts. He has a game piece that is placed on the center od the grid at the beginning of the game. At each move he moves the piece one step horizontally or vertically. Bertil places a wall along each move any of an optional four squares. Arne is not allowed to move his piece through a wall. Arne wins if he manages to move the pice out of the board, while Bertil wins if he manages to prevent Arne from doing that. Who wins if from the beginning there are no walls on the game board and both players play optimally?

In a row the numbers $1,2,...,2010$ have been written. Two players, taking turns, write $+$ or $\times$ between two consecutive numbers whenever possible. The first player wins if the algebraic sum obtained is divisible by $3$; otherwise, the second player wins. Find a winning strategy for one of the players.

[b]p1.[/b] Show that for every $n \ge 6$, a square in the plane may be divided into $n$ smaller squares, not necessarily all of the same size. [b]p2.[/b] Let $n$ be the $4018$-digit number $111... 11222...2225$, where there are $2008$ ones and $2009$ twos. Prove that $n$ is a perfect square. (Giving the square root of $n$ is not sufficient. You must also prove that its square is $n$.) [b]p3.[/b] Let $n$ be a positive integer. A game is played as follows. The game begins with $n$ stones on the table. The two players, denoted Player I and Player II (Player I goes first), alternate in removing from the table a nonzero square number of stones. (For example, if $n = 26$ then in the first turn Player I can remove $1$ or $4$ or $9$ or $16$ or $25$ stones.) The player who takes the last stone wins. Determine if the following sentence is TRUE or FALSE and prove your answer: There are infinitely many starting values n such that Player II has a winning strategy. (Saying that Player II has a winning strategy means that no matter how Player I plays, Player II can respond with moves that lead to a win for Player II.) [b]p4.[/b] Consider a convex quadrilateral $ABCD$. Divide side $AB$ into $8$ equal segments $AP_1$, $P_1P_2$, $...$ , $P_7B$. Divide side $DC$ into $8$ equal segments $DQ_1$, $Q_1Q_2$, $...$ , $Q_7C$. Similarly, divide each of sides $AD$ and $BC$ into $8$ equal segments. Draw lines to form an $8 \times 8$ “checkerboard” as shown in the picture. Color the squares alternately black and white. (a) Show that each of the $7$ interior lines $P_iQ_i$ is divided into $8$ equal segments. (b) Show that the total area of the black regions equals the total area of the white regions. [img]https://cdn.artofproblemsolving.com/attachments/1/4/027f02e26613555181ed93d1085b0e2de43fb6.png[/img] [b]p5.[/b] Prove that exactly one of the following two statements is true: A. There is a power of $10$ that has exactly $2008$ digits in base $2$. B. There is a power of $10$ that has exactly $2008$ digits in base $5$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].