Vlad draws 100 rays in the Euclidean plane. David then draws a line $\ell$ and pays Vlad one pound for each ray that $\ell$ intersects. Naturally, David wants to pay as little as possible. What is the largest amount of money that Vlad can get from David? [i]Proposed by Vlad Spătaru[/i]

On her blackboard, Alice has written $n$ integers strictly greater than $1$. Then, she can, as often as she likes, erase two numbers $a$ and $b$ such that $a \neq b$, and replace them with $q$ and $q^2$, where $q$ is the product of the prime factors of $ab$ (each prime factor is counted only once). For instance, if Alice erases the numbers $4$ and $6$, the prime factors of $ab = 2^3 \times 3$ and $2$ and $3$, and Alice writes $q = 6$ and $q^2 =36$. Prove that, after some time, and whatever Alice's strategy is, the list of numbers written on the blackboard will never change anymore. [i]Note: The order of the numbers of the list is not important.[/i]

On a table are given $1999$ red candies and $6661$ yellow candies. The candies are indistinguishable due to the same packing. A gourmet applies the following procedure as long as it is possible: (i) He picks any of the remaining candies, notes its color, eats it and goes to (ii). (ii) He picks any of the remaining candies, and notes its color: if it is the same as the color of the last eaten candy, eats it and goes to (ii); otherwise returns it upon repacking and goes to (i). Prove that all the candies will be eaten and find the probability that the last eaten candy will be red.

From a deck of playing cards, four [i]threes[/i], four [i]fours[/i] and four [i]fives[/i] are selected and put down on a table with the main side up. Players $A$ and $B$ alternately take the cards one by one and put them on the pile. Player $A$ begins. A player after whose move the sum of values of the cards on the pile is (a) greater than 34; (b) greater than 37; loses the game. Which player has a winning strategy?

On the blackboard, there are $K$ blanks. Alice decides $N$ values of blanks $(0-9)$ and then Bob determines the remaining digits. Find the largest possible integer $N$ such that Bob can guarantee to make the final number isn't a power of an integer.

Two players in turn paint cells of the $7\times7$ table each using own color. A player can't paint a cell if its row or its column contains a cell painted by the other player. The game stops when one of the players can't make his turn. What maximal number of the cells can remain unpainted when the game stops?

Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$; or by $q-1$; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$. The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations. [b]a)[/b] Show that if the initial number written is $0$, then Azambuja cannot reach his goal. [b]b)[/b] Find all initial numbers for which Azambuja can achieve his goal.

The Devil and the Man play a game. Initially, the Man pays some cash $s$ to the Devil. Then he lists some $97$ triples $\{i,j,k\}$ consisting of positive integers not exceeding $100$. After that, the Devil draws some convex polygon $A_1A_2...A_{100}$ with area $100$ and pays to the Man, the sum of areas of all triangles $A_iA_jA_k$. Determine the maximal value of $s$ which guarantees that the Man receives at least as much cash as he paid. [i]Proposed by Nikolai Beluhov, Bulgaria[/i]

On a blackboard, there are $17$ integers not divisible by $17$. Alice and Bob play a game. Alice starts and they alternately play the following moves: $\bullet$ Alice chooses a number $a$ on the blackboard and replaces it with $a^2$ $\bullet$ Bob chooses a number $b$ on the blackboard and replaces it with $b^3$. Alice wins if the sum of the numbers on the blackboard is a multiple of $17$ after a finite number of steps. Prove that Alice has a winning strategy. (Daniel Holmes)

Three boxes with at least one marble in each are given. In each step we double the number of marbles in one of the boxes, taking the required number of boxes from one of the other two boxes. Is it always possible to have one of the boxes empty after several steps?

There are $11$ empty boxes. In one move, a player can put one coin in each of some $10$ boxes. Two people play, taking turns. The winner is the player after whose move in one of the boxes there will be $21$ coins. Who has a winning strategy?

We have a square of side $1$ and a number $\ell$ such that $0 <\ell <\sqrt2$. Two players $A$ and $B$, in turn, draw in the square an open segment (without its two ends) of length $\ell $, starts A. Each segment after the first cannot have points in common with the previously drawn segments. He loses the player who cannot make his play. Determine if either player has a winning strategy.

Sixty-four dice with the numbers ”one” to ”six” are placed on one table and formed into a square with eight horizontal and eight vertical rows of cubes pushed together. By rotating the dice, while maintaining their place, we want to finally have all sixty-four dice the "one" points upwards. Each dice however, may not be turned individually, but only every eight dice in a horizontal or vertical row together by $90^o$ to the longitudinal axis of this row may turn. Prove that it is always possible to solve the dice by repeatedly applying the permitted type of rotation to the required end position.

Players $A$ and $B$ play the following game. Each of them throws a dice, and if the outcomes are $x$ and $y$ respectively, a list of all two digit numbers $10a + b$ with $a,b\in \{1,..,6\}$ and $10a + b \le 10x + y$ is created. Then the players alternately reduce the list by replacing a pair of numbers in the list by their absolute difference, until only one number remains. If the remaining number is of the same parity as the outcome of $A$’s throw, then $A$ is proclaimed the winner. What is the probability that $A$ wins the game?

At the beginning of a two-player game, the number $2004!$ is written on the blackboard. The players move alternately. In each move, a positive integer smaller than the number on the blackboard and divisible by at most $20$ different prime numbers is chosen. This is subtracted from the number on the blackboard, which is erased and replaced by the difference. The winner is the player who obtains $0$. Does the player who goes first or the one who goes second have a guaranteed win, and how should that be achieved?

The expression $*3^5*3^4*3^3*3^2*3*1$ is given. Ana and Branka alternately change the signs $*$ to $+$ or $-$ (one time each turn). Can Branka, who plays second, do this so as to obtain an expression whose value is divisible by $7$?

The number "3" is written on a board. Ana and Bernardo take turns, starting with Ana, to play the following game. If the number written on the board is $n$, the player in his/her turn must replace it by an integer $m$ coprime with $n$ and such that $n<m<n^2$. The first player that reaches a number greater or equal than 2016 loses. Determine which of the players has a winning strategy and describe it.

The figure shows a game board with $16$ squares. At the start of the game, two cars are placed in different squares. Two players $A$ and $B$ alternately take turns, and A starts. In each turn, the player chooses one of the cars and moves it one or more squares to the right. The left-most car may never overtake or land on the same square as the right-most car. The first player which is unable to move loses. [img]https://cdn.artofproblemsolving.com/attachments/1/b/8d6f40fac4983d6aa9bd076392c91a6d200f6a.png[/img] (a) Prove that A can win regardless of how $B$ plays, if the two cars start as shown in the figure. (b) Determine all starting positions in which $B$ can win regardless of how $A$ plays.

Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$; Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$. Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins. [i]Proposed by Hans Zantema, Netherlands[/i]

Players $A$ and $B$ play a game with $N \geq 2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B,A,B,A,\ldots $ by the following rules: [b](a)[/b] On every move of his $B$ passes $1$ coin from every box to an adjacent box. [b](b)[/b] On every move of hers $A$ chooses several coins that were [i]not[/i] involved in $B$'s previous move and are in different boxes. She passes every coin to an adjacent box. Player $A$'s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.

You are given positive integers $m, n>1$. Vasyl and Petryk play the following game: they take turns marking on the coordinate plane yet unmarked points of the form $(x, y)$, where $x, y$ are positive integers with $1 \leq x \leq m, 1 \leq y \leq n$. The player loses if after his move there are two marked points, the distance between which is not a positive integer. Who will win this game if Vasyl moves first and each player wants to win? [i]Proposed by Mykyta Kharin[/i]

Andy, Bess, Charley and Dick play on a $1000 \times 1000$ board. They make moves in turn: Andy first, then Bess, then Charley and finally Dick, after that Andy moves again and so on. At each move a player must paint several unpainted squares forming $2 \times 1, 1 \times 2, 1 \times 3$, or $3 \times 1$ rectangle. The player that cannot move loses. Prove that some three players can cooperate to make the fourth player lose.

Two players take turns alternatively and remove a number from $1,2,\dots,1000$. Players can not remove a number that differ with a number already removed by $1$ also they can not remove a number such that it sums up with another removed number to $1001$. The player who can not move loses. Determine the winner.

Let $\alpha, \beta$ be two rational numbers strictly between 0 and 1. Alice and Bob play a game. At the start of the game, Alice chooses a positive integer $n$. Knowing that, Bob then chooses a positive integer $T$. They then do the following for $T$ rounds: at the $i$th round, Bob chooses a set $X_i$ of $n$ positive integers that form a complete residue system modulo $n$. Then Alice chooses a subset $Y_i$ of $X_i$ such that the sum of elements in $Y_i$ is at most $\alpha$ times the sum of elements in $X_i$. After the $T$ rounds, Alice wins if it is possible to pick an integer $s$ between 0 and $n-1$ such that there are at least $\beta T$ positive integers among the elements in $Y_1, Y_2, . . . , Y_T$ (counted with multiplicities) that are equal to $s \pmod n$, and Bob wins otherwise. Find all pairs $(\alpha, \beta)$ of rational numbers strictly between 0 and 1 such that Alice has a winning strategy. [i]Proposed by Hans[/i]

There are red and blue balls in an urn : $1024$ in total. In one round, we do the following: we draw the balls from the urn two by two. After all balls have been drawn, we put a new ball back into the urn for each pair of drawn balls: the colour of the new ball depends on that of the drawn pair. For two red balls drawn, we put back a red ball. For two blue balls, we put back a blue ball. For a red and a blue ball, we put back a black ball. For a red and a black ball, we put back a red ball. For a blue and a black ball, we put back a blue ball. Finally, for two black balls we put back a black ball. Then the next round begins. After $10$ rounds, a single ball remains in the urn, which is red. What is the maximal number of blue balls that might have been in the urn at the very beginning?