Found problems: 304
A dragon gave a captured knight $100$ coins. Half of them are magical, but only dragon knows which are. Each day, the knight should divide the coins into two piles (not necessarily equal in size). The day when either magic coins or usual coins are spread equally between the piles, the dragon set the knight free. Can the knight guarantee himself a freedom in at most
(a) $50$ days?
(b) $25$ days?
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?
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.
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 $N=29$
There are $36$ cards in a deck arranged in the sequence spades, clubs, hearts, diamonds, spades, clubs, hearts, diamonds, etc. Somebody took part of this deck off the top, turned it upside down, and cut this part into the remaining part of the deck (i.e. inserted it between two consecutive cards). Then four cards were taken off the top, then another four, etc. Prove that in any of these sets of four cards, all the cards are of different suits.
(A Merkov, Moscow)
Seated in a circle are $11$ wizards. A different positive integer not exceeding $1000$ is pasted onto the forehead of each. A wizard can see the numbers of the other $10$, but not his own. Simultaneously, each wizard puts up either his left hand or his right hand. Then each declares the number on his forehead at the same time. Is there a strategy on which the wizards can agree beforehand, which allows each of them to make the correct declaration?
There is an empty table with $2^{100}$ rows and $100$ columns. Alice and Eva take turns filling the empty cells of the first row of the table, Alice plays first. In each move, Alice chooses an empty cell and puts a cross in it; Eva in each move chooses an empty cell and puts a zero. When no empty cells remain in the first row, the players move on to the second row, and so on (in each new row Alice plays first).
The game ends when all the rows are filled. Alice wants to make as many different rows in the table as possible, while Eva wants to make as few as possible. How many different rows will be there in the table if both follow their best strategies?
Proposed by Denis Afrizonov
There are a million numbered chairs at a large round table. The Sultan has seated a million wise men on them. Each of them sees the thousand people following him in clockwise order. Each of them was given a cap of black or white color, and they must simultaneously write down on their own piece of paper a guess about the color of their cap. Those who do not guess will be executed. The wise men had the opportunity to agree on a strategy before the test. What is the largest number of survivors that they can guarantee?
Nicholas and Peter are dividing $(2n+1)$ nuts. Each wants to get more. Three ways for that were suggested. (Each consist of three stages.) First two stages are common.
1 stage: Peter divides nuts onto $2$ heaps, each contain not less than $2$ nuts.
2 stage: Nicholas divides both heaps onto $2$ heaps, each contain not less than $1$ nut.
3 stage:
1 way: Nicholas takes the biggest and the least heaps.
2 way: Nicholas takes two middle size heaps.
3 way: Nicholas takes either the biggest and the least heaps or two middle size heaps, but gives one nut to the Peter for the right of choice.
Find the most and the least profitable method for the Nicholas.
There are two bowls on the table, in one there are $p$, in the other $q$ stones ($p, q \in N*$ ). Two players $A$ and $B$ take turns playing, starting with $A$.
Who's turn:
$\bullet$ takes a stone from one of the bowls
$\bullet$or removes one stone from each bowl
$\bullet$ or puts a stone from one of the bowls into the other.
Whoever takes the last stone wins.
Under what conditions can $A$ and under what conditions can $B$ force the win?
The answer must be justified.
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 $N=29$
Let $m$ and $n$ be positive integers. Player $A$ has a field of $m \times n$, and player $B$ has a $1 \times n$ field (the first is the number of rows). On the first move, each player places on each square of his field white or black chip as he pleases. At each next on the move, each player can change the color of randomly chosen pieces on your field to the opposite, provided that in no row for this move will not change more than one chip (it is allowed not to change not a single chip). The moves are made in turn, player $A$ starts. Player $A$ wins if there is such a position that in the only row player $B$'s squares, from left to right, are the same as in some row of player's field $A$.
Prove that player $A$ has the ability to win for any game of player $B$ if and only if $n <2m$.
The numbers $1,2,3,\dots,1000$ are written on the board. Patya and Vassya are playing a game. They take turn alternatively erasing a number from the board. Patya begins. If after a turn all numbers (maybe one) on the board be divisible by a natural number greater than $1$ the player who last played loses. If after some number of steps the only remaining number on the board be $1$ then they call it a draw. Determine the result of the game if they both play their best.
Two pirates divide the loot, consisting of two bags of coins and a diamond, according to the following rules. First the first pirate takes take a few coins from any bag and transfer them from this bag in the other the same number of coins. Then the second pirate does the same (choosing the bag from which he takes the coins at his discretion) and etc. until you can take coins according to these rules. The pirate who takes the coins last gets the diamond. Who will get the diamond if is each of the pirates trying to get it? Give your answer depending on the initial number of coins in the bags.
In the centre of a square swimming pool is a boy, while his teacher (who cannot swim) is standing at one corner of the pool. The teacher can run three times as fast as the boy can swim, but the boy can run faster than the teacher . Can the boy escape from the teacher?
Pasha and Vova play the following game, making moves in turn; Pasha moves first. Initially, they have a large piece of plasticine. By a move, Pasha cuts one of the existing pieces into three(of arbitrary sizes), and Vova merges two existing pieces into one. Pasha wins if at some point there appear to be $100$ pieces of equal weights. Can Vova prevent Pasha's win?
Two players play alternatively on an infinite square grid. The first player puts an $X$ in an empty cell and the second player puts an $O$ in an empty cell. The first player wins if he gets $11$ adjacent $X$'s in a line - horizontally, vertically or diagonally. Show that the second player can always prevent the first player from winning.
If there are four numbers $(a,b,c,d)$ in four registers of the calculating machine, they turn into $(a-b,b-c,c-d,d-a)$ numbers whenever you press the button. Prove that if not all the initial numbers are equal, machine will obtain at least one number more than $1985$ after some number of the operations.
Initially, the numbers $1, 2, \dots, 2024$ are written on a blackboard. Trixi and Nana play a game, taking alternate turns. Trixi plays first.
The player whose turn it is chooses two numbers $a$ and $b$, erases both, and writes their (possibly negative) difference $a-b$ on the blackboard. This is repeated until only one number remains on the blackboard after $2023$ moves. Trixi wins if this number is divisible by $3$, otherwise Nana wins.
Which of the two has a winning strategy?
[i](Birgit Vera Schmidt)[/i]
A nonnegative real number is written at every cube's vertex. The sum of those numbers equals to $1$. Two players choose in turn faces of the cube, but they cannot choose the face parallel to already chosen one (the first moves twice, the second -- once). Prove that the first player can provide the number, at the common for three chosen faces vertex, to be not greater than $1/6$.
To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.
Let $n\geq 3$ be a positive integer. Alice and Bob are playing a game in which they take turns colouring the vertices of a regular $n$-gon. Alice plays the first move. Initially, no vertex is coloured. Both players start the game with $0$ points.
In their turn, a player colours a vertex $V$ which has not been coloured and gains $k$ points where $k$ is the number of already coloured neighbouring vertices of $V$. (Thus, $k$ is either $0$, $1$ or $2$.)
The game ends when all vertices have been coloured and the player with more points wins; if they have the same number of points, no one wins. Determine all $n\geq 3$ for which Alice has a winning strategy and all $n\geq 3$ for which Bob has a winning strategy.
Given a cube and two colours. Two players paint in turn a triple of arbitrary unpainted edges with his colour. (Everyone makes two moves.) The first wins if he has painted all the edges of some face with his colour. Can he always win?
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]