There are 365 cards with 365 different numbers. Each step, we can choose 3 cards $a_{i},a_{j},a_{k}$ and we know the order of them (examble: $a_{i}<a_{j}<a_{k}$). With 2000 steps, can we order 365 cards from smallest to biggest??

Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.

Arne and Berit are playing a game. They have chosen positive integers $m$ and $n$ with $n\geq 4$ and $m \leq 2n + 1$. Arne begins by choosing a number from the set $\{1, 2, \dots , n \}$, and writes it on a blackboard. Then Berit picks another number from the same set, and writes it on the board. They continue alternating turns, always choosing numbers that are not already on the blackboard. When the sum of all the numbers on the board exceeds or equals $m$, the game is over, and whoever wrote the last number has won. For which combinations of $m$ and $n$ does Arne have a winning strategy?

A student is playing computer. Computer shows randomly 2002 positive numbers. Game's rules let do the following operations - to take 2 numbers from these, to double first one, to add the second one and to save the sum. - to take another 2 numbers from the remainder numbers, to double the first one, to add the second one, to multiply this sum with previous and to save the result. - to repeat this procedure, until all the 2002 numbers won't be used. Student wins the game if final product is maximum possible. Find the winning strategy and prove it.

The leader of the Gnome country wants to print banknotes in $12$ different denominations (each with an integer number) in such a way that it is possible to pay an arbitrary amount from $1$ to $6543$ with these banknotes without a balance, using a maximum of $8$ banknotes. (Several bills with the same denomination can be used during payment.) Can the leader of the land of Gnomes do it?

[b]p1.[/b] On the island of Nevermind some people are liars; they always lie. The remaining habitants of the island are truthlovers; they tell only the truth. Three habitants of the island, $A, B$, and $C$ met this morning. $A$ said: “All of us are liars”. $B$ said: “Only one of us is a truthlover”. Who of them is a liar and who of them is a truthlover? [b]p2.[/b] Pinocchio has $9$ pieces of paper. He is allowed to take a piece of paper and cut it in $5$ pieces or $7$ pieces which increases the number of his pieces. Then he can take again one of his pieces of paper and cut it in $5$ pieces or $7$ pieces. He can do this again and again as many times as he wishes. Can he get $2004$ pieces of paper? [b]p3.[/b] In Dragonland there are coins of $1$ cent, $2$ cents, $10$ cents, $20$ cents, and $50$ cents. What is the largest amount of money one can have in coins, yet still not be able to make exactly $1$ dollar? [b]p4.[/b] Find all solutions $a, b, c, d, e$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & d \\ + & a & c & a & c \\ \hline c & d & e & b & c \\ \end{tabular}$ [b]p5.[/b] Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

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]

Given is a positive integer $n$. There are $2n$ mutually non-attacking rooks placed on a grid $2n \times 2n$. The grid is splitted into two connected parts, symmetric with respect to the center of the grid. What is the largest number of rooks that could lie in the same part?

Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).

A total of $2010$ coins are distributed in $5$ boxes. At the beginning the quantities of coins in the boxes are consecutive natural numbers. Martha should choose and take one of the boxes, but before that she can do the following transformation finitely many times: from a box with at least 4 coins she can transfer one coin to each of the other boxes. What is the maximum number of coins that Martha can take away?

Let $G$ be a simple graph with $100$ edges on $20$ vertices. Suppose that we can choose a pair of disjoint edges in $4050$ ways. Prove that $G$ is regular.

Prove that if $n$ is large enough, in every $n\times n$ square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by $1391$.

In a country every 2 cities are connected either by a direct bus route or a direct plane flight. A $clique$ is a set of cities such that every 2 cities in the set are connected by a direct flight. A $cluque$ is a set of cities such that every 2 cities in the set are connected by a direct flight, and every 2 cities in the set are connected to the same number of cities by a bus route. A $claque$ is a set of cities such that every 2 cities in the set are connected by a direct flight, and every 2 numbers of bus routes from a city in the set are different. Prove that the number of cities of any clique is at most the product of the biggest possible number of cities in a cluque and the the biggest possible number of cities in a claque. Tuymaada 2017 Q3 Juniors

In how many ways can Irena can write the integers $1$ through $7$ in a line such that whenever she looks at any three consecutive (in the line) numbers, the largest is not in the rightmost position? [i]Proposed by Noah Kravitz[/i]

Let $ n$ and $ k$ be positive integers. Please, find an explicit formula for \[ \sum y_1y_2 \dots y_k,\] where the summation runs through all $ k\minus{}$tuples positive integers $ (y_1,y_2,\dots,y_k)$ satisfying $ y_1\plus{}y_2\plus{}\dots\plus{}y_k\equal{}n$.

$N$ unit squares on the infinite sheet of cross-lined paper are painted with black colour. Prove that you can cut out the finite number of square pieces and satisfy two conditions all the black squares are contained in those pieces the area of black squares is not less than $1/5$ and not greater than $4/5$ of every piece area.

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

Two players in turns color the sides of an $n$-gon. The first player colors any side that has $0$ or $2$ common vertices with already colored sides. The second player colors any side that has exactly $1$ common vertex with already colored sides. The player who cannot move, loses. For which $n$ the second player has a winning strategy?

There are $100$ people standing in a line from left to right. Half of them are randomly chosen to face right (with all ${100 \choose 50}$ possible choices being equally likely), and the others face left. Then, while there is a pair of people who are facing each other and have no one between them, the leftmost such pair leaves the line. Compute the expected number of people remaining once this process terminates.

In a group of $k$ people, some are acquainted with each other and some are not. Every evening, one person invites all his acquaintances to a party and introduces them to each other(if they have not already acquainted). Suppose that after each person has arranged at least one party, some two people do not know each other. Prove that they do not meet each other in the next party.

A kingdom consists of $12$ cities located on a one-way circular road. A magician comes on the $13$th of every month to cast spells. He starts at the city which was the 5th down the road from the one that he started at during the last month (for example, if the cities are numbered $1–12$ clockwise, and the direction of travel is clockwise, and he started at city #$9$ last month, he will start at city #$2$ this month). At each city that he visits, the magician casts a spell if the city is not already under the spell, and then moves on to the next city. If he arrives at a city which is already under the spell, then he removes the spell from this city, and leaves the kingdom until the next month. Last Thanksgiving the capital city was free of the spell. Prove that it will be free of the spell this Thanksgiving as well.

In a qualification group with $15$ volleyball teams, each team plays with all the other teams exactly once. Since there is no tie in volleyball, there is a winner in every match. After all matches played, a team would be qualified if its total number of losses is not exceeding $N$. If there are at least $7$ teams qualified, find the possible least value of $N$.

Let $P$ be a regular $99$-gon. Assign integers between $1$ and $99$ to the vertices of $P$ such that each integer appears exactly once. (If two assignments coincide under rotation, treat them as the same. ) An [i]operation[/i] is a swap of the integers assigned to a pair of adjacent vertices of $P$. Find the smallest integer $n$ such that one can achieve every other assignment from a given one with no more than $n$ operations. [i]Proposed by Zhenhua Qu[/i]

On the planet Automory, there are infinitely many inhabitants. Every Automorian loves exactly one Automorian and honours exactly one Automorian. Additionally, the following can be noticed: $\bullet$ each Automorian is loved by some Automorian; $\bullet$ if Automorian $A$ loves Automorian $B$, then also all Automorians honouring $A$ love $B$, $\bullet$if Automorian $A$ honours Automorian $B$, then also all Automorians loving $A$ honour $B$. Is it correct to claim that every Automorian honours and loves the same Automorian?

An $8 \times 11$ rectangle of unit squares somehow becomes disassembled into $21$ contiguous parts . Prove that at least two of these parts, except for rotations and reflections have the same shape.