In a culturing of bacteria, there are two species of them: red and blue bacteria. When two red bacteria meet, they transform into one blue bacterium. When two blue bacteria meet, they transform into four red bacteria. When a red and a blue bacteria meet, they transform into three red bacteria. Find, in function of the amount of blue bacteria and the red bacteria initially in the culturing, all possible amounts of bacteria, and for every possible amount, the possible amounts of red and blue bacteria.

How many subsets of $\{1, 2,...,9\}$ do not contain $2$ adjacent numbers?

In a game of rock-paper-scissors with $n$ people, the following rules are used to determine a champion: (a) In a round, each person who has not been eliminated randomly chooses one of rock, paper, or scissors to play. (b) If at least one person plays rock, at least one person plays paper, and at least one person plays scissors, then the round is declared a tie and no one is eliminated. If everyone makes the same move, then the round is also declared a tie. (c) If exactly two moves are represented, then everyone who made the losing move is eliminated from playing in all further rounds (for example, in a game with $8$ people, if $5$ people play rock and $3$ people play scissors, then the $3$ who played scissors are eliminated). (d) The rounds continue until only one person has not been eliminated. That person is declared the champion and the game ends. If a game begins with $4$ people, what is the expected value of the number of rounds required for a champion to be determined? [i]In the game of rock-paper-scissors, two players each choose one of rock, paper, or scissors to play. Rock beats scissors, scissors beats paper, and paper beats rock. If the players play the same thing, the match is considered a draw.[/i]

Let $k$ and $N$ be integers such that $k > 1$ and $N > 2k + 1$. A number of $N$ persons sit around the Round Table, equally spaced. Each person is either a knight (always telling the truth) or a liar (who always lies). Each person sees the nearest k persons clockwise, and the nearest $k$ persons anticlockwise. Each person says: ''I see equally many knights to my left and to my right." Establish, in terms of $k$ and $N$, whether the persons around the Table are necessarily all knights. Sergey Berlov, Russia

In Skinien there 2009 towns where each of them is connected with exactly 1004 other town by a highway. Prove that starting in an arbitrary town one can make a round trip along the highways such that each town is passed exactly once and finally one returns to its starting point.

[b]p1.[/b] A tennis net is made of strings tied up together which make a grid consisting of small squares as shown below. [img]https://cdn.artofproblemsolving.com/attachments/9/4/72077777d57408d9fff0ea5e79be5ecb6fe8c3.png[/img] The size of the net is $100\times 10$ small squares. What is the maximal number of sides of small squares which can be cut without breaking the net into two separate pieces? (The side is cut only in the middle, not at the ends). [b]p2.[/b] What number is bigger $2^{300}$ or $3^{200}$ ? [b]p3.[/b] All noble knights participating in a medieval tournament in Camelot used nicknames. In the tournament each knight had combats with all other knights. In each combat one knight won and the second one lost. At the end of tournament the losers reported their real names to the winners and to the winners of their winners. Was there a person who knew the real names of all knights? [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $10$ rocks in the first pile and $12$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p5.[/b] There is an interesting $5$-digit integer. With a $1$ after it, it is three times as large as with a $1$ before it. What is the number? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer. Suppose that we have disks of radii $1, 2, . . . , n.$ Of each size there are two disks: a transparent one and an opaque one. In every disk there is a small hole in the centre, with which we can stack the disks using a vertical stick. We want to make stacks of disks that satisfy the following conditions: $i)$ Of each size exactly one disk lies in the stack. $ii)$ If we look at the stack from directly above, we can see the edges of all of the $n$ disks in the stack. (So if there is an opaque disk in the stack,no smaller disks may lie beneath it.) Determine the number of distinct stacks of disks satisfying these conditions. (Two stacks are distinct if they do not use the same set of disks, or, if they do use the same set of disks and the orders in which the disks occur are different.)

Suppose that each point on a plane is colored with one of the colors red or yellow. Show that exist a convex pentagon with three right angles and all vertices are with same color.

With pearls of different colours form necklaces, it is said that a necklace is [i]prime[/i] if it cannot be decomposed into strings of pearls of the same length, and equal to each other. Let $n$ and $q$ be positive integers. Prove that the number of prime necklaces with $n$ beads, each of which has one of the $q^n$ possible colours, is equal to $n$ times the number of prime necklaces with $n^2$ pearls, each of which has one of $q$ possible colours. Note: two necklaces are considered equal if they have the same number of pearls and you can get the same colour on both necklaces, rotating one of them to match it to the other.

A round table is surrounded by $n\geqslant 2$ people, each assigned one of the integers $0, 1,\ldots , n-1$ such that no two people have the same number. In each round, everyone adds their number to their right neighbour’s number, and their new number becomes the remainder of the sum when divided by $n{}.$ We call an initial configuration of integers [i]glorious[/i] if everyone’s number remains the same after some finite number of rounds, never changing again. [list=a] [*]For which integers $n\geqslant 2$ is every initial configuration glorious? [*]For which integers $n\geqslant 2$ is there no glorious initial configuration at all? [/list]

Let $a_n$ be the number of ways to express $n$ as an ordered sum of powers of $3.$ For example $a_4=3,$ since $$4=1+1+1+1=1+3=3+1.$$ Let $b_n$ denote the remainder upon dividing $a_n$ by $3.$ Evaluate $$\sum_{n=1}^{3^{2025}} b_n.$$

Let $n$ blue points $A_i$ and $n$ red points $B_i \ (i = 1, 2, \ldots , n)$ be situated on a line. Prove that \[\sum_{i,j} A_i B_j \geq \sum_{i<j} A_iA_j + \sum_{i<j} B_iB_j.\]

A chess-board $6\times 6$ is tiled with the $2\times 1$ dominos. Prove that you can cut the board onto two parts by a straight line that does not cut dominos.

A rectangular field consists of $1520$ unit squares. How many rectangles $6\times1$ at most can be cut out from this field?

There are two circles of perimeter $1979$. Let $1979$ points be marked on the first circle, and several arcs with the total length of $1$ on the second. Show that it is possible to place the second circle onto the first in such a way that none of the marked points is covered by a marked arc.

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

The shape of a military base is an equilateral triangle of side $10$ kilometers. Security constraints make cellular phone com­munication possible only within $2.5$ kilometers. Each of $17$ soldiers patrols the base randomly and tries to contact all others. Prove that at each moment at least two soldiers can communicate.

Treasure was buried in a single cell of an $M\times N$ ($2\le M$, $N$) grid. Detectors were brought to find the cell with the treasure. For each detector, you can set it up to scan a specific subgrid $[a,b]\times[c,d]$ with $1\le a\le b\le M$ and $1\le c\le d\le N$. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up $Q$ detectors, which may only be run simultaneously after all $Q$ detectors are ready. In terms of $M$ and $N$, what is the minimum $Q$ required to gaurantee to determine the location of the treasure?

Let $A$ be a set of $N$ residues $\pmod{N^{2}}$. Prove that there exists a set $B$ of of $N$ residues $\pmod{N^{2}}$ such that $A + B = \{a+b|a \in A, b \in B\}$ contains at least half of all the residues $\pmod{N^{2}}$.

Each cell of a $100$ x $100$ board is painted in one of $100$ different colors so that there are exactly $100$ cells of each color. Prove that there is a row or column in which there are at least $10$ cells of different colors.

There is a unit circle that starts out painted white. Every second, you choose uniformly at random an arc of arclength $1$ of the circle and paint it a new color. You use a new color each time, and new paint covers up old paint. Let $c_n$ be the expected number of colors visible after $n$ seconds. Compute $\lim_{n\to \infty} c_n$.

Some participants of the tournament are friends with each other, and everyone has at least one friend. Each participant of the tournament was given a T-shirt with the number of his friends at the tournament written on it. Prove that at least one participant has the arithmetic mean of the numbers written on his friends' T-shirts, not less than the arithmetic mean of the numbers on all T-shirts. [i] From Czech-Slovak Olympiad 1991 [/i]

A finite family of finite sets $F$ is given, satisfying two conditions: (i) if $A, B \in F$, then $A \cup B \in F$; (ii) if $A \in F$, then the number of elements $| A |$ is not a multiple of $3$. Prove that you can specify at most two elements so that every set of the family $F$ contains at least one of them.

Suppose $a_i, b_i, c_i, i=1,2,\cdots ,n$, are $3n$ real numbers in the interval $\left [ 0,1 \right ].$ Define $$S=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k<1 \right \}, \; \; T=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k>2 \right \}.$$ Now we know that $\left | S \right |\ge 2018,\, \left | T \right |\ge 2018.$ Try to find the minimal possible value of $n$.

What is the largest possible number of vertices one can have in a graph that satisfies the following conditions: each vertex is connected to exactly $3$ other vertices, and there always exists a path of length less than or equal to $2$ between any two vertices?