[b]p1.[/b] A Slavic dragon has three heads. A knight fights the dragon. If the knight cuts off one dragon’s head three new heads immediately grow. Is it possible that the dragon has $2018$ heads at some moment of the fight? [b]p2.[/b] Peter has two squares $3\times 3$ and $4\times 4$. He must cut one of them or both of them in no more than four parts in total. Is Peter able to assemble a square using all these parts? [b]p3.[/b] Usually, dad picks up Constantine after his music lessons and they drive home. However, today the lessons have ended earlier and Constantine started walking home. He met his dad $14$ minutes later and they drove home together. They arrived home $6$ minutes earlier than usually. Home many minutes earlier than usual have the lessons ended? Please, explain your answer. [b]p4.[/b] All positive integers from $1$ to $2018$ are written on a blackboard. First, Peter erased all numbers divisible by $7$. Then, Natalie erased all remaining numbers divisible by $11$. How many numbers did Natalie remove? Please, explain your answer. [b]p5.[/b] $30$ students took part in a mathematical competition consisting of four problems. $25$ students solved the first problem, $24$ students solved the second problem, $22$ students solved the third, and, finally, $21$ students solved the fourth. Show that there are at least two students who solved all four problems. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Can one paint four points in the plane red and another four points black so that any three points of the same colour are vertices of a parallelogram whose fourth vertex is a point of the other colour? (NB Vassiliev)

Let $n > 10$ be an odd integer. Determine the number of ways to place the numbers $1, 2, \ldots , n$ around a circle so that each number in the circle divides the sum its two neighbors. (Two configurations such that one can be obtained from the other per rotation are to be counted only once.)

[b]p1.[/b] Find the minimum value of $x^4 +2x^3 +3x^2 +2x+2$, where x can be any real number. [b]p2.[/b] A type of digit-lock has $5$ digits, each digit chosen from $\{1,2, 3, 4, 5\}$. How many different passwords are there that have an odd number of $1$'s? [b]p3.[/b] Tony is a really good Ping Pong player, or at least that is what he claims. For him, ping pong balls are very important and he can feel very easily when a ping pong ball is good and when it is not. The Ping Pong club just ordered new balls. They usually order form either PPB company or MIO company. Tony knows that PPB balls have $80\%$ chance to be good balls and MIO balls have $50\%$ chance to be good balls. I know you are thinking why would anyone order MIO balls, but they are way cheaper than PPB balls. When the box full with balls arrives (huge number of balls), Tony tries the first ball in the box and realizes it is a good ball. Given that the Ping Pong club usually orders half of the time from PPB and half of the time from MIO, what is the probability that the second ball is a good ball? [b]p4.[/b] What is the smallest positive integer that is one-ninth of its reverse? [b]p5.[/b] When Michael wakes up in the morning he is usually late for class so he has to get dressed very quickly. He has to put on a short sleeved shirt, a sweater, pants, two socks and two shoes. People usually put the sweater on after they put the short sleeved shirt on, but Michael has a different style, so he can do it both ways. Given that he puts on a shoe on a foot after he put on a sock on that foot, in how many dierent orders can Michael get dressed? [b]p6.[/b] The numbers $1, 2,..., 2015$ are written on a blackboard. At each step we choose two numbers and replace them with their nonnegative difference. We stop when we have only one number. How many possibilities are there for this last number? [b]p7.[/b] Let $A = (a_1b_1a_2b_2... a_nb_n)_{34}$ and $B = (b_1b_2... b_n)_{34}$ be two numbers written in base $34$. If the sum of the base-$34$ digits of $A$ is congruent to $15$ (mod $77$) and the sum of the base $34$ digits of $B$ is congruent to $23$ (mod $77$). Then if $(a_1b_1a_2b_2... a_nb_n)_{34} \equiv x$ (mod $77$) and $0 \le x \le 76$, what is $x$? (you can write $x$ in base $10$) [b]p8.[/b] What is the sum of the medians of all nonempty subsets of $\{1, 2,..., 9\}$? [b]p9.[/b] Tony is moving on a straight line for $6$ minutes{classic Tony. Several finitely many observers are watching him because, let's face it, you can't really trust Tony. In fact, they must watch him very closely{ so closely that he must never remain unattended for any second. But since the observers are lazy, they only watch Tony uninterruptedly for exactly one minute, and during this minute, Tony covers exactly one meter. What is the sum of the minimal and maximal possible distance Tony can walk during the six minutes? [b]p10.[/b] Find the number of nonnegative integer triplets $a, b, c$ that satisfy $$2^a3^b + 9 = c^2.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

7. There are 100 chess bishops on white squares of a $100 \times 100$ chess board. Some of them are white and some of them are black. They can move in any order and capture the bishops of opposing color. What number of moves is sufficient for sure to retain only one bishop on the chess board?

$a_1, a_2,...,a_{2006}$ is a permutation of $1,2,...,2006$. Prove that $\prod_{i = 1}^{2006} (a_{i}^2-i) $ is a multiple of $3$. ($0$ is counted as a multiple of $3$)

Find the smallest real number \(a\) such that for any positive integer number \(n > 2\) and any arrangement of the numbers from 1 to \(n\) on a circle, there exists a pair of adjacent numbers whose ratio (when dividing the larger number by the smaller one) is less than \(a\). [i]Proposed by Mykhailo Shtandenko[/i]

The number $328$ is written on the board. Two players alternate writing positive divisors of $328$ on the board, subject to the following rules: $\bullet$ No divisor of a previously written number may be written. $\bullet$ The player who writes 328 loses. Who has a winning strategy, the first player or the second player?

There are $27$ cards, each has some amount of ($1$ or $2$ or $3$) shapes (a circle, a square or a triangle) with some color (white, grey or black) on them. We call a triple of cards a [i]match[/i] such that all of them have the same amount of shapes or distinct amount of shapes, have the same shape or distinct shapes and have the same color or distinct colors. For instance, three cards shown in the figure are a [i]match[/i] be cause they have distinct amount of shapes, distinct shapes but the same color of shapes. What is the maximum number of cards that we can choose such that non of the triples make a [i]match[/i]? [i]Proposed by Amin Bahjati[/i]

An Indian raga has two kinds of notes: a short note, which lasts for $1$ beat and a long note, which lasts for $2$ beats. For example, there are $3$ ragas which are $3$ beats long; $3$ short notes, a short note followed by a long note, and a long note followed by a short note. How many Indian ragas are 11 beats long?

On a rectangular piece of paper there are (a) several marked points all on one straight line, (b) three marked points (not necessarily on a straight line). We are allowed to fold the paper several times along a straight line not containing marked points and then puncture the folded paper with a needle. Show that this can be done so that after the paper has been unfolded all the marked points are punctured and there are no extra holes. (A Shapovalov)

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.

There are two-way flights between some of the $2017$ cities in a country, such that given two cities, it is possible to reach one from the other. No matter how the flights are appointed, one can define $k$ cities as "special city", so that there is a direct flight from each city to at least one "special city". Find the minimum value of $k$.

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

A rectangular piece of paper is divided into square cells by lines parallel to the sides of the rectangle. $n$ (horizontal) rows of $m$ cells have emerged and $m$ (vertical) columns of $n$ cells have also been formed. There is a number in each cell. Find the largest number in each of the $n$ rows. The smallest maxima of those $n$ rows is called $A$. We also look for the smallest number in each of the $m$ columns. The largest minima of those $m$ columns is called $B$. Prove that $A$ is greater than or equal to $B$. Can you give a simple example where $A = B$?

Consider an infinite plane divided into unit squares by horizontal and vertical lines. A coloring of some cells in this grid is called a $\emph{net coloring}$ if the centers of the colored squares coincide with the intersection points of an infinite family of equally spaced parallel lines and another directed and equally spaced infinite family of lines. The distance between the centers of the nearest colored squares is called the size of the $\emph{net coloring}.$ Determine all natural numbers \(N\) for which it is possible to color all these unit squares using \(N\) colors such that the following conditions are met: $\bullet$ Each color is used to color at least one square. $\bullet$The coloring for every color forms a $\emph{net coloring}.$ $\bullet$ The sizes of each of the \(N\) $\emph{net colorings}$ are equal. [i]Proposed by Aleksandre Saatashvili, Georgia [/i]

When the unit squares at the four corners are removed from a three by three squares, the resulting shape is called a cross. What is the maximum number of non-overlapping crosses placed within the boundary of a $ 10\times 11$ chessboard? (Each cross covers exactly five unit squares on the board.)

There is a safe that can be opened by entering a secret code consisting of $n$ digits, each of them is $0$ or $1$. Initially, $n$ zeros were entered, and the safe is closed (so, all zeros is not the secret code). In one attempt, you can enter an arbitrary sequence of $n$ digits, each of them is $0$ or $1$. If the entered sequence matches the secret code, the safe will open. If the entered sequence matches the secret code in more positions than the previously entered sequence, you will hear a click. In any other cases the safe will remain locked and there will be no click. Find the smallest number of attempts that is sufficient to open the safe in all cases.

Let $n \geq 2$ be a natural. Define $$X = \{ (a_1,a_2,\cdots,a_n) | a_k \in \{0,1,2,\cdots,k\}, k = 1,2,\cdots,n \}$$. For any two elements $s = (s_1,s_2,\cdots,s_n) \in X, t = (t_1,t_2,\cdots,t_n) \in X$, define $$s \vee t = (\max \{s_1,t_1\},\max \{s_2,t_2\}, \cdots , \max \{s_n,t_n\} )$$ $$s \wedge t = (\min \{s_1,t_1 \}, \min \{s_2,t_2,\}, \cdots, \min \{s_n,t_n\})$$ Find the largest possible size of a proper subset $A$ of $X$ such that for any $s,t \in A$, one has $s \vee t \in A, s \wedge t \in A$.

Let $m$ be an integer, $m \ge 2$. Each student in a school is practising $m$ hobbies the most. Among any $m$ students there exist two students who have a common hobby. Find the smallest number of students for which there must exist a hobby which is practised by at least $3$ students .

Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

A pirate has five purses with 30 coins in each. He knows that one purse contains only gold coins, another one contains only silver coins, the third one contains only bronze coins, and the remaining two ones contain 10 gold, 10 silver and 10 bronze coins each. It is allowed to simultaneously take one or several coins out of any purses (only once), and examine them. What is the minimal number of taken coins that is necessary to determine for sure the content of at least one purse? [i]Mikhail Evdokimov[/i]

On a line, $44$ points are marked and numbered $1, 2, 3,…,44$ from left to right. Various crickets jump around the line. Each starts at point $1$, jumping on the marked points and ending up at point $44$. In addition, each cricket jumps from a marked point to another marked point with a greater number. When all the crickets have finished jumping, it turns out that for pair $i, j$ with ${1}\leq{i}<{j}\leq{44}$, there was a cricket that jumped directly from point $i$ to point $j$, without visiting any of the points in between the two. Determine the smallest number of crickets such that this is possible.

Some number of coins is firstly separated into 200 groups and then to 300 groups. One coin is [i]special[/i], if on the second grouping it is in a group that has less coins than the previous one, in the first grouping, that it was in. Find the least amount of [i]special[/i] coins we can have.

A cube is cut into 99 smaller cubes, exactly 98 of which are unit cubes. Find the volume of the original cube. (V Proizvolov)