Six lines are drawn in the plane. Determine the maximum number of points, through which at least $3$ lines pass.

Let $S$ be a subset of $\{1,2,\dots,2017\}$ such that for any two distinct elements in $S$, both their sum and product are not divisible by seven. Compute the maximum number of elements that can be in $S$.

Some $1\times 2$ dominoes, each covering two adjacent unit squares, are placed on a board of size $n\times n$ such that no two of them touch (not even at a corner). Given that the total area covered by the dominoes is $2008$, find the least possible value of $n$.

Per and Kari each have $n$ pieces of paper. They both write down the numbers from $1$ to $2n$ in an arbitrary order, one number on each side. Afterwards, they place the pieces of paper on a table showing one side. Prove that they can always place them so that all the numbers from $1$ to $2n$ are visible at once.

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.

There are $12$ candies on the table, four of which are rare candies. Chad has a friend who can tell rare candies apart from regular candies, but Chad can’t. Chad’s friend is allowed to take four candies from the table, but may not take any rare candies. Can his friend always take four candies in such a way that Chad will then be able to identify the four rare candies? If so, describe a strategy. If not, prove that it cannot be done. Note that Chad does not know anything about how the candies were selected (e.g. the order in which they were selected). However, Chad and his friend may communicate beforehand.

Initially the numbers 1 through 2005 are marked. A finite set of marked consecutive integers is called a block if it is not contained in any larger set of marked consecutive integers. In each step we select a set of marked integers which does not contain the first or last element of any block, unmark the selected integers, and mark the same number of consecutive integers starting with the integer two greater than the largest marked integer. What is the minimum number of steps necessary to obtain 2005 single integer blocks?

Alina and Bogdan play the following game. They have a heap and $330$ stones in it. They take turns. In one turn it is allowed to take from the heap exactly $1$, exactly $n$ or exactly $m$ stones. The player who takes the last stone wins. Before the beginning Alina says the number $n$, ($1 < n < 10$). After that Bogdan says the number $m$, ($m \ne n, 1 < m < 10$). Alina goes first. Which of the two players has a winning strategy? What if initially there are 2018 stones in the heap? adapted from a Belarus Olympiad problem

For the Dürer final results announcement, four loudspeakers are used to provide sound in the hall. However, there are only two sockets in the wall from which the power comes. To solve the problem, Ádám got two extension cords and two power strips. One plug can be plugged into an extension cord, and two plugs can be plugged into a power strip. Gábor, in his haste before the announcement of the results, quickly plugs the $8$ plugs into the $8$ holes. Every possible way of plugging has the same probability, and it is also possible for Gábor to plug something into itself. What is the probability that all $4$ speakers will have sound at the results announcement? For the solution, give the sum of the numerator and the denominator in the simplified form of the probability. A speaker sounds when it is plugged directly or indirectly into the wall.

There are cities in country, and some cities are connected by roads. Not more than $100$ roads go from every city. Set of roads is called as ideal if all roads in set have not common ends, and we can not add one more road in set without breaking this rule. Every day minister destroy one ideal set of roads. Prove, that he need not more than $199$ days to destroy all roads in country.

A convex polygon on a plane contains at least $m^2+1$ points with integer coordinates. Prove that it contains $m+1$ points with integer coordinates that lie on the same line.

[b]p1.[/b] Is there an integer such that the product of all whose digits equals $99$ ? [b]p2.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p3.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/9/f/359d3b987012de1f3318c3f06710daabe66f28.png[/img] [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $5$ rocks in the first pile and $6$ 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] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We are given a natural number $k$. Let us consider the following game on an infinite onedimensional board. At the start of the game, we distrubute $n$ coins on the fields of the given board (one field can have multiple coins on itself). After that, we have two choices for the following moves: $(i)$ We choose two nonempty fields next to each other, and we transfer all the coins from one of the fields to the other. $(ii)$ We choose a field with at least $2$ coins on it, and we transfer one coin from the chosen field to the $k-\mathrm{th}$ field on the left , and one coin from the chosen field to the $k-\mathrm{th}$ field on the right. $\mathbf{(a)}$ If $n\leq k+1$, prove that we can play only finitely many moves. $\mathbf{(b)}$ For which values of $k$ we can choose a natural number $n$ and distribute $n$ coins on the given board such that we can play infinitely many moves.

How many $5\times5$ grids are possible such that each element is either $0$ or $1$ and each row sum and column sum is $4$? $\textbf{(A)}~64$ $\textbf{(B)}~32$ $\textbf{(C)}~120$ $\textbf{(D)}~96$

There are $16$ points marked on the circle. Find the greatest possible number of acute triangles with vertices at the marked points.

How many ways can a set of $ n $ items be partitioned into two sets?

In a group of 12 persons, among any 9 there are 5 which know each other. Prove that there are 6 persons in this group which know each other

Let $ n$ be an even number, and $ S$ be the set of all arrays of length $ n$ whose elements are from the set $ \left\{0,1\right\}$. Prove that $ S$ can be partitioned into disjoint three-element subsets such that for each three arrays $ \left(a_i\right)_{i \equal{} 1}^n$, $ \left(b_i\right)_{i \equal{} 1}^n$, $ \left(c_i\right)_{i \equal{} 1}^n$ which belong to the same subset and for each $ i\in\left\{1,2,...,n\right\}$, the number $ a_i \plus{} b_i \plus{} c_i$ is divisible by $ 2$.

Let there be three cups $A, B$ and $C$, which start with $a, b$ and $c$ (all of them are natural numbers) units of gallium filled. It is also believed that all cups are large enough to contain the total amount of gallium available. It is now allowed to move gallium from one cup to another cup, provided that the contents of the latter cup are exactly double. (a) For which starting positions is it possible to empty one of the cups? (b) For which starting positions is it possible to put all of the gallium in one cup?

Consider the main diagonal and the cells above it in an \( n \times n \) grid. These cells form what we call a tabular triangle of length \( n \). We want to place a real number in each cell of a tabular triangle of length \( n \) such that for each cell, the sum of the numbers in the cells in the same row and the same column (including itself) is zero. For example, the sum of the cells marked with a circle is zero. It is known that the number in the topmost and leftmost cell is $1.$ Find all possible ways to fill the remaining cells.

A convex polygon $P$ can be partitioned into $27$ parallelograms. Prove that it can also be partitioned into $21$ parallelograms.

Find the maximum number of increasing arithmetic progressions that can have a finite sequence of real numbers $a_1<a_2<\cdots<a_n$ of $n\ge 3$ real numbers.

The teacher wrote on a blackboard: $$x^2 + 10x + 20$$ Then all the pupils in the class came up in turn and either decreased or increased by $1$ either the free coefficient or the coefficient at $x$, but not both. Finally they have obtained: $$x^2 + 20x + 10$$ Is it true that some time during the process there was written the square polynomial with the integer roots?

Stierlitz wants to send an encryption to the Center, which is a code containing $100$ characters, each a "dot" or a "dash". The instruction he received from the Center the day before about conspiracy reads: i) when transmitting encryption over the radio, exactly $49$ characters should be replaced with their opposites; ii) the location of the "wrong" characters is decided by the transmitting side and the Center is not informed of it. Prove that Stierlitz can send $10$ encryptions, each time choosing some $49$ characters to flip, such that when the Center receives these $10$ ciphers, it may unambiguously restore the original code.

Let $A$ be a family of subsets of $\{1,2,\ldots,n\}$ such that no member of $A$ is contained in another. Sperner’s Theorem states that $|A|\leq{n\choose{\lfloor\frac{n}{2}\rfloor}}$. Find all the families for which the equality holds.