Let $A_1,A_2,...$ be a sequence of sets such that for any positive integer $i$, there are only finitely many values of $j$ such that $A_j\subseteq A_i$. Prove that there is a sequence of positive integers $a_1,a_2,...$ such that for any pair $(i,j)$ to have $a_i\mid a_j\iff A_i\subseteq A_j$.

A cube with edge of length $n$ ($n\ge 3$) consists of $n^3$ unit cubes. Prove that it is possible to write different $n^3$ integers on all the unit cubes to provide the zero sum of all integers in the every row parallel to some edge.

Antonio plays a game where he continually flips a fair coin to see the sequence of heads ($H$) and tails ($T$) that he flips. Antonio wins the game if he sees on four consecutive flips the sequence $TTHT$ before he sees the sequence $HTTH$. The probability that Antonio wins the game is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $G$ be a simple graph with $11$ vertices labeled as $v_{1} , v_{2} , ... , v_{11}$ such that the degree of $v_1$ equals to $2$ and the degree of other vertices are equal to $3$.If for any set $A$ of these vertices which $|A| \le 4$ , the number of vertices which are adjacent to at least one verex in $A$ and are not in $A$ themselves is at least equal to $|A|$ , then find the maximum possible number for the diameter of $G$. (The distance between two vertices of graph is the number of edges of the shortest path between them and the diameter of a graph , is the largest distance between arbitrary pairs in $V(G)$. ) [i]Proposed by Alireza Haqi[/i]

Prove that there exists a four-coloring of the set $M = \{1, 2, \cdots, 1987\}$ such that any arithmetic progression with $10$ terms in the set $M$ is not monochromatic. [b][i]Alternative formulation[/i][/b] Let $M = \{1, 2, \cdots, 1987\}$. Prove that there is a function $f : M \to \{1, 2, 3, 4\}$ that is not constant on every set of $10$ terms from $M$ that form an arithmetic progression. [i]Proposed by Romania[/i]

For $n>2$, an $n\times n$ grid of squares is coloured black and white like a chessboard, with its upper left corner coloured black. Then we can recolour some of the white squares black in the following way: choose a $2\times 3$ (or $3\times 2$) rectangle which has exactly $3$ white squares and then colour all these $3$ white squares black. Find all $n$ such that after a series of such operations all squares will be black.

The $12$ numbers on a clock face are rearranged. Show that we can still find three adjacent numbers whose sum is $21$ or more.

The $28$ students of a class are seated in a circle. They then all claim that 'the two students next to me are of different genders'. It is known that all boys are lying while exactly $3$ girls are lying. How many girls are there in the class?

It is known that $a_1, a_2,...a_{108}$ are $108$ different positive integers not exceeding $2015$. Prove that there is a positive integer $k$ such that there are at least four different pairs $(i, j) $satisfying $a_i-a_j =k$.

In a graph with $2n-1$ vertices throwing out any vertex the remaining graph has a complete subgraph with $n$ vertices. Prove that the initial graph has a complete subgraph with $n+1$ vertices.

Consider the segment $[0; 1]$. At each step we may split one of the available segments into two new segments and write the product of lengths of these two new segments onto a blackboard. Prove that the sum of the numbers on the blackboard never will exceed $1/2$. [i]Mikhail Lukin[/i]

A conference is attended by $n (n\ge 3)$ scientists. Each scientist has some friends in this conference (friendship is mutual and no one is a friend of him/herself). Suppose that no matter how we partition the scientists into two nonempty groups, there always exist two scientists in the same group who are friends, and there always exist two scientists in different groups who are friends. A proposal is introduced on the first day of the conference. Each of the scientists' opinion on the proposal can be expressed as a non-negative integer. Everyday from the second day onwards, each scientists' opinion is changed to the integer part of the average of his/her friends' opinions from the previous day. Prove that after a period of time, all scientists have the same opinion on the proposal.

A circle is divided into $n$ sectors ($n \geq 3$). Each sector can be filled in with either $1$ or $0$. Choose any sector $\mathcal{C}$ occupied by $0$, change it into a $1$ and simultaneously change the symbols $x, y$ in the two sectors adjacent to $\mathcal{C}$ to their complements $1-x$, $1-y$. We repeat this process as long as there exists a zero in some sector. In the initial configuration there is a $0$ in one sector and $1$s elsewhere. For which values of $n$ can we end this process?

Compute the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $$a!b!c!d!=24!$$$\textbf{(A)}~4$ $\textbf{(B)}~4!$ $\textbf{(C)}~4^4$ $\textbf{(D)}~\text{None of the above}$

In a $2\times n$ array we have positive reals s.t. the sum of the numbers in each of the $n$ columns is $1$. Show that we can select a number in each column s.t. the sum of the selected numbers in each row is at most $\frac{n+1}4$.

In a class of $14$ boys, each boy was asked how many classmates had the same first name. and how many colleagues had the same last name as them. The numbers $0, 1, 2, 3, 4, 5$ and $6$. Proves that there are two colleagues with the same first name and the same last name

We deal $n-1$ cards in some way to $n$ people sitting around a table. From then on, in one move a person with at least $2$ cards gives one card to each of his/her neighbors. Prove that eventually a state will be reached where everyone has at most one card.

On an initially colourless plane three points are chosen and marked in red, blue and yellow. At each step two points marked in different colours are chosen. Then one more point is painted in the third colour so that these three points form a regular triangle with the vertices coloured clockwise in ''red, blue, yellow". A point already marked may be marked again so that it may have several colours. Prove that for any number of moves all the points containing the same colour lie on the same line.

In how many ways can a $2\times n$ grid be covered by (a) 2 monominoes and $n-1$ dominoes (b) 4 monominoes and $n-2$ dominoes.

[url=https://artofproblemsolving.com/community/c675547][b]Greece JBMO TST 2017[/b][/url] [url=http://artofproblemsolving.com/community/c6h1663730p10567608][b]Problem 1[/b][/url]. Positive real numbers $a,b,c$ satisfy $a+b+c=1$. Prove that $$(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \geq 8(ab+bc+ca).$$ Also, find the values of $a,b,c$ for which the equality happens. [url=http://artofproblemsolving.com/community/c6h1663731p10567619][b]Problem 2[/b][/url]. Let $ABC$ be an acute-angled triangle inscribed in a circle $\mathcal C (O, R)$ and $F$ a point on the side $AB$ such that $AF < AB/2$. The circle $c_1(F, FA)$ intersects the line $OA$ at the point $A'$ and the circle $\mathcal C$ at $K$. Prove that the quadrilateral $BKFA'$ is cyclic and its circumcircle contains point $O$. [url=http://artofproblemsolving.com/community/c6h1663732p10567627][b]Problem 3[/b][/url]. Prove that for every positive integer $n$, the number $A_n = 7^{2n} -48n - 1$ is a multiple of $9$. [url=http://artofproblemsolving.com/community/c6h1663734p10567640][b]Problem 4[/b][/url]. Let $ABC$ be an equilateral triangle of side length $a$, and consider $D$, $E$ and $F$ the midpoints of the sides $(AB), (BC)$, and $(CA)$, respectively. Let $H$ be the the symmetrical of $D$ with respect to the line $BC$. Color the points $A, B, C, D, E, F, H$ with one of the two colors, red and blue. [list=1] [*] How many equilateral triangles with all the vertices in the set $\{A, B, C, D, E, F, H\}$ are there? [*] Prove that if points $B$ and $E$ are painted with the same color, then for any coloring of the remaining points there is an equilateral triangle with vertices in the set $\{A, B, C, D, E, F, H\}$ and having the same color. [*] Does the conclusion of the second part remain valid if $B$ is blue and $E$ is red? [/list]

In [i]Terra Brasilis[/i] there are $n$ houses where $n$ goblins live, each in a house. There are one-way routes such that: - each route joins two houses, - in each house exactly one route begins, - in each house exactly one route ends. If a route goes from house $A$ to house $B$, then we will say that house $B$ is next to house $A$. This relationship is not symmetric, that is: in this situation, not necessarily house $A$ is next to house $B$. Every day, from day $1$, each goblin leaves the house where he is and arrives at the next house. A legend of [i]Terra Brasilis[/i] says that when all the goblins return to the original position, the world will end. a) Show that the world will end. b) If $n = 98$, show that it is possible for elves to build and guide the routes so that the world does not end before $300,000$ years.

Let $K$ denote the set $\{a, b, c, d, e\}$. $F$ is a collection of $16$ different subsets of $K$, and it is known that any three members of $F$ have at least one element in common. Show that all $16$ members of $F$ have exactly one element in common.

There are $n$ cards such that for each $i=1,2, \cdots n$, there are exactly one card labeled $i$. Initially the cards are piled with increasing order from top to bottom. There are two operations: [list] [*] $A$ : One can take the top card of the pile and move it to the bottom; [*] $B$ : One can remove the top card from the pile. [/list] The operation $ABBABBABBABB \cdots $ is repeated until only one card gets left. Let $L(n)$ be the labeled number on the final pile. Find all integers $k$ such that $L(3k)=k$.

Suppose that every integer has been given one of the colours red, blue, green or yellow. Let $x$ and $y$ be odd integers so that $|x| \neq |y|$. Show that there are two integers of the same colour whose difference has one of the following values: $x,y,x+y$ or $x-y$.

Show that a path on a rectangular grid which starts at the northwest corner, goes through each point on the grid exactly once, and ends at the southeast corner divides the grid into two equal halves: (a) those regions opening north or east; and (b) those regions opening south or west. [img]https://cdn.artofproblemsolving.com/attachments/b/e/aa20c9f9bc44bd1e5a9b9e86d49debf0f821b7.png[/img] (The figure above shows a path meeting the conditions of the problem on a $5 \times 8$ grid. The shaded regions are those opening north or east while the rest open south or west.)