For an integer $n \ge 3$ we consider a circle with $n$ points on it. We place a positive integer at each point, where the numbers are not necessary need to be different. Such placement of numbers is called [i]stable [/i] as three numbers next to always have product $n$ each other. For how many values of $n$ with $3 \le n \le 2020$ is it possible to place numbers in a stable way?

Is it possible to put $\binom{n}{2}$ consecutive natural numbers on the edges of a complete graph with $n$ vertices in a way that for every path (or cycle) of length $3$ where the numbers $a,b$ and $c$ are written on its edges (edge $b$ is between edges $c$ and $a$), $b$ is divisible by the greatest common divisor of the numbers $a$ and $c$? [i]Proposed by Morteza Saghafian[/i]

Let us choose arbitrarily $n$ vertices of a regular $2n$-gon and color them red. The remaining vertices are colored blue. We arrange all red-red distances into a nondecreasing sequence and do the same with the blue-blue distances. Prove that the two sequences thus obtained are identical.

Let there be given real numbers $x_i>1~(i=1,2,\ldots,2n)$. Prove that the interval $[0,2]$ contains at most $\binom{2n}n$ sums of the form $\alpha_1x_1+\ldots+\alpha_{2n}x_{2n}$, where $\alpha_i\in\{-1,1\}$ for all $i$.

At a chess tournament the winner gets 1 point and the defeated one 0 points. A tie makes both obtaining $\frac{1}{2}$ points. 14 players, none of them equally aged, participated in a competition where everybody played against all the other players. After the competition a ranking was carried out. Of the two players with the same number of points the younger received the better ranking. After the competition Jan realizes that the best three players together got as many points as the last 9 players obtained points together. And Joerg noted that the number of ties was maximal. Determine the number of ties.

Through the points with integer coordinates in the right-angled coordinate system $Oxyz$ are constructed planes, parallel to the coordinate planes and in this way the space is divided to unit cubes. Find all triples ($a, b, c$) consisting of natural numbers ($a \le b \le c$) for which the cubes formed can be coloured in $abc$ colors in such a way that every palellepiped with dimensions $a \times  b \times c$, having vertices with integer coordinates and sides parallel to the coordinate axis doesn't contain unit cubes in the same color.

Construct a set of $k$ integer weights that allows you to get any total integer weight from $1$ up to $55$ grams even if some of the weights of the set are lost. Consider two versions: (a) $k = 10$, and any one of the weights may be lost; (b) $k = 12$, and any two of the weights may be lost. (D Zvonkin) (In both cases prove that the set found has the property required.)

In how many ways we can choose 3 non empty and non intersecting subsets from $ (1,2,\ldots,2008)$.

a) Given $555$ weights: of $1$ g, $2$ g, $3$ g, . . . , $555$ g, divide them into three piles of equal mass. b) Arrange $81$ weights of $1^2, 2^2, . . . , 81^2$ (all in grams) into three piles of equal mass.

In a certain state there were 2004 cities connected by roads so that from any city one could get to any other. It is known that when it is prohibited to travel on any of the roads, the least of them any city could be reached to any other. The Minister of Transport and the Minister of Internal Affairs take turns introducing restrictions on the roads while there is possibility, one-way traffic (on one road per turn), and minister, after whose move it became impossible to leave any city to reach any other, immediately resigns. First the Minister of Transport walks. Can any of the ministers force the resignation of another, regardless of his performance? [hide=original wording]В некотором государстве было 2004 города, соединенных дорогами так, что из любого города можно было добраться до любого другого. Известно, что при запрещенном проезде по любой из дорог, по-прежнему из любого города можно было добраться до любого другого. Министр транспорта и министр внутренних дел по очереди вводят на дорогах, пока есть возможность, одностороннее движение (на одной дороге за ход), причем министр, после хода которого из какого-либо города стало невозможно добраться до какого-либо другого, немедленно уходит в отставку. Первым ходит министр транспорта. Может ли кто-либо из министров добиться отставки другого независимо от его игры?[/hide]

For three arbitrary crossroads $A,B,C$ in a certain city there exist a way from $A$ to $B$ not coming through $C$. Prove that for every couple of the crossroads there exist at least two non-intersecting ways connecting them. (there are at least two crossroads in the city)

On her blackboard, Alice has written $n$ integers strictly greater than $1$. Then, she can, as often as she likes, erase two numbers $a$ and $b$ such that $a \neq b$, and replace them with $q$ and $q^2$, where $q$ is the product of the prime factors of $ab$ (each prime factor is counted only once). For instance, if Alice erases the numbers $4$ and $6$, the prime factors of $ab = 2^3 \times 3$ and $2$ and $3$, and Alice writes $q = 6$ and $q^2 =36$. Prove that, after some time, and whatever Alice's strategy is, the list of numbers written on the blackboard will never change anymore. [i]Note: The order of the numbers of the list is not important.[/i]

The points $P = (a, b)$ and $Q = (c, d)$ are in the first quadrant of the $xy$ plane, and $a, b, c$ and $d$ are integers satisfying $a < b, a < c, b < d$ and $c < d$. A route from point $P$ to point $Q$ is a broken line consisting of unit steps in the directions of the positive coordinate axes. An allowed route is a route not touching the line $x = y$. Tetermine the number of allowed routes.

A page of an exercise book is painted with $23$ colours, arranged in squares. A pair of colours is called [i]good [/i] if there are neighbouring squares painted with these colours. What is the minimum number of good pairs?

Julian and Johan are playing a game with an even number of cards, say $2n$ cards, ($n \in Z_{>0}$). Every card is marked with a positive integer. The cards are shuffled and are arranged in a row, in such a way that the numbers are visible. The two players take turns picking cards. During a turn, a player can pick either the rightmost or the leftmost card. Johan is the first player to pick a card (meaning Julian will have to take the last card). Now, a player’s score is the sum of the numbers on the cards that player acquired during the game. Prove that Johan can always get a score that is at least as high as Julian’s.

Finite number of $2017$ units long sticks are fixed on a plate. Each stick has a bead that can slide up and down on it. Beads can only stand on integer heights $( 1, 2, 3,..., 2017 )$. Some of the bead pairs are connected with elastic bands. $The$ $young$ $ant$ can go to every bead, starting from any bead by using the elastic bands. $The$ $old$ $ant$ can use an elastic band if the difference in height of the beads which are connected by the band, is smaller than or equal to $1$. If the heights of the beads which are connected to each other are different, we call it $valid$ $situation$. If there exists at least one $valid$ $situation$, prove that we can create a $valid$ $situation$, by arranging the heights of the beads, in which $the$ $old$ $ant$ can go to every bead, starting from any bead.

Leticia has a $9\times 9$ board. She says that two squares are [i]friends[/i] is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each square a number will be written based on the following rules: - If the square is green, write the number of red friends plus twice the number of blue friends. - If the square is red, write the number of blue friends plus twice the number of green friends. - If the square is blue, write the number of green friends plus twice the number of red friends. Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.

Let $A$ be a non-empty set of integers with the following property: For each $a \in A$, there exist not necessarily distinct integers $b,c \in A$ so that $a=b+c$. (a) Proof that there are examples of sets $A$ fulfilling above property that do not contain $0$ as element. (b) Proof that there exist $a_1,\ldots,a_r \in A$ with $r \ge 1$ and $a_1+\cdots+a_r=0$. (c) Proof that there exist pairwise distinct $a_1,\ldots,a_r$ with $r \ge 1$ and $a_1+\cdots+a_r=0$.

In a country there are $4^9$ schoolchildren living in four cities. At the end of the school year a state examination was held in 9 subjects. It is known that any two students have different marks at least in one subject. However, every two students from the same city got equal marks at least in one subject. Prove that there is a subject such that every two children living in the same city have equal marks in this subject. [i]Fedor Petrov[/i]

Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.

For a positive integer $n$, denote $p(n)$ to be the number of nonnegative integer tuples $(x,y,z,w)$ such that $x+y+2z+3w=n-1$. Also, denote $q(n)$ to be the number of nonnegative integer tuples $(a,b,c,d)$ such that (i). $a+b+c+d=n$. (ii). $a \ge b$, $c \ge d$, $a \ge d$. (iii). $b < c$. Prove that for all $n$, $p(n) = q(n)$.

Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations. [i]Proposed by Warut Suksompong, Thailand[/i]

Given $n$ integers $a_1 = 1, a_2,..., a_n$ such that $a_i \le a_{i+1} \le 2a_i$ ($i = 1, 2, 3,..., n - 1$) and whose sum is even. Find whether it is possible to divide them into two groups so that the sum of numbers in one group is equal to the sum of numbers in the other group.

[b]p1.[/b] In triangle $ABC$, the median $BM$ is drawn. The length $|BM| = |AB|/2$. The angle $\angle ABM = 50^o$. Find the angle $\angle ABC$. [b]p2.[/b] Is there a positive integer $n$ which is divisible by each of $1, 2,3,..., 2018$ except for two numbers whose difference is$ 7$? [b]p3.[/b] Twenty numbers are placed around the circle in such a way that any number is the average of its two neighbors. Prove that all of the numbers are equal. [b]p4.[/b] A finite number of frogs occupy distinct integer points on the real line. At each turn, a single frog jumps by $1$ to the right so that all frogs again occupy distinct points. For some initial configuration, the frogs can make $n$ moves in $m$ ways. Prove that if they jump by $1$ to the left (instead of right) then the number of ways to make $n$ moves is also $m$. [b]p5.[/b] A square box of chocolates is divided into $49$ equal square cells, each containing either dark or white chocolate. At each move Alex eats two chocolates of the same kind if they are in adjacent cells (sharing a side or a vertex). What is the maximal number of chocolates Alex can eat regardless of distribution of chocolates in the box? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the least integer $k$ such that for any $2011 \times 2011$ table filled with integers Kain chooses, Abel be able to change at most $k$ cells to achieve a new table in which $4022$ sums of rows and columns are pairwise different.