The numbers from 1 to 360 are partitioned into 9 subsets of consecutive integers and the sums of the numbers in each subset are arranged in the cells of a $3 \times 3$ square. Is it possible that the square turns out to be a magic square? Remark: A magic square is a square in which the sums of the numbers in each row, in each column and in both diagonals are all equal.

In the plane, $n$ lines are drawn in general position (that is, there are neither two of them parallel nor three of them passing through the same point). Prove that it is possible to put a positive integer in each region (finite or infinite) determined by these lines so that for each line the sum of the numbers in the regions of a sdemiplane is equal to the sum of the numbers in the regions of the other semiplane. Note: A region is a set of points such that the straight line connecting any two of them it does not intersect any of the lines. For example, a line divides the plane into $2$ infinite regions and three lines into general position divide the plane into $7$ regions, some finite(s) and others infinite.

Misha came to country with $n$ cities, and every $2$ cities are connected by the road. Misha want visit some cities, but he doesn`t visit one city two time. Every time, when Misha goes from city $A$ to city $B$, president of country destroy $k$ roads from city $B$(president can`t destroy road, where Misha goes). What maximal number of cities Misha can visit, no matter how president does?

Kostas and Helene have the following dialogue: Kostas: I have in my mind three positive real numbers with product $1$ and sum equal to the sum of all their pairwise products. Helene: I think that I know the numbers you have in mind. They are all equal to $1$. Kostas: In fact, the numbers you mentioned satisfy my conditions, but I did not think of these numbers. The numbers you mentioned have the minimal sum between all possible solutions of the problem. Can you decide if Kostas is right? (Explain your answer).

In Miskolc there are two tram lines: line $1$ runs between Tiszai railway station and UpperMajláth, while line $2$ runs between Tiszai railway station and the Ironworks. The timetable for trams leaving Tiszai railway station is as follows: tram $ 1$ leaves at every minute ending in a $0$ or $6$, and tram $2$ leaves at every minute ending in a $3$. There are three types of passengers waiting for the trams: those who will take tram $ 1$ only, those who will take tram $2$ only and those who will take any tram. Every minute there is a constant number of passengers of each type arriving at the station. (This number is not necessarily the same for the different types.) Also, every tram departs with an equal number of passengers from Tiszai railway station. How many passengers are there on a departing tram, if we know that every minute there are $3$ passengers arriving at the station who will take tram $2$ only?

On a circle, 2022 points are chosen such that distance between two adjacent points is always the same. There are $k$ arcs, each having endpoints on chosen points, with different lengths. Arcs do not contain each other. What is the maximum possible number of $k$?

Let $G$ be a connected simple graph. When we add an edge to $G$ (between two unconnected vertices), then using at most $17$ edges we can reach any vertex from any other vertex. Find the maximum number of edges to be used to reach any vertex from any other vertex in the original graph, i.e. in the graph before we add an edge.

How many rooks can be placed in an $n\times n$ chessboard such that each rook is threatened by at most $2k$ rooks? (15 points) [i]Proposed by Mostafa Einollah zadeh[/i]

Let $n>1$ be a given integer. Let $G$ be a finite simple graph with the property that each of its edges is contained in at most $n$ cycles. Prove that the chromatic number of the graph is at most $n+1$.

There are four basketball players $A,B,C,D$. Initially the ball is with $A$. The ball is always passed from one person to a different person. In how many ways can the ball come back to $A$ after $\textbf{seven}$ moves? (for example $A\rightarrow C\rightarrow B\rightarrow D\rightarrow A\rightarrow B\rightarrow C\rightarrow A$, or $A\rightarrow D\rightarrow A\rightarrow D\rightarrow C\rightarrow A\rightarrow B\rightarrow A)$.

Let $N \ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$. Per move one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pairwise non-similar sequences of length $N$.

Let $n$ be a positive integer, set $S_n = \{ (a_1,a_2,\cdots,a_{2^n}) \mid a_i=0 \ \text{or} \ 1, 1 \leq i \leq 2^n\}$. For any two elements $a=(a_1,a_2,\cdots,a_{2^n})$ and $b=(b_1,b_2,\cdots,b_{2^n})$ of $S_n$, define \[ d(a,b)= \sum_{i=1}^{2^n} |a_i - b_i| \] We call $A \subseteq S_n$ a $\textsl{Good Subset}$ if $d(a,b) \geq 2^{n-1}$ holds for any two distinct elements $a$ and $b$ of $A$. How many elements can the $\textsl{Good Subset}$ of $S_n$ at most have?

The girl continues the sequence of letters $ASSARA... $, adding one of the three letters $A$, $R$ or $S$. When adding the next letter, the girl makes sure that no two written sevens of consecutive letters coincide. At some point it turned out that it was impossible to add a new letter according to these rules. What letter could be written last?

The colonizers of a spherical planet have decided to build $N$ towns, each having area $1/1000$ of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of $N$?

Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$. If $n \geq 5$ and $e(G_{n}) \geq \frac{n^{2}}{4}+2$, prove that $G_{n}$ contains two triangles that share exactly one vertex.

Let $n$ be a natural number. A sequence $x_1,x_2, \cdots ,x_{n^2}$ of $n^2$ numbers is called $n-\textit{good}$ if each $x_i$ is an element of the set $\{1,2,\cdots ,n\}$ and the ordered pairs $\left(x_i,x_{i+1}\right)$ are all different for $i=1,2,3,\cdots ,n^2$ (here we consider the subscripts modulo $n^2$). Two $n-$good sequences $x_1,x_2,\cdots ,x_{n^2}$ and $y_1,y_2,\cdots ,y_{n^2}$ are called $\textit{similar}$ if there exists an integer $k$ such that $y_i=x_{i+k}$ for all $i=1,2,\cdots,n^2$ (again taking subscripts modulo $n^2$). Suppose that there exists a non-trivial permutation (i.e., a permutation which is different from the identity permutation) $\sigma$ of $\{1,2,\cdots ,n\}$ and an $n-$ good sequence $x_1,x_2,\cdots,x_{n^2}$ which is similar to $\sigma\left(x_1\right),\sigma\left(x_2\right),\cdots ,\sigma\left(x_{n^2}\right)$. Show that $n\equiv 2\pmod{4}$.

The given sequences are $ (x_1, x_2, \ldots, x_n) $, $ (y_1, y_2, \ldots, y_n) $ with positive terms. Prove that there exists a permutation $ p $ of the set $ \{1, 2, \ldots, n\} $ such that for every real $ t $ the sequence $$ (x_{p(1)}+ty_{p(1)}, x_{p(2)}+ty_{p(2)}, \ldots, x_{p(n)}+ty_{p(n) })$$ has the following property: there is a number $ k $ such that $ 1 \leq k \leq n $ and all non-zero terms of the sequence with indices less than $ k $ are of the same sign and all non-zero terms of the sequence with indices not less than $ k $ are the same sign.

A journalist wants to report on the island of scoundrels and knights, where all inhabitants are either scoundrels (and they always lie) or knights (and they always tell the truth). The journalist interviews each inhabitant exactly once and gets the following answers: $A_1$: On this island there is at least one scoundrel, $A_2$: On this island there are at least two scoundrels, $...$ $A_{n-1}$: On this island there are at least $n-1$ scoundrels, $A_n$: On this island everybody is a scoundrel. Can the journalist decide whether there are more scoundrels or more knights?

In certain country, the coins have the following values: $2^0, 2^1, 2^2,\dots 2^{10}$. A cash machine has $1000$ coins of each value and give the money using each coin(of each value) at most once. The customers order all the positive integers: $1,2,3,4,5,\dots$ (in this order) in coins. a) Determine the first integer, such that the cash machine cannot provide. b) In the moment that the first customer can not be attended, by the lack of coins, what are the coins which are not available in the cash machine?

Every phone number in an area consists of eight digits and starts with digit $ 8$. Mr Edy, who has just moved to the area, apply for a new phone number. What is the chance that Mr Edy gets a phone number which consists of at most five different digits?

$(GDR 3)$ Find the number of permutations $a_1, \cdots, a_n$ of the set $\{1, 2, . . ., n\}$ such that $|a_i - a_{i+1}| \neq 1$ for all $i = 1, 2, . . ., n - 1.$ Find a recurrence formula and evaluate the number of such permutations for $n \le 6.$

Six teams participate in a hockey tournament. Each team plays exactly once against each other team. A team is awarded $3$ points for each game they win, $1$ point for each draw, and $0$ points for each game they lose. After the tournament, a ranking is made. There are no ties in the list. Moreover, it turns out that each team (except the very last team) has exactly $2$ points more than the team ranking one place lower. Prove that the team that finished fourth won exactly two games.

Find the maximum number of Permutation of set {$1,2,3,...,2014$} such that for every 2 different number $a$ and $b$ in this set at last in one of the permutation $b$ comes exactly after $a$

A mathematical competition was attended by 120 participants from several contingents. At the closing ceremony, each participant gave 1 souvenir each to every other participants from the same contingent, and 1 souvenir to any person from every other contingents. It is known that there are 3840 souvenirs whom were exchanged. Find the maximum possible contingents such that the above condition still holds? [i]Raymond Christopher Sitorus, Singapore[/i]

Suppose a class contains $100$ students. Let, for $1\le i\le 100$, the $i^{\text{th}}$ student have $a_i$ many friends. For $0\le j\le 99$ let us define $c_j$ to be the number of students who have strictly more than $j$ friends. Show that \begin{align*} & \sum_{i=1}^{100}a_i=\sum_{j=0}^{99}c_j \end{align*}