Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

Let \( n \geq 4 \) be an integer. Show that at a party of \( n \) people, it is possible for each person to have greeted exactly three other people if and only if \( n \) is even.

There are several counters of various colours and sizes. No two of them have, simultaneously, the same colour and the same size. On each counter $F$ two numbers are written. One of them is the number of counters that have the same colour as $F$ but a different size than $F$. The other number is the number of counters that have the same size as $F$ but a different colour. It is known that each of the $101$ numbers $0,1,\ldots,100$ is written at least once. Determine the smallest number of counters for which this is possible.

Is it possible to fill a $3 \times 3$ grid with each of the numbers $1,2,\ldots,9$ once each such that the sum of any two numbers sharing a side is prime?

What is the least positive integer $k$ such that, in every convex $101$-gon, the sum of any $k$ diagonals is greater than or equal to the sum of the remaining diagonals?

Suppose $F$ is a family of finite subsets of $\mathbb{N}$ and for any 2 sets $A,B \in F$ we have $A \cap B \not= \O$. (a) Is it true that there is a finite subset $Y$ of $\mathbb{N}$ such that for any $A,B \in F$ we have $A\cap B\cap Y \not= \O$? (b) Is the above true if we assume that all members of $F$ have the same size?

Let $n$ be a natural number. Given a chessboard sized $m \times n$. The sides of the small squares of chessboard are not on the perimeter of the chessboard will be colored so that each small square has exactly two sides colored. Prove that a coloring like that is possible if and only if $m \cdot n$ is even.

Let $n$ be a positive integer. We are given a regular $4n$-gon in the plane. We divide its vertices in $2n$ pairs and connect the two vertices in each pair by a line segment. What is the maximal possible amount of distinct intersections of the segments?

In a meeting of $4042$ people, there are $2021$ couples, each consisting of two people. Suppose that $A$ and $B$, in the meeting, are friends when they know each other. For a positive integer $n$, each people chooses an integer from $-n$ to $n$ so that the following conditions hold. (Two or more people may choose the same number). [list] [*] Two or less people chose $0$, and if exactly two people chose $0$, they are coupled. [*] Two people are either coupled or don't know each other if they chose the same number. [*] Two people are either coupled or know each other if they chose two numbers that sum to $0$. [/list] Determine the least possible value of $n$ for which such number selecting is always possible.

A planar country has an odd number of cities separated by pairwise distinct distances. Some of these cities are connected by direct two-way flights. Each city is directly connected to exactly two ther cities, and the latter are located farthest from it. Prove that, using these flights, one may go from any city to any other city

$101$ distinct numbers are chosen among the integers between $0$ and $1000$. Prove that, among the absolute values ​​of their pairwise differences, there are ten different numbers not exceeding $100$.

Five points are chosen uniformly at random on a segment of length $1$. What is the expected distance between the closest pair of points?

There are given the integers $1 \le m < n$. Consider the set $M = \{ (x,y);x,y \in \mathbb{Z_{+}}, 1 \le x,y \le n \}$. Determine the least value $v(m,n)$ with the property that for every subset $P \subseteq M$ with $|P| = v(m,n)$ there exist $m+1$ elements $A_{i}= (x_{i},y_{i}) \in P, i = 1,2,...,m+1$, for which the $x_{i}$ are all distinct, and $y_{i}$ are also all distinct.

In the training of a state, the coach proposes a game. The coach writes four real numbers on the board in order from least to greatest: $a < b < c < d$. Each Olympian draws the figure on the right in her notebook and arranges the numbers inside the corner shapes, however she wants, putting a number on each one. Once arranged, on each segment write the square of the difference of the numbers at its ends. Then, add the $4$ numbers obtained. [img]https://cdn.artofproblemsolving.com/attachments/9/a/ea348c637ae266c908e0b97e64605808b3b1d2.png[/img] For example, if Vania arranges them as in the figure on the right, then the result would be $$ (c - b)^2 + (b- a)^2 + (a - d)^2 + (d - c)^2.$$ [img]https://cdn.artofproblemsolving.com/attachments/8/b/9c5375d66a4a6344b2bce333534fa7fac2ad6c.png[/img] The Olympians with the lowest result win. In what ways can you arrange the numbers to win? Give all the possible solutions.

The set $S$ is a subset of $\{1, 2, \dots, 2025\}$ such that no two elements of $S$ differ by $2$ or by $7$. What is the largest number of elements that $S$ can have?

A regular $1997$-gon is divided into triangles by non-intersecting diagonals. Prove that exactly one of them is acute-angled.

Initially, the point-like particles $A, B$ and $C{}$ are located respectively at the points $(0,0), (1,0)$ and $(0,1)$ in the coordinate plane. Every minute some two particles repel each other along the straight line connecting their current positions, moving the same (positive) distance. [list=a] [*]Can the particle $A{}$ be at the point $(3,3)$? What about the point $(2,3)$? [*]Can the particles $B{}$ and $C{}$ be at the same time at the points $(0,100)$ and $(100,0)$ respectively? [/list] [i]Proposed by K. Krivosheev[/i]

There are 2012 lamps arranged on a table. Two persons play the following game. In each move the player flips the switch of one lamp, but he must never get back an arrangement of the lit lamps that has already been on the table. A player who cannot move loses. Which player has a winning strategy?

Points on a spherical surface with radius $4$ are colored in $4$ different colors. Prove that there exist two points with the same color such that the distance between them is either $4\sqrt{3}$ or $2\sqrt{6}$. (Distance is Euclidean, that is, the length of the straight segment between the points)

Consider the following operation on positive real numbers written on a blackboard: Choose a number $ r$ written on the blackboard, erase that number, and then write a pair of positive real numbers $ a$ and $ b$ satisfying the condition $ 2 r^2 \equal{} ab$ on the board. Assume that you start out with just one positive real number $ r$ on the blackboard, and apply this operation $ k^2 \minus{} 1$ times to end up with $ k^2$ positive real numbers, not necessarily distinct. Show that there exists a number on the board which does not exceed kr.

Given an integer $ k > 1.$ We call a $ k \minus{}$digits decimal integer $ a_{1}a_{2}\cdots a_{k}$ is $ p \minus{}$monotonic, if for each of integers $ i$ satisfying $ 1\le i\le k \minus{} 1,$ when $ a_{i}$ is an odd number, $ a_{i} > a_{i \plus{} 1};$ when $ a_{i}$ is an even number, $ a_{i}<a_{i \plus{} 1}.$ Find the number of $ p \minus{}$monotonic $ k \minus{}$digits integers.

The set $\{1, 2, 3, 4\}$ can be partitioned into two subsets $A = \{1, 4\}$ and $B = \{3, 2\}$ with no common elements and such that the sum of the elements of $A$ is equal to the sum of the elements of B. Such a partition is impossible for the set $\{1, 2, 3, 4, 5\}$ and also for the set $\{1, 2, 3, 4, 5, 6\}$. Determine all values of $n$ for which the set of the first $n$ natural numbers can be partitioned into two subsets with no common elements such that the sum of the elements of each subset is the same.

In Skinien there 2009 towns where each of them is connected with exactly 1004 other town by a highway. Prove that starting in an arbitrary town one can make a round trip along the highways such that each town is passed exactly once and finally one returns to its starting point.

[b]Problem 2[/b] : $k$ number of unit squares selected from a $99 \times 99$ square grid are coloured using five colours Red, Blue, Yellow, Green and Black such that each colour appears the same number of times and on each row and on each column there are no differently coloured unit squares. Find the maximum possible value of $k$.

Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: [list=1] [*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. [*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell. [/list] At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. [i]Proposed by Warut Suksompong, Thailand[/i]