Do there exist $5$ points in the space, such that for all $n\in\{1,2,\ldots,10\}$ there exist two of them at distance between them $n$?

Let $M_1, M_2, . . ., M_{11}$ be $5-$element sets such that $M_i \cap M_j \neq {\O}$ for all $i, j \in \{1, . . ., 11\}$. Determine the minimum possible value of the greatest number of the given sets that have nonempty intersection.

Find all natural numbers n with the following property: there exists a permutation $(i_1,i_2,\ldots,i_n)$ of the numbers $1,2,\ldots,n$ such that, if on the circular table there are $n$ people seated and for all $k=1,2,\ldots,n$ the $k$-th person is moving $i_n$ places in the right, all people will sit on different places. [i]V. Drenski[/i]

There are $18$ students in a class. Each student is asked two questions: how many other students have the same first name as you and how many other students have the same surname as you. The answers $0, 1, 2, . . ., 7$ all occur. Prove that there are two students with the same first name and last name.

Numbers $1$ to $22$ are written on a board. A "move" is a procedure of picking two numbers $a,b$ on the board such that $b \geq a+2$, then erasing $a$ and $b$ to be replaced with $a+1$ and $b-1$. Determine the maximum possible number of moves that can be done on the board.

Let $n \geq 2$ be a natural number. Find a way to assign natural numbers to the vertices of a regular $2n$-gon such that the following conditions are satisfied: (1) only digits $1$ and $2$ are used; (2) each number consists of exactly $n$ digits; (3) different numbers are assigned to different vertices; (4) the numbers assigned to two neighboring vertices differ at exactly one digit.

In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is[i] color-identifiable[/i] if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$. For all positive integers $n$ and $t$, determine the maximum integer $g(n, t)$ such that: In any sports league with exactly $n$ distinct colors present over all teams, one can always find a color-identifiable set of size at least $g(n, t)$.

At every vertex $A_k(1\le k\le n)$ of a regular $n$-gon, $k$ coins are placed. We can do the following operation: in each step, one can choose two arbitrarily coins and move them to their adjacent vertices respectively, one clockwise and one anticlockwise. Find all positive integers $n$ such that after a finite number of operations, we can reach the following configuration: there are $n+1-k$ coins at vertex $A_k$ for all $1\le k\le n$.

Nirajan is trapped in a magical dungeon. He has infinitely many magical cards with arbitrary MPs(Mana Points) which is always an integer $\mathbb{Z}$. To escape, he must give the dungeon keeper some magical cards whose MPs add up to an integer with at least $2024$ divisors. Can Nirajan always escape? [i]( Proposed by Vlad Spǎtaru, Romania)[/i]

$1991$ students sit around a circle and play the following game. Starting from some student $A$ and counting clockwise, each student on turn says a number. The numbers are $1,2,3,1,2,3,...$ A student who says $2$ or $3$ must leave the circle. The game is over when there is only one student left. What position was the remaining student sitting at the beginning of the game?

Each student in a classroom has $0,1,2,3,4,5$ or $6$ pieces of candy. At each step the teacher chooses some of the students, and gives one piece of candy to each of them and also to any other student in the classroom who is friends with at least one of these students. A student who receives the seventh piece eats all $7$ pieces. Assume that for every pair of students in the classroom, there is at least one student who is friend swith exactly one of them. Show that no matter how many pieces each student has at the beginning, the teacher can make them to have any desired numbers of pieces after finitely many steps.

Let $\pi$ be a randomly chosen permutation of the numbers from $1$ through $2012$. Find the probability that$$ \pi (\pi(2012)) = 2012.$$

Two players $A$ and $B$ play the following game: $A$ chooses a point, with integer coordinates, on the plane and colors it green, then $B$ chooses $10$ points of integer coordinates, not yet colored, and colors them yellow. The game always continues with the same rules; $A$ and $B$ choose one and ten uncolored points and color them green and yellow, respectively. a. The objective of $A$ is to achieve $111^2$ green points that are the intersections of $111$ horizontal lines and $111$ vertical lines (parallel to the coordinate axes). $B$'s goal is to stop him. Determine which of the two players has a strategy that ensures you achieve your goal. b. The objective of $A$ is to achieve $4$ green points that are the vertices of a square with sides parallel to the coordinate axes. $B$'s goal is to stop him. Determine which of the two players has a strategy that will ensure that they achieve their goal.

$p$ passengers get on a train with $n$ wagons. Find the probability of being at least one passenger at each wagon.

Let $n$ be a positive integer. Ana and Beto play a game on a $2 \times n$ board (with 2 rows and $n$ columns). First, Ana writes a digit from 1 to 9 in each cell of the board such that in each column the two written digits are different. Then, Beto erases a digit from each column. Reading from left to right, a number with $n$ digits is formed. Beto wins if this number is a multiple of $n$; otherwise, Ana wins. Determine which of the two players has a winning strategy in the following cases: $\bullet$ (a) $n = 1001$. $\bullet$ (b) $n = 1003$.

Consider an arrangement of tokens in the plane, not necessarily at distinct points. We are allowed to apply a sequence of moves of the following kind: select a pair of tokens at points $A$ and $B$ and move both of them to the midpoint of $A$ and $B$. We say that an arrangement of $n$ tokens is [i]collapsible[/i] if it is possible to end up with all $n$ tokens at the same point after a finite number of moves. Prove that every arrangement of $n$ tokens is collapsible if and only if $n$ is a power of $2$.

Find the formula to express the number of $ n\minus{}$series of letters which contain an even number of vocals (A,I,U,E,O).

prove that if graph $G$ is a tree, then there is a vertex that is common between all of the longest paths. [i]proposed by Sina Rezayi[/i]

In a dance class there are $10$ boys and $10$ girls. It is known that for each $1\le k\le 10$ and for each group of $k$ boys, the number of girls who are friends with at least one boy in the group is not less than $k$. Prove that it is possible to pair up the boys and the girls for a dance so that each pair consists of a boy and a girl who are friends.

At a party, $25$ elves give each other presents. No elf gives a present to herself. Each elf gives a present to at least one other elf, but no elf gives a present to all other elves. Show that it is possible to choose a group of three elves including at least two elves who give a present to exactly one of the other two elves in the group.

In a country there are $n$ major cities, $n \ge 4$, connected by railroads, such that each city is directly connected to each other city. Each railroad company in that country has but only one train, which connects a series of cities, at least two, such that the train doesn’t pass through the same city twice in one shift. The companies divided the market such that any two cities are directly$^1$ connected only by one company. Prove that among any $n+1$ companies, there are two which have no common train station or there are two cities that are connected by two trains belonging to two of these $n+1$ companies. $^1$ directly connected means that they are connected by a railroad, without no other station between them

Show that, in a chessboard, it is possible to traverse to any given square from another given square using a knight. (A knight can move in a chessboard by going two steps in one direction and one step in a perpendicular direction as shown in the given figure)

You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.

Using each of the eight digits $1,3,4,5,6,7,8$ and $9$ exactly once, a three-digit number $A$, two two-digit numbers $B$ and $C$, $B<C$, and a one digit number $D$ are formed. The numbers are such that $A+D=B+C=143$. In how many ways can this be done?

There are $63$ houses at the distance of $1, 2, 3, . . . , 63 \text{ km}$ from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants, $63$ elfs named $E_1, E_2, . . . , E_{63}$ , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf $E_n$ takes $n$ minutes to travel $1 \text{ km}$ for each $n = 1,2,...,63$ , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery?