Let $n$ be a positive integer. There is a pawn in one of the cells of an $n\times n$ table. The pawn moves from an arbitrary cell of the $k$th column, $k \in \{1,2, \cdots, n \}$, to an arbitrary cell in the $k$th row. Prove that there exists a sequence of $n^{2}$ moves such that the pawn goes through every cell of the table and finishes in the starting cell.

Let $k$ be a positive integer. At the European Chess Cup every pair of players played a game in which somebody won (there were no draws). For any $k$ players there was a player against whom they all lost, and the number of players was the least possible for such $k$. Is it possible that at the Closing Ceremony all the participants were seated at the round table in such a way that every participant was seated next to both a person he won against and a person he lost against. [i]Proposed by Matija Bucić.[/i]

There are $2020$ groups, each of which consists of a boy and a girl, such that each student is contained in exactly one group. Suppose that the students shook hands so that the following conditions are satisfied: [list] [*] boys didn't shake hands with boys, and girls didn't shake hands with girls; [*] in each group, the boy and girl had shake hands exactly once; [*] any boy and girl, each in different groups, didn't shake hands more than once; [*] for every four students in two different groups, there are at least three handshakes. [/list] Prove that one can pick $4038$ students and arrange them at a circular table so that every two adjacent students had shake hands.

Let $n$ be a positive integer. There is a pawn in one of the cells of an $n\times n$ table. The pawn moves from an arbitrary cell of the $k$th column, $k \in \{1,2, \cdots, n \}$, to an arbitrary cell in the $k$th row. Prove that there exists a sequence of $n^{2}$ moves such that the pawn goes through every cell of the table and finishes in the starting cell.

In the unit squares of a transparent $1 \times 100$ tape, numbers $1,2,\cdots,100$ are written in the ascending order.We fold this tape on it's lines with arbitrary order and arbitrary directions until we reach a $1 \times1$ tape with $100$ layers.A permutation of the numbers $1,2,\cdots,100$ can be seen on the tape, from the top to the bottom. Prove that the number of possible permutations is between $2^{100}$ and $4^{100}$. ([i]e.g.[/i] We can produce all permutations of numbers $1,2,3$ with a $1\times3$ tape) [i]Proposed by Morteza Saghafian[/i]

In the unit squares of a transparent $1 \times 100$ tape, numbers $1,2,\cdots,100$ are written in the ascending order.We fold this tape on it's lines with arbitrary order and arbitrary directions until we reach a $1 \times1$ tape with $100$ layers.A permutation of the numbers $1,2,\cdots,100$ can be seen on the tape, from the top to the bottom. Prove that the number of possible permutations is between $2^{100}$ and $4^{100}$. ([i]e.g.[/i] We can produce all permutations of numbers $1,2,3$ with a $1\times3$ tape) [i]Proposed by Morteza Saghafian[/i]

Determine all graphs $G$ with the following two properties: $\bullet$ G contains at least one Hamilton path. $\bullet$ For any pair of vertices, $u, v \in G$, if there is a Hamilton path from $u$ to $v$ then the edge $uv$ is in the graph $G$

There are two airlines A and B and finitely many airports. For each pair of airports, there is exactly one airline among A and B whose flights operates in both directions. Each airline plans to develop world travel packages which pass each airport exactly once using only its flights. Let $a$ and $b$ be the number of possible packages which belongs to A and B respectively. Prove that $a-b$ is a multiple of $4$. The official statement of the problem has been changed. The above is the form which appeared during the contest. Now the condition 'the number of airports is no less than 4'is attached. Cite the following link. [url]https://artofproblemsolving.com/community/c6h2923697p26140823[/url]

There are $2024$ cities in a country, some pairs of which are connected by bidirectional flights. For any distinct cities $A, B, C, X, Y, Z$, it is possible to fly directly from some of the cities $A, B, C$ to some of the cities $X, Y, Z$. Prove that it is possible to plan a route $T_1\to T_2 \to \ldots \to T_{2022}$ that passes through $2022$ distinct cities. [i]Proposed by Lior Shayn[/i]