There are six ports on a lake. Is it possible to organize a series of routes satisfying the following conditions ? [i](i)[/i] Every route includes exactly three ports; [i](ii)[/i] No two routes contain the same three ports; [i](iii)[/i] The series offers exactly two routes to each tourist who desires to visit two different arbitrary ports.

Given some $m\times n$ table, and some numbers in its fields. You are allowed to change the sign in one row or one column simultaneously. Prove that you can obtain a table, with nonnegative sums over each row and over each column.

For even positive integer $n$ we put all numbers $1, 2, \cdots , n^2$ into the squares of an $n \times n$ chessboard (each number appears once and only once). Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that it is possible to have $\frac{S_1}{S_2}=\frac{39}{64}$.

Every positive integer is marked with a number from the set $\{ 0,1,2\}$, according to the following rule: $$\text{if a positive integer }k\text{ is marked with }j,\text{ then the integer }k+j\text{ is marked with }0.$$ Let $S$ denote the sum of marks of the first $2019$ positive integers. Determine the maximum possible value of $S$. [i]Proposed by Ivan Novak[/i]

Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine the minimum number of integers in a complete sequence of $n$ numbers.

For any integer $n\geq1$, we consider a set $P_{2n}$ of $2n$ points placed equidistantly on a circle. A [i]perfect matching[/i] on this point set is comprised of $n$ (straight-line) segments whose endpoints constitute $P_{2n}$. Let $\mathcal{M}_{n}$ denote the set of all non-crossing perfect matchings on $P_{2n}$. A perfect matching $M\in \mathcal{M}_{n}$ is said to be [i]centrally symmetric[/i], if it is invariant under point reflection at the circle center. Determine, as a function of $n$, the number of centrally symmetric perfect matchings within $\mathcal{M}_{n}$. [i]Proposed by Mirko Petrusevski[/i]

Maisy is at the origin of the coordinate plane. On her first step, she moves $1$ unit up. On her second step, she moves $ 1$ unit to the right. On her third step, she moves $2$ units up. On her fourth step, she moves $2$ units to the right. She repeats this pattern with each odd-numbered step being $ 1$ unit more than the previous step. Given that the point that Maisy lands on after her $21$st step can be written in the form $(x, y)$, find the value of $x + y$. Proposed by Audrey Chun

Let $A_1,A_2,...$ be a sequence of sets such that for any positive integer $i$, there are only finitely many values of $j$ such that $A_j\subseteq A_i$. Prove that there is a sequence of positive integers $a_1,a_2,...$ such that for any pair $(i,j)$ to have $a_i\mid a_j\iff A_i\subseteq A_j$.

In CMI, each person has atmost $3$ friends. A disease has infected exactly $2023$ peoplein CMI . Each day, a person gets infected if and only if atleast two of their friends were infected on the previous day. Once someone is infected, they can neither die nor be cured. Given that everyone in CMI eventually got infected, what is the maximum possible number of people in CMI?

Indicate whether it is possible to write the integers $1, 2, 3, 4, 5, 6, 7, 8$ at the vertices of an regular octagon such that the sum of the numbers of any $3$ consecutive vertices is greater than: a) $11$. b) $13$.

Cards numbered 1 to 9 are arranged at random in a row. In a move, one may choose any block of consecutive cards whose numbers are in ascending or descending order, and switch the block around. For example, 9 1 $\underline{6\ 5\ 3}$ $2\ 7\ 4\ 8$ may be changed to $9 1$ $\underline{3\ 5\ 6}$ $2\ 7\ 4\ 8$. Prove that in at most 12 moves, one can arrange the 9 cards so that their numbers are in ascending or descending order.

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

Every cell of a $m \times n$ chess board, $m\ge 2,n\ge 2$, is colored with one of four possible colors, e.g white, green, red, blue. We call such coloring good if the four cells of any $2\times 2$ square of the chessboard are colored with pairwise different colors. Determine the number of all good colorings of the chess board. [i]Proposed by N. Beluhov[/i]

Three integers are written on a blackboard. At every step one of them is erased and the sum of the other two decreased by $1$ is written instead. Is it possible to obtain the numbers $17,75,91$ if the three initial numbers were: $\textbf{(a)}~2,2,2$; $\textbf{(b)}~3,3,3$?

On the board there are written in a row, the integers from $1$ until $2030$ (included that) in an increasing order. We have the right of ''movement'' $K$: [i]We choose any two numbers $a,b$ that are written in consecutive positions and we replace the pair $(a,b)$ by the number $(a-b)^{2020}$.[/i] We repeat the movement $K$, many times until only one number remains written on the board. Determine whether it would be possible, that number to be: (i) $2020^{2020}$ (ii)$2021^{2020}$

Start with a stack of $100$ cards. Now repeat the following: choose a stack of at least $2$ cards and split them into two smaller piles (at least $1$ card of each). Continue this until there are finally $100$ stacks of $1$ card each. Every time you split a pile into two stacks you get a number of points that is equal to the product of the number of cards in the two new stacks. What is the maximum number of points that you can earn in total?

In a country there are $2014$ airports, no three of them lying on a line. Two airports are connected by a direct flight if and only if the line passing through them divides the country in two parts, each with $1006$ airports in it. Show that there are no two airports such that one can travel from the first to the second, visiting each of the $2014$ airports exactly once.

It is possible to write the numbers $111$, $112$, $121$, $122$, $211$, $212$, $221$ and $222$ at the vertices of a cube, so that the numbers written in adjacent vertices match at most in one digit?

Nair has puzzle pieces shaped like an equilateral triangle. She has pieces of two sizes: large and small. [img]https://cdn.artofproblemsolving.com/attachments/a/1/aedfbfb2cb17bf816aa7daeb0d35f46a79b6e9.jpg[/img] Nair build triangular figures by following these rules: $\bullet$ Figure $1$ is made up of $4$ small pieces, Figure $2$ is made up of $2$ large pieces and $8$ small, Figure $3$ by $6$ large and $12$ small, and so on. $\bullet$ The central column must be made up exclusively of small parts. $\bullet$ Outside the central column, only large pieces can be placed. [img]https://cdn.artofproblemsolving.com/attachments/5/7/e7f6340de0e04d5b5979e72edd3f453f2ac8a5.jpg[/img] Following the pattern, how many pieces will Nair use to build Figure $20$?

The map of a country is in the form of a convex polygon with $n (n\geq5)$ sides, such as any 3 diagonals do not concur inside the polygon. Some of the diagonals are roads, which allow movement in both directions and the other diagonals are not roads. There are cities on the intersection points of any two diagonals inside the polygon and at least one of the two diagonals is a road. Prove that you can move from any city to any other city using at most 3 roads.

An infinite sequence of digits $1$ and $2$ is determined by the following two properties: i) The sequence is built by writing, in some order, blocks $12$ and blocks $112.$ ii) If each block $12$ is replaced by $1$ and each block $112$ by $2$, the same sequence is again obtained. In which position is the hundredth digit $1$? What is the thousandth digit of the sequence?

Three coins lie on integer points on the number line. A move consists of choosing and moving two coins, the first one $ 1$ unit to the right and the second one $ 1$ unit to the left. Under which initial conditions is it possible to move all coins to one single point?

Find all elements $n \in A = \{2,3,...,2016\} \subset \mathbb{N}$ such that: every number $m \in A$ smaller than $n$, and coprime with $n$, must be a prime number

Let $ T$ be a triangulation of an $ n$-dimensional sphere, and to each vertex of $ T$ let us assign a nonzero vector of a linear space $ V$. Show that if $ T$ has an $ n$-dimensional simplex such that the vectors assigned to the vertices of this simplex are linearly independent, then another such simplex must also exist. [i]L. Lovasz[/i]

Let $n > 1$ be an odd natural number. The squares of an $n \times n$ chessboard are alternately colored white and black so that the four corner squares are black. An $L$-triomino is an $L$-shaped piece that covers exactly three squares of the board. For which values ​​of $n$ is it possible to cover all black squares with $L$-triominoes, so that no two $L$-triominos overlap? For these values ​​of $n$ determine the smallest possible number of $L$-triominoes that are necessary for this.