Given a set of $4n$ positive numbers such that any distinct choice of ordered foursomes of these numbers constitutes a geometric progression. Prove that at least $4$ numbers of the set are identical.

There is $n\times n$ chessboard. Each square has a number between $0$ and $k$. There is a button for each row and column, which increases the number of $n$ numbers of the row or column the button represents(if the number of the square is $k$, then it becomes $0$). If certain button is pressed, call it 'operation.' And we have a chessboard which is filled with 0(for all squares). After some 'operation's, the numbers of squares are different now. Prove that we can make all of the number $0$ within $kn$ 'operation's.

Sofía has many boxes where she keeps candies. Every morning, she chooses two of these boxes and places one candy in each. However, each night, a thief selects one box and steals all the candies inside it. Sofía dreams of waking up one day and finding a box with 2024 candies. Prove that Sofía can always fulfill her dream if she has enough boxes.

In a $m\times n$ chessboard ($m,n\ge 2$), some dominoes are placed (without overlap) with each domino covering exactly two adjacent cells. Show that if no more dominoes can be added to the grid, then at least $2/3$ of the chessboard is covered by dominoes. [i]Proposed by DVDthe1st, mzy and jjax[/i]

$3n$ points are given ($n\ge 1$) in the plane, each $3$ of them are not collinear. Prove that there are $n$ distinct triangles with the vertices those points.

Each person stands on a whole number on the number line from $0$ to $2022$ . In each turn, two people are selected by a distance of at least $2$. These go towards each other by $1$. When no more such moves are possible, the process ends. Show that this process always ends after a finite number of moves, and determine all possible configurations where people can end up standing. (whereby is for each configuration is only of interest how many people stand at each number.) [i](Birgit Vera Schmidt)[/i] [hide=original wording]Bei jeder ganzen Zahl auf dem Zahlenstrahl von 0 bis 2022 steht zu Beginn eine Person. In jedem Zug werden zwei Personen mit Abstand mindestens 2 ausgewählt. Diese gehen jeweils um 1 aufeinander zu. Wenn kein solcher Zug mehr möglich ist, endet der Vorgang. Man zeige, dass dieser Vorgang immer nach endlich vielen Zügen endet, und bestimme alle möglichen Konfigurationen, wo die Personen am Ende stehen können. (Dabei ist für jede Konfiguration nur von Interesse, wie viele Personen bei jeder Zahl stehen.)[/hide]

The schoolchildren sat down on chairs located in transverse and longitudinal rows. The tallest student was chosen from each transverse row, and the lowest was chosen among them. Then the lowest student was selected from each longitudinal row, and the tallest was chosen among them. Which of these two students is higher?

Let $N$ be the number of ways to place $4$ bishops on a $5 \times 5$ chessboard such that no $3$ are on the same diagonal. Find the remainder when $N$ is divided by $100$. (Note: the length of a diagonal on a $5 \times 5$ chessboard can be 2, 3, 4, or 5.)

We consider the $n$ vertices of a regular polygon with $n$ sides. There is a set of triangles with vertices at these $n$ points with the property that for each triangle in the set, the sides of at least one are not the side of any other triangle in the set. What is the largest amount of triangles that can have the set? [hide=original wording]Consideramos los n vértices de un polígono regular de n lados. Se tiene un conjunto de triángulos con vértices en estos n puntos con la propiedad que para cada triángulo del conjunto, al menos uno de sus lados no es lado de ningún otro triángulo del conjunto. ¿Cuál es la mayor cantidad de triángulos que puede tener el conjunto?[/hide]

The numbers from $1$ to $2n$ are arranged in a certain order in the cell of the strip $1 \times 2n$. Let's call a [i]flip[/i] an operation that takes one cell from the left half of the strip and one cell from the right half of the strip, after which it swaps the numbers written in them, but only if the larger of these numbers is located to the left of the smaller one. Prove that if all $n^2$ possible flips are performed in any order, then all numbers from $1$ to $n$ will be written in the left, and all numbers from $n + 1$ up to $2n$ — in the right half of the strip.

Let $U$ be a set of $m$ triangles. Prove that there exists a subset $W$ of $U$ which satisfies the following. (i). The number of triangles in $W$ is at least $0.45m^{\frac{4}{5}}$ (ii) There are no points $A, B, C, D, E, F$ such that triangles $ABC$, $BCD$, $CDE$, $DEF$, $EFA$, $FAB$ are all in $W$.

How many integers belong to ($a,2008a$), where $a$ ($a > 0$) is given.

The house SEAT recommends to the users, for the correct conservation of the wheels, periodic replacements of the same in the form $R \to 3 \to 2 \to 1 \to 4 \to R$, according to the numbering of the figure. Calling $G$ to this change of wheels, $G^2 = GG$ to making this change twice, and so on for the other powers of $G$, a) Show that the set of these powers forms a group, and study it. b) Each puncture of one of the wheels is also equivalent to a substitution in which said wheel is replaced by the spare one $ (R)$ and, once repaired, it comes to occupy the place of this obtained $G$ as a product of prick transformations. Do they form a group? [img]https://cdn.artofproblemsolving.com/attachments/4/a/712fede88321c67753417fda828a08ba528b4f.png[/img]

At Port Aventura there are $16$ secret agents, each of whom is watching one or more other agents. It is known that if agent $A$ is watching agent $B$, then $B$ is not watching $A$. Moreover, any $10$ agents can be ordered so that the first is watching the second, the second is watching the third, etc, the last is watching the first. Show that any $11$ agents can also be so ordered.

The squares of an $8 \times 8$ board are colored black and white in such a way that every row and every column contains exactly four black squares. Prove that the number of pairs of neighboring white squares is the same as the number of pairs of neighboring black squares. (Two squares are neighboring if they have a side in common.)

A pupil is writing on a board positive integers $x_0,x_1,x_2,x_3...$ after the following algorithm which implies arithmetic progression $3,5,7,9...$.Each term of rank $k\ge2$ is a difference between the product of the last number on the board and the term of arithmetic progression of rank $k$ and the last but one term on the bord with the sum of the terms of the arithemtic progression with ranks less than $k$.If $x_0=0 $ and $x_1=1$ find $x_n$ according to n.

The cells of a $100 \times 100$ table are colored white. In one move, it is allowed to select some $99$ cells from the same row or column and recolor each of them with the opposite color. What is the smallest number of moves needed to get a table with a chessboard coloring? [i]S. Berlov[/i]

We are given two positive integers $p$ and $q$. Step by step, a rope of length $1$ is cut into smaller pieces as follows: In each step all the currently longest pieces are cut into two pieces with the ratio $p:q$ at the same time. After an unknown number of such operations, the currently longest pieces have the length $x$. Determine in terms of $x$ the number $a(x)$ of different lengths of pieces of rope existing at that time.

Let $m$ and $n$ be positive integers. A circular necklace contains $mn$ beads, each either red or blue. It turned out that no matter how the necklace was cut into $m$ blocks of $n$ consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair $(m, n)$. [i]Proposed by Rishabh Das[/i]

There are $169$ non-zero digits written around a circle. Prove that they can be split into $14$ non-empty blocks of consecutive digits so that among the $14$ natural numbers formed by the digits in those blocks, at least $13$ of them are divisible by $13$ (the digits in each block are read in clockwise direction).

Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and all $i \in \{1, 2, \ldots, k\}$ there exist two distinct elements of $A_i$ whose sum is $n.$ [i]Proposed by Igor Voronovich, Belarus[/i]

Aidan rolls a pair of fair, six sided dice. Let$ n$ be the probability that the product of the two numbers at the top is prime. Given that $n$ can be written as $a/b$ , where $a$ and $b$ are relatively prime positive integers, find $a +b$. [i]Proposed by Aidan Duncan[/i]

Let $k\leq n$ be two positive integers. $n$ points are marked on a line. It is known that for each marked point, the number of marked points at a distance $\leq 1$ from it (including the point itself) is divisible by $k$. Show that $k$ divides $n$ (without remainder).

Marko lives on the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\tfrac{2}{9}$, $1$ unit right with probability $\tfrac{2}{9}$, $1$ unit up and $1$ unit right with probability $\tfrac{4}{9}$, and he doesn’t move with probability $\tfrac{1}{9}$. After $2019$ seconds, Marko ends up on the point $(A, B)$. What is the expected value of $A\cdot B$?

There is an ant on every vertex of a unit cube. At the time zero, ants start to move across the edges with the velocity of one unit per minute. If an ant reaches a vertex, it alternatively turns right and left (for the first time it will turn in a random direction). If two or more ants meet anywhere on the cube, they die! We know an ant survives after three minutes. Prove that there exists an ant that never dies!