a) A game for two. The first player writes two rows of ten numbers each, the second under the first. He should provide the following property: if number $b$ is written under $a$, and $d$ -- under $c$, then $a + d = b + c$. The second player has to determine all the numbers. He is allowed to ask the questions like "What number is written in the $x$ place in the $y$ row?" What is the minimal number of the questions asked by the second player before he founds out all the numbers? b) There was a table $m\times n$ on the blackboard with the property: if You chose two rows and two columns, then the sum of the numbers in the two opposite vertices of the rectangles formed by those lines equals the sum of the numbers in two another vertices. Some of the numbers are cleaned but it is still possible to restore all the table. What is the minimal possible number of the remaining numbers?

Let $F_n$ be the set of all bijective functions $f:\left\{1,2,\ldots,n\right\}\rightarrow\left\{1,2,\ldots,n\right\}$ that satisfy the conditions: 1. $f(k)\leq k+1$ for all $k\in\left\{1,2,\ldots,n\right\}$ 2. $f(k)\neq k$ for all $k\in\left\{2,3,\ldots,n\right\}$ Find the probability that $f(1)\neq1$ for $f$ randomly chosen from $F_n$.

Prove that for any integer $n\ge 6$ we can divide an equilateral triangle completely into $n$ smaller equilateral triangles.

Determine whether or nor there exist two disjoint infinite sets $ A$ and $ B$ of points in the plane satisfying the following conditions: a.) No three points in $ A \cup B$ are collinear, and the distance between any two points in $ A \cup B$ is at least 1. b.) There is a point of $ A$ in any triangle whose vertices are in $ B,$ and there is a point of $ B$ in any triangle whose vertices are in $ A.$

On the plane painted $101$ points in brown and another $101$ points in green so that there are no three lying on one line. It turns out that the sum of the lengths of all $5050$ segments with brown ends equals the length of all $5050$ segments with green ends equal to $1$, and the sum of the lengths of all $10201$ segments with multicolored equals $400$. Prove that it is possible to draw a straight line so that all brown points are on one side relative to it and all green points are on the other.

Each girl among $100$ girls has $100$ balls; there are in total $10000$ balls in $100$ colors, from each color there are $100$ balls. On a move, two girls can exchange a ball (the first gives the second one of her balls, and vice versa). The operations can be made in such a way, that in the end, each girl has $100$ balls, colored in the $100$ distinct colors. Prove that there is a sequence of operations, in which each ball is exchanged no more than 1 time, and at the end, each girl has $100$ balls, colored in the $100$ colors.

The integers $1 , 2,..., n$ are rearranged in such a way that if the integer $k, 1 \le k\le n$, is not the first term, then one of the integers $k + 1$ or $k-1$ occurs to the left of $k$ . How many arrangements of the integers $1 , 2,..., n$ satisfy this condition? (A. Andjans, Riga)

On an infinite (in both directions) strip of squares, indexed by the integers, are placed several stones (more than one may be placed on a single square). We perform a sequence of moves of one of the following types: (a) Remove one stone from each of the squares $n - 1$ and $n$ and place one stone on square $n + 1$. (b) Remove two stones from square $n$ and place one stone on each of the squares $n + 1$, $n - 2$. Prove that any sequence of such moves will lead to a position in which no further moves can be made, and moreover that this position is independent of the sequence of moves. [i]D. Fon-der-Flaas[/i]

An $8\times 8$ square is drawn on the board divided into $64$ $1\times1$ squares by lines parallel to the sides. Gustavo erases some segments of length $ 1$ so that every $1\times 1$ square he erases $0, 1$ or $2$ sides. Gustavo states that he erased $6$ segments of length $1$ from the edge of the $8\times 8$ square and that the amount of $1\times 1$ squares that have exactly $ 1$ side erased is equal to $5$. Decide if what Gustavo said it may be true.

$2 \ 000 \ 000$ points with integer coordinates are marked on the numeric axis. Segments of lengths $97$, $100$ and $103$ with ends at these points are considered. What is the largest number of such segments?

In a football tournament, $n$ teams play one round ($n \vdots 2$). In each round should play $n / 2$ pairs of teams that have not yet played. Schedule of each round takes place before its holding. For which smallest natural $k$ such that the following situation is possible: after $k$ tours, making a schedule of $k + 1$ rounds already is not possible, i.e. these $n$ teams cannot be divided into $n / 2$ pairs, in each of which there are teams that have not played in the previous $k$ rounds. PS. The 3 vertical dots notation in the first row, I do not know what it means.

Let $n$ be a positive integer. The Georgian folk dance team consists of $2n$ dancers, with $n$ males and $n$ females. Each dancer, both male and female, is assigned a number from 1 to $n$. During one of their dances, all the dancers line up in a single line. Their wish is that, for every integer $k$ from 1 to $n$, there are exactly $k$ dancers positioned between the $k$th numbered male and the $k$th numbered female. Prove the following statements: a) If $n \equiv 1 \text{ or } 2 \mod{4}$, then the dancers cannot fulfill their wish. b) If $n \equiv 0 \text{ or } 3 \mod{4}$, then the dancers can fulfill their wish. [i]Proposed by Giorgi Arabidze, Georgia[/i]

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?

Claudia and Adela are betting to see which one of them will ask the boy they like for his telephone number. To decide they roll dice. If none of the dice are a multiple of 3, Claudia will do it. If exactly one die is a multiple of 3, Adela will do it. If 2 or more of the dice are a multiple of 3 neither one of them will do it. How many dice should be rolled so that the risk is the same for both Claudia and Adela?

There are $2n+1$ segments on the line. Any segment intersects at with at least $n$ others. Prove that there is a segment that intersects all the others.

We play the following game in a Cartesian coordinate system in the plane. Given the input $(x,y)$, in one step, we may move to the point $(x,y\pm 2x)$ or to the point $(x\pm 2y,y)$. There is also an additional rule: it is not allowed to make two steps that lead back to the same point (i.e, to step backwards). Prove that starting from the point $\left(1;\sqrt 2\right)$, we cannot return to it in finitely many steps.

$k,n$ are two arbitrary positive integers. Prove that there exists at least $(k-1)(n-k+1)$ positive integers that can be produced by $n$ number of $k$'s and using only $+,-,\times, \div$ operations and adding parentheses between them, but cannot be produced using $n-1$ number of $k$'s. [i]Proposed by Aryan Tajmir[/i]

There are $n$ students $ A_1 , A_2 , \cdots , A_n $ and some of them shaked hands with each other. ($ A_i $ and $ A_j$ can shake hands more than one time.) Let the student $ A_i $ shaked hands $ d_i $ times. Suppose $ d_1 + d_2 + \cdots + d_n > 0 $. Prove that there exist $ 1 \le i < j \le n $ satisfying the following conditions: (a) Two students $ A_i $ and $ A_j $ shaked hands each other. (b) $ \frac{(d_1 + d_2 + \cdots + d_n ) ^2 }{n^2 } \le d_i d_j $

[b]p1.[/b] To every face of a given cube a new cube of the same size is glued. The resulting solid has how many faces? [b]p2.[/b] A father and his son returned from a fishing trip. To make their catches equal the father gave to his son some of his fish. If, instead, the son had given his father the same number of fish, then father would have had twice as many fish as his son. What percent more is the father's catch more than his son's? [b]p3.[/b] A radio transmitter has $4$ buttons. Each button controls its own switch: if the switch is OFF the button turns it ON and vice versa. The initial state of switches in unknown. The transmitter sends a signal if at least $3$ switches are ON. What is the minimal number of times you have to push the button to guarantee the signal is sent? [b]p4.[/b] $19$ matches are placed on a table to show the incorrect equation: $XXX + XIV = XV$. Move exactly one match to change this into a correct equation. [b]p5.[/b] Cut the grid shown into two parts of equal area by cutting along the lines of the grid. [img]https://cdn.artofproblemsolving.com/attachments/c/1/7f2f284acf3709c2f6b1bea08835d2fb409c44.png[/img] [b]p6.[/b] A family of funny dwarfs consists of a dad, a mom, and a child. Their names are: $A$, $R$, and $C$ (not in order). During lunch, $C$ made the statements: “$R$ and $A$ have different genders” and “$R$ and $A$ are my parents”, and $A$ made the statements “I am $C$'s dad” and “I am $R$'s daughter.” In fact, each dwarf told truth once and told a lie once. What is the name of the dad, what is the name of the child, and is the child a son or a daughter? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose in the plane 10 pairwise nonparallel lines intersect one another. What is the maximum possible number of polygons (with finite areas) that can be formed?

In a country, there are $2018$ cities, some of which are connected by roads. Each city is connected to at least three other cities. It is possible to travel from any city to any other city using one or more roads. For each pair of cities, consider the shortest route between these two cities. What is the greatest number of roads that can be on such a shortest route?

The cells of an $n\times n$ board with $n\ge 5$ are coloured black or white so that no three adjacent squares in a row, column or diagonal are the same colour. Show that for any $3\times 3$ square within the board, two of its corner squares are coloured black and two are coloured white.

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 $a_i=N$).

In a game, two players control an adventurer in a dungeon. The adventurer starts with $H{}$ hit points, an integer greater than one. The dungeon consists of several chambers. There are some passageways in the dungeon, each leading from a chamber to a chamber. These passageways are one-way, and a passageway may return to its starting chamber. Every chamber can be exited through at least one passageway. There are 5 types of chambers: [list] [*]Entrance: the adventurer starts here, no passageway comes in here; [*]Hollow: nothing happens; [*]Spike: the adventurer loses a hit point; [*]Trap: the adventurer gets shot by an arrow; [*]Catacomb: the adventurer loses hit points equal to the total number of times they have been hit by an arrow. [/list] The two players control the adventurer alternatively. At a turn, a player can move him through one passageway. A player loses if the adventurer’s hit points fall below zero due to their action (at 0 hit points, the character is alive). Construct a dungeon map, which has at most 20 chambers in total and exactly one entrance, with the following condition: the first player has a winning strategy if $H{}$ is a prime, and the second player has a winning strategy if $H{}$ is composite. [i]Note: If the game doesn’t end after a finite number of moves, neither player wins.[/i]

A $9\times 1$ rectangle is divided into unit squares. A broken line, from the lower left to the upper right corner, goes through all $20$ vertices of the unit squares and consists of $19$ line segments. How many such lines are there?