Let $n$ be a positive integer. There is an infinite number of cards, each one of them having a non-negative integer written on it, such that for each integer $l \geq 0$, there are exactly $n$ cards that have the number $l$ written on them. A move consists of picking $100$ cards from the infinite set of cards and discarding them. Find the least possible value of $n$ for which there is an infinitely long series of moves such that for each positive integer $k$, the sum of the numbers written on the $100$ chosen cards during the $k$-th move is equal to $k$.

The [i][b]duchess[/b][/i] is a chess piece such that the duchess attacks all the cells in two of the four diagonals which she is contained(the directions of the attack can vary to two different duchesses). Determine the greatest integer $n$, such that we can put $n$ duchesses in a table $8\times 8$ and none duchess attacks other duchess. Note: The attack diagonals can be "outside" the table; for instance, a duchess on the top-leftmost cell we can choose attack or not the main diagonal of the table $8\times 8$.

An apartment building consists of 20 rooms number 1, 2,..., 20 arranged clockwise in a circle. To move from one room to another, one can either walk to the next room clockwise (i.e. from room $i$ to room $(i+1)\pmod{20}$) or walk across the center to the opposite room (i.e. from room $i$ to room $(i+10)\pmod{20}$). Find the number of ways to move from room 10 to room 20 without visiting the same room twice.

A table has $3 000 000$ rows and $100$ columns, divided into unit squares. Each row contains the numbers from $1$ to $100$, each exactly once, and no two rows are the same. Above each column, the number of distinct entries in that column is written in red. Find the smallest possible sum of the red numbers.

A [i]triminó[/i] is a rectangular tile of $1\times 3$. Is it possible to cover a $8\times8$ chessboard using $21$ triminós, in such a way there remains exactly one $1\times 1$ square without covering? In case the answer is in the affirmative, determine all the possible locations of such a unit square in the chessboard.

A chessboard $1000 \times 1000$ is covered by dominoes $1 \times 10$ that can be rotated. We don't know which is the cover, but we are looking for it. For this reason, we choose a few $N$ cells of the chessboard, for which we know the position of the dominoes that cover them. Which is the minimum $N$ such that after the choice of $N$ and knowing the dominoed that cover them, we can be sure and for the rest of the cover? (Bulgaria)

In a circle of diameter $1$ consider $65$ points, no three of them collinear. Prove that there exist three among these points which are the vertices of a triangle with area less than or equal to $\frac{1}{72}$.

Suppose we have a $n$-gon. Some $n-3$ diagonals are coloured black and some other $n-3$ diagonals are coloured red (a side is not a diagonal), so that no two diagonals of the same colour can intersect strictly inside the polygon, although they can share a vertex. Find the maximum number of intersection points between diagonals coloured differently strictly inside the polygon, in terms of $n$. [i]Proposed by Alexander Ivanov, Bulgaria[/i]

$1993$ points are arranged in a circle. At time $0$ each point is arbitrarily labeled $+1$ or $-1$. At times $n = 1, 2, 3, ...$ the vertices are relabeled. At time $n$ a vertex is given the label $+1$ if its two neighbours had the same label at time $n-1$, and it is given the label $-1$ if its two neighbours had different labels at time $n-1$. Show that for some time $n > 1$ the labeling will be the same as at time $1.$

Given a $n\times m$ grid we play the following game . Initially we place $M$ tokens in each of $M$ empty cells and at the end of the game we need to fill the whole grid with tokens.For that purpose we are allowed to make the following move:If an empty cell shares a common side with at least two other cells that contain a token then we can place a token in this cell.Find the minimum value of $M$ in terms of $m,n$ that enables us to win the game.

Austin is at the Lincoln Airport. He wants to take $5$ successive flights whose destinations are randomly chosen among Indianapolis, Jackson, Kansas City, Lincoln, and Milwaukee. The origin and destination of each flight may not be the same city, but Austin must arrive back at Lincoln on the last of his flights. Compute the probability that the cities Austin arrives at are all distinct.

What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?

For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares. [b]a.)[/b] Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$. [b]b.)[/b] Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$. [b]c.)[/b] Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$

Natural $p$ and $q$ are relatively prime. The $[0,1]$ is divided onto $(p+q)$ equal segments. Prove that every segment except two marginal contain exactly one from the $(p+q-2)$ numbers $$\{1/p, 2/p, ... , (p-1)/p, 1/q, 2/q, ... , (q-1)/q\}$$

A social club has $2k+1$ members, each of whom is fluent in the same $k$ languages. Any pair of members always talk to each other in only one language. Suppose that there were no three members such that they use only one language among them. Let $A$ be the number of three-member subsets such that the three distinct pairs among them use different languages. Find the maximum possible value of $A$.

Prove that there exists a positive integer $n$, such that if an equilateral triangle with side lengths $n$ is split into $n^2$ triangles with side lengths 1 with lines parallel to its sides, then among the vertices of the small triangles it is possible to choose $1993n$ points so that no three of them are vertices of an equilateral triangle.

$300$ parliament members are divided into $3$ chambers, each chamber consists of $100$ members. For every $2$ members, they either know each other or are strangers to each other.Show that no matter how they are divided into these $3$ chambers, it is always possible to choose $2$ members, each from different chamber such that there exist $17$ members from the third chamber so that all of them knows these two members, or all of them are strangers to these two members.

[b]p1.[/b] A rectangular pool has diagonal $17$ units and area $120$ units$^2$. Joey and Rachel start on opposite sides of the pool when Rachel starts chasing Joey. If Rachel runs $5$ units/sec faster than Joey, how long does it take for her to catch him? [b]p2. [/b] Alice plays a game with her standard deck of $52$ cards. She gives all of the cards number values where Aces are $1$’s, royal cards are $10$’s and all other cards are assigned their face value. Every turn she flips over the top card from her deck and creates a new pile. If the flipped card has value $v$, she places $12 - v$ cards on top of the flipped card. For example: if she flips the $3$ of diamonds then she places $9$ cards on top. Alice continues creating piles until she can no longer create a new pile. If the number of leftover cards is $4$ and there are $5$ piles, what is the sum of the flipped over cards? [b]p3.[/b] There are $5$ people standing at $(0, 0)$, $(3, 0)$, $(0, 3)$, $(-3, 0)$, and $(-3, 0)$ on a coordinate grid at a time $t = 0$ seconds. Each second, every person on the grid moves exactly $1$ unit up, down, left, or right. The person at the origin is infected with covid-$19$, and if someone who is not infected is at the same lattice point as a person who is infected, at any point in time, they will be infected from that point in time onwards. (Note that this means that if two people run into each other at a non-lattice point, such as $(0, 1.5)$, they will not infect each other.) What is the maximum possible number of infected people after $t = 7$ seconds? [b]p4.[/b] Kara gives Kaylie a ring with a circular diamond inscribed in a gold hexagon. The diameter of the diamond is $2$ mm. If diamonds cost $\$100/ mm ^2$ and gold costs $\$50 /mm ^2$ , what is the cost of the ring? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider all possible sets of natural numbers $(x_1, x_2, ..., x_{100})$ such that $1\leq x_i \leq 2017$ for every $i = 1,2, ..., 100$. We say that the set $(y_1, y_2, ..., y_{100})$ is greater than the set $(z_1, z_2, ..., z_{100})$ if $y_i> z_i$ for every $i = 1,2, ..., 100$. What is the largest number of sets that can be written on the board, so that any set is not more than the other set?

Let $S=\{1,2,3,\ldots,n\}$, $n$ an odd number. Find the parity of number of permutations $\sigma : S \Rightarrow S$ such that the sequence defined by \[a(i)=|\sigma(i)-i|\] is monotonous.

$10$ persons sit around a circular table and on the table there are $22$ vases. Two persons can see each other if and only if there are no vases aligned with them. Prove that there are at least two people who can see each other.

There are $2018$ boxes $C_1$, $C_2$, $C_3$,..,$C_{2018}$. The $n$-th box $C_n$ contains $n$ balls. A move consists of the following steps: a) Choose an integer $k$ greater than $1$ and choose $m$ a multiple of $k$. b) Take a ball from each of the consecutive boxes $C_{m-1}$, $C_m$, $C_{m+1}$ and move the $3$ balls to the box $C_{m+k}$. With these movements, what is the largest number of balls we can get in the box $2018$?

$25$ lightbulbs are distributed in the following way: the first $24$ are placed on a circumference, placing a bulb at each vertex of a regular $24$-gon, and the remaining bulb is placed on the center of said circumference. At any time, the following operations may be applied: [list] [*] Take two vertices on the circumference with an odd amount of vertices between them, and change the state of the bulbs on those vertices and the center bulb. [*] Take three vertices on the circumference that form an equilateral triangle, change the state of the bulbs on those vertices and the center bulb.[/list] Prove from any starting configuration of on and off lightbulbs, it is always possible to reach a configuration where all the bulbs are on.

There are $100$ cities in a country, some of them being joined by roads. Any four cities are connected to each other by at least two roads. Assume that there is no path passing through every city exactly once. Prove that there are two cities such that every other city is connected to at least one of them.

An invisible tank is on a $100 \times 100 $ table. A cannon can fire at any $60$ cells of the board after that the tank will move to one of the adjacent cells (by side). Then the progress is repeated. Can the cannon grantee to shoot the tank?