It is given $2018$ points in plane. Prove that it is possible to cover them with circles such that: $i)$ sum of lengths of all diameters of all circles is not greater than $2018$ $ii)$ distance between any two circles is greater than $1$

Consider a set of $n$ points in plane. Prove that the number of isosceles triangles having their vertices among these $n$ points is $\mathcal O (n^{\frac{7}{3}})$. Find a configuration of $n$ points in plane such that the number of equilateral triangles with vertices among these $n$ points is $\Omega (n^2)$.

A square of side $1$ is covered with $m^2$ rectangles. Show that there is a rectangle with perimeter at least $\frac{4}{m}$.

We say that a group of $25$ students is a [i]team[/i] if any two students in this group are friends. It is known that in the school any student belongs to at least one team but if any two students end their friendships at least one student does not belong to any team. We say that a team is [i]special[/i] if at least one student of the team has no friend outside of this team. Show that any two friends belong to some special team.

There are lamps in every field of $n\times n$ table. At start all the lamps are off. A move consists of chosing $m$ consecutive fields in a row or a column and changing the status of that $m$ lamps. Prove that you can reach a state in which all the lamps are on only if $m$ divides $n.$

Let $n \geq 6$ and $3 \leq p < n - p$ be two integers. The vertices of a regular $n$-gon are colored so that $p$ vertices are red and the others are black. Prove that there exist two congruent polygons with at least $[p/2] + 1$ vertices, one with all the vertices red and the other with all the vertices black.

Show that the edges of a strongly connected bipolar graph can be oriented in such a way that for any edge $ e$ there is a simple directed path from pole $ p$ to pole $ q$ containing $ e$. (A strongly connected bipolar graph is a finite connected graph with two special vertices $ p$ and $ q$ having the property that there are no points $ x,y,x \not \equal{} y$, such that all paths from $ x$ to $ p$ as well as all paths from $ x$ to $ q$ contain $ y$.) [i]A. Adam[/i]

You are given a row made by $2018$ squares, numbered consecutively from $0$ to $2017$. Initially, there is a coin in the square $0$. Two players $A$ and $B$ play alternatively, starting with $A$, on the following way: In his turn, each player can either make his coin advance $53$ squares or make the coin go back $2$ squares. On each move the coin can never go to a number less than $0$ or greater than $2017$. The player who puts the coin on the square $2017$ wins. ¿Who is the one with the wining strategy and how should he play to win?

On a plane are given finitely many red and blue lines, no two parallel, such that any intersection point of two lines of the same color also lies on another line of the other color. Prove that all the lines pass through a single point.

How many coins can be placed on a $10 \times 10$ board (each at the center of its square, at most one per square) so that no four coins form a rectangle with sides parallel to the sides of the board?

One-round hockey tournament is finished (each plays with each one time, the winner gets $2$ points, looser -- $0$, and $1$ point for draw). For arbitrary subgroup of teams there exists a team (may be from that subgroup) that has got an odd number of points in the games with the teams of the subgroup. Prove that there was even number of the participants.

assume that we have a n*n table we fill it with 1,...,n such that each number exists exactly n times prove that there exist a row or column such that at least $\sqrt{n}$ diffrent number are contained.

Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.

During a break, $n$ children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule: He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on. Determine the values of $n$ for which eventually, perhaps after many rounds, all children will have at least one candy each.

$12$ schoolchildren are engaged in a circle of patriotic songs, each of them knows a few songs (maybe none). We will say that a group of schoolchildren can sing a song if at least one member of the group knows it. Supervisor the circle noticed that any group of $10$ circle members can sing exactly $20$ songs, and any group of $8$ circle members - exactly $16$ songs. Prove that the group of all $12$ circle members can sing exactly $24$ songs.

An [i]animal[/i] with $n$ [i]cells[/i] is a connected figure consisting of $n$ equal-sized cells[1]. A [i]dinosaur[/i] is an animal with at least $2007$ cells. It is said to be [i]primitive[/i] it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur. (1) Animals are also called [i]polyominoes[/i]. They can be defined inductively. Two cells are [i]adjacent[/i] if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.

Ryan Murphy is playing poker. He is dealt a hand of $5$ cards. Given that the probability that he has a straight hand (the ranks are all consecutive; e.g. $3,4,5,6,7$ or $9,10,J,Q,K$) or $3$ of a kind (at least $3$ cards of the same rank; e.g. $5, 5, 5, 7, 7$ or $5, 5, 5, 7,K$) is $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$. [i]Proposed by Aditya Rao[/i]

Let $n$ be a positive integer and $(x_1, x_2, ..., x_{2n})$, $x_i = 0$ or $1, i = 1, 2, ... , 2n$ be a sequence of $2n$ integers. Let $S_n$ be the sum $S_n = x_1x_2 + x_3x_4 + ... + x_{2n-1}x_{2n}$. If $O_n$ is the number of sequences such that $S_n$ is odd and $E_n$ is the number of sequences such that $S_n$ is even, prove that $$\frac{O_n}{E_n}=\frac{2^n - 1}{2^n + 1}$$

Two babies A and B are playing a game with $2022$ bottles of milk. Each bottle has a maximum capacity of $200$ml, and initially each bottle holds $30$ml of milk. Starting from A, they take turns and do one of the following: (1) Pick a bottle with at least $100$ml of milk, and drink half of it. (2) Pick two bottles with less than $100$ml of milk, pour the milk of one bottle into the other one, and toss away the empty bottle. Whoever cannot do any operations loses the game. Who has a winning strategy? [i] Proposed by Chu-Lan Kao and usjl[/i]

Let $a_1,a_2,..., a_n$ be different integers and let $(b_1,b_2,..., b_n),(c_1,c_2,..., c_n)$ be two of their permutations, different from the identity. Prove that $$(|a_1-b_1|+|a_2-b_2|+...+|a_n-b_n| , |a_1-c_1|+|a_2-c_2|+...+|a_n-c_n| ) \ge 2$$ where $(x,y)$ denotes the greatest common divisor of the numbers $x,y$

Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

We have big multivolume encyclopaedia about dogs on the shelf in alphabetical order, each volume in its specially selected place. Near each place there is an instruction that prescribes one of four actions: to rearrange this volume is one or two places left or right. If you simultaneously run all instructions, volumes will be placed in the same places in another order. The cynologist Dima performs all the instructions every morning. Once he discovered, that the volume of "Bichons" stands still, which was initially occupied by the volume of "Terriers". Prove , that after some time the volume of "Mudies" will stand on the original place of the volume "Poodles".

There are $315$ marbles divided into three piles of $81, 115$ and $119$. In each move Ah Meng can either merge several piles into a single pile or divide a pile with an even number of marbles into $2$ equal piles. Can Ah Meng divide the marbles into $315$ piles, each with a single marble?

Let $A$ be a finite set of positive reals, let $B = \{x/y\mid x,y\in A\}$ and let $C = \{xy\mid x,y\in A\}$. Show that $|A|\cdot|B|\le|C|^2$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

Maker and Breaker are building a wall. Maker has a supply of green cubical building blocks, and Breaker has a supply of red ones, all of the same size. On the ground, a row of $m$ squares has been marked in chalk as place-holders. Maker and Breaker now take turns in placing a block either directly on one of these squares, or on top of another block already in place, in such a way that the height of each column never exceeds $n$. Maker places the first block. Maker bets that he can form a green row, i.e. all $m$ blocks at a certain height are green. Breaker bets that he can prevent Maker from achieving this. Determine all pairs $(m,n)$ of positive integers for which Maker can make sure he wins the bet.