Let $n \ge 2$ and $k \ge1$ be positive integers. In a country there are $n$ cities and between each pair of cities there is a bus connection in both directions. Let $A$ and $B$ be two different cities. Prove that the number of ways in which you can travel from $A$ to $B$ by using exactly $k$ buses is equal to $\frac{(n - 1)^k - (-1)^k}{n}$ .

Given three automates that deal with the cards with the pairs of natural numbers. The first, having got the card with ($a,b)$, produces new card with $(a+1,b+1)$, the second, having got the card with $(a,b)$, produces new card with $(a/2,b/2)$, if both $a$ and $b$ are even and nothing in the opposite case; the third, having got the pair of cards with $(a,b)$ and $(b,c)$ produces new card with $(a,c)$. All the automates return the initial cards also. Suppose there was $(5,19)$ card initially. Is it possible to obtain a) $(1,50)$? b) $(1,100)$? c) Suppose there was $(a,b)$ card initially $(a<b)$. We want to obtain $(1,n)$ card. For what $n$ is it possible?

Does there exist a hexagon (not necessarily convex) with side lengths $1, 2, 3, 4, 5, 6$ (not necessarily in this order) that can be tiled with a) $31$ b) $32$ equilateral triangles with side length $1?$

Pasha and Vova play the game crossing out the cells of the $3\times 101$ board by turns. At the start, the central cell is crossed out. By one move the player chooses the diagonal (there can be $1, 2$ or $3$ cells in the diagonal) and crosses out cells of this diagonal which are still uncrossed. At least one new cell must be crossed out by any player's move. Pasha begins, the one who can not make any move loses. Who has a winning strategy?

Is it possible to arrange 1400 positive integer ( not necessarily distinct ) ,at least one of them being 2021 , around a circle such that any number on this circle equals to the sum of gcd of the two previous numbers and two next numbers? for example , if $a,b,c,d,e$ are five consecutive numbers on this circle , $c=\gcd(a,b)+\gcd(d,e)$

A V-tromino is a diagram formed by three unit squares.(As attachment.) (a)Is it possible to cover a $3\times 2013$ table by $3\times 671$ V-trominoes? (b)Is it possible to cover a $5\times 2013$ table by $5\times 671$ V-trominoes?

The princess wishes to have a bracelet with $r$ rubies and $s$ emeralds arranged in such order that there exist two jewels on the bracelet such that starting with these and enumerating the jewels in the same direction she would obtain identical sequences of jewels. Prove that it is possible to fulfill the princess’s wish if and only if $r$ and $s$ have a common divisor.

For a finite set $E$ of cardinality $n \geq 3$, let $f(n)$ denote the maximum number of $3$-element subsets of $E$, any two of them having exactly one common element. Calculate $f(n)$.

A town-planner has built an isolated city whose road network consists of $2N$ roundabouts, each connecting exactly three roads. A series of tunnels and bridges ensure that all roads in the town meet only at roundabouts. All roads are two-way, and each roundabout is oriented clockwise. Vlad has recently passed his driving test, and is nervous about roundabouts. He starts driving from his house, and always takes the first edit at each roundabout he encounters. It turns out his journey incluldes every road in the town in both directions before he arrives back at the starting point in the starting direction. For what values of $N$ is this possible?

Several people came to the congress, each of whom has a certain number of tattoos on both hands. There are $n{}$ types of tattoos, and each of the $n{}$ types is found on the hands of at least $k{}$ people. For which pairs $(n, k)$ is it always possible for each participant to raise one of their hands so that all $n{}$ types of tattoos are present on the raised hands?

$1989$ equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.

Prove that there exist positive constants $ c$ and $ n_0$ with the following property. If $ A$ is a finite set of integers, $ |A| \equal{} n > n_0$, then \[ |A \minus{} A| \minus{} |A \plus{} A| \leq n^2 \minus{} c n^{8/5}.\]

The following game is played with a group of $n$ people and $n+1$ hats are numbered from $1$ to $n+1.$ The people are blindfolded and each of them puts one of the $n+1$ hats on his head (the remaining hat is hidden). Now, a line is formed with the $n$ people, and their eyes are uncovered: each of them can see the numbers on the hats of the people standing in front of him. Now, starting from the last person (who can see all the other players) the players take turns to guess the number of the hat on their head, but no two players can guess the same number (each player hears all the guesses from the other players). What is the highest number of guaranteed correct guesses, if the $n$ people can discuss a common strategy? [i]Proposed by Viktor Kiss, Budapest[/i]

Baron Munchausen took several cards and wrote a positive integer on each one (some numbers may be the same). The baron reports that he has used only two distinct digits to do that. He also reports that among the leftmost digits of the sums of integers on each pair of these cards there are all the digits from 1 to 9 . Can it occur that the baron is right? Maxim Didin

An equilateral triangle $T$ with side $111$ is partitioned into small equilateral triangles with side $1$ using lines parallel to the sides of $T$. Every obtained point except the center of $T$ is marked. A set of marked points is called $\textit{linear}$ if the points lie on a line, parallel to a side of $T$ (among the drawn ones). In how many ways we can split the marked point into $111$ $\textit{linear}$ sets?

Each point of the plane is coloured in one of two colours. Given an odd integer number $n \geq 3$, prove that there exist (at least) two similar triangles whose similitude ratio is $n$, each of which has a monochromatic vertex-set. [i]Vasile Pop[/i]

Let $n$ be a positive integer. A regular hexagon with side length $n$ is divided into equilateral triangles with side length $1$ by lines parallel to its sides. Find the number of regular hexagons all of whose vertices are among the vertices of those equilateral triangles. [i]UK - Sahl Khan[/i]

An ant crawls along the edges of a cube turning only at its vertices. It has visited one of the vertices $25$ times. Is it possible that it has visited each of the other $7$ vertices exactly $20$ times? (S Tokarev)

Given a finite sequence with terms in the set $A=\{0,1,…,121\}$ , it is allowed to replace each term by a number from the set $A$ so that like terms are replaced by like numbers, and different terms by different numbers. (Terms may remain without replacement.) The objective is to obtain, from a given sequence, through several such changes, a new sequence with sum divisible by $121$ . Show that it is possible to achieve the objective for every initial sequence. [hide=original wording]Dada una secuencia finita con términos en el conjunto A={0,1,…,121} , está permitido reemplazar cada término por un número del conjunto A de modo que términos iguales se reemplacen por números iguales, y términos distintos por números distintos. (Pueden quedar términos sin reemplazar.) El objetivo es obtener, a partir de una sucesión dada, mediante varios de tales cambios, una nueva sucesión con suma divisible por 121 . Demostrar que es posible lograr el objetivo para toda sucesión inicial.[/hide]

There are 4 countries: Argentina, Brazil, Peru and Uruguay. Each country consists of 4 islands. There are bridges going back and forth between some of the 16 islands. Carlos noted that whenever he travels between some of the islands using the bridges, without using the same bridge twice, and ending in the island where he started his journey, he will necessarily visit at least one island of each country. Determine the maximum number of bridges there can be.

Petya, Vasya and Tolya play a game on a $100\times 100$ board. They take turns (starting from Petya, then Vasya, then Tolya, then Petya, etc.) paint the boundary cells of the board (i.e., having a common side with the boundary of the board.) It is forbidden to paint the cell that is adjacent to the already painted one. In addition, it’s also forbidden to paint the cell which is symmetrical to the painted one, with respect to the center of the board. The player who can’t make the move loss. Can Vasya and Tolya, after agreeing, play so that Petya loses?

Prove that an $m \times n$ rectangle can be constructed using copies of the following shape if and only if $mn$ is a multiple of $8$ where $m>1$ and $n>1$ [asy] draw ((0,0)--(0,1)); draw ((0,0)--(1.5,0)); draw ((0,1)--(.5,1)); draw ((.5,1)--(.5,0)); draw ((0,.5)--(1.5,.5)); draw ((1.5,.5)--(1.5,0)); draw ((1,.5)--(1,0)); [/asy]

Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.

$n$ consecutive positive integers are put down in a row (not necessarily in order) so that the sum of any three successive integers in the row is divisible by the leftmost number in the triple. What is the largest possible value of $n$ if the last number in the row is odd? (A Shapovalov)

Consider $2011^2$ points arranged in the form of a $2011 \times 2011$ grid. What is the maximum number of points that can be chosen among them so that no four of them form the vertices of either an isosceles trapezium or a rectangle whose parallel sides are parallel to the grid lines?