Let $M$ be a map made of several cities linked to each other by flights. We say that there is a route between two cities if there is a nonstop flight linking these two cities. For each city to the $M$ denote by $M_a$ the map formed by the cities that have a route to and routes linking these cities together ( to not part of $M_a$). The cities of $M_a$ are divided into two sets so that the number of routes linking cities of different sets is maximum; we call this number the cut of $M_a$. Suppose that for every cut of $M_a$ , it is strictly less than two thirds of the number of routes $M_a$ . Show that for any coloring of the $M$ routes with two colors there are three cities of $M$ joined by three routes of the same color.

In a town there are $n$ avenues running from south to north. They are numbered $1$ through $n$ (from west to east). There are $n$ streets running from west to east – they are also numbered $1$ through $n$ (from south to north). If you drive through the junction of the $k$th avenue and the $\ell$th street, you have to pay $k\ell$ kroner. How much do you at least have to pay for driving from the junction of the $1$st avenue and the $1$st street to the junction of the nth avenue and the $n$th street? (You also pay for the starting and finishing junctions.)

You want to color the side faces of a cube in such a way that each face is colored evenly. Six colors are available: [i]red, white, blue, yellow, purple, orange[/i]. Two cube colors are called the same if one arises from the other by a rotation of the cube. (a) How many different cube colorings are there, using six colors? (b) How many different cube colorings are there, using exactly five colors?

Let $k \in \mathbb{N},$ $S_k = \{(a, b) | a, b = 1, 2, \ldots, k \}.$ Any two elements $(a, b)$, $(c, d)$ $\in S_k$ are called "undistinguishing" in $S_k$ if $a - c \equiv 0$ or $\pm 1 \pmod{k}$ and $b - d \equiv 0$ or $\pm 1 \pmod{k}$; otherwise, we call them "distinguishing". For example, $(1, 1)$ and $(2, 5)$ are undistinguishing in $S_5$. Considering the subset $A$ of $S_k$ such that the elements of $A$ are pairwise distinguishing. Let $r_k$ be the maximum possible number of elements of $A$. (i) Find $r_5$. (ii) Find $r_7$. (iii) Find $r_k$ for $k \in \mathbb{N}$.

a) There are $2n$ rays marked in a plane, with $n$ being a natural number. Given that no two marked rays have the same direction and no two marked rays have a common initial point, prove that there exists a line that passes through none of the initial points of the marked rays and intersects with exactly $n$ marked rays. (b) Would the claim still hold if the assumption that no two marked rays have a common initial point was dropped?

Given a regular $26$-gon. Prove that for any $9$ vertices of that regular $26$-gon, then there exists three vertices that forms an isosceles triangle.

Several numbers are written in a row. In each move, Robert chooses any two adjacent numbers in which the one on the left is greater than the one on the right, doubles each of them and then switches them around. Prove that Robert can make only a finite number of moves.

Find the smallest number $n \geq 5$ for which there can exist a set of $n$ people, such that any two people who are acquainted have no common acquaintances, and any two people who are not acquainted have exactly two common acquaintances. [i]Bulgaria[/i]

[b]p1. [/b](a) Show that for every positive integer $m > 1$, there are positive integers $x$ and $y$ such that $x^2 - y^2 = m^3$. (b) Find all pairs of positive integers $(x, y)$ such that $x^6 = y^2 + 127$. [b]p2.[/b] (a) Let $P(x)$ be a polynomial with integer coefficients. Suppose that $P(0)$ is an odd integer and that $P(1)$ is also an odd integer. Show that if $c$ is an integer then $P(c)$ is not equal to $0$. (b) Let P(x) be a polynomial with integer coefficients. Suppose that $P(1,000) = 1,000$ and $P(2,000) = 2,000.$ Explain why $P(3,000)$ cannot be equal to $1,000$. [b]p3.[/b] Triangle $\vartriangle ABC$ is created from points $A(0, 0)$, $B(1, 0)$ and $C(1/2, 2)$. Let $q, r$, and $s$ be numbers such that $0 < q < 1/2 < s < 1$, and $q < r < s$. Let D be the point on $AC$ which has $x$-coordinate $q$, $E$ be the point on AB which has $x$-coordinate $r$, and $F$ be the point on $BC$ that has $x$-coordinate $s$. (a) Find the area of triangle $\vartriangle DEF$ in terms of $q, r$, and $s$. (b) If $r = 1/2$, prove that at least one of the triangles $\vartriangle ADE$, $\vartriangle CDF$, or $\vartriangle BEF$ has an area of at least $1/4$. [b]p4.[/b] In the Gregorian calendar: (i) years not divisible by $4$ are common years, (ii) years divisible by $4$ but not by $100$ are leap years, (iii) years divisible by $100$ but not by $400$ are common years, (iv) years divisible by $400$ are leap years, (v) a leap year contains $366$ days, a common year $365$ days. From the information above: (a) Find the number of common years and leap years in $400$ consecutive Gregorian years. Show that $400$ consecutive Gregorian years consists of an integral number of weeks. (b) Prove that the probability that Christmas falls on a Wednesday is not equal to $1/7$. [b]p5.[/b] Each of the first $13$ letters of the alphabet is written on the back of a card and the $13$ cards are placed in a row in the order $$A,B,C,D,E, F, G,H, I, J,K, L,M$$ The cards are then turned over so that the letters are face down. The cards are rearranged and again placed in a row, but of course they may be in a different order. They are rearranged and placed in a row a second time and both rearrangements were performed exactly the same way. When the cards are turned over the letters are in the order $$B,M, A,H, G,C, F,E,D, L, I,K, J$$ What was the order of the letters after the cards were rearranged the first time? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There were made 7 golden, 7 silver and 7 bronze for a tournament. All the medals of the same material should weigh the same and the medals of different materials should have different weight. However, it so happened that exactly one medal had a wrong weight. If this medal is golden, it is lighter than a standard golden medal; if it is bronze, it is heavier than a standard bronze one; if it is silver, it may be lighter or heavier than a standard silver one. Is it possible to find the nonstandard one for sure, using three weighings on a balance scale with no weights?

Let $n$ be a positive integer $\geq 2$ . Consider a $n$ by $n$ grid with all entries $1$. Define an operation on a square to be changing the signs of all squares adjacent to it but not the sign of its own. Find all $n$ such that it is possible after a finite sequence of operations to reach a $n$ by $n$ grid with all entries $-1$

For an upcoming international mathematics contest, the participating countries were asked to choose from nine combinatorics problems. Given how hard it usually is to agree, nobody was surprised that the following happened: [b]i)[/b] Every country voted for exactly three problems. [b]ii)[/b] Any two countries voted for different sets of problems. [b]iii)[/b] Given any three countries, there was a problem none of them voted for. Find the maximal possible number of participating countries.

Determine the smallest integer $j$ such that it is possible to fill the fields of the table $10\times 10$ with numbers from $1$ to $100$ so that every $10$ consecutive numbers lie in some of the $j\times j$ squares of the table. Czech Republic

Let $U=\{1, 2,\ldots, 2014\}$. For positive integers $a$, $b$, $c$ we denote by $f(a, b, c)$ the number of ordered 6-tuples of sets $(X_1,X_2,X_3,Y_1,Y_2,Y_3)$ satisfying the following conditions: (i) $Y_1 \subseteq X_1 \subseteq U$ and $|X_1|=a$; (ii) $Y_2 \subseteq X_2 \subseteq U\setminus Y_1$ and $|X_2|=b$; (iii) $Y_3 \subseteq X_3 \subseteq U\setminus (Y_1\cup Y_2)$ and $|X_3|=c$. Prove that $f(a,b,c)$ does not change when $a$, $b$, $c$ are rearranged. [i]Proposed by Damir A. Yeliussizov, Kazakhstan[/i]

In each square of the $100\times 100$ square table, type $1, 2$, or $3$. Consider all subtables $m \times n$, where $m = 2$ and $n = 2$. A subtable will be called [i]balanced [/i] if it has in its corner boxes of four identical numbers boxes . For as large a number $k$ prove, that we can always find $k$ balanced subtables, of which no two overlap, i.e. do not have a common box.

Find the smallest real $\alpha$, such that for any convex polygon $P$ with area $1$, there exist a point $M$ in the plane, such that the area of convex hull of $P\cup Q$ is at most $\alpha$, where $Q$ denotes the image of $P$ under central symmetry with respect to $M$.

Let $n$ cards are placed in a circle. Each card has a white side and a black side. On each move, you pick one card with black side up, flip it over, and also flip over the two neighboring cards. Suppose initially, there are only one black-side-up card. (a)If $n=2015$ , can you make all cards white-side-up through a finite number of moves? (b)If $n=2016$ , can you make all cards white-side-up through a finite number of moves?

Starting at $(0, 0)$, Richard takes $2n+1$ steps, with each step being one unit either East, North, West, or South. For each step, the direction is chosen uniformly at random from the four possibilities. Determine the probability that Richard ends at $(1, 0)$.

15 boxes are given. They all are initially empty. By one move it is allowed to choose some boxes and to put in them numbers of apricots which are pairwise distinct powers of 2. Find the least positive integer $k$ such that it is possible to have equal numbers of apricots in all the boxes after $k$ moves.

[u]Round 1[/u] [b]p1.[/b] Three snails – Alice, Bobby, and Cindy – were racing down a road. Whenever one snail passed another, it waved at the snail it passed. During the race, Alice waved $3$ times and was waved at twice. Bobby waved $4$ times and was waved at $3$ times. Cindy waved $5$ times. How many times was she waved at? [b]p2.[/b] Sherlock and Mycroft are playing Battleship on a $4\times 4$ grid. Mycroft hides a single $3\times 1$ cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser? [b]p3.[/b] Thirty girls – $13$ of them in red dresses and $17$ in blue dresses – were dancing in a circle, hand-in-hand. Afterwards, each girl was asked if the girl to her right was in a blue dress. Only the girls who had both neighbors in red dresses or both in blue dresses told the truth. How many girls could have answered “Yes”? [b]p4.[/b] Herman and Alex play a game on a $5\times 5$ board. On his turn, a player can claim any open square as his territory. Once all the squares are claimed, the winner is the player whose territory has the longer border. Herman goes first. If both play their best, who will win, or will the game end in a draw? [img]https://cdn.artofproblemsolving.com/attachments/5/7/113d54f2217a39bac622899d3d3eb51ec34f1f.png[/img] [b]p5.[/b] Is it possible to find $2014$ distinct positive integers whose sum is divisible by each of them? [u]Round 2[/u] [b]p6.[/b] Hermione and Ron play a game that starts with 129 hats arranged in a circle. They take turns magically transforming the hats into animals. On each turn, a player picks a hat and chooses whether to change it into a badger or into a raven. A player loses if after his or her turn there are two animals of the same species right next to each other. Hermione goes first. Who loses? [b]p7.[/b] Three warring states control the corner provinces of the island whose map is shown below. [img]https://cdn.artofproblemsolving.com/attachments/e/a/4e2f436be1dcd3f899aa34145356f8c66cda82.png[/img] As a result of war, each of the remaining $18$ provinces was occupied by one of the states. None of the states was able to occupy any province on the coast opposite their corner. The states would like to sign a peace treaty. To do this, they each must send ambassadors to a place where three provinces, one controlled by each state, come together. Prove that they can always find such a place to meet. For example, if the provinces are occupied as shown here, the squares mark possible meeting spots. [img]https://cdn.artofproblemsolving.com/attachments/e/b/81de9187951822120fc26024c1c1fbe2138737.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $N$ be an integer greater than $1$ and let $T_n$ be the number of non empty subsets $S$ of $\{1,2,.....,n\}$ with the property that the average of the elements of $S$ is an integer.Prove that $T_n - n$ is always even.

Find the number of positive integers $n$ less than $10000$ such that there are more $4$’s in the digits of $n + 1$ than in the digits of $n$.

$2021$ copies of each of the number from $1$ to $5$ are initially written on the board.Every second Alice picks any two f these numbers, say $a$ and $b$ and writes $\frac{ab}{c}$.Where $c$ is the length of the hypoteneus with sides $a$ and $b$.Alice stops when only one number is left.If the minnimum number she could write was $x$ and the maximum number she could write was $y$ then find the greatest integer lesser than $2021^2xy$. [hide=PS]Does any body know how to use floors and ceiling function?cuz actuall formation used ceiling,but since Idk how to use ceiling I had to do it like this :(]

How many paths are there from$ A$ to the line $BC$ if the path does not go through any vertex twice and always moves to the left? [img]https://cdn.artofproblemsolving.com/attachments/e/6/a4bc3a9decc06eaeed6f7e99cb58f7b2524471.jpg[/img]

What is the largest number of knights that can be put on a $5 \times 5$ chess board so that each knight attacks exactly two other knights? (M Gorelov)