Some cities of Russia are connected with some cities of Ukraine with international airlines. The Interstate Council for the Promotion of Migration intends to introduce a one-way traffic on each airline so that, by taking off from the city, it could no longer be returned in this city (using other one-way airlines). Prove that the number of ways to do this is not divided by $3$.

Let $k$ be a positive integer. There are $2k$ pieces arranged in a row. A [i]move[/i] consists of swapping two adjacent pieces. Several moves must be made so that each piece passes through both the first and the last position. What is the minimum number of moves required to achieve this?

Let $P$ be a cyclic polygon with circumcenter $O$ that does not lie on any diagonal, and let $S$ be the set of points on 2D plane containing $P$ and $O$. The $\textit{Matcha Sweep Game}$ is a game between two players $A$ and $B$, with $A$ going first, such that each choosing a nonempty subset $T$ of points in $S$ that has not been previously chosen, and such that if $T$ has at least $3$ vertices then $T$ forms a convex polygon. The game ends with all points have been chosen, with the player picking the last point wins. For which polygons $P$ can $A$ guarantee a win? [i]Proposed by Anzo Teh Zhao Yang[/i]

In a $10$ by $10$ square grid (which we call “the bay”) you are requested to place ten “ships”: one $1$ by $4$ ship, two $1$ by $3$ ships, three $1$ by $2$ ships and four $1$ by $1$ ships. The ships may not have common points (even corners) but may touch the “shore” of the bay. Prove that (a) by placing the ships one after the other arbitrarily but in the order indicated above, it is always possible to complete the process; (b) by placing the ships in reverse order (beginning with the smaller ones), it is possible to reach a situation where the next ship cannot be placed (give an example). (KN Ignatjev)

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

There are two bowls on the table, in one there are $p$, in the other $q$ stones ($p, q \in N*$ ). Two players $A$ and $B$ take turns playing, starting with $A$. Who's turn: $\bullet$ takes a stone from one of the bowls $\bullet$or removes one stone from each bowl $\bullet$ or puts a stone from one of the bowls into the other. Whoever takes the last stone wins. Under what conditions can $A$ and under what conditions can $B$ force the win? The answer must be justified.

[b]p1.[/b] You are on a flat planet. There are $100$ cities at points $x = 1, ..., 100$ along the line $y = -1$, and another $100$ cities at points $x = 1, ... , 100$ along the line $y = 1$. The planet’s terrain is scalding hot, and you cannot walk over it directly. Instead, you must cross archways from city to city. There are archways between all pairs of cities with different $y$ coordinates, but no other pairs: for instance, there is an archway from $(1, -1)$ to $(50, 1)$, but not from $(1, -1)$ to $(50, -1)$. The amount of “effort” necessary to cross an archway equals the square of the distance between the cities it connects. You are at $(1, -1)$, and you want to get to $(100, -1)$. What is the least amount of effort this journey can take? [b]p2.[/b] Let $f(x) = x^4 + ax^3 + bx^2 + cx + 25$. Suppose $a, b, c$ are integers and $f(x)$ has $4$ distinct integer roots. Find $f(3)$. [b]p3.[/b] Frankenstein starts at the point $(0, 0, 0)$ and walks to the point $(3, 3, 3)$. At each step he walks either one unit in the positive $x$-direction, one unit in the positive $y$-direction, or one unit in the positive $z$-direction. How many distinct paths can Frankenstein take to reach his destination? [b]p4.[/b] Let $ABCD$ be a rectangle with $AB = 20$, $BC = 15$. Let $X$ and $Y$ be on the diagonal $\overline{BD}$ of $ABCD$ such that $BX > BY$ . Suppose $A$ and $X$ are two vertices of a square which has two sides on lines $\overline{AB}$ and $\overline{AD}$, and suppose that $C$ and $Y$ are vertices of a square which has sides on $\overline{CB}$ and $\overline{CD}$. Find the length $XY$ . [img]https://cdn.artofproblemsolving.com/attachments/2/8/a3f7706171ff3c93389ff80a45886e306476d1.png[/img] [b]p5.[/b] $n \ge 2$ kids are trick-or-treating. They enter a haunted house in a single-file line such that each kid is friends with precisely the kids (or kid) adjacent to him. Inside the haunted house, they get mixed up and out of order. They meet up again at the exit, and leave in single file. After leaving, they realize that each kid (except the first to leave) is friends with at least one kid who left before him. In how many possible orders could they have left the haunted house? [b]p6.[/b] Call a set $S$ sparse if every pair of distinct elements of S differ by more than $1$. Find the number of sparse subsets (possibly empty) of $\{1, 2,... , 10\}$. [b]p7.[/b] How many ordered triples of integers $(a, b, c)$ are there such that $1 \le a, b, c \le 70$ and $a^2 + b^2 + c^2$ is divisible by $28$? [b]p8.[/b] Let $C_1$, $C_2$ be circles with centers $O_1$, $O_2$, respectively. Line $\ell$ is an external tangent to $C_1$ and $C_2$, it touches $C_1$ at $A$ and $C_2$ at $B$. Line segment $\overline{O_1O_2}$ meets $C_1$ at $X$. Let $C$ be the circle through $A, X, B$ with center $O$. Let $\overline{OO_1}$ and $\overline{OO_2}$ intersect circle $C$ at $D$ and $E$, respectively. Suppose the radii of $C_1$ and $C_2$ are $16$ and $9$, respectively, and suppose the area of the quadrilateral $O_1O_2BA$ is $300$. Find the length of segment $DE$. [b]p9.[/b] What is the remainder when $5^{5^{5^5}}$ is divided by $13$? [b]p10.[/b] Let $\alpha$ and $\beta$ be the smallest and largest real numbers satisfying $$x^2 = 13 + \lfloor x \rfloor + \left\lfloor \frac{x}{2} \right\rfloor +\left\lfloor \frac{x}{3} \right\rfloor + \left\lfloor \frac{x}{4} \right\rfloor .$$ Find $\beta - \alpha$ . ($\lfloor a \rfloor$ is defined as the largest integer that is not larger than $a$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For which natural $ n \geq 3 $ numbers from 1 to $ n $ can be arranged by a circle so that each number does not exceed $60$ % of the sum of its two neighbors?

Given a board of $3 \times 3$ you want to write the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and a number in their boxes positive integer $M$, not necessarily different from the above. The goal is that the sum of the three numbers in each row be the same $a)$ Find all the values of $M$ for which this is possible. $b)$ For which of the values of $M$ found in $a)$ is it possible to arrange the numbers so that no only the three rows add the same but also the three columns add the same?

Albert and Betty are playing the following game. There are $100$ blue balls in a red bowl and $100$ red balls in a blue bowl. In each turn a player must make one of the following moves: a) Take two red balls from the blue bowl and put them in the red bowl. b) Take two blue balls from the red bowl and put them in the blue bowl. c) Take two balls of different colors from one bowl and throw the balls away. They take alternate turns and Albert starts. The player who first takes the last red ball from the blue bowl or the last blue ball from the red bowl wins. Determine who has a winning strategy.

$p$ parallel lines are drawn in the plane and $q$ lines perpendicular to them are also drawn. How many rectangles are bounded by the lines?

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

A rectangular piece of paper is divided into square cells by lines parallel to the sides of the rectangle. $n$ (horizontal) rows of $m$ cells have emerged and $m$ (vertical) columns of $n$ cells have also been formed. There is a number in each cell. Find the largest number in each of the $n$ rows. The smallest maxima of those $n$ rows is called $A$. We also look for the smallest number in each of the $m$ columns. The largest minima of those $m$ columns is called $B$. Prove that $A$ is greater than or equal to $B$. Can you give a simple example where $A = B$?

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

Some positive integers, sum of which is $23$, are written in sequential form. Neither one of the terms nor the sum of some consecutive terms in the sequence is equal to $3$. [b]a) [/b]Is it possible that the sequence contains exactly $11$ terms? [b]b)[/b]Is it possible that the sequence contains exactly $12$ terms?

Chip and Dale play the following game. Chip starts by splitting $1001$ nuts between three piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $1001$. Then Chip moves nuts from the piles he prepared to a new (fourth) pile until there will be exactly $N$ nuts in any one or more piles. When Chip accomplishes his task, Dale gets an exact amount of nuts that Chip moved. What is the maximal number of nuts that Dale can get for sure, no matter how Chip acts? (Naturally, Dale wants to get as many nuts as possible, while Chip wants to lose as little as possible).

Let $n$ be positive integer and fix $2n$ distinct points on a circle. Determine the number of ways to connect the points with $n$ arrows (oriented line segments) such that all of the following conditions hold: [list] [*]each of the $2n$ points is a startpoint or endpoint of an arrow; [*]no two arrows intersect; and [*]there are no two arrows $\overrightarrow{AB}$ and $\overrightarrow{CD}$ such that $A$, $B$, $C$ and $D$ appear in clockwise order around the circle (not necessarily consecutively). [/list]

There are $63$ houses at the distance of $1, 2, 3, . . . , 63 \text{ km}$ from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants, $63$ elfs named $E_1, E_2, . . . , E_{63}$ , deliever the vaccine to each house; each elf will deliever vaccine to exactly one house and never return. Suppose that the elf $E_n$ takes $n$ minutes to travel $1 \text{ km}$ for each $n = 1,2,...,63$ , and that all elfs leave the north pole simultaneously. What is the minimum amount of time to complete the delivery?

Consider the string of length $6$ composed of three characters $a, b, c$. For each string, if two $a$s are next to each other, or two $b$s are next to each other, then replace $aa$ by $b$, and replace $bb$ by $a$. Also, if $a$ and $b$ are next to each other, or two $c$s are next to each other, remove all two of them (i.e. delete $ab, ba, cc$). Determine the number of strings that can be reduced to $c$, the string of length $1$, by the reducing processes mentioned above.

A finite graph $G(V,E)$ on $n$ points is drawn in the plane. For an edge $e$ of the graph, let $\chi(e)$ denote the number of edges that cross over edge $e$. Prove that \[\sum_{e\in E}\frac{1}{\chi(e)+1}\leq 3n-6.\][i]Proposed by Dömötör Pálvölgyi, Budapest[/i]

[b]Problem 3.[/b] Let $n\geq 3$ is given natural number, and $M$ is the set of the first $n$ primes. For any nonempty subset $X$ of $M$ with $P(X)$ denote the product of its elements. Let $N$ be a set of the kind $\ds\frac{P(A)}{P(B)}$, $A\subset M, B\subset M, A\cap B=\emptyset$ such that the product of any 7 elements of $N$ is integer. What is the maximal number of elements of $N$? [i]Alexandar Ivanov[/i]

Given two distinct points $A_1,A_2$ in the plane, determine all possible positions of a point $A_3$ with the following property: There exists an array of (not necessarily distinct) points $P_1,P_2,...,P_n$ for some $n \ge 3$ such that the segments $P_1P_2,P_2P_3,...,P_nP_1$ have equal lengths and their midpoints are $A_1, A_2, A_3, A_1, A_2, A_3, ...$ in this order.

Two positive integers $m$ and $n$ are written on the board. We replace one of two numbers in each step on the board by either their sum, or product, or ratio (if it is an integer). Depending on the numbers $m$ and $n$, specify all the pairs that can appear on the board in pairs. (Radovan Švarc)

In a country of $n$ cities, an express train runs both ways between any two cities. For any train, ticket prices either direction are equal, but for any different routes these prices are different. Prove that the traveler can select the starting city, leave it and go on, successively, $n-1$ trains, such that each fare is smaller than that of the previous fare. (A traveler can enter the same city several times.)

Let $m$ be a positive integer, and $A_1, A_2, \ldots, A_m$ (not necessarily different) be $m$ subsets of a finite set $A$. It is known that for any nonempty subset $I$ of $\{1, 2 \ldots, m \}$, \[ \Big| \bigcup_{i \in I} A_i \Big| \ge |I|+1. \] Show that the elements of $A$ can be colored black and white, so that each of $A_1,A_2,\ldots,A_m$ contains both black and white elements.