There are $2n-1$ lamps in a row. In the beginning only the middle one is on (the $n$-th one), and the other ones are off. Allowed move is to take two non-adjacent lamps which are turned off such that all lamps between them are turned on and switch all of their states from on to off and vice versa. What is the maximal number of moves until the process terminates?

Each entry in a $2000\times 2000$ array is $0$, $1$, or $-1$. Show that it's possible for all $4000$ row sums and column sums to be distinct.

For a set $S$ of $n$ points, let $d_1 > d_2 >... > d_k > ...$ be the distances between the points. A function $f_k: S \to N$ is called a [i]coloring function[/i] if, for any pair $M,N$ of points in $S$ with $MN \le d_k$ , it takes the value $f_k(M)+ f_k(N)$ at some point. Prove that for each $m \in N$ there are positive integers $n,k$ and a set $S$ of $n$ points such that every coloring function $f_k$ of $S$ satisfies $| f_k(S)| \le m$

An equilateral triangle of integer side length $n \geq 1$ is subdivided into small triangles of unit side length, as illustrated in the figure below for $n = 5$. In this diagram a subtriangle is a triangle of any size which is formed by connecting vertices of the small triangles along the grid lines. [img]https://cdn.artofproblemsolving.com/attachments/e/9/17e83ad4872fcf9e97f0479104b9569bf75ad0.jpg[/img] It is desired to color each vertex of the small triangles either red or blue in such a way that there is no subtriangle with all of its vertices having the same color. Let $f(n)$ denote the number of distinct colorings satisfying this condition. Determine, with proof, $f(n)$ for every $n \geq 1$

In chess tournament participates $n$ participants ($n >1$). In tournament each of participants plays with each other exactly $1$ game. For each game participant have $1$ point if he wins game, $0,5$ point if game is drow and $0$ points if he lose game. If after ending of tournament participant have at least $ 75 % $ of maximum possible points he called $winner$ $of$ $tournament$. Find maximum possible numbers of $winners$ $of$ $tournament$.

Chip and Dale play the following game. Chip starts by splitting $222$ nuts between two piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $222$. Then Chip moves nuts from the piles he prepared to a new (third) pile until there will be exactly $N$ nuts in any one or two 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).

Given scales and a set of $n$ different weights. We take weights in turn and add them on one of the scales sides. Let us denote "$L$" the scales state with the left side down, and "$R$" -- with the right side down. a) Prove that you can arrange the weights in such an order, that we shall obtain the sequence $LRLRLRLR...$ of the scales states. (That means that the state of the scales will be changed after putting every new weight.) b) Prove that for every $n$-letter word containing $R$'s and $L$'s only you can arrange the weights in such an order, that the sequence of the scales states will be described by that word.

2011 storage buildings are connected by roads so that it is possible to reach any building from any other building, possibly using multiple roads. The buildings contain $x_1,\dots,x_{2011}$ kilogram of cement. In one move, it is possible to relocate any quantity of cement from one building to any other building that is connected to it. The target is to have $y_1,\dots,y_{2011}$ redistributed across storage buildings and \[x_1+x_2+\dots+x_{2011}=y_1+y_2+\dots+y_{2011}.\] What is the minimal number of moves that the redistribution can take regardless of values of $x_i$ and $y_i$ and of the road plan? (Author: P. Karasev)

Consider an integer \(n \ge 2\) and write the numbers \(1, 2, \ldots, n\) down on a board. A move consists in erasing any two numbers \(a\) and \(b\), then writing down the numbers \(a+b\) and \(\vert a-b \vert\) on the board, and then removing repetitions (e.g., if the board contained the numbers \(2, 5, 7, 8\), then one could choose the numbers \(a = 5\) and \(b = 7\), obtaining the board with numbers \(2, 8, 12\)). For all integers \(n \ge 2\), determine whether it is possible to be left with exactly two numbers on the board after a finite number of moves. [i]Proposed by China[/i]

Let $n$ be a positive integer. Denote by $a(n)$ the ways of expression $n=x_1+x_2+\dots$ where $x_1\leqslant x_2 \leqslant\dots$ are positive integers and $x_i+1$ is a power of $2$ for each $i$. Denote by $b(n)$ the ways of expression $n=y_1+y_2+\dots$ where $y_i$ is a positive integer and $2y_i\leqslant y_{i+1}$ for each $i$. Prove that $a(n)=b(n)$.

On a square board with $9$ squares (three by three), nine elements of the set $S=\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ must be placed, different from each other, so that each one is in a box and the following conditions are met: $\bullet$ The sums of the numbers in the second and third rows are, respectively, double and triple the sum of the numbers in the first row. $\bullet$ The sum of the numbers in the second and third columns are, respectively, double and triple the sum of the numbers in the first column. Show all the possible ways to place elements of $S$ on the board, fulfilling the indicated conditions.

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

5)Players $A$ and $B$ play the following game: a player writes, in a board, a positive integer $n$, after this they delete a number in the board and write a new number where can be: i)The last number $p$, where the new number will be $p - 2^k$ where $k$ is greatest number such that $p\ge 2^k$ ii) The last number $p$, where the new number will be $\frac{p}{2}$ if $p$ is even. The players play alternately, a player win(s) if the new number is equal to $0$ and player $A$ starts. a)Which player has the winning strategy with $n = 40$?? b)Which player has the winning strategy with $n = 2012$??

In a regular $ 6n\plus{}1$-gon, $ k$ vertices are painted in red and the others in blue. Prove that the number of isosceles triangles whose vertices are of the same color does not depend on the arrangement of the red vertices.

[u]Round 5[/u] [b]p13.[/b] Jason flips a coin repeatedly. The probability that he flips $15$ heads before flipping $4$ tails can be expressed as $\frac{a}{2^b}$ where $a$ and $b$ are positive integers and $a$ is odd. Find $a +b$. [b]p14.[/b] Triangle $ABC$ has side lengths $AB = 3$, $BC = 3$, and $AC = 4$. Let D be the intersection of the angle bisector of $\angle B AC$ and segment $BC$. Let the circumcircle of $\vartriangle B AD$ intersect segment $AC$ at a point $E$ distinct from $A$. The length of $AE$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p15.[/b] The sum of the squares of all values of $x$ such that $\{(x -2)(x -3)\} = \{(x -1)(x -6)\}$ and $\lfloor x^2 -6x +6 \rfloor= 9$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a +b$. Note: $\{a\}$ is the fractional part function, and returns $a -\lfloor a \rfloor$ . [u]Round 6[/u] [b]p16.[/b] Maisy the Polar Bear is at the origin of the Polar Plane ($r = 0, \theta = 0$). Maisy’s location can be expressed as $(r,\theta)$, meaning it is a distance of $r$ away from the origin and at a angle of $\theta$ degrees counterclockwise from the $x$-axis. When Maisy is on the point $(m,n)$ then it can jump to either $(m,n +1)$ or $(m+1,n)$. Maisy cannot jump to any point it has been to before. Let $L(r,\theta)$ be the number of paths Maisy can take to reach point $(r,\theta)$. The sum of $L(r,\theta)$ over all points where $r$ is an integer between $1$ and $2020$ and $\theta$ is an integer between $0$ and $359$ can be written as $\frac{n^k-1}{m}$ for some minimum value of $n$, such that $n$, $k$, and $m$ are all positive integers. Find $n +k +m$. [b]p17.[/b] A circle with center $O$ and radius $2$ and a circle with center $P$ and radius $3$ are externally tangent at $A$. Points $B$ and $C$ are on the circle with center $O$ such that $\vartriangle ABC$ is equilateral. Segment $AB$ extends past $B$ to point $D$ and $AC$ extends past $C$ to point $E$ such that $BD = CE = \sqrt3$. A line through $D$ is tangent to circle $P$ at $F$. Find $DF^2$. [img]https://cdn.artofproblemsolving.com/attachments/2/7/0ee8716cebd6701fcae6544d9e39e68fff35f5.png[/img] [b]p18.[/b] Find the number of trailing zeroes at the end of $$\prod^{2021}_{i=1} (2021i -1) = (2020)(4041)...(2021^2 -1).$$ [u]Round 7[/u] [b]p19.[/b] A function $f (n)$ is defined as follows: $$f (n) = \begin{cases} \frac{n}{3} \,\,\, if \,\,\, n \equiv 0 (mod \, 3) \\ n^2 +4n -5 \,\,\,if \,\,\,n \equiv 1 (mod \, 3) \\ n^2 +n -2 \,\,\, if \,\,\,n \equiv 2 (mod \, 3) \end{cases}$$ Find the number of integer values of $n$ between $2$ and $1000$ inclusive such that $f ( f (... f (n))) = 1$ for some number of applications of $f (n)$. [b]p20.[/b] In the diagram below, the larger circle with diameter $AW$ has radius $16$. $ABCD$ and $WXY Z$ are rhombi where $\angle B AD = \angle XWZ = 60^o$ and $AC = CY = YW$. $M$ is the midpoint of minor arc $AW$, as shown. Let $I$ be the center of the circle with diameter $OM$. Circles with center $P$ and $G$ are tangent to lines $AD$ and $WZ$, respectively, and also tangent to the circle with center $I$ . Given that $IP \perp AD$ and $IG \perp WZ$, the area of $\vartriangle PIG$ can be written as $a +b\sqrt{c}$ where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of a prime. Find $a +b +c$. [b]p21.[/b] In a list of increasing consecutive positive integers, the first item is divisible by $1$, the second item is divisible by $4$, the third item is divisible by $7$, and this pattern increases up to the seventh item being divisible by $19$. Find the remainder when the least possible value of the first item in the list is divided by $100$. [u]Round 8[/u] [b]p22.[/b] Let the answer to Problem $24$ be $C$. Jacob never drinks more than $C$ cups of coffee in a day. He always drinks a positive integer number of cups. The probability that he drinks $C +1-X$ cups is $X$ times the probability he drinks $C$ cups of coffee for any positive number $X$ from $1$ to $C$ inclusive. Find the expected number of cups of coffee he drinks. [b]p23.[/b] Let the answer to Problem $22$ be $A$. Three lines are drawn intersecting the interior of a triangle with side lengths $26$, $28$, and $30$ such that each line is parallel and a distance A away from a respective side. The perimeter of the triangle formed by the three new lines can be expressed as $\frac{a}{b}$ for relatively prime integers $a$ and $b$. Find $a +b$. [b]p24.[/b] Let the answer to Problem $23$ be $B$. Given that $ab-c = bc-a = ca-b$ and $a^2+b^2+c^2 = B +2$, find the sum of all possible values of $|a +b +c|$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166489p28814241]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166500p28814367]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A convex polygon has the property that its vertices are coloured by three colors, each colour occurring at least once and any two adjacent vertices having different colours. Prove that the polygon can be divided into triangles by diagonals, no two of which intersect in the [b]interior[/b] of the polygon, in such a way that all the resulting triangles have vertices of all three colours.

Four boys and four girls each bring one gift to a Christmas gift exchange. On a sheet of paper, each boy randomly writes down the name of one girl, and each girl randomly writes down the name of one boy. At the same time, each person passes their gift to the person whose name is written on their sheet. Determine the probability that [i]both[/i] of these events occur: [list] [*] (i) Each person receives exactly one gift; [*] (ii) No two people exchanged presents with each other (i.e., if $A$ gave his gift to $B$, then $B$ did not give her gift to $A$).[/list]

There are $2009$ cities in country, and every two are connected by road. Businessman and Road Ministry play next game. Every morning Businessman buys one road and every evening Minisrty destroys 10 free roads. Can Business create cyclic route without self-intersections through exactly $75$ different cities?

Find the smallest $n$ such that in an $8\times 8$ chessboard any $n$ cells contain two cells which are at least $3$ knight moves apart from each other.

Let $S = \{1, 2, 3, ..., 100\}$ and $P$ is the collection of all subset $T$ of $S$ that have $49$ elements, or in other words: $$P = \{T \subset S : |T| = 49\}.$$ Every element of $P$ is labelled by the element of $S$ randomly (the labels may be the same). Show that there exist subset $M$ of $S$ that has $50$ members such that for every $x \in M$, the label of $M -\{x\}$ is not equal to $x$

Along a circle-shaped path are $100$ boys and $100$ girls. The distance between two points on the path is defined as the length of the smaller arc through which it is possible to get from one point to the other. Prove that the sum of distances between pairs of the same gender is always less than or equal to the sum of distances between all pairs of a boy and a girl.

$13$ fractions are corrected by using each of the numbers $1,2,...,26$ once.[b]Example:[/b]$\frac{12}{5},\frac{18}{26}.... $ What is the maximum number of fractions which are integers?

Each $1 \times 1$ square of an $n \times n$ table contains a different number. The smallest number in each row is marked, and these marked numbers are in different columns. Then the smallest number in each column is marked, and these marked numbers are in different rows. Prove that the two sets of marked numbers are identical. (V Klepcyn)

There are $ n \ge 3 $ puddings in a room. If a pudding $ A $ hates a pudding $ B $, then $ B $ hates $ A $ as well. Suppose the following two conditions holds: 1. Given any four puddings, there are two puddings who like each other. 2. For any positive integer $ m $, if there are $ m $ puddings who like each other, then there exists $ 3 $ puddings (from the other $ n-m $ puddings) that hate each other. Find the smallest possible value of $ n $.

Let $G$ be a simple graph with $3n^2$ vertices ($n\geq 2$). It is known that the degree of each vertex of $G$ is not greater than $4n$, there exists at least a vertex of degree one, and between any two vertices, there is a path of length $\leq 3$. Prove that the minimum number of edges that $G$ might have is equal to $\frac{(7n^2- 3n)}{2}$.