The squares of an $8 \times 8$ board are colored black and white in such a way that every row and every column contains exactly four black squares. Prove that the number of pairs of neighboring white squares is the same as the number of pairs of neighboring black squares. (Two squares are neighboring if they have a side in common.)

Find the smallest integer $k \geq 2$ such that for every partition of the set $\{2, 3,\hdots, k\}$ into two parts, at least one of these parts contains (not necessarily distinct) numbers $a$, $b$ and $c$ with $ab = c$.

For which natural number $n$ is it possible to draw $n$ line segments between vertices of a regular $2n$-gon so that every vertex is an endpoint for exactly one segment and these segments have pairwise different lengths?

Let $n$ be some natural number. One boss writes $n$ letters a day numerated from 1 to $n$ consecutively. When he writes a letter he piles it up (on top) in a box. When his secretary is free, she gets the letter on the top of the pile and prints it. Sometimes the secretary isn’t able to print the letter before her boss puts another one or more on the pile in the box. Though she is always able to print all of the letters at the end of the day. A permutation is called [i]“printable”[/i] if it is possible for the letters to be printed in this order. Find a formula for the number of [i]“printable”[/i] permutations.

Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=n^{17}$ .Find the maximum possible value of $n$ .

$ m$ and $ n$ are positive integers. In a $ 8 \times 8$ chessboard, $ (m,n)$ denotes the number of grids a Horse can jump in a chessboard ($ m$ horizontal $ n$ vertical or $ n$ horizontal $ m$ vertical ). If a $ (m,n) \textbf{Horse}$ starts from one grid, passes every grid once and only once, then we call this kind of Horse jump route a $ \textbf{H Route}$. For example, the $ (1,2) \textbf{Horse}$ has its $ \textbf{H Route}$. Find the smallest positive integer $ t$, such that from any grid of the chessboard, the $ (t,t\plus{}1) \textbf{Horse}$ does not has any $ \textbf{H Route}$.

Every city in the certain state is connected by airlines with no more than with three other ones, but one can get from every city to every other city changing a plane once only or directly. What is the maximal possible number of the cities?

There are $n$ cards such that for each $i=1,2, \cdots n$, there are exactly one card labeled $i$. Initially the cards are piled with increasing order from top to bottom. There are two operations: [list] [*] $A$ : One can take the top card of the pile and move it to the bottom; [*] $B$ : One can remove the top card from the pile. [/list] The operation $ABBABBABBABB \cdots $ is repeated until only one card gets left. Let $L(n)$ be the labeled number on the final pile. Find all integers $k$ such that $L(3k)=k$.

Determine the smallest positive integer $n$ with the following property: on a board $n \times n$, whose squares are painted in checkerboard pattern (that is, for any two squares with a common edge, one of them is black and the other is white), it is possible to place the numbers $1,2,3 , ... , n^2$, a number in each square, so if $B$ is the sum of the numbers written in the white squares and $P$ is the sum of the numbers written in the black squares, then $\frac {B}{P} = \frac{2021}{4321}$.

Rosemonde is stacking spheres to make pyramids. She constructs two types of pyramids $S_n$ and $T_n$. The pyramid $S_n$ has $n$ layers, where the top layer is a single sphere and the $i^{th}$ layer is an $i\times $i square grid of spheres for each $2 \le i \le n$. Similarly, the pyramid $T_n$ has $n$ layers where the top layer is a single sphere and the $i^{th}$ layer is $\frac{i(i+1)}{2}$ spheres arranged into an equilateral triangle for each $2 \le i \le n$.

A triangular pie has the same shape as its box, except that they are mirror images of each other. We wish to cut the pie in two pieces which can t together in the box without turning either piece over. How can this be done if [list][b](a)[/b] one angle of the triangle is three times as big as another; [b](b)[/b] one angle of the triangle is obtuse and is twice as big as one of the acute angles?[/list]

We number a hundred blank cards on both sides with the numbers $1$ to $100$. The cards are then stacked in order, with the card with the number $1$ on top. The order of the cards is changed step by step as follows: at the $1$st step the top card is turned around, and is put back on top of the stack (nothing changes, of course), at the $2$nd step the topmost $2$ cards are turned around, and put back on top of the stack, up to the $100$th step, in which the entire stack of $100$ cards is turned around. At the $101$st step, again only the top card is turned around, at the $102$nd step, the top most $2$ cards are turned around, and so on. Show that after a finite number of steps, the cards return to their original positions.

The numbers $1, 2,..., 2023$ are written on the board in this order. We can repeatedly perform the following operation with them: We select any odd number of consecutively written numbers and write these numbers in reverse order. How many different orders of these $2023$ numbers can we get? [i]Example[/i]: If we start with only the numbers $1, 2, 3, 4, 5, 6$, we can perform the following steps $$1, 2, 3, 4, 5, 6 \to 3, 2, 1,4, 5, 6 \to 3, 6, 5, 4, 1, 2 \to 3, 6, 1, 4, 5, 2 \to ...$$

[u]Round 1[/u] [b]p1.[/b] What is the remainder when $2021$ is divided by $102$? [b]p2.[/b] Brian has $2$ left shoes and $2$ right shoes. Given that he randomly picks $2$ of the $4$ shoes, the probability he will get a left shoe and a right shoe is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p + q$. [b]p3.[/b] In how many ways can $59$ be written as a sum of two perfect squares? (The order of the two perfect squares does not matter.) [u]Round 2 [/u] [b]p4.[/b] Two positive integers have a sum of $60$. Their least common multiple is $273$. What is the positive diffeerence between the two numbers? [b]p5.[/b] How many ways are there to distribute $13$ identical apples among $4$ identical boxes so that no two boxes receive the same number of apples? A box may receive zero apples. [b]p6.[/b] In square $ABCD$ with side length $5$, $P$ lies on segment $AB$ so that $AP = 3$ and $Q$ lies on segment $AD$ so that $AQ = 4$. Given that the area of triangle $CPQ$ is $x$, compute $2x$. [u]Round 3 [/u] [b]p7.[/b] Find the number of ordered triples $(a, b, c)$ of nonnegative integers such that $2a+3b+5c = 15$. [b]p8.[/b] What is the greatest integer $n \le 15$ such that $n + 1$ and $n^2 + 3$ are both prime? [b]p9.[/b] For positive integers $a, b$, and $c$, suppose that $gcd \,\,(a, b) = 21$, $gcd \,\,(a, c) = 10$, and $gcd \,\,(b,c) = 11$. Find $\frac{abc}{lcm \,\,(a,b,c)}$ . (Note: $gcd$ is the greatest common divisor function and $lcm$ is the least common multiple function.) [u]Round 4[/u] [b]p10.[/b] The vertices of a square in the coordinate plane are at $(0, 0)$, $(0, 6)$, $(6, 0)$, and $(6, 6)$. Line $\ell$ intersects the square at exactly two lattice points (that is, points with integer coordinates). How many such lines $\ell$ are there that divide the square into two regions, one of them having an area of $12$? [b]p11.[/b] Let $f(n)$ be defined as follows for positive integers $n$: $f(1) = 0$, $f(n) = 1$ if $n$ is prime, and $f(n) = f(n - 1) + 1$ otherwise. What is the maximum value of $f(n)$ for $n \le 120$? [b]p12.[/b] The graph of the equation $y = x^3 + ax^2 + bx + c$ passes through the points $(2,4)$, $(3, 9)$, and $(4, 16)$. What is $b$? PS. You should use hide for answers. Rounds 5- 8 have been posted [url=https://artofproblemsolving.com/community/c3h2949415p26408227]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P_1, ... , P_{2556}$ be distinct points in a regular hexagon $ABCDEF$ with unit side length. Suppose that no three points in the set $S = \{A, B, C, D, E, F, P_1, ... , P_{2556}\}$ are collinear. Show that there is a triangle whose vertices are in $S$ and whose area is less than $\frac{1}{1700}$ .

A [i]beautiful configuration[/i] of points is a set of $n$ colored points, such that if a triangle with vertices in the set has an angle of at least $120$ degrees, then exactly 2 of its vertices are colored with the same color. Determine the maximum possible value of $n$.

Prove that there exists a positive constant $c$ such that the following statement is true: Consider an integer $n > 1$, and a set $\mathcal S$ of $n$ points in the plane such that the distance between any two different points in $\mathcal S$ is at least 1. It follows that there is a line $\ell$ separating $\mathcal S$ such that the distance from any point of $\mathcal S$ to $\ell$ is at least $cn^{-1/3}$. (A line $\ell$ separates a set of points S if some segment joining two points in $\mathcal S$ crosses $\ell$.) [i]Note. Weaker results with $cn^{-1/3}$ replaced by $cn^{-\alpha}$ may be awarded points depending on the value of the constant $\alpha > 1/3$.[/i] [i]Proposed by Ting-Feng Lin and Hung-Hsun Hans Yu, Taiwan[/i]

$n\ge 3$ guests met at a party. Some of them know each other but there is no quartet of different guests $a, b, c, d$ such that in pairs $\lbrace a, b \rbrace, \lbrace b, c \rbrace, \lbrace c, d \rbrace, \lbrace d, a \rbrace$ guests know each other but in pairs $\lbrace a, c \rbrace, \lbrace b, d \rbrace$ guests don't know each other. We say a nonempty set of guests $X$ is an [i]ingroup[/i], when guests from $X$ know each other pairwise and there are no guests not from $X$ knowing all guests from $X$. Prove that there are at most $\frac{n(n-1)}{2}$ different ingroups at that party.

A [i]uniform covering[/i] of the integers $1,2,...,n$ is a finite multiset of subsets of $\{1,2,...,n\}$, so that each number lies in the same amount of sets from the covering. A covering may contain the same subset multiple times, it must contain at least one subset, and it may contain the empty subset. For example, $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})$ is a uniform covering of $1,2,3,4$ (every number occurs in two sets). The covering containing only the empty set is also uniform (every number occurs in zero sets). Given two uniform coverings, we define a new uniform covering, their [i]sum[/i] (denoted by $\oplus$), by adding the sets from both coverings. For example: $(\{1\},\{1\},\{2,3\},\{3,4\},\{2,4\})\oplus(\{1\},\{2\},\{3\},\{4\})=$ $(\{1\},\{1\},\{1\},\{2\},\{3\},\{4\},\{2,3\},\{3,4\},\{2,4\})$ A uniform covering is called [i]non-composite[/i] if it's not a sum of two uniform coverings. Prove that for any $n\geq1$, there are only finitely many non-composite uniform coverings of $1,2,...,n$.

Let $n$ be a positive integer. There are $2018n+1$ cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city $C$ and integer $1\le i\le 2018,$ there are exactly $n$ cities that are a distance $i$ away from $C.$ (The [i]distance[/i] between two cities is the least number of roads on any path between the two cities.) For which $n$ is it possible for Mark to achieve this? [i]Proposed by Michael Ren[/i]

In the figures, the vertices are marked with a circle. The segments that join vertices are called paths. Non-negative integers are distributed to the vertices and, to the paths, the differences between the numbers at their ends. [img]https://cdn.artofproblemsolving.com/attachments/d/6/e6fce93719a5b35dbf34d58652b01a8631de57.gif[/img] We will say that a distribution of numbers is [i]graceful [/i] if all the numbers from $1$ to $n$ appear in the paths, where $n$ is the number of paths. The following is an example of graceful distribution: [img]https://cdn.artofproblemsolving.com/attachments/1/1/a8c2b4fde673ca902b655804c4f5321f9666e9.gif[/img] Give -if possible- a graceful distribution for the following figures. If you can't do it, show why.

Six people, with distinct weights, want to form a triangular position where there are three people in the bottom row, two in the middle row, and one in the top row, and each person in the top two rows must weigh less than both of their supports. How many distinct formations are there?

Anselmo and Claudio are playing alternatively a game with fruits in a box. The box initially has $32$ fruits. Anselmo plays first and each turn consists of taking away $1$, $2$ or $3$ fruits from the box or taking away $\frac{2}{3}$ of the fruits from the box (this is only possible when the number of the fruits left in the box is a multiple of $3$). The player that takes away the last fruit from the box wins. Which of these two players has a winning strategy? How should that player play in order to win?

Color every vertex of $2008$-gon with two colors, such that adjacent vertices have different color. If sum of angles of vertices of first color is same as sum of angles of vertices of second color, than we call $2008$-gon as interesting. Convex $2009$-gon one vertex is marked. It is known, that if remove any unmarked vertex, then we get interesting $2008$-gon. Prove, that if we remove marked vertex, then we get interesting $2008$-gon too.

3 countries $A, B, C$ participate in a competition where each country has 9 representatives. The rules are as follows: every round of competition is between 1 competitor each from 2 countries. The winner plays in the next round, while the loser is knocked out. The remaining country will then send a representative to take on the winner of the previous round. The competition begins with $A$ and $B$ sending a competitor each. If all competitors from one country have been knocked out, the competition continues between the remaining 2 countries until another country is knocked out. The remaining team is the champion. [b]I.[/b] At least how many games does the champion team win? [b]II.[/b] If the champion team won 11 matches, at least how many matches were played?