In town $ A,$ there are $ n$ girls and $ n$ boys, and each girl knows each boy. In town $ B,$ there are $ n$ girls $ g_1, g_2, \ldots, g_n$ and $ 2n \minus{} 1$ boys $ b_1, b_2, \ldots, b_{2n\minus{}1}.$ The girl $ g_i,$ $ i \equal{} 1, 2, \ldots, n,$ knows the boys $ b_1, b_2, \ldots, b_{2i\minus{}1},$ and no others. For all $ r \equal{} 1, 2, \ldots, n,$ denote by $ A(r),B(r)$ the number of different ways in which $ r$ girls from town $ A,$ respectively town $ B,$ can dance with $ r$ boys from their own town, forming $ r$ pairs, each girl with a boy she knows. Prove that $ A(r) \equal{} B(r)$ for each $ r \equal{} 1, 2, \ldots, n.$

An $11\times 11$ chess board is covered with one $\boxed{ }$ shaped and forty $\boxed{ }\boxed{ }\boxed{ }$ shaped tiles. Determine the squares where $\boxed{}$ shaped tile can be placed.

Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$. Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have \[ n\mid a_i \minus{} b_i \minus{} c_i \] [i]Proposed by Ricky Liu & Zuming Feng, USA[/i]

Philipe has two congruent regular $4046$-gons. He invites Philomena to a trick he's planning: he'll give her one of the $4046$-gons and she will paint in red $2023$ vertices of her polygon. Without knowing which vertices she chose, he'll paint $k$ vertices of the remaining polygon. After this, he wants to rotate the polygons so that each painted vertices from Philomena's polygon is corresponding to some painted vertice from Philipe's polygon. What is the minimal value of $k$ for which he can choose his vertices so that, no matter how she paints her polygon, the trick is always possible.

There are two families of convex polygons in the plane. Each family has a pair of disjoint polygons. Any polygon from one family intersects any polygon from the other family. Show that there is a line which intersects all the polygons.

Let $S$ be the set of all n-tuples $(X_1,...,X_n)$ of subsets of the set $\{1,2,..,1000\}$, not necessarily different and not necessarily nonempty. For $a = (X_1,...,X_n)$ denote by $E(a)$ the number of elements of $X_1\cup ... \cup X_n$. Find an explicit formula for the sum $\sum_{a\in S} E(a)$

[b]p1.[/b] Is there an integer such that the product of all whose digits equals $99$ ? [b]p2.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p3.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/9/f/359d3b987012de1f3318c3f06710daabe66f28.png[/img] [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $5$ rocks in the first pile and $6$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p5.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

It is possible to write the numbers $111$, $112$, $121$, $122$, $211$, $212$, $221$ and $222$ at the vertices of a cube, so that the numbers written in adjacent vertices match at most in one digit?

Suppose that every integer is colored using one of $4$ colors. Let $m, n$ be distinct odd integers such that $m + n \ne 0$. Prove that there exist integers $a$, $ b$ of the same color such that $ a - b$ equals one of the numbers $m$, $n$, $m - n$, $m + n$.

Let $M$ be a point which has coordinates $(p\times 1994,7p\times 1994)$ in the Cartesian plane ($p$ is a prime). Find the number of right-triangles satisfying the following conditions: (1) all vertexes of the triangle are lattice points, moreover $M$ is on the right-angled corner of the triangle; (2) the origin ($0,0$) is the incenter of the triangle.

1) Is it possible to place five circles on the plane in such way that each circle has exactly 5 common points with other circles? 2) Is it possible to place five circles on the plane in such way that each circle has exactly 6 common points with other circles? 3) Is it possible to place five circles on the plane in such way that each circle has exactly 7 common points with other circles?

There are $2014$ balls with $106$ different colors, $19$ of each color. Determine the least possible value of $n$ so that no matter how these balls are arranged around a circle, one can choose $n$ consecutive balls so that amongst them, there are $53$ balls with different colors.

How many $5\times5$ grids are possible such that each element is either $0$ or $1$ and each row sum and column sum is $4$? $\textbf{(A)}~64$ $\textbf{(B)}~32$ $\textbf{(C)}~120$ $\textbf{(D)}~96$

The numbers from $1$ to $64$ must be written on the small squares of a chessboard, with a different number in each small square. Consider the $112$ numbers you can make by adding the numbers in two small squares which have a common edge. Is it possible to write the numbers in the squares so that these $112$ sums are all different?

Eight rooks are placed on a chessboard so that no two rooks attack each other. Prove that one can always move all rooks, each by a move of a knight so that in the final position no two rooks attack each other as well. (In intermediate positions several rooks can share the same square).

On a \(4 \times 4\) board, each cell is colored either black or white such that each row and each column have an even number of black cells. How many ways can the board be colored?

Let $n\geq 2$ be a fixed positive integer. Let $\{a_1,a_2,...,a_n\}$ be fixed positive integers whose sum is $2n-1$. Denote by $S_{\mathbb{A}}$ the sum of elements of a set $A$. Find the minimal and maximal value of $S_{\mathbb{X}}\cdot S_{\mathbb{Y}}$ where $\mathbb{X}$ and $\mathbb{Y}$ are two sets with the property that $\mathbb{X}\cup \mathbb{Y}=\{a_1,a_2,...,a_n\}$ and $\mathbb{X}\cap \mathbb{Y}=\emptyset.$ [i]Note: $\mathbb{X}$ and $\mathbb{Y}$ can have multiple equal elements. For example, when $n=5$ and $a_1=...=a_4=1$ and $a_5=5$, we can consider $\mathbb{X}=\{1,1,1\}$ and $\mathbb{Y}=\{1,5\}$. Moreover, in this case, $S_\mathbb{X}=3$ and $S_{\mathbb{Y}}=6.$[/i]

Three friends Archie, Billie, and Charlie play a game. At the beginning of the game, each of them has a pile of $2024$ pebbles. Archie makes the first move, Billie makes the second, Charlie makes the third and they continue to make moves in the same order. In each move, the player making the move must choose a positive integer $n$ greater than any previously chosen number by any player, take $2n$ pebbles from his pile and distribute them equally to the other two players. If a player cannot make a move, the game ends and that player loses the game. $\hspace{5px}$ Determine all the players who have a strategy such that, regardless of how the other two players play, they will not lose the game. [i]Proposed by Ilija Jovčeski, Macedonia[/i]

Consider a $8\times 8$ chessboard where all $64$ unit squares are at the start white. Prove that, if any $12$ of the $64$ unit square get painted black, then we can find $4$ lines and $4$ rows that have all these $12$ unit squares.

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]

Merlin summons the $n$ knights of Camelot for a conference. Each day, he assigns them to the $n$ seats at the Round Table. From the second day on, any two neighbours may interchange their seats if they were not neighbours on the first day. The knights try to sit in some cyclic order which has already occurred before on an earlier day. If they succeed, then the conference comes to an end when the day is over. What is the maximum number of days for which Merlin can guarantee that the conference will last?

This season, there are $3n + 1$ teams in the MLS (Major League Soccer). As of now, each team has played exactly $n -1$ matches. Prove that there exist $4$ teams such that none of the $4$ teams have faced each other.

A graph has $100$ points. Given any four points, there is one joined to the other three. Show that one point must be joined to all $99$ other points. What is the smallest number possible of such points (that are joined to all the others)?

Let $U=\{1,2,\ldots ,n\}$, where $n\geq 3$. A subset $S$ of $U$ is said to be [i]split[/i] by an arrangement of the elements of $U$ if an element not in $S$ occurs in the arrangement somewhere between two elements of $S$. For example, 13542 splits $\{1,2,3\}$ but not $\{3,4,5\}$. Prove that for any $n-2$ subsets of $U$, each containing at least 2 and at most $n-1$ elements, there is an arrangement of the elements of $U$ which splits all of them.

Given a (fixed) positive integer $N$, solve the functional equation \[f \colon \mathbb{Z} \to \mathbb{R}, \ f(2k) = 2f(k) \textrm{ and } f(N-k) = f(k), \ \textrm{for all } k \in \mathbb{Z}.\] [i](Dan Schwarz)[/i]