Given $n$ sets $A_i$, with $| A_i | = n$, prove they may be indexed $A_i = \{a_{i,j} \mid j=1,2,\ldots,n \}$, in such way that the sets $B_j = \{a_{i,j} \mid i=1,2,\ldots,n \}$, $1\leq j\leq n$, also have $| B_j | = n$. (Anonymous)

Consider the lattice $ L$ of the contradictions of a simple graph $ G$ (as sets of vertex pairs) with respect to inclusion. Let $ n \geq 1$ be an arbitrary integer. Show that the identity \[ x \bigwedge \left( \bigvee_{i\equal{}0}^n y_i \right) \equal{} \bigvee_{j\equal{}0}^n \left( x \bigwedge \left( \bigvee_{0 \leq i \leq n, \;i\not\equal{}j\ } y_i \right)\right)\] holds if and only if $ G$ has no cycle of size at least $ n\plus{}2$. [i]A. Huhn[/i]

Consider an $m \times n$ grid of unit squares. Let $R$ be the total number of rectangles of any size, and let $S$ be the total number of squares of any size. Assume that the sides of the rectangles and squares are parallel to the sides of the $m \times n$ grid. If $\frac{R}{S} =\frac{759}{50}$ , then determine $mn$.

Mother baked $15$ pasties. She placed them on a round plate in a circular way: $7$ with cabbage, $7$ with meat and one with cherries in that exact order and put the plate into a microwave. All pasties look the same but Olga knows the order. However she doesn't know how the plate has been rotated in the microwave. She wants to eat a pasty with cherries. Can Olga eat her favourite pasty for sure if she is not allowed to try more than three other pasties?

Drawing all diagonals in a regular $2013$-gon, the regular $2013$-gon is divided into non-overlapping polygons. Prove that there exist exactly one $2013$-gon out of all such polygons.

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 $3$.

An ant walks along the lines of a grid made up of $55$ horizontal lines and $45$ vertical lines. You want to paint some sections of lines so that the ant can go from any intersection to any other intersection, walking exclusively along painted sections. If the distance between consecutive lines is $10$ cm, what is the least possible number of centimeters that should be painted? What is the higher value?

Some towns in a country are connected by two–way roads, so that for any two towns there is a unique path along the roads connecting them. It is known that there is exactly 100 towns which are directly connected to only one town. Prove that we can construct 50 new roads in order to obtain a net in which every two towns will be connected even if one road gets closed.

$m\times n$ grid is tiled by mosaics $2\times2$ and $1\times3$ (horizontal and vertical). Prove that the number of ways to choose a $1\times2$ rectangle (horizontal and vertical) such that one of its cells is tiled by $2\times2$ mosaic and the other cell is tiled by $1\times3$ mosaic [horizontal and vertical] is an even number.

$234$ viewers came to the cinema. Determine for which$ n \ge 4$ the viewers could be can be arranged in $n$ rows so that every viewer in $i$-th row gets to know just $j$ viewers in $j$-th row for any $i, j \in \{1, 2,... , n\}, i\ne j$. (The relationship of acquaintance is mutual.) (Tomáš Jurík)

Two sets of intervals $A ,B$ on the line are given. The set $A$ contains $2m-1$ intervals, every two of which have an interior point in common. Moreover, every interval from $A$ contains at least two disjoint intervals from $B$. Show that there exists an interval in $B$ which belongs to at least $m$ intervals from $A$ .

A clock hand is attached to every face of a convex polyhedron. Each hand always points towards a neighboring face and every minute, exactly one of the hands turns clockwise to point at the next face. Suppose that the hands on neighboring faces never point towards one another. Show that one of the hands makes only finitely many turns.

Matthias covered a $7 \times 7$ square board, divided into $1 \times 1$ squares, with pieces of the following three types without gaps or overlaps, and without going off the board. [img]https://cdn.artofproblemsolving.com/attachments/9/9/8a2e63f723cbdf188f22344054f364f1924d47.gif[/img] Each type $1$ piece covers exactly $3$ squares and each type $2$ or type $3$ piece covers exactly $4$ squares. Determine the number of pieces of type $1$ that Matías could have used. (Pieces can be rotated and flipped.)

Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible. [i]Proposed by Horst Sewerin, Germany[/i]

On a board 12 × 12 are placed some knights in such a way that in each 2 × 2 square there is at least one knight. Find the maximum number of squares that are not attacked by knights. (A knight does not attack the square in which it is located.)

Let $n \geq 3$ be an integer. A labelling of the $n$ vertices, the $n$ sides and the interior of a regular $n$-gon by $2n + 1$ distinct integers is called [i]memorable[/i] if the following conditions hold: (a) Each side has a label that is the arithmetic mean of the labels of its endpoints. (b) The interior of the $n$-gon has a label that is the arithmetic mean of the labels of all the vertices. Determine all integers $n \geq 3$ for which there exists a memorable labelling of a regular $n$-gon consisting of $2n + 1$ consecutive integers.

Let \[M=\{1, 2, 3, \ldots, 2022\}\] Determine the least positive integer $k$, such that for every $k$ subsets of $M$ with the cardinality of each subset equal to $3$, there are two of these subsets with exactly one common element.

There are $30$ apparently equal balls, $15$ of which have the weight $a$ and the remaining $15$ have the weight $b$ with $a \ne b$. The balls are to be partitioned into two groups of $15$, according to their weight. An assistant partitioned them into two groups, and we wish to check if this partition is correct. How can we check that with as few weighings as possible?

Alma and Bertha play the following game. There are $100$ round, $200$ triangular and $200$ square pieces on a table. In each move a player must remove two pieces, but it cannot be a triangle and a square. Alma starts, and one loses if one is unable to move or if there are no pieces left when it is one’s turn. Which player has a winning strategy?

In a $7\times 7\times 7$ cube, the unit cubes are colored white, black and gray colors so that for any two colors the number of cubes of these two colors are different. In this case, $N$ parallel rows of $7$ cubes were found, each of which there are more white cubes than gray and than black. Likewise, there were $N$ parallel rows of $7$ cubes, each of which contained gray there are more cubes than white and than black, and there are also N parallel rows of $7$ cubes, each of which contains more black cubes than white ones and than gray ones. What is the largest $N$ for which this is possible?

Let $K$ be a cube with edge $n$, where $n>2$ is an even integer. Cube $K$ is divided into $n^3$ unit cubes. We call any set of $n^2$ unit cubes lying on the same horizontal or vertical level a layer. We dispose of $\frac{n^3}4$ colors, in each of which we paint exactly $4$ unit cubes. Prove that we can always select $n$ unit cubes of distinct colors, no two of which lie on the same layer.

Q. For integers $m,n\geq 1$, Let $A_{m,n}$ , $B_{m,n}$ and $C_{m,n}$ denote the following sets: $A_{m,n}=\{(\alpha _1,\alpha _2,\ldots,\alpha _m) \colon 1\leq \alpha _1\leq \alpha_2 \leq \ldots \leq \alpha_m\leq n\}$ given that $\alpha _i \in \mathbb{Z}$ for all $i$ $B_{m,n}=\{(\alpha _1,\alpha _2,\ldots ,\alpha _m) \colon \alpha _1+\alpha _2+\ldots + \alpha _m=n\}$ given that $\alpha _i \geq 0$ and $\alpha_ i\in \mathbb{Z}$ for all $i$ $C_{m,n}=\{(\alpha _1,\alpha _2,\ldots,\alpha _m)\colon 1\leq \alpha _1< \alpha_2 < \ldots< \alpha_m\leq n\}$ given that $\alpha _i \in \mathbb{Z}$ for all $i$ $(a)$ Define a one-one onto map from $A_{m,n}$ onto $B_{m+1,n-1}$. $(b)$ Define a one-one onto map from $A_{m,n}$ onto $C_{m,n+m-1}$. $(c)$ Find the number of elements of the sets $A_{m,n}$ and $B_{m,n}$.

There are two kinds of creatures living in the flatland of Pentagonia: the Spires ($S$) and the Bones ($B$). They all have the shape of an isosceles triangle: the Spiers have an apical angle of $36^o$ and the bones an apical angle of $108^o$. Every year on [i]Great Day of Division[/i] (September 11 - the day this Olympiad was held) they divide into pieces: each $S$ into two smaller $S$'s and a $B$; each $B$ in an $S$ and a $B$. Over the course of the year they then grow back to adult proportions. In the distant past, the population originated from one $B$-being. Deaths do not occur. Investigate whether the ratio between the number of Spires and the number of Bones will eventually approach a limit value and if so, calculate that limit value.

Let $\mathbb{N} = \{1, 2, 3, \dots\}$ be the set of all positive integers, and let $f$ be a bijection from $\mathbb{N}$ to $\mathbb{N}$. Must there exist some positive integer $n$ such that $(f(1), f(2), \dots, f(n))$ is a permutation of $(1, 2, \dots, n)$?

A maths teacher has $10$ cards with the numbers $1$ to $10$ on them, one number per card. She places these cards in some order in a line next to each other on the table. The students come to the table, one at a time. The student whose turn it is goes once through the line of cards from left to right and removes every card she encounters that is (at that moment) the lowest card on the table. This continues till all cards are removed from the table. For example, if the line is in order $3$, $1$, $4$, $5$, $8,$ $6$, $9$, $10$, $2$, $7$ from left to right, the first student takes cards $1$ and $2$. Then the second student comes who, in our example, takes the cards $3$, $4$, $5$, $6$, and $7$. The third student then takes the cards $8$, $9$, and $10$. Let $A$ be the number of sequences of cards that the teacher can choose so that exactly nine students get a turn to pick cards. Let $B$ be the number of sequences of cards that the teacher can choose so that exactly two students get a turn to pick cards. Prove that $A = B$.