Let $n$ be an integer greater than $3$. Show that it is possible to divide a square into $n^2 + 1$ or more disjointed rectangles and with sides parallel to those of the square so that any line parallel to one of the sides intersects at most the interior of $n$ rectangles. Note: We say that two rectangles are [i]disjointed [/i] if they do not intersect or only intersect at their perimeters.

Let $A = \{1,2,3,4,5,6,7,8\}$, $B = \{9,10,11,12,13,14,15,16\}$ and $C =\{17,18,19,20,21,22,23,24\}$. Find the number of triples $(x, y, z)$ such that $x \in A, y \in B, z \in C $ and $x + y + z = 36$.

Grid the plane forming an infinite board. In each cell of this board, there is a lamp, initially turned off. A permitted operation consists of selecting a square of \(3\times 3\), \(4\times 4\), or \(5\times 5\) cells and changing the state of all lamps in that square (those that are off become on, and those that are on become off). (a) Prove that for any finite set of lamps, it is possible to achieve, through a finite sequence of permitted operations, that those are the only lamps turned on on the board. (b) Prove that if in a sequence of permitted operations only two out of the three square sizes are used, then it is impossible to achieve that at the end the only lamps turned on on the board are those in a \(2\times 2\) square.

Among the points corresponding to number $1,2,...,2n$ on the real line, $n$ are colored in blue and $n$ in red. Let $a_1,a_2,...,a_n$ be the blue points and $b_1,b_2,...,b_n$ be the red points. Prove that the sum $\mid a_1-b_1\mid+...+\mid a_n-b_n\mid$ does not depend on coloring , and compute its value. :roll:

At least $3$ players take part in a tennis tournament. Each participant plays exactly one match against each other participant. After the tournament has ended, we find out that each player has won at least one match. (There are no ties in tennis). Show that in the tournament, there was at least one trio of players $A,B,C$ such that $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$.

Let $m$ be a nonnegative integer. Show that the number of tilings of a $(2m + 2) \times (2m + 2)$ grid of squares by $1 \times 2$ or $2 \times 1$ rectangles is at least $$2 \cdot 2^{\frac{5}{2}m} \cdot 5120^{\frac{1}{8}m^2}.$$ [i]Proposed by Milan Haiman[/i]

Andrew, Benji, and Carlson want to split a pile of $5$ indistinguishable left shoes and $7$ indistinguishable right shoes. Andrew is practical and wants the same number of left and right shoes. Benji is greedy and wants the most shoes out of the three of them. Carlson is a trickster and wants Benji to have a different number of left and right shoes. How many ways are there to split up the shoes in a way that suits everyone’s desires?

Let $n$ be a positive integer. Determine the number of sequences $a_0, a_1, \ldots, a_n$ with terms in the set $\{0,1,2,3\}$ such that $$n=a_0+2a_1+2^2a_2+\ldots+2^na_n.$$

a) A castle is equilateral triangle with the side of $100$ metres. It is divided onto $100$ triangle rooms. Each wall between the rooms is $10$ metres long and contain one door. You are inside and are allowed to pass through every door not more than once. Prove that you can visit not more than $91$ room (not exiting the castle). b) Every side of the triangle is divided onto $k$ parts by the lines parallel to the sides. And the triangle is divided onto $k^2$ small triangles. Let us call the "chain" such a sequence of triangles, that every triangle in it is included only once, and the consecutive triangles have the common side. What is the greatest possible number of the triangles in the chain?

A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$? [i]Proposed by Dylan Toh[/i]

There are $1 000 000$ soldiers in a line. The sergeant splits the line into $100$ segments (the length of different segments may be different) and permutes the segments (not changing the order of soldiers in each segment) forming a new line. The sergeant repeats this procedure several times (splits the new line in segments of the same lengths and permutes them in exactly the same way as the first time). Every soldier originally from the first segment recorded the number of performed procedures that took him to return to the first segment for the first time. Prove that at most $100$ of these numbers are different.

Let $n \geq 1$ be an integer, and let $S$ be a set of integer pairs $(a,b)$ with $1 \leq a < b \leq 2^n$. Assume $|S| > n \cdot 2^{n+1}$. Prove that there exists four integers $a < b < c < d$ such that $S$ contains all three pairs $(a,c)$, $(b,d)$ and $(a,d)$.

It is known that from these five segments it is possible to form four different right triangles. Find the ratio of the largest segment to the smallest.

Let $n \geq 2$ be a natural number, and let $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ be a permutation of $\left(1;\;2;\;...;\;n\right)$. For any integer $k$ with $1 \leq k \leq n$, we place $a_k$ raisins on the position $k$ of the real number axis. [The real number axis is the $x$-axis of a Cartesian coordinate system.] Now, we place three children A, B, C on the positions $x_A$, $x_B$, $x_C$, each of the numbers $x_A$, $x_B$, $x_C$ being an element of $\left\{1;\;2;\;...;\;n\right\}$. [It is not forbidden to place different children on the same place!] For any $k$, the $a_k$ raisins placed on the position $k$ are equally handed out to those children whose positions are next to $k$. [So, if there is only one child lying next to $k$, then he gets the raisin. If there are two children lying next to $k$ (either both on the same position or symmetric with respect to $k$), then each of them gets one half of the raisin. Etc..] After all raisins are distributed, a child is unhappy if he could have received more raisins than he actually has received if he had moved to another place (while the other children would rest on their places). For which $n$ does there exist a configuration $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ and numbers $x_A$, $x_B$, $x_C$, such that all three children are happy?

Let $m\ge 2$ be an integer, $A$ a finite set of integers (not necessarily positive) and $B_1,B_2,...,B_m$ subsets of $A$. Suppose that, for every $k=1,2,...,m$, the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $\dfrac{m}{2}$ elements.

Two persons, A and B, set up an incantation contest in which they spell incantations (i.e. a finite sequence of letters) alternately. They must obey the following rules: i) Any incantation can appear no more than once; ii) Except for the first incantation, any incantation must be obtained by permuting the letters of the last one before it, or deleting one letter from the last incantation before it; iii)The first person who cannot spell an incantation loses the contest. Answer the following questions: a) If A says '$STAGEPREIMO$' first, then who will win? b) Let $M$ be the set of all possible incantations whose lengths (i.e. the numbers of letters in them) are $2009$ and containing only four letters $A,B,C,D$, each of them appearing at least once. Find the first incantation (arranged in dictionary order) in $M$ such that A has a winning strategy by starting with it.

Place in the cells the boards measuring: a) $2 \times 2$, b) $4 \times 4$, c) $2n \times 2n$, numbers $0$, $1$ and $-1$ so that in each case all the sums of numbers in rows and columns are different.

We call a subset $B$ of natural numbers [i]loyal[/i] if there exists natural numbers $i\le j$ such that $B=\{i,i+1,\ldots,j\}$. Let $Q$ be the set of all [i]loyal[/i] sets. For every subset $A=\{a_1<a_2<\ldots<a_k\}$ of $\{1,2,\ldots,n\}$ we set \[f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i}\qquad\text{and}\qquad g(A)=\max_{B\subseteq A, B\in Q} |B|.\] Furthermore, we define \[F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A)\qquad\text{and}\qquad G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A).\] Prove that there exists $m\in \mathbb N$ such that for each natural number $n>m$ we have $F(n)>G(n)$. (By $|A|$ we mean the number of elements of $A$, and if $|A|\le 1$, we define $f(A)$ to be zero). [i]Proposed by Javad Abedi[/i]

Chess piece called [i]skew knight[/i], if placed on the black square, attacks all the gray squares. [img]https://i.ibb.co/HdTDNjN/Kyiv-MO-2021-Round-1-11-2.png[/img] What is the largest number of such knights that can be placed on the $8\times 8$ chessboard without them attacking each other? [i]Proposed by Arsenii Nikolaiev[/i]

All sides and diagonals of a convex $n$-gon, $n\ge 3$, are coloured one of two colours. Show that there exist $\left[\frac{n+1}{3}\right]$ pairwise disjoint monochromatic segments. [i](Two segments are disjoint if they do not share an endpoint or an interior point).[/i]

There is $n$ black points in the plane.We do the following algorithm: Start from any point from those $n$ points and colour it red. Then connect this point to the nearest black point available and colour this new point red. Then do the same with this point but at any step , but you are never allowed to draw a line which intersects on of the current drawn segments. If you reach an intersection , the algorithm is over. Is it true that for any $n$ and at any initial position , we can start from a point st in the algorithm , we reach all the points?

Take any 26 distinct numbers from {1, 2, ... , 100}. Show that there must be a non-empty subset of the $ 26$ whose product is a square. [hide] I think that the upper limit for such subset is 37.[/hide]

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?

The points $A_1,A_2, \ldots , A_{1983}$ are set on the circumference of a circle and each is given one of the values $\pm 1$. Show that if the number of points with the value $+1$ is greater than $1789$, then at least $1207$ of the points will have the property that the partial sums that can be formed by taking the numbers from them to any other point, in either direction, are strictly positive.

Find the greatest number of depicted pieces composed of $4$ unit squares that can be placed without overlapping on an $n \times n$ grid (where n is a positive integer) in such a way that it is possible to move from some corner to the opposite corner via uncovered squares (moving between squares requires a common edge). The shapes can be rotated and reflected. [img]https://cdn.artofproblemsolving.com/attachments/b/d/f2978a24fdd737edfafa5927a8d2129eb586ee.png[/img]