In the cells of a square table $n$ x $n$ the numbers $1,2,...,n^2$ are written in an arbitrary way. Prove that there exist two adjacent cells, for which the difference between the numbers written in them is no lesser than $n$.

A 3x3 magic square, with magic number $m$, is a $3\times 3$ matrix such that the entries on each row, each column and each diagonal sum to $m$. Show that if the square has positive integer entries, then $m$ is divisible by $3$, and each entry of the square is at most $2n-1$, where $m=3n$. An example of a magic square with $m=6$ is \[\left( \begin{array}{ccccc} 2 & 1 & 3\\ 3 & 2 & 1\\ 1 & 3 & 2 \end{array} \right)\]

Consider the set $A=\{1,2,3\ldots ,2^n\}, n\ge 2$. Find the number of subsets $B$ of $A$ such that for any two elements of $A$ whose sum is a power of $2$ exactly one of them is in $B$. [i]Aleksandar Ivanov[/i]

How many triangles are in this figure? [asy] import olympiad; pair A = (0,0); pair B = (0,1); pair C = (0,2); pair D = (0,3); pair E = (0,4); pair F = (1,0); pair G = (2,0); pair H = (3,0); pair I = (4,0); pair J = (1,4); pair K = (2,4); pair L = (3,4); pair M = (4,4); pair N = (4,3); pair O = (4,2); pair P = (4,1); draw(A--E--I--A); draw(M--E--I--M); draw(B--F); draw(C--G); draw(D--H); draw(L--N); draw(O--K); draw(P--J); draw(B--F); draw(B--F); draw(H--P); draw(G--O); draw(F--N); draw(B--L); draw(C--K); draw(D--J); draw(A--M); [/asy] $(A) 56$ $(B) 60$ $(C) 64$ $(D) 68$

Consider $m$ segments on the real line. Each segment has its two endpoints in the set of integers $\{1, 2, \ldots, 2024\}$, and no two segments have the same length. No segment is entirely contained in another segment, but two segments may partially overlap each other. What is the maximum value of $m$?

A group of $n$ people play a board game with the following rules: 1) In each round of the game exactly $3$ people play 2) The game ends after exactly $n$ rounds 3) Every pair of players has played together at least at one round Find the largest possible value of $n$

$10$ people attended a party. For every $3$ people, there exist at least $2$ people who don’t know each other. Prove that there exist $4$ people who don’t know each other.

[b]p1.[/b] $2011$ distinct points are arranged along the perimeter of a circle. We choose without replacement four points $P$, $Q$, $R$, $S$. What is the probability that no two of the segments $P Q$, $QR$, $RS$, $SP$ intersect (disregarding the endpoints)? [b]p2.[/b] In Soviet Russia, all phone numbers are between three and six digits and contain only the digits $1$, $2$, and $3$. No phone number may be the prefix of another phone number, so, for example, we cannot have the phone numbers $123$ and $12332$. If the Soviet bureaucracy has preassigned $10$ phone numbers of length $3$, $20$ numbers of length $4$, and $77$ phone numbers of length $6$, what is the maximum number of phone numbers of length $5$ that the authorities can allocate? [b]p3.[/b] The sequence $\{a_n\}_{n\ge 1}$ is defined as follows: we have $a_1 = 1$, $a_2 = 0$, and for $n \ge 3$ we have $$a_n = \frac12 \sum\limits_{\substack{1\le i,j\\ i+j+k=n}} a_ia_ja_k.$$ Find $$\sum^{\infty}_{n=1} \frac{a_n}{2^n}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We have many $\text{three-element}$ subsets of a $1000\text{-element}$ set. We know that the union of every $5$ of them has at least $12$ elements. Find the most possible value for the number of these subsets.

Ana and Banana are playing a game. First Ana picks a word, which is defined to be a nonempty sequence of capital English letters. (The word does not need to be a valid English word.) Then Banana picks a nonnegative integer $k$ and challenges Ana to supply a word with exactly $k$ subsequences which are equal to Ana's word. Ana wins if she is able to supply such a word, otherwise she loses. For example, if Ana picks the word "TST", and Banana chooses $k=4$, then Ana can supply the word "TSTST" which has 4 subsequences which are equal to Ana's word. Which words can Ana pick so that she wins no matter what value of $k$ Banana chooses? (The subsequences of a string of length $n$ are the $2^n$ strings which are formed by deleting some of its characters, possibly all or none, while preserving the order of the remaining characters.) [i]Proposed by Kevin Sun

An equilateral triangle is partitioned into smaller equilateral triangular pieces. Prove that two of the pieces are the same size.

Group of $10$ people are buying books. We know the following: $i)$ Every person bought four different books $ii)$ Every two persons bought at least one book common for both of them Taking in consideration book which was bought by maximum number of people, determine minimal value of that number

There are two round tables with $n{}$ dwarves sitting at each table. Each dwarf has only two friends: his neighbours to the left and to the right. A good wizard wants to seat the dwarves at one round table so that each two neighbours are friends. His magic allows him to make any $2n$ pairs of dwarves into pairs of friends (the dwarves in a pair may be from the same or from different tables). However, he knows that an evil sorcerer will break $n{}$ of those new friendships. For which $n{}$ is the good wizard able to achieve his goal no matter what the evil sorcerer does? [i]Mikhail Svyatlovskiy[/i]

Six straight lines are given in space. Among any three of them, two are perpendicular. Show that the given lines can be labeled $\ell_1,...,\ell_6$ in such a way that $\ell_1, \ell_2, \ell_3$ are pairwise perpendicular, and so are $\ell_4, \ell_5, \ell_6$.

There are $n$ matches lying on a table, forming $n$ one-match piles. Adam wants to combine them into a single pile of $n$ matches. He will do this using $n-1$ operations, each of which consists of combining two piles into one. Adam has made a deal with Bartek that every time he combines a pile of $a$ matches with a pile of $b$ matches, he will receive $a\cdot b$ candies from Bartek. What is the greatest number of candies that Adam can receive after performing $n-1$ operations? Justify your answer.

[u]Round 5 [/u] [b]p13.[/b] Circles $\omega_1$, $\omega_2$, and $\omega_3$ have radii $8$, $5$, and $5$, respectively, and each is externally tangent to the other two. Circle $\omega_4$ is internally tangent to $\omega_1$, $\omega_2$, and $\omega_3$, and circle $\omega_5$ is externally tangent to the same three circles. Find the product of the radii of $\omega_4$ and $\omega_5$. [b]p14.[/b] Pythagoras has a regular pentagon with area $1$. He connects each pair of non-adjacent vertices with a line segment, which divides the pentagon into ten triangular regions and one pentagonal region. He colors in all of the obtuse triangles. He then repeats this process using the smaller pentagon. If he continues this process an infinite number of times, what is the total area that he colors in? Please rationalize the denominator of your answer. p15. Maisy arranges $61$ ordinary yellow tennis balls and $3$ special purple tennis balls into a $4 \times 4 \times 4$ cube. (All tennis balls are the same size.) If she chooses the tennis balls’ positions in the cube randomly, what is the probability that no two purple tennis balls are touching? [u]Round 6 [/u] [b]p16.[/b] Points $A, B, C$, and $D$ lie on a line (in that order), and $\vartriangle BCE$ is isosceles with $\overline{BE} = \overline{CE}$. Furthermore, $F$ lies on $\overline{BE}$ and $G$ lies on $\overline{CE}$ such that $\vartriangle BFD$ and $\vartriangle CGA$ are both congruent to $\vartriangle BCE$. Let $H$ be the intersection of $\overline{DF}$ and $\overline{AG}$, and let $I$ be the intersection of $\overline{BE}$ and $\overline{AG}$. If $m \angle BCE = arcsin \left( \frac{12}{13} \right)$, what is $\frac{\overline{HI}}{\overline{FI}}$ ? [b]p17.[/b] Three states are said to form a tri-state area if each state borders the other two. What is the maximum possible number of tri-state areas in a country with fifty states? Note that states must be contiguous and that states touching only at “corners” do not count as bordering. [b]p18.[/b] Let $a, b, c, d$, and $e$ be integers satisfying $$2(\sqrt[3]{2})^2 + \sqrt[3]{2}a + 2b + (\sqrt[3]{2})^2c +\sqrt[3]{2}d + e = 0$$ and $$25\sqrt5 i + 25a - 5\sqrt5 ib - 5c + \sqrt5 id + e = 0$$ where $i =\sqrt{-1}$. Find $|a + b + c + d + e|$. [u]Round 7[/u] [b]p19.[/b] What is the greatest number of regions that $100$ ellipses can divide the plane into? Include the unbounded region. [b]p20.[/b] All of the faces of the convex polyhedron $P$ are congruent isosceles (but NOT equilateral) triangles that meet in such a way that each vertex of the polyhedron is the meeting point of either ten base angles of the faces or three vertex angles of the faces. (An isosceles triangle has two base angles and one vertex angle.) Find the sum of the numbers of faces, edges, and vertices of $P$. [b]p21.[/b] Find the number of ordered $2018$-tuples of integers $(x_1, x_2, .... x_{2018})$, where each integer is between $-2018^2$ and $2018^2$ (inclusive), satisfying $$6(1x_1 + 2x_2 +...· + 2018x_{2018})^2 \ge (2018)(2019)(4037)(x^2_1 + x^2_2 + ... + x^2_{2018}).$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2784936p24472982]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $d$ be an even positive integer. John writes the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2 $ on the blackboard and then chooses three of them, let them be ${a_1}, {a_2}, {a_3}$, erases them and writes the number $1+ \displaystyle\sum_{1\le i<j\leq 3} |{a_i} -{a_j}|$ He continues until two numbers remain written on on the blackboard. Prove that the sum of squares of those two numbers is different than the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2$. (Albania)

Fibonacci numbers are defined as follows: $F_0 = F_1 = 1, F_{n+2} = F_{n+1}+F_n, n \geq 0$. Let $a_n$ be the number of words that consist of $n$ letters $0$ or $1$ and contain no two letters $1$ at distance two from each other. Express $a_n$ in terms of Fibonacci numbers.

The $1 \times 1$ cells located around the perimeter of a $4 \times 4$ square are filled with the numbers $1, 2, \ldots, 12$ so that the sums along each of the four sides are equal. In the upper left corner cell is the number $1$, in the upper right - the number $5$, and in the lower right - the number $11$. [img]https://i.ibb.co/PM0ry1D/Kyiv-City-MO-2021-Round-1-10-2.png[/img] Under these conditions, what number can be located in the last corner cell? [i]Proposed by Mariia Rozhkova[/i]

Let $n$ be the answer to this problem. Find the minimum number of colors needed to color the divisors of $(n - 24)!$ such that no two distinct divisors $s, t$ of the same color satisfy $s | t$.

A square of side $1$ is covered with $m^2$ rectangles. Show that there is a rectangle with perimeter at least $\frac{4}{m}$.

In a grid of $100\times 100$ squares, there is a mouse on the top-left square, and there is a piece of cheese in the bottom-right square. The mouse wants to move to the bottom-right square to eat the cheese. For each step, the mouse can move from one square to an adjacent square (two squares are considered adjacent if they share a common edge). Now, any divider can be placed on the common edge of two adjacent squares such that the mouse cannot directly move between these two adjacent squares. A placement of dividers is called "kind" if the mouse can still reach the cheese after the dividers are placed. Find the smallest positive integer $n$ such that, regardless of any "kind" placement of $2023$ dividers, the mouse can reach the cheese in at most $n$ steps.

Let $n\ge 2$ be an integer. Show that there exists a subset $A\in \{1,2,\ldots ,n\}$ such that: i) The number of elements of $A$ is at most $2\lfloor\sqrt{n}\rfloor+1$ ii) $\{ |x-y| \mid x,y\in A, x\not= y\} = \{ 1,2,\ldots n-1 \}$ [i]Radu Todor[/i]

A convex polygon is dissected into a finite number of triangles with disjoint interiors, whose sides have odd integer lengths. The triangles may have multiple vertices on the boundary of the polygon and their sides may overlap partially. [list=a] [*]Prove that the polygon's perimeter is an integer which has the same parity as the number of triangles in the dissection. [*]Determine whether part a) holds if the polygon is not convex. [/list] [i]Proposed by Marius Cavachi[/i] [i]Note: the junior version only included part a), with an arbitrary triangle instead of a polygon.[/i]

For a natural number $ n, $ a string $ s $ of $ n $ binary digits and a natural number $ k\le n, $ define an $ n,s,k$ [i]-block[/i] as a string of $ k $ consecutive elements from $ s. $ We say that two $ n,s,k\text{-blocks} , $ namely, $ a_1a_2\ldots a_k,b_1b_2\ldots b_k, $ are [i]incompatible[/i] if there exists an $ i\in\{1,2,\ldots ,k\} $ such that $ a_i\neq b_i. $ Also, for two natural numbers $ r\le n, l, $ we say that $ s $ is $ r,l $ [i]-typed[/i] if there are, at most, $ l $ pairwise incompatible $ n,s,r\text{-blocks} . $ Let be a $ 3,7\text{-typed} $ string $ t $ consisting of $ 10000 $ binary digits. Determine the maximum number $ M $ that satisfies the condition that $ t $ is $ 10,M\text{-typed} . $ [i]Cătălin Gherghe[/i]