The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

Consider a table whose entries are integers. Adding a same integer to all entries on a same row, or on a same column, is called an [i]operation[/i]. It is given that, for infinitely many positive integers $n$, one can obtain, through a finite number of operations, a table having all entries divisible by $n$. Prove that, through a finite number of operations, one can obtain the table whose all entries are zeroes.

Show for any positive integer $n$ that there exists a circle in the plane such that there are exactly $n$ grid points within the circle. (A grid point is a point having integer coordinates.)

Pete chose a positive integer $ n$. For each (unordered) pair of its decimal digits, he wrote their difference on the blackboard. After that, he erased some of these differences, and the remaining ones are $ 2,0,0,7$. Find the smallest number $ n$ for which this situation is possible.

A brick has the shape of a cube of size $2$ with one corner unit cube removed. Given a cube of side $2^{n}$ divided into unit cubes from which an arbitrary unit cube is removed, show that the remaining figure can be built using the described bricks.

Alexandre and Bernado are playing the following game. At the beginning, there are $n$ balls in a bag. At first turn, Alexandre can take one ball from the bag; at second turn, Bernado can take one or two balls from the bag, and so on. So they take turns and in $k$ turn, they can take a number of balls from $1$ to $k$. Wins the one who makes the bag empty. For each value of $n$, find who has the winning strategy.

Twenty players participated in a chess tournament. Each player faced every other player exactly once and each match ended with either player winning or a draw. In this tournament, it was noticed that for every match that ended in a draw, each of the other $18$ players won at least one of the two players involved in it. We also know that at least two games ended in a draw. Show that it is possible to name the players as $P_1, P_2, ...., P_{20}$ so that player $P_k$ beat player $P_{k+1}$, to each $k \in \{ 1, 2, 3,... , 19\}$.

Five points $A_1,A_2,A_3,A_4,A_5$ lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles $\angle A_iA_jA_k$ can take where $i, j, k$ are distinct integers between $1$ and $5$.

Consider a partition of $\{1,2,\ldots,900\}$ into $30$ subsets $S_1,S_2,\ldots,S_{30}$ each with $30$ elements. In each $S_k$, we paint the fifth largest number blue. Is it possible that, for $k=1,2,\ldots,30$, the sum of the elements of $S_k$ exceeds the sum of the blue numbers?

Given a $2025 \times 2025$ board and $k$ chips lying on the table. Initially, the board is empty. It is allowed to place a chip from the table on any free square $A$ if two conditions are met simultaneously: – the cell next to $A$ from below has a chip (or $A$ is on the bottom edge of the board); – the cell next to $A$ on the left has a chip (or $A$ is on the left edge of the board). In addition, it is allowed to remove any chip from the board and put it on the table. At what minimum $k$ can a chip appear in the upper-right corner of the board?

Given positive integers$ m,n$ such that $ m < n$. Integers $ 1,2,...,n^2$ are arranged in $ n \times n$ board. In each row, $ m$ largest number colored red. In each column $ m$ largest number colored blue. Find the minimum number of cells such that colored both red and blue.

There are several bears living on the $2025$ islands of the Arctic Ocean. Every bear sometimes swims from one island to another. It turned out that every bear made at least one swim in a year, but no two bears made equal swams. At the same time, exactly one swim was made between each two islands $A$ and $B$: either from $A$ to $B$ or from $B$ to $A$. Prove that there were no bears on some island at the beginning and at the end of the year. [i]A. Kuznetsov[/i]

Each vertex of an $m\times n$ grid is colored blue, green or red in such a way that all the boundary vertices are red. We say that a unit square of the grid is properly colored if: $(i)$ all the three colors occur at the vertices of the square, and $(ii)$ one side of the square has the endpoints of the same color. Show that the number of properly colored squares is even.

How many rectangles are there on an $8 \times 8$ checkerboard? [img]https://cdn.artofproblemsolving.com/attachments/9/e/7719117ae393d81a3e926acb567f850cc1efa9.png[/img]

A $4\times4$ grid has its 16 cells colored arbitrarily in three colors. A [i]swap[/i] is an exchange between the colors of two cells. Prove or disprove that it always takes at most three swaps to produce a line of symmetry, regardless of the grid's initial coloring. [i]Proposed by Matthew Babbitt[/i]

[u]Set 7[/u] [b]p19.[/b] The polynomial $x^4 + ax^3 + bx^2 - 32x$, where$ a$ and $b$ are real numbers, has roots that form a square in the complex plane. Compute the area of this square. [b]p20.[/b] Tetrahedron $ABCD$ has equilateral triangle base $ABC$ and apex $D$ such that the altitude from $D$ to $ABC$ intersects the midpoint of $\overline{BC}$. Let $M$ be the midpoint of $\overline{AC}$. If the measure of $\angle DBA$ is $67^o$, find the measure of $\angle MDC$ in degrees. [b]p21.[/b] Last year’s high school graduates started high school in year $n- 4 = 2017$, a prime year. They graduated high school and started college in year $n = 2021$, a product of two consecutive primes. They will graduate college in year $n + 4 = 2025$, a square number. Find the sum of all $n < 2021$ for which these three properties hold. That is, find the sum of those $n < 2021$ such that $n -4$ is prime, n is a product of two consecutive primes, and $n + 4$ is a square. PS. You should use hide for answers.

For three arbitrary crossroads $A,B,C$ in a certain city there exist a way from $A$ to $B$ not coming through $C$. Prove that for every couple of the crossroads there exist at least two non-intersecting ways connecting them. (there are at least two crossroads in the city)

Consider the set $L$ of binary strings of length less than or equal to $9$, and for a string $w$ define $w^{+}$ to be the set $\{w,w^2,w^3,\ldots\}$ where $w^k$ represents $w$ concatenated to itself $k$ times. How many ways are there to pick an ordered pair of (not necessarily distinct) elements $x,y\in L$ such that $x^{+}\cap y^{+}\neq \varnothing$?

For which $n \in N$ is it possible to arrange a tennis tournament for doubles with $n$ players such that each player has every other player as an opponent exactly once?

Let $n > 4$ be an integer. A square is divided into $n^2$ smaller identical squares, in some of which were [b]1’s[/b] and in the other – [b]0's[/b]. It is not allowed in one row or column to have the following arrangements of adjacent digits in this order: $101$, $111$ or $1001$. What is the biggest number of [b]1’s[/b] in the table? (The answer depends on $n$.)

On the name day of a man there are $5$ people. The men observed that of any $3$ people there are $2$ that knows each other. Prove that the man may order his guests around circular table in such way that every man have on its both side people that he knows. [i]N. Nenov, N. Hazhiivanov[/i]

A set $S$ of natural numbers is called [i]good[/i], if for each element $x \in S, x$ does not divide the sum of the remaining numbers in $S$. Find the maximal possible number of elements of a [i]good [/i]set which is a subset of the set $A = \{1,2, 3, ...,63\}$.

A game is played with two heaps of $p$ and $q$ stones. Two players alternate playing, with $A$ starting. A player in turn takes away one heap and divides the other heap into two smaller ones. A player who cannot perform a legal move loses the game. For which values of $p$ and $q$ can $A$ force a victory?

The cells of the $9 \times 9$ table are colored black and white. It turned out, that there were $k$ rows, in each of which there are more black cells than white ones white, and there were $k$ columns, each of which contained more than black. At what highest $ k$ is this possible?

A $4$ x $4$ square board is called $brasuca$ if it follows all the conditions: • each box contains one of the numbers $0, 1, 2, 3, 4$ or $5$; • the sum of the numbers in each line is $5$; • the sum of the numbers in each column is $5$; • the sum of the numbers on each diagonal of four squares is $5$; • the number written in the upper left box of the board is less than or equal to the other numbers the board; • when dividing the board into four $2$ × $2$ squares, in each of them the sum of the four numbers is $5$. How many $"brasucas"$ boards are there?