Several squares on a $15 \times 15$ chessboard are marked so that a bishop placed on any square of the board attacks at least two of marked squares. Find the minimal number of marked squares.

A polygon is given in which any two adjacent sides are perpendicular. We call its two vertices non-friendly if the bisectors of the polygon emerging from these vertices are perpendicular. Prove that for any vertex the number of vertices that are not friends with it is even.

Let $n$ be an even positive integer. We say that two different cells of a $n \times n$ board are [b]neighboring[/b] if they have a common side. Find the minimal number of cells on the $n \times n$ board that must be marked so that any cell (marked or not marked) has a marked neighboring cell.

For non-empty subsets $A,B \subset \mathbb{Z}$ define \[A+B=\{a+b:a\in A, b\in B\},\ A-B=\{a-b:a\in A, b\in B\}.\] In the sequel we work with non-empty finite subsets of $\mathbb{Z}$. Prove that we can cover $B$ by at most $\frac{|A+B|}{|A|}$ translates of $A-A$, i.e. there exists $X\subset Z$ with $|X|\leq \frac{|A+B|}{|A|}$ such that \[B\subseteq \cup_{x\in X} (x+(A-A))=X+A-A.\]

Given a positive integer $N$ (written in base $10$), define its [i]integer substrings[/i] to be integers that are equal to strings of one or more consecutive digits from $N$, including $N$ itself. For example, the integer substrings of $3208$ are $3$, $2$, $0$, $8$, $32$, $20$, $320$, $208$, $3208$. (The substring $08$ is omitted from this list because it is the same integer as the substring $8$, which is already listed.) What is the greatest integer $N$ such that no integer substring of $N$ is a multiple of $9$? (Note: $0$ is a multiple of $9$.)

There are $n$ people with hats present at a party. Each two of them greeted each other exactly once and each greeting consisted of exchanging the hats that the two persons had at the moment. Find all $n \ge 2$ for which the order of greetings can be arranged in such a way that after all of them, each person has their own hat back.

[b]p1[/b] How many two-digit positive integers with distinct digits satisfy the conditions that 1) neither digit is $0$, and 2) the units digit is a multiple of the tens digit? [b]p2[/b] Mirabel has $47$ candies to pass out to a class with $n$ students, where $10\le n < 20$. After distributing the candy as evenly as possible, she has some candies left over. Find the smallest integer $k$ such that Mirabel could have had $k$ leftover candies. [b]p3[/b] Callie picks two distinct numbers from $\{1, 2, 3, 4, 5\}$ at random. The probability that the sum of the numbers she picked is greater than the sum of the numbers she didn’t pick is $p$. $p$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd (a, b) = 1$. Find $a + b$.

Each of the $n$ students in a class sent a card to each of his $m$ colleagues. Prove that if $2m + 1 > n$, then at least two students sent cards to each other.

Let $n$ be a fixed positive integer. Ana and Banana are playing a game. First, Ana picks a subset $S$ of $\{1,2,\ldots,n\}$. Then for each $k=1,2,\ldots,n$, she tells Banana how many numbers from $k-1$ to $k+1$ she has picked (i.e. $\lvert S \cap \{k-1,k,k+1\}\rvert$). Then Banana guesses $S$; she wins if her guess is correct and she loses otherwise. (a) Determine all $n$ for which Banana will win regardless of what Ana chooses. (b) For the values of $n$ for which Ana can win, determine the number of sets $S$ she can choose so as to do so.

Let $ABCDEFG$ be a regular heptagon. We'll call the sides $AB$, $BC$, $CD$, $DE$, $EF$, $FG$ and $GA$ opposite to the vertices $E$, $F$, $G$, $A$, $B$, $C$ and $D$ respectively. If $M$ is a point inside the heptagon, we'll say that the line through $M$ and a vertex of the heptagon intersects a side of it (without the vertices) at a $\textit{perfect}$ point, if this side is opposite the vertex. Prove that for every choice of $M$, the number of $\textit{perfect}$ points is always odd.

Let $n$ be a natural number. Given a chessboard sized $m \times n$. The sides of the small squares of chessboard are not on the perimeter of the chessboard will be colored so that each small square has exactly two sides colored. Prove that a coloring like that is possible if and only if $m \cdot n$ is even.

Pairwise distinct points $P_1,\dots,P_{1024}$, which lie on a circle, are marked by distinct reals $a_1,\dots,a_{1024}$. Let $P_i$ be $Q-$good for a $Q$ on the circle different than $P_1,\dots,P_{1024}$, if and only if $a_i$ is the greatest number on at least one of the two arcs $P_iQ$. Let the score of $Q$ be the number of $Q-$good points on the circle. Determine the greatest $k$ such that regardless of the values of $a_1,\dots,a_{1024}$, there exists a point $Q$ with score at least $k$.

Arnaldo and Bernardo play a Super Naval Battle. Each has a board $n \times n$. Arnaldo puts boats on his board (at least one but not known how many). Each boat occupies the $n$ houses of a line or a column and the boats they can not overlap or have a common side. Bernardo marks $m$ houses (representing shots) on your board. After Bernardo marked the houses, Arnaldo says which of them correspond to positions occupied by ships. Bernardo wins, and then discovers the positions of all Arnaldo's boats. Determine the lowest value of $m$ for which Bernardo can guarantee his victory.

Let $A_n$ be the number of ways to tile a $4 \times n$ rectangle using $2 \times 1$ tiles. Prove that $A_n$ is divisible by 2 if and only if $A_n$ is divisible by 3.

Let $a_1<a_2$ be two given integers. For any integer $n\ge 3$, let $a_n$ be the smallest integer which is larger than $a_{n-1}$ and can be uniquely represented as $a_i+a_j$, where $1\le i<j\le n-1$. Given that there are only a finite number of even numbers in $\{a_n\}$, prove that the sequence $\{a_{n+1}-a_{n}\}$ is eventually periodic, i.e. that there exist positive integers $T,N$ such that for all integers $n>N$, we have \[a_{T+n+1}-a_{T+n}=a_{n+1}-a_{n}.\]

Let $n \ge 2$ be an integer. In each cell of a $4n \times 4n$ table we write the sum of the cell row index and the cell column index. Initially, no cell is colored. A move consists of choosing two cells which are not colored and coloring one of them in red and one of them in blue. Show that, however Alex perfors $n^2$ moves, Jane can afterwards perform a number of moves (eventually none) after which the sum of the numbers written in the red cells is the same as the sum of the numbers written in the blue ones.

$2021$ points are marked on a circle. $2021$ segments with marked endpoints are drawn. After that one counts the number of different points where some $2$ drawn segments intersect(endpoints of segments do [b]not[/b] count as intersections) Find the maximum number one can get.

In a very long corridor there is an infinite number of cabinets, which start with $1,2,3,...$ numbered and initially all are closed. There is also a horde of QEDlers, whose number lies in set $A \subseteq \{1, 2,3,...\}$ . In ascending order, the QED people now cause chaos: the person with number $a \in A$ visits the cabinet with the numbers $a,2a,3a,...$ opening all of the closed ones and closes all open. Show that in the end the cabinet has never exactly the same numbers from $A$ open.

. Anthony denotes in sequence all positive integers which are divisible by $2$. Bertha denotes in sequence all positive integers which are divisible by $3$. Claire denotes in sequence all positive integers which are divisible by $4$. Orderly Dora denotes all numbers written by the other three. Thereby she puts them in order by size and does not repeat a number. What is the $2017th$ number in her list? [i]¨Proposed by Richard Henner[/i]

The vertices of a regular $n$-gon are initially marked with one of the signs $+$ or $-$ . A [i]move[/i] consists in choosing three consecutive vertices and changing the signs from the vertices , from $+$ to $-$ and from $-$ to $+$. [b]a)[/b] Prove that if $n=2015$ then for any initial configuration of signs , there exists a sequence of [i]moves[/i] such that we'll arrive at a configuration with only $+$ signs. [b]b)[/b] Prove that if $n=2016$ , then there exists an initial configuration of signs such that no matter how we make the [i]moves[/i] we'll never arrive at a configuration with only $+$ signs.

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

Each point on a circle is colored with one of $10$ colors. Is it true that for any coloring there are $4$ points of the same color that are vertices of a quadrilateral with two parallel sides (an isosceles trapezoid or a rectangle)?

Initially on the blackboard all the integers from $1$ to $2002$ inclusive are written in one line and in some order, without repetitions. In each step, the first and second numbers of the line are deleted and the absolute value of the subtraction of the two numbers that have just been deleted is written at the beginning of the line; the other numbers are not modified in that step, and there is a new line that has one less number than the previous step. After completing $2001$ steps, only one number remains on the board. Determine all possible values of the number left on the board by varying the order of the $2002$ numbers on the initial line (and performing the $2001$ steps).

Two persons are playing the following game on a $n\times m$ table, with drawn lines: Person $\#1$ starts the game. Each person in their move, folds the table on one of its lines. The one that could not fold the table on their turn loses the game. Who has a winning strategy?

Suppose we have a hexagonal grid in the shape of a hexagon of side length $4$ as shown at left. Define a “chunk” to be four tiles, two of which are adjacent to the other three, and the other two of which are adjacent to just two of the others. The three possible rotations of these are shown at right. [img]https://cdn.artofproblemsolving.com/attachments/a/7/147d8aa2c149918ab855db1e945d389433446a.png[/img] In how many ways can we choose a chunk from the grid?