A $8 \times 8$ board is given, with sides directed north-south and east-west. It is divided into $1 \times 1$ cells in the usual manner. In each cell, there is most one [i]house[/i]. A house occupies only one cell. A house is [i] in the shade[/i] if there is a house in each of the cells in the south, east and west sides of its cell. In particular, no house placed on the south, east or west side of the board is in the shade. Find the maximal number of houses that can be placed on the board such that no house is in the shade.

In a $2025 \times 2025$ table, several cells are marked. At each move, Cyril can get to know the number of marked cells in any checkered square inside the initial table, with side less than $2025$. What is the minimal number of moves, which allows to determine the total number of marked cells for sure? (5 marks)

Dexter is running a pyramid scheme. In Dexter's scheme, he hires ambassadors for his company, Lie Ultimate. Any ambassador for his company can recruit up to two more ambassadors (who are not already ambassadors), who can in turn recruit up to two more ambassadors each, and so on (Dexter is a special ambassador that can recruit as many ambassadors as he would like). An ambassador's downline consists of the people they recruited directly as well as the downlines of those people. An ambassador earns executive status if they recruit two new people and each of those people has at least $70$ people in their downline (Dexter is not considered an executive). If there are $2020$ ambassadors (including Dexter) at Lie Ultimate, what is the maximum number of ambassadors with executive status?

Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? $\textbf{(A)}\ 12\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 40$

[u]Round 1[/u] [b]p1.[/b] Goldilocks enters the home of the three bears – Papa Bear, Mama Bear, and Baby Bear. Each bear is wearing a different-colored shirt – red, green, or blue. All the bears look the same to Goldilocks, so she cannot otherwise tell them apart. The bears in the red and blue shirts each make one true statement and one false statement. The bear in the red shirt says: “I'm Blue's dad. I'm Green's daughter.” The bear in the blue shirt says: “Red and Green are of opposite gender. Red and Green are my parents.” Help Goldilocks find out which bear is wearing which shirt. [b]p2.[/b] The University of Washington is holding a talent competition. The competition has five contests: math, physics, chemistry, biology, and ballroom dancing. Any student can enter into any number of the contests but only once for each one. For example, a student may participate in math, biology, and ballroom. It turned out that each student participated in an odd number of contests. Also, each contest had an odd number of participants. Was the total number of contestants odd or even? [b]p3.[/b] The $99$ greatest scientists of Mars and Venus are seated evenly around a circular table. If any scientist sees two colleagues from her own planet sitting an equal number of seats to her left and right, she waves to them. For example, if you are from Mars and the scientists sitting two seats to your left and right are also from Mars, you will wave to them. Prove that at least one of the $99$ scientists will be waving, no matter how they are seated around the table. [b]p4.[/b] One hundred boys participated in a tennis tournament in which every player played each other player exactly once and there were no ties. Prove that after the tournament, it is possible for the boys to line up for pizza so that each boy defeated the boy standing right behind him in line. [b]p5.[/b] To celebrate space exploration, the Science Fiction Museum is going to read Star Wars and Star Trek stories for $24$ hours straight. A different story will be read each hour for a total of $12$ Star Wars stories and $12$ Star Trek stories. George and Gene want to listen to exactly $6$ Star Wars and $6$ Star Trek stories. Show that no matter how the readings are scheduled, the friends can find a block of $12$ consecutive hours to listen to the stories together. [u]Round 2[/u] [b]p6.[/b] $2013$ people attended Cinderella's ball. Some of the guests were friends with each other. At midnight, the guests started turning into mice. After the first minute, everyone who had no friends at the ball turned into a mouse. After the second minute, everyone who had exactly one friend among the remaining people turned into a mouse. After the third minute, everyone who had two human friends left in the room turned into a mouse, and so on. What is the maximal number of people that could have been left at the ball after $2013$ minutes? [b]p7.[/b] Bill and Charlie are playing a game on an infinite strip of graph paper. On Bill’s turn, he marks two empty squares of his choice (not necessarily adjacent) with crosses. Charlie, on his turn, can erase any number of crosses, as long as they are all adjacent to each other. Bill wants to create a line of $2013$ crosses in a row. Can Charlie stop him? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $T$ be a finite set of positive integers greater than 1. A subset $S$ of $T$ is called [i]good[/i] if for every $t \in T$ there exists some $s \in S$ with $\gcd(s, t) > 1$. Prove that the number of good subsets of $T$ is odd.

On a chess board ($8*8$) there are written the numbers $1$ to $64$: on the first line, from left to right, there are the numbers $1, 2, 3, ... , 8$; on the second line, from left to right, there are the numbers $9, 10, 11, ... , 16$;etc. The $\"+\"$ and $\"-\"$ signs are put to each number such that, in each line and in each column, there are $4$ $\"+\"$ signs and $4$ $\"-\"$ signs. Then, the $64$ numbers are added. Find all the possible values of this sum.

$n$ vertices from a regular polygon with $2n$ sides are chosen and coloured red. The other $n$ vertices are coloured blue. Afterwards, the $\binom{n}{2}$ lengths of the segments formed with all pairs of red vertices are ordered in a non-decreasing sequence, and the same procedure is done with the $\binom{n}{2}$ lengths of the segments formed with all pairs of blue vertices. Prove that both sequences are identical.

The cells of a $100 \times 100$ table contain non-zero numbers. It turned out that all $100$ hundred-digit numbers written horizontally are divisible by 11. Could it be that exactly $99$ hundred-digit numbers written vertically are also divisible by $11$?

A number of $17$ workers stand in a row. Every contiguous group of at least $2$ workers is a $\textit{brigade}$. The chief wants to assign each brigade a leader (which is a member of the brigade) so that each worker’s number of assignments is divisible by $4$. Prove that the number of such ways to assign the leaders is divisible by $17$. [i]Mikhail Antipov, Russia[/i]

A game is played on a $n \times n$ chessboard. In the beginning Bars the cat occupies any cell according to his choice. The $d$ sparrows land on certain cells according to their choice (several sparrows may land in the same cell). Bars and the sparrows play in turns. In each turn of Bars, he moves to a cell adjacent by a side or a vertex (like a king in chess). In each turn of the sparrows, precisely one of the sparrows flies from its current cell to any other cell of his choice. The goal of Bars is to get to a cell containing a sparrow. Can Bars achieve his goal a) if $d=\lfloor \frac{3\cdot n^2}{25}\rfloor$, assuming $n$ is large enough? b) if $d=\lfloor \frac{3\cdot n^2}{19}\rfloor$, assuming $n$ is large enough? c) if $d=\lfloor \frac{3\cdot n^2}{14}\rfloor$, assuming $n$ is large enough?

There are $2020$ groups, each of which consists of a boy and a girl, such that each student is contained in exactly one group. Suppose that the students shook hands so that the following conditions are satisfied: [list] [*] boys didn't shake hands with boys, and girls didn't shake hands with girls; [*] in each group, the boy and girl had shake hands exactly once; [*] any boy and girl, each in different groups, didn't shake hands more than once; [*] for every four students in two different groups, there are at least three handshakes. [/list] Prove that one can pick $4038$ students and arrange them at a circular table so that every two adjacent students had shake hands.

Milo rolls five fair dice which have 4, 6, 8, 12, and 20 sides respectively (and each one is labeled $1$-$n$ for appropriate $n$. How many distinct ways can they roll a full house (three of one number and two of another)? The same numbers appearing on different dice are considered distinct full houses, so $(1,1,1,2,2)$ and $(2,2,1,1,1)$ would both be counted. [i]Proposed by Robert Trosten[/i]

There are $100$ boxes that were labeled with the numbers $00$, $01$, $02$,$…$, $99$ . The numbers $000$, $001$, $002$, $…$, $999$ were written on a thousand cards, one on each card. Placing a card in a box is permitted if the box number can be obtained by removing one of the digits from the card number. For example, it is allowed to place card $037$ in box $07$, but it is not allowed to place the card $156$ in box $65$.Can it happen that after placing all the cards in the boxes, there will be exactly $50$ empty boxes? If the answer is yes, indicate how the cards are placed in the boxes; If the answer is no, explain why it is impossible

For positive integers $m,n \geq 2$, let $S_{m,n} = \{(i,j): i \in \{1,2,\ldots,m\}, j\in \{1,2,\ldots,n\}\}$ be a grid of $mn$ lattice points on the coordinate plane. Determine all pairs $(m,n)$ for which there exists a simple polygon $P$ with vertices in $S_{m,n}$ such that all points in $S_{m,n}$ are on the boundary of $P$, all interior angles of $P$ are either $90^{\circ}$ or $270^{\circ}$ and all side lengths of $P$ are $1$ or $3$.

There are $32$ boxers in a tournament. Each boxer can fight no more often than once per day. It is known that the boxers are of different strength, and the stronger man always wins. Prove that a $15$ day tournament can be organised so as to determine their classification (put them in the order of strength). The schedule of fights for each day is fixed on the evening before and cannot be changed during the day. (A. Andjans, Riga)

Let $A_1, A_2, ... , A_k$ be the subsets of $\left\{1,2,3,...,n\right\}$ such that for all $1\leq i,j\leq k$:$A_i\cap A_j \neq \varnothing$. Prove that there are $n$ distinct positive integers $x_1,x_2,...,x_n$ such that for each $1\leq j\leq k$: $$lcm_{i \in A_j}\left\{x_i\right\}>lcm_{i \notin A_j}\left\{x_i\right\}$$ [i]Proposed by Morteza Saghafian, Mahyar Sefidgaran[/i]

Compute the number of sets $S$ such that every element of $S$ is a nonnegative integer less than 16, and if $x\in S$ then $(2x\bmod{16})\in S$.

Consider $2n+1$ points in space, no four of which are coplanar where $n>1$. Each line segment connecting any two of these points is either colored red, white or blue. A subset $M$ of these points is called a [i]connected monochromatic[/i] subset, if for each $a,b \in M$, there are points $a=x_0,x_1, \dots, x_l = b$ that belong to $M$ such that the line segments $x_0x_1, x_1x_2, \dots, x_{l-1}x_1$ are all have the same color. No matter how the points are colored, if there always exists a connected monochromatic $k-$subset, find the largest value of $k$. ($l > 1$)

A square $ 600 \times 600$ divided into figures of $4$ cells of the forms in the figure: In the figures of the first two types in shaded cells The number $ 2 ^ k $ is written, where $ k $ is the number of the column in which this cell. Prove that the sum of all the numbers written is divisible by $9$.

In 11 cells of a square grid there live hedgehogs. Every hedgehog divides the number of hedgehogs in its row by the number of hedgehogs in its column. Is it possible that all the hedgehogs get distinct numbers? (V.Brayman)

Six equal disks are placed on a plane so that their centers lie at the vertices of a regular hexagon with sides equal to the diameter of the disks. How many revolutions will a seventh disk of the same size make when rolling in the same plane externally over the disks before returning to its initial position?

There are $n$ husbands and wives at a party in the palace. The husbands sit at a round table, and the wives sit at another round tables. The king and queen (not included in the $n$ couples) are going to shake hands with them one by one. Assume that the king starts from a man, and the queen starts from his wife. Consider the following two ways of shaking hands: (i) The king shakes hands with the men one by one clockwise. Each time when the king shakes hands with a man, the queen moves clockwise to his wife and shakes hands with her. Assume that at last when the king gets back to the man he begins with, the queen goes around the table $a$ times. (ii) The queen shakes hands with the women one by one clockwise. Each time when the queen shakes hands with a woman, the king moves clockwise to her husband and shakes hands with him. Assume that at last when the queen gets back to the woman she begins with, the king goes around the table $b$ times. Determine the maximum possible value of $|a-b|$.

Let $n$ be a positive integer. We are given a $3n \times 3n$ board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a $2 \times 2$ square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all $n$ for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black. Proposed by [i]Boris Stanković and Marko Dimitrić, Bosnia and Herzegovina[/i]

We call a permutation $(a_1, a_2,\cdots , a_n)$ of the set $\{ 1,2,\cdots, n\}$ "good" if for any three natural numbers $i <j <k$, $n\nmid a_i+a_k-2a_j$ find all natural numbers $n\ge 3$ such that there exist a "good" permutation of a set $\{1,2,\cdots, n\}$.