1.1 Let $f(A)$ denote the difference between the maximum value and the minimum value of a set $A$. Find the sum of $f(A)$ as $A$ ranges over the subsets of $\{1, 2, \dots, n\}$. 1.2 All cells of an $8 × 8$ board are initially white. A move consists of flipping the color (white to black or vice versa) of cells in a $1\times 3$ or $3\times 1$ rectangle. Determine whether there is a finite sequence of moves resulting in the state where all $64$ cells are black. 1.3 Prove that for all positive integers $m$, there exists a positive integer $n$ such that the set $\{n, n + 1, n + 2, \dots , 3n\}$ contains exactly $m$ perfect squares.

The tennis federation has assigned numbers to $1024$ sportsmen, participating in the tournament, according to their skill. (The tennis federation uses the olympic system of tournaments. The looser in the pair leaves, the winner meets with the winner of another pair. Thus, in the second tour remains $512$ participants, in the third -- $256$, et.c. The winner is determined after the tenth tour.) It comes out, that in the play between the sportsmen whose numbers differ more than on $2$ always win that whose number is less. What is the greatest possible number of the winner?

In the centre of a square swimming pool is a boy, while his teacher (who cannot swim) is standing at one corner of the pool. The teacher can run three times as fast as the boy can swim, but the boy can run faster than the teacher . Can the boy escape from the teacher?

Every square of a $2020 \times 2020$ chess table is painted in red or white. For every two columns and two rows, at least two of the intersection squares satisfies that they are in the same column or row and they are painted in the same color. Find the least value of number of columns and rows that are completely painted in one color.

A government’s land with dimensions $n \times n$ are going to be sold in phases. The land is divided into $n^2$ squares with dimension $1 \times 1$. In the first phase, $n$ farmers bought a square, and for each rows and columns there is only one square that is bought by a farmer. After one season, each farmer could buy one more square, with the conditions that the newly-bought square has a common side with the farmer’s land and it hasn’t been bought by other farmers. Determine all values of n such that the government’s land could not be entirely sold within $n$ seasons.

A set of five different positive integers is called [i]virtual[/i] if the greatest common divisor of any three of its elements is greater than $1$, but the greatest common divisor of any four of its elements is equal to $1$. Prove that, in any virtual set, the product of its elements has at least $2020$ distinct positive divisors. [i]Proposed by Víctor Almendra[/i]

Consider a $2021$-by-$2021$ board of unit squares. For some integer $k$, we say the board is tiled by $k$-by-$k$ squares if it is completely covered by (possibly overlapping) $k$-by-$k$ squares with their corners on the corners of the unit squares. What is the largest integer k such that the minimum number of $k$-by-$k$ squares needed to tile the $2021$-by-$2021$ board is exactly equal to $100$?

An enormous party has an infinite number of people. Each two people either know or don't know each other. Given a positive integer $n$, prove there are $n$ people in the party such that either they all know each other, or nobody knows each other (so the first possibility means that if $A$ and $B$ are any two of the $n$ people, then $A$ knows $B$, whereas the second possibility means that if $A$ and $B$ are any two of the $n$ people, then $A$ does not know $B$).

There is a 100 x100 square table, a real number being written in each cell.$A$ and $B$ play the following game. They choose, turn by turn, some row of the table (if it has not been chosen before). When $A$ and $B$ have $50$ rows chosen each, they sum the numbers in the corresponding cells of the chosen rows, and then sum the squares of all $100$ obtained numbers and compare the results. $A$ player who has the greater result wins. Player $A$ begins. Show that $A$ can avoid a defeat.

Four heaps contain $38,45,61$ and $70$ matches respectively. Two players take turn choosing any two of the heaps and take some non-zero number of matches from one heap and some non-zero number of matches from the other heap. The player who cannot make a move, loses. Which one of the players has a winning strategy ?

There are nine boxes. In the first there is $1$ stone, in the second there are $2$ stones, in the third there are $3$ stones, and thus continuing, in the eighth there are $8$ stones and in the ninth there are $9$ stones. The allowed operation is to remove the same number of stones from two different boxes and place them in a third box. The goal is that all stones are in a single box. Describe how to do it with the minimum number of operations allowed. Explain why it is impossible to achieve it with fewer operations.

Nine points are placed on a circle. Show that it is possible to colour the $36$ chords connecting them using four colours so that for any set of four points, each of the four colours is used for at least one of the six chords connecting the given points

Denote $S$ as the subset of $\{1,2,3,\dots,1000\}$ with the property that none of the sums of two different elements in $S$ is in $S$. Find the maximum number of elements in $S$.

Write either $1$ or $-1$ in each of the cells of a $(2n) \times (2n)$-table, in such a way that there are exactly $2n^2$ entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be $M$. Determine the largest possible value of $M$.

Let $n$ be a natural number and suppose that $ w_1, w_2, \ldots , w_n$ are $n$ weights . We call the set of $\{ w_1, w_2, \ldots , w_n\}$ to be a [i]Perfect Set [/i]if we can achieve all of the $1,2, \ldots, W$ weights with sums of $ w_1, w_2, \ldots , w_n$, where $W=\sum_{i=1}^n w_i $. Prove that if we delete the maximum weight of a Perfect Set, the other weights make again a Perfect Set.

On the blackboard, there are $K$ blanks. Alice decides $N$ values of blanks $(0-9)$ and then Bob determines the remaining digits. Find the largest possible integer $N$ such that Bob can guarantee to make the final number isn't a power of an integer.

(a) The numbers $1 , 2, 4, 8, 1 6 , 32, 64, 1 28$ are written on a blackboard. We are allowed to erase any two numbers and write their difference instead (this is always a non-negative number). After this procedure has been repeated seven times, only a single number will remain. Could this number be $97$? (b) The numbers $1 , 2, 22, 23 , . . . , 210$ are written on a blackboard. We are allowed to erase any two numbers and write their difference instead (this is always a non-negative number) . After this procedure has been repeated ten times, only a single number will remain. What values could this number have? (A.Shapovalov)

We call a $6\times 6$ table consisting of zeros and ones [i]right[/i] if the sum of the numbers in each row and each column is equal to $3$. Two right tables are called [i]similar[/i] if one can get from one to the other by successive interchanges of rows and columns. Find the largest possible size of a set of pairwise similar right tables.

A table $2022 \times 2022$ is divided onto the tiles of two types: $L$-tetromino and $Z$-tetromino. Determine the least amount of $Z$-tetromino one needs to use.

Given 10 light switches, each can be in two states: on and off. For each pair of switches there is a light bulb which is on if and only if when both switches are on (45 bulbs in total). The bulbs and the switches are unmarked so it is unclear which switches correspond to which bulb. In the beginning all switches are off. How many flips are needed to find out regarding all bulbs which switches are connected to it? On each step you can flip precisely one switch

Without lifting pen from paper, we draw a polygon in such away that from every two adjacent sides one of them is vertical. In addition, while drawing the polygon all vertical sides have been drawn from up to down. Prove that this polygon has cut itself.

For what smallest $n$ can a $n \times n$ square be cut into squares $40 \times 40$ and $49 \times 49$ so that squares of both types are present?

[b]p1.[/b] Twelve tables are set up in a row for a Millenium party. You want to put $2000$ cupcakes on the tables so that the numbers of cupcakes on adjacent tables always differ by one (for example, if the $5$th table has $20$ cupcakes, then the $4$th table has either $19$ or $21$ cupcakes, and the $6$th table has either $19$ or $21$ cupcakes). a) Find a way to do this. b) Suppose a Y2K bug eats one of the cupcakes, so you have only $1999$ cupcakes. Show that it is impossible to arrange the cupcakes on the tables according to the above conditions. [b]p2.[/b] Let $P$ and $Q$ lie on the hypotenuse $AB$ of the right triangle $CAB$ so that $|AP|=|PQ|=|QB|=|AB|/3$. Suppose that $|CP|^2+|CQ|^2=5$. Prove that $|AB|$ has the same value for all such triangles, and find that value. Note: $|XY|$ denotes the length of the segment $XY$. [b]p3.[/b] Let $P$ be a polynomial with integer coefficients and let $a, b, c$ be integers. Suppose $P(a)=b$, $P(b)=c$, and $P(c)=a$. Prove that $a=b=c$. [b]p4.[/b] A lattice point is a point $(x,y)$ in the plane for which both $x$ and $y$ are integers. Each lattice point is painted with one of $1999$ available colors. Prove that there is a rectangle (of nonzero height and width) whose corners are lattice points of the same color. [b]p5.[/b] A $1999$-by-$1999$ chocolate bar has vertical and horizontal grooves which divide it into $1999^2$ one-by-one squares. Caesar and Brutus are playing the following game with the chocolate bar: A move consists of a player picking up one chocolate rectangle; breaking it along a groove into two smaller rectangles; and then either putting both rectangles down or eating one piece and putting the other piece down. The players move alternately. The one who cannot make a move at his turn (because there are only one-by-one squares left) loses. Caesar starts. Which player has a winning strategy? Describe a winning strategy for that player. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

(a) On each of the $1 \times 1$ squares of the top row of an $8 \times 8$ chessboard there is a black pawn, and on each of the $1 \times 1$ squares of the bottom row of this chessboard there is a white pawn. On each move one can shift any pawn vertically or horizontally to any adjacent empty $1 \times 1$ square. What is the smallest number of moves that are needed to move all white pawns to the top row and all black pawns to the bottom one? (b) The same question for a $7 \times 7$ board. (A Shapovalov_

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?