Consider 26 letters $A,..., Z$. A string is a finite sequence consisting of those letters. We say that a string $s$ is nice if it contains each of the 26 letters at least once, and each permutation of letters $A,..., Z$ occurs in $s$ as a subsequences the same number of times. Prove that: (a) There exists a nice string. (b) Any nice string contains at least $2022$ letters.

Is it possible to write natural numbers at all points of the plane with integer coordinates so that three points with integer coordinates lie on the same line if and only if the numbers written in them had a common divisor greater than one?

Let us put pieces on some squares of $2n \times 2n$ chessboard in such a way that on every horizontal and vertical line there is an odd number of pieces. Prove that the whole number of pieces on the black squares is even.

Consider a game on an \( n \times n \) board, where each square starts with exactly one stone. A move consists of choosing $5$ consecutive squares in the same row or column of the board and toggling the state of each of those squares (removing the stone from squares with a stone and placing a stone in squares without a stone). For which positive integers \( n \geq 5 \) is it possible to end up with exactly one stone on the board after a finite number of moves?

The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.

There are $40$ members of jury, that want to choose problem for contest. There are list with $30$ problems. They want to find such problem, that can be solved at least half members , but not all. Every member solved $26$ problems, and every two members solved different sets of problems. Prove, that they can find problem for contest.

A hundred of silver coins are laid down in a line. A wizard can convert silver coin into golden one in $3$ seconds. Each golden coin, which is near the coin being converted, reduces this time by $1$ second. What minimal time is required for the wizard to convert all coins to gold?

[b]p1.[/b] A boy has as many sisters as brothers. How ever, his sister has twice as many brothers as sisters. How many boys and girls are there in the family? [b]p2.[/b] Solve each of the following problems. (1) Find a pair of numbers with a sum of $11$ and a product of $24$. (2) Find a pair of numbers with a sum of $40$ and a product of $400$. (3) Find three consecutive numbers with a sum of $333$. (4) Find two consecutive numbers with a product of $182$. [b]p3.[/b] $2008$ integers are written on a piece of paper. It is known that the sum of any $100$ numbers is positive. Show that the sum of all numbers is positive. [b]p4.[/b] Let $p$ and $q$ be prime numbers greater than $3$. Prove that $p^2 - q^2$ is divisible by $24$. [b]p5.[/b] Four villages $A,B,C$, and $D$ are connected by trails as shown on the map. [img]https://cdn.artofproblemsolving.com/attachments/4/9/33ecc416792dacba65930caa61adbae09b8296.png[/img] On each path $A \to B \to C$ and $B \to C \to D$ there are $10$ hills, on the path $A \to B \to D$ there are $22$ hills, on the path $A \to D \to B$ there are $45$ hills. A group of tourists starts from $A$ and wants to reach $D$. They choose the path with the minimal number of hills. What is the best path for them? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are four boats on one of the river banks; their names are Eight, Four, Two and One, because that is the number of hours it takes each of them to cross the river. One boat can be tied to another, but not more than one, and then the time it takes to cross is equal to that of the slower of the two boats. A single sailor must take all the boats to the other shore. What is the least amount of time you need to complete the move?

On sport games there was 1991 participant from which every participant knows at least n other participants(friendship is mutual). Determine the lowest possible n for which we can be sure that there are 6 participants between which any two participants know each other.

Prove that there are infinitely many positive integers $n$ for which it is possible for a knight, starting at one of the squares of an $n \times n$ chessboard, to go through each of the squares exactly once.

Beto has rectangular chessboard where the number of rows and columns are consecutive numbers (for example, $30$ and $31$). Ana has tiles of two colors and different sizes: the red tiles are $5 \times 7$ rectangles and the blue tiles are $3 \times 5$ rectangles. Ana realized that she can cover all the squares of Beto’s board using only red tiles, which can be rotated, but must not overlap or extend beyond the board. Then, she realized she can also do the same using only blue tiles. What is the minimum number of squares that Beto’s board can have?

For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.

Determine all pairs $(m, n)$ of positive integers, for which cube $K$ with edges of length $n$, can be build in with cuboids of shape $m \times 1 \times 1$ to create cube with edges of length $n + 2$, which has the same center as cube $K$.

We fill in an $n\times n$ table with real numbers such that the sum of the numbers in each row and each coloumn equals $1$. For which values of $K$ is the following statement true: if the sum of the absolute values of the negative entries in the table is at most $K$, then it's always possible to choose $n$ positive entries of the table such that each row and each coloumn contains exactly one of the chosen entries. [i]Proposed by Dávid Bencsik, Budapest[/i]

There are $8$ cards on which the numbers $1$, $2$, $\dots$, $8$ are written respectively. Alice and Bob play the following game: in each turn, Alice gives two cards to Bob, who must keep one card and discard the other. The game proceeds for four turns in total; in the first two turns, Bob cannot keep both of the cards with the larger numbers, and in the last two turns, Bob also cannot keep both of the cards with the larger numbers. Let $S$ be the sum of the numbers written on the cards that Bob keeps. Find the greatest positive integer $N$ for which Bob can guarantee that $S$ is at least $N$.

Let $\mathcal P$ be the set of all points in $\mathbb R^n$ with rational coordinates. For the points $A,B \in \mathcal l{P}$, one can move from $A$ to $B$ if the distance $AB$ is $1$. Prove that every point in $\mathcal l{ P}$ can be reached from any other point in $\mathcal{P}$ by a finite sequence of moves if and only if $n \geq 5$.

A pawn is a chess piece which attacks the two squares diagonally in front if it. What is the maximum number of pawns which can be placed on an $n \times n$ chessboard such that no two pawns attack each other?

An $8\times8$ chessboard is completely tiled by $2\times1$ dominoes. Prove that we can place positive integers in all cells of the table in such a way that the sums of numbers in every domino are equal and the numbers placed in two adjacent cells are coprime if and only if they belong to the same domino. (Two cells are called adjacent if they have a common side.) Well this can belong to number theory as well...

2. Let $A_1,A_2,...,A_n$be distinct subsets of an n-element set $ X$ ($n \geq 2$). Show that there exists an element $x$ of $X$ such that the sets $A_1\setminus \{x\}$ ,:......., $A_n\setminus \{x\}$ are all distinct.

$n$ balls are placed independently uniformly at random into $n$ boxes. One box is selected at random, and is found to contain $b$ balls. Let $e_n$ be the expected value of $b^4$. Find $$\lim_{n \to \infty}e_n.$$

Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ with $c_1 = a, c_k = b$ such that $c_i, c_{i+1}$ are adjacent for $i=1,2,\ldots,k-1$. \\ \\ Find the maximal number $M$ such that there exists a coloring admitting $M$ pairwise disconnected squares.

Given are $N$ points on the plane such that no three of them are collinear,which are coloured red,green and black.We consider all the segments between these points and give to each segment a [i]"value"[/i] according to the following conditions: [b]i.[/b]If at least one of the endpoints of a segment is black then the segment's [i]"value"[/i] is $0$. [b]ii.[/b]If the endpoints of the segment have the same colour,re or green,then the segment's [i]"value"[/i] is $1$. [b]iii.[/b]If the endpoints of the segment have different colours but none of them is black,then the segment's [i]"value"[/i] is $-1$. Determine the minimum possible sum of the [i]"values"[/i] of the segments.

There are $1998$ rectangular pieces $2$ cm wide and $3$ cm long and with them squares are assembled (without overlapping or gaps). What is the greatest number of different squares that can be had at the same time?

Two players play the following game. The first player starts by writing either $0$ or $1$ and then, on his every move, chooses either $0$ or $1$ and writes it to the right of the existing digits until there are $1999$ digits. Each time the first player puts down a digit (except the first one) , the second player chooses two digits among those already written and swaps them. Can the second player guarantee that after his last move the line of digits will be symmetrical about the middle digit? (I Izmestiev)