Elisa has $2023$ treasure chests, all of which are unlocked and empty at first. Each day, Elisa adds a new gem to one of the unlocked chests of her choice, and afterwards, a fairy acts according to the following rules: [list=disc] [*]if more than one chests are unlocked, it locks one of them, or [*]if there is only one unlocked chest, it unlocks all the chests. [/list] Given that this process goes on forever, prove that there is a constant $C$ with the following property: Elisa can ensure that the difference between the numbers of gems in any two chests never exceeds $C$, regardless of how the fairy chooses the chests to unlock.

Consider a box of dimensions $10$ cm $\times 16$ cm $\times 1$ cm. Determine the maximum number of balls of diameter $ 1$ cm that the box can contain.

Let $n$ be a positive integer. For a set of points in the plane $M$, we call $2$ distinct points $A,B \in M$ [i]connected[/i] if the line $AB$ contains exactly $n+1$ points from $M$. Find the minimum value of a positive integer $m$ such that there exists a set $M$ of $m$ points in the plane with the property that any point $A \in M$ is connected with exactly $2n$ other points from $M$.

What is the minimum number of cells that can be colored black in white square $ 300 \times 300 $ so that no three black cells formed a corner, and after painting any white cell this condition violated?

Let $M$ be a subset of the set of $2021$ integers $\{1, 2, 3, ..., 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$ we have $|a + b - c | > 10$. Determine the largest possible number of elements of $M$.

John and Mary each have a white $8 \times 8$ square divided into $1 \times 1$ cells. They have painted an equal number of cells on their respective squares in blue. Prove that one can cut up each of the two squares into $2 \times 1 $ dominoes so that it is possible to reassemble John's dominoes into a new square and Mary's dominoes into another square with the same pattern of blue cells. (A Shapovalov)

The checkered plane (infinitely large house paper) is given. For which pairs (a,, b) one can color each of the squares with one of $a \cdot b$ colors, so that each rectangle of size $ a \times b$ or $b \times a$, placed appropriately in the checkered plane, always contains a unit square with each color ?

Kelvin has a set of eight vertices. For each pair of distinct vertices, Kelvin independently draws an edge between them with probability $p \in (0,1).$ A set $S$ of four distinct vertices is called [i]good[/i] if there exists an edge between $v$ and $w$ for all $v,w \in S$ with $v \neq w.$ The variance of the number of good sets can be expressed as a polynomial $f(p)$ in the variable $p.$ Find the sum of the absolute values of the coefficients of $f(p).$ (The [i]variance[/i] of a random variable $X$ is defined as $\mathbb{E}[X^2]-\mathbb{E}[X]^2.$)

[b]7.1 / 6.3[/b] All integers from 1 to 1966 are written on the board. Allowed is to erase any two numbers by writing their difference instead. Prove that repeating such an operation many times cannot ensure that There are only zeros left on the board. [b]7.2 [/b] Prove that the radius of a circle is equal to the difference between the lengths of two chords, one of which subtends an arc of $1/10$ of a circle, and the other subtends an arc in $3/10$ of a circle. [b]7.3[/b] Prove that for any natural number $n$ the number $ n(2n+1)(3n+1)...(1966n + 1) $ is divisible by every prime number less than $1966$. [b]7.4[/b] What number needs to be put in place * so that the next the problem had a unique solution: [i]“There are n straight lines on the plane, intersecting at * points. Find n.” ?[/i] [b]7.5 / 6.4[/b] Black paint was sprayed onto a white surface. Prove that there are three points of the same color lying on the same line, and so, that one of the points lies in the middle between the other two. [b]7.6 [/b] There are $n$ points on the plane so that any triangle with vertices at these points has an area less than $1$. Prove that all these points can be enclosed in a triangle of area $4$. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here[/url].

There are $N\ge3$ letters arranged in a circle, and each letter is one of $L$, $T$ and $F$. For a letter, we can do the following operation: if its neighbors are the same, then change it to the same letter too; otherwise, change it so that it is different from both its neighbors. Show that for any initial state, one can perform finitely many operations to achieve a stable state. Here, a stable state means that any operation does not change any of the $N$ letters. (ltf0501)

Vanya was asked to write on the board an expression equal to $10$, using only the numbers $1$, the signs $+$ and $-$ and brackets (you cannot make up the numbers $11$, $111$, etc., as well as $(-1)$). He knows that the bully Anton will then correct all the $+$ signs to $-$ and vice versa. Help Vanya compose the required expression, which will remain equal to $10$ even after Anton's actions.

Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.

The numbers $1$ and $2$ are written on an empty blackboard. Whenever the numbers $m$ and $n$ appear on the blackboard the number $m + n + mn$ may be written. Can we obtain : (1) $13121$, (2) $12131$?

In a certain game, the set $\{1, 2, \dots, 10\}$ is partitioned into equally-sized sets $A$ and $B$. In each of five consecutive rounds, Alice and Bob simultaneously choose an element from $A$ or $B$, respectively, that they have not yet chosen; whoever chooses the larger number receives a point, and whoever obtains three points wins the game. Determine the probability that Alice is guaranteed to win immediately after the set is initially partitioned.

In a $5 \times 5$ checkered square, the middle row and middle column are colored gray. You leave the corner cell and move to the cell adjacent to the side with each move. For each transition from a gray cell to a gray one you need to pay a ruble. What is the smallest number of rubles you need to pay to go around all the squares of the board exactly once (it is not necessary to return to the starting square)?

Sets $A_0, A_1, \dots, A_{2023}$ satisfy the following conditions: [list] [*] $A_0 = \{ 3 \}$ [*] $A_n = \{ x + 2 \mid x \in A_{n - 1} \} \ \cup \{x(x+1) / 2 \mid x \in A_{n - 1} \}$ for each $n = 1, 2, \dots, 2023$. [/list] Find $|A_{2023}|$.

Ziquan makes a drawing in the plane for art class. He starts by placing his pen at the origin, and draws a series of line segments, such that the $n^{th}$ line segment has length $n$. He is not allowed to lift his pen, so that the end of the $n^{th}$ segment is the start of the $(n + 1)^{th}$ segment. Line segments drawn are allowed to intersect and even overlap previously drawn segments. After drawing a finite number of line segments, Ziquan stops and hands in his drawing to his art teacher. He passes the course if the drawing he hands in is an $N$ by $N$ square, for some positive integer $N$, and he fails the course otherwise. Is it possible for Ziquan to pass the course?

Let $A$ be the number of 2019-digit numbers, that is made of 2 different digits (For example $10\underbrace{1...1}_{2016}0$ is such number). Determine the highest power of 3 that divides $A$.

Suppose $S= \{1,2,\dots,n\}$ and $n \geq 3$. There is $f:S^k \longmapsto S$ that if $a,b \in S^k$ and $a$ and $b$ differ in all of elements then $f(a) \neq f(b)$. Prove that $f$ is a function of one of its elements.

A positive integer is said to be a “palindrome” if it reads the same from left to right as from right to left. For example 2002 is a palindrome. Find the sum of all 4-digit palindromes.

Fred and George play a game, as follows. Initially, $x = 1$. Each turn, they pick $r \in \{3,5,8,9\}$ uniformly at random and multiply $x$ by $r$. If $x+1$ is a multiple of 13, Fred wins; if $x+3$ is a multiple of 13, George wins; otherwise, they repeat. Determine the probability that Fred wins the game.

A token is placed in one square of a $m\times n$ board, and is moved according to the following rules: [list] [*]In each turn, the token can be moved to a square sharing a side with the one currently occupied. [*]The token cannot be placed in a square that has already been occupied. [*]Any two consecutive moves cannot have the same direction.[/list] The game ends when the token cannot be moved. Determine the values of $m$ and $n$ for which, by placing the token in some square, all the squares of the board will have been occupied in the end of the game.

Given a positive integer $n>1$. In the cells of an $n\times n$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least two cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.

Finite set $\{a_1, a_2, ..., a_n, b_1, b_2, ..., b_n\}=\{1, 2, …, 2n\}$ is given. If $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$, show that $$\sum_{i=1}^n |a_i-b_i|=n^2$$

There are $ n \geq 5$ pairwise different points in the plane. For every point, there are just four points whose distance from which is $ 1$. Find the maximum value of $ n$.