The vertices of a regular $2005$-gon are colored red, white and blue. Whenever two vertices of different colors stand next to each other, we are allowed to recolor them into the third color. (a) Prove that there exists a finite sequence of allowed recolorings after which all the vertices are of the same color. (b) Is that color uniquely determined by the initial coloring?

$X$ is a set with $100$ members. What is the smallest number of subsets of $X$ such that every pair of elements belongs to at least one subset and no subset has more than $50$ members? What is the smallest number if we also require that the union of any two subsets has at most $80$ members?

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

[b]8.1[/b] A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. Draw another straight line so that after that the rectangle was divided into the largest possible number of parts. [b]8.2[/b] Engineers always tell the truth, but businessmen always lie. F and G are engineers. A declares that, B asserts that, C asserts that, D says that, E insists that, F denies that G is an businessman. C also announces that D is a businessman. If A is a businessman, then how much total businessmen in this company? [b]8.3 [/b]There is a straight road through the field. A tourist stands on the road at a point ?. It can walk along the road at a speed of 6 km/h and across the field at a speed of 3 km/h. Find the locus of the points where the tourist can get there within an hour's walk. [b]8.4 / 7.5 [/b] Let $ [A]$ denote the largest integer not greater than $A$. Solve the equation: $[(5 + 6x)/8] = (15x-7)/5$ . [b]8.5.[/b] In some state, every two cities are connected by a road. Each road is only allowed to move in one direction. Prove that there is a city from which you can travel around everything. state, having visited each city exactly once. [b]8.6[/b] Find all eights of prime numbers such that the sum of the squares of the numbers in the eight is 992 less than their quadruple product. [hide=original wording]Найдите все восьмерки простых чисел такие, что сумма квадратов чисел в восьмерке на 992 меньше, чем их учетверенное произведение.[/hide] PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].

Consider $n$ different points lying on a circle, such that there are not three chords defined by that point that pass through the same interior point of the circle. a) Find the value of $n$, if the numbers of triangles that are defined using $3$ of the n points is equal to $2n$ b) Find the value of $n$, if the numbers of the intersection points of the chords that are interior to the circle is equal to $5n$.

A $7 \times 7$ grid is given. Julián colors $29$ cells black. Pilar must then place an $L$-shaped piece, covering exactly three cells (oriented in any direction, as shown in the figure). Pilar wins if all three cells covered by the $L$-shaped piece are black. Can Julián color the grid in such a way that it is impossible for Pilar to win? [asy] size(1.5cm); draw((0,1)--(1,1)--(1,2)--(0,2)--(0,1)--(0,0)--(1,0)--(2,0)--(2,1)--(1,1)--(1,0)); [/asy]

Consider two disjoint finite sets of positive integers, $A$ and $B$, have $n$ and $m$ elements, respectively. It is knows that all $k$ belonging to $A \cup B$ satisfies at least one of the conditions $k + 17 \in A$ and $k - 31 \in B$. Prove that $17n = 31m$.

There are 20 buns with jam and 20 buns with treacle arranged in a row in random order. Alice and Bob take in turn a bun from any end of the row. Alice starts, and wants to finally obtain 10 buns of each type; Bob tries to prevent this. Is it true for any order of the buns that Alice can win no matter what are the actions of Bob? [i]Alexandr Gribalko[/i]

Consider a $100 \times 100$ table, and identify the cell in row $a$ and column $b$, $1 \leq a, b \leq 100$, with the ordered pair $(a, b)$. Let $k$ be an integer such that $51 \leq k \leq 99$. A $k$-knight is a piece that moves one cell vertically or horizontally and $k$ cells to the other direction; that is, it moves from $(a, b)$ to $(c, d)$ such that $(|a-c|, |b - d|)$ is either $(1, k)$ or $(k, 1)$. The $k$-knight starts at cell $(1, 1)$, and performs several moves. A sequence of moves is a sequence of cells $(x_0, y_0)= (1, 1)$, $(x_1, y_1), (x_2, y_2)$, $\ldots, (x_n, y_n)$ such that, for all $i = 1, 2, \ldots, n$, $1 \leq x_i , y_i \leq 100$ and the $k$-knight can move from $(x_{i-1}, y_{i-1})$ to $(x_i, y_i)$. In this case, each cell $(x_i, y_i)$ is said to be reachable. For each $k$, find $L(k)$, the number of reachable cells.

A cube side $5$ is divided into $125$ unit cubes. $N$ of the small cubes are black and the rest white. Find the smallest $N$ such that there must be a row of $5$ black cubes parallel to one of the edges of the large cube.

There are $c$ red, $p$ blue, and $b$ white balls on a table. Two players $A$ and $B$ play a game by alternately making moves. In every move, a player takes two or three balls from the table. Player $A$ begins. A player wins if after his/her move at least one of the three colors no longer exists among the balls remaining on the table. For which values of $c,p,b$ does player $A$ have a winning strategy?

Given are the real line and the two unique marked points $0$ and $1$. We can perform as many times as we want the following operation: we take two already marked points $a$ and $b$ and mark the reflection of $a$ over $b$. Let $f(n)$ be the minimum number of operations needed to mark on the real line the number $n$ (which is the number at a distance $\left| n\right|$ from $0$ and it is on the right of $0$ if $n>0$ and on the left of $0$ if $n<0$). For example, $f(0)=f(1)=0$ and $f(-1)=f(2)=1$. Find $f(n)$.

Prove that in a tournament with 799 teams, there exist 14 teams, that can be partitioned into groups in a way that all of the teams in the first group have won all of the teams in the second group.

A sequence $a_1, a_2, a_3, . . . , a_{100}$ of $100$ (not necessarily distinct) positive numbers satisfy that the$ 99$ fractions$$\frac{a_1}{a_2},\frac{a_2}{a_3},\frac{a3}{a_4}, ... ,\frac{a_{99}}{a_{100}}$$ are all distinct. How many distinct numbers must there be, at least, in the sequence $a_1, a_2, a_3, . . . , a_{100}$?

(a) On each square of a squared sheet of paper of size $20 \times 20$ there is a soldier. Vanya chooses a number $d$ and Petya moves the soldiers to new squares in such a way that each soldier is moved through a distance of at least $d$ (the distance being measured between the centres of the initial and the new squares) and each square is occupied by exactly one soldier. For which $d$ is this possible? (Give the maximum possible $d$, prove that it is possible to move the soldiers through distances not less than $d$ and prove that there is no greater $d$ for which this procedure may be carried out.) (b) Answer the same question as (a), but with a sheet of size $21 \times 21$. (SS Krotov, Moscow)

For any $n$-tuple $a=(a_1,a_2,\ldots,a_n)\in\mathbb{N}_0^n$ of nonnegative integers, let $d_a$ denote the number of pairs of indices $(i,j)$ such that $a_i-a_j=1.$ Determine the maximum possible value of $d_a$ as $a$ ranges over all elements of $\mathbb{N}_0^n.$

A table $10 \times 10$ was filled according to the rules of the game “Bomb Squad”: several cells contain bombs (one bomb per cell) while each of the remaining cells contains a number, equal to the number of bombs in all cells adjacent to it by side or by vertex. Then the table is rearranged in the “reverse” order: bombs are placed in all cells previously occupied with numbers and the remaining cells are filled with numbers according to the same rule. Can it happen that the total sum of the numbers in the table will increase in a result?

The numbers $1, 2, ..., 2012$ are written on a blackboard, in some order, each of them exactly once. Between every two neighboring numbers the absolute value of their difference is written and the original numbers are deleted. This process is repeated until only a number remains on the board. What is the largest number that can stay on the board?

Let $P$ be a set of people. For two people $A$ and $B$, if $A$ knows $B$, $B$ also knows $A$. Each person in $P$ knows $2$ or less people in the set. $S$, a subset of $P$ with $k$ people, is called [i][b]k-independent set[/b][/i] of $P$ if any two people in $S$ don’t know each other. $X_1, X_2, …, X_{4041}$ are [i][b]2021-independent set[/b][/i]s of $P$ (not necessarily distinct). Show that there exists a [i][b]2021-independent set[/b][/i] of $P$, $\{v_1, v_2, …, v_{2021}\}$, which satisfies the following condition: [center] For some integer $1 \le i_1 < i_2 < \cdots < i_{2021} \leq 4041$, $v_1 \in X_{i_1}, v_2 \in X_{i_2}, \ldots, v_{2021} \in X_{i_{2021}}$ [/center] [hide=Graph Wording] Thanks to Evan Chen, here's a graph wording of the problem :) Let $G$ be a finite simple graph with maximum degree at most $2$. Let $X_1, X_2, \ldots, X_{4041}$ be independent sets of size $2021$ [i](not necessarily distinct)[/i]. Prove that there exists another independent set $\{v_1, v_2, \ldots, v_{2021}\}$ of size $2021$ and indices $1 \le t_1 < t_2 < \cdots < t_{2021} \le 4041$ such that $v_i \in X_{t_i}$ for all $i$. [/hide]

[b]p1.[/b] A Slavic dragon has three heads. A knight fights the dragon. If the knight cuts off one dragon’s head three new heads immediately grow. Is it possible that the dragon has $2018$ heads at some moment of the fight? [b]p2.[/b] Peter has two squares $3\times 3$ and $4\times 4$. He must cut one of them or both of them in no more than four parts in total. Is Peter able to assemble a square using all these parts? [b]p3.[/b] Usually, dad picks up Constantine after his music lessons and they drive home. However, today the lessons have ended earlier and Constantine started walking home. He met his dad $14$ minutes later and they drove home together. They arrived home $6$ minutes earlier than usually. Home many minutes earlier than usual have the lessons ended? Please, explain your answer. [b]p4.[/b] All positive integers from $1$ to $2018$ are written on a blackboard. First, Peter erased all numbers divisible by $7$. Then, Natalie erased all remaining numbers divisible by $11$. How many numbers did Natalie remove? Please, explain your answer. [b]p5.[/b] $30$ students took part in a mathematical competition consisting of four problems. $25$ students solved the first problem, $24$ students solved the second problem, $22$ students solved the third, and, finally, $21$ students solved the fourth. Show that there are at least two students who solved all four problems. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We call a sequence of length $n$ of zeros and ones a "string of length $n$" and the elements of the same sequence "bits". Let $m,n$ be two positive integers so that $m<2^n$. Arik holds $m$ strings of length $n$. Giora wants to find a new string of length $n$ different from all those Arik holds. For this Giora may ask Arik questions of the form: "What is the value of bit number $i$ in string number $j$?" where $1\leq i\leq n$ and $1\leq j\leq m$. What is the smallest number of questions needed for Giora to complete his task when: [b]a)[/b] $m=n$? [b]b)[/b] $m=n+1$?

The $28$ students of a class are seated in a circle. They then all claim that 'the two students next to me are of different genders'. It is known that all boys are lying while exactly $3$ girls are lying. How many girls are there in the class?

The set $M$ was formed from the plane by removing three points $A, B, C$, which are vertices of the triangle. What is the smallest number of convex sets whose union is $M$? [hide=original wording] Množina M Vznikla z roviny vyjmutím tří bodů A, B, C, které jsou vrcholy trojúhelníka. Jaký je nejmenší počet konvexních množin, jejichž sjednocením je M?[/hide]

An array $ (a_1,a_2,...,a_{13})$ of $ 13$ integers is called $ tame$ if for each $ 1 \le i \le 13$ the following condition holds: If $ a_i$ is left out, the remaining twelve integers can be divided into two groups with the same sum of elements. A tame array is called $ turbo$ $ tame$ if the remaining twelve numbers can always be divided in two groups of six numbers having the same sum. $ (a)$ Give an example of a tame array of $ 13$ integers (not all equal). $ (b)$ Prove that in a tame array all numbers are of the same parity. $ (c)$ Prove that in a turbo tame array all numbers are equal.

An infinite number of participants gathered for the Olympiad, who were registered under the numbers $1, 2,\ldots$. It turns out that for every $n = 1, 2,\ldots$ a participant with number $n{}$ has at least $n{}$ friends among the remaining participants (note: friendship is mutual). There is a hotel with an infinite number of double rooms. Prove that the participants can be accommodated in double rooms so that there is a couple of friends in each room. [i]Proposed by V. Bragin, P. Kozhevnikov[/i]