At a circular track, $10$ cyclists started from some point at the same time in the same direction with different constant speeds. If any two cyclists are at some point at the same time again, we say that they meet. No three or more of them have met at the same time. Prove that by the time every two cyclists have met at least once, each cyclist has had at least $25$ meetings.

A graph has $ 30$ vertices, $ 105$ edges and $ 4822$ unordered edge pairs whose endpoints are disjoint. Find the maximal possible difference of degrees of two vertices in this graph.

There are 2010 students and 100 classrooms in the Olympiad High School. At the beginning, each of the students is in one of the classrooms. Each minute, as long as not everyone is in the same classroom, somebody walks from one classroom into a different classroom with at least as many students in it (prior to his move). This process will terminate in $M$ minutes. Determine the maximum value of $M$.

There is a finite number of towns in a country. They are connected by one direction roads. It is known that, for any two towns, one of them can be reached from another one. Prove that there is a town such that all remaining towns can be reached from it.

In a country there are $100$ cities, some of which are connected by roads. For each four cities there are at least two roads between them. Also, there is no path that passes through each city exactly one time. Prove that one can choose two cities among those $100$, such that each of the $98$ remaining cities would be connected by a road with at least one of the two chosen cities.

Let $n$ ($\ge 4$) be an even integer. We label $n$ pairwise distinct real numbers arbitrarily on the $n$ vertices of a regular $n$-gon, and label the $n$ sides clockwise as $e_1, e_2, \ldots, e_n$. A side is called [i]positive[/i] if the numbers on both endpoints are increasing in clockwise direction. An unordered pair of distinct sides $\left\{ e_i,e_j \right\}$ is called [i]alternating[/i] if it satisfies both conditions: (i) $2 \mid (i+j)$; and (ii) if one rearranges the four numbers on the vertices of these two sides $e_i$ and $e_j$ in increasing order $a < b < c < d$, then $a$ and $c$ are the numbers on the two endpoints of one of sides $e_i$ or $e_j$. Prove that the number of alternating pairs of sides and the number of positive sides are of different parity.

Let $M=\{1,2,\cdots,n\}$, each element of $M$ is colored in either red, blue or yellow. Set $A=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n$, $x,y,z$ are of same color$\},$ $B=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n,$ $x,y,z$ are of pairwise distinct color$\}.$ Prove that $2|A|\geq |B|$.

Find the smallest positive integer $n$ that satisfies the following condition: For every finite set of points on the plane, if for any $n$ points from this set there exist two lines containing all the $n$ points, then there exist two lines containing all points from the set.

$n$ teams take part in a tournament, in which every two teams compete exactly once, and that no draws are possible. It is known that for any two teams, there exists another team which defeated both of the two teams. Find all $n$ for which this is possible.

The microcalculator ''MK-97'' can work out the numbers entered in memory, perform only three operations: a) check whether the selected two numbers are equal; b) add the selected numbers; c) using the selected numbers $a$ and $b$, find the equation $x^2 +ax+b = 0$, and if there are no roots, display a message about this. The results of all actions are stored in memory. Initially, one number $x$ is stored in memory. How to use ''MK-97'' to find out whether is this number one?

There are $n$ points on a circle ($n>1$). Define an "interval" as an arc of a circle such that it's start and finish are from those points. Consider a family of intervals $F$ such that for every element of $F$ like $A$ there is almost one other element of $F$ like $B$ such that $A \subseteq B$ (in this case we call $A$ is sub-interval of $B$). We call an interval maximal if it is not a sub-interval of any other interval. If $m$ is the number of maximal elements of $F$ and $a$ is number of non-maximal elements of $F,$ prove that $n\geq m+\frac a2.$

Find the largest $K$ satisfying the following: Given any closed intervals $A_1,\ldots, A_N$ of length $1$ where $N$ is an arbitrary positive integer. If their union is $[0,2021]$, then we can always find $K$ intervals from $A_1,\ldots, A_N$ such that the intersection of any two of them is empty.

Suppose there are several juice boxes, one of which is poisoned. You have $n$ guinea pigs to test the boxes. The testing happens in the following way: [list] [*] At each round, you can have the guinea pigs taste any number of juice boxes. [*] Conversely, a juice box can be tasted by any number of guinea pigs. [*] After the round ends, any guinea pigs who tasted the poisoned juice die. [/list] Suppose you have to find the poisoned juice box in at most $k$ rounds. What is the maximum number of juice boxes such that it is possible?

Let $n \geq 2$ be a natural number. For any two permutations of $(1,2,\cdots,n)$, say $\alpha = (a_1,a_2,\cdots,a_n)$ and $\beta = (b_1,b_2,\cdots,b_n),$ if there exists a natural number $k \leq n$ such that $$b_i = \begin{cases} a_{k+1-i}, & \text{ }1 \leq i \leq k; \\ a_i, & \text{} k < i \leq n, \end{cases}$$ we call $\alpha$ a friendly permutation of $\beta$. Prove that it is possible to enumerate all possible permutations of $(1,2,\cdots,n)$ as $P_1,P_2,\cdots,P_m$ such that for all $i = 1,2,\cdots,m$, $P_{i+1}$ is a friendly permutation of $P_i$ where $m = n!$ and $P_{m+1} = P_1$.

The expressions $ a \plus{} b \plus{} c, ab \plus{} ac \plus{} bc,$ and $ abc$ are called the elementary symmetric expressions on the three letters $ a, b, c;$ symmetric because if we interchange any two letters, say $ a$ and $ c,$ the expressions remain algebraically the same. The common degree of its terms is called the order of the expression. Let $ S_k(n)$ denote the elementary expression on $ k$ different letters of order $ n;$ for example $ S_4(3) \equal{} abc \plus{} abd \plus{} acd \plus{} bcd.$ There are four terms in $ S_4(3).$ How many terms are there in $ S_{9891}(1989)?$ (Assume that we have $ 9891$ different letters.)

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

Misha and Sahsa play a game on a $100\times 100$ chessboard. First, Sasha places $50$ kings on the board, and Misha places a rook, and then they move in turns, as following (Sasha begins): At his move, Sasha moves each of the kings one square in any direction, and Misha can move the rook on the horizontal or vertical any number of squares. The kings cannot be captured or stepped over. Sasha's purpose is to capture the rook, and Misha's is to avoid capture. Is there a winning strategy available for Sasha?

Rishabh has $2024$ pairs of socks in a drawer. He draws socks from the drawer uniformly at random, without replacement, until he has drawn a pair of identical socks. Compute the expected number of unpaired socks he has drawn when he stops.

[b]7.1. / 6.5[/b] Prove that out of any six people there will always be three pairs of acquaintances or three pairs of strangers. [b]7.2[/b] Given a circle $O$ and a square $K$, as well as a line $L$. Construct a segment of given length parallel to $L$ and such that its ends lie on $O$ and $K$ respectively [b]7.3[/b] The three-digit number $\overline{abc}$ is divisible by $37$. Prove that the sum of the numbers $\overline{bca}$ and $\overline{cab}$ is also divisible by $37$.[b] (typo corrected)[/b] [b]7.4.[/b] Point $C$ is the midpoint of segment $AB$. On an arbitrary ray drawn from point $C$ and not lying on line $AB$, three consecutive points $P$, $M$ and $Q$ so that $PM=MQ$. Prove that $AP+BQ>2CM$. [img]https://cdn.artofproblemsolving.com/attachments/f/3/a8031007f5afc31a8b5cef98dd025474ac0351.png[/img] [b]7.5.[/b] Given $2n+1$ different objects. Prove that you can choose an odd number of objects from them in as many ways as an even number. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here[/url].

A piece is placed in the lower left-corner cell of the $15 \times 15$ board. It can move to the cells that are adjacent to the sides or the corners of its current cell. It must also alternate between horizontal and diagonal moves $($the first move must be diagonal$).$ What is the maximum number of moves it can make without stepping on the same cell twice$?$

The Macedonian Mathematical Olympiad is held in two rooms numbered $1$ and $2$. At the beginning all of the competitors enter room No. $1$. The final arrangement of the competitors to the rooms is obtained in the following way: a list with the names of a few of the competitors is read aloud; after a name is read, the corresponding competitor and all of his/her acquaintances from the rest of the competitors change the room in which they currently are. Hence, to each list of names corresponds one final arrangement of the competitors to the rooms. Show that the total number of possible final arrangements is not equal to $2009$ (acquaintance between competitors is a symmetrical relation).

Do there exist such a triangle $T$, that for any coloring of a plane in two colors one may found a triangle $T'$, equal to $T$, such that all vertices of $T'$ have the same color. [b] proposed by S. Berlov[/b]

Mariana plays with an $8\times 8$ board with all its squares blank. She says that two houses are [i]neighbors [/i] if they have a common side or vertex, that is, two houses can be neighbors vertically, horizontally or diagonally. The game consists of filling the $64$ squares on the board, one after the other, each with a number according to the following rule: she always chooses a house blank and fill it with an integer equal to the number of neighboring houses that are still in White. Once this is done, the house is no longer considered blank. Show that the value of the sum of all $64$ numbers written on the board at the end of the game does not depend in the order of filling. Also, calculate the value of this sum. Note: A house is not neighbor to itself. [hide=original wording]Mariana brinca com um tabuleiro 8 x 8 com todas as suas casas em branco. Ela diz que duas casas s˜ao vizinhas se elas possu´ırem um lado ou um v´ertice em comum, ou seja, duas casas podem ser vizinhas verticalmente, horizontalmente ou diagonalmente. A brincadeira consiste em preencher as 64 casas do tabuleiro, uma ap´os a outra, cada uma com um n´umero de acordo com a seguinte regra: ela escolhe sempre uma casa em branco e a preenche com o n´umero inteiro igual `a quantidade de casas vizinhas desta que ainda estejam em branco. Feito isso, a casa n˜ao ´e mais considerada em branco. Demonstre que o valor da soma de todos os 64 n´umeros escritos no tabuleiro ao final da brincadeira n˜ao depende da ordem do preenchimento. Al´em disso, calcule o valor dessa soma. Observa¸c˜ao: Uma casa n˜ao ´e vizinha a si mesma[/hide]

Let $n\ge 2$ be a natural number. An $n \times n$ grid is drawn on a blackboard and each field with one of the numbers $-1$ or $+1$ labeled. Then the $n$ row and also the $n$ column sums calculated and the sum $S_n$ of all these $2n$ sums determined. (a) Show that for no odd number $n$ there is a label with $S_n = 0$. (b) Show that if $n$ is an even number, there are at least six different labels with $S_n = 0$.

We colored the $n^2$ unit squares of an $n\times n$ square lattice such that in each $2\times 2$ square, at least two of the four unit squares have the same color. What is the largest number of colors we could have used? [i]Based on a problem of the Dürer Competition[/i]