A regular tetrahedron with unit edge length is given. Prove that: (a) There exist four points on the surface $S$ of the tetrahedron, such that the distance from any point of the surface to one of these four points does not exceed $1/2$; (b) There do not exist three points with this property. The distance between two points on surface $S$ is defined as the length of the shortest polygonal line going over $S$ and connecting the two points.

Let \(p\) a prime number, and \(N\) the number of matrices \(p \times p\) \[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1p}\\ a_{21} & a_{22} & \ldots & a_{2p}\\ \vdots & \vdots & \ddots & \vdots \\ a_{p1} & a_{p2} & \ldots & a_{pp} \end{array}\] such that \(a_{ij} \in \{0,1,2,\ldots,p\} \) and if \(i \leq i^\prime\) and \(j \leq j^\prime\), then \(a_{ij} \leq a_{i^\prime j^\prime}\). Find \(N \pmod{p}\).

(a) Let be given $2n$ distinct points on a circumference, $n$ of which are red and $n$ are blue. Prove that one can join these points pairwise by $n$ segments so that no two segments intersect and the endpoints of each segments have different colors. (b) Show that the statement from (a) remains valid if the points are in an arbitrary position in the plane so that no three of them are collinear.

Some cities of a country consisting of $n$ cities are connected by round trip flights so that there are at least $k$ flights from any city and any city is reachable from any city. Prove that for any such flight organization these flights can be distributed among $n-k$ air companies so that one can reach any city from any city by using of at most one flight of each air company.

Peter and Bob play a game on a $n\times n$ chessboard. At the beginning, all squares are white apart from one black corner square containing a rook. Players take turns to move the rook to a white square and recolour the square black. The player who can not move loses. Peter goes first. Who has a winning strategy?

An $ n \times n$ matrix whose entries come from the set $ S \equal{} \{1, 2, \ldots , 2n \minus{} 1\}$ is called a [i]silver matrix[/i] if, for each $ i \equal{} 1, 2, \ldots , n$, the $ i$-th row and the $ i$-th column together contain all elements of $ S$. Show that: (a) there is no silver matrix for $ n \equal{} 1997$; (b) silver matrices exist for infinitely many values of $ n$.

In a kingdom, there are roads open between some cities with lanes both ways, in such a way, that you can come from one city to another using those roads. The roads are toll, and the price for taking each road is distinct. A minister made a list of all routes that go through each city exactly once. The king marked the most expensive road in each of the routes and said to close all the roads that he marked at least once. After that, it became impossible to go from city $A$ to city $B$, from city $B$ to city $C$, and from city $C$ to city $A$. Prove that the kings order was followed incorrectly.

A $45 \times 45$ grid has had the central unit square removed. For which positive integers $n$ is it possible to cut the remaining area into $1 \times n$ and $n\times 1$ rectangles?

Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches. What is the largest possible number of matches?

A subset $S$ of the set of integers 0 - 99 is said to have property $A$ if it is impossible to fill a crossword-puzzle with 2 rows and 2 columns with numbers in $S$ (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in the set $S$ with the property $A.$

Two players play the following game: there are two heaps of tokens, and they take turns to pick some tokens from them. The winner of the game is the player who takes away the last token. If the number of tokens in the two heaps are $A$ and $B$ at a given moment, the player whose turn it is can take away a number of tokens that is a multiple of $A$ or a multiple of $B$ from one of the heaps. Find those pair of integers $(k,n)$ for which the second player has a winning strategy, if the initial number of tokens is $k$ in the first heap and $n$ in the second heap. [i]Proposed by Dömötör Pálvölgyi, Budapest[/i]

Ati has $ 7$ pots of flower, ordered in $ P_1,P_2,P_3,P_4,P_5,P_6,P_7$. She wants to rearrange the position of those pots to $ B_1,B_2,B_2,B_3,B_4,B_5,B_6,B_7$ such that for every positive integer $ n<7$, $ B_1,B_2,\dots,B_n$ is not the permutation of $ P_1,P_2,\dots,P_7$. In how many ways can Ati do this?

A grid of cells is tiled with dominoes such that every cell is covered by exactly one domino. A subset $S$ of dominoes is chosen. Is it true that at least one of the following 2 statements is false? (1) There are $2022$ more horizontal dominoes than vertical dominoes in $S$. (2) The cells covered by the dominoes in $S$ can be tiled completely and exactly by $L$-shaped tetrominoes.

Let $n > 1$ be a natural number. There are $n$ bulbs in a line, each of which can be on or off. Every minute, simultaneously, all the lit bulbs turn off and the unlit bulbs that were adjacent to exactly one lit bulb turn on. Determine for what values of $n$ there is an initial arrangement such that if this process is followed indefinitely, all the lights will never be off.

[b]6.1[/b] The capacities of cubic vessels are in the ratio 1:8:27 and the volumes of liquid poured into them are 1: 2: 3. After this, from the first to a certain amount of liquid was poured into the second vessel, and then from the second in the third so that in all three vessels the liquid level became the same. After this, 128 4/7 liters were poured from the first vessel into the second, and from the second in the first back so much that the height of the liquid column in the first vessel became twice as large as in the second. It turned out that in the first vessel there were 100 fewer liters than at first. How much liquid was initially in each vessel? [b]6.2[/b] How many times a day do all three hands on a clock coincide, including the second hand? [b]6.3.[/b] Prove that in Leningrad there are two people who have the same number of familiar Leningraders. [b]6.4 / 7.4[/b] Each of the eight given different natural numbers less than $16$. Prove that among their pairwise differences there is at least at least three are the same. [b]6.5 / 7.6[/b] The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].

An underground burrow consists of an infinite sequence of rooms labeled by the integers $(\dots, -3, -2, -1, 0, 1, 2, 3,\dots)$. Initially, some of the rooms are occupied by one or more rabbits. Each rabbit wants to be alone. Thus, if there are two or more rabbits in the same room (say, room $m$), half of the rabbits (rounding down) will flee to room $m-1$, and half (also rounding down) to room $m+1$. Once per minute, this happens simultaneously in all rooms that have two or more rabbits. For example, if initially all rooms are empty except for $5$ rabbits in room $\#12$ and $2$ rabbits in room $\#13$, then after one minute, rooms $\text{\#11--\#14}$ will contain $2$, $2$, $2$, and $1$ rabbits, respectively, and all other rooms will be empty. Now suppose that initially there are $k+1$ rabbits in room $k$ for each $k=0, 1, 2, \ldots, 9, 10$, and all other rooms are empty. [list=a] [*]Show that eventually the rabbits will stop moving. [*] Determine which rooms will be occupied when this occurs. [/list]

Prair writes the letters $A,B,C,D$, and $E$ such that neither vowel are written first, and they are not adjacent; such that there exists at least one pair of adjacent consonants; and such that exactly five pairs of letters are in alphabetical order. How many possible ways could Prair have ordered the letters?

A rectangle of dimensions $2024 \times 2023$ is filled with pieces of the following types: [asy] size(200); // Figure (A) draw((0,0)--(4,0)--(4,1)--(0,1)--cycle); draw((1,0)--(1,1)); draw((2,0)--(2,1)); draw((3,0)--(3,1)); // Figure (B) draw((6,0)--(8,0)--(8,2)--(6,2)--cycle); draw((7,0)--(7,2)); draw((6,1)--(8,1)); // Figure (C) draw((10,0)--(12,0)--(12,1)--(11,1)--(11,2)--(9,2)--(9,1)--(10,1)--cycle); draw((10,0)--(10,1)); draw((11,0)--(11,1)); draw((10,1)--(11,1)); draw((9,1)--(9,2)); draw((10,1)--(10,2)); draw((11,0)--(12,0)); draw((10,1)--(12,1)); // Labeling label("(A)", (2, -0.5)); label("(B)", (7, -0.5)); label("(C)", (10.5, -0.5)); [/asy] Each piece can be turned arround, and each square has side length $1$. Is it possible to use exactly 2023 pieces of type $(A)$?

On each side of an equilateral triangle of side $n \ge 1$ consider $n - 1$ points that divide the sides into $n$ equal segments. Through these points draw parallel lines to the sides of the triangles, obtaining a net of equilateral triangles of side length $1$. On each of the vertices of the small triangles put a coin head up. A move consists in flipping over three mutually adjacent coins. Find all values of $n$ for which it is possible to turn all coins tail up after a finite number of moves. Colombia 1997

Let $S = \{-17, -16, ..., 16, 17\}$. We call a subset $T$ of $S$ a good set if $-x \in T$ for all $x \in T$ and if $x, y, z \in T (x, y, z$ may be equal) then $x + y + z \ne 0$. Find the largest number of elements in a good set.

Given some cards with either "$-1$" or "$+1$" written on the opposite side. You are allowed to choose a triple of cards and ask about the product of the three numbers on the cards. What is the minimal number of questions allowing to determine all the numbers on the cards ... a) for $30$ cards, b) for $31$ cards, c) for $32$ cards. (You should prove, that you cannot manage with less questions.) d) Fifty above mentioned cards are lying along the circumference. You are allowed to ask about the product of three consecutive numbers only. You need to determine the product af all the $50$ numbers. What is the minimal number of questions allowing to determine it?

A spider built a web on the unit circle. The web is a planar graph with straight edges inside the circle, bounded by the circumference of the circle. Each vertex of the graph lying on the circle belongs to a unique edge, which goes perpendicularly inward to the circle. For each vertex of the graph inside the circle, the sum of the unit outgoing vectors along the edges of the graph is zero. Prove that the total length of the web is equal to the number of its vertices on the circle.

Call a number unremarkable if, when written in base $10$, no two adjacent digits are equal. For example, $123$ is unremarkable, but $122$ is not. Find the sum of all unremarkable $3$-digit numbers. (Note that $012$ and $007$ are not $3$-digit numbers.)

There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$. [i]Proposed by Dusan Dukic, Serbia[/i]

There are infinitely many people registered on the social network Mugbook. Some pairs of (different) users are registered as friends, but each person has only finitely many friends. Every user has at least one friend. (Friendship is symmetric; that is, if $A$ is a friend of $B$, then $B$ is a friend of $A$.) Each person is required to designate one of their friends as their best friend. If $A$ designates $B$ as her best friend, then (unfortunately) it does not follow that $B$ necessarily designates $A$ as her best friend. Someone designated as a best friend is called a $1$-best friend. More generally, if $n> 1$ is a positive integer, then a user is an $n$-best friend provided that they have been designated the best friend of someone who is an $(n-1)$-best friend. Someone who is a $k$-best friend for every positive integer $k$ is called popular. (a) Prove that every popular person is the best friend of a popular person. (b) Show that if people can have infinitely many friends, then it is possible that a popular person is not the best friend of a popular person. [i]Romania (Dan Schwarz)[/i]